ty Hele near ity wit n . f Rerene nt a sage A M By : is sia, ne Hatt bs j ea pie ht ce si ha) th rt Bs ba ave Hitt a bet Ste ae hf ie N ahah jit se ace . fate ie apa nf “ wit eeas = ne Heh Hi a pi ! ate it ry PEARL ITLS PAR EE Hy 9} Ls it a ti Ha eat i i ni i! " if ia es tn Pe RE ashy its) at es Bi ae PO! Ta eh reste is La vA alae hey maa fake HNihaiae te bat aa tai ate ae = aes Tape, sists = rs= = T Sees = = ee arene Ne leyeiae: oe Dee = aeaiee pe eee Sed eee Hn) ti 4 an ‘ta baat th, i i i tae Hel ie hae M) i ? Nita iba it oh Seine “hE 1 bei sri itt Hane oe jan i Neate ap) 07) San 4 Bayt Ohh : wel a aie ii i iF ze = SSeS, aS See siarece= . = Sse snes: SS Fane oe — = eae: i ie oe fy pagit ifeitt Ast niin iste i” Harri We rae end iy HA = = == Ae = Seer —— sa a a bo -~= Dy, i ee =A ce 4 hal (arent) a Ox bdo! ose iy) = (as Seer = ry x W pra fi & iy) ris ay SMM MMMM MMAWMMMVMAMMMwATKww 11— 8625 GOVERNMENT PRENTING OFFICE THE ——_ a a LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. & CONDUCTED BY SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.B.S. SIR JOSEPH JOHN THOMSON, O.M., M.A., Sc.D., LL.D., F.R.S. JOHN JOLY, M.A., D.Sc., F.R.S., F.G.S. RICHARD TAUNTON FRANCIS AND WILLIAM FRANCIS, F.L.S. “ Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster vilior quia ex alienis libamus ut apes.” Just. Itps. Polit. lib. i. cap. 1. Not. VOL. XLITI.—SIXATH SERIES. JANUARY—JUNE 1922. LONDON: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. SOLD BY SMITH AND SON, GLASGOW ;— HODGES, FIGGIS. AND CO., DUBLIN —- AND VEUVE J. BOYVEAU, PARIS. > r \ ‘‘Meditatiunis est perscrutari occulta; contemplationis est admirari perspicua .... Admiratio generat queestionem, queestio 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 ccelo, Quo micet igne Iris, superos quis conciat orbes Tam yario motu.” J. B. Pinelli ad Mazonium. Fi vo Td CONTENTS OF VOL. XLIII. (SIXTH SERIES). NUMBER CCLITI.—JANUARY 1922. Prof. J. W. Nicholson on Zonal Harmonies of the Second Type Prof. F. D. Murnaghan on the Derivation of Symmetrical Gravita- Sue RUBINO ON OeY cle os bee ee tin w bey ae ee Ae eS Profs. J. N. Bronsted and G. Hevesy on the Ses cf the arpa OMT CEE IIEY. 55) casa ciieelo clas wpe sims och e's albedo 604 Mr. A. R. McLeod on the Lags of Thermometers .............. Dr. T. J. Ya. Bromwich on Kinetic ROUAL OL LViel sn arr mn Saere es 3° Step cet Mr. V. Lough on the Beating Tones of Overblown Organ Pipes. (Plate I. re + a Sse SoG ines bie 0 er ar eee en ge eee Messrs. C. N. Hinshelwood and H. Hartley on the Probability of Spontaneous Crystallization of Super-cooled Liquids .......... Mr. J. J. Manley on the Insulation of Highly Attenuated Wires in Platinum Resistance Thermometers .... 00. .0.secce cee e eee. Dr. John Prescott on the Iquations of Equilibrium of an Elastic femme Normal Pressure 0. 0's scree ve cee = he eee Prof. S. P. Timoshenko on the Transverse Vibrations of Bars of PEERS TOSS SE CHION teri c or vis sso cine plete Cia gie cis ooje ei eae ® Prof. A. Anderson on Scalar and Vector Potentials due to Moving See OT AE SES 7s eit os es alg GN ce Sates oe bce po ge sy Lee Prof, 0. W. Richardson on Gravitation. With an Appendix by wo ays SEER oo od ieee Se ern i eee rere aneee Prof. A. Ll. Hughes on Characteristic X-Rays from Boron and SIN ee ee oes hg ere dee sepa sole nw «sae aims wai es Prof. O. W. Richardson and Mr. F. 8. Robertson on the Effect of Gases on the Contact Difference of Potential between Metals at Petetshota, WMP EEALUECS goo. stasis bs bieyae ie be see oe meen Set Prof. A. S. Eddington on the significance of Finstein’s Gravita- tional Equations in terms of the Curvature of the World ...... Mr. I. V. Appleton and Dr. Balth. van der Pol on a Type of Oscillation-Hysteresis in a Simple Triode Generator .......... Mr. S. Ratner on Polarization Phenomena in X-Ray Bulbs ...... Messrs. A. F’. Joffe and M. V. Kirpitcheva on Rontgenograms of Pee On tysndis. (Pave Bl). ae as ws hee ve hacen ed Dr. Alfred A. Robb on a Graphical Solution of a Class of Differen- tial quations occurring in Wireless Telegraphy ; with Note by ENMU Wet ca ss Oye Yi GS Le o's gp bo meta tease ee & Profs. D. N. Mallik and A. B. Das on certain Types of Electric SE DEEL 2p Spe Re ea i arn ER oa Mr. W. Sucksmith on the Application of the Ultra-Micrometer to the Measurement of Small Increments of Temperature ........ Mr. H. P. Waran on an Interesting Case of Mechanival Disintegra- L101) eer StS VOUS 605 oe Sees cise eet eee ee Dr. Leonard B. Loeb on the Relative Affinity of some Gas Mole- Biles fereeeemmeaten tes ete che eS eee ce te Page ii 19 162 174 1Me7d 193 204 206 216 223 226 tie et re <<. hae 1V CONTENTS OF VOL. XLIJI.—SIXTH SERIES. Notices respecting New Books :— Messrs. G. W. C. Kaye and T. H. Laby’s Tables of Physical and Chemical Constants (2... 220... eae eee Prof. H.S. Carslaw’s Introduction to the Theory of Fourier Seuies and Integrals 20... 5a ieuece oi eee ene Dr. S. Brodetsky’s ‘The Mechanical Principles of the Aeroplane . Prof. D. N. Mallik’s Optical Theories eeoeceeecevee se ee ee Be Oe ee NUMBER CCLIV.—FEBRUARY. Prof. F. Y. Edgeworth on the Application of Probabilities to the Movement.of Gas Molecules 1.55... hate) ee eee Mr. A. S. Percival: Method of Tracing Caustic Curves.......... Dr. J. 8. G. Thomas on the Forced Convection of Heat from a Pair of fine: heated, Wares 2.1.0.0. aa ae che oc Mr. G. A. Hemsalech and the Comte de Gramont :. Observations and Experiments on the Occurrence of Spark Lines (Enhanced Lines) in the Arc.—Part I. Lead and Tin. (Plates I1I.-V.).... Mr. 8. I. Vavilov on the Dependence of the Intensity of the Fluores- cence of Dyes upon the Wave-leugth of the Hxeiting Light .... Prof. W. B. Morton and Mr, L. J. Close on Hertz’s Theory of the Contact of Blastie Bodies... 21: o..2 coe Sacco ee ee Mr. A. H. Davis on Natural Convective Cooling of Wires ........ Mr, L. St. C. Broughall on the Frequency of the Electrons in the Neon Atom 2.6 0. dia ale oes cee a tame oe ee cle eee Dr. J. H. J. Poole on an Attempt to determine whether a Minimum ‘lime is necessary to excite the Human Retina eeoee eee es ow oe ® Mr. H. E. Roaf on the Analysis of Sound Waves by the Cochlea.. ¢ Mr. Frank M. Lidstone on the Measurement of Absolute Viscosity . Prof. C. V. Raman on the Phenomenon of the “ Radiant Spectrum ” observed by Sir David Brewster ............5. ee ols Prof. Kk. Taylor Jones, Mr. J. D. Morgan, and Prof. R. V. Wheeler on the Form of the Temperature Wave spreading by Conduction from Point and Spherical Sources ; with a suggested application Page 237 238 240 240 to the Problem of Spark Vemition.2 25s... ae eee eee 309 Prof. J. R. Partington and Mr. L. J. Cant on the Specific Heats of Ammonia, Sulphur dioxide, and Carbon dioxide .............. 369 Mr. P. W. Burbidge on the Absorption of the K X-rays of Silver in Gases'and Gaseous Mixtures 0.0.0 cocci eo el) eee 381 Mr. P. W. Burbidge on the Absorption of Narrow X- “ray Beams .. 389 Dr. J. L. Glasson on Beta Rays and Atomic Number ............ 395 Prof. H. J. Priestley: Reply to Sir Oliver Lodge’s paper on the Hinstein Spectral Shitt 40 fon... sees ol. cia ee 596 Dr. Ncrman R. Campbell: The Fundamental Principles of Scien- PIWCHY GIO (UN EI COMO ME rin MRA NE RAny Ad ogo 00 oo oa: 597 Drs. D. M. Wrinch and Harold Jeffreys’s Reply to the above note. . 398 Notices respecting New Books :— Dr. A. I. M. Geddes’s Meteorology: an Introductory Treatise. 398 Mr. K. Cunningham’s Relativity, the Mlectron Theory, and GAWA AMON 2. cance wletecsieone © ape ayounidics Ao syevete wicleusia soi aaa eee 599 M. E. M. Lémeray’s Legons Elémentaires sur la Gravitation dlapres la Theorie d Himstetn sai. 00 oot ee 399 Dr, J; Johnstone’s The Mechanism of lite Goo. va ae 399 Prof. O. W. Richardson’s The Emission of Electricity from ROG BOK S pis hs nate fia oe ous om deers wiele WOKE 6) | op er 400 Bibliotheca) Chemico-Mathematica.....5.........1-. ae 400 CONTENTS OF VOL. XLIII.—SIXTH SERIES. v NUMBER CCLV.—MARCH. Page J. R. Ashworth on the Theory of the Intrinsic Field of a ; eee and the relation of its Magnetic to its characteristic Electric and Thermal IBROPE LUGS ct min coleteaye, Aetna ileus outceds. + 401 Dr. Ludwik Silbersteiv on the Relation between the Projective and the Metrical Scales, and its bearing on the Theory of Parallels... 420 Messrs. H. Hartley, A. O. Ponder, E. J. Bowen, and Dr. T. R. Merton: An Attempt to Separate the Isotopes of Chlorine .... 480 Mrs. K. Stratton and Prof. J. R. Partington on Latent Heats of Fusion.—Part I. Benzophenone, Phenol, GING SUN CTE NS Aine a oe 436 ’ Prof. C. V. Raman and Dr. Nihal Karan Sethi on the Convection of te (Fizeau Effect) in Moving Gases. (Plates VI. & VII.) .. 447 Dr. i. H. Newman on Active Modifications of Hydrogen and ee ETROONTCE CONE A LCAYSit. i aNern eat SNe Se aks edd shee sukls Pea 0 4 455 Mr. J. B. Dale on the Analysis of Microseismograms ............ 463 Mr. H. P. Waran on an Interferometer Method of determining the Phase Difference resulting from Metallic Reflexion. (Plate VILL. ) 471 Mr. J. L. Glasson on Stopping Power and Atomic Number ...... 477 Mr. L. C. Jackson on the Dielectric Constants of some of the Esters PRM CINE NATUEEOS «0.9 sree gh yeie.e Sie ¢ Welders «(ale widens wales 48] Mr. G. L. Addenbroole: A Study of Franklin’s Iixperiment on the Beyden Jar with Movable Coatings 2.0... 0.0... eset cee 489 Sir J, Alfred Ewing on a New Model of Ferromagnetic Induction.. 493 | Mr. N. C. Krishnaiyar on the Amplitude of Vibrations maintained Imetionceaor Double Prequency jo... ceils tees anes oe ene 503 Prof. C. V. Raman and Mr. V.S. Tamma on a New Optical Property pact Piel Tay SNS ce Rega teee oe ok ci ilele ose ada vs ved oleh oe 510 Dy. F. W. Aston and Mr. R. H. Fowler on some Problems of the 2 Bibs SCE CCE RB SE CR PR ne ar 514 Mr. E. H. Synge: A Definition of Simultaneity and the Aither ., 528 Prof. KX. T. Compton on Lonization by Cumulative Action ........ d31 Mr. John J. Dowling and Miss Katharine M. Preston on the Resist- ance of Electrol ytes Atala louie HTOQUEMCIES |. c aes ss giv las Stee nas 537 Prof. Frederick Slate on a Graphical Synthesis of ‘the Linear ee MEMO ay ta ees ae Sees, Se ves sic io ole LES ae SAL OS 545 Prof. O. W. Richardson and Mr. F. 8. Robertson on Contact Dif- ference of Potential and Thermionic Emission ................ dof Mr 5. N. Chuckerbutti on the Deformation of the “ Rings and Brushes,” as observed through a Spath Hemitrope. (Plate IX.) . 560 Prof. A. L, Narayan on Coupled Vibrations by means of a Double Te 5c? LEDUC (LEI re at. [Se ec ee 567 Prof. A. L. Narayan: Mechanical Illustration of three Magnetically (oupled Oscillating Circuits. «(Plate XII.) ..............000, 575 Prof. F, D. Murnaghan on the Deflexion of a Ray of Light in the DILAD (Cheng tere] ae) 0 re ae 580 Dr. W. J. Walker on the Effect of Variable Specific Heat on the Discharge of Gases through Orifices or Nozzles .............. 589 Prof. J. S. Townsend and Mr. V. A. Bailey on the Motion of Elec- EOUILS Gi AMBOGI Bh OR aa eee ee 598 Dr. G5.) effery : Identical Relations in Einstein’s Theory ...... 600 Dr. D. N. Mallik on the Mutual Induction between two Circular Cert a adey Lee eke seis saat osc kreie coh aa isielelstéae se aes 604 Mr. Bernard A. M. Cavanagh on Molecular Thermodynamics...... 606 Notices respecting New Books : — Mr. J. Edwards’s A Treatise on the Integral Caleculus........ 636 iProccedings of the Geological SOGIeLy. . . 0. eee ieee ecto se ces 639 v1 CONTENTS OF VOL. XLIII.—SIXTH SERIES. NUMBER CCLVI.—APRIL. Mr. C. G. Darwin on the Theory of Radiation .-................ 641 Mr. Shizuwo Sano: Thermodynamical Theory of Surface Tension.. 649 Drs. Paul D. Foote, F. L. Mohler, and W. F. Meggers on a Sie- nificant Exception to the Principle of Selection. (Plate XIII.).. 659 Prof. A. L. Narayan on a Modified Form of Double Slit Spectro- photometer. (Plate XIV. z sb odaeida cihGls oe. seth ool 662 Messrs. A. L, Narayan and G. Subrahmanyam on Surface Tension of Soap Solutions for Differ ane Concentrations. (Plate XV.).... 663 Prof. B. M. Sen on the Kinetic Theory of Solids (Metals) and the Partition‘of Thermal! Hnerey.— Part do. 22.22. eee 672 Prof. B. M. Sen on the Kinetic Theory of Solids (Metals) and the Partition of Thermal Hnerey._-Part 1. 9). ae 683 Dr. J. 8. G. Thomas on the Thermometric Anemometer .......... 688 Mr. Gerald A. Newgass on a Possible Physical Interpretation of Lewis and Adams’ Relationship between 4, c, ande .......... 698 Dr. Balth. van der Pol on Oscillation Hysteresis in a Triode Generator with Two Degrees of Freedom .................--- 700 Prof. R. Whiddington on Polarization Phenomena in X-ray Bulbs. 720 Sir J.J. Thomson: Application of the Hiectron Theory of Chemistry to Solids ois a he) ere a latent Ie ee er 721 Prof. R. W. Wood on Fluorescence and Photo-Chemistry. (Plate OVD) a ean alah seal any al Wg aoe ece' lala etece euete itz (elses eae 797 Mr. F. Ian G. Rawlins on a possible Relationship between the Focal Length of Microscope Objectives and the Number of Fringes seen in Convergent Polarized hight.) ...5 000. oe ee 766 Dr. J. W. Nicholson on Products of Legendre Functions ........ 768 NUMBER CCLVII.—MAY. Mr. R. H. Fowler on the Kinetic Theory of Gases. Sutherland’s Constant S and van der Waals’ @ and their relations to the intermoleewlar field) (00. ioc. ee eo sae ee 785 Mr. C. G. Darwin on the Reflexion of X-Rays from Imperfect Onystale.-o. 2 oon ee ie setae la eb mip eee) ale reren neonate een 800 Mr. U. Doi on Scattering and Dispersion of Light .............. 829 Mr. G. A. Hemsalech and the Comte de Gramont: Observations and Experiments on the Occurrence of Spark Lines (Hnhanced Lines) in the Are.—Part II. Magaesium, Zinc, and Cadmium. (Plates VAD RRR eae aia. Seale ee ston lie dete tee oan aa 834 Mr. H. Carrington on Young’s Modulus and Poisson’s Natio for Spruce... (Plate XXII) volo vices. tie be) Pee ee 871 Dr. Charles Davison on the Diurnal Periodicity of Harthquakes .. 878 Dr. Albert C. Crehore on Atoms and Molecules.—II............. 886 The Research Staff of the General Hiectric Company Ltd., London, on the Disappearance of Gas in the Hlectric Discharge. cee ., OU Mr. A. G. Shenstone on an Attempt to detect Induced Radioactiv ity results, trom é-hay Bombardment) ph... ccs. 0. eee 938 Mr. A. M. Mosharrafa on the Appearance of Unsymmetrical Com- ponents in the Starke Witeet. 2 ale sae oe Se eee Oona ee $43 Prof. G. D. Birkhoff on Circular Plates of Variable Thickness .... 9538 Dr. Gregory Breit on the Effective Capacity of Multilayer Coils with square and, Circular Section |. 2100 0h..-): 0s 963 CONTENTS OF VOL. XLIII.—-SIXTH SERIES. vil Page Dr. Dorothy Wrinch on the Orbits in the Field of a Doublet...... 993 Dr. E. H. Kennard on a Simplified Proof for the Retarded Poten- Pere rluyohenas PTIMCiple™ . 05. ue kee cele tee eens 1014 Prof. 8. P. Timoshenko on the Forced Vibrations of Bridges...... 1018 Prof, Frank Horton and Miss A. C. Davies on the Occurrence of ieaon by Cumulative Kffects 22.0... ca ieee cee ee 1020 Prof. S. Timoshenko on the Buckling of Deep Beams............ 1023 Mr. Frank M. Lidstone on the Measurement of Absolute Viscosity . 1024 Intelligence and Miscellaneous Articles :— On the Measurement of Absolute Viscosity, by Jobn Satterly . 1024 NUMBER CCLVIII.—JUNE. Dr. Albert C. Crehore on the Hydrogen Molecule—III. ........ 1025 Mr. Allen G. Shenstone and Prof. Herman Schlundt on a Deter- mination of the Number of a-Particles per Second emitted by eorum © of known y-Ray Activity ..0. 60. eee eee ee 1¢38 Mr. Arthur Fairkourne on Restricted Movements of Molecules at very low pressures: A Limit of Applicability of the Second eters WHC OMON PNOPMCS ees kk cok cis ee Ge be eee ewe ds 1047 Mr. E. J. Hartung on the Construction and Use of the Steele-Grant To TE RSPR TERR ce ona i fee Td oe 1056 Mr. C. F. Bickerdike on the Interaction between Radiation and VINES (8S alle Sigs ee mens nan aera an crea Ye ae ety oon ol 1064 Mr. D. Coster on the Spectra of X-rays and the Theory of Atomic mma, (edate MOREE) OW ere fk ea blew swine « 1070 Mr. V. A. Bailey on a Development of Maxwell’s Capacity Bridge . 1107 Prof. N. Bohr on the Selection Principle of the Quantum Theory . 1112 Prof. R. Whiddington on X-ray Electrons. (Plate XXIV.) .... 1116 Messrs. J. S. Townsend and V. A. Bailey on the abnormally long Free Paths of Electrons in Argon 1127 Mr. Satyendra Ray on Viscosity of Air in a Transverse Electric 23 2 ee ee es Oi nA Dike eek aE oes eee 1129 Notices respecting New Books :— Prof. I. H. Neville’s Multilinear Functions of Direction and their uses in differential geometry. .........0.c00.000e- 1130 Prof. H. 8. Carslaw’s Introduction to the Mathematical Theory of the Conduction of Heat in Solids Sir George Beilby’s The Ageregation and Flow of Solids.... 1131 Prof. E. F. Burton’s The Physical Properties of Colloidal SCLITISTT 1S 0 Sete ae eg On nm a gr ye 1132 Dr. Norman R. Campbell’s Series Spectra Dr. Felix Auerbach’s Moderne Magnetik Proceedings of the Geological Society :— Mr. Henry Brewer Milner on the Nature and Origin of the Pliocene Deposits of the County of Cornwall and their bearing on the Pliocene Geography of the South-West of ee) * {ea She) 2; (0) ¢e tee 6 Yolts) (a) 608 © Ww © ee © 6 MMPS er nee on Ss ye eS ede ad Wore as @e os 1135 Mr. Launcelot Owen on the Phosphate Deposit of Ocean Potente 7c o's... Re sai cae ene ss sie) wre ede gw « 1186 PLATES. I. Illustrative of Mr. V. Lough’s Paper on the Beating Tones of Overblown Organ Pipes. II. Illustrative of Messrs. A. F. Joffe and M. V. Kirpitcheva’s Paper on Rontgenograms of Strained Crystals. III.-V. Illustrative of Mr. G. A. Hemsalech and the Comte de Gramont’s Paper on the Occurrence of Spark Lines (En- hanced Lines) in the Are. VI. & VII. Illustrative of Prof. C. V. Raman and Mr. Nihal Karan Sethi’s Paper on the Convection of Light (Fizeau Effect) in Moving Gases. VIII. Illustrative of Mr. H. P. Waran’s Paper on an Interferometer Method of determining the Phase Difference resulting from Metallic Reflexion. TX. Illustrative of Mr. B. N. Chuckerbutti’s Paper on the Defor- mation of the ‘‘ Rings and Brushes,” as observed through a Spath Hemitrope. X. & XI. Illustrative of Prof. A. L. Narayan’s Paper on Coupled Vibra- tions by means of a Double Pendulum. XII. Illustrative of Prof. A. L. Narayan’s Paper on Mechanical Illustration of three Magnetically Coupled Oscillating Cir- cuits. XIII. Illustrative of Drs. Paul D. Foote, F. L. Mohler, and W. F. Meggers’s Paper on a Significant Exception to the Principle of Selection. XIV. Illustrative of Prof. Av L. Narayan’s Paper on a Modified Form of Double Slit Spectrophotometer. XV. Illustrative of Messrs. A. L. Narayan and G. Subrahmanyam’s Paper on Surface Tension of Soap Solutions for Different Concentrations. XVI. Illustrative of Prof. R. W. Wood’s Paper on Fluorescence and Photo-Chemistry. XVII.-XXI. Illustrative of Mr. G. A. Hemsalech and the Comte de Gramont’s Paper on the Occurrence of Spark Lines (Kn- hanced Lines) in the Arc. XXII. Illustrative of Mr. H. Carrington’s Paper on Young’s Modulus and Poisson’s Ratio for Spruce. XXIII. Illustrative of Mr. D. Coster’s Paper on the Spectra of X-rays and the Theory of Atomic Structure. XXIV. Illustrative of Prof. R. Whiddington’s Paper on X-ray Electrons. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCLENCE. 3 SIXTH SERIES.] \ . <, \ 8g pages HOSEA NU AR Y 1922, — I. Zonal Harmones of the Second Type. By J. W. Nicuotson, £.R.S.* HE function Q,() has received but little study in some of its aspects. In fact, nowhere in the literature of the subject has the writer been able to find the value of the integral a LQn(H) ]’dp, _which is clearly fundamental. Together with certain other properties of the function, it was found necessary recently to evaluate this integral in connexion with the solution of an integral equation arising in the problem of two parallel con- ducting disks. The capacity of an electrical condenser of this type depends on these properties, and can be expressed without approximation as an integral containing ( functions. A short account of these properties which, though apparently not on record or perhaps unknown, can be obtained in an elementary way, is given in the present note. The function Q,(«), the second solution of Legendre’s equation, may be defined by dr Qu() =4P (a) log FIT — & ayaa when p is less than unity, where the coefficients a, are numerical. * Communicated by the Author. Phil. Mag. 8. 6. Vol. 43. No. 253. Jan. 1922. B \ 2 Prof. J. W. Nicholson on Zonal In the first place, by a usual procedure in the theory of differential equations, we can show at once that (m—n)(m+n-+1) (" Qn (He) Qn(m) de mee = [ (1—p’) (QO; Qn — Qn Qi) | 45 the accent denoting differentiation with respect to uw. We shall suppose throughout that m and n are integers, and that w lies between the values +1. The right-hand side of this equation becomes, by an easy reduction, (1 —p”) ~ Or laren = Us Page 23 =i (all — 1") > Ay Ph olae 2 As Pees r=0 s=0 | Ihe l+yp. aoe 41 —") en log i. Cy peo a 2 (1 ee ji) ee log = > ds Pian il The functions P and P’ are polynomials, and finite for all the values of « concerned. Thus the first row vanishes by virtue of its factor l—p”. The second also vanishes because the limit of y (le) log > aes ie is zero when p=+1. We deduce that 1 (m—n)(m+ 2+ »f OF Ode — 0) i and if m and n are different, 1 ( Qin, OQ, dz—0. oe a as (Ge etait It is more difficult to discuss the case m=n. The generating function / of the series a h” Qn (mw) =F (H, h), which also defines Q,(w) when n is an integer, is too com- plicated to permit of use after the usual manner adopted for the zonal harmonic P,. Instead, we shall expand Q,(w) in a series of harmonics of type P,(mu). The possibility of the expansion is evident from the facts that (1) the product P(t) Pale) 1s so expunsible by Adams’s theorem, and l+p a (2) log; - except at w=+1, must admit such an Harmonics 07 the Second. T: ype. 3 ‘expansion because the integral 1a Pan) logy Bd exists, and in fact can nt determined readily. For we know that if p is any integer, v1 Po(w) du=0. Moreover, sea 1 ba) iD Ce Re Qriet [09 e=— [ay OG GO MS 2 eda Pease! Ge G22) ol age? Near b= +1, rf is infinite like ae or like an ae Thus | f Q,(p) dp= — sey, if n is even eee n(n+1) , —2 te: Se if n is odd. Accordingly, cite to the definition of Q,, 3 Ht ee | { P.ot0e log T= == n+ 1) if n is odd, =( if n js even. ‘This is equivalent to ; =4 { Pw Qo(w) du = n(n-+ 1) USE and enables us to write down the expansion of Q (jw) at once in the form 3 7 ar a —Qo(u) = moet gre pubs. 2) This result is quite elementary. For the general expan— sion, we return to the differential equation, and can show readily that (m—n)(m+n+1) 4 Pa(z) Qn(H) dp “yp 1 = [ (114) (Qn Pa! =P, Qu’) ] y (3) where the only terms in the bracket, not-obviousiy zero as B2 4. Prof. J. W. Nicholson on Zonal polynomials with factor 1—y’, are (l=?) P,P, log — 4(1—p’)P, Py'log gee d Lae fe = Em Ein dae ) dee ies = The first and second of these three expressions tend to zero at the limits, and thus (man omtn gf 7, OO oe Ee ; a) = (eee and is zero if m+n is even, and equal to —2 if m+n is odd. We deduce the two formule j ae) lh = ou onze _Renle) Pon41 (1) Ne (2m S| De i (Qm+ ie ae (4) 1 ‘, | : cS ee anh) it (2m—2n+1)(2m+2n+ 2)’ & where m and n are integers. Since when n is odd, Q, is an even function of w and P, an odd function, and wice versd, [ 2@ewa=0. in all cases in which p and g are both odd or both even. Similar integrals from zero to unity are of importance in the applications mentioned. For these, we require Q,(0), On (Om shies By eee pia Or EP, logy? — (e277 2n—5 m sey 5 pot e ae (7) [leew we readily find Q)=0, QAO)=—1, Q(0)=%, Q(0)=— corresponding to PxO)=1, P\O)=0, P,(0)=—3, P(0)=0. The values of P,,'(0) are coefficients of powers of h in the expansion of h(1+?)~?/?, the general value being ay f hah OMe. Seem wo age PO) a) aa eee Bo Or Harmonics of the Second Type. as against 1.3....2n—1 Fe, 1(0) = 0, = (= ee DAT Ty eu DR Ox (9) The Q’s admit a recurrence formula (n + 1) ayes (2n + 1) BQ» + n er — 05 so that ts (0) | Ones (0) — n | (n+ 1) On Qn+2(0) /Q,.(0) =—(n+1) / (n+2). We thus find without difficulty, that BMAN di 2in Goan OT F ee) Me a one T a From another recurrence formula, (27+ 1) wQn' = (n+ 1) Qn-1+ 2 Qs, we have Q'n41(0) / Q',-1(0) = —(n +1) | n or Q'n+2(0 al Ge (Oj = —(n-+ 2) Na (n+1). But il Qo (u) rar t=? ) (Qo (0) = L whence a) =8, QYni()=0, Yn(=(—)* a. (a1) Returning to the formula (m—n)(m+n+ ye Q d= (1 — 27 )\(Qn Pn —Pr Qn’), which is true between any two limits, we can evaluate the whole set of integrals between limits zero and unity. Thus ul (2m— 2n)(2m + an 0 »| Pon Qomd uw 0 = —Po(1) Pon(1)— {Qem(0) P'n(0) — Pax(0) Q'2m(0)} pie coe oe le en ea x DI ee — \nm+n an Pa Se, Ce Ie lee om ke (12) 1 (2m—2n—1)(2m+ 2n +2) | Pon+1 Qem dpe 0 = —1—{Qom(0) P’on41(0) —Pan41(0) Q'2m(0)}= —1. (13) 6 Prof. J. W. Nicholson on Zonal Thus ] ie ‘h Pon+i() Qom( mw) dw = — 1 / (2m — 2n —1)(2m + 2n + 2) : (14) for all integer values of m and n. In the same way, | ( Pan () Qom+i(u) du=— 1] (2m— 2n + 1)(2m + Qn + 2), (15) and the fourth case can be worked out at once. EKapansions of Harmonec Functions. If, returning to (4), we have Qom(H) ae Ex an Pon 4i(u). Then BS eas An Ay a Qen(H) on+1( fe) fad 2 aoe (2m —2n —1)(2m +4 2n+ 2) and we obtain the expansion med a (An a 3) Ponti) : Qon (Hw) = eS (2m—2n—1)(2m+ 2n +2)’ Co, which is convergent when pw is between +1, both exclusive. Similarly, with the same convergence, ON A ie (4n +1) Po(p) Qan+i(H) = 4 (Qm—2n +1)(2m+2n+ ays (17) The saine expansions are valid for the ‘associated Legendre functions ” dt ha Pin()s Qn” (Hw) =i @ = fh) he du” (P,,, Qn) by operating on both sides. ‘Thus Tr ht ae (4n+ 3) Pons1(p) : Qem ae (2m —2n—1)(2m+2n+2)’ (189 UN 0 (4n+ IED) Qen+t (Mw)= Xo (2m—2n 4- 1)(2m+2n+ 2) £ ve Harmonics of the Second Type. i From (16) we obtain, on squaring, v ea Quon (u) dp (An+3)? | =3; (2m —2n—1)?(2m+2n+ 2)? « [Paaal) ee. the limits not requiring delicate considerations. Thus 1 ie An+-3 Wale? an, 6 {0.601 d= a>, (2m—2n—1)?(2m+ 2n+2)? ae SG if if ~ 4m4+170 Spear —: (20) This can be expressed in Gamma functions. For the special case m=(), the value is 2( 5 1 1 Ml if i ae ed 12 . For m=1, it becomes es eee aaa ma 1? 32 sy In general, the value may be written as 2 if i 1 hie 1 ac a ee oa “peg, _.ad inf. Sear ; | or r = an (2m +4)? + ...ad inf. ) - 1 1 pail? (a+ 5 3 2 Gopal == aay ef. or boii at r +7 dz slog T(e Igoe ‘Finally, ‘ib [Qon() ]2dp= moi; t oylog r@} rn (2) z=2m 8 Prof. J. W. Nicholson on Zonal By the use of Binet’s formula when z is large,. we readily find that the second term of the bracket is of order : when m is large, so that the integral tends, for large values of m, to oe This is important in regard to the convergency of series of Q functions obtained as representations of given functions. The case of functions of odd order may be noticed briefly. We show without difficulty that | [Qom4i(p) | ; dp _9s ance 0 (2m—2n4+1)?(2m+ 2n + 2)? oe 1 1 ’ ~ 4m+3~0 Von: (2m +2n+2)? “-3{ (tat + ociy)t(etqet ~adine) a pay sae (2 Be ee (2m +1)? io 32 ... ad Int. if = vA “i ee 3 a I ear me Bey eo Wee \ Am+3) 4 (2m+ 2)? Qm+3)? °° and finally : 2g y qe Ge het Ne [ Qom+i(+) | ee ae A zs ee 02 Olin (22) The formule are clearly the same whether the order of the function Q be odd or even. Catalan gave the following expansion :— TT 2O fl vos ees Lie . which is typical of other useful results not apparently noticed. If this be differentiated, we find, since d Porat Res d ‘Pow i dp. dp sin = (An + 1) Jess 1 To? 1.3....2n—1)/ at Feat Pho et | Fae ee ) Pos) Harmonics of the Second Type. The simplest proof is probably by use of the formula { P,,(cos @ cos 6’ +sin 6 sin @ cos ) dd 0 = P,(cos 8) P,(cos @ ). Writing 0=0'= 5, P, (cos $) dp=m[P, (0) which is zero when n is odd. pepe: cos f=, we have (oe {Bp (0) 2, Sale ie leading immediately to the expansion. If a direct proof be adopted as follows, we obtain important properties of the Q functions. For, whether P stands for P or Q, we have ——ad P dP n n+1)\P, dp = — »/1—p?—— Hts ety | ( )} Pn (m) du peel tent a: and therefore, applying this formula twice, if a Us we have Vv1—p n(n+1)uU,—(n—2)(n--1) un _2 ds ee dP -S=") os H(G2- Nay -\4.(% Ue de ) == 2-1) WI= WP. On [EP dy (an —1)4/ J — py? 2 P,-) +n Unt (n—1L)tn-2 or Wur—(n—1)?u,_2= —(2n—1) V1 —p? Pan. Between limits +1, the right side vanishes whether the Pp or @ function is eaccenedl and Un/Un-2=(n—1/n)?. ee ALD Prof. J. W. Nicholson on Zonal Phas. |.) \ Pon(p) d =“ 21 dp 1NV1— EN WE eae ) ro Vl—p? leading at once to the result. Moreover, ee | 2.4....2n ab Qi) lp (23) 1 Tee vl 3.5... 20-+ 1p ae The last integral is readily evaluated. For l+yp Qi(w) =4p log —1, (ee) = 3h cia oe and therefore 1 : Qi(u) acne 1l+cos0é ) oF ea (3 G08 7108 rare ay ae dé =) "cos 6 log cots dO—m 0 : a /2 j ‘ =2h cos 28 (log cos 0—log sin 0) d0—T. : 0 Now when + is odd, it igs known from a familiar result in the integral calculus that m/2 ( cos 2rA log cot @dd= 5 Q PB a’ So that, in the present case r=1, we find Qiu) ie dp=2 5TH. It follows that for all integer values of n, 270, rs mii J1l— a somewhat surprising result, which is a special case of a formula in the next section. It can be expressed also in the form . ( Q, (cos. 0) d0=0.. ~s eee 0 flarmonics of the Second Type. 1l Relation to Fourier sertes. The Fourier series representing P,(w) is well known, and can be found at once. For P,(cos@) is the coefficient of h” in the expansion of (1—2hp + h?)-?=(1 —he®)-2(1 —he-®)-3 and expanding each bracket and multiplying directly, we- find Pein 1 2a Palco) =2 «ani, 18 8089+ 9-9, cos (n—2) 0 Nese .o2n.. 2a — 2 ieee Zn—1l.2n— CE ome This invoives the fact that x P, (cos 6) cos (n—2r) 6 dé 0 i: 2n ! 2H) C2 ease | a aero 2 (27) BRR n Pye 2 !)?” 2n—1.....2n—2r4+1, when n—2r is a positive or negative integer, the integral being zero otherwise when n and vr are separately integers.. On reduction, with n—2r=m, under these circumstances nee yr) T D ( y » i) Remens eas mig se N28 4 p(“+24")r nae) eles This is the same as the integral t SEC HTS + P.,,(u) cos (meos™! K ) ( H) V1—- pw? Thus we have a) cos (m cos 3 }) 3A. Pita), where. Vie ae 7 . A= Bes, i ( P..() cos (m cos~? p) pelle . (n>m), a Ni1-p EY Prof. J. W. Nicholson on Zonal Let m-be even. Then n is even, and writing 2m, 2n for them, | cos(2mcos"*) Tin+m+5)Ca—m+3) ip 1 2 Pope) Vip =2>.,, (40+ Tene 2n(#) (29) The case m=O gives the old expansion for (1—y’)™. Integrating between zero and p, sin (2m cos~!p) cot oe n—m+4 a } . =mz,, Tintm+)P (n—m-+1) Pon+1(H = Ea (30) ‘of which the case m=O is Catal: an’s theorem. We also have , cos (2m+1. cos pm) A ai? ware Ponti (“), ~where d An= a Ponea(H) ner 1) cos~* pw} Vie _ 4n+3T(n+m+ 8) T(n—m+3) ~ 2) Dnt+m+2)Cm—m+1)’ and therefore cos(2m+1.cos7!p) Ving T(n+m+3)C(n-—m+ 4) D(n+m+2) D(n—m +1) Ponsa (u) ee _(4n +3) and sin (2m+1.cos~!p) oD(nt+m+3)Tin—m+4 ™D(n+m+2) Cin—m+1) =(m+4)> } Ponsa) —Pan(u) b. (31) In order io obtain corresponding developments relating to Q functions, we first find the Fourier series-by another Harmonics of the Second Type. 13. method. We have it Q, (cos 0) {cos mA —cos (m+ 2)0} dO =2(" om (cos @) sin @ sin (m+1)0d@ i: 2sin(m+1)6 in? 9)" : =| n(n +1) de Io 2(m+1) “n(n+1) = +16. n? oe oe elk n(n+ 1) ayy x i. sin 6 cos (m+1)@ 77 de: 2(m+1) n(n+1) 2 NOW es = rar ( {cos 8 cos (m+1)6 0 — (m+1)sin @sin (m+1)0} Q, dé. The integrated terms vanish. If Min O.cos (m+1)é. Qe | 0 v= |" Qa (cos 8) cos ma da dulce ae) 0 (m being an integer), the equation is readily reduced to Umt+2 __ (n—m) (n +m+ 1) Gi a) (a me Oy (33), In particular, if m=7, Uni2=0, so that te (,, (cos 0) cos (n+ 2) 0 .dA@=0 0 for all integer values of n. This involves the ‘further consequence that {7@ (cos 8) cos (n+27r)0.d@=0 0 if n and rare both positive integers. It is, in fact, odd in. e0s@. Again, if m=n—1, u,,=0, so that 1 Geos) cos (n—1)0.db=0;° 4. (34), -and since = ‘it follows that 14 Prof. J. W. Nicholson on Zonal Mies (n—m +2) (n+m—1) Ure (n—m+1)\(n+m) ’ we have ; Um —2 = 0 = Un—3- Thus i Q,, {cos 8) cos (n—1—27r)0.d0=0 . (85) ‘for integer values of n and r. Now let 2 be even, and written as 2n. If m is also even, and written 2m, Hon= | i Qon (cos A) cos 2mO dé 0 (2n— 2m + 2) (20-4 2m — (2n—2m+1)(2n+2m) ” = U2m—2 and so down to uw, which is zero as might be foreseen, Qo, being odd, and cos 2m@ even in p. Thus (" (on (cos 8) cos 2mé dd =0 e 0 ‘for all integer values of mand xn. Similarly, 1p (Qon+1 (cos 8) cos (2m+ 1)0dd=0. 0 The equation (35) has a remarkable consequence. For “since : | ” Q, (cos 0) cos (n—1—2r)6.dd=0, Q -and also, the integrand being an odd function, { "(con 6) cou rye Jo | i " Q, (cos @)\cos md) = 0. ee ‘ for all cases in which m,4; admits the Fourier series Qon+1 (cos 0) 2n+1!) 1 4n+4 epee 9) 9 et =2 wee cos (2n+2)0+ 5.7 pee tee Sean” OT ae ee cos (2n+ 46) 4 cos (2n-+6)0+ ... b, . (43) which ceases to converge at the limits. We now consider the integral \ Qon (cos @) cos (2n+1)0d8. 0 Pint. Mago S.6-V ol. 43. No. 253. Jan. 1922. °° C (* Qon(cos @) cos (2n4+1+2r)0d0=7 2/9 tonal Zonal Harmonics of the Second Type. This, by successive reduction of the order of Q, is equiva- lent to 2 84a On In 2h A eee tes ond) aeons (“ Q (cos 0) cos (2n+1) 0dé@. Jo By the other reduction formula, we find (* > (cos @) cos (2n +1)0d0 0 #13 4.2. 2n—1 Zs 4b 77 oA ae ee a sf Qy (cos 0) cos 6 dé and therefore (" Qon (cos A) cos (2n+ i 6 dé 0 eee = eae Qp (cos A) cos 6 dé. The last integral is Gal ay m/2 9 C08 0 log cot 3 de= aN cos 2¢ log cot dé dd=z, On and thus ™ 2n !)? { Qen(cos 8) cos (2n+1)0d0=7. 2” oe . (44) Moreover, ‘' Q., (cos @) cos (2n + 1+ 2r)0d0 ny aro: Petia An+2.4n+4.... 4n+2r gary wir OREO 2) An+3. Op aa | “Qo,cos (2n + 1) 0d, whence 0 2r! (2n+r!)? (r!)?4n+2r4+11 ? Derivation of Symmetrical Gravitational Fields. . 19 the corresponding Fourier series being (>, (cos 8) | (2n' 1)? i es 2 — Jintl one 9 : = a 108 Qn+1)e+4 3 008 ( n+3)0 1.3 4n4+2.4n+4 ey ane cos (2n+5)0+ ae ie (46) Both series can be combined into the single formula, whether n be odd or even, (,, (cos @) (n !) 1 2n+2 = 92n+1 TET OM) O45 On ape aan) 1.3 2n4+2.2n+4 _ | are Wee S apSCe SOF f. tte GQ) The limitation Oe @ between 0 and zw is clearly not _ necessary. Il. The Derivation of Symmetrical Gravitational Fields. By F. D. Murnacuan, Ph.D., Assoc. Prof. Applied ane Johns Hopkins University *. i a recent issue of the Philosophical Magazine, Hill and Jeffery f call attention to a symmetrical gravita- tional field which differs somewhat from the classical one due to Schwarzschild and Hinstein. In the usual treatment of this problem a field is said te be symmetrical about a point if the form for (ds)? is invariant under linear ortho- gonal transformations of the “Cartesian coordinates ” (21, 2, #3); then a transformation is made to “ polar co- ordinates” rv, 0, 6, where r= /a?+a,?+ 2,7 etc. and an appeal is made, in choosing a form for (ds)?, to the corresponding form in Huclidean space. Now the essential assumption in the relativity theory of a permanent gravita- tional field is that the physical space-time continuum of four dimensions is non-lKuclidean ; the term “ Cartesian coordinates ”’ for a non-Huclidean space requires definition, and the equation for r given above assumes an underlying * Communicated by Prof. J. S. Ames. + Phil. Mag. May 1921. ‘he result of this paper has already appeared i in a paper by Weyl, Ann. der Physik, liv. p. 132 (1917). C2 20 Prof. F. D. Murnaghan on the Pythagorean theory that is not tenable for non-Huclidean spaces in general. Since the only existing experimental verifications of the relativity gravitational theory are based. on the expression for (ds)? in a permanent symmetrical gravitational field, it is desirable that the assumptions in the mathematical treatment should be clearly stated. In the following discussion Hinstein and Schwarzschild’s form as well as others are derived on certain definitely stated assumptions as to the meaning of the term symmetry, and, peculiarly enough, the differential equations of the theory prove easily ‘integrable, without making any use- of the form of the Huclidean element of length on the unit sphere. According to Hinstein the fundamental space which has physical reality (that is, with reference to which the. laws of physics must have the tensor form) is of four dimensions. All coordinate systems are, without doubt,. equally valid tor the description of physical phenomena, but it is reasonable to suppose that for a given observer a certain coordinate system may have a direct and simple relationship to the measurements he makes ; such a system: is termed a natural coordinate system for that observer.. For the problem under discussion, one of the natural co- ordinates 2x, 1s a time coordinate which is such that: the coefficients gi4, 921, gs, in the expression for (ds)?, (ds)” = gr5da,dx,... (7, s umbral or summation symbols) vanish identically whilst the other coefficients grs do not involve a,. Accordingly (ds)? = g,-.dx,da.+gas(da,) - (r, s=1, 2, 3). ' Now in’ a space of. three (hiss we can always find an orthogonal system of coordinates. (71, Yo, Y3) Say, for we have merely three differential equations g” ou Gin = 0. O.-. (7, s = 1) 2) 3. ame smart symbols whilst / and m are different numbers of the set 1, 2,3) (the g’ being the coefficients of the reciprocal form for (ds)*) for the three unknown functions y;, y, y3 of #1, v2, #3. Hence, using these coordinates instead of Eee 2, (OUL keeping the notation «# instead of y) we see that there is no lack of generality in writing (ds)? = gui (421)? + 922 (daz)? es +gs4(dx,)?, where the coefficients g,, are functions of (2, #., x3) at most. [It- will be observed in passing that this statical field is very special; it is not in general possible to find orthogonal coordinates in space of four dimensions, there being sw oe Ge 9 te GN Cape ta aa £) for the four eouations g PLE Derwation of Symmetrical Gravitational Fields. 21 unknown functions y, and these equations are not in general consistent.! It will be convenient to drop the double label, which is now useless, and so we write (ds)? = gi (da)? + g2(dae)’ + gs (das)? + 94 (day)? ‘Since g,,=0 if rs the Riemann-Christoffel symbols take comparatively simple forms. If 17, s, ¢ are distinct numbers of the set 1, 2, 3, 4, oat Qe Bee Be) a irs, t} = g* | rs, k] = g# (rs, t] = 0, since g*=0 if the umbral symbol & is different from ¢. and {rs, 7} = to, 7} = 9" [r5, K] = gy" [vs, 7] SoU since g” = aa Ir Similarly, | {rr, 7 99, Sa {rr, sl = ea The Riemann four-index symbol of the second kind {pq, rs} is defined by 1pq: 785 = 2. {pr, 9} — {ps qs + {pr, 1} {ls, 9} —{ps, U} {lr, gq}... (1 an umbral symbol). Hence, if the four letters p, g, 7, s have all different numerical values, {pq, rs} = 0, since, for example, in the summation {pr, /}{/s, g} the first factor {pr,1/} of any term vanishes unless /=p or yr, in which cases the second factor vanishes. The remaining types are readily found to be {14 4S a 0794 1 O9o O92 __ 1 0% 09 299 OLsOLy Aged OxpOLs AQnGq Oxe 02, el 09s 09. A9;9q OLp O&s where g is not intended to be an umbral or summation symbol. 22 Prof. F. D. Murnaghan on Hie 1 Ooi k Ome ue: ge 1 O9p 2G, 4 Pq, = 204 Quy? 294 Ova Ag¢ (5, Age Om, Om, __ 1 (Og O97 092) 1 1 Og, Oo meee cat ae eas Oz, ; 49 i Of oz gs Oks Se where p and q are distinct non-umbral symbols and 7, s are different from p and gq. In order to obtain the differential equations G,.=0, which Hinstein assumes the g, must satisfy, it is necessary to contract this four-index tensor. Thus 4 4 Gpe= Zing gsh=0 and Grp => 1p, api =O are the ten differential equations. Now in the expression for IE qs} it will be noticed that differentiation with respect to x, occurs in every term, telling us at once that ‘pq, q4+ =0 since none of the coefficients g, involve 2... Hence the results Gu=d; Gu =0; G = 0 are a consequence of the assumed properties of a statical field. In the remaining calculations a will simplify matters to observe that in the summation | pq, gs¢ it is sufficient = to give g the two values dimeieat from p and s. For, from the definition of the four-index symbols of the jirst kind, [ pp, gs] =O (p not umbral) and, in our system of orthogonal coordinates, ‘pp, qst = g?*| pk, qs| ... (& umbral) = g?? | pp, qs| = 0, as is similarly {pgq, ss}. (In this argument p, g, s take any numerical values distinct or not.) Thus Gy,= {13, 32$+{14, 42}. We shall now suppose that one set of coordinate lines, x, varying say, are geodesics through the space-time point to be specified and the gravitating centre with the same value of x4, and we may conveniently take v, as the are distance from the gravitating centre, which is, then, a singularity of the coordaneie system 3; for the knowledge of merely two coordinates 7,=0 and its 2% is sufficient to determine it—the other two coordinates x, and xz, being indeterminate (for the sake of an analogy #, may be compared with the axial distance in ordinary cylindrical coordinates). The equations of the geodesics in any space: Derivation of Symmetrical Gravitational Fields. 23 are known to be Pein Fr} 2,4,—0...(r=1,...45 1, m umbral symbols), dots denoting differenti ation oul aes to the are length of the geodesic. Putting x, #3, all constant, Ain we find Oey Oe wor lege 4) yielding g,= constant, which constant is in fact unity since ds=dz, when 2, x3, w, are constant (by definition of x). This argument shows, conversely, that if g;=constant the coordinate lines x, are geodesics. It is now easily seen that it is sufficient for this that g, should be a function of x, alone, since a change of coordinates n=\Vgida ; Yo=#o5 Ys= X35 Yar leaves the ccordinate lines 2, unaltered. The arc length along these coordinate geodesics is in this case, of course, not x, but { Vode. As an assumption of symmetry we now say that g, is a function of x, alone and not of 2, nor #3, and we observe that this together with g=1 makes {14, 42}=0, so that Gy = {13, 32} i 0°93 1 093 0493 1 092 O93 mle 021022 Ags” Oxy On. 49093 02, Oe vind to a a 1 093 093 1 O92 O45 » 20a, Js Oe Ags” OX, O02. 49093 041 0&2. Hquating this to zero and writing momentarily lg = log go; ls = 93, it is immediately integrable with respect to «, and we have 2 log 2° +1 —l, = a function independent of a. The formule become more symmetrical from this on if we write g,=H,’ and express our results in terms of the H,. Thus the result just obtained may be written 2 1 oe) is independent of 2, 9293 \OX2 or equivalently (on extracting the square root) a es is independent of z." . - . (A) 2 ? 24 ! Prof. F. D). Murnaghan on the Writing out Gy= {12, 21}+{13, 31}+ 414, 41} with the simplification g,=1, we obtain era (Rey Lid’ge i (2) ab 2g Ox? Age \Oa,) * 293 dx 49? \Oay ony sy ay, Ox) ° and on introducing the H, and sea this to zero there results 11H, een, al een Jaks arene Oui ish One uF : (B) The next in simplicity is Gus = (41, 14} 4 {42, 24} + {438, 34}. Remembering that g, is a function of a, alone and that none of the coefficients involve #,, we have on equating this to zero Ih opgn al ie _ 1 O44 O92, 1 0% Ogs eZ Ox? 49, 02; © dey 0X2 021 49s Ot, Oran = or H ae = = 1 Qaf 4 Oa, LH, on, 1, oe which is immediately integrable with respect to w,, yielding oH, HH gon independent of 2... =) s.u)eyane) Now the two-dimensional spreads 2,=const. #,=const. are what may be termed geodesic spheres. On one of these let us suppose that x; is a “longitude ” coordinate whilst aw, is a latitude coordinate. As a demand of symmetry we say that the expression for (ds}? on this sphere cannot have its coefficients g, and g3 involving the longitude coordinate x3, whilst the are differential along a “meridian” curve ds=/g.da, cannot depend on the latitude x,. Thus gz is a function of 2, alone, whilst gz is a function of w, and w, at most. Just as the non-appearance of the time-coordinate xv, in the coefficients made G,,=0, Goy=0, Ga1=0, so now the absence of w3 from the co- efficients gives in addition Gij3=0, Go23=0. Differentiating Derivation of Symmetrical Gravitational Fields. 25 (C) with respect to wx, and eliminating Lae to is independent of 2,; but it by means H of (A), we see that Heo 2 eannot involve any of the variables but x, since H, and H, are functions of x, alone, and so we conclude that it is a constant. , oH, 2 OL, = a, where a is an arbitrary constant. . (C’) Let’ us now write the two remaining equations Goo=0, G33=0. These take relatively simple forms under the hypotheses just made. ‘Goo = {21, 12} + (23, 32} + {24, 42} oe a a ts) 07930)! (ee 20a" 49.\021 293 eae 493° O02» 1 O42 093 , 1 O92 O94 Ags 021 O21 4g, Ox; 02; ar By 7 /OHs | Eigse, ee ‘ re — a ee (D) primes denoting as usual differentiations with regard to the argument of the function. G53 = 131, 13} + 432, 23$ 4 $34, 43} mL 295! 1 Ses) 1 0793 1 a) ~ 20a," Ags\Oay 292 023" ~ Agegs 02s of) (992) 4 1 O93 094 490\0a1) \Oay 49,02; O24 0’Hz , H3 07H; ae ole Gl OH, | 1 OH, ah 0. = BH" + = |5 Oe? PH? oul OM fee O21 Sy Or, Now wz, and «, are the only variables that can enter ke coefficients g,=H,?, so that (A) gives 2 Ss = el) where g@ isa function of x alone, ¢' being its derivative ; whence H3= H.d+f, where i is a function of 2, alone. 2 is a function of x, alone, Then (B) shows that ee ae so that its derivative with respect to x vanishes identically, yielding pb’ { log eddy, ae 0. 26 Prof. F. D. Murnaghan on the Hence either ¢ is a constant, so that all the coetticients depend on 2, alone, or f H,'—/" H, is a constant giving pee idly We shall discuss in some detail the case when f=0, as this leads immediately to the Hinstein-Schwarzschild form. We have Jale — Had, und on substituting this in (D) we get 7 = constant x H, ye I" HH,” +H,” + HH,’ H, +f == (|). Since the first three terms involve av, alone and the last «, alone, they must separately be constants which cancel each other. If the constants are zero, we derive $!=constant. Otherwise, by at most a mere linear transformation on the variable w, alone (which is unessential to the argument, since it merely changes the origin of measurement and the size of the unit in which it is measured) we may sa Ois= Sin ay... 3 a W On substituting 2 et inetne expression just obtained from (D) we find | lake iahy H "+ HHL ot Ug Phan) ae ; H, a BL, ( ) and (EK) yields exactly the same equation connecting H, and H,. On writing H3= He sin a», (B) becomes ep Se aa aR TT eae Oe oe, ete ! et oe (B’) Differentiating (C’) and eliminating H,H." we obtain | lal NeVaeel ole Malas which yields on integration Ho’ = BH,, where 8 is an arbitrary constant. . 0”) From (C'’) we have Hy" = Hy = 25, ae lala = (447), where ¥ is an arbitrary constant. Derivation of Symmetrical Gravitational Fields. 27 Now on putting H,= 5H, in (D’) we get 2s He By = 0, so that 1=2y*, and our solution of the — set of differential equations is : i H,=1 ; H3= Ho S1N @o 5 Hia= 4 He ; where. Hy satisfies the simple differential equation i H,’? = 2af8 It is now convenient to introduce a new variable 7 which is a function of 2, alone, and which is defined in such a way that Ho=r; then Pe, 9 ee Hy = 5 = he aa8 and (ds)? = (da)? + H.?(dxe)? + H;?(daz)? + H,?(da,)? dr)? , Ly eee : ( an + r?(day)? +1? sin? x2 (das3)? TB (te =f) (div4)"t- di With an obvious modification of the notation this is the classical result. In the paper referred to at the beginning of this paper, what is done is to determine 2, such a function of r that mo — 7H. adr 2a eM tre de = Hide Hy es : Lf jal So ti at dr = — 35 2a Lis Ew ey ee whence He—aBt+ rane = Cr, * If we put ¢”=0, ¢'=constant, and y=0. ti Anoikes schition (corresponding to ¢’=constant) is ene): 2a3 fds)? = Y 442 L (dee P+ 2'2?(da'3)” i, (da'4)°. 28 Prof. F. D. Murnaghan on the where C is a constant, or D(He =e) ene ae Writing a8=m, C=2, we obtain 2 or He = r(1+2) : 2 H.—m = rath a 2r Ay? ‘whence ji = (14 il i. it 1” OH; dr 1 my. 1.-- eo ] * Bde,” B or de” B\” 4r? A eae and then 7 2r (ds) = (14 a [ (dv)? +7? dag? + r? sin? a2 (dus)? ] wee eee, alas i : 2r which is essentially the result of the paper quoted. It may be interesting, in conclusion, to say a few words about the case where ¢’=0, so that IB @ is a function of a alone, as are all the other coefficients. This will be interesting in connexion with certain axial - symmetric solutions of G,;=0 if the coordinate lines x, are interpreted as geodesics though not necessarily through a fixed point in the sub-space of three dimensions z,=const. Gy: is now automatically zero for the same reason as are all the other components Gys., rs; (B) yields Fee ee Jal, I Hn, foe where primes denote as before differentiations with respect to the only variable occurring, 2. From (C) we have H,H3H,’=constant=c, (say), where the constant c, may be zero (taking care of the case Hy =0 tacitly omitted on division across el before ntegration). Derivation of Symmetrical Gravitational Fields. 29: (D) gives on division by H,H,! and integration i He = constanti—c,, where as before cy may be zero. — (E) yields similarly Fisiiishi, = es. These equations may be written symmetrically Ce eae 1 Gel see vesHs HEH? it being understood that if c, for example, is zero we omit this member of the equations and write instead «=O or Hz = constant. It is apparently advisable to write l,=log H, ete., and we have bee ts Le! é Gill tee bs en eH El = JT IT JT bi £3 5 I, 1 2h Ss passe’ at eg A ! ibe LEG Suppose, for example, c,0 and eliminate J; and i, and. obtain 1p! = —1,' CG == ee l' 3 2 Cg whence z oe Co + ¢3+ 4 Ee ( C2 Choosing the origin of measurement of «x, so that this. additive constant is zero, we obtain ) (x, + const.). ae ee RACE. i e i Z Cotegt+c, a ee 108 £ “i fe aU eemras eT sre Co . c3 or lae = Coa c2tates : Ee = Ca ac2t cate : c4 H, = C422 F3t 44, (There is an exceptional case, co+cs+ci1=0. Here Ee 0), lo’ = const. = ¢,k (say). 30 Derivation of Symmetrical Gravitational Fields. Then | Is — Cok + % 5 le = c3ka, +B; la = cykay+y, where a, 8, y are constants.) The constants Cs, Cs, Cy or a, B, y respectively are not independent. We have CyC3Cq = cotegste, or a+8+yt+logk =0. There remains the equation ae Te! Ey” ia H, " Hs TED) sab 4 Ia! + [y+ Uy” + I? +1 +1? = 0. or Again eliminating /; and lj, 2 2 2 Te Oe wlio oa q L Tat es et Vite e This shows (a) that if c,+e;+¢e,=0 then J,’=0 (giving the trivial Euclidean case) unless ¢)?+ ¢3?+ cy?=0, in which case all the equations are satisfied. Hence we have the solution (ds)? = (day)? + 602 (dary)? + 628% (dang)? + e201" (dary)? IL = ie (dr)? +772 (day)? + 7° (das)? + 174 (day)?, where r=e, and ¢g, ¢3, c, are three constants satisfying the two equations 6p Ga Gn — Ol 6p" es i (We have suitably modified the units in which a, 2%, ws, v4 ; are measured.) (b) If cotes+c40 we have / le co? + ¢3?7 +6" €2+€3 +4 ln'2 a (2 + €3 + C4) C2 from a previous equation ; yielding C203 + C3Ca + CyCg = O. (This relation is equivalent to - tet == (). .1f none of the c’s =0.) If, then, we choose three constants ¢y, ¢3, Cy On the Separation of the Isotopes of Mercury. a satisfying this relation CoC3 + €3C4 + C4Cg = O and whose sum is unity * we have that (ds)? = (dz)? + a? (dag)? + 2 1°°3(d3)? + @174(des)? will satisfy Hinstein’s equations G,,=0. However, a detailed examination of all possibilities is out of place here. Tt will suffice if we have shown that the WHinstein differential equations are of a regular character and are exactly integrable in a large variety of cases. III, On the Separation of the [sotopes of Mercury. By J. N. Bronstep and G. HEvEsy ft. 1. Introduction. ee hypothesis, based on various considerations, that isotopy, hitherto found only within the domain of the radioactive elements, may be tinally proved to be a general property of matter, has been established by Aston’sf{ brilliant experiments. The question of the separation of isotopes, already much discussed in radio-chemistry, has become thereby a problem of still more general importance. We have already elsewhere § given brief accounts of the results of some experiments made in this laboratory on the separation of isotopes. The present paper contains a closer description of the principles and the methods used, together with the results of further experiments. 2. On some methods of separating of Isotopes. On account of the chemical properties of the atom being materially independent of its mass, only a few methods of separation come into consideration—chiefly, those which make use of the difference in the molecular velocities (atomic * The units of measurement of x,, v3, «, can be so chosen that C,, C,, 2 C, in the equations H,=C, 2,2'%*% etc. above all equal unity and this necessitates c,+-e,;+ci:=1 since C.0;Ci=c2+e3 +c. + Communicated by the Authors. ¢ Aston, Phil. Mag. xxxviii. p. 707 (1919) ; xxxix. pp. 449, 611 (1920); xl. p, 628 (1920). § ‘Nature,’ cyi. p. 144 (1920); evii. p. 619 (1921). 32 ~. Prof. J. N. Bronsted and Prof. G. Hevesy on - velocities) appearing as a consequence of the mass difference- of the isotopes. That is for instance the case with an incom- plete reaction between a solid or liquid and a highly diluted gaseous phase *, and thus also with a partial evaporation or condensation which belong to the most simple heterogeneous reactions. If a liquid is in equilibrium with its vapour, then in unit time a certain number of vapour.molecules will hit its surface and adhere; in order to keep up the state of equilibrium, it is necessary that in unit time the same number of molecules pass from the liquid into the vaporous phase. The velocity of exchange of molecules between the two phases. depends upon the density of the vapour and the velocity of the vapour molecules; the quicker the latter, the more vapour molecules will reach the surface in the time-unit and the: livelier will be the exchange of molecules between the two. phases. : We will now consider a liquid containing two isotopes in equal amounts. As in this case the vapour-pressure of the two isotopes is equal, the vapour phase will have the same. composition as the liquid. Notwithstanding this equality in. the composition of the two phases, a livelier exchange will take place between the lighter than between’ the heavier atoms in the two phases. The former have, namely, in accordance with their smaller mass (m,), a velocity (2) Ve times as great as the latter—the mass and velocity of 1 which we will denote by m,, respectively v,—and so in the e . Mo ° ° time-unit Ape times as many molecules of the lighter as. 1 of the heavier isotope will be transferred from the liquid in the vaporous stage, and vice versa. As long as an ° * A concrete example best explains these conditions. Jf chlorine- gas is passed through a silver tube and only a small fraction of the molecules is assumed to react with the metal, then on account of the- greater molecular velocity of the lighter isotope the molecules of this will have a greater probability of hitting the wall of the silver tube than the molecules of the heavier one have. More of the lighter chlorine will therefore be bound as silver chloride than of the heavier, and—as a simple estimate shows—the ratio will under the most favourable conditions equal / 32 times the ratio of the isotopes in the gaseous phase. By this estimate, however, the possibility of the result cf an impact being influenced by the velocity of the molecules is neglected, as well as the fact that besides (Cl*’). and (Cl*’),, also (Cl°’C1°”) molecules are- present in the gaseous phase. ' q q the Separation of the Isotopes of Mercury. 33d evaporation under usual conditions is considered, we are not in the position to utilize the quicker movements of the lighter molecules ror the purpose of separating isotopes, because the quicker evaporation tn the case of the lighter molecules is just compensated by a quicker re- -condensation. We can, however, prevent this compensation by suppressing one of the two compensating processes ; and on the grounds of this principle we are able to reach a simple method for a partial separation of the isotopes. This is most easily accomplished by allowing the liquid to evaporate in a vacuum and placing a highly-cooled glass plate over is surface. Now when the vapour-pressure of the liquid is sufficiently slight, each molecule which leaves the liquid will reach the cooled wall before it has had an opportunity of meeting other molecules and being thrown back into the liquid. Having reached the cooled wall, it will be held by it, transferred into the solid state, and hindered in re-evaporation. It follows from the above that in the “ideal distillate’? obtained in this way there will be Mo — times as much of the lighter isotope as in the initial My , substance, and through repeated “ideal distillations” of the tractions thus obtained it is possible to carry the partial separation further. The applicability of the evaporation method depends on two conditions: (a) a not too great vapour-pressure of the liquid and a very good vacuum, in order to avoid collision of the evaporating melenles with each other or with other molecules contingently present which would cause refiexion and return to the liquid phase; (6) equal composition of the surface and interior of the liquid phase. If the velocity of the exchange of atoms between the liquid surface and the liquid body be insufficient, then the heavier isotope will be concentrated in the surface layer and soon render all further separation impossible. As we shall see later on (§ 3), these conditions in most cases do not in any way render the evaporation method impracticable as long as the substance to be separated is in a liquid state; while, on the other hand, the requirement of the condition (6) practically excludes the use of this method in the case of a solid substance. The second separation method used by us is the effusion process. If a molecular flow * of an isotopic vapour. takes * Knudsen, Ann. der Physik, (4) xxviii. p. 999 (1909). R. W. Wood, Phil. Mag. xxx. p. 300 (1915). Phil. Mag. 8. 6. Vol. 43. No. 258. Jan. 1922. D 34 Prof. J. N. Bronsted and Prof. G. Hevesy on place through a narrow opening, then the lighter isotope (molecular weight m,), which in consequence of its greater molecular velocity hits the opening comparatively more often than the heavier, will have a greater probability for penetrating the opening than the latter (molecular weight =m,). As the molecules which pass through the opening are soon condensed on the other side at low . temperature, the probability that they are able to go the reverse way is very slight. If in the initial substance the two isotopes were present in the proportion 1:1, and only a small fraction pass through the narrow opening, we must expect the two isotopes on the other side of the opening to be present in the proportion = 1 As in the case of the evaporation method, the method here mentioned therefore admits under the mest favourable conditions a separation proportional to the square root of the ratio of the two molecular weights. Although not favourable from a practical point of view on account of the slowness of the effusion process, this method is of theoretical importance, and was therefore also used by us in our experiments with mercury. 3. Separation of the Mercury Isotopes based on the difference in their evaporation velocity. A number of reasons have induced us to begin the series of mixed elements, to be separated into pure elements, with mercury. he vapour-pressure of this substance can be varied in the particularly handy temperature-range 0°-100° within wide limits, embracing a practically negligible pressure as well as the already considerable one of about 0°3 mm.; besides this, the free mean path. and other inagnitudes important for the kinetic theory of gases are well known in the ease of mercury. There is hardly another substance which can be produced in a pure state with so little labour, and no substance of which the density can be determined so easily and exactly. A density deter- mination is certainly the most easily approachable and the most exact manner for determining a partial or complete separation of isotopes. The volume of the isotopic atoms is, as is well known, equal”; so that by comparing the density of the “normal” and “ separated’? mercury, a separation can be easily detected and its degree measured. * Compare the agreement found by Th. W. Richards and Ch. Wads- worth (Journ. Amer. Chem. Soc. xxxvill. p. 221, 1916), F. Soddy (‘Nature,’ cviil. p. 41, 1921), and especially the conclusion following from Bohr’s spectral theory. the Separation of the [sotopes of Mercury. 35 (a) The experimental method. In separating the mercury isotopes the following arrange- ment was used:—In the space H between the outer and - Bio. J: inner flask of the Dewar vessel (fig. 1) about 300 c. cm. mercury were placed, H thoroughly evacuated by means of a Vollmer pump which was connected at B, and the inner D2 “5 Prof. J. N. Bronsted and Prof. G. Hevesy on vessel A filled with liquid air. The distance between the mercury and the cooled glass surface was 1-2c.cm. As evaporation proceeds rather slowly at room-temperature, the mercury was heated to about 40°-60° by means of the surrounding oil-bath C. After a suitable portion of the mereury was evaporated, we removed the remaining “heavier” part by evacuating the receiver HE and opening © the cock D, whereby the mercury by its own weight flowed into the receiver and could be taken out through the cock G. The liquid air was now allowed to evaporate, whereafter the solid mercury melted, dropped into the bottom of H, and was removed from there in just the same way as the ‘“‘heavier’’ mercury. The operation was now repeated with another sample of ordinary mereury, and so on until a total of about 2700 c. cm. was separated in a distilled (D,). and a residual fraction (R). In order to obtain heavier mercury, R, and the following R-fractions were subjected to further similar separations. When lighter mercury was desired, io and the following D-fractions were treated in the same wa For smaller quantities of mercury, as available in the later part of our work, we used two apparatus, as shown in fig. 2, with a capacity of 40 and 8 c. em. and working as Lollows :— | After having finished the evaporation, the residual mercury is removed by turning the ground-joint at D and thus the whole apparatus nour 50°, which permits the mercury to flow into the bulb B. Now the vacuum is destroyed, the point of the capillary tube broken off, and the outflowing mercury caught. The solid mercury, which after evaporation drips off and gathers in the lower space, is then removed in a similar manner. By using the arrangement shown in fig. 1 at 45°, about 6c. cm. mercury evaporated per Hone, corresponding to an evaporation of 0°35 c. cm. per cm.? At this temperature the vapour-pressure of the mercury amounts to about 0-01 mm., and the mean free path, in so far as the vapour can be considered as stationary, about 10 mm. Ideal separating conditions were thus nearly fulfilled at this temperature. Yet the evaporating process can be carried out more swiftly without thereby reducing the degree of the’ separation. For instance, by working at 120° and using the apparatus represented in fig. 2, the large amount of 1:5 c.cm. per hour and per square cm. evaporated ; yet the degree of separation achieved was found to be only slightly smaller than the one obtained under “ ideal ” conditions. At 120° the vapour-pressure of mercury the Separation of the Isotopes 07 Mercury. oe already attains the value of 0°7 mm., and the mean free path is accordingly only 0°15 mm.—only quite a small fraction of the distance between the mercury and the cold glass surface. That the separation succeeded in spite ‘of this is explained by the fact that the current of the mercury vapour from the warm to the cold surface is onesidedly directed, and that the presence of the steep Fig. 2. gradient hinders the retreat of the evaporated molecules into the liquid. Under these conditions “the mean free 99 . path” surmounts many times the mean free path of the mercury molecules experienced under normal conditions. From the fact that even by rather rapid evaporation a partial separation of the mercury isotopes can be achieved, we are justified in concluding that the exchange of the aire tian 38 Prof. J. N. Brgnsted and Prof. G. Hevesy on mercury molecules between the surface-layer and the interior of the liquid is also rapid; for it is clear that a slow exchange of molecules would lead to an accumulation of the heavier isotope in the surface-layer, thus frustrating the separation. In so far as the exchange is only accom- plished by diffusion, the time required can be approximately calculated. We can, namely, determine approximately the velocity with hie mercury diffuses in mercury (seli- diffusion) from the known diffusion-rate of lead in mercury”. ‘The latter + at the temperature of 100° equals about 3.10~° em.?/sec.~1, and as the mean displacement square of the molecule per sec. depends on the diffusion-constant (D) as in the equation = 210). then the mean tinea (r) of the mercury molecules in the liquid mercury is about 5.10-° cm./sec.~7. It follows from this calculation that if not more than 5.107% c¢. em. per cm.? surface evaporates during the time-unit, no dis- turbing accumulation of the heavier isotope in the surface- layer takes place. (b} Lhe experimental results. As initial material, 2700 c.cm. of the purest mercury was used. This was submitted, in portions of about 300 ec. em., to an “ideal distillation”? in the large apparatus (fig. 1), and the process continued until about a fourth of the mercury was distilled over. A total of 2062 c.-cm. residual and 642 ec. em. distilled substance, which we will denote by R, and D,, was acquired. In order to obtain heavy mercury > Ry, was submitted to a similar treatment as the initial material, and 1602 c. cm. residual (R,) and 460 c. cm. distillate (R,D,) acquired. By proceeding in the same manner, continuously decreasing remainder volumes (Cai oe) of continuously i increasing density were obtained. Beginning from Ryo, the volume of the mercury became too small to be treated in the large apparatus ; so first the middle-sized, and from Ry, onwards the small apparatus was used. The progression of the experiment in the case of the residual mercury is shown in Table I., which contains the densities found with the density of ordinary mercury as unity. As seen from the table, the density of the R-fractions increases gradually as the residual volume decreases, and pee in the case of the heaviest mercury becomes 2/o0 larger than the density of the normal “substance. * Compare Groh & Hevesy, Ann d. Phys. \xiii. p. 92 (1920). + M. Knudsen, Ann. d. Physik, (4) xxix. p. 179 (1909). the Separation of the Isotopes of Mercury. ag Tasue I.. Fraction. Volume. Density. {| Fraction. Volume. Density. 10) ome 642 0:999977 hy wae 2062 an BeBe... 461 0-999989 renee 1602 1000016 Rep o.. 320 Le ee 1283 2 Bee 243 2. Eee Wane 1030 1000024 ae 235 s aera fie ay: 791 Ue Hep... 202 == | DE ee 585 1000034: Bae)... 94 ae Reet 489 2: ee. 115 = Rear a. 382 1000043 ee. 105 a Ry go t's 268 0 ee 82 sss Tey ee 185 pune B,D, 35 ae Roe ney 150 Ae De... 29 ze Bis 128 1-000079 eS. 87 1-000060 Bie coe 2 a eee Bio FOCOOS0. FRY... 103 1.000134 ioe 48 a ee ao 1000153 ET... 4-4 yea Aotiind vheareaie 11 ue ee. Osby ae Ls tray hae) 05 Be ee 2. OM on ett ey te. 02 1:00023 Fie. 3. Looe loo / 000 050 —10.0 mie me on =O. - GO These results are also shown in fig. 3, where the density is plotted as ordinate and the abscissa represents the natural a ee AQ) Prof. J. N. Brensted and Prof. G. Hevesy on logarithm of the ratio of the remainder and the initial volume. The significance of the rectilinear course as shown in the figure is explained by equation (7) in § 5. In order to obtain hght mercury, D, has been submitted to a further distillation, and thus separated into a distillate D, and a residue D,R,; D, has been treated in the same way, and so on. The results of the first five experiments of this series appear in Table LI. TaBue Il. Fraction. Volume. Density. | Fraction. Volume. Density. 1D. one ee 642 0:999977 TM A Si 472 0-999979 dB hie duce 154 0999953 DERE 101 0999961 a HN 50 0-999933 DAs aa. 35°5 0999941 Data 13°5 0999911 DAR ya cece ae 10 0-999914 Dee tee oO 0999881 As was to be expected, the D-fractions exhibit continuously decreasing densities. From D; onwards the systematic treatment described above could no longer be applied favourably on account of the smail amount ‘of the material left. It was therefore found profitable to make residues of distillates of high order the material of departure for further distillations, and to unite the various portion of similar density obtained in this way. It was thus possible to in- crease essentially the otherwise quickly decreasing quantities of the higher-order fractions. The results are given in Table LILI. (Derbi UE, Fraction. Volume. Pensity. 1D ie asta Mes 8 70 0-999874 Dede) asker g! 39” = ID) anes rch ta 2°3 0-499824 AD ae an pea 0-9 0999779 Dea oe ete 3 O07 == Die orien eetenie 0°55 ze Dire as He 0-5 a Dama ries O-+4 am DDT aa tie 0-2 099974 The densities are seen to be steadily decreasing. The lightest mercury possesses a density deviating trom ‘that of the ordinar y peneuny by about 7 4 °foo—the same deviation as shown by the heaviest mercury in opposite direction. The density difference between the heaviest and lightest mercury therefore amounts to nearly 4 °/oo. the Separation of the Isotopes of Mercury. 4} 4. The density determination. As has already been mentioned, in determining the partial separation of the mercury isotopes, density measurements of the fractions gained through the described separation method were used. The atomic volume of the isotopes being equal, the change in density offers a simple measure of the change in the element weight * as achieved by separation. Tig. 4 shows the arrangement used for the density determinations. The mercury is brought into the reservoir Fig. 4. R, which ends eles in a capillary tube K sufficiently narrow to prevent the mercury from running out when left alone. The capillary is inserted into the0°3mm. narrow neck of the little bottle P. When tube V is connected with the Gaede pump, the over- pressure resulting drives the mercury through the capillary, and the pyknometer is slowly filled with mercury. R is now removed, the pyknometer placed in a ther- mostat for about an hour anda half, and after removing the drop of mercury, expelled by the heating, weighed. We employed two pyknometers, of which the volumes amounted to about 5 and 1°3 cm.? respectively. By using the firstthe density could be determined with an accuracy of 2: 10°, with the second the accuracy was correspondingly less. Lor sacar nine quite small volumes (about 0°2 cm.*) of the heaviest and lightest fractions, a thick- walled vlass tube of O'1 mm. inner dia- meter, with both ends drawn out very fine, was nee asa pyknometer. The mercury was sucked into the pyknometer when held in vertical position, and as it could not run out by itself into the vacuum through the very fine upper. opening, the tube could be placed in reversed position in the thermostat. On account of the slight volume of this capillary pykometer, such exact measurements cannot be made with it as with the one of larger capacity. The capillary pyknometer, however, was only used in cases in which the change in density was already very considerable. * The term element weight is used for the mean atomic weight of the mix-element: see Harkins, J. Am. Ch. Soc. xliii. p. 1058 (1921). 49 Prof. J. N. Bronsted and Prof, G. Hevesy on The mercury used in the density determination was present in the purest state. In each fraction examined we made sure that any further vacuum-distillation made in the usual way had no influence on the density of mercury within the limit of our errors of measurement. 5. Calculation of the separation. In the case of a mix-element composed of two isotopes, to be separated into its components by means of the evapo- ration method described, the change which the element weight suffers through a single evaporation can be calculated in the following way. If Nj, and N, are the number of molecules originally present, m and ny are those of the evaporated molecules of the first resp. second isotope, M, and M, the corresponding molecular weights (atomic weights) ; then, granted the above- mentioned ideal conditions “(p. oe) the differential equation fundamental to our calculation, dny N, S| M, dng SONS =n. My (1) follows ; from which, by integration, we obtain ] eet | /M. ae N,—n, nwiNG Mo ae In the special ease, which we will deal with first, where the initial ratio of the two isotopes is equal to unity, we can put N, = No» aa it and further where M; indicates the element weight of the mix-element in its original state. We then cain In(L—m) _ os +a Pea) Nae or with great approximation In(1—n,) = A. : : Se : 2 In(1l—ng) lone @) Taking into consideration that the element weight of the fraction remaining after distillation is expressed by Gl. =) MS n)M, 3) M, = ————————— Za Hit) the Separation of the Isotopes of Mercury. 43 and the ratio between the initial volume and that remaining after distillation by we obtain MM, 0: ; M = I+ (m—72) 5 0, ( ) or yu 4 Gees 1—n, i DS aoe a } 5 Vi ; L (6) | | i Introducing the two latter equations into (2), and denoting the ratio of the element weights MW which is equal to the corresponding densities, by d,—the meaning of this magnitude being nothing but the density of the remaining part expressed in terms of that of the standard substance— we obtain Ur? =1—d, In aor K ed Ut A or, transformed, oN Dp 1l—-d,= Dea Per oh ESA A Tea gia eae tule ia) By means of this equation from the known ratio 2 and the relative density of the residue d,, A and thus the molecular weight (atomic weight) of both the pure elements can be calculated. The validity of*the above equation can be tested by proving that it can be satisfied for all corre- : V, bia sponding d, and — values by a single A value. This is v to possible when calculating the density of our mercury residue (d,) from equation (7), on the assumption that A has the constant value 0:0070. The results of this calculation, as well as the densities experimentally determined, are shown in Table IV. 44. Prof. J. N. Bronsted and Prof. G. Hevesy on TaBueE LV, ) Density Density Fraction. oe Density Fraction. i. ound. cale, | found. : cale. TR se vane 1:000016 10000138 | Da aa ORS 5 S)9 fi 0999579 EU prsanee 1:000024 1000024 ID ena 0999953 0999956 LR ae ea: 1:000034 1:000038 Doan ate) DO eae 0:999935 NG ieee 1:000053 1000048 BOice:. MURS SI 0999913 1 a eee 1000079 1-000075 DE pacers 0-999881 0°999890 TR apse 10001384 1:C00137 Bae te es 1000153 1:000152 | He eae 1:00023 1:000234 | As the figures found and calculated agree very well, we may conclude, as will be more elaborately explained later,that the evaporation of the mercury in our experiments took place according to the theoretical supposition (reversed proportion of the evaporation velocity to the square root of the atomic weight). Further, that the separating process of the mercury proceeds like that of a mixture consisting of equal parts of two pure elements with the atomic weights : M, = M;(1+A4) = 200°6-.1:0070)— 202 a0: M, = M,(1—A) = 200°6., 0°9930:== 19a; The results are also represented in fig. 4, exhibiting clearly the fulfilment of the requirement of a rectilinear interdependancy between the density and In “* as follows from equation (7). o Sy As already mentioned, the above calculation presupposes that the two constituting pure elements are present in the mix-element in equal atomic proportions. If this sup- position is omitted, there is an infinite number of M, and M, values which are compatible with the value of M,, and corresponding to each of these cases a separate shape of the separation curve (representing the interdependance between the density d, and the volume of the residue) is furnished. For evaluating this fact in determining the atomic weights, it is necessary, however, to consider a longer portion of the curve than the one corresponding to our experimental results. If we only consider the result of the first separating operations, practically coincident curves may be found also where the number of atoms and therefore also the atomic weights of the two pure elements in the mixed element vary. In order to further illuminate this point, we proceed from the expressions (8) and (9) repre- senting the atomic weight of the unchanged mix-element, - the Separation of the Isotopes of Mercury. 45 and that of the first distillate, in so far as the distilled amount is a small one: : N,M,+N.\, M,; = CEN . . . . ° ° (8) ee Ni/M, ae N.v/ M, ¢ Way NET. enc eed yes 28 VM, VMs N, Ny When we write No and Mee, we find M, ) of8? lp oad aes Ee eta) M,~ Bat (B+) which, when the atomic weights M, and M, are not too unlike, is practically identical with the following : i Pigg wot=g7 = fee “ap 8 Fy ak ieee) where dz denotes the density of the first distillate. The value of dz can either be directly determined from the result of a single distillation, or with greater accuracy from the A value with the aid of the equation : This expression inserted in equation (11) gives finally 2a 5 rN or for small values of A: A2 om 2a 12 (ae. Cae ee (te If A is given, then in equation (14) to each value of a there is only one single 6 value—that is, when in addition to the experimentally determined A the proportion in which the pure elements are present in the mix-element is known, the atomic weight of the pure elements can be determined * * The equation (14) lends to the magnitude A (defined initially as relating to the case where the mix-element contains the two pure elements in equal amounts) a more general significance when interpreting separation results. 46 Prof. J. N. Bronsted and Prof. G. Hevesy Ap In the case of mercury, where, as already mentioned, A=0-:0070, the following values of 8, M, and M, corre- spond to the values of « given in the first column of the table. TApiE Ve a. B. M,. “M,. “eae ile pee amet 1 CORO ees 2020 199-2 = 38 Eis Soe 3 LOS taper ee: 201:3.—-«197°8 3:5 Gee a LOMA eine, Lae 2011 196-7 4-7 Consequently we are able to interpret our experimental results by assuming the ordinary mercury as made up of the pure elements 202°0 and 199-2 present in equal amounts or by the elements 201°3 and 197°8, where the former is four times as strongly represented as the latter, and soon. Our results agree with those of Aston’s, found by means of his mass-spectrograph, in the fact that Aston found a strong line corresponding to the atomic weight 202 and an un- dissolved band the centre of which corresponds to about 199. A very slight quantity of an isotope with the atomic weight of 204, the presence of which follows from Aston’s experiments, cannot be found by our method if the separation is not to be pursued very considerably farther. For the numeric calculation stated above, we used the results of density measurements made on residual fractions. The calculation can also be carried out by using density data of the distilled fractions, or reversely the course of the distillation process can be predicted by means of the values furnished by measurements on residual fractions. Whilst the production of the neavy fraction consisted in a steady evaporation of the initial volume 2700 c. em. until only 0°2 c. cm. was left, we were for practical reasons— dimensions of our apparatus avd working economy— obliged to mix distillates of different quality when endea- vouring to produce the most valuable light fraction. The calculation of the separation proceedings on the basis of the density of the lighter fractions is therefore rendered some- what complicated. In spite of this, the calculated dq values as shown in Table IV. agree fairly well with the ones deter- mined, though the conformity is not as good as in the case of the d values of the residual fractions. | the Separation of the Isotopes of Mercury. 47 Separation of the Mercury Isotopes using the Ejusion Method. Fig. 5 represents an apparatus used for the separation of the isotopes based on the method described on p. 34. The Fig. 5. a main quantity of the mercury vapour passes from the heated bulb A into tube C cooled with ice, and only a fraction of the vapour has the opportunity of entering the equally cooled tube B by passing one of the numerous holes in a platinum foil inserted in the glass tube at H. The diameter of the platinum foil, as well as that of the glass tube, is 2 cm. and it has 1000 holes of 0°15 mm. diameter each. The whole apparatus was thoroughly exhausted. As found by Knudsen*, the rule of the molecular flow through a narrow opening is only strictly valid when the * Ann. d. Physik, (4) xxviii. p. 999 (1909) ; xxix. p. 179 (1909). 48 On the Separation of the Isotopes of Mercury. mean free path of the molecules is more than ten times as great as the diameter of the opening. Knudsen, however, by using a 2°4 mim. wide opening and working at 59°°8, noted a deviation of only about 8 per cent. from what was calculated from the rule of the molecular flow, and it was to be expected that also by using essentially larger openings or a smaller mean free path the partial separation of mercury would be successful. By heating the bulb A to 105°, corresponding to a mean free path of the mereury molecules of about 0°3 mm. at H, the product condensed in B showed the density 0°999987. In this connexion we might mention that diffusion methods have been repeatedly employed in order to attain separation of the isotopes. Aston™* was the first to use this method and got, in the case of neon, considerable changes in density. By means of a similar method, Stern and Volmer f have tried to separate the elements hydrogen and oxygen, yet without result, in accordance with their nature as pure elements. A trial to separate the isotopes of uranium through diffusion in aqueous solution was equally unsuccessful ft Summary. 1. A partial separation of the isotopes of mercury based on the two methods following was successful. | (a) Evaporation method (“ideal distillation”) based on the difference in the evaporation velocities of isotopes. The distillate was found richer, the remainder poorer in the lighter isotope, than the initial substance. (>) Effusion method. A fraction of the mercury vapour penetrates through narrow openings into the con- densation space, ‘where the lighter isotope 1s found in a relatively larger amount than im ordinary mercury. The results of the experiments agree with the theory, ae to which the evapor ation—as well as the effusion velocity ‘of. the isotopes——is inversely proportional to the square root of their molecular weights; they are further in conformity with Aston’s results Averell by means of his mass spectrograph. * Brit. Assoc. Report, 1915; Phil.. Mag. xxxix. p. 450 (1920). Compare also the following attempts : :—D, Harkins, ‘Nature,’ ev. p. 220 (1920); E. Kohlweiler, Leitschr. f. Phys. Chem. xev. p. 95 (1920); and H. Grimm (K. Fajans, Radioaktitdt, 3 Aufl., footnote p. 112). + Ann. d. Phystk, lix. p. 226 (1919). { Hevesy & Putnoky, Phys. Zettschr. xiv. p. 63 (1913). On the Lags of Thermometers. 49 3. The partial separation achieved was proved by density measurements. The density difference found between the heaviest and lightest mercury amounts to 0°49 °/o, corre- sponding to a difference of 01 unit in the element weight of mercury. In carrying out the experiments described here we are indebted to Mr. V. Thal-—Jantzen for his valuable assistance. Physico-Chemical Laboratory of the Polytechnic Institute of Copenhagen. IV. On the Lags of Thermometers. By A. R. McLeop, MLA., Fellow of Gonville and Caius College, Cambridge *. N a former paper { expressions have been given for the steady lags in the mean temperatures of thermometers with spherical and cylindrical bulbs, when the external temperature increased at a constant rate and the initial tem- perature was zero eee Under the same conditions, Dr. T. J. VA. Bromwich f has since given formule for ty mean lags at all times until the steady value is reached. The present paper deals among other things with the lags for such thermometers, when ‘the initial and surface con- ditions are somewhat different. The conditions were suggested to the author by Mr. EH. Gold, and are based on experience in taking air temperatures by aeroplane in France. . The new conditions are obtained by warming up the thermometer initially to a fixed temperature, the external temperature being thereby initially unchanged, and by having a discontinuous surface condition, the temperature of the air (or other medium) changing linearly with time until a certain time ¢,, and then changing linearly at a new mrate. Numerical values for mereury and alcohol agree with those given in the former paper. Using the same thermo- meter stem with each bulb, the alcohol cylinder is better than the mercury cylinder and both are decidedly superior to spheres. Owing to the fact that much finer capillary tubes may be used with mercury than with alcohol, the possibility exists of constructing mercury thermometers with much less * Communicated by the Author. + Phil. Mag. xxxvi. Jan. 1919, p. 134. t Phil. Mag. April 1919, p. 407, Phil. Mag. 8. 6. Vol. 43. No. 253. Jan. 1922. EK 50 Mr. A. R. Mcleod on the lag than alcohol thermometers which have the same » accuracy of reading. a general expression is given for the steady lag of a bi- metallic thermometer, used in air. ‘The lags of some bimetallic recording fics mometers considered are of the same order as those of. the cylinders considered, but the com- parison is not based on an equality of temperature scale. The lag of the Dines ‘ballon sonde” thermometer is very small. Experiment shows that the mean surface conductivity for a bimetallic thermometer in the form of a flat spiral which is wound ina cylindrical tube, may be increased 34 per cent. by turning the thermometer from the “end on” position to the “ broadside on” position in a wind of 5—10 m.p.h. Radiation, convection, and the effects of the glass wall, in the case of liquid-in-glass thermometers, are not considered. 1. Steady Mean Lag of a Metallic Thermometer. The steady lag in the mean temperature of a thermometer may be written in the form pa oN. Gpo( x = ve ae eee ee SY (1) where G is the constant temperature-time gradient in the medium, p is the density, o the specific heat, K the thermal conductivity, and / the surface conductivity of the thermo- metric substance, and M and N are constants whose values depend upon the shape and dimensions of the bulb. Under certain circumstances the internal lag, which is represented by the first term in (1), may be neglected leaving only the surface lag, represented by the second term and due to the passage of heat across the surface, to be considered. This may be done if K is large compared with h, which is the case for most metals used in air or some other medium in which the value of h is not too great. It’ may also be done if one of the dimensions of the thermo- meters is very small, since M depends upon the square of the linear dimensions while N depends only upon the first power. In such cases the steady lag may be written in the form NGpo [hye 2... re) and the value of N is given by the ratio volume of thermometer ~ area of exposed surface (3) Lags of Thermometers. D1 It is easily seen that this ratio must occur as a factor in N; for the surface conductivity term only takes account of the passage of heat across the surface, and if the area is doubled, the heat is lost or gained twice as rapidly and so the lag is halved ; while if the volume is doubled, there is twice as much heat to be lost or gained and so the lag is doubled. Any other non-dimensional factor must be unity, since N=c/3 for a sphere, and c/2 for a cylinder, c being the radius. 2. Case of a Cylindrical Bulb. (Infinite Length.) If a?=K/pc, we require a solution of Our Ou) Ou eae Ce Or” ed Gal, PRET sel cs (4) which satisfies the conditions ie — oh (G Or o_O hry) aay en coe ne Zap TU= OO HOVE P= Oy ctece ar Nps se (Ot) ¢ being the radius of the cylinder, u the temperature, and t the time. The solution satisfying condition (6) is given by Fourier in his Théorie Analytique de la Chaleur, p. 309. The fol- lowing treatment is set out here, hewesee. as it is much more concise than Fourier’s* and follows different lines. The general solution of the form required is oe) i > eh o( &,7"/ OC) emay tn ee: OO ie ees te (7) "i where A, and a, are arbitrary constants to be determined by the conditions, and Jy is the Bessel function of zero order. The basis of the solutions is the well-known expansion (cf. Whittaker, ‘ Modern Analysis,’ p. 374, Kx. 20) | £E)I(BEDdE ()= ay saa ee oan where Baotea) =F HJ (Bn) = 0, lai => ch] K. 5 : (9) * See also, Carslaw, ‘Introduction to the Theory of Fourier’s Series and Integrals,’ p. 208. EK 2 D2 Mr. A. R. McLeod on the Take 7 (rjc) =1.- Then, sinee i £5o(@ré)dé=HI (Bx) |? (°£{ 518.) }P0e= (2 +12) {To()}*/ 280 we have 4) QE (CC are) L= a 5 a gia ee eae as >, (B2+ HIB) Ly, Write foo) 9 wie ( ‘) saree > HJ (8B, r/c) e~ PB, tle? t>0O n=1 (orm ag H?)Jo(Bn) Mr) — Oe CE Ok Consider, for ¢>0, aS 14H( a We see that > 23 o(Bur/e)e~ Parl n=" (8,7 a HP) J (Bn) i WAG Hee ir, S10), w(c,t)=0, t<0. Further, as r—>c, the ratio of W(r,t)—1 to | ee +y—I1 and we define tends to unity. Now by the usual method *, the function (, i) (90) oiey hed reduces to the value ¢(t) when r=c. Performing the differentiation, this becomes 2a tee Sar Jo(Bnr/eje % Pudi TeV Bre dr oe a) (8,.? + H?)J o( Bx) i becuase | Phones by inspection, the solution corresponding to x, which is the one we require for condition (6), 1s the second of the terms on the right of (11), and the solution we require is 28,23 ,( Byn|een 2A? (7 ) 0 > EERE CBOE |, PPIs ! (11) 4 20H S Batol Barl de (oc omceitge | > (B.?-+ H2)Jg(B.) \, oO) a * Phil, Mag. Jan. 1919, or Weber, Partielle Differentialgleichungen, Band ii. Lags of Thermometers. 53 As a special case take F(r) =u,=constant, and $(¢) = Ge where G is a constant. Then the solution becomes way 20H o(Bar/e)_ nan (Bn? + H*) J o( Bn) r+ (12) i oo 2G HI o(Bnr/c) Ps aa Ne ee 2t/e2 1) | qo. (8,7 at A?) J (Bn) rt Gap } 3 To evaluate the series which occur, we equate the expan- Ss vf . » e sions of Jo ((«)+ HJo(w) as an infinite product and as an infinite series. 2 Las Me tgucs H (1-3 = Hl 1-25 an (2 Rinala Wan ] ba ae) ea Oe Won arias) obs le (13) 9 e—*B,7t/e? | Writing r=c in (10), we get and with the aid of (14) we find, after equating coefficients ay (13), > Se ic St 4H? ut al n=1 Br? Br! + Et) iy p= ety | (Sir ee lale) oh 8 2H i Taking the mean temperature we find from (12) the value for the mean lag Goce a) ie Sa eHN | i Ge? (15) ==) 2 ee) | Ug +t —=zaz )- a” \3 “LP oH caulerd(etmaetal a) (0) OB Bne On writing uy=0, we get the lag given by Dr. Bromwich’s formula. In the next case to be considered, we have the solution (12) until t=¢,, when the surface condition becomes OG) = Girt G'(i—1,) fon ¢> ty. 5. (6) Writine ¢’=t—t,. we have, after time ¢,. a new problem : : = ecks ? I> p in which the conditions are = S 2H ((Bnr/e) —a2B, 2ty/c? | Br) = & B24 HIB.) [we | 2 Paced (ela + G {ts = ZB 5} (e- V8 2h/e? yt] | ( ) $(é)= Gt, + Gt’. = 54. Mr. A. R. McLeod on the Substituting in (11), we find, since the series are uniformly convergent, as Fu = 2HIo(Bn7/c) (G—G' ee “= Gi, +G't'— = (B2+ H2)Jo(Bn) rp 23 2 G'?? Ge? ; co EB a (vot ee) oe a] : : cry a. the mean temperature, we get for the mean lag or (G+ on) +S 4H? —- G') ¢? o— 2? Bt! |e? 8 * 2H Beet) | @Be C 2 2 ne ne el ey ne (19} 3. Case of a Spherical Bulb. To preserve the analogy, the solution will be obtained by the use of Bessel functions. We require a solution, satis- fying conditions (5) and (6) (in which ¢ is now the radius of the sphere) of the equation Ou 2 Ou oe & a BB 2t fee 2 ree VG ne Mer pipes (20) Substituting w=r-2ve—Vantl?, g=a,r/c, we get 2a) ) = Me + (1-4: Oe # AL Hence the general solution of (20) of the type required is u= > Ar iJy(a,r[e) e-em n=l To determine the constants «,, we arrive at the condition Onda! (Am) + (H—4)Ja(a,)=0, . . . (22) or (1— 9) tan a, (a7. The basic expansion is 1 } EMO Tientlaé (= > ( Np J2( (an? |c) 5) (23) a Tie e £1 Ta (amb) jade Lags of Thermometers. 5D and since ae EJ, (a,£)dE= HI; (a,) /an”, [Et en) PAE = fen? + HH — 19}, (25) Pa 0 we have, calling the constant coefficient Aj, Be 2 (r/c) U3(anr/e) Le = > a,2+ HC 1)}J a(n) => Alle" 2Jx(anr/c). (24) m= (pak Defining x(7, t) to be zero for t<0, and xv(r, t)=1— S A,(7/c) ~“aJ)a(agr/eje 7 ante, t>QO, 7 we have for the pean W(r,t), when ¢>0, Wr, = J1— S A, (ofc) dette Ke ie (anr/c) ea tu{-3 5. Ja(onrle) )t ands '( Qn nie) V) And, as before, the function oe takes the value o(t) for 7—C 18 given by oO- LS A072 0,267 Conte —— | Tuan] c) pane (rfc) ; 1 mh Sees + yf — EM Garfd) + anh Coarle) } ] [penetra The solution for wu satisfying (6) is then found to be the second term on the right of (25), and the solution required is : = Qa,2e- Van tle? Ji(a,r/c) (7 ZaFFHE=D Heyy lor J, BPCD mba ——— == = 2 Ha? Oe= a2an?t/C J2(anr/c CLan +H(H—-1)jSi(4n) (r/o) Taking F(r)=wuo, and $(t)=Gt, we get ws 2uyH (r/c) ~2d2(anr/c) nah ton? + H(H— 1) $d (an) «o 2GH( (rjc) Ser ip 1(ay7/c) | Or Sa) opted | +>, fs Hoa ats hes Lara sue no’ 1) | r ) t b(r er tM dr, J —a2aq..2t/e2 aa),-t/C 56 Mr. A. R. Mcleod on the Using the series ‘s 2H pe aes Sa 6H? ert H(BH1) 9 Sie a. oH i ent { on” +H(H—1)! 15 3H’ we find for the lag in the mean temperature Ges 1 )- S Ones? he Ge? ae 3H) Davie oy (w+ 35 —) ey On writing w=0, we get the lag given by Dr. Bromwich’s formula. In the case of the temperature inversion, in which the solution (26) holds until t=¢,, and new conditions then set in, the initial condition being given by writing t=#¢, in (26), and the surface condition by (16), we find without difficulty = 2H(r/c)Wyer/c) [(G-G)e et rie ie, NY IS | a eae EPH, 78'/c2 - u=Gt,+G't > (a2 + aE 1)}Si(a,) ara,? GiG Ge? Shenae | +a — (ot pes pee |. (8) A hy a The lag of the mean temperature is found to be CeO 6H? ss (GG) a aq ‘i 3H “i > attare H(H—1)} Oe Ge? Mae | 4 — ara, *t/e2 (ve oe oe 4A, Numerical Results. We consider the same bulbs that were dealt with in the former paper (Phil. Mag. Jan. 1919). These have the same sensitiveness for a given thermometer stem, the expansion of alcohol being taken as six times that of mercury. Par- ticulars are given below. The surface conductivity is taken from the same paper and is given (in ©.G.S. units) by h=-0000515 VV... . 2 em The air density for which (30) holds is that near the ground, VY being the velocity of the aeroplane in miles per hour. The data on which this value is based were obtained from work on radiators and engine cooling done at the Royal —a2a,2t'/c2 e a?a,,-t'/¢ Lags of Thermometers. D7 Aircraft Establishment. As a rough check, values of h were calculated from the cooling of mercury and alcchol thermometers in the Wind Channel, the air velocity being 34 m.p.h. The value obtained from the cooling of the mercury thermometer (c=1 cm. nearly) for the constant in (30) was about ‘000047, and the alcohol sphere ‘e='5 em.) yielded a value nearly equal to ‘0001, but the result was much less reliable than that with the mercury thermometer as only one term of tlie series was used in the calculation. For convenience the symbols used below are collected here with their meanings. Up= Initial constant temperature of thermometer. V =velocity of wind past the thermometer in miles per hour. h=surface conductivity of thermometer bulb. K=thermal conductivity of thermometric substance. p=density of thermometric substance. o=specific heat of thermometric substance. = K/po. c=radius of sphere or cylinder (inner radius of glass bulb). H=ch/K. (;=temperature-time gradient before the temperature inver- sion (time=f). G'=temperature-time gradient after inversion (time=?’ =t—t,.) 41, 81, %, etc. are constants occurring in formule (19) and (29) which give the mean lags. ‘L,, L,, L3, are lags occurring with the pairs of values of G and G’ which are given below. The following data are given :— tig). V =70 ov.p:b. h=-00360. Mercury. (At temperature 15°-20° C.) = ye (eae Aue Sphere : C= em: ass “= “1276 a, =4°934. Cylinder : e= °365 cm. H= ‘0667 (10 cm. long) B,= °3622 S2=a' S49. Alcohol. K=-)0043 i= 000905. Sphere : €=3 “900 cm, *. bt =4- B= U2 Dao k ==). Cylinder== ¢="149 cm... . H=1-248 (10 em. long) 8,;=1:365 Spe 58 Mr. A. R. McLeod on the Substituting in the formule (19) and (29), we have the values of the lags in the mean temperature of the bulbs which are given below. The temperatures are in degrees centigrade, the times in seconds, and %, is large so that the inversion occurs only when the steady lag on the first vra- dient has been attained. If ¢em. The steel was ‘03 cm. thick and the brass ‘07 cm. The lag, when G=-028, for the steel strip is °16°, and for the brass °37° C.; so the thermometer would have a mean lag of about :30° C. As a further example, consider the case of the Dines ‘ballon sonde” thermometer, In this instrument the pen is supported at one point by an invar tube, and at a second point by a german-silver strip *02 cm. thick and about ‘5 em. wide. The lag of the invar is immaterial. For the german-silver strip po=°84, and the value of N in for- mula (2) is simply half the thickness, viz.: N=-O01. Hence the lag of the thermometer is *0084G/a, If G='028° C./sec. and h='00360 the lag is 065°C. But if the thermometer is carried by a balloon ascending at the rate of 750 feet per minute, V=8°5 m.p.h., and from (30) we get h=:00044. Then if the lapse rate is 1°68 C. per thousand feet (5°5 C. per km.) the lag is 40°C. We do not consider here the variation of surface conductivity with air density. In the same case (h=*00044) the lag of a Teisserenc de Bort thermometer as used at Trappes has a mean lag of 1°73 ©. The bimetallic’ strip has dimensions =i fem. x°62 cm.x11°0 cm. and consists of brass and iron, the thicknesses being in the ratio 3: 2. 9? } 7. Displacement of Temperature Maximum due to Lag. One of the effects of lag is to displace the position of the maximum or minimum reading of the thermometer from Lags of Thermometers. 65 the true time-position on the chart. Suppose the tempera- ‘ture inversion in the accompanying figure is under con- sideration, times being measured to the right, and temperature Fig. 3. towards the top. On the left, the temperature is increasing at the constant rate G, shown by the full line, and the ther- mometer reading lags behind, as shown by the dotted line. Atter the inversion at P, the air temperature decreases at the constant rate G', shown by the full line on the right. The true maximum is at P, but the thermometer maximum is at Q, where the dotted line crosses the second straight line. In the neighbourhood of P, the lag is given by for- mula (19)—in the case of the cylinder—and we have on writing ¢=~* in that formula, and retaining only the first term of the series, GE 6 a 1 4H?(G—G")c? e-12, — (49) ~ a \8* 2H) * Be(BP+ He t' being the time reckoned from the inversion P. The value, 6, of ¢’, at which the dotted line crosses the full line in the figure, gives the displacement of the temperature maximum and for this value the lag L is zero. Hence to determine 6 we have the equation 1) £ a) 4H?(G—G') — 028126 /c2 — : i (s+ on) + B62 F Hy” eR ILO) Now for cases in which the first term of the series gives the lag with sufficient accuracy the value of 4H°8,~?(878,+ H*)~ is very nearly equal to unity. For example, we have the Phil. Mag. 8. 6. Vol, 43. No. 253. Jan. 1922. i 66 Mr. A. R. McLeod on the following values for the cylindrical bulbs considered, and the values of 6H? ~*{a,? + H(H—1)} for the spherical bulbs considered in para. 4 :— Mercury sphere, °999; alcohol sphere, -865 (3 terms given); Mer- cury cylinder, 1:000 ; alcohol cylinder, -978. If H is small we have very nearly 6;= /2H, and so we find from (50) eure ‘G'—G 8= a2 ee ). 21a) SSbiy faes 5 (51) But when H is very small, the value of the steady lag after the inversion is passed is G'e? L= ono O . 5 ° . 6 oD Hence iby G'—G | 5= Glog. (Gr). Saul ie hee Note that 6 depends only on the ratio of G to G’, and not on the absolute values of the gradients. In fact the value of 6 as given by (51) is For the mercury cylinder considered in para. 4, we found H=-0667, and so (53) should apply. Calculation yields the same values for 6 as were found in para. 4. ' As a numerical example we may consider the Kew Recording Thermometer. To obtain the values of the temperature time gradients, a series of monthly mean, diurnal temperature neaaInSS (Kew Records) was examined, The maximum value of the oradient occurs in the mornings (March—Sept.) and is 2°-0 F. per hour, or in degrees centi- grade and seconds, G,=-000306. The mean gradient for an hour preceding or following the maximum was found to be "25° F'./hour or Ga=" 0000389. The surface conductivity is really given by a relation of the form h=h,+°0000515 V, where the constant term hy is negligible for aeroplane speeds, so that equation (30) then holds. In general, therefore, it will not be permissible to use (30) with small values of V. But as an example we shall take the cases in which Lags of Thermometers. 67 VY=2 m.p.h. and V=10 m.p.h. The values of the lags and displacements obtained will then be upper limits. The following data are given for the Kew Recording Thermometer (Mercury cylinder) :— se Ue a? ="0437. ¢='425 em.=radius of bulb. /=10 cm.=length of bulb. mem. V2 m.p.h. = velocity of air. h=:000103 = surface conductivity. ii "O022 21: The formula for the steady lag becomes L=933G. feo, —o,— 000306, Reo: Ti G=G,=:0000389, ie=2045 °C; These are, respectively, the greatest lag and the lag at the temperature maximum. ‘The value of the displacement of the temperature maximum viven by (53) is foand to be == LS, eee Y= 1() mph, 2—-000515, H=-01111. ~The values of the lags and of 6 are 1/5th those in Case I. Ee—c, L=-06°C. o-c, L=-01°C. C= OAS” As previously stated, the true value of Land § may be less than those given here, but they cannot exceed them. 8. Cooling of Thermometers. Surface Conductivity and Orientation of Thermometer in Wind. The formula for the cooling of a thermometer bulb is much simplified when the first term of the series alone need be retained. In the case of a cylindrical bulb, we have for the mean temperature when the initial temperature (at t=0) has a constant value wo, and the air temperature is zero every- where, the effects of the ends of the cylinder being neglected : pees ENE bois 2) (5A) gE IB (Bn AEN?) 7 ~vhere 8, is defined by equation (9). When the first term Be | | | 68 Mr. A. R. McLeod on the alone suffices we have approximately (see discussion of equation (50) ) WU Om Oem ie eee eed 4. (595) since 8,= “2H when H is small. For a spherical bulb, when the first term of the series suffices, the cooling equation, which is eo a Bu He — @2a,,2t/c? ao Dee (56) nai tn 1% + H(H—1)} becomes Ub = Uy e VE ge hes eee since #,= ” 3H when H is small. Now the reading of a thermometer whose temperature is changing depends upon the mean value of the surface con— ductivity, h, over its surface. If the mean value of h varies when the thermometer is oriented differently in the wind, the mean temperature must also vary in consequence. Let us consider only metallic thermometers. For these, if H=ch/K is small, we have for the cooling of the thermo- meter from a uniform initial value w: U — Ue Ne e . e e e ° (© 8), «u being the mean temperature; and X is proportional to the surface conductivity, h, and is otherwise constant for a given thermometer. The time, T, required for the mean tempera- ture to drop from wp to w/p is given by T=1/\.. log. pee) When comparing the cooling of a thermometer in two orientations in the wind, for a given value of T, the values of X are proportional to the corresponding values of log, p,. 2. e. for the same time of cooling, hy log.p he) loo ebon where the suffixes refer to the two orientations in the wind. The following experiment was carried out to determine the ratio h,/h, for the Short & Mason bimetallic thermometer. described in para. 6. The strip in this case was *5 cm. wide. -and made 4} turns with an outer radius of 3°0 cm. ‘The gap between successive turns of the spiral varied from °05 to ‘10 cm. The surface was silver-plated to prevent rust. Lags of Thermometers. 69 This thermometer was mounted by itself on the top of a wooden block, the centre being 5 cm. above the top and near the edge of the block. An aluminium pointer 19 cm. long was fitted, and a scale, marked off in degrees, was mounted on the side of the block under the pointer. ‘The seale could easily be réad to ‘1 division (*3°C.). The therinometer was heated in the dry inner vessel of a copper calorimeter, heated water being in the outer vessel and the block resting on the rim. When the temperature of the spiral was about 81° C. (room temperature 17° C.) the block was quickly removed, and the thermometer placed on the axis of a 30 em. fan at a distance of 120 cm. from it. The air velocity at this distance, measured by manometer, was from 5 to 10 m.p.h. Times measured from the instant of -removal and positions of the pointer were simultaneously recorded. In the first orientation the spiral cylinder was broadside to the fan draught. In the second it was end-on. In the latter position, about half the opening at each end was blocked by standards supporting the axis of the spiral cylinder, but there was a clearance between these and the spiral of at least 0'6 cm. The following table gives the data obtained, ¢ being the time, 1/p, the fractional drop in the “broadside” position, and 1/p, the same in the “ end-on”’ position. The values of p, and p. are means from two experiments in each position. Hxperiments in the same pesition showed good agreement. t Py Po loge p,/loge p,. 0 1-00 1-00 at 15 1:73 1:54 1:27 30 2:92 2-38 1:24 45 5:29 3°58 1-31 60 8-46 5:18 1:30 90 20-9 10-02 1:32 120 49:1 16-2 1:40 150 80 23:2 1:39 180 113 35°4 1:32 210 200 55 1:32 240 310 75-2 1:33 270 620 93 1:42 MIGANI ceansac,. lie Hence /y/hy=1°34, that is, turning the thermometer from 70 Dr. T. J. Pa. Bromwich on Kinetic Stability. ‘“‘end-on” to “ broadside-on ” increases the surface conduc- tivity by 34 per cent. The thanks of the author are due to Mr. Gold for sug- gesting some of tbe problems, and to Mr. G. M. B. Dobson for many valuable suggestions and kind cooperation in numerous ways. Royal Aircraft Establishment, Farnborough, Sept. 1919. V. Kinetic Stability. To the Editors of the Philosophical Magazine. GENTLEMEN,— A LTHOUGH the idea of Kinetic Stability appears to be of considerable importance in certain Dynamical investigations, yet there are misunderstandings to be found in even some of the best text-books ; and at the moment I am aware of only one book * which contains a good account of the theory accompanied by two or three well-chosen examples. Still I should hardly have troubled you with these remarks, if I had not chanced to come across a very misleading statement on the subject of Kinetic Stability in Mr. — Jeans’s recent Adams Prize Hssay +. On account of the author’s reputation, and of his very attractive presentation of the subject-matter, it seems likely that this Hssay will soon be one of the standard sources of information on cosmica] mechanics. It therefore seems desirable to warn readers against the oversight, so that they may not be misled in similar investigations. The passage in question will be found in Art. 27f. Summing it up into a brief statement, it implies that when a dynamical system is referred to axes rotating uniformly, an instability is introduced ; but this is plainly absurd, for if we make the change of variables X=2#cosot—ysinot, Y=wxsin wt +y cos at, where w= A,sin(ort+e,), y=2B,sin (o,t+ 8,), then the new variables will remain as simple harmonic * Lamb’s ‘ Higher Mechanics,’ (See Ch. xi. Arts. 98, 99). t ‘Problems in Cosmogony and Stellar Dynamics.’ ' Cambridge, t See the foot of p. 29 and the first half of p. 30. Dr. T. J. Pa. Bromwich on Kinetic Stability. fils functions of the time; although of course any single term in ot will be replaced by a pair of terms containing (o +o)t and (c—o)t. What appears to have been overlooked by Mr. Jeans is that these systems are not rendered unstable until frictional terms are introduced, the amount of the friction being pro- portional to the relative velocity; the facts are clearly brought out in two examples given by Lamb ™*. A similar oversight appears to occur in the associated gyrostatic problem “(of constant angular momentum) men- tioned in Art. 29: incidentally, the equations there used are attributed to Schwarzschild (quoting papers dated 1896 and 1897). They appear te me to be simply a special example of Routh’s general process + for ignoration of coordinates, published in his Adams ieee Essay of 1877. But before leaving the topic of stability, it may be worth while to refer to the fact that errors have been made by attempiing to infer instability from the method of small oscillations. The classical instance is the top, sleeping upright with such a spin as to satisfy exactly the critical condition of stability: then (using the method of small oscillations), Routh deduced that this top would be really unstable (Adams Prize Essay). About twenty years later Klein proved in his Princeton lectures that this top is really stable f. It does not appear to be possible to give any coinplete test to settle the question of stability in these critical cases ; but a number of general considerations are given in a paper of my own on this topic§, together with a detailed examination ot the allied problem of a solid moving through liquid, accompanied by circulation round the solid; in this problem the critical cases may be either stable or unstable, according to a rather elaborate additional criterion. It may be useful here to quote from Klein’s paper, in reference to the general question of trying to discuss problems of stability by means of the method of small oscillations :— “From the start this method of small oscillations lies open to severe criticism. In the so-called unstable case it is directly self-contradictory, since the quantities, which in the * * Proc. Roy. Soc. (A) vol. xxx. p. 168 (1908) ; or ‘Higher Mechanics,’ Art. 99, exs. 2, 3. + See, for instance, Lamb’s ‘ Higher Mechanics,’ Art. 83. { Bulletin of the American Mathematical Society, vol. iii. p. 129 (1897). § Proc. Lond. Math. Soe. (1) vol. xxxiii. p. 331 (1902), ae Mr. V. Lough on the Beating construction of the differential equations are assumed to be small, become, after its integration, large. There is no reason whatever, therefore, for regarding the results as an approximation to the actual conditions. Even in the stable case the method lacks an accurate basis.” However, these remarks seem to me to be unduly severe, after having borne them in mind for the past twenty-four years. After a careful examination of a large number of special problems, I have never found the method lead to erroneous results, except in these critical cases (such as the top-problem settled by Klein); and it must be remem- bered that (even in the problems of Statics) the critical cases are inevitably associated with the examination of terms of higher order than is usually necessary. I am, Gentlemen, Your obedient servant, T. J. Pa. BRomwicu. *St. John’s College, Cambridge, 4 August, 1921. VI. On the Beating Tones of Overblown Organ Pipes. By V. Loves, B.Se., A. RC rSes [Plate I.| 1. Introduction. NE of the most interesting problems in Acoustics, on which much has’ been written without arriving at a complete solution, is the mechanism of the excitation of “flue” or “flute” pipes by blowing. According to Helmholtz, this is a simple matter of the wafting from side to side of the blade-shaped air-jet into and out of the mouth of the pipe under the influence of the oscillating motion of the air within the pipe itself. In his book on “The Sensations of Tone’ he explains how the air-jet maintains these oscillations; he suggests that, having no appreciable stiffness, the jet follows the inward motion of the air inside the pipe, snd so delivers a puff of air at or just after the period of maximum condensation. From the dis- continuous nature of these impulses he deduces that the forced vibrations maintained by them should, in the case of narrow pipes having free periods in nearly harmonic relation, produce a tone rich in upper partials, similar to that of. stringed instruments: a deduction which is fully supported by experience. * Communicated by Prof. C. V. Raman, M.A. i il studied experimentally by Raps | Tones of Overblown Organ Pipes. 13 The action of the jet, however, is in reality not such a simple matter as would appear from the above suggestion. Helmholtz’s theory fails to account for the initiation of the vibration, for the well-known rise of pitch with wind pressure, quantitative determinations of which have been made by Blaikley, Rayleigh, and others, and also for the jumping of the pitch to a higher frequency which results trom overblowing. A summary of subsequent work on the subject, with full references, may be found in Winkelmann’s Hand- buch™. Attention may be drawn especially to the investigations of Hensen tf, Wachsmutht, and others on the behaviour of the air-jet, its breaking up into vortices, and the relation of the tones of the pipe to the so-called “ edge-tones”’ resulting from the wind-rush against a wedge- shaped obstacle. Reference should also be made to a more recent paper by A. C. Lunn §, who has suggested that the rise of pitch with wind-pressure may be due to the kinetic stiffness or quasi-elasticity of the air-jet, and adduces mea- surements in support of his theory. The character of the vibrations cf the air in organ pipes and the influence on it of the wind-pressure have been . The photographic records which he obtained by an interference method show that the vibration which at low pressures is comparatively simple and consists mainly of the fundamental, develops overtones as the pressure rises, the octave becoming more and more pronounced as the fundamental diminishes and finally dis- appears. In certain cases he noticed a peculiar “ rolling ”’ tone produced at an intermediate stage, but did not pursue bis investigations in this direction. Now in its practical application for musical purposes, the organ pipe has been developed exclusively as an instrument for producing a steady tone, and any other condition, such as the ‘‘ rolling” or wavy tones noticed by Raps, is carefully avoided in voicing, and previous investigations have been conducted on the usual steady tones. But these wavy tones so carefully avoided in practice are nevertheless interesting from a theoretical aspect, as suggesting a departure from the harmonic overtone relation in forced vibrations and a comparison with the more familiar wolf- note phenomenon in stringed instruments. In view of the * 2nd ed. vol. i. pp. 435-446. + Ann. der Phys. vol. xxi. p. 786 (1906). q Ibid. vol. xiv. p. 467 (1904). § Physical Review, May 1920, p. 446. il Ann. der Phys. vol. 1. p. 193 (1893). 74. Mr. V. Lough on the Beating importance of these wolf-notes in, the theory of bowed instruments, as demonstrated by C. V. Raman™* in his monograph on the subject, it was thought that an investi- | gation of the corresponding ‘phenomenon in organ pipes might lead to useful results. 2. Keperimental methods and Results. Having collected a number of wooden pipes of the ordinary open-ended type, each pipe was tested separately at varlous wind-pressures, and it was found that instead of Jumping suddenly to the octave as generally supposed, in all or nearly all cases the transition is perfectly gradual and usually includes an intermediate stage extending over a considerable range of pressure in which regular beats. occur before settling down to the regular octave tone. In one particular instance, selected on account of the promi- nence of the beats, the pipe having a pitch of approximately 300 vibs. per sec. and being voiced for a pressure of about 6-8 em. of water, perfectly regular beats of a frequency of 8-10 per sec. were audible over a range of about 11-15 cm. pressure. The frequency of the beats is nearly constant throughout this range. From the results of analysis by ear with the help of spherical resonators, it seems that the beating or variation of intensity takes place mainly in the first and second overtones, the fundamental remaining steady or nearly so throughout its gradual decrease. The character of the phenomena evidently depends on the voicing of the pipe, but the relation between them has not been established. In order to verify and enlarge these observations, a series of photographic records of the vibrations was taken, each under similar conditions but at different pressures, and these records are reproduced on the accompanying Plate. The eight upper curves are records taken with the particular pipe above referred to ; the lowest one is a single record for another pipe selected at random. These records were taken by means of the special phonodeik devised by Dr. P. H. Edwards and described by Prof. Raman in a recent paper f. In view of the unavoidable effect of the horn and membrane of the phonodeik and no doubt also of the acoustic properties of the room in tending to distort the form of the record obtained with the instrument, the * Bulletin No. 15, Indian Association for the Cultivation of Science, Calcutta (1918). Tt Phil. Mag. Jan. 1920, p. 145. Ver) it~ Wave. iS aa cares, : “w a RY Re DS Qa Chi ee ae Ry OP 8 5 S OuinwcBice : S, Denese S Aa tale os Ss as. Te ae = Ome ~ SS) sts aan a ; > haaes % CRieae. : S Cie nates. 32 N Cie ae see C cei ae Ae ae Cet Sadie : ogee ree Be Pressure. 10 em. 11 12 Oereen ee eee ee etee Summary of Analysis Fundamental. First Overtone. “Tell of Wave-forms for Beating Pipe. Second Overtone. Har (aided by appropriate resonators). Overtones feeble; all steady, Overtones stronger ; still steady. Beats audible in first and second overtones. Fundamental (?) and overtones all beating, but especially the second and third, Fundamental steady and reduced, octave beating strongly, twelfth less strongly, Fundamental and octave steady, slight beats in twelfth, Fundamental and twelfth disappearing ; octaye steady, Steady octave. Mowe Mr. V. Lough on the Beating results PRs be relied on quantitatively, but they show clearly enough the cyclical character of the variation and the changing proportions of the partial tones with alteration of pressure. The results of the harmonic analysis from measurements of selected waves on each of these curves are exhibited for comparison with the results of the corre- sponding ear-tests in the accompanying table. Jt will be seen from the figures in this table that instead of increasing steadily and then falling abruptly as might be expected, the amplitude of the fundamental after rising to a maximum falls off gradually with increasing wind-pressure. If beats occur at all in this partial, which is doubtful, they occur only over a short range. . The octave increases gradually and develops well-marked beats before settling down to the final steady tone, while the twelfth increases to a maximum and then falls off and disappears with the fundamental ; the beats are most prominent in this partial. 3. Discussion of the Results. - Following on the analogy pointed out by Helmholtz between the form of the vibration of the air in narrow flue pipes and that of a violin string under the action of the bow, and in view of the comparison already suggested between the beating tones of overblown pipes and the similar effects obtained with bowed stringed instruments in certain cases, we should be led to seek an explanation of the beating tones in the departure of the free periods of vibration of the air-column from harmonic relation and a failure of the air-jet to supply sufficient energy for the continuous maintenance of all the possible modes of vibration. There are, however, certain difficulties in accepting this view in its entirety. It is true that the end corrections being not quite independent of wave-length have the effect of rendering the partials slightly inharmonic, and a similar effect may also arise from the elasticity of the walls of the pipe. But beyond this point the analogy does not seem to help us, for it is difficult to suppose that the air-jet which at normal pressure exhausts only a small fraction of its energy in sustaining the full tone of the pipe, can become less effective at higher pressures. Again, though the records of the actual vibrations show a cyclical variation, they do not at any stage show the very strongly marked fluctuation amounting almost to periodic disappearance of the fundamental vibration which is cha- racteristic of the records with bowed instruments obtained at the wolf-note pitch. We are therefore forced to seek an explanation of the Tones of Overblown Organ Pipes. té phenomenon in the special characteristics of the air-jet in relation to the natural frequencies of vibration of the air enclosed in the pipe. Unlike the bow, the air-jet has, as we have seen above, certain periodic tendencies of its own arising from the formation of vortices. It has been observed by Wachsmuth * that the excitation of the pipe cannot be maintained steadily unless the lip of the pipe is beyond the breaking-up stage of the jet, and then only when the pitch of the edge-tones approximates to that of the pipe. If so, we may suppose the initiation of the vibration in the pipe to. be due to the deflexion of the jet by vortex reaction, its maintenance to the controlling of the deflexions by the combined effect of the pulsations of the air-column and of the vortices forcing each other into step. Then, if by raising the wind-pressure the frequency of the vortex pulsations tends to rise and finally breaks out of step, leaving the pulsations of the air-column to control the defiexions of the air-jet, though weakened by the discordance, the quicker vortex pulsations might act independently to. excite that mode of vibration of the air-column corresponding to its next higher free period, which would be nearly but not quite the octave. In this way we should have virtually two. independent generators, each exciting a composite forced tone consisting of harmonic partials but slightly out of tune. with each other. This state of affairs would continue until the rise of pressure brought the frequency of the vortex pulsations into approximate agreement with the octave pitch, when the pipe would again give a steady tone at the. higher pitch. It may be noted that on the above assumption the frequency of the beats would approximate to the difference between the frequencies of the natural and the harmonic overtones, and would remain nearly constant during the. whole range of pressure, as seems to be the case in practice. Direct observation of the behaviour of the air-jet may be expected to furnish a test of the correctness of the sugges-. tions made above. It is proposed at an early opportunity to undertake the experimental work necessary for this. ‘purpose. In conclusion, the author wishes to express his cordial thanks to Prof. C. V. Raman, in whose laboratory and at whose suggestion this investigation was carried out, for the- facilities placed at his disposal. University College of Science, Calcutta, May 3rd, 1921. * Toc. ett. Pe fal VIL. The Probability of Spontaneous Crystallization of Super- cooled Liquids. By C.N. Htnsgeiwoon, Fellow of Trinity College, Oxford, and HaroLp Harriny, Fellow of Balliol College, Oxford *. T is well known that a supercooled liquid, even in the absence of a crystal nucleus, usually crystallizes spontaneously if the temperature is lowered sufficiently. Ostwald put forward the view that there exists a definite line of demarcation between the “ metastable’ region of temperature, in which crystallization cannot take place in the absence of a crystal nucleus, and the “labile”’ region, in which spontaneous crystallization is possible. Failure to crystallize in the metastable region is attributed to the non-survival of the minute crystals which may be formed momentarily as the result of favourable molecular en- counters. It can be shown thermodynamieally, and has been verified experimentally by Pavlov +, that the smaller the dimensions of a solid particle the lower is its melting- point. The first tormed minute crystals, therefore, may possess a melting-point below the temperature of the liquid, even though this is some degrees supercooled with respect to a large plane crystal face, and hence they have but a transitory existence. If, however, the temperature of the liquid is lower than their melting-point they continue to grow, and crystallization spreads throughout the system. The dividing line between the regions of non-survival and of continued growth is the metastable limit. The experiments of Miers and Isaac{, Hartley §, and others showed that various substances and solutions, when cooled, crystallized at more or less definite temperatures— often about 10 degrees below the melting-point or satu- ration temperature. Miers || considered that this supported Ostwald’s theory of a metastable limit, but de Coppet 4] ‘pointed out that the distinction between the metastable and labile conditions was arbitrary, and that the probability of anucleus large enough to survive and grow varies con- tinuously with the temperature. The metastable limit * Communicated by the Authors. + Pavlov, J. Russ. Phys. Chem. Soc. xl. p. 1022 (1908). + Trans. Chem. Soc. Ixxxix. p. 413 (1906); Proc. Roy. Soc. A. Ixxix. yp. 322 (1907). § Trans. Chem. Soc. xciii. p. 825 (1908), | Loc. cit. ‘@ Ann. chim. phys. x. p. 457 (1907). Spontaneous Crystallization of Supercooled Liquids. 79 merely represents a stage, in principle somewhat ill-defined, at which the probability of crystallization becomes very great. De Coppet found * that after the lapse of varying intervals of time, supercooled liquids crystallized within the region which the metastable limit was supposed to bound. The experiments just quoted, however, show that in practice the probability of crystallization nied vary very rapidly in the neighbourhood of this point. The object of the experiments to be described in this paper was to determine the mode of variation with temperature of the probabilitv of crystallization, in the hope of throwing some light on the nature of the causes which lead to the formation of a crystal nucleus in a supercooled liquid. Tammann } has measured the way in which the linear soe of crystallization varies with temperature, and also the number of nuclei formed in a inass frozen suddenly at different: temperatures, but not the probability of crystallization—that is, the probability of the formation of a single nucleus Frond an initially homogeneous liquid. It is clear that to obtain results of quantitative signi- ficance a statistical method must be adopted. A large number of similar tubes must be filled with equal volumes of the liquid under investigation, and the number counted whose contents have crystallized at the end of various intervals of time at a constant known temperature. A number of organic substances were investigated— salol, phenol, p.-toluidine, ap ay and 0.-nitro- phenol, as these melt at convenient temperatures. The procedure was to fill about a hundred tubes, to seal them, and tie them to a board ; or in some experiments where very light and small tubes were employed they were sewn to a piece of nickel gauze. The contents were then melted by immersion in a bath of water at a known temperature above the melting-point, sufficient time being allowed to ensure that all crystalline nuclei dissolved. The tubes were then transferred to a thermostat, and the number of crystallizations observed from time to time. When one experiment was completed, those tubes in which crystal- lization had not taken place were induced to crystallize as far as possible by immersion in cold water. Then the whole set were melted up again and a fresh experiment begun. * Loe. cit. . + ‘ Krystallisieren und Schmelzen,’ Leipzig, 1903, p. 131. 80 Messrs. Hihshelwood and Hartley on the Probability of The catalytic nature of the process of nucleus formation. The study of the phenomenon is complicated from the ~ outset by the fact that the formation of a nucleus is a Number OF CLV SEM TSAEIONS catalytic process. Previous observers have stated that the tendency of a supercooled liquid to crystallize depends upon the tempe- | rature to which it has been previously heated *, on the number of times fusion and solidification have taken place f, and on the length of time it has been maintained in the fused state f. These influences were very evident in our first series of experiments, which were made with salol contained in steamed-out soda-glass tubes 5em.x0°8em.. Hqual volumes . (about 2 c.c.) of salol were placed in all the tubes, which were numbered so that an individual record of each might be kept. The results are given in Table I. (see p. 92). With seventy tubes fairly smooth curves were obtained for the relation between time and number of erystallizations. Fig. 1.—Spontaneous crystallization of Para-toluidine. ra | 20 40 60 core 100 Minutes This justifies the application of the laws of probability to the results. The form of the curves is explained in the next section. (The salol results are not plotted here, but the similar series with p.-toluidine is shown in fig. 1.) * de Coppet (oc. cit.). + Karl Schaum, Zeitschr. phys. Chem. xxv. p. 722 (1898). t Ibid. % Spontaneous Crystallization of Supercooled Liquids. 81 De Coppet’s statement that salol shows less tendency to erystallize the higher the temperature to which it has been heated is confirmed, and in addition a rough quan- titative measure of this influence is obtained. Tvelative probability of: crystallization at 32°°3 C. (The method of calculating this number is explained in a later section.) exbed. G09 1) gece 1000 Maximum temperature to which the salol has been heated after sealins of tubes. eine a Omer eo eee 570) A ae eT Pee 240 Dis Peat ree ae Oe Fas esa anor ¢) 2 It must be emphasized tnat the diminution in the tendency to crystallize is governed by the highest temperature to which the salol has been previously heated, independently of whether it has been allowed to crystallize or not in the meantime. ‘Thus, for example, after the tendency to erystallize had been decreased by heating at 80°, tlie contents of the tubes were made to solidify; they were then meited at 50°, when they behaved almost exactly as in the experiment immediately preceding, having regained none of their power of crystallizing. This is ‘Important because it disposes of two possibilities: (1) that the diminished probability of crystallization aiter heating is merely due to the more complete removal of residual crystal nuclei, and (2) that it may be due to some change in the molecular state of the liquid brought about by the he: ating. Although the most marked and rapid diminution in the pendency “to crystallize is produced by heating, a_ slow ‘ageing ” effect gradually becomes evident, even in a series of experiments in which ‘the melting-up process is effected at a constant temperature. The tendency of some tubes to crystallize diminishes progressively. ‘Possible explanations of these facts are: (1) that gradual solution of alkali from the glass has a disturbing influence, (2) that gradual decomposition takes place, or (8) that some catalytic agency provoking crystallization gradually loses 1ts activity. That the last explanation is the correct one is shown con- elusively by opening the tubes, exposing the salol to the air, and re-sealing. After a few days’ exposure to the air in a peas the salol had regained its lost power of crystallizing, when the tubes were treated exactly as before. Phil. Mag. Ser. 6. Vol. 43. No. 253. Jan. 1922. G 82 Messrs. Hinshelwood and Hartley on the Probability of It sees clear, therefore, that the fine organic dust particles which are always floating in the air (Tyndall) are effective agents in provoki ing crystallization. This conclusion is supported by the observation of Jaffé* that repeated and intensive filtration of a liquid reduces its power of crystal- lization. Crystallization of an organic liquid seems, therefore, to be analogous to the condensation of a supersaturated vapour. Since these particles lose their activ ity as a result of heating, and also, but less markedly, as the result of ageing, they are probably colloidal in nature. The fone eal of a nucleus thus appears to be due to heterogeneous catalysis, the effect of heating and ageing being to change the degree of dispersion of the allaiee | s\ Seem, and ender its activity. Sinular behaviour after heating was also found with phenol (Table IT.), p.-toiuidine (Table 1II.), and diphenylamine. The following results relating to the “temperature of spontaneous erystallization ” illustrate the same point :— p--Toluidine heated in a tube for some time at 100° crystallized when cooled at 18°5 and 17°°5. After opening and exposing to the air, re-sealing, and melting at 50°, crystallization took place on cooling at 28°°5 amd 23. De a-Napl hthylamine in a tub e heated to 100° erystallized on See at 14°-9, 13°°6, 16°-0, and 132°35m sANter exposure to air, crystallization took place at 28°, 33°, Ama 206s The relation between Time and Number of Crystallizations. Suppose there are ng tubes, and the number whose contents have crystallized atter a time ¢ be n; then the relation between n and ¢ can give some information about the nature of the phenomenon of nucleus formation. If, for instance, a nucleus were formed as a result of slow consecutive changes in the o dn 5 = ‘ solution, then a7 would have a maximum at a certain point. ‘ a There is, however, no sign of such a maximum. If, on the Ore: han the formation were the result of raneoin chances, and the chance of formation were the same for each tube, then m and ¢ would be connected by the same exponential ioe as is found for a unimolecular chemical reaction n=n,(1—e-*), where y is a constant whose mag- nitude measures the probability of erystallizaticn. * Zeitschr, phys. Chem. xiii. p. 565 (1903). Spontaneous Crystallization of Supercooled Liquids. . 83 The experimental results show, however, that an abnormally large number of tubes tend to crystallize in the very early stages, and that the later stages are always drawn out much longer than the exponential relation given above allows (see curves in fig.1). In other words, the tendency to crystallize is different in different tubes. For example, in the salol experiments tube 9 almost invariably crystallized very early while tube 18 was as frequently still unchanged at the con- clusion of the experiment. As the phenomenon appears to be catalytic this is not surprising, as the amount and activity of the catalyst may vary from tube to tube. In this case we might assume that the exponential ex- pression held, but that y instead of being constant had values distributed about a most probable value—e.g., that the number of tubes for which y lay within a certain range was given by some such distribution law as a ayre—" Xl, so that ‘ No =( ax? eX dy. Then fi n =i) are" X(1—e—X*) dy : so that ik, 9 ora) 2/4 fy2 a ats — fp272 at — pene Ge s+ 5a) { eae da + 7. a a) t/2h2 4h This gives a curve steeper initially than the monomolecular curve aed reaching its limiting value more slowly. » The curves (e. g. those in fig. 1) cannot, however, be represented even by an expression like this with distributed values of y, being far too steep in the first short interval of time. Experiments were made with p.-toluidine, diphenylamine, and o.-nitrophenol in very thin small tubes, Wee 1mm. by 20 mm., which very rapidly attained thermal equilibrium when plunged in the thermostat. The results obtained with these (Tables III. and IV.) show that at a given temperature a certain proportion of the tubes ery stallize almost instanta- neously, and that among the remainder there is a distribution of activities something. like that suggested above. The probable explanation seems to be as follows :—Nucleus formation occurs round the colloidal particles which the liquid has derived from the air. An adsorbed layer is presumably G 2 84 Messrs. Hinshelwood and Hartley on the Probability of formed in which the molecules are orientated. If the radius of the particle is equal to or greater than that of the particles of solid which would be in equilibrium with the liquid at.the temperature of the experiment, then the nucleus will grow and crystallization will take place immediately. But if the radius of the dust particle is smaller than this equilibrium radius, then crystallization will not take place until, as the result ol a series of fortunate chances, the nucleus hae been built up to the necessary size. The average time which must elapse before this happens obviously depends upon the discrepancy between the radii of the largest active dust particles present and the radius of the nucleus which is first capable of continued growth. The Influence of Temperature. The curve in fig. 1, which refers to p.-toluidine, and the numbers in Tables III. and LV. show that as the temperature is lowered, the number of crystallizations taking place almost instantaneously i increases largely, but that the remainder of ihe curve resembles exactly some portion of a curve for a higher temperature. The radius of the particles in equilibrium with the liquid diminishes with temperature. Hence, some of the dust particles too small to effect imaricdeme crystallization at one temperature may be able to do so at a lower temperature.. As the temperature falls the average life of each tube diminishes, and some reach a critical temperature at which crystallization takes place at once. This is in @ sense a compromise between the views of Ostwald and de Coppet. The dependence of average life upon size of particle / ge ul P present. ¥ The attempt to determine this is rendered more difficult by the gradually changing degree of dispersion of the colloidal particles throusl hout a series of experiments. Madinaveitia and Acuirréche * and Rocasolano + have shown that as a platinum sol ages and its degree of dispersion diminishes, its catalytic activity first increases slightly and then decreases. A_ precisely similar effect is found in the formation of crystal nuclei, but only rarely is there an initial increase. It is found that with diphenyl- amine the tendency to crystallize at first increases and then falls off--the increase being due a to the initial * Anal. Fis. Quim. xix. p. 124 (1921). ' Anal. Fis. Quim. xix. p. e (ee ee Spontaneous Crystallization of Supercooled Liquids. 85 increase in the size of the colloidal particles present, and the decrease to the subsequent destruction of their activity owing to coagulation or some other cause. The loss of activity is the generai rule, and may be so serious as to make one experiment incomparable quantitatively with the next. This was unfortunately the case with diphenyl- amine. o0.-nitrophenol was more satisfactory, while a set of tubes filled with p.-toluidine was sufficiently ‘‘ stable” for a whole series of experiments to be carried out under approxi- mately constant internal conditions. The details of these are now given. The procedure is as foliows :— (a) To calculate the variation with temperature of the radius of p.-toluidine particles in equilibrium with the liquid. (6) To determine the number of tubes of p.-toluidine which crystallize instantaneously at a given temperature. By combining (a) and (0), we find how many tubes contain active dust particles whose radius is greater than a certain value. (c) From the n,¢ curves at different temperatures to find the average life of tubes containing particles of radius between certain limits. (a) Ostwald * showed that this could be calculated from the surface energy of the solid. Jones and Partington t have made a similar calculation. The following is a modification of Ostwald’s calculation, introducing Nernst’s Heat Theorem :—— Let A be the free energy of the process of crystallization of a supercooled liquid with respect to a plane surface of the solid. Then, by Nernst’s Heat Theorem f, A = U,—8T’, where Up and 8 have the conventional meanings. Let r be the radius of small particles of the solid which would be in equilibrium with the liquid at the temperature T; then, if o be the surface energy of the solid and p its density, the work done when one gram of the solid is changed from particles of radius r to radius «© (2. e. a plane surface) is Ae, er * ZLertschr. phys. Chem. xxxiv. p. 495 (1900). + Phil. Mag. xxix. p. 35 (1915). { Nernst, Theor. Chem. 4th Eng. ed., p. 750. ¢ a 86 Messrs. Hinshelwood and Hartley on the Probability of This must be equal to A ; | 2a il 2 e p (U,»—BT’)J co is not known ea but since tor the small tempe- rature differences of a few degrees with which we are concerned we may regard it as constant, the variation of r is chiefly governed by at Ee eee oe or From the seg ey point of p.-toluidine, 43°°3, and its latent heat of fusion, 39 cals. per gram, Ug ang) 8 are calculated in the usual way ; whence 20 ik Ao p (19°5—-000195T?) J p= ws J = Mech. Equiv. of Heat. This allows us to calculate the following values of 7, which are plotted against temperature in fig. 3 :— Variation with temperature of radius of particle in equilibrium with liquid. Temp. F, 26 oo. : aes ee nee Ot Ue em. BOS eee ees 0-15 DD Ty hieaeets sacs Ol DO Ee will, derdee 0-086 AO) aoe Ene oes 0-072 (6) Fig. 2 gives the number of tubes of para-toluidine out of the 122 used, which crystallized instantaneously at different temperatures. The points are numbered in the order in which the measurements were made, to illustrate the comparative absence of progressive changes in activity. Wenow suppose that those tubes in which crystallization occurred instanta- neously contain active particles of radius equal to or greater than the equilibrium radius for this temperature, and thus by eliminating the temperature between curves 2 and 3 we obtain the distribution of particles of various radii among the tubes. Curve 4 gives the number of tubes containing active particles with radii greater than a certain limit. Spontaneous Crystallization of Supercooled Liquids. 87 Fig. 2.—Number of tubes of Para-toluidine crystallizing within one minute at various temperatures. / + The points are numbered in the order in which they were determined. | 80 60 rN re) Number 20 (c) Now from the statistical n, ¢ curves (fig. 1) we can proceed to find the average life of tubes containing particles not quite large enough to provoke immediate crystallization. The working out of an actual example will make the method of procedure clear. Consider the curve referring to 122 tubes of p.-toluidine at 31°25. The number crystallizing within the first minute is 19.° On the smoothed curve (in fig. 2) 19 corresponds to a temperature of 29°-8, the difference being due to variations in the activity of colloidal matter from one experiment to another. We regard, therefore, 2¥°8 as the corrected temperature. The equilibrium radius at this temperature is 0:147x constant. We can now estimate the average life of tubes containing particles within 20 per cent. of this size—i. e¢., between 0-147 and 0-118. Fig. 4 shows that there are 32 tubes with particles greater than 0°118—that is, 32—19=13 tubes with particles whose radius is not more than 20 per cent. less than that of the particles provoking immediate crystallization. From the n, ¢ curves (fig. 1) we Radius x Constant Nurnber 88 Messrs. Hinshelwood and Hartley on the Probability of Fig. 3.—Para-toluidine. Variation of equilibrium radius with temper tie Bec ea se : SES RETA oa s eA eae Fig. 4.— Para-toluidine. Number of tubes containing ee of radius greater than a in value. F ; e : i | a ie | | scalar 25 -20 15 40 ‘05 Radius x Constant Spontaneous Crystallization of Supercooled Liquids. 89 see that 17 minutes were required for the number of erystal- lizations to increase from 19 to 32—thiat is, the average life is in this case about 8 minutes *. Similar caleulations yield a fairly consistent picture for x whole series of experiments ; thus :-— Average life, in minutes, of tubes containing particles whose radius differs from the equilibrium size by Vemperature of ,. ————— A = — experiment. 0-10 p.'c. 16-20 p.c. 20-30p.c. 30-10 p. ¢. [1 ae 3 15 79 360 approx. 21 5G ee a 2 10 32 290 approx. BETO. sas sicken 2 9 52 670 approx. SS Ea 4 80 approx. It is evident, therefore, thatas the discrepancy between the radius of the particle of equilibrium size and the radius of the particle present increases hy equal fractions, the average life increases more or less geometrically. As we have said, the tubes of diphenylamine were of variable activity and showed an initial increase. Although their behaviour was in accordance with the theory sug gested, it made a series of comparative measurements difficult. Exactly the same general relations however appear, as is shown by the following numbers, which refer to an experiment at 34°-0 with 133 tubes :—- Time, Number of in minutes. erystallizations. JL} Sse eae te at Lo sata oO” iol oe basen ae 47 i ahs eee ‘Ag oN hice ag ha D4 SPN aetna ae 56 1a! Ut Cae 63 The calculations of particle size, etc., are not given, but lead for the above experiment to the following result :— Discrepancy between equilibrium size and size of particle present. ae Eta i = SELES PRI U-10 per cent. 10-20 per cent. 20-30 per cent. Average life, in minutes... 6 approx. 50 approx. > 700 2. e., the same sort of relation as previously found with p.-toluidine. * The justification for calculating the average life in this simplified way is the rapidity with which ‘the number of crystallizations in successive intervals of time falls off. JO Messrs. Hinshelwood and Hartley on the Probability oj o.-Nitrophenol was found to behave in a similar manner, both as to the form of the curves showing the relation between number of crystallizations and time, and as to the average life relationships. There was, however, a con- sidewall variability in the tendency to erystallize from experiment to experiment and in the distribution of sizes such as to hinder accurate calculation. An experiment at 38°°7, the first of the series carried out (see Table [V.), gave the following approximate results :-- Discrepancy between equilibrium size and size of particle present. Cae reas A —__— aac oe 0-10 p.c. 10-20 p.c. 20-30p.c. 30-40 p. e. Average li fe, in minutes . 5 30 ES 290 Lifect of Heating on the form of the erystallization-tume curves for a given set of tubes. he effect of heating in causing a general diminution in the tendency to erystallize has already been dealtwith. | But the effect of heating fora given period at a given temperature does not necessarily influence the activity of all the tubes to the same extent. For example, the most active tubes may at first be sterilized most rapidly, so that the curves change somewhat in form. This may be seen from one of the experiments on p.-toluidine. If it were possible to bring all the tubes into exactly the same state, the curve would become a unimolecular curve ; and it was by applying a unimolecular formula as an approximation to a portion of the salol curves that the relative values for the tendency to crystallize (p. 81) were calculated. Discussion. To estimate the actual size of the particles about which crystallization takes place it is necessary to know the value of the surface energy of the solid. Little information is available about the numerical magnitude of this property, but the experiments of Hulett on the dependence of the solubility of certain sparingly soluble salts on the size of particles indicates that it is the neighbourhood of 10° (¢7. Partington and Jones, doc. cit.), and numbers of the same order may be deduced from the experiments of Pavlov on salol. In these series of experiments (p.-toluidine, diphenyl- amine, and ortho-nitrophenol) the contents of half of the tubes crystallized instantaneously at nee where es “ee the equilibrium radius was of the order 0°1 to 0°2 x 107 Spontaneous Crystallization of Supercooled Liquids. 91 Taking pas | to 2, and o as 10°, this gives a value for r of the order of 2x 10~° cm., which is equivalent to about three hundred molecular diameters. Consisting of a foreign substance which does not dissolve, the particle acts as a centre upon which successive layers of molecules may be deposited. If it is 30 per cent. too small to bring about immediate crystallization, successive layers are deposited, but continually re-dissolve until by chance a succession of depositions carry it past the critical size. This appears to happen after an average life of about an hour in the case of p.-toluidine, where this 30 per cent. increment seems to represent about 100 layers of molecules. Theoreticaily a particle of sufficient size should relieve supercooling, even in quite close proximity to the melting- point, but a limit is set by the well-known fact that colloidal particles become less active when their size increases beyond a certain value. The reason for this is not clearly understood, but the magnitude of this limit is probably as specific as all other catalytic phenomena. It would be rash to assume that the relationships described in this paper are entirely general, but a well-defined class of cases seem to come within the scope of the principles suggested. We hope to carry investigations further by ultra-micro- scopic observation, as this seems to be the direction in which results of some interest in connexion with the problems of surface energy might be found. Summary. The statistical investigation of the spontaneous crystal- lization of several supercooled organic liquids leads to the following conclusions :— 1. In the case ef these substances crystallization is provoked by the colloidal organic dust particles from the air. 2. The activity of these particles diminishes in general as the result of heating or ageing. ©. Their effectiveness depends upon their radius. If this is equal to the radius of a small particle of the solid which should thermodynamically be in equilibrium with the super- cooled liquid, then crystallization occurs at once. If the radius is less than this, the supercooled liquid has an average life depending on the discrepancy between the equilibrium radius and the radius of the particles present. The magnitudes of these quantities are discussed. 92 Messrs. Hinshelwood and Hartley on the Probability of TABLE IL Salol. 70 tubes. 16 experiments were made, of which the following are typical. M.P. of Salol = 42°95, © 2. Melted at about 65°. Temp. of thermostat, 32°:3. 1. Melted at 60°. Temp. of thermostat 32°:3. Time Number of in hours, crystallizations, Ol 21 0-2 29 0-3 32 0-5 3D 13 40 a0 39 8:0 37 23 62 3 Melted at 75°, allowed most to crystallize, and melted again atl ot?) Thermostat, 32°°3: Time Number of in hours. crystallizations. 0-5 19 2°5 29 15 38 22 42, 39 48 5. All tubes except one made to crystallize. Then melted at 49°—50°. Temp. of thermo- Stal tooeres Time Number of in hours. erystallizaticns. 05 1 10 19 5 29 19 30 43 34 67 40 100 44 140 49 . 165 50 212 52 236 DO 260 a7 ) | Time in hours. can) PAINE OS C1 OO Or GN OD! 7 Ye Oa bo The experiments are numbered in the order in which they were carried out :— Number of crystallizations. 4. Heated to 85°. Temp. of thermostat, 32°:3. Time in hours. 0:5 2-0 15 AT 72, 136 158 208 279 Number of crystallizations. 6. Melted again at 50°. Thermostat, 36°°5. Tinae in hours. 5 21 125 216 305 453 Number of crystallizations. Spontaneous Crystallization of Supercooled Liquids. 7. Melted at 50°. Thermostat, : 20°:5. Time Number of in hours. erystallizations. 2 36 26 45 150 48 300 51 9. Tubes opened and exposed to the air in a cupboard for Re-sealed 36 tubes. Temp. -of 4 days. Melted at.50°. thermostat, 3é°. Number of : | | 93 8. Heated the whole batch of tubes to 95°-100° for 4 hours. In thermostat at 32°. [ Cf. experiments 1-5.] Time Number of in hours. erystallizations. 48 15 216 18 792 24 1050 24. 10. Melted at 50°. Temp. of thermostat, 28°°8. Number of erystallizations cale. for 70 tubes. Time in hours. erystallizations | 6 a Time cale. for 70 tubes | 20 63 in hours. (7. é. actual number X 70/36). |. 0-5 “op 1) | 8 35 | Ze 39 7 49 TABLE II. Phenol. 41 tubes. typical :— 1. Melted at 50°. Temp. of thermostat, 18°:0. Time in minutes, Number of erystallizations. 10 3 25 14 40 20 230 32 1200 38 3. Melted at 50°. Temp. of thermostat, 20°-0. Time Number of in minutes. erystallizations, 30 10 900 26 2300 29 (12 expts. made.) The following are 2. Melted at 50°. None crystallized in 15 days at 25°. 4. Tubes heated at 100° for 2 hours. Temp. of thermostat, 18°:0. Time Number of in minutes. ery:tallizations. 10 3 200 12 14v0 17 Re pe a ae eo 94 Spontaneous Crystallization of Supercooled Liquids. TABLE IIT. Para-toluidine. 122 tubes, sufficiently small to attain the temperature of the bath very rapidly. Melted at 50° in each ease. ¢ = time in minutes; 2 = number of crystallizations. Meo 87. | ROe eT ONS. | Mas ele os mate A. 17°98; Gs 1. | ¢ nN. bs N. t. n 1 300 1 63 L 67 i 76 4 Ps kane 65 4 1 3 79 8 SU im cane 67 8 79 Z 86 20 G2 hia G4 79 16 86 27 95 32 Gomi eno 82 32 92 53 98 AG TO | ae 85 66 96 , 95. 1100 54 71 72 88 82 99 285 104 345 84 100 92 OO) 1000 —s«114 1320 6 ba0u 100 1S aaion | pe BOD, CES De! Oe. 6. 31°26. rey. ees” ler 8. After heating t 1 t n pee 2 | Cae 1 35 1 19 1 11 ae 3 3 3 24 5 15 F i‘ 12 45 14 31 27 a ol 1 1s 30 Do 70 1S meen mele 27 5 20 50 60 120 51 DD 34 16 26 80 65 210 53 345 3 Math 37 115 68 370 56° | . 1380 BE DG 62 395 79 1545 2 | 1020 84 1350 oe | 1365 87 | 2490 93 Tasue LV. Ortho-nitrophenol. 130 tubes. Tubes once heated to 70°, contents allowed to solidify, and then re-melted at 50° in each ease. = time in minutes; #2 = number of erystallizations. il. B8o72% | ON GUO. 3. 86°07. | 4. 38°52 yaeede ibe Fea ha I. be N. ibe f- | "fe nN. OT Aiea OOeun ein ie anon | 585 ees Be OAcl Pte. vena OuN maa hea ROS 3 64 4 66 Hoos ly lommaaol 16. 35 | MO: OR ey ees 30 BO ers ye 9 as) 30: P22 82-9" Se aI 45 42 900 MOGs alGSe 72 4 94) Oh 00 Salon 75 46 | | 270 80 10 1OlY fa nmaiene 115 49 379 89 130) “08h 4 240 58) 1420 105 310 86112 315 60 | 630 67 | 1350 70 | | Numbers of tubes crystallizing within ] minute at various temperatures :— 26°°2, 126; 28°:9; 1135) 31°:6, 80; 34°20, 86; 35°05 iG eee emo. 39°°85; 40; 42°°0,05) 4050, 19; 37°:0, 48 5 39°:8, 04 ; 347-0162 ere ocamaoe 292-5 3605) 20 nO wate: : if i J ¥ ea y VILL. On the Insulation of Highly Attenuated Wires in Pla- tinum Resistance Thermometers. By J.J. Manury, M.A., Research Fellow, Magdalen College, Oxford”. ae construction of platinum thermometers having bulbs with a diameter of 2 mm. and resistance wires the thinness of which ranges from ‘06 to ‘(02 mm., presents some difficulties of an exceptional order. Of these the greatest is probably that encountered when we proceed to support and insulate the highly attenuated wire. The cross frame of mica introduced by Griffiths, being from lack of space inadmissible, the difficulty can only be surmounted by having recourse to other methods. By trial it was ultimately found that in addition to the plan first described by Burstall tf and sub- sequently by. Callendar and Nicholson + three others were available : these we now briefly describe. Mi | NE = x x 1. In all cases where a resistance of from 3 to 5 ohms suffices, Burstall’s simple plan may with advantage be followed, the wire being arranged as a loop as shown in fig. a. Placed in situ, much of the loop makes contact with the containing tube: this is helpful, for the thermometer is thereby enabled all the more readily to acquire the tempe- rature of the surrounding medium. If the loop is well * Communicated by the Author. + Phil. Mag. Oct. 1895. t Proc. Inst. C. E. 1898. 96 Insulation of Wires in Platinum Resistance Thermometers. formed and carefully arranged within its tube, short- circuiting becomes impossible. 2. When for a given purpose a short wire offers too small a resistance, Burstall’s plan becomes for highly attenuated wires, imperfect ; ; this is due to the absence of the necessary - stiffness in the wire. The defect is successfully met by inserting a thin slip of mica between the two limbs of the loop, as shown in fig. 8. The upper end of the slip is held between the two stouter pieces of mica to which are fastened the thermometer leads, and the other end is notched for the reception and retention of the lowest part of theloop. Asan alternative the slip may be pierced with a needle near the lower end, the resistance wire being then threaded through the hole drawn taut, and finally fused to the leads. Loopsot any des Adee ee h may be mounted and highly insulated in this way. . For a thermometer having a high resistance, the sup- por ine insulator consists of dee plates of mica each about 1 mm. thick, and cut as shown in fig. y. The wide and shorter portions of tle plates are held between two others which are used for supporting the platinum, gold, or silver leas, and all four firmly bound together wana te nickel wire. QOne oe is made to terminate just below the com- pound block ; but the second lead is longer and its lower portion is Nepne flat and placed between the two narrow limbs of the central slips of mica, with its end slghtly projecting. The resistance wire having been fused to the longest lead, is wound spiral-wise, as shown in fig. 6, upon twin slips, and fin: ally fused to the second and shorter lead. It is then ready to be introduced into its protecting tube. 4. More recently we have adopted the following convenient and very efficient plan for insulating and supportine highly attenuated wire loops of any required length. A olass. rod about 1 or 2 mm. in diameter is prepared. ‘and a needle- -eye formed at ona end. Having fused one end of the resistance wire to the lead, the other end is drawn through the eye in the glass rod anal brought up and secured to its enn Holding the leads vertically, the rod will, if of appropriate weight, pull and maintain the loop taut. The wires are now placed within their protecting glass tube, the lower end of which is open and of such dimensions that the rod suspended from the resistance wire can just pass through. a keeping the thermometer vertical, and applying a suitable blowpipe-flame, the lower end of the tube is closed by fusing it to the glass rod. When cold, the superfluous glass is cut off and the end rounded in the flame. Fi ig.e showsa completed thermometer- bulb‘ of this form. Daubeny reer Oxford, Se piensa | IX. The Equations of Equilibrium of an Elastic Plate under Normal Pressure. By Joun Prescott, M.A., D.Sce., Lecturer in Mathematics in the Faculty of Technology of Manchester University *. HE earliest writers who had any success with the problem of the bending of an elastic plate followed the analogy of the beam so faithfully that they missed altogether one very important difference between a plate and a beam or rod. ‘hey were so obsessed with the idea that only the curvatures of the plate mattered that they ignored the effect of the stretching of the middle surface that must accompany any bending of the surface. It is a very well- known fact that a plane area cannot be made io fit on the surface of a sphere, for example, without a stretching or contraction of some of its parts. Yet this strain has nearly always been neglected by writers on the bending of thin plates. It is true that there are many cases in which it is justifiable to neglect these stretchings and shortenings, but there are so many practical problems in which it is not justifiable that no theory of thin plates is satisfactory which does not take them into account, or else point out the limitations of the restricted theory. The usual Poisson-Kirchhoff equations need no alteration when the maximum deflexion of any particle of the middle surface is small in comparison with the thickness of the plate, this deflexion being measured from a plane which touches the middle surface at any convenient point, or from any developable surface which touches the middle surface at any point and nearly coincides with it elsewhere. But when this deflexion is of the same order as the thickness, then Poisson’s equations may be very much in error. In such cases the mean tensions over sections perpendicular to the middle surface play as great a part in supporting a pressure on the plate as do those stresses of which the usual theory takes account. In fact, to make Poisson’s theory correct there must be quantities of three different orders of magni- tude. In ascending order these quantities are (1) the maximum deflexion perpendicular to the unstrained middle surface or perpendicular to a developable surface ; (2) the thickness of the plate ; (3) the lateral dimensions of the plate. * Communicated by the Author. Phil. Mag. 8. 6. Vol. 43. No. 253. Jan. 1922. H 98 Dr. J. Prescott on the Equations of Equilibrium The only previous reference to this limitation that I am aware of occurs in Thomson and Tait’s ‘ Natural Philosophy.’ Thev state (Art. 632) that “the deflexion is nowhere, within finite distance of the point of reference, more than an infinitely small fraction of the thickness.” They had already worked out (Art. 629) the mean circumferential strain of a small circular portion of the middie surface of a plate which is bent into a surface with principal curvatures p: and p, when the radial lines are unstrained. From their result they justify the limitation that they impose. The footnote added to the above quotation from Thomson and Tait by Professor Karl Pearson in the ‘ History of the Theory of Elasticity * shows how little the point was under- stood. ‘The footnote runs thus: “The pressure of the finger on the bottom of a round tin canister seems to produce a deflexion which is far from being an infinitely small fraction of the thickness, and w hich might, I think, be fairly discussed by the ordinary theory.” I regard this footnote, written by the author of such a complete history of the modern develop- ments of the subject, as very good evidence that no other writers have laid any stress on this very important limitation of the usual theory. We shall now produce a theory in which the only restrictions are that the thickness of the plate and the maximum displacement just mentioned are of smaller order than the radii of curvature of the bent middle surface. This will bring the plate theory to the same level as the Bernoulli- Kulerian theory of beams. Let us assume that the xy plane touches the kent middle surface of the plate at some point, and that a particle of the middle surface which, in the unstrained state, would be at x, y, 0, is dispiaced to e+u, y+v, w. Let dx, dy denote the components of an element of length ds in the unstrained middle surface, and let ds, be the length of this element after strain. Then (ds)? = (da)* + (dy;* and (ds,)?=(dx+ du)? + (dy+dv)?+ (dw). Now we may neglect (du)? and (dv)? since we shall retain the more important quantities du and dv, but we have no such reason for neglecting (dw)?. Then (ds,)? =(da)? + (dy)? + 2dadu+ 2dydv+ (dw)? = (ds)? + 2dxdu+2dydv + (dw)?. of an Elastic Plate under Normal Pressure. 99 The extensional strain of ds, is ds,—ds _ (ds,)*—(ds)? ds —— ds(ds,+ds) es (ds,)?—(ds)? + 2(dsy _dadu , dydv , 1(dw 2 "lasds. dsids "2 <=) (1) Now let’ds make an angle @ with the z-axis. dx te dy hau Then de O88 0, 75 sin 0. Ou Ou d= = di Also = Oi fe ae dy, ; du Ow Ou whence a= x, cos @+ Se 0. dv a There are similar expressions for —- and —-, and when ds these are substituted in equation (1), that es gives the complete expression for the strain in any direction in the middle surface. Putting @=0, we find that the strain in the direction of the w-axis 1s OU Ow Ox + — (Se) e . . e e es 2 (2) Putting 0= ae we find that the strain in the y direction is p= 2+ 3(S) ae Oar eng Again, the strains in the directions making angles 45° and 45° with the z-axis ure (ee or) - sie + Ps) + (oe + ey, a= 2\0z Oy ANGE | OY A\Ow Oy Oe oy ehrow “ou, jew. dw\ ” x -S") ALY: Sy) als peri Therefore «—6,= e +2 -- » ‘ oe pater. (4) But it is easy to mia aa (4;— 8) is the shear strain of the element which was originally a rectangle with sides dw and dyin the middle sur face. Let us call this shear strain +. H 2 100. Dr. J. Prescott on the Equations of Equilibrium Now let | H= Young’s modulus for the material of the plate, o = Poisson’s ratio, n=the modulus of rigidity, 2h =the thickness of the plate, Py P,.==the tensional stresses in the middle surface in the directions of dx and dy, S=the shear stress in the middle surface on the faces on which P, and P, act. The stresses P,, P,, 8, are shown in fig. 1. It is under- stood that, as dz and dy approach zero, the stresses 8;', S.’, Shea Sb all approach the limit 8. Fig. 1. x Now, assuming that the tensional stress in the z-direction is negligible in comparison with P,; and P,, the usual equations of elasticity give fo P,-oP,=Ba= { &” +5(S") 1 Mit Ow\? v P.—oP,=HR=8 4S” +5(5") (>> eee (6) eT ae Ju. Ov. Qw Ow | lela Onn cee Oy AE) Since the stresses P;, P., 8, can be proved to be the mean values of the stresses at x, y, from z=—h to z= +h, and since we are assuming that there are no external forces ‘of an Elastic Plate under Normal Pressure. 101 parallel to the middle surface of the plate, the equilibrium of the element dz x dy (fig. 1) requires that 2h(P,"—P,' )dy +2h (S."— Sh )dx=0 and 2h( P2" — Ps')da+ 2h(8;""—S8y')dy=0. , i But Py’ —Py'= S1do, fol» SB —§,/= = Oy dy: Therefore the above equations give OPiS 08 =0 (8 Oa 7 OY and aaa Baten diese GE) If we now choose a function ¢, such that ian sS=— 0D Sse ae eee Cea (10) then equations (8) and (9) show that p=EO?. Eo ?. Sen i P= Ea’ ee eee (CE) Writing er 5 for n in equation (7), and then elimin- ating wand v from (5), (6), (7), we get gE P,—aP,) pe a, Et Ow vi id Ow =n {S, sey +s =) 2,2 (2 Salt That is, 0*d Od oe 2 ON Ow Ge Ox* Be 02"0y’ 4 Oy? alee) Gm Ogee Writing V? for oot es this last equation may be written eo Ore J kG ea a e e ° 12 Var (5 a 2) On OY? - This is one of the equations of equilibrium of the plate. The other equation is a modified form of Poisson’s equation, which form we shall now find. 102 Dr. J. Prescott on the Equations of Equilibrium . Let p denote the external force per unit area perpendicular to the plate. Then Poisson obtains the equation 2 twp. . ee 3 This equation gives the pressure supported by transverse shearing forces in the plate in the same way that the load on a beam is supported by the transverse shearing force. ‘The stresses which we have denoted by P,, P., 8, are assumed to be zero in arriving at equation (13), but it is the object of this paper to show that these stresses cannot usually be zero. We must now correct this equation by taking into account the effect of the mean stresses P,, P,, and S. Let P,' (fig. 2) make a small angle —,' with the plane of vay ; and let P;"’ make an angle w,'’ with the same plane. Fig. 2. P} Then, since these two stresses act on an area 2hdy, the component, in the direction ‘of the z-axis, of the force due to P,’ and P,"' acting on the small rectangle is = / / Phdy(Py!ay' —Py!p,) = Bhdy oe di But y,'= 9 i aol (Pa! IE, ee a Therefore aha as "— Pia) = 2h ic aa $") dx dy. Phe fie Lammomen force per a area on the small rectangle in the direction of the z-axis due to P,’ and P,’’ is fe) Ow | aS (Ps Oa Likewise, due to P,’ and P," there is a force per unit. -area in the same ee of amount a are =P (57): of an Elastic Plate under Normal Pressure. LO Again, let —¢,' denote the small angle which the force due to 8,’ makes with the zy plane, and ¢,/ av the corresponding angle tor §,".- Chen the component force parallel to the z-axis due to these two forces is 2h dy(S1"o," = Sidr) => Qh dy Ott) ae But bi=S, and 5,/=S,=S. _ Therefore Z2hdy(31"'61" —8)/¢y') = 2h 2(s - ) dedy. Likewise, the corresponding force due to 8.) and S," is 2h =i ge ) dady. Therefore the total force per unit area in the direction of the z-axis due to all the mean stresses acting on the edges of the element dz x a is af 2. (v.22) +2 (v.28) +2 (ad4)+ 2(52)}, which, by means of equations (10) and (11), simplifies to 2h foe oe O2w 07> 4 07w Leis i OF, Oye eae (OMe EE BOY ,OLOy ds This is the expression that must be added to p in equation (13), since, in that equation, p is the force per unit area in the direction of the z-axis which has to be supported by the transverse shear stresses. Thus the corrected form of Poisson’s equation is 4 Hh? peels (0°wd°d , 0’w 07h ae Py Ba? 0y’ +t 3y? On pO Oe ACE ‘Oy OxoyS 2” CS p being, in this equation, as in Poisson’s equation, the external force per unit area. This last equation takes the place of Poisson’s equation. Since it contains two unknown functions, w and @, it must be conibined with equation (12), which we rewrite here, Vib= (5-) - oe eS iid cGl2) The residual stresses parallel to the middle surface after the mean stresses P,, P,, S, are taken away, are proportional ee ee ee ee on SS 104 Dr. J. Prescott on the Equations of Equilibrium to the distance from the middle surface. The residual tensions are equivalent to couples M,, My, per unit length of Fig. 3. @) sections perpendicular to dw and dy respectively. The residual shear stresses give rise to equal couples Q per unit length about normals to the same two sections. These couples are represented by vectors in fig. 3, the right-handed screw system of representation being understood. The z-axis points upwards—that is, towards the reader. The expressions for these couples, which are correctly given by Poisson’s theory, are M,=C = oot as Ly M,=C oe +00 (16) OF Pn = a eee sae ag ae ca. * (18) It should be observed that the vector representing Q is along the outward normal for the pair of faces perpendicular to dz, and along the inward normal for the pair of faces perpendicular to dy. : of an Elastic Plate under Normal Pressure. 105 In addition to the above couples there are mean shear * stresses F, and F,, on the faces on which P and Py act, these stresses acting in the direction of the z-axis. The magnitudes of these stresses are F,=— BERG, Liv) x Dear a uamad lien ae) x OY The directions of these stresses are shown in fig. 4 Fig. 4. \. If the edge of the plate is perpendicular to the x-axis, and if this edge is free, the boundary conditions, according to Kirchhoff, are 0. S=0, | 2hF,— 08 0.) ) M0) c vee oy J Poisson thought that F,; and Q could be made to vanish separately at the boundary, hut Kirchhoff showed that the solution of Poisson’s differential equation does not contain enough arbitrary functions to satisfy all Poisson’s boundary conditions. The reason why Poisson’s boundary conditions cannot be satisfied is because his differential equation is derived from assumptions which are not even approximately 106 Dr. J. Prescott on the Equations of Equilibrium true near the edge of the plate. The best physical explan- ation of this difficulty has been given by Professor Lamb in his paper “On the Flexure of an Elastic Plate” | Proceedings. of the London Math. Soc. vol. xxi.). Our new differential equations may require new boundary conditions. I think, however, that it is unlikely that we can satisfy more than the four boundary conditions of Kirchhoff, though this number depends upon the number of arbitraty functions, involving only real quantities, that are contained in the general values of w and @ satisfying equations (12) and (14). How many such functions there- are I have to admit that I am unable to discover. Although some rules are given in the theory of partial differential equations concerning the number of arbitrary functions in * the solution of a partial differential equation, these rules are: not, I think, of any use for our purpose, because they make no distinction between functions with reali arguments and functions with imaginary arguments. The theory tells us,. for example, that the solution of the equation av Vv oe Om contains two arbitrary functions. True, it does. But when we are confined to real quantities, these only amount to one,. because the solution V=/(e@+ y) +F(e—iy) de generates into V=/le+iy) +fle—¥y). I believe, however, that the number of boundary conditions. is four in all cases, because it seems to be definitely four in some cases. Suppose, for example, that an unstrained plate has its edge completely fixed and clamped in its unstrained’ position. Then let any transverse forces be applied to the plate while the edge is still held. The problem in this case appears to be completely determinate, and the conditions at Ow the edge are u=0, v=0, w=0, aa =(, dy being an element of the outward normal to the edge. This suggests that the functions wand @ are completely determined by the differ- ential equations and four boundary conditions in one case, and therefore also in all other cases as well. If we could solve equations (12) and (14) completely, we should have no trouble in deciding how many boundary conditions could be satisfied. But these equations are of an Elastic Plate under Normal Pressure. 107 probably much too difficult to be solved completely. The introduction of the function @ has spoilt the linearity of Poisson’s equations. It will be observed, however, that the equations are linear as far as @ is concerned. The physical interpretation of this is that, when one set of values of the stresses has been found for a given state of bending of the plate, any set of mean stresses P,, Pz, and 8, which would be in equilibrium if the plate were not bent, could be super- posed on the set already obtained, and thease new stresses would satisfy the differential equations. The pressure p is also altered since the added mean stresses, because they exist in a bent plate, could support a pressure on the surface, just as the tension in a membrane can support a pressure. Owing to the complexity of the equations there are very few problems that we can solve directly; that is to say, there are few cases in which we can find w and ¢ from a given value of p. The inverse process of finding p when w is assumed offers, however, no great difficulty. We shall first express our equations in a form that applies to symmetry about the z-axis, and then make use of these equations to find p corresponding toa given w. We shall also solve the problem of a rectangular plate bent into a surface of revolution by forces applied at one pair of edges only. The second of the following problems illustrates the importance of the maximum displacement w in its effect on the pressure. Symmetry about the z-axis. Let 2?+y?=,r", and let us suppose that w and ¢ are functions of 7 only. Let P,, Mj, etc., bear the same relation to.the radius as they bore in the earlier equations to the #-axis. Then, u being the radial displacement, the two longitudinal strains are de, 1(dey eR ON dy je 2 and the shear strain is zero on an element with sides along and perpendicular to a radius. The expressions for the mean stresses in terms of ¢ are 2 P=E-S; p,-pS*, s=0. . . (22) 108 Dr. J. Prescott on the Equations of Equilibrium Also the principal curvatures of the middle surface are d*w 1 dw an — dr? r dr Consequently sug (GE Ged i 1 - me i ee oy 1 dw ot w 9 =—C 24 oh r dr Cdr? | aah 2) Q=0 Again ny ad , 1d ee dr? rar ld/( dw 9 eae (pr ete 20 1 ae 7) r C2) Ly lb | 2 == —- — —_— ee sak It follows that | Cd hi —— oy aph VY) } | Ghd Ae dw Ver | See a) — Qhdretr al? ra) | B= 0 Weare a) Finally, the differential ae 12) and (14) become 4 ida, dae Vi a r dr’ dr that is, dw d’w ps a Se Me oe AG v)=- "Tea Pa ee pee Sebere | Nae , “5 "adr and 2 EM opp) (Cw dd , dw wy | Bee Noh aie (Sa dr ' dr dr?) 2° that is, 2 HR, 2hb id dw of) = 3 Ge ee (a dp} oe am) Problem 1.—To find the stresses in a circular plate which is bent into the form of a piece of a spherical surface, of small solid angle, bounded by a circle. of an Elastic Plate under Normal Pressure. 109: Taking the zy plane touching the middle surface of the bent plate at its middle point, the deflexion is approximately Fae? ¢ being the radius of the sphere. Let a be written for the radius of the plate. Then equation (27) becomes 5 (°*)=-4. Oe a oe : Sean = Be ee Integrating this once, we get dy ne (Ro Sa ae dr 2¢ The constant H must be zero because the other two terms. in the equation are clearly zero when r=0. Therefore db dy 7. Be? i whence w =>=—_— gg" — ©); b? being the constant of integration. Now we have got 1d(/(_d¢ tae ca (Gn) =n age + + (29). Integrating again, and omitting the new constant, which is obviously zero, we get d lo pb = — rtd). Cet Mae as (30) Now the mean tensions are Ed E Bi pe yoke ha Cat) ' : P,= BLE ‘ [bee oi yen te) Osan (ae) _ Let us now suppose that the radial tension is zero at the rim of the plate. 'I'hat is, P,=0 where r=a, or 267 —a’?=0, which gives the constant 0”, 110 Dr. J. Prescott on the Equations of Equilibrium | E Thus i 162 (7), 2. E P= 1G (a?—3r’). c - > > c (34) Since V4*w=0, equation (28) gives 2h ad vr dp ears L lon Gra’). . . |. eee It should be observed that the total thrust on the plate, namely, : 2arpdr, ("2nep is zero, and also that the pressure p changes sign at the circle where 277=a?. Moreover, the circumferential tension a rE a thrust, where 7 is greater than eS The mean shear stress F, is ey al A ee PS 3 ot dae =0: i. 1.04 Sees The bending moments per unit length along and perpen- dicular to the radius are P, is positive where * is less than and negative, that is, dw adw i, te =— (a +et Cae oe ee hal ia = re 3(1—a)e M, = Ml. eee ee (38) We thus find that jthe middle surface of a circular plate _ will take the form} of a portion of a spherical surface of radius ¢ provided that a constant bending moment given of an Elastic Plate under Normal Pressure. 111 by equation (37) is applied about each unit length of the rim, and that a variable pressure p given by equation (35) is applied to that surface of the plate which becomes the convex surface in the bent state. Wig. 5 shows the way the pressure acts across a diametral: section, and shows also the direction of the bending moment on the rim, M, The usual Poisson or Kirchhoff method would make the pressure p and the tensions P; and P, zero, but would give the same values of M, and My. In the solution we have just obtained we assumed that P, was zero at the edge of the plate, and this gave b?=2a7. If, however, P; is not zero, but has a positive value at the rim, we have only to leave 4” in our equations, and for this case b2 will be greater than 2a. In this way a constant term will be added to each of the quantities P,, P,, and p. The added quantities are just the stresses and pressure in a stretched membrane which has no flexural rigidity. Problem 2.—To find the pressure required, on the present theory, to produce the deflexion which a circular disk assumes according to Poisson’s theory when subjected toa uniform pressure and supported without clamping at its edge. ‘According to Love (‘Theory of Elasticity,’ 3rd edition, Art. 314) or Morley (‘Strength of Materials,’ Art. 149), the deflexion of a disk of radius a due to a uniform pressure p; is Be 20 7 a te es iy aw COD) where igie= TORMRRT, a ie (40) ae and Ga re one lian Ml tomes SR el tal ie (41) and C is a constant which depends on the position of the origin. If the zy plane touches the middle surface at the centre, then the constant C is zero. We shall find the pressure on the present theory which will produce the deflexion given in (39). 112 Dr. J. Prescott on the Equations of Equilibrium ape (27) oo re) = _ dw Pw al dr dr dr” whence, by integration, WN aw, i (= q “dr 7 eo a == Ol (On 9)”, the constant of integration being clearly zero. Integrating again, p= SEP (Sh soit or *) +B, that is, dg tro fb : Los, bea S(0 Gp) = SEE (5 608 — 94? + ar") + Br, Therefore | db _ fo ere ae es a 1 rf = 8H (so ae yas Br, (42) Thus the radial tension is \ 1ld¢ Ea ye = -SHe(5 bt pity E 5) +5 2 By eae 'To make P, zero at the rim we must have ib 1 05 4 2 B= ¢ Hiat(6s'—4b'a? + a de 51 apa cannes) Again, by equation (28), 2) ee, 2th d (dw dd‘ PT 8 ear an ae a} ie 2Hh d /dw ~ in: oar ae SHENG — 5 HE l®(120° —200"r? + 20024 — Sr) (45) Denoting the difference of the deflexions at the middle and the rim of the plate by wo, and serene the pressure at ofan Elastic Plate under Normal Pressure. 113 the centre by po, we get Wy = Ha?( 2b? — Ztihs cM (SE) ir pa fect ey.) og 3 LEO) 6? 3+oe where SiaaieT AL ot: Live Sane esau gl (e279) Also Po= pit SHBENOC? =n + : H?Bha’(6s?—4s? +s) odbsie, , 6s? — 45+ 8 =—1 -- 3 HEhw, “(9s n? 3 a | wo” 65° —457 +5). = oy els ee a If c=i, this becomes po=p {141-07(Fr) t tte hae sO which shows that pp is approximately twice as great as p, if the maximum detlexion wy is equal to the thickness of the plate. Moreover, at the rim of the plate, where r=a, we find that Wo \7 | - p=p {1- asa (5) J CE es so that, if wy=2h, the pressure at the edge of the plate is little more than half of yy. The preceding numerical results show how inaccurate are the usual Poisson equations when the maximum deflexion of the plate is of the same order as the thickness. As long as the maximum value of w is less than one-tenth of the thickness, it isclear that Poisson’s equations give results that are as accurate as any that we can ever hope to get in such calculations. But for a body like a piece of tin bent so that the maximum deflexion is three or four times the thickness of the sheet, Poisson’s method does not even give a pressure of the right order. In the example we have just worked out, if wo were only four times as great as 2h, the pressure would vary from nearly —7p, at the rim to 17p, at the centre, whereas Poisson’s equations give a uniform pressure p, over the disk. Problem 3.—A rectangular plate is bent into a part of a surface of revolution, which is nearly nearly cylindrical, by forces applied at one pair of opposite edges, the other pair Phil. Mag. 8. 6. Vol. 43. No. 253. Jan. 1922. 1 114 Dr. J. Prescott on the Equations of Equilibrium of edges being circular arcs. To find the form of this surface. Let the wy plane be parallel to the tangent plane to the bent surtace at its middle point. It is less troublesome to take the wy plane parallel to this tangent plane than actually coincident with it. Also let the y-axis be parallel to the axis of the surface of revolution, and the z-axis drawn towards the concave side of the cylinder. The sections of the middle surface perpendicular to the y-axis are circular arcs with nearly equal radi. Let p be the radius of the are at distance y from the az plane, and let a denote the distance of the axis of revolution from the xy plane. Thus a is an approximate value of the radius of the circular ares, since it is understood that the xy plane nearly coincides with the tangent plane at the centre of the bent middle surface. Now let P=GHWi 3 .- i ee where, clearly, w, is a function of y only. Then the displacement of any point of the bent plate (fig. 6) at a small angular distance from the origin is approximately | of an Elastic Plate under Normal Pressure. 115 Now it is clear that the mean shear force S is zero because, in such a surface of revolution as we have assumed, the circumferential shear force on the circular sections is just as likely to act in one direction as in the other ; and since it cannot act in both at once, it must be zero. That is, OP. Se ae whence Fen Gals oedts SER) Moreover, the mean tensions P,; and P, cannot be functions of x. But “po? _ER(2)- (5 PhS SNE) 3 Caen OP eS) and since this cannot be a function of 2 it can only bea constant or zero. Then let '@)=A. It follows that Ge) Ag Bia oy) ee EL O9) There can be no term eontaining the first power of x on account of the symmetry about the z-axis. Since P, is constant, it has the same value at the edges of the plate as it has at any other point of the plate. As we are assuming that there are no actions on the circular edges of the plate, the stress P, must be zero, and therefore A must be zero. Moreover, the constant B can be merged into f(y), so that Lew ah tae Capen et (OW) The equation for p now is 2 El d'e, 2Eh dd tet 8 Tee dak a age -» Also the equation connecting w, and ¢ is Cee es dy* ae: dy? dh 1 from which ae 7 beat ©) ee ve pe ee Oe) since symmetry requires no term involving the first power of y. The constant C depends only on the position of the ay plane, and since we lett this position indefinite in that it had only to be parallel to a certain tangent plane, we may put 12 116 Dr. J. Prescott on the Equations of Equilibrium zero for C, provided we find finally that this leaves the wy plane somewhere very near the tangent plane mentioned. Then 22 7 - ee 23) ay ee and, substituting this in equation (61), we ce af 2 Eh d‘w, 2Eh 28 eae Be a oe Now let p=0. Then | . (64) 4 a= Ant DO ete Rees where dint ee 2s) GL eee The complete solution for w, is w,=H cosh my cos my+ K sinh my sin my +H, cosh my sin my+K, sinh my cosmy; . (67) but since w; is an even function.of y we know that Hy and K, are zero. Therefore w;=H cosh my cosmy+K sinhmy sin my. . , (68) Now let the width of the plate, that is, the dimension parallel to the y-axis, he 2/. Then, in order to make the stresses zero at the circular edges of the bent plate, we must make Roo pvhere ysl ee ee 2=0 But 2 M,=C{ SP 4055} On —C 1 2m? (H sinh my sin my — K cosh my eos my) + . ec! (70) and 3 Cm*°( H(cosh my sin my + sinh my cos my) : f= ak C ° F (71) h ( — K(sinh my cos my —cosh my sin my) Writing 6 for ml, the conditions that M, and F, should be zero where y= are — H sinh @sin 0+ K eosh 6 cos @= —— . ; ZANV A H(cosh @ sin 6 + sinh @ cos 0) = K(sinh @ cos 8 —cosh @ sin 0). of an Elastic Plate under Normal Pressure. TG Solving these for H and K, we get — _ o@ sinh @cos @—cosh 6 sin 0 ~ ma sinh 20+sin 26 ‘ ao eosh@sin O+ sinh @ cos @ ~ ma sinh 20 + sin 20 b) wherein it should be noticed that o oh ie a ae. ace ; Thus 1) ual Bere yt x ma sinh 20 + sin 20 | (sinh 8 cos @— cosh @ sin 8) cosh my cos Ly (74) + (cosh @ sin @+sinh @ cos 6)sinh my sin my)’ and. «. Waa Wy ay es PAPO) As a particular case suppose mi, and therefore also my, is a small fraction. Then an approximate value of w,, obtained by. expanding the hyperbolic and circular functions, is ha le Ne w= = oat . Mare ioraa tl ae (76) This shows that the section of the middle surface by the yz plane is nearly a circle of radius = . Thus our result gives the well-known anticlastic curvature of a bent rect- angular rod. , Let us next suppose that m/ is large, so that we may assume cosh @=sinh 0=te*; sinh 20=4¢e. Then Wy = ae-®( (cos @—sin @) cosh my cos my (77) ma +(cos @+sin @) sinh my sin ny vy, In order to find the form of the surface near the free edges we can put y=l—y’, and assume that y’, the distance from the nearest free edge, 118 Dr. J. Prescott on the Equations of Equilibrium — is small compared with /. On doing this we get pee io 4(cos oad ees cee ee ma | +4(cos@+sin @)e-"Y sin (@—my') / . / — —— e-™Y'(cos my —sin my’ Ima ( J J ) 2 ; T =oh VJ sas emmy eos (my! u eS) Thus we see that, when a broad rectangular plate is bent into a surface of revolution, small corrugations are formed on the bent surface, each crest and each trough forming a circular arc about the axis of revolution. However wide the plate may be, the maximum amplitude of the corrugations oF is approximately Ns 3 oh, and the amplitude is smaller the nearer the wave lies to the middle circular are of the bent plate. When the plate is narrow, so that it approaches what we may call a beam or rod, only one small length of the oscillatory curve is comprised on the breadth of the plate, . ; CaO and this may be regarded as a circle of radius o The extreme cases to which we have referred occur when ml is large and when ml is small respectively ; that is, when /? is large and when. /? is small compared with ah. When / and ah are of the same order the complete expression for w, must be used. Tf we had started with a sheet whose unstrained middle surface was that of a piece of a cylinder bounded by two. generators and two circular arcs, it is easy to see that the preceding investigation can be adapted to give the strain when this cylinder is bent still further by forces and couples — applied along the edges of the generators. If the radius of the cylinder before strain was 6, and‘the approximate radius after strain a, then we have only to replace : in the foregoing work by (; = aI If 7? is small in comparison with ee a db b—a the longitudinal section of the bent middle surface is. Ca eae Taal approximately a circie with curvature o(—~—-—}; whereas a b pneme : ; : abh if 7? is large in comparison with ;——, there are many —a b of an Elastic Plate under Normal Pressure. 119 corrugations on the sheet, and the maximum amplitude is, ) 2 as for a plane sheet, oha/ slo). In a paper read before the London Mathematical Society (Proceedings, vol. xxi. p. 142), Professor Lamb.has obtained results for the bent cylinder which differ only in one respect from the results I have just indicated. The difference is that his results do not show the relation between / and the amplitude of the corrugations, and this is due to his tacit assumption that (6—a) is small in comparison with 0, a restriction which is quite unnecessary. He has a boundary condition which, written in terms of the symbols of this paper, has the form 2 d (p—8) bal = dy” b? Tf this had been written in the more correct form d*(p —b) ( i : =) dy” < 5) and if, then, p had been replaced in the second term by its approximate value a, the condition would have been d(p—b) Lie INN gs dy? +o(,- |) =0 and this would have given the results indicated in the present paper. In a later paper, read before the Manchester Literary and Philosophical Society *, Professor Lamb worked out the problem of the flat plate bent into a surface of revolution by using an elementary method which has the advantage of showing clearly the physical actions taking place. His method was, however, a special one which did not show the connexion between this problem and the general theory in which the stretching of the middle surface is taken into account. Problem 4.—The stretching of a circular membrane by a uniform pressure p over its area. If the terms having a factor hare omitted from equation (14), we fall back on Poisson’s equations. If, however, the term containing the factor h? were omitted, this would amount to assuming that the plate is much more effective in supporting pressure by means of its stretching than by * See also Phil. Mag. [5] vol. xxxi. p. 182. 120 Dr. J. Prescott on the Equations of Equilibrium means of its bending. That is to say, weshould be assuming that the flexural rigidity is negligible and that the plate behaves as a membrane with a non-uniform tension. We are justified in neglecting the terms containing the factor h when w is every where small in comparison with h, and we are justified in neglecting the terms containing h° when h is small compared with the inaximum value of w, unless the bent middle surface nearly coincides with a developable surface, in which case h must be smali in comparison with the maximum deflexion measured from this developable surface. We are now going to assume that a bent plate is symmetrical about the z-axis and that the terms containing h? in (14) are negligible. With these assumptions, equations (12) and (14) become d 5 dw d?w a a (V's) b= de ee dw * dd te dr de inte mie One integration a each of these equations gives i \ ena *b) = OE aS aren em ns, AY De —3(@) (my dy | 2. 2) and mdi i uiTipece an? No aL dr dr AhK The constant of integration 1s zero in each case because dw . : : 0B a is zero where r is zero for a circular plate with no dr : central hole. Now putting dd Bit ROO aarrey ane E=r— Oar s=T, equations (81) and (82) become PE ala 1 2 83 : Ao ae PPT CS. CUE Mle and Ge= TURE (S+) When @ is eliminated from these the equation for € is — Ce dey Ro? 1)/1%5? Gate ale HP? ee of an Elastic Plate under Normal Pressure, Pk Next, putting : 9 aad He | | ey Ee 0 ea and hte = €, we find that . Deut S17 on Ge ea ee a (86) The complete solution of this equation contains two arbitrary constants, and we can see immediately how one of these constants is involved. For, putting £5 Oil OF MIT esta foie) we find that (86) becomes * Cy wen ie ; Oe ne ve an equation which does not contain C. It follows that C is one of the arbitrary constants. Now there is only one boundary condition in our present problem. It might be, for example, that P, has a given value when r=a, the radius of the boundary circle. The one constant C which occurs in the relations between 2, y, sj, £1, 1s quite enough to enable us to satisfy this one condition. Then we need only a particular integral of equation (88) of the right type to solve our present problem. Equation (88) has a solution expressing y in an infinite series of integral powers of « starting with the first power. This solution is ee ee om Lee) OG. 200g ee ep ae8” 80a". 36288". ©”) It is interesting to notice that, as long as @ is a fairly small fraction, the series for y differs very little from ih aa 50° 7 a8 ot 39— : that is, from —(1—w) log, (1—2). The radial tension is pdb _ HE Ei edn eres. ed 2 r2R2\t0 4 — i 1 2 13, 4 P, being, of course, the value of Pi at the centre of the 122 Dr. J. Prescott on the Equations of Equilibrium plate. Its value in terms of C is Py=s Chey. > 2 eee Dusan Liens eat mame = pee “ 3 Z 9 5 cS . (92). Let the radius of the plate be a, and let P})=T at the rim. Then Ta? saa spelics atel cor [=P, |\l-5p.-g pe ue If Py is given, this equation determines T directly; but if T is given, it determines P, indirectly. Inverting the series to give Po in terms of T, we get Tha PGS OO) teas 3 Banc 2°73 2S m6 a: m9 aeccece i" (94), The first approximation, namely Pos, . ui a ees eo makes P, constant over the plate, which is the usual assumption in dealing with stretched membranes. The second approximation is ik Jele? . Po= T+ 5 5 wel hl Ly Geet eee and the corresponding value of the radial tension at any point is 1 ibe ae Veta —T+ : (ear) 2 L pk ~* 1024 #E The third approximations are 1 Metal Wears 218) ots dk 3 and eis 1+ 9 (er) — : ae (eo a (99), Po=Pe= (Ga) ee P,=T+ (98): Equations (94) to (99) are valid only on the assumption of an Elastic Plate under Wormal Pressure. 123 that T and Py are nearly equal. We shall now consider the possibility of a zero value of Pj. dy hee d. Hquation (88) shows that “7 decreases as x increases for dx all values of g and y. It follows that y must finally become zero, and it is easy to see that, starting with finite values of d at: y and a y becomes zero for a finite value of x, and therefore Ax for a finite value of r. Since P, is proportional to ~. 1t follows that P, vanishes for a finite value of r, and equation (82) shows that = is infinite when P, is zero. This is, of course, a physical impossibility, and, moreover, the whole theory fails when = is not a small fraction. Nevertheless it is interesting to consider this extreme case. Tf P, is zero, the following equation must be satisfied : ee ee ee ee 1 5% 6” jaa” — 988 ~=0. . (100) Now we have already noticed the good agreement between the earlier terms in the series for y and those in the expansion of —(1—2)log(1—z). If we could be sure that all the remaining coefficients of the series in equation (100) were less than the corr esponding coefficients in the expansion of ~ 1 = “log; (1—2), as they certainly promise to be, then we could safely assert that the root x, of (100) is less ‘han the root of —— Se 0 ge(1—#)=0, which latter root is a It is not very difficult to get a closer approximation to the value of x, as can be seen in the following process. a7 3/7 If the expression 2(1— : «) be expanded in powers of e, it will be seen that the first three powers of 2 have the same coefficients as the corresponding terms in equation (89). Moreover, the curve represented by 7 3/7 y= =2(1-i2) Porras 132% CLO) Se LE PS A 124 Hquations of Hquilibrium of Hlastic Plate under Pressure. has the same general characteristics as the curve represented by (89); for equation (88) shows that the curve deter- mined by tbat equation meets the z-axis perpendicularly at the point where y is zero and & is not zero, and the curve represented by (101) has this same characteristic. Now, assuming that y is correctly given by (101), and substituting this value for 4 y on the ‘right-hand side of equation (88), we get AUR (1 : ff ie Ga ae ae ; Solving this and adjusting the constants so that y and dy dx we find have the same values when x=0 as are given by (89), 8/7 y=5-3(1-22) — 5 illie eis The expansion of the right-hand side of this last equation gives | 1 LO A 174 ell 89 4 1826 y="— 9% 6" 14a” 988" — 864” eee (103) an equation which coincides almost exactly with (89). For values of w less than unity, the difference between the two values of y is certainly very small. Then we may assume that the value of y given by (89) is approximately the same as that given by (102). To find an approximate value of «,; we have therefore to solve the lic. 9a Dee =5(1- 5] See | is: A close approximation to the root is 21 == 05803. It follows then that P, is zero or very small when + has the value 7, given by the equation eeey Dili) leneles Thus we see that the complete membrane determined by our differential equations (in which certain physical difficulties are ignored) has an asymptotic cylinder, the radius of which is given by (105). A section through the middle of the (883). 1, Gey Cree Transverse Vibrations of Bars of Unitorm Cross-Section. 125: membrane would have some resemblance to the curve of a ee of uniform strength. If @ is zero or negligible the eqtations in this paper reduce to the usual Poisson equations for thin plates. Now it follows from equation (12) that @ can be zero prov ided that | 02w ) O-w 510 ee te anny chess crt NOG Oxy 02? Oy’ : oe But this is precisely the condition that the bent middle surface should be a developable surface. When ¢ is zero the middle surface is unstretched, and now we find from our equations, what is quite obvious from geometry, that a plane sheet can be bent into a developable surface without the stretching or shrinking of any of its elements. If the expression on the left-hand side of equation (106) is small but not zero, it is still possible that @ be so small that Poisson’s equations are approximately true. The examples we have worked out indicate that this condition is satisfied if w 1s everywhere small in comparison with the thickness of the plate. Since the expression in (106) is of the second order in w it is clear that @ depends on w? rather than on w. The condition that @ should be negligible will still be satisfied if the deflexion of the middle surface, measured from some developable surface, is small compared with the thickness of the plate. The third problem worked out above supplies an example of this. X. On the Transverse Vibrations of Bars of Uniform Cross- Section. By Prof. 8. P. TrwosHenKo * § 1. PN a paper recently published in this Magazine f, I have dealt with the corrections which must be introduced into the equation for transverse vibrations of a prismatic bar, viz., me Os EY Oat g Oe in order that the effects of ‘“‘rotatory inertia’ and of the deflexion due to shear may be taken into account. Ae Pee ate. CL * Communicated by Mr. R. V. Southwell, M.A. T Phil. Mag. vol. xli. pp. 744-746. oe 126 Prof. 8. P. Timoshenko on the Transverse In equation (1) EI denotes the flexural rigidity of the bar, Q, the area & the cross-section and F the density of the material. 9g j It was shown that the correction for shear, in a repre- sentative example, was four times as important as the correction for ‘‘ rotatory inertia,” and that both corrections are unimportant if the wave-length of the transverse vibrations is large in comparison with the dimensions of the cross-section. In the present paper, an exact solution of the problem is given in the case of a beam of rectangular section, of which the breadth is great or small compared with the depth, so that the problem is virtually one of plane strain or of plane stress. The results are compared with those of my former paper, and confirm the conclusions which were there obtained. § 2. When the problem is one of plane strain (so that w is constant), we have to solve the equations * (A+ 2u)V7A + pp’A = 0, ) and pyatppa = 0, | where A denotes the cubical dilatation (e +2), | OM Ou) ‘ oO 5 the rotation : ($°-$), | Cea Ok , 2 Or Meas amd V2 ,, | the operator (<3 an al | @T aking the w-axis in the direction of the central line, and choosing for A an even and for o an uneven function of y, we may write A = Asin az sinh my cos pt, 2a = B cos aa cosh ny cos pe, } (3) where A and B are undetermined coefficients, * A. E. H. Love, ‘ Theory of Elasticity,’ §§ 14 (d) and 204. Vibrations of Bars of Uniform Cross-Section. Li ae 5) a og V1 ae a nee 1(y, | V= S denotes the velocity of waves of t J = | (4) transverse vibration, —<—___— a v= ne denotes the velocity of p waves of dilatation, and Vo= oe, denotes the velocity of waves p of distortion. It is easily verified that equations (3) are satisfied by the expressions u = cos av (M sinh my+N sinh ny) cos pt, ] 3 7” : a = fi v= sinar (a cosh my + N- cosh ny) COS pt, | Oo 7t and the conditions that the boundaries (y=+c) are free from traction give us ee [A+ 2)m?—ra*] M sinh me + 2u2?N sinh ne = 0, and 2mnM cosh me+(a?+n?) N cosh ne = 0, ; Hence, by eliminating the ratio M/N, we obtain the “frequency equation ” in the form * Auc* mn tanh ne = (a? +n?) (A+ 2h) m?—)x?| tanh me ; whence, denoting the length of the waves by J, and putting WVo=7, V/V.=h, we haye 4 V¥(1—/7?)(1—h?) tanh (= V1 — i?) : 27¢ j)——; i =(2—28)? tanh (=F Te (5) If V be given, the corresponding value of the ratio 1/2c * This “frequency equation” was fuund by Prof. P. Ehrenfest and myself in collaboration. The solution given in this paper is my own. 128 Prof. 8. P. Timoshenko on the Transverse can be calculated from this equation. Some results are given in the table below *. | a 0°5 0-7 0:9 09165 09192 4 ai Li 26 (= 4:8 2°50 0-79 0-47 0:30 § 3. In the case of long waves, the velocity V is inversely proportional to the length /, but as / diminishes it can be seen, trom equation (5), to approach a limit which is lower than V., and can be found from the relation 1 /A- POs Ses This limit is the velocity of the “Rayleigh waves” f. If we put AX=yp, we obtain, in the limit, V=0°9194 V.. As the wave-length / increases, the arguments of the hyperbolic functions in equation (5) decrease, and we can > employ the ascending series. If we limit our attention to the first three terms, we may write equation (5) in the form a(ac)* + b(ac)?+d =0, . 3 (6) where | | a= ~(4(1—/?)? —(2—A?)?(1—/7?)?], ) b = —1(4(1—1?)?—-(2—’)*2(1—f?)], &. - @) 7 ae J When ac is very small, we can obtain a first approximation by neglecting the first term in (6) and putting (acy? = —", a(S} This approximation will be (a? = GO. whence ee Oe) :. A = 3 (a) 1 2ny: (10) pe == BN — oa eee te aoe * In these calculations, « has been taken to be 0°25. + A. E. H. Love, op. eit. § 214. Vibrations of Bars of Uniform Cross-Section. 129 Replacing X by the quantity ~ as ry aoe we should obtain the ae aneocinate solution for the case of plane stress, and it is easily verified that this is equivalent to the solution of equation (1). Proceeding to a further approximation in the case of plane strain, we observe that the quantity d in (6) is small in comparison with a and 0, and that the solution of this equation can therefore be written as follows: oy See —S (1g. a HTH Ge) The second term in the bracket on the right-hand side is a small correction. In calculating it, we may take for z the first approximation (8), and in the expression for afd we may retain the terms of the order A? and /? only. We then have : | Gad Tr . WK = 2 (ae)? aNERire Wn Ete Nhe) Ls (14) The first term on the right of (13) must be calculated more exactly. Retaining in the expression for } terms of the order f* and h*, and substituting for V the first approximation (10), we have OO Sef ah Wits. Io Ae | epialll eee a +H | (15) 3 arene: Substituting from (14) and (15) in (13), and neglecting . small quantities of higher order, we obtain 2 é 2 2 (ac)? = gH 4 4 (eo) (S EZ), Noe, oD or 5 ee An(N+ EB) [ a (ot 2u =) Sree nse Wout a) it (16) §4. The square brackets in (16) contain the required corrections to (11). These correspond principally to the effects of “rotatory inertia” and of shearing force, and could have been obtained with sufficient accuracy from the Phil. Mag. 8. 6. Vol. 43. No. 253. Jan. 1922... K 130 Transverse Vibrations of Bars of Uniform Cross-Section. equation for transverse vibrations of rods, if this is supple- mented in the manner explained in my previous paper, where the correction to the frequency p was given in the form of a multiplying term of amount * Je gre? E\] [1-5 Fe (1450) |: in es In this expression, 7/L is equivalent to the @ of the present paper, and 4?=c?/3. 2X is a constant relating the shearing force with the angle of shear at the section con- sidered : its value for a rectangular section (if we make use of the experimental results of L. N. G. Filon t) is 8/9. The ratio H/C must be replaced, for comparison with our present 4 A+P) problem of plane strain, by the quantity Vee (in the notation of the present paper). We then have from (17), in our present notation, as the correcting factor required in the expression for the square of the frequency, pene eye [1 Mee ca With o=0°25, or X=p, the correction to p*, given by (17), is thus [1 =e)? |, 6 oe ee ee ee) [1—81(ee)?|; ee § 5. This close agreement gives us some confidence in applying the approximate solution of my earlier paper to other shapes of cross-section. We take, for example, the case of a circular cross-section [: the exact solution may be written in the form age BK di —#)| save vam / ett a Gt, Aire (240) where c denotes the radius of the ae whilst the methods of my earlier paper would give 5 Hr 2G K where « is the “shear constant” previously denoted by » whilst (16) gives * Of, equation (13) of the paper referred to. + A. E. H. Love, op. crt. § 245. t See Pochhammer, Journal f. d. reine u. angw. Math, Bd. |xxxi, p. 385 (1876). Potentials due to Moving Electric Charges. 131 If we put Me jo elie = 3x 1-40 = 1-05, mde Tee 22 and from (21) will be 1°81 Thus we have again the correcting factor found from (20) will be 1°87 obtained very satisfactory results from the approximate formula (17). § 6. In this investigation of transverse vibrations we have taken the expressions (3) for A and w. By taking the even function of y for A and the uneven function for a, we may obtain the solution of the problem of longitudinal vibrations. Zagreb, oD January 26, 1921. XI. On Scalar ane Vector Potentials due to Moving Electric Charges. By Prof. A. ANDERSON *. FEN the Philosophical Magazine (March, 1921) Prof. A. Liénard criticises my paper on “ A method of finding Scalar and Vector Potentials due to Motion of Electric Charges” (Phil. Mag. August 1920). The criticism, though just, touches only a slight error in notation ; the results of the paper are not affected thereby. The last part of my paper is very much condensed and perhaps, on that account, unintelligible, and the object of the present communication is to make the argument fuller, and, I hope, clearer. Fig.’ 1, = xz > In fig. 1, Q is the position at time t of » moving charge e, and (' is a point moving in the path of Q—the companion * Communicated by the Author. ae ‘Prof. A. Anderson on Scalar and Vector point of Q, we may callit. The position of Q' is the same as that of Q ata time ¢— , where r is the distance of Q’ from a point P in the field whose coordinates are «, y, 2. I proved in the paper réferred to that, if , Us ° > . f where P(e) is any function of i—— and (ne G $ resolved part along Q'P of the velocity of Q when at Q’, 107A 2 RRsat Rail TN V7A C2 0 ='() abies The velocity of Q' is, of course, not the same as the. velocity of Q when at Q’, the latter is a funetion ‘of ¢— the former is not. ig Re Suppose now that there is a distribution of electricity jin motion. The whole distribution may be conceived as divided up into elemental charges. Let dg be the element charge at Q. It is clear that, if el Lael i (1- | i where r is the distance of Q’ the companion point of Q from any point P, and w, the resolved part of the velocity of dq when at ()' along Q’P, i 1 o’A VA 3 Wa at dee if the density of the moving electric distribution be zero. at P. If the distribution is a volume and surface distri- bution on moving bodies unaltered by the motion of the bodies, we may write where dT and dS are elements of volume and surface, p and Potentials due to Moving Electric Charges. 133 being the volume and surface densities, and the denominators of the integrands having the same meanings as before. But if there is Sa at the point P either at rest or ofan 2 On find what its ie is. If we imagine a small sphere of electricity removed, the expression becomes zero at P. Hence its actual value at P is that due to’ this sphere of moving: electricity. But we shall see that, if the radius of the sphere diminishes without finite limit both A and o7A ila is the limiting, value of V7?A at P due to the sphere, when its radius is made to diminish indefinitely. Inside this small sphere the velocity of the electricity may be regarded as uniform and as having the value that it has at P. in motion, VAs % =e no longer vanishes, and we must tend to zero. Hence the value of the expression at P Fig. 2. P Let an element of electricity. dq at Q at the time ¢ be moving with uniform velocity u in the straight line Q,Q ’ (fig. 2). Its companion point Q, is the position of dq at the QiP time _haren and the element contributed to the integral A by dq is dg dy ages PQ,—PQ, Bees ee, — -QQ,cosd PQ cos a’ But sna QQ, u sini ok @; me 134 Prof. A. Anderson on Scalar and Vector Hence the element contributed to the integral A is. dq ue a 2 raf ole C where r=PQ is the distance of the actual charge from P and @ is the angle which the direction of motion makes with QP. With the aid of this expression we must now find the value of A for any point P inside a sphere of radius a whose centre is O, the electricity in it being uniformly distributed and of density p, with a uniform velocity u parallel to the axis of 2. ; Fig. 3. In fig. 3, let OP=6, and let the direction cosines of OP be 1, m,n. Also, let axes be drawn through P parallel to- the rectangular axes through O. Draw a cone of small solid angle dw having its vertex at P and intersecting the sphere at Land M. We first find the part of the integral A contributed by this cone. fs This is le ( prdr Rear ibe dw ea wu? e ay p 2 : u? e : r 1— —=sin?@ | 1— ysin’ 0 (Ga where PL=7,, PM=1,, and @=angle LPz. Let the plane Potentials due to Moving Electric Charges. 135 LPz intersect the plane 2Py in a line which makes an angle @ with Pz. Denoting the angle OPM by a, we have ry? + 7.2 = 4b? cos? « + 2(a? —b*) =46?[ncos 9+ /sind cos$+m sin@sin¢ |?+ 2(a?—0?). Hence +a?—b*}sin Odd dd A=p| | (22 cos0+/sin @cos#+msin 6 sin d)? “ 2 é ve - sin? 7 or, if the co-ordinates of P in reference to the axes through O ALC ey U2, : +a? —2?—y?—27t sin 0 d0 dd i fy ees cos $+ sin 0 cos f +y sin @ sin } )” wu? a LS = 2 sin’ 0 the limits of @ being a and 7 and those of 6, 0 and zr. Hence A, ce 2A tops sind dbdd SS 5 SS SS = Oe” rad \ , If in the expression for A we make v=0, y=0, -=0, we get for the value of A at O T pare low ot u aT aay 2 which yanishes with a. ae having the same _ linear dimensions as A, also vanishes with a. It follows, therefore, that Aad dg Up Ls | Qaer (1- “) 136 Prot. A. Anderson ‘on Scalar and Vector is the solution of 10°A Cy. €+u 2 ea sta cea ae Vis co Ot? pee Goa p being the density and u the velocity of the electricity at any point. If, now, we write a) where w is the velocity that dg had when at its companion point, then 10°A * Ot at any point where there is no electricity, since u is a V2A— 555 =0 A i 7 function of ¢— of But at any point where there is electricity it satisfies the equation | 1) eX VA Go itn dds If this is not evident it can easily be shown to be true by using the method of the small-sphere as before. In this integral « means the velocity of the element dq when at its companion point. The companion point of P coincides with P,and therefore there the actual value of u is the velocity at the companion point. In like manner Ne ={— uu, dq Zn¢ log- a ge el is the solution of In this latter equation wv, means the « component of the actual velocity at P: in the integral it means the # com- ponent of the velocity of ue when at its companion point, which is a function of t—“. Similar expressions may be G written down for the solutions of ‘ Ore Uy 107A U, V7A— oo orm ae and VAS 4 > ee Potentials due to Moving Electric Charges. 137 Thus we have determined the scalar potential and the components of the vector potential at P. These expressions being true for every point and for every distribution of moving electricity, it would seem to follow that for a single charge e at Q moving in any manner the scalar potential at any point P is CHul u ‘ 2mc log r( = a) 19Gb te é and the components of vector potential CUUy eu Uy i 3 5 9 ¢ 2mc* log ada | r( — | Q7rc? log ot [r(a- “)] C—U C—U C ewu, ty e+tu , (oN i ee hd) C—u c/) If the square of ~ be neglected, we get C Sad Pegs a Sa a | 1ba\ © Uy\’ Aqr(1— ) Ancr| 1— — ) ros c i r e Uy eu, | WO Ne Uy Arc ( ~*) Arc (155) Though I have some misgivings as to the validity of the last step of the above argument, I think the expressions given have some advantages over those obtained from them by neglecting the square of =. Take the case of a point charge e in uniform motion in a straight line. If the potential be taken to be é Anr(1— *), bj ¢ i it will be found that the potential at any point in the line : Ait 2 : | is ,—, where a is the distance of the point from the fl Ara | 4 moving charge. Thus this potential is independent of the i velocity ; but it would seem that, if the velocity approaches 138 Prof. O. W. Richardson : that of light, the potential at any point of the line in front of the par oe should tend to zero. Again, if a sphere be described round the moving charge as centre, and the integral of normal force due to the moving charge be taken over its surface, it will be found that, with the simpler expression for the potential, the value of the integral is — lo og ox S 2u above, it is e, in iL cae with Saree s Theorem. , whereas with the expression given XII. Note on Gravitation. By O. W. Ricwarpson, /.R.S., Wheatstone Professor of Physics, University of London, King’s College. With an Appendix, by L. Stmons, D.Sc., Lecturer in Physics, University of Capetown *. pee note is mainly a brief account of experiments. undertaken ‘to seek a connexion between gravitation and electricity. The line of attack is largely empirical and follows a train of thought developed prior to the current theories of gravitational relativity +. Although no positive results have been obtained it seems desirable that a short account of the experiments should be published, both on account of the importance of the subject and also because the results may set limitations on the possible scope of other gravitational theories. The central idea was that gravita- tion might be due to a slight modification of the law of force between the positive and negative electrons out of which matter is built up, such modification arising out of the different amounts of energy associated with the respective positive or negative electrons as evidenced by their different masses. This point of view is to a certain extent anti- pathetic to the relativity theory of gravitation, and its comparative failure may be considered in some degree a success for the latter. One consequence of the point of view referred to is that the ratio between weight and mass should not be quite the same for different elementary substances. The experiments of Bessel and Hoétvés show that this ratio is extraordinarily constant for a number of substances. Nevertheless experi- ments on this point were commenced by the writer in 1914, were abandoned during the war, and have now been recom- menced by Mr. H. H. Potter. The present note does not * Communicated by the Author. + O. W. Richardson, Phys. Rev. vol. xxxi. pp. 610, “ee (1910) ; ‘ Blectron Theory of } Matter,’ chap. xxi. (1914). -Note on Gravitation. 139 refer to those experiments which are still uncompleted. As their subject-matter forms the whole basis of gravitational relativity, their importance is obvious apart from the line of attack here considered, and they will be dealt with separately on completion. One direction in which we might expect to find effects of the kind here contemplated is in the displacement of the spectral lines as between different isotopes. There is of course such a displacement to be expected on Bohr’s theory on account of the dynamical effect of the changed mass of the nucleus. There might, however, be an additional effect due to the modification of the field of force arising from the nuclear charge owing to the change in the mass associated with it. Asa matter of fact, Aronberg * and Merton + both find a small displacement for a given line as between uranium lead—ordinary lead—thorium lead. For example, Merton. for X=4058 A.U. finds » (Ur. lead) —~ (ord. lead) =0°0050 A+0:0007 A r (ord. lead) —r (Th. lead) =0°0022 A +0-0008 A. Whilst these displacements are small they are nevertheless. many times larger than the Bohr shift calculated from the dynamical effect of the change in the mass of the nucleus. Merton also finds a displacement as between thallium from pitchblende and ordinary thallium. On the other hand, ordinary lithium (atomic weight 6°94), which has been claimed to be a mixture of isotopes (atomic weight 7 with a small proportion of 6), shows no evidence of the expected duplicity in its spectral lines. However, there is no evidence of the Bohr shift, which should be large in this case ; so that the lithium observation does not seem very helpful in this connexion. There are, of course, other possible causes for the displace-. ment of the spectral lines of isotopes such as, for example, a small variation in the electrostatic field of the nucleus arising from a difference in the configuration of its electronic con- stituents ; but if, for the sake of argument, we assume it arises from a modification of the law of force of the more fundamental kind now under consideration, the connexion between such modification and the change of wave-length may be calculated as follows :— Using the notation of ‘Electron Theory of Matter,’ 2nd edn. p. 616, let capital letters denote positive and small * Astrophys. Journ. vol. xlvii. p. 96 (1918). T Roy. Soe. Proc. A, vol. xevi. p. 388 (1920). 140 Prof. O. W. Richardson: letters negative charges. Tor an electron charge e' mass m! revolving round a nucleus containing positive electrons of total charge E and mass M and negative electrons of total charge e and mass m, the force at distance r apart is is =e +e)e!-+a[(E+e)m! +e'(M +m)] 12 2 2 +e | E+e)5 +e! (= + alll + g(M-+m)m’ }. For isotopes E+e, e’, and m’ have the same value but M, m, E, and e are different. The difference in F for two isotopes is thus it M? 2 . OF qe ; ae'd(M + m) + ce 3(ar “F | + gm'o (M+ m) } . To a first approximation M/E and m/e are respectively constant, being the oxygen nucleus and negative electron values, also m/M is small, so that ol (2 AoW, where A=ae'+ce'; ie wt and W=M-4m™m. The forces will thus be the same as if the nuclear charge were increased by an amount M m («+ pets )sW=Bew. According to Bohr’s theory the frequency of any pee ‘spectral line is yon Oa M,m,H,’e,? Oop! & ; Mone a ‘ €, and m, being the charge and mass of the circulating elec- tron and HK, and M, the charge and mass of the nucleus and electron system about which it circulates. 7, and 7, are positive integers and fh is Planck’s constant. The corre- sponding frequency for the isotope is (M,+6W)m, (KE, + weet 1 =| Meee isto rer ae (M,+m,+6W)A? Te | ers Note on Gravitation. 141 Thus ae y! r 4 B | ee ae Xr na V hi 1+6M AN on) a i, ; my 5 B my, is lee Saha | wat, em aN M,(M, + ™,) ; Hy, oe Nd and the relative shift on wt A ae CE, eau Rane +m) Ne oa ‘ my B m4 al ia Lai (Ml, +a) ea M,(M, +m,) M6 Ky Stee The atomic weights of the different leads tested by Merton do not appear to have been measured, but we need not go beyond the order of magnitude of the different terms, and for this accuracy it will be safficient to put —8r/rHl-2x 10-8, 8W=0°5, M,=207, = 1860. 1 The first term on the right represents the Bohr shift. It amounts to about 6x 10~° and therefore forms only a small fraction of the observed 6A/X. It appears that all the other terms are small except the second which has to account for practically the whole of 6\/A. On the type of view here considered then it appears that B/E, must be of the order LO-*, ¢ If B/E, were as large as this it would have several notable consequences. In terms of the coefficients the value of B/E, m! 1 M is —l(at+—sct+9q-—). he k "essi is 5 (« BC p=) The last term in this expression / represents the ordinary Newtonian gravitational force due to the mass m’ of the electron. On substituting the / numerical data the value of ae is found to be 2°5 x 107%, 1 Tt is thus a very minute fraction of the whole of B/E,. This would mean that in a gravitational field positive and nega- tive electrons would be acted on by opposite torces of nearly - equai magnitude but much larger than the Newtonian forces for neutral particles of equal masses. In an insulator at rest this would give rise to an electric polarization proportional to the gravitational intensity. Ina conductor in equilibrium we should expect a separation of the charges giving rise to an equilibrating electric field. In the earth’s gravitational 142 Prof. O. W. Richardson : field, if B/E,=10~°, this field should be equal to 2°13 x 107? eae We might also expect currents to be generated in a circuit of which part was falling under gravity relative to the rest. I have made experiments to test this point. About 200 lb. of mercury were allowed to fall in a continuous stream from a container through a short tube 9 mm. in diameter for a distance of about.a metre to a container at a lower level. The containers were electrically connected by mercury-filled glass tubes (to avoid thermoelectric disturbances) with a galvanometer having a resistance of 600 ohms and a‘sensi- tivity of 1 division for 2°6x10~ ampere. The resistance of the rest of the circuit was negligible in comparison. No deflexion amounting to a scale-division could be observed when the mercury was running steadily. At the start and finish rather irregular deflexions were observed, but there are a number of ordinary explanations which might account for them. However, I do not consider that these experi- ments rule out the possibility of such effects at the start and finish. In order to settle this point definitely more elaborate apparatus would be required than [ happened to have available. On the view we are considering the force on an electron in a gravitational field would be quite different from the ordinary Newtonian attraction on its mass. The accelera- tion of an electron in the earth’s gravitational field, for. example, would be about 10” cm. sec.~*. A correlated effect is that the apparent weight of equal electric charges should, to a very close approximation, change sign with the sion of the charge. (The approximation neglects the gravi- tational term in. B, which is about 1071 of the whole.) This can be tested by placing a very light earth-connected plate in a median position in a flat insulated metal box and finding if the apparent weight of the plate varies with the sign of the charge on the ‘pox. Very careful experiments on this principle have been made in the Wheatstone Laboratory by Dr. L. Simons, and as the result is important they are described in the appendix. Dr. Simons finds no displace- ment, although he concludes that he could have detected a deflexion equal to about one-third of that to be expected if iB — Oe Oy The charges which would develop on a conducting sphere of gravitating matter are apparently much too small to account for the earth’s magnetic field. I find that the equations are solved by an electric distribution of uniform volume density throughout the sphere, and if the sphere is uncharged an Note on Gravitation. 3 143 equal and opposite total charge ef uniform surface density over the surface. If this rotates uniformly in a periodic time + the magnetic field at external points is the same as that of a smal] magnet along the axis of rotation having a magnetic moment as Ag BM). ET pl tet ii. where M is the mass and R the radius of the sphere and ¢ is the velocity of light. Putting B=10~° EH, and the values of M and R for the earth, this gives the magnetic intensity at the pole as 5-9 x 107”. One positive conclusion which may be drawn from these notes is that the displacement of spectral lines as between isotopes cannot be regarded as furnishing evidence of the existence of cross terms in the law of force between electric charges and mass elements. Its explanation is probably to be sought in an electrostatic effect of some variation in the geometrical structure of the elements of the nucleus. APPENDIX by Dr. L. Simons. The apparatus consisted of a heavy brass box of internal dimensions 34 x 14 x 2°5 cm., the lid, resting on three screws, being adjustable. A silica framework consisting of two squares of 10 cm. side were mounted coplanar with their centres at a distance of 20 cm. apart. and a silica cross-piece arranged to carry brass ferrules at its ends through which there passed pointed steel screws which rested on steel planes. Aluminium foil 0°002 em. thick was fixed over the two squares, and the whole balanced symmetrically within the box which was lined with the same aluminium in order to avoid irregularities arising from contact e.m.f. The total mass of the moving parts of the balance was approximately 4gm. The sensibility could be very finely adjusted either by means of the weights (w) or by means of the screw pivots. The motion of the balance could be observed by means of a mirror attached to the cross-bar and a telescope and millimetre scale at a distance of about 1 metre, the adjustment for zero being made with the flag (f), which was a short strand of silica movable by a lever from the outside of the wooden box containing the apparatus. The yanes could be charged from the outside vid thin copper. wires joining them to the brass ferrules, thence across the steel pivots to the steel planes on which they rested, which in turn were connected to terminals outside the box. 144 Note on Gravitation. The sensibility was found directly. A number of small weights were cut from the aluminium foil mentioned above, the mass of each being 2°18x107~> gm. These could be picked up on a small brush and pushed through the holes (h) directly over the centre of each vane. Independent deter- minations gave the sensibility to be 1:42x10~° om. per millimetre deflexion on the seale, and as this was con- siderably magnified by the telescope a motion of one-tenth of a division could easily be observed. Fig, 1. S------ /0 cm. ---~---~ > is ee y oF | i paar ae RE Tic oe Lr, 2 ti) gs Z| i a ee ee g a | ——— on i = i A ee HL RU) = an! 4 i ma rs " 90-442 x< 10>). 25. Del 17 65 463 1000 Voltage for half value...... — "25 —°30 —'3d —'41 —*45 It will be seen that the voltage displacements for a given increment in pressure are greater when the pressure is low. This condition, of which figs. 4 and 5 are typical examples, appeared to be stable and persisted to the end of the experi- ments, in the course of which the wire was heated some hundreds of hours. The displacements on reducing the pres- sure were generally somewhat smaller than those obtained when the pressure was increased between the same limits. 732 Gases on Contact Potential Difference between Metals, 167 This is probably due to the time required to attain equi- librium between the gas and the metal. Fig. 4. 400 360 320 280 4 Division = 2x410°7° amperes. 240 DEFLECTION. 160 _ 120 GALVANOMETER 80 40 42 ‘8 oh 0) +6 “3 12 4-6 Pr THRORY OF THE PLATINUM DISPLACEMENTS. Whilst a complete test of the theory of these effects will require a more elaborate experimental investigation than the present, the results are in some respects definite enough to merit a theoretical discussion on broad lines. A theory of the connexion between contact eléctromotive force and electron emission from contaminated surfaces has been developed by one of the authors*, and is discussed in connexion with H. A. Wilson’s theories of the effect of hydrogen on the * Richardson, Roy. Soc. Proc. A, vol. xci. p. 524 (1915). 168 Prof. Richardson and Mr. Robertson : fect of emission from platinum 1 in the ‘ Emission of Electricity fror Hot Bodies,’ Chap. iv. pp: 108 e¢ seq., to which the following Tip. 5. 400 ” 360 ce = o* Si Ss al ft 3 280 = = a 240 . 2 2 - ie) uu! 200 is ul a Apressure oF H, o-0065 mm, Hg 160 / ° ” " " Q-0463 tej - -O5 oe] co) 0-0589 = 420 g ce) 0-089 < 3 5B « o “ 0-100 2 < ‘ Bo VOLTS references relate. In general (equation 34’ p. 109), 1 contact potential difference between any two surfaces a temperature T’; is kT me vy KY 1 Vii , log = ae - 4 fy (1) where 7 and 2’ are the saturation currents per unit area of the two surfaces at T,. If the dashed variables refer to the clean metal and the undashed to the metal in presence of gas, it follows from (13) p. 112 that i= apy ~), Mae! Sah whence Via (Fe =i og (1 +ap > if (3) é Gases on Contact Potential Difference between Metals. 169 For platinum when it is sensitive to hydrogen the constants me ange have the values a—1:27 104, ¢c=0-73, and a= 2°43 x 10° when p is in millimetres of mercury. The contact potential which is operative during the experi- ments is not V,, which refers to a gap between two metal surfaces both at temperature T, when one is in a gas-free space and the other in hydrogen at pressure », but it is the potential difference between two surfaces at temperatures T, (that of the hot wire) and 1’) (that of the cold cylinder), both in an atmosphere of hydrogen at pressure p. Let us call this potential difference QO. To find the value of © consider the gap between two blocks of the metal, one maintained at temperature T, and the other at To, both immersed in hydrogen at pressure p and connected together by a platinum wire. Next consider the work done in taking an electron round a closed circuit starting inside the metal at T,, across the boundary of TI, across the gap, then acress the boundary of Ty, and finally down the wire to the starting point. ‘This consists of the algebraic sums of the amounts of work done in crossing five surfaces, viz. that between the pure and contaminated metal at T, (7,), that between the contaminated metal at T, and the gas (¢,), that across the gap (—eQ), that from the gas to the contaminated metal at Ty (@y), and that between the pure and contaminated metal at Ty (jm). Thus by the energy principle CO Oita — (hot tae 20-0 C4) The notation is the same as that in ‘Hmission of Electr icity etc.’ p. 109.. In the same notation (p. 109 equation 34') we have Ve eh (Ort ss | 22) eVo=h0 +1) — (ho +70): Teese (0) Now 7,'=7)'=0 ; so that eO=e(Vo— V,) + ,'— Do - c - ° (7) Now, from loc. cit. p. 33 equation (16) (equation (15) p- 32 is more accurate but the differ ence is hardly material) d,' — dy! = 3k (T,- Days ° ° e . (16) and substituting for V) and V, from (8) above and the similar equation for Vy k O=7(T,=T,) {5 tlog L+ap*) |, er ar or, a little more accurately, k Oey qn —_— _ a =i Cc Ee ee ~ O=—(T,—T) 15 + log (1+ aps) \ jar. a8) and similarly 170 ~=Prof. Richardson and Mr. Robertson : Liffect of Now we have seen (p. 164, ante) that T, varied in the different comparative experiments, being a function of the temperature I, of the hot wire in the high vacuum experi- ments and the pressure p of the gas. It was in fact the temperature which at the pressure p gave the emission its high vacuum value and is therefore’ determined by the equation Ate. = A'T 3-0 In this equation the factors T,? and T,? can be treated with sufficient exactness as equal ; so that to such approximation as we require fee Pore ae : j us b+T,log A/A’ ” and using equations equivalent to loc. cit. p. 112 (15) and (17), b' — "log (1 +ap*) T= 7 (21) —1T, log (1+ap’) ee and the operative contact potential in volts is bf — Be CIs: ape) | | o— Ty r Se) k (3 te x {22 +log anon — | | eat. (22) Kquations (21) and (22) express V as a function of the temperature T, of the het wire in the high vacuum experi- ments, I’) that of the cold electrode, the pressure p of the hydrogen, the universal constants k and e, the constants a, c, and a defined already, the emission constant b' for the metal ina vacuum, and the specific heat of electricity o in platinum. The term in o is unimportant, but is added for the sake of completeness. If this theory is correct the ohdeered: ‘displacements ot the curves will be equal to the differences in the values of V For a given value of T,, corresponding to the gas pressures used. Neglecting the unimportant term in o and substituting the known values of the constants, the values of V given by equation (22) are :— . For T,=1200° K: when p=0:112 mm. V=0-452 volt, and when p=0:0013 mm. V=0°374 volt. Grases on Contact Potential Difference between Metals. 171 The displacement for this limit of pressure would thus be dV ='078 volt. For T,=1460° K : when p=0-112 mm. V=0°629 volt, and when p=0:0013 mm. V=0°504 volt. For this value of T,, 6V =0°125 volt. . The experimental value of the displacement over this range of pressure for T,=1460°K. was 0°16 volt; so that the observed effect is in fair accordance with the theory. It should also be added that it is in the prescribed direction. There is one pointin this discussion which perhaps has not received sufficient emphasis, namely, that it appears vital in order to reconcile the results with the type of theory dealt with to assume that the hydrogen is in equilibrium not merely with the surface of the hot platinum wire but with that of the cold platinum foil as well. If the hydrogen did not modify the cold platinum surface this would act as a reference surface of constant potential, and a calculation along similar lines for this case shows that the effects should be considerably greater than those observed and in the opposite direction. ‘This is in accordance with the require- ment that the hydrogenated platinum surface should be electropositive toa clean platinum surface at all temperatures. The physical reason for the opposite direction of the effect as given by the experiments and supported by the calculations, liesin the fact that the temperature coefficient of the electron work function is negative at all pressures and increases with the pressure of the hydrogen. EXPERIMENTS WITH TUNGSTEN. We have made a large number of experiments with tungsten filaments using the type of tube shown in fig. 6. The anodes were of copper foil and were 5:0 cm. long and 1-9 cm. in diameter. The tungsten filaments were 3:0 em. long and 0°127 mm. in diameter except in the data shown in fig. 7 where the diameter was 0076 mm. The saturation currents were of the same order as before, 107-7 amp. The wires were kept taut along the axis of the cylinder by means of the molybdenum wire springs shown at the ends. These were under slight tension and took up the slack due to expansion on heating, For some reason we have found it more difficult to get consistent results with tungsten than with platinum. This may be due partly to inexperience in controlling the condi- tions as the tungsten experiments were, in point of fact, 172 Prof. Richardson and Mr. Robertson: Lect of made first. It is probable, however, that the main factor in this is the superior chemical activity of tungsten and ~ copper as compared with platinum. Another difficulty arises from the very marked “clean up ”’ effects with tungsten, which for example when working with hydrogen cause the pressure of the gas to vary considerably in a single experiment. It would take too much space to discuss the whole mass of data which have been obtained in detail, and we shall merely point out the more important conclusions which follow from PF the experiments. The first point tested was as to whether this admission of small amounts of mereury vapour at a pressure of the order ‘001 mm. would displace the curves along the voltage axis. No displacement was observable, in- dicating that mereury vapour does not affect the contact potential between hot tungsten and cold copper. This is in agreement with the observation * that the electron emission from tungsten is not affected by mercury vapour. In these experiments the mercury vapour was only admitted for short intervals. No doubt if enough were admitted to. amalgamate the surface of the cold electrode some displacements would be anticipated. The problem of the effect of hydrogen was attacked in a similar way to that in the case of platinum. The experiments show that any genuine effect of small quantities of hydrogen is small, and we have not been able to convince ourselves that there is such an effect. The most complete set of data gave the following values for the potentials V on the voltage axis at which the currents had attained half the saturation value at the pressures stated :— p (mm.)—> "1795 1745. °1452 132 “1275. -123 0805) FeOre Om 0 22 V (volts.)—> —1:38 —137 —132 —141 —142 —145 —141 —140 —1-43 —1-43 Means. — ee p(mm.)> ‘0155 -0085 -00625 -00385 177 +182 0777 008 V Wolts)> {157-051 <1:5t- Sie. =1:375 = 1-40 =e ees —_ % Richardson, Phil. Mag. vol. xxvi, p. 347 (1913). Gases on Contact Potential Difference between Metals. 173 These figures suggest that if the effect is real it is in the opposite direction to that observed with platinum, 7.e. the hydrogen tends to make the hot tungsten more electro- positive with reference to the cold electrode. However, further tests were inconclusive, the gas sometimes apparently eausing a small displacement in one direction and at other times in another. All that we are prepared to state definitely at present is that there is no effect of an ofder of magnitude exceeding that indicated by the figures given above. 440 400 a 360 & J a = < J i = 3290 x v1 i = 2 280 9 2 S =“ 240 ® Pressure -157 MM. Hg 1H, admitted to = thoroughly clean tuse = © Ww “0002 « “ FILAMENT AND PLATE CLEAN e) 200 4} AD ote 0002 - = FIL.CLEAN PLATE DIRTY. i ra) - a 00025 * ~ FILLAND PLATE DIRTY, 160 [og ul = us = 120 © . 2 Ciel ss CN Ol es) ° OTL Oke i W hen the first derivatives vanish, the Riemann- Christoffel tensor simplifies to Bm 3( Rib 4 tee — Bite _ Btn BO OL 08,. « 02, Ox, OLi0e, =>, (Gy Siren Fro ve) ; : It will be seen that the six <,’s contribute terms to ice which are simply additive. Our subsequent formule will 176 = Significance of Hinstein’s Gravitational Equations. involve B,,,.) linearly ; so we need consider only one z ata time, and shall accordingly shorten the notation by dropping the r. Thus Bi vop = UuvAap— UtaArp- Hence Gv = 97 Buvop = — Gun Ay + gy + Gag + yy) + Myer Aye by substituting the Huclidean values of the g’s at the origin given above. In particular Gy =— Gay (Gnaiat G9 te Ose ote Cupane Chg = a + Ay,” 3a Se Gee) he Ay 455) at hei Oh gy) ace) Also C—O (a,, + Oneal Qos a, (ak + Cane + HS Fes + Biba + as = —-21(@) — 4,0) a. (SIX terms) Pee Hence, sinceg,, = —1, Co 9 AO = a (ase = Foo Ga,) st (Oy ane ay + (yg? — Mgsttaa) be eta) Consider the three-dimensional continuum, which is the section of the world by the plane z,=0. This is described by three coordinates #9, #3, v, and its Gaussian curvature, which we shall denote by GQ, is formed by dropping the terms in G which contain the suffix 1. We see by (2) and ()) that) | 30) = Gy, a9: De Since the ¢,’s contribute linearly to each term in this equation, it holds for six 2’s as well as for one z. The radius of (spherical) curvature of a manifold is defined as the radius of a sphere which has the same Gaussian curvature as the manifold. It is easily shown that for the Gaussian curvature Gi) of a three-dimensional manifold, the corresponding radius p, is given by 6 G j=: Be oe Thus the result (4) may be written 3 G,j—-39,G= py . miacane See ian G (3) Consider the quadric (Gia sG 0 dada, =2) aes Oscillation-Hysteresis in Simple Triode Generators. 177 Setting dz,=(p,, 0, 0, 0) we see by (5) that the equation is satisfied; that is to say, p, is the radius of the quadric in the 2, direction. The quadric is seen by inspection of its equation to be invariant; consequently we may take w, in any direction we please. Hence the radius of the quadric (6) in any direction is equal to the radius of (spherical) eurvature of the corresponding section of the world. If G,,=Aguy, then G=4), and the quadric reduces to —VAGy AL, Ax, =3 or —ds =3/2X, showing that the quadric is a sphere of redius ,/(3/d). Conversely, if the radii ef spherical curvature of sections of the world at all points and in all directions are equal to ,/(3/X), Hinstein’s equations G,,=Ag,, will be satisfied. This demonstrates the theorem. Ii may be uoticed that in this general proof we have substituted spherical curvature for the normal curvature considered in the Appendix to the previous paper. This is necessary because in the general case normal curvature becomes meaningless. XVI. Ona Type of Oscillation-Hysteresis in a Simple Triode Generator. By EH. V. Appteton, M.A., WSc., Fellow of St. John’s College, Cambridge, and BALTH. VAN DER PoL junr., D.Sc., Conservator Physical Laboratory of Teyler’s Institute, Haarlem (Holland) *. HE conditions for the production of free infinitesimal oscillations in various triode circuits have been worked out in great detail during the last few years, but the question of the stability and maintenance of oscillations of finite amplitude does not appear to have received equal attention. In a recent paper f we have dealt with the calculation of the amplitude finally attained in a simple case of free triode vibrations in which use was made of a non-linear “ oscillation characteristic’ easily determined by experiment for any particular tube and circuit. This oscillation characteristic, which represents the relation between the variations of anode potential and of anode current, may be regarded as expressing the electrical properties of an imaginary non-reactive re- sistance connected in parallel with the inductance of the * Communicated by Professor Dr. H. A. Lorentz, For.Mem.R.8. t+ Phil. Mag. vol. xlii. p. 201, August 1921. Phul. Mag. 8. 6. Vol. 43. No. 253. Jan. 1922. N == ae ———= alee sesame SS SS oe SS et — 178 Mr. Appleton and Dr. B. van der Pol on a Type of oscillatory circuit. The form of the oscillation characteristic may be varied within fairly large limits by adjusting the applied electrode potentials of the triode and the reaction coupling, and thus many interesting problems arise. For example, we may ask ourselves: What must be the shape of the oscillation characteristic such that for some particular set of circuit conditions there may be more than one possible stable amplitude? In such a case we should expect the particular amplitude obtaining to depend on the method of approach to the set of conditions in question. We shall deal here only with the answer to the question for a case in which two stable amplitudes are possible for any given set of circuit constants, but it would not be difficult fa extend the analysis to more complicated cases. For example, we find that for two stable amplitudes to be possible, one of which is zero, the expression for the oscillation characteristic, when developed as a Maclaurin series, must have a positive fifth differential coefficient and a negative third differential coefficient. In the case of ordinary receiving triodes we have found it quite easy to obtain characteristics of this particular type, and thus have been able to compare the theoretical results with experiment. The circuit used throughout the experiments was of the simple type shown in fig. 1 A. Fig. iP As is well known, the anode current of a triode is a function of both the anode and grid potentials with respect to the filament. °But in the circuit of fig. 1 there is a definite relation between the variable parts (v, and v,) of the anode Oscillation-Hysteresis in Simple Triode Generators. 179 and grid potentials, namely Ug - Vas SURG Copies Uap ae) vary et (1) where grid currents are assumed to be negligible. ‘Thus in such a case the variable part (7,) of the anode current may be expressed as a function of v, only, and it is precisely this relation which is represented by the oscillation characteristic mentioned above. In this way we are able to leave out of account the retroactive action of the control electrode, and deal simply with the problem of a conductor possessing a characteristic relation 7,=f(v,) connected to an oscillatory circuit, as shown in fig. 1 B. 7 It will be seen that the problem we have to discuss is exactly the same as that arising in the case of the dynatron generator of A. W. Hull *, so that the general theory given below will apply in toto to both triode and dynatron, if the analogy between the oscillation characteristic of the former and the direct characteristic of the latter is borne in mind. We shall be concerned in general with the determination of the possible stationary amplitudes, and also with the stability of those amplitudes when the characteristic of either generator is given. The application of Kirchhoff’s laws to such a circuit as is shown in fig. 1 B leads to L Z = Rio+ (itt — Ug, , 4) +29=ta=W(V), which together give Va d? : d R r(va) dt (va+ Raa) + ar {; Va + ea } vy GL But in the practical case of a high-frequency circuit, Ri, is small compared with v,, and thus (2) may be written d*v dt? where the subscripts of i, and v, have been omitted, wy? has =. 4 (2) d 2 + yx) ie wev= Os RP oh rina a (3) been written for oe and y(v) has been written for Gr Py a We have been unable to obtain a direct solution of (3), but * Hull, Proc. Inst. Radio Eng. 6, i. Feb, 1918. NZ 180 Mr. Appleton and Dr. B. van der Pol on a Type of an approximate solution for conditions which closely resemble those of a practical case is possible. Thus in the case of a triode we may consider v as approximately sinusoidal, and equal to a sin wt since y(v) may be regarded as small compared with wov. In this way the higher harmonics may be neglected, and an approximate value for the fundamental amplitude in the Fourier expansion for v obtained, where a is regarded as 2 a function of the time %, but such that = O. . . - (7%) «0 Now by partial differentiation, (6) can further be written . a) Tiida) cos wt.adi—0, . . (6a) 0 with the aid of which we are able to write (7) more simply im £( al Pa cm wildie 0.0 2 2. (8) 9 In the case of a triode, the oscillation characteristic | 182. Mr. Appleton and Dr. B. van der Pol on a Type of t=wW(v) cannot in general be represented by a simple expression, but use may be made of a power series *. If this series be written as W(v) =av + Bv? + yr? + dut+er?..., chem Cy) = (« aa 5) v+ Pv? +yv?+ou'+ev"... On substituting for y(v) in (6) and (8), we find that the possible amplitudes are given by the real roots of: R slot at ot +. ead on BU ay ie) erates 2n—1 2n=1 2 +73 4.600 on re while any amplitude a is stable for which (e+ a oe 2S gee © eat tas L 2 : See eee Oy ==) L he + 9.4.6. ....2052 oe If the oscillation characteristic is of such a simple form that the power series can be limited to the first three terms, it may be shown that only one stable amplitude is possible for any given set of triode and circuit parameters. Thus in this case (9) reduces to p(e+ at * a? =0, 2 L the solutions of which are given by a, =O; Al a= =] ; 3 L } a But from (10) we see that for an amplitude a to be stable we must have (a+* Tea a a), Thus in this simple case ge one stable finite ea Aly is possible, the conditions being such that (2+* + ) <0); * Of. van der Pol, Radio Review, vol. i. p. 701, Nov. 1920; and Tudschrift van het Nederlandsch Radiogenootschap, Deel i. Oct. 1920. Oscillation-Aysteresis in Simple Triode Generators. 183 This is in agreement with the result of the paper (Radio Review, Nov. 1920) mentioned above. In such a case the limiting conditions for starting oscil- lations (i. ey +a=0) are exactly the same as the limiting conditions for stopping an oscillation. That is to say, the stationary amplitude is a single valued function of the circuit parameters, and thus, if R or © is varied, the whole process is reversible. Our experiments have shown, however, that it is often necessary to take at least two more terms in the series to represent the type of oscillation characteristic met with in a practical case, and we shall now consider the amplitude and stability of the oscillations tor such a type of characteristic. From equations (5) and (9) we have at once a + (5 rat i at 2 ow =0 gat EO 8 6 a CR Cy Were ne) or Coa + (2+) a+ qyet gen =), Spa Gh) while for any amplitude a to be stable, we must have Cle tot ig Lar oy (at =) +50 + ea SU ae ieee ei) 2) A solution of (11) can be immediately obtained, but for the present discussion such a solution is unnecessary. For example, the stationary amplitudes are given by the real roots of CR 2 3 4 5) Go (a+ re + 74 + 3 a =al)p These may be written ay’ =0, i —A-+ V/V A?—B, Ean remy Se ey Die ae CR \ where 1 ae amc) = a and the square roots are taken as having a positive real part. The conditions of stability are given respectively for 3 eh oi eerey eis —ea;’>0, ‘wile es, ~ ~N : 184 Mr. Appleton and Dr. B. van der Pol on a Type of If we now consider all possible combinations of signs of a, y, and e, the possibility and stability of the three ampli- tudes can be tabulated as follows, where it is assumed that A?>B. When this is not the case, stationary amplitudes a, and a3 are obviously impossible. 7 Sign | Sign ae Sign | Sign re | Aes ae ofe. | of y. Jat a OL Gt uhOla.2. a 2 Alerts he + | -F + | = — | stabie| unstable} stable Dai as a + — fs = ae [stable] stable Shahsee + — + | + + | stable | [stable| unstable Ase a — | + | = |imstable [stable| stable Dene om + + + — [stable | unstable | unstable Ones: = + _ + + {unstable | unstable [stabiel bere — — + + — | stable | unstable | unstable Cai: — ~ -- — — |unstable! stable | unstable (In this table, atuplitudes which are both possible and stable are emphasized by thick rectangles.) We thus see that the only combination giving two possible and stable amplitudes is (3),-in which case e is positive, CR be ie y negative, and (= 4 x) positive. For such a combination L of oscillation characteristic and circuit constants the system is stable when not oscillating (a4;=0), and also when oscil- lating with an amplitude ag. In this case the circuit possesses a positive initial damping coefficient, so that oscillations will not build up automatically, although a stationary amplitude of value aj, when once produced, will be maintained. These analytical considerations can further be elucidated in the following way. In (3) Te) can be written as dv) do 4 dx(e) Z ieee. dp a kind of damping factor which may be either positive or negative, and which, unlike the damping factor used in the ordinary linear treatment of oscillations, is a function of the dy(v) dv may be regarded as representing and v for cases 2 amplitude. The relation between Oscillation-Hysteresis in Simple Triode Generators. 185 3, 4, and 6 of the table have been drawn in fig. 2, where the possible amplitudes have also been indicated. Fig. 2. Nors3: No. 6. For any amplitude to be stationary, it follows from energy considerations that the damping term for a cycle of v must be partially positive and partially negative. In this way we _ see that the amplitudes a, and a; are possible. In order that the amplitude should be stable, however, it is necessary that, if the amplitude of v increases, the damping factor ey) should be positive over. a greater part of the cycle of v and vice versa. Thus az is stable in Nos. 2, 3, and 4 but not in No. 6. In the same way a; is stable in No. 6 but notin No. 3. In fig. 2 the function aie) has been made symmetrical with respect to the axis v=0, sinee from (9) and (10) it is seen that, so far as our present approximations go, the even terms (¢.9., 8, 6, etc.) of the series for the oscillation characteristic can be neglected. We may obtain a derived 186 Mr. Appleton and Dr. B.van der Pol on a Type of oscillation characteristic in which these terms are excluded — by means of a simple graphical method. A typical oscil- lation characteristic obtained experimentally is shown in fig. 3,.for which the value of we is 0°57 and the steady electrode potentials such as to give a case of marked oscil- lation hysteresis. nae volés. = -8 wo/ts Anoge current in Milli-amperes >» No -90 -80 -60 -40 -20 0 20 80 990 z A) / /t ~ Variable Anode Potential (v) in Volts c. = We have eliminated the even part in the function repre- senting this curve by folding the curve about the 7 axis and then again about the » axis, and taking the mean of the two ordinates occurring in the quadrants Il.and IV. The result is shown in fig. 3A. The general agreement between the differential coefficient of this symmetrical curve which re- presents the relation z= (v), and the curve of No. 3, fig. 2, is clearly apparent if we bear in mind that y‘(v) is equal to Rv). jee ne We are now prepared to discuss in greater detail the quantitative results previously obtained for our example of Case 3 of the table and No. 3 of fig. 2. Vig. 4 shows a graphical representation of these results, in which the square of the amplitude is plotted as a v function i) v WS N g : a ~~ 8 2 N BS) O u v ° c N Oscillation-Hysteresis in Simple Triode Generators. 187 of B. It may here be noted that in a practical case, B may be varied continuously by varying the resistance R or the capacity C of the oscillatory circuit. an an | Anode Potential in Volts, Fig. 4, —> 0 N Sh ee em em om em em oot ey OP On ee oe me ee owe ee fe ce en } 188 Mr. Appleton and Dr. B.van der Pol on a Type of For ail values of B for which B> A? (Region 1), we see that the roots are- either zero or imaginary, so that no sustained oscillation is possible. In Region 2, however, for which we have A?>B>0, various possibilities arise according to the initial value of a?. If the initial amplitude is small d so that a.2>a;7>a?, we see from (11) that Ae <9 and thus the amplitude will tend to decrease to zero as indicated by the arrows. If, however, the initial amplitude is such that ay* >a*>a;”, then a a? >0 and the amplitude will increase dt to ad. Further, if a?>a,’>a,7, then ye <9 and the ampli- W tude again finally reaches az. We thus see that in Region 2 no finite stable amplitude will be reached automatically unless by some external agency we succeed in producing in the system an amplitude a such that a?>a;’, in which case. the amplitude automatically builds up to the stable value a. For any gradual variation of B after such an amplitude has once built up, the amplitude value a, is maintained. Fig. 5. ———> Bs As a method of testing this theory, we may imagine a cycle of operations in which B is varied from a large positive value to a negative one, and then the process reversed. We see from the above discussion that the cycle will be irreversible and of the type illustrated in fig. 5, where the relation between a? and B is shown. The value Galvanometer Deflection Oscillation-Hysteresis in Simple Triode Generators. 189 of B may be most easily varied in a practical case by means of a variation of C the capacity, or R the resistance of the oscillatory circuit. A eycle of the type indicated in fig. 5 was accordingly carried out for a circuit of the type shown in fig. 1, A, the amplitude of the oscillations being indicated by means of a crystal detector and galvanometer connected in series with a coil which was loosely coupled with the oscillator circuit. Fig. 6 shows the relation obtained between the galvanometer deflexion and the value of R. The general agreement with the theoretical curve of fig. 5 is at once apparent. For certain types of oscillation characteristic we found evidence of the possibility of two stable amplitudes of the same frequency, both differing from zero for a given set of circuit parameters. For example, it was sometimes found that on decreasing the resistance R a small oscillation started before the resistance value equivalent to B=0 was reached. This eifect is illustrated by the dotted line in fig. 6. It was also found that if the resistance was not Fig. 6. t ct © ~—— Sr ree ae reer erst | 1S0 200 210 220 230 240 250 Pesistance FR 17 ohis. allowed to become less than that represented by B=0 (fig. 5), the relation between amplitude and resistance was reversible. It is clear that two more terms are necessary in the series for y(v). to account for this effect. We have thus evidence of a phenomenon depending on the value of the seventh differential coefficient of the oscillation characteristic. -190 Mr. Appleton and Dr. B. van der Pol on a Type of In the theoretical discussion given above, it was found that for conditions represenied by Region 2 (see fig. 4) it should be possible to start a stable oscillation by means of an electrical impulse greater than a certain amount. This theoretical prediction was experimentally verified in a qualitative way by using transient induced electromotive forces produced with the aid of a coil and a bar magnet. From the point of view of certain practical applications of triodes (such as triode relays and quiescent aerial telephony), it is of interest to consider how the difference between the limiting parameters necessary for starting an oscillation and those necessary for stopping one vary with the values of the maintained electrode potentials of the triode. Experiments were therefore carried out to test this point. A triode that had been previously carefully tested for steadiness was used in the circuit of fig. 1, care being taken to allow the thermal conditions of the triode and its supports to become steady before measurements were made. ‘The resistance R of the oscillatory circuit was non-inductive (CuSO, solution) and continuously variable. The values of the limiting resistances for starting and stopping an oscilla- tion were found for various values of the applied steady grid potential v,,. The existence of a sustained oscillation was indicated both by means of an autoheterodyne circuit some metres away and also by means of a loosely-coupled senx thus becomes ais , J(p) dp i ae (F.) + ee m? m dw which on ... gives = (oR tor 5, | Moer=e . OE e where C is some constant. We now take a piece of transparent squared paper and, taking rectangular axes OX and OY, we plot on it in Cartesian coordinates the curve ae | +2 \ serayae. A Nn este a era Let DE be the curve thus drawn. We next take a piain sheet of paper fastened to a drawing-board and mark a point P on it with a dot. Having selected a particular direction on the plain sheet of paper with respect to which the angle » is to be measured, and taking P as pole, we plot upon it in polar coordinates the curve om ah o(S)ao. le i le, The constant of integration may be given any value we please, but it is convenient. to give such a value as may render the curve as simple as possible. eZ 212 Dr. A. A. Robb: Graphical Solution of Differential We shall then have | : i str+ { A(p)ap=C. Let the transparent squared paper now be placed on the top of the plain sheet so that one of the lines on the squared paper parallel to OX passes through the point P and let this line intersect the curve DE in the point Ro. pale 7 Dw So 7 (OL \ Cd Ne a Oy chet Sar C7 Let the perpendicular through P on OX intersect OX in N, and the curve (5) in To, and let the perpendicular through Ro on OX intersect OX in Mo. Let a length M,)Ay equal to PT) (or 7) be measured off .along the axis of w, and let it be measured from Mo in the negative direction if 79 be positive, or in the positive direc- Bonn if 7 be negative ; and let a pin be stuck through Ao into the drawing-board. Let the point of the lower paper thus marked be denoted by Ao. "F urther, let a pin-prick be made through N, into the lower paper and let the point of it thus marked be denoted by Np. Next let the squared paper be turned through a very small angle about the pin through Ay until another of the lines on it parallel to OX passes through the point P. Equations occurring in Wireless Telegraphy. 213 Let this line intersect the curve DE in the point R, which will be very near to Ro. Let the perpendicular through P on OX in its new position intersect OX in N, and the curve (5) in Tj, and let the perpendicular through R,; on OX in its new position inter- sect OX in M,. Leia length M,A, equal to PT, (or 1) be measured off along the axis of w as before and let a pin be stuck through A, -into the drawing-board and the pin through Ag be removed. Let the point of the lower paper marked by the pin through A, be denoted by Aj. Also let a pin-prick be made through Ny, into the lower paper and let the point of it thus marked be denoted by Nj. Let the process, as described, be repeated an indefinite number of times and we obtain two sets of pin-pricks in the meeps A, Aj, A, ....A, and N,, Ny, No --.- Na, and these approximate indefinitely to two curved lines as we take the small angles of rotation of the squared paper more and more minute. We shall call these two curved lines the A curve and the N curve respectively. Tt is evident that esa ee + A, An= PAA Age Pe and, if the A curve has its curvature of one sign in the interval, then this = AyA, so that AjA, tends in the limit to the length of the stretch of the A curve. Also PN,, PN;, PN.,....PN, are the perpendiculars from P on the various positions of the line OX which in the limit becomes the tangent to the A curve; so that PN,, PN,, PN. .... PN,» become successive values of p for this. curve. Until we have shown that the A curve is the one required we shall distinguish the p and s of this curve by accents. Now we have PN, = EM, and this is the ordinate of the curve (4) corresponding to the point M, on the axis of 2, while AvAn + A,M, = AoMn = OM, aT OAo, or AyAntm=a—OAy. 214 Dr, A. A. Robb: Graphical Solution of Differential Thus since A,An=6s’, we have for the A curve ee ib a lit és’ — oo o(2)ao+ UG \dp! =a + (addy — Oo But the right-hand side of this equation is constant, and so for the A curve, the relation (3) holds. It is thus evident that the A curve is one whose p and equation satisfies the differential equation (2). On the other hand, the N curve is one whose polar equation with P as pole satisfies the differential equation (2), and this curve is more convenient than the A curve for giving the relation between p and o directly. In the special case where | p(t) =a sin mt, the curve (5) becomes a r= — —, sin do, m a =-—,cos@+ const. m If the constant of integration be taken as zero, we get the simplest curve, which is f= = COS @. m This is clearly a circle passing through P and of which the diameter is Ll m Thus in this particular case, which is one of special importance, the curve (5) is very easily drawn. The curve (5) might be drawn with a difterent pole from : P in case we desire that it should not be in too close prox- imity to the N curve. The lines on the squared paper parallel to the axis of y would still give the proper direction of the radius vector. It will be observed that if the function ¢@ is zero the points Ay, A,, Ao,....A» coincide respectively with the points My, M,, M,,....M, and we get the same result as previously obtained. In conclusion, I wish to express my best thanks to Mr. E. V. Appleton, who called my attention to the im- portance of this problem. Note on the above by E. V. APPLETON, M.A. The production of undamped- electrical oscillations is nowadays accomplished in various ways. But in most cases the assembly of the generator may very simply be reduced Equations occurring in Wireless Telegraphy. 215 to two elements, (1) an oscillatory circuit containing self- inductance, capacity, and small resistance, and (2) some type of conductor possessing a non-linear voltage-current charac- teristic. Sometimes the conductor is placed in series with the capacity in the oscillatory circuit as in the Poulsen arc, but more often it is included as a shunt across the condenser as in the Dynatron* and the ordinary triode generatort. (See accompanying figures where the conductor is shown as possessing a characteristic current-voltage relation [= F(V).) Dynatron or Triode In either case the fundamental equation of the generator representing the relation between the voltage V, across the condenser, and time ¢, may be reduced to d?V dV, pee PION) oa Fr +m*V,=0, Sana SLY) where f( V;) wi m? include the circuit constants C, 7, and L. For many purposes it is sufficient to consider the characteristic [= F(V) as linear (e. g.in deducing the limiting conditions for instability), but it is clear that the inclusion of the non-linear terms is essential in a discussion of the maintained amplitude of the oscillations or the production of harmonics. A certain amount of progress towards the solntion of (1) for the somewhat limited cases in which f(V,) may be simply expressed as a power series has been made by Dr. van der Pol and the writer ¢, but no extension has been made to the case where the siohit. hand side of the equation is not zero. The latter case (dealt with in Part II. of Dr. Robb’s paper) is of practical importance as representing the case of a sinusoidal electromotive force impressed on a crystal or triode receiver circuit. * Hull, Proc. Rad. Inst. Eng. vol. vi. p. 5 (1918). + Appleton and van der Pol, Phil. Mae. vol. xl. p. 201 (Aug. 1921). t¢ Phil. Mag. supra, p. 177. [2a XX. On Certain Types of Electric Discharge. “By Prot. D. N. Maus, #.4.S.E., and Prof. A. B. Das, M.Se.* 1. JN previous papers (Phil. Mag. Oct. 1908, Oct. 1912, July 1916) we studied the behaviour of electric discharge in a De La Rive tube, under gradually decreasing. pressure. In the present series of experiments, tubes of - ordinary pattern (with electrodes consisting of thin rods) were used, in order to verify the various theoretical deduc- tions, previously arrived at. 2. Three tubes (T,;=34-4 cm. in length, T,=14°5 cm., and T;=4:°2 cm.) were used in parallel, in order that the effects of length and decreased pressure on the character of the discharge should be clearly observable. The inducticn-coil used gave a spark-length of 19°94 cm. in air, with a spherical electrode of 3 cm. in diameter. : 3. At a pressure of 73mm. no discharge passes in Ty, it is spindle-shaped in T, while in T3 it is in the form of a band. As the pressure is further reduced at 20°5 mm. the discharge in ‘T, is spindle-shaped, T, shows a band which was the stage reached in T; at the higher pressure, while in T; the band is changing into a glow discharge. 4, On the theory previously worked out, the spindle-shape is due to the mutual repnision between the various streams of discharge, which is seen to be operative, in spite of the symmetry of the electrodes. 5. Moreover, during the band stage, according to theory previously worked out, the result of ionization is annulled by recombination. Now ionization depends on a where X is the electric intensity along the discharge and p the pressure. ‘This quantity, therefore, depends on jp where lis the length of the tube. The effect of decreasing the length must, therefore, correspond to that of increasing the pres- sure. As, however, the potential difference itself depends on length and pressure (arts. 15 and 16), there is not as yet sufficient material for working out the law of this corre- spondence. 6. Itis found that it is only when the discharge is in the form of a thin band or a single stream of discharge, that there 1s an action (attraction or repulsion) when the discharge- tube is placed in a transverse magnetic field. In the actual experiment, a rectangular coil of wire, carrying current, two * Communicated by the Authors. On Certain Types of Electric Discharge. 204 YE ‘ of whose sides are parallel to the tube, was used to produce such a field, fig. 1 pL de ree) 6 —_—__ i ieE|]“_ _ —_—_———— | Se To induction A =Y 4 ZA, From coil Gia z S ca eT BA induction cait . 5 —_— 7. According to the theory previously worked out, this usual electrodynamic action between currents will be observable only when the discharge is in the form of a band, the number of corpuscles in this case being equal to that of positive lons. The present experiment, therefore, is in confirmation of this theory. 8. When the pressure is further reduced (‘03 mm.) T, shows fine striatory discharge. This tube thus does not at all show the band stage—which alone is affected by a magnetic field—with the induction- coil used in the above experiment. 9. At this pressure, the discharge in T, is striatory but the striz are thicker ; while in T, the positive column is entirely absent, the cathode glow odhenine over the whole length of the tube. 10. It will be seen that the characteristic changes in these discharge-tubes and the characteristic differences between the three tubes arising from a difference in length correspond to the peculiarities exhibited by the curves connecting pres- sure and pe geial difference in them. Curve II, fig. 2, gives the relation between pressure and potential difference in Ate 11. In T,, at high pressure down to nearly 55 cm., the showery stage is indicated by the curved line. Tacrn this pressure to nearly 25 cm., the discharge is in the form of a band. This is indicated by a straight portion in the curve. The discharge behaves during this stage as an ordinary flexible wire carrying current. As the pressure is further reduced, the curve bends aw ay trom the line of pressure, the difference of potential increasing enormously with decreased pressure, when the pressure is lowered below 2 mm. In the case of Ts, the first stage is absent, as well as the first portion of the curve. (Curve I, fig. 2.) 218 Profs. D. N. Mallik and A. B. Das on In T, the second stage is absent, and the second portion of the curve with it. (Curve III, fig. 2.) —~ Pressure in mm. ——> Spark length in mm. Spark-length of the induction-coil= 19:94 mm, 12. When, however, the voltage of the induction-coil is: increased, the discharge in T; also passes through all the various stages and the curve I (fig. 3) exhibits the usual peculiarities in their entirety. 13. These characteristics are the same as those exhibited by De La Rive tubes. They are, therefore, common to all electric discharges, as they should be on theoretical grounds. 14. It is possible to work out an approximate theory giving a relation between pressure, potential difference, and the length of the discharge-tube, but the formula Certain Types of Electrie Discharge. 219: obtained is much too complicated for discussion to serve any: useful purpose, at the present stage. —— > Pressure in mm. » Spark length in m.m. Spark-length of the induction-coil=46 mm. _ 15. In the particular case in which the curve is a straight line, the formula obtained in a previous paper was rn ce Oe xX q q iN y pat” where Vg is the difference of potential between the terminals of the induction-coil when the circuit is open, X’ the electric intensity, assumed to be constant throughout the tube, gq, q’, velocities of +ions and corpuscles, and A, d’, their mean free paths, p being the pressure. 16. Now experiment shows that X itself depends on the ie 220 - Profs. D. N. Mallik and A. B. Das on discharge circuit, at any rate with an ordinary induction- coil provided with a metallic interrupter, which is controlled by a spring. Thus, with such an induction-coil of spark- length in air equal to about 20 mm., when the discharge was passed through the tube T,, the interrupter worked with much greater frequency than when the tube T; was in circuit. This is verified, by the curves I and II, fig. 4. Fig. 4. 5O 40. % Pressure in mrt. a La ‘4 X=/(p). If in the above formula X were independent of | the discharge-tube, the straight portions of the two curves ought to have been nearly coincident, if, at any rate, we could admit qe; : qe het, Were ; : 17. The interdependence of the various factors that enter into the phenomena is clearly brought out by the curves 1, Certain Types oy Electric Discharge. IO 2,3, 4 (fig. 5). Thus, it is seen that for the same E.M.F. in the primary, the current as well as the character of the discharge depend on the pressure. There is one feature of this phenomenon brought out by these curves, which is worthy of notice. When the discharge is in the form of a band (curve 1, fig. 5) (pressure 24 mm.). ——> Voltage on the primary 406 600 Current in deflections of the galvenometer within a certain range, the current is proportional to the K.M.¥. of the primary. Now, according to Ohm’s law, the current in a wire circuit is proportional to the E.M.F. when the circuit is open. As we may reasonably take the E.M.F. of the secondary, when no discharge passes, proportienal to. the H.M.F. in the primary, we conclude that the band discharge satisfies the criterion of a current in a wire circuit, in respect of Ohm’s law. This is of interest in view of art. 6. 18. When the striatory discharge (in tube T,) is placed in the centre of a rectangular coil carrying current, the number of striz increases, and in the actual experiment they become inclined to their original directions, while as the current is reversed, the inclination is reversed also. 19. A tentative theory of these experimental results: (art. 18) may be given as follows :— : The Faraday dark space is, as we have seen (‘‘Hlectric: Discharge in a Transverse Magnetic Field,’’ Philosophical Magazine, July 1916), a region practically devoid ofions. If this is so, all the dark spaces in the striatory column must be held to be also regions containing but few ions, and per contra, the illuminated portions to mark regions where there 1s copious ionization. 20. A corpusele being thrown off from the negative electrodes will be in a condition to ionize the gas almost. directly, if the pressure of the gas in the tube is sufficiently 222 On Certain Types of Liectric Discharge. high ; for the electric intensity near the cathode, in this case, being great, the energy required by a corpuscle for ionizing the gas will be acquired at once. 21. As the pressure decreases, the electric intensity near the cathode decreases also, and the corpascle has to move through a finite distance under the action of the electric force before it acquires sufficient energy for producing ionization. The space through which a corpuscle shot off from the cathode has to traverse, before itis in a condition to ionize the gas, would be the Faraday dark space. The energy so acquired, however, is used up in producing ionization in the first illuminated area. A corpuscle issuing from _ this illuminated area has, therefore, to move through some distance before it is again in a condition to ionize the gas. But as the average electric force at the striz is greater than along the Faraday dark space (H. A. Wilson, Proc. Camb. Phil. Soc. xi.) a corpuscle has to move ‘through a much shorter distance than the Faraday dark space, in order to recover the minimum energy required (Phil. Mag. Feb. 1920) for ionization. Accordingly, the dark space along the strize is of shorter length than the Faraday -dark space. 22. When a radial magnetic field is introduced, a -corpuscle issuing from the cathode tends to move under the electric force along a line of discharge and undergoes at the same time an angular displacement under the magnetic force. The resultant velocity acquired attains to the minimum value required for ionization at a shorter distance from the cathode than when the magnetic field was not on. The effect of a magnetic field is, therefore, to shorten the Faraday dark space and all the other dark spaces and, pari passu, to increase the number of strie. Moreover, the ions in the striatory dis- charge, under the joint effect of electric and magnetic fields, move in a spiral, the tangent to the spiral at any point being inclined to the axis of the tube, one way or the other, according to the direction of the magnetic lines of force in the field. Remembering that the movement of ions under the electric field itself produces a magnetic field (necessarily comparatively weak in comparison with the extraneous field that may be introduced), we conclude that ions in a striatory discharge in general move in spirals, more or less inclined to the axis of the tube, even without the application of a magnetic field. Measurements are being undertaken for a quantitative verification of the above theory. Our thanks are due to the authorities of the Presidency College, Calcutta, for facilities for carrying on our work there. iy DOSE XXI. The Application of the Ultra-Micrometer to the Mea- surement of Small Increments of Temperature. By W. Sucxsmirs, B.Sc., University of Leeds*. ) eS Whiddington’s ultra-micrometer apparatus f two oscil- lating valve circuits were set up. In the valve-anode circuit of the second was inserted a three-stage amplifier, together with a loud-speaking telephone, to magnify suitably the heterodyne note produced. The condenser in the first oscillating circuit consisted of two parallel plates, the variations in the distance of which produced variations, in the note emitted from the telephone. A tuning-fork was set up close to a small telephone, in order to provide a standard of pitch to which the note from the interfering circuits could be adjusted. The oscillations of this fork are amplified into the loud-speaking telephone mentioned above. This differed from the original experiment in that pre- viously a third oscillating valve circuit had been set up inducing into an amplifier, with capacity and inductances so large as to produce an audible note in the telephone. Fig. 1. M Loi Me | XK Pp The method used was to attach a metal bar to one of the condenser plates, and measure change in the temperature by the change in the note produced in the telephone. Fig. 1 shows a geometrical slide carrying two rigid vertical * Communicated by Prof. R. Whiddington, M.A., D.Sc. t+ Whiddington, “The Ultra-~Micrometer,” Phil. Mag. ser. 6. vol. xl. pp. 634-9. 224 Mr. W. Sucksmith : Application of Ultra-Micrometer to rods. To the left-hand one is attached a polished steel con- denser-plate A, 17 sq. cm. in area and about 5 mm. thick, and insulated from the supporting rod by a thin sheet of mica. The distance separating the two condenser-plates can be varied by a bending couple applied by placing weights in the pan K which hangs from the graduated quartz rod MUL. The right-hand condenser-plate B is made of polished aluminium 2 mm. thick, and is rigidly attached to the end of the copper rod CD (which is shown enclosed in non-conducting materials) the change in temperature of which is to be measured. The method is as follows. Initially the scale-pan P is empty. The heterodyne note in the telephone is adjusted by means of an auxiliary condenser in the second cireuit to be within 2 beats per second of the note produced by the fork. The copper rod, which is hollow, is heated electrically, thus bringing the aluminium plate B nearer*to A. - Weights are then placed in the pan until the same number of beats as before are heard in the telephone. | A knowledge of the weight required to produce a certain lateral shift of the plate A, together with the expansion coefficient of the specimen of copper used, enables the rise of temperature to be determined. Fig. 2. A cross-section of the expansion part of the apparatus is shown in fig. 2. Fis the hollow copper tube, which was 4 inches in length and embodying as little material as was consistent with rigidity. Two pieces of quartz QQ were Measurement of Small Increments of Temperature. 225 fitted tightly over the ends of this tube. Inside the ends of the quartz tube were fitted short lengths of invar C C, drilled as shown to admit the leads connected to the heating wire XY. Hard-setting sealing-wax was used to make the Joints tight. | “The heating wire XY was of constantan, soldered to thicker copper leads, and connected through a low-reading ammeter A and an adjustable rheostat to a battery. The whole of this wire was carefully insulated from the rest of the apparatus, — the spaces between the quartz and the heating wire being plugged with cotton-wool to prevent convection currents. The temperature of the copper was determined by a copper- -constantan couple, which had been previously calibrated by an accurate thermometer. A constantan wire, C, soldered along the length of the copper tube, 7. e. along GH, formed the hot junction. ‘The cold junction was kept some distance away in a water-bath protected from air currents. Thé thermo-couple leads were connected directly to a suitable galvanometer, the scale of which was so adjusted that ‘01° C. difference of temperature between the junctions could be read easily. A series of preliminary experiments were performed, and it was found that the rise in temperature of the copper tube followed the ordinary laws for increments up to 3° C., which was greater than required for the experiment. Further, the System maintained that temperature over an interval of time sufficient to allow the plate A to be readjusted. The heat equivalent of the system was found to be 3°74 calories per degree ©. To determine the lateral displacement of the plate A, pro- duced by placing weights in the pan, the method used in the. original experiment was repeate’. Large bending moments were applied to the quartz rod, and the displacement measured by a micrometer, contact being indicated electrically. It was found that 1°8 grams placed one inch along the rod displaced the centre of the rod 107° inch. The copper tube was heated to exactly 1° C. above the cold junction. 15°3 grams were required to restore the note to that emitted by the fork. This gives the coefficient of linear expansion of the copper used as ‘17 x 10~4 per degree U., the actual expansion being °68 x 107+ inches. Thus the arrangement is capable of detecting a change of 4-3x10-° 4 68x 10-2 = 000063° Ce or about i Gy. Phil. Mag. 8. 6. Vol. 48. No. 253. Jan. 1922. Q temperature of 226 | Mr. H. P. Waran on Mechanical since the smallest distance measurable by the ultra-micrometer was 4°3 x 1072 inch. It was observed that whilst the heating current was fowing, the note emitted by the telephone changed very smoothly, showing that no discontinuity in expansion coma be detected even with an apparatus capable of measuring =), millionth of an inch. My grateful thanks are due to Professor Whiddington for suggesting the experiment and allowing me to use hisoriginal apparatus, and also for his advice during the work, which was carried out in the Univ ersity Physics Laboratories. Leeds University. XXII. An Interesting Case of Mechanical Disintegration caused by Positive Ions. By H. P. Waran, M.A., Government of India Scholar of the University of Madras*. | W HILE studying the effect of a transverse magnetic field on the spectrum of an electric discharge through a rarefied gas, it occurred to the writer to examine the extent of the disintegration of the walls of the tube caused by the bombardment of ions deflected on to them by the magnetic field. An ordinary Pliicker discharge-tube containing pure nitrogen at a few millimetres pressure was arranged with its capillary between the conical poles of an electromagnet, a shown in fig. 1, and excited by the unrectified current from the secondary of an induction coil. It is easy to see that under the influence of the field transverse to the direction of the discharge, the discharge gets split into two streams, as shown in fig. 2, corresponding to the make and break currents which travel in opposite directions. Further, it isalso evident that the corresponding ions in the two streams travel in opposite directions and get deflected to the opposite sides of the tube. Ordinarily the current through the tube is about 2 m.a. and the field about 5000 cas... and very little of any local disintegration opposite the pole-pieces is noted. But by screwing the break of the coil and increasing the current through the magnet coils it is possible to pass a heavy dis- charge of about 15 m.a. ina field of the order of 10,000 c.e.s. momentarily, thus intensifying the disintegration consider- ably. Under such circumstances, in a few seconds a grey streaky patch developed on either side of the walls of the capillary, indicating a corrosion of the glass under the * Communicated by Prof. A. W. Porter, F.R.S. * Disintegration caused by Positive Ions. Q2 228 Mechanical Disintegration caused by Positive lons. sand-blast action of the ions. This region, when examined with a low-power microscope, revealed the presence of a series of short little grooves cut into the glass, all converging in a direction parallel to the axis of the tube and presenting a pretty pattern, as illustrated in fig. 3 (a). Fig. 3 \ \y y\ \) y a Qs | om ei vi ‘a \ i / ; fe : I \/ (a) (0) On examining the similar streak diametrically opposite, caused by the impact of ions from the reverse current, it was interesting to note that they all gave a pattern of grooves quite similar, as shown in fig. 3 (6), but with the grooves converging in a direction exactly opposite. There can be very little doubt that this grooving of the glass was effected by the bombardment of the massive positive ions which travel in opposite directions in the two streams causing grooves that converge in opposite directions. A closer examination, taking into consideration the direction of the currents and the deflecting field, confirmed the truth of the above explanation. ‘To make the point sure, with a point. and plate in series with the tube the reverse current was practically cut off, and the corresponding streak disappeared accordingly. When the upper electrode was the cathode, and the lines of force passed from left to right, the grooving was found to be on the side away from the observer and converging upwards, as would be the case if the positive ions were the cause of this peculiar corrosion. Fig. 4 is an enlarged view through the capillary of a dis- charge-tube in which the grooves have been formed, taken Relative Affinity of some Gas Molecules for Electrons. 229 especially to show both the sets of grooves side by side con- verging in opposite directions. Bio, 4, (2) The reasons why these grooves are on the average of about the same length and occur evenly spaced, start from about the same azimuth and converge, are other noteworthy points of special interest under further investigation. University College, London. XXII. The Relative Affinity of some Gas Molecules for Elec- trons. By Leonard B. Lorn, Ph.D., National Research Fellow, University of Chicago *. 1 es a recent paper the writer has shown that in some gases the electron must on the average make a number of impacts m with the molecules of the gas before it can strike one in such a manner as to attach to it to form the negative ion. In this same paper it was also shown that this quantity n varied with the chemical nature of the gas. It consequently becomes of interest to determine the value of » for a number of the common gases. It is the purpose of this note to discuss briefly the hearing of a recent study of electronic mobilities in N, on the determination of this constant, and on the basis of this to evaluate n for a series of gases from the experimental measurements made by the writer and Mr. Wahlin. * Communicated by Prof. R. A. Millikan. + L. B. Loeb, Phys. Rev. xvii. No. 2, pp. 89-115 (Feb. 1921); Proc. Nat. Acad, Sci. vii. No. 1 (Jan. 1921). 230 Dr. L. B. Loeb on the Relative Afinity It has been shown * that the current 2 between two con- denser plates as a function of the alternating potential difference V between them when one of them is illuminated by ultra-violet light is represented by the equation ecole * Gia) N a Hae (1) 29 with a fair degree of approximation. In this equation p is the pressure of the gas in mm., V is the alternating potential difference of square wave-form in volts, d is the plate dis- tance, N is the frequency of alternation, K is the mobility of the negative ions at 760 mm., K’ is the electronic mobility, d’ the mean free path of the electron at 760 mm., W the energy of thermal agitation of the electron, i/ip the fraction of the maximum current possible existing under the given conditions, and n the constant of attachment. It was also shown that if the mobility of the electron K’' be assumed equal to 200 T em./sec. volt/em., and if it be assumed that the mean free path of the electron is 4,/2 of that of the gas molecules, then by inserting the value of W, the velocity of thermal agitation of the electron, into the equation, it became ~ capable of experimental verification in the form __ 1:08 x 109 4°) ‘i iu) ey i Mi a 3 a Leokioiire (2) lo n may be evaluated by choosing a point nea1 the feet of one of the current-voltage curves and placing the values of the various quantities into the equation above. Then so obtained was found to be constant in order of magnitude for a con- siderable range of pressures and frequencies for any given gas ; in fact, n for oxygen was found to lie between 2 x 10# and 3x 10° in gases which varied in oxygen content from that of pure oxygen to a mixture of 745 parts of nitrogen to one part of oxygen. The range of pressures covered in this set of determinations varied fon 15 mm. to 750 mm., while N varied from 70 alternations per second to 750 alternations. This variation in n between 2x104 and 3x10° seems to be a definite variation in with some of the experimental * L. B. Loeb, loc. cit. + This was the most reliable value of electronic mobility in nitrogen available at the time of publication of the original paper. of some Gas Molecules tor Electrons. 231 factors. In general it was observed that for a set of curves taken at different pressures and frequencies, the values of n were uniformly greater the lower the pressure. However, the parts of the curves from which n was taken under these conditions also lay at progressively lower values of the voltage as the values of n increased. It was therefore a priort difficult to determine which of the variables, if either, was responsible for the changes inn. As has been pointed out before, this variation in n does not necessarily _ invalidate the theory of electron attachment which assumes n to be a constant; for the change in n may be only apparent. A real change in K’' of the equation (1) (about whose behaviour little is mee could be the real cause of the variation. In the hope of gaining some insight into this situation, the values of n were determined for mixtures of oxygen ‘and nitrogen. It was found that as the nitrogen became very pure ‘the value of. K, the mobility of the carriers, became quite large. In pure nite ogen, values of K were obtained of the order of 1100 em./sec. It was further found that the variation in 2 due to pressure observed when the quantities of oxygen were small, was greater than with air or pure oxygen. For example, it was found that in order to change n from 1 x 10° to 9 x 10° in air the pressure-change required was from 150 mm. down to 20 mm. Ina mixture of ] mm. oxygen to 745 mm. nitrogen it required a change from 508 mm. to 222 mm. to cause a change in n (computed back to air) of from 1:0 x 10° to 15x 10°. The change of n with the voltage was about the same in the two cases. The effect is then obviously not produced by a variation in the pressure alone, and is perhaps influenced by the value of the field strength. An investigation of the electronic mobilities became im- perative, in view of the fact that measurements had revealed negative carriers in pure nitrogen whose mobility was five times as great as those assumed for K', from the work of earlier observers * The mobilities of electrons in pure nitrogen gas were deteriiined, using high-frequency oscilla- tions from an audion oscillator Th ese results are being published elsewhere +. The measurements showed that the mobility of the electron in nitrogen was far higher than had been anticipated. It was further found that K’ was not * J. Franck, Verh. d. Deut. Phys. Ges. xii. p. 613 (1913); W. B. Haines, of ey (6) xxx. p. 003 (1915); KE. M. Wellisch, Am. Journ, Sci. July E7 ep. i. + To appear in the Physical Review, 1921. oe Dr. L. B. Loeb on the Relative Affinity a constant, but that it varied according to the equation 571,000 het 2S60N pd.) or (3) Here V is the voltage, p is the pressure in mm., and d is the plate distance. This result at first sight seems strange. Townsend *, however, had pointed out that for electrons makiny elastic collisions with gas molecules W, the velocity of the electrons among the gas molecules, must increase as a function of the field strength. If this occurs the mobility K’ of the electron will, according to him, decrease as the field increases. ‘These results obtained on electrons in N, show that it is possible to ascribe the apparent variations in n to real variations in K'; for they show that K' does in one instance vary as a function of the pressure and the field. It is very doubtful whether the above results in N, may be rigorously carried over to other gases for the purpose of evaluating n. Since, however, K’ has been shown to be a function of V/d, and of p, it is certain that in comparing the values of n for different gases, these values of » must be chosen from data for which the field strength and the pres- sure are as nearly the same as possible. In determining the values of n to be used in the table, care has been taken to choose values cbtained when the field strength is close to 20 volts per cm. and the pressure is as near 7) mm. as practicable. In the recent work on the mobility of electrons in nitrogen it was found that the value of K’ at a field strength of 20 volts/em. and a pressure of 75 mm. lies close to 2800 cm./sec. This requires that the absolute values of n heretofore determined on the basis of K'=200 must be modified. Furthermore, it is obvious that in order to apply the equation to the evaluation of n in other gases, where the value of K' is not known at all, some tentative value of K’ must be assumed. It is known that the mobility of the negative ion in different gases does vary by small amounts. In the evaluation of n from thesé experiments the value of K' for the electrons in the different gases will be taken as the value of the electron mobility in N, (e. g. 2800 cm./sec.) multiplied by the ratios of the mobilities of the negative ions in the gas to the mobilities of negative ions in N, K gas (aC ae gas=K'y, where K represents the ‘onic PASS K mobilities). The values of n, the constant of attachment * J. S. Townsend, Elect. in Gases, p. 124 (Oxford, 1914); Phil. Mag. {6) xl. p. 505 (1920). of some Gas Molecules jor Electrons. 233 of electrons given in column 4 of the table, were calculated on the basis of equation (1), using values of K’ estimated in the manner suggested above *. lf the equation of electron mobility suggested by Townsend t be accepted as correct it becomes evident that one is not justified in computing n according to the equation (1); for in Townsend’s equation the velocity of agitation of the electron among the gas molecules is nota constant. Also, ff from the view-point of this equation the value of \’ be computed from the results of the electron mobility measurements in N,, a value of 2’ which is several times greater than that estimated on the basis of kinetic theory results. The uncertain quantities \’ and W may be eliminated from the equation by substituting the value of W/)’ obtained from the Townsend equation, 815 en! jee | Oe TON in equation (1). In this equation ¢ is the electronic charge, m is the mass of the electron, and the factor 300 is put in to reduce ithe mobilities to volts per cm. The equation (1) then becomes 815 ¢ iB (ey K (a) & he te 300 nm(K')2 AV; Prom y (4) The factor K’ is then the only uncertain factor present in the equation besides the n which is to be determined. In the table of results the values of n computed on the basis of this equation are given in column 2. The preparation and purification of air, No, O, and H,, used in the determinations has been reported elsewhere f. The CO, was prepared through the action of HCl on pure marble. It was passed through a tube packed with NaHCO, then over heated CuO, and heated, finely divided, Cu. From there it passed through a drying train of CaCl, and P.O; tubes into the chamber through a trap cooled to —20° C. by frozen CCl, Absorption tests made with CaCl, failed to reveal the presence of any water-vapour. On absorbing the CO, in NaOH a residue of 1 ¢.c. of an inert gas was found present in about 250 c.c. of the gas. Tests with alkaline pyrogallol indicated that this residue was free from oxygen. * This changes the value of N for O, to 3-6 10° instead of 50,000 given in the original paper. + J. S. Townsend, J. ec. { ‘L. B. Loeb, Phys. Rev. xvii. No. 2, pp. 89-115 (Feb. 1921); Proc. Nat. Acad. Sci. vii. No. | (Jan. 1921). Lo 234 — Dr. L. B. Loeb on the Relative Affinity It was presumably nitrogen which had been entrapped in the pores of the marble. The N,O used came from a small commercial tank such as is used by the dental profession. It was bubbled through two flasks containing a concentrated fresh solution of FeSO,. From there it passed through two CaCl, drying tubes and a tube of P,O;, through a trap kept at —70° C. to remove the more condensible impurities, and finally into anotlier trap where it was condensed out by frozen alcohol. From this trap the liquid gas was fractionated, the intermediate por- tion only being admitted to the measuring chamber. The gas finally went through a system of NaOH, CaCl, and P.O; tubes before passing into the measuring chamber. In all cases the apparatus was filled by exhausting it to 20 mm. and then running in the gas to atmospheric pressure. When this process had been repeated from four to seven times, depending on what gas had been in the chamber before, the filling was considered complete. The measure- ments were carried out precisely as described in the earlier work on air. The methods employed by Mr. Wahlin as well as a detailed discussion of his results will be published sepa- rately by Mr. Wahlin. The values of n obtained are given in the Table on p. 235. The results show that this quantity » varies through an enormous range of values. In N, and H, it is doubtful whether the electron may ever attach permanently to these molecules. In Cl, as far as experiments can show, the elec- tron attaches within very few impacts to form chlorine ions. For other gases, n has all values lying between these limits. It is consequently obvious that in spite of the difficulties encountered in determining its value accurately, the constant of electron attachment n is, even in. its order of magnitude, a marked and characteristic property of the different kind of gas molecules. In a paper on the ‘‘ Arrangement of Hlectrons in Atoms and Molecules,” Langmuir * suggests that owing to the fact that the structure of N,O and CO, molecules is quite alike, their properties as influenced by this structure should be the same. ‘This is shown to be wrong as far as the value of m is concerned. The value of n for CO, when freshly prepared is 1°5x 10’, while that for N,O. is 671 x 10°. Furthermore, the value of » for CO, undergoes a rapid decrease with time after preparation. This change is not shown for N,O or for any of the other gases studied. ‘The * I, Langmuir, Journ. Am. Chem. Soc. xli. No. 6, p. 898 (June 1919): 239 of some Gas Molecules for Electrons. TaBLE oF ReEsuuts. Constant with time. Constant with time. Constant with time. Constant with time. Constant with time. Constant with time. Constant with time. Constant with time. Constant with time. Constant with time. n K N Limits of fom ionic from variation of 2 on Observer. Comments. Gas, equation (4), mobility. equation (1). equation (1). NE Serene es Infinite. 2°50 Infinite. =a Loeb. H, ee eae ee ee Infinite. wa AoW Hii Cs ees area me eee 38 Loeb. 010 Mar eer ee or 1:6 x 108 1:20 3°1 x 107 (1-6) x 107 Wablin. INSEL Gc victor danannn ch cities. «3 9°9 x 104 0°78 13 x 10% (0°7-1°6) x 107 Wahlin. ©; Hoear. aes tae eater 4°7 x 10° 091 Pixie (‘18-1°6) x 107 Wablin. (Oe 3 Uaeen tence a eee 78x 10° 1-15 15x 10° (0°8-2 3) x 10° Wahlin. OL ri. cag ss os Sansa 25 x 10° 1°30 5:3 x 10° (2'5-8'8) x 10° Wahlin. (KO gp niet Sots onreriaPrs ic 1:5 x 107 1:22 2°9 « 10° (2°3-4'3) x 10° Loeb. Freshly prepared. OOS. gibursace or 3°5 x 10° 1-22 6°9 x 10° a Loeb. 4 hours old. OOmSe ee. senders ek XL" 1:22 BOE SC = ae Loeb. 22 hours old. NAO eoireg.cSes aie. Senin os 61 x 10° 1°33 1:4 10° (0°8-2'3) x 10° Loeb. Freshly prepared. INE Or sings mabe ttew Het scart 3°6 x 10° 1°33 TES SCO tas Stim TR Loeb. 24 hours old. OPE AO a cens Sdusrase 377 X 10° 0°30 esc L0- (13-23) x 104 Wablin. Dee ee are ae 4-3 x 10# 2°50 1-8 x 104 (O'7—-6°4) x 104 Loeb. Oe ar ident Pre aie 8:7 x 10° 2°50 3°6 X 10° (1'4-5°7) x 108 Loeb. Toga se ae ty age ah oe ae Spi Wablin. a i | veel 236 Relative Affinity of some Gas Molecules jor Electrons. change takes place in the dark, and is similar to an effect found by Wellisch* when he studied the mobilities of electrons in CO,. He found that the mobility rapidly decreased in the gas with an increase of time after pre- paration. An even more marked effect of this nature was tound by Wellisch t in the vapour of petroleum ether. The small change in the value of x for N,O with time was accompanied with an increase in pressure of 13 mm. Such an increase in pressure was possibly due to a decomposition of the N,O. If this were the case the change in n would be quantitatively aceounted for by the oxygen produced in the process. It has been shown by Mr. Wahlin that the relative energy of impact between electron and molecule does not influence attachment. One may then conclude from the nature of the values of n obtained, that n is dependent on either the elec- trons striking a particular point in the atom in order to attach, or in its striking a molecule in some particular state of chemical or physical activity. Summary. 1. The results obtained in a recent study of the mobilities of electrons in pure nitrogen are discussed relative to the theory of the formation of negative ions from electrons and molecules developed by the writer. . 2. An equation is arrived at which gives the values of n, the constant of attachment, with a greater degree of certainty than the one previously given. The equation is, however, not yet satisfactory, owing to the lack of knowledge of the electronic mobilities in the various gases. ; 3. On the basis of the equation derived, the values of n for a number of different gases are given as computed from recent experiments. 4. The results show that in spite of the uncertainties in the determination of n it varies through such an enormous range of values for the different gases, that the order of magnitude of the quantity alone furnishes a characteristic constant of the gas. In concluding, the writer desires to acknowledge with thanks the assistance of Mr. Wahlin in the preparation and measurements made on the gases CO, and N,O. The writer’s thanks are also due to Prof. R. A. Millikan for his kind criticism of this paper. Ryerson Physical Laboratory, July 15th, 1921. * BM. Wellisch, Am. Journ. Sci. xliv. p. 11 (July 1917). t+ E. M. Wellisch, eed. p. 15. XXIV. Notices respecting New Books. Tables of Physical and Chenucal Constants. G. W. C. Kaye and T. H. Lapy. Fourth Edition. 160 pages. (Longmans, Green & Co., 1921; price 14 shillings.) oe test of the utility of these ables is the call for a fourth edition. Alterations and additions have been made relating to the figure of the earth, acceleration of gravity, and the gravi- tation constant; and the ehemical data have been recalculated, using the international atomic weights. The conventional niggling C.G.8. units are employed through- out. But it will do no harm to point out certain advantages in the M.K.S. system (metre-kilogramme-second), tacitly employed in electrical work units more usual for commercial purposes. A metre cube (m*) of water is here the metric tonne (t) of 1000 kg, and density is reckoned in kg/m’. The theoretician’s oljection to the M.K.S. system of making 1000 the density of water is a practical asset, as the last unit figure of the tabulated density of a substance is then affected by the density of the air, to the extent of about 1:25; the density given in the table being absolute density in a vacuum, from which this deduction must be made in a careful weighing carried out in air actually by the human machine, and not merely the mechanical balance alone. It is curious how this Hospitalier notation should be so long in making its way into use, recommended as far back as 1883 in an International Electrical Congress. No need then for that confusing chapter on Units and Dimensions; these are always kept in view by Hospitalier. In the M.K.S. units the joule replaces the erg, but no name has been found yet to replace the dyne, equivalent of 100,000 dynes ; this force would represent the weight of about 100 g, say 10 pennies, so 1t cannot be described as large, while the dyne and erg are too microscopic to be used except with high powers of 10 in the index notation. In matter relating to the figure of the Earth, the circumference should be stated, and the radius or diameter never be mentioned. It is not right in a treatise on Trigonometry for the use of the naval officer to give the radius of the Earth as about 4000 miles, and then not specifying whether it is the military land mile, or the nautical geographical (G) mile that is intended. Only the G mile should be implied, and then the circumference of the Earth is 21,600 miles, as laid down by Roger Bacon in the Opus Majus, and this makes the radius 3438 G miles, a radian arc of 57°-3, or 3438’. In the Metric System the circumference was made into forty thousand kilometres; and then in these units the slight varia- tions from the true sphere are expressed in a small variation from a large round number. 238 Notices respecting New Books. The mass of the Earth is given, and also of an estimate of the ocean; the mass of the atmosphere may be stated as the mass of an ocean of mercury covering the Earth, of mean depth the average height of the barometer. The chief value of these tables is in giving the latest measured value of a physical quantity to the number of figures warranted by experiment, and not to wander off i the region of the dishonest decimal. These accurate statements appear more elognene and convincing as a small correction on a large round number representing the accepted average value; and risk of error is diminished in the large leading fioures oi a result in this enormous mass of data. Stated in logical order, the length 7 of the seconds’ pendulum is the measured quantity, and g is derived from it by the relation - g=rl. Introduction to the Theory of Fourier’s Series and Integrals. By H. S. Carstaw, Professor of Mathematics in the University of Sydney. (Macmillan, 1921, 320 pages; price +40 shillings.) ALTHOUGH classing itself modestly as a mere Introduction to the subject, the work extends to over 300 pages, and costs 30 shillings; and a sequel seems promised in the future to a complete treatise. This is very complete at present in a rigorous examination of the nature of the convergency and continuity of the Fourier series, a succession of fluctuating harmonic terms, equivalent of the harmonies of the monochord, with frequency in the ratio of the integer i, 2, 3,...., or wave-length in the harmonic pro- gression of the reciprocal. These are the commensurable periods in the terms of a Fourier series. But in the Lunar and Planetary theory the various anomalies are resoived into terms of different incommensurable period, such as evection and variation. The Fourier series is a summation of the harmonics of commen- surable period such as those produced on the monochord, and the theory was born of the representation of any arbitrary musical vibration of the chord by a resolution into the pure harmonics. The legitimacy was disputed by the originators of the idea, d’Alembert, Euler, Bernoulli; but Lagrange settled the question on the modern point of view, although the difficulties and objections raised by Huler still exist, and form the bulk of the discussion in this treatise. The new mathematician is interested chiefly in what may be called the morbid pathology of the series, in its behaviour at points of discontinuity where a differentiation may be expected to break down, and the convergence requires examination, whether recular, uniform, or not. But the man who employs the series in an application to an electrical alternating current, or to the balance of quick running Notices respecting New Books. 239 machinery, will look at the question in a reverse order. This is the outlook contemplated by Professor Perry. He takes a few well chosen harmonic terms at the beginning of a Fourier series, and examines the shape of the resulting graph, to see if it is a zood imitation of the mechanical result under examination ; in a problem such as machinery balance or electrical surging. To raise the peak of a curve and to sharpen it if too blunt, an odd harmonic would be introduced; the effect of the even harmonic is felt in enlarging the haunches of a wave-arch. The figures in Chap. VII will give a lead in this treatment, how to smooth out ripples if too prominent. Starting. with an arbitrary Fourier series, such as on page 3, and denoting it by 34,+ .... +a, cos prv/l+b, sin praji+....=f(«)=y, then, in the interval ]>«>—l, the coefficient a, is the average y cos prax/l, and b, is the average y sin prv/l; and so Fourier’s theorem is stated without using the language of the Integral Calculus, in a form that appeals to a practical mathematician ; and the operation of averaging can be carried out mechanically by wrapping the curve: of y round a cylinder, once, twice, thrice, .... Simple statement of this kind is required in an appeal to tie interest of the beginner. In the applications, the w and 6} coefficients appear usually as rational numbers, as for instance in the representation of y=1—«2. But the Jacobian zeta funetion, zn cK, is penultimate to this curve, as «—>1; the curve, a straight line, is imitated witheut any ripples or discontinuities, and the Gibbs phenomenon of p. 268 is absent. | So too the curves of sn xK or en «K may be investigated in the penultimate form of «x—+>], in their representation of broken straight lines. Here the a and 0 coefficients are transcendental numbers. This is the method of the average, described and resumed in $ 95, but a further discussion is inaugurated, as the statement of the average is considered quite incomplete and inconclusive ly the author; and Dirichlet’s conditions of $91 are to be satisfied in the interval, leading to very delicate and abstruse consideration, in which the author spreads himself to his own delight. He will not allow a proper comprehension of Fourier’s Series and Integral without a knowledge of what is involved in the convergence of infinite series, and integrals; and the Definite Integral is treated in Chapter 1V from Riemann’s point of view. The volume is called an Introduction because the original work is to be divided into two parts, the second part devoted to a inathematical discussion of Heat Conduction, where the most interesting applications are to be found, sequel to Fourier’s Théorie de la Chaleur, of 1822. 240 Notiees respecting New Books. The Mechanical Principles of the Aeroplane. By 8. Bropgtsxy, Reader in Applied Mathematics, University of Leeds. (J. & A. Churchill, 1921, 270 pages; price 21 shillings.) Tuts treatise goes very deeply and thoroughly into the mechanical aud mathematical calculations involved in the design of the flying machine. | An introduction discusses the possibility of flight of a machine heavier than air, and might well be amplified to show the main facts involved, atter the resistance was taken to vary as the simple sine law instead of the squared sine, as required on Newton’s method of treating air as a cloud of dust particles, ignoring the equation of continuity. But as soon as the theoretical work of Heimholtz and Kirchhoff on discontinuous flow of a fluid pointed out the simple sine law as much more appropriate, it was seen that mechanical fight was in the region of possibility, and had only to await the arrival of an engine of weight per horse-power low enough. This was provided happily by the advent of the motor vehicle and its internal combustion engine as a present ready made, and then man could really take to the air, and not merely dream of it in the realm of poetical fancy. Such an introduction need not go beyond calculations to appeal to the young scholar, and would serve to attract him further into the deeper treatment given here, as in other complete treatises, such as Bairstow’s ‘ Applied Aerodynamics.’ Optical Theorres. By D. N. Matuix, Professor, Presidency College, Calcutta. Second Edition revised. 200 pages ; 16 shillings. (Cambridge; at the University Press, 1921.) A course of lectures delivered before the Calcutta University to advanced students on Optical Theories. Early speculations, on the corpuscnlar and the undulatory theory, are discussed in Chap. I. The elastic solid theory, electro-magnetic theory, and electron theory follow in Chaps. II, Ill, 1V. In this new edition a discussion is introduced in Chap. V of the new theory of Relativity, in its bearing on the modification of former theories of the ether. These various theories are summarised and compared, in a valuable discussion in eloquent words in Chap. VI, of the nature of the electro-magnetic field. In this way the true inner significance is brought to light of the accurate interpretation of the results concealed in the differ- ential equations and triple integrals of the rigorous mathematical treatment. Phil. Mag. Ser. 6, Vol. 43. Pl. I. a Hb) +4) 6 3 Gq @) E © frst Pressures , if ff nf N\A \ \ \\ \ \ NNW f | /\ / \ f Ny A f rH N MH) Fo Wi a ATR A SLATE ERRAND RAR RI RARE TN A JOFFE. Phil. Mag. Ser. 6, Vol. 43. Pl. II. Viera Lee ys ih 5 THE LONDON, EDINBURGH, AND DUBLIN PHILOSOPHICAL MAGAZINE AND Pee ae ee ZORA R py JOURNAL OF SCIENCE. Ps so rave 6 yt DEE pt Bip be UF A Fe ¥ G22, XXV. On the Application of Probabilities to the Movement of Gas-Molecules. By Prof. F. Y. Epgzeworts, F.B.A.* HIS is a sequel to the paper bearing the same title in ° the Philosophical Magazine for September 1920. The three arguments there employed to determine the distribution of velocities in a molecular medley are here reinforced. T. Laruace’s THEory oF Error. The first argument was based on the leading property of the law of error first stated by Laplace. The enunciation was facilitated by the fiction of molecular movement in one dimension. Two sets of perfectly elastic piston-shaped particles were supposed to be moving in one and the:same right line under stated conditions (loc. cit. p. 249). 1. The form of the frequency-junction.—It was shown that, whatever the initial distribution of the velocities U and wu pertaining to particles of mass M and m respectively, the system would through repeated collisions be ultimately distributed according to the normal error function gis fla) exp —(NU*E au). 2, . (1) That is, presuming that the two sets of velocities fluctuate independently. Otherwise the ultimate distribution can be written (N,/ Aaja 1—r?) exp —(AU*—27 V/ Aa+au*)/(1--1r?). (2) ~* Communicated by the Author. Phil. Mag. S. 6. Vol. 43. No. 254. Feb. 1922. N 242 Prof. F. Y. Edgeworth on the Application of This may mean that, if we select all the members of one class (in a unit of the medley), say those between U and U-t AU, and observe the velocities of those molecules belong- ing to a different set which are nearest to a member of the selected class, the distribution of the array thus presented will not be that of the u-velocities in general, but one more closely congregated about a centre, rU. The essential condition for this consummation is the linear relation between the velocities U and u of two particles about to collide and the resulting velocities U' and w’: U'=(M—m)U 4+ 2mu wor? 7. eee w =2MU0+(m—M)u if M+m=1 (loc. cit. pp. 250, 251). When we go on to two dimensions, considering first perfectly elastic circular disks moving over a_pertectly smooth plane, the essential condition still persists. But the coefficients of the linear function are not now as in the simplest case (3) the same for every collision. They involve a variable parameter, the sine or cosine of an angle 6, made by the line of centres at the moment of collision with the direction of the relative velocity (or with a fixed axis) (J. c. p. 259). But the presence in the coefficients of such a parameter hovering in random fashion about an average is not fatal to the genesis of the law of error. This was pointed out in the original paper (p. 262) with reference to a later stage, the impact between molecules with several degrees of freedom. The principle should have been intro- duced at an earlier stage ; and it might have been employed at a still later stage, namely with reference to encounters as distinguished from collisions. Consider molecules with several degrees of freedom such » as chains—bars linked by hinges—moving over a perfectly smooth plane. Let the generalized components of momentum Pi, Po... Pys Dy Po:.. ps be hanged’: by” amy impaemene Py, Pe’ .-.3 pis Po... Hach of the latter setqemayaao= regarded as a linear function of the original components ; the coefficients now depending not only (as in the simple case of disks) on the relative position of the mass-cenires (before the encounter), but also on the co-ordinates. Still the coefficients of the linear functions connecting the P’s with the P’s may be regarded as hovering about a mean value ; and accordingly the law of error will be set up. This reasoning reinforces the proof which was before offered in the case of encounters (/.c. p. 268); based on the mean eel ee Probabilities to the Movement of Gas-Molecules. 248 powers of a magnitude which is a function other than linear of numerous independently fluctuating elements. These conclusions and those which follow are readily extended from two to three dimensions. 2. The constants of the frequency-function.—The form of the sought function having been ascertained, we have next to evaluate the constants. [or this purpose we might obtain aid from one of the other arguments, from the third as before (1. ¢. pp. 252, 263), or from the second (below II. 1) if employed to determine the constants, the function being given. Otherwise, supposing the ultimate stable distribution of velocities to have been reached, observe the velocities in any assigned direction, e. g. U, for numerous specimens presented at random by molecules of the type M. Their average velocity in that direction ought to be the same as the average of any other large set of U-velocities, in parti- cular the set which is formed by molecules of the selected class meeting with molecules of the m type. This condition requires certain relations between the coefficients of the frequency-function. Inthe simplest case where the function is of the form (1) the U of any M particle colliding with any m particle becomes by (3) U’, where U'=(M—m)U+4+ 2mu: Since [U’?]=[U?|, square brackets denoting mean value, we have [2 == CM an)? (U2) 4m ws oe (4) the mean vaiue | Uw] being zero Gf the mass-centre of the system is zero). Now from (1) we have (W2[ =F 2A and “(a7 | = ly2a. Substituting in (4) and multiplying by 2A, we have 1=(M—m)?+4m’?A/a GE M+m=1). Whence a/m=A/M. Thus the index of (1) is of the form —)(MU?+ mu’). Shs incite spe) If the frequency-function is of the form (2) substitute 1/A(1—7?) and 1/a(1—r?) for 1/A and 1/a in the preceding argument ; and a similar conclusion will be reached. The index will now be of the form —A(MU?—2r /MmUu+mu’?)/(1—7?). . . (6) For the general case of molecules with several degrees of freedom we may use the formule for the components of momentum after impact between such molecules (/. ¢. p. 261). R 2 ~ 244 Prof. F. Y. Edgeworth on the Application of Transform the system by a linear (preferably orthogonal) substitution so that 'T the kinetic energy assumes the form BT? + BLT.? +... + 6)7y?7 + bore? +...5 2. « (7) where the B’s and 0’s are functions of the co-ordinates, the II’s and m’s are “momentoids.” The general expression for II,’, what II, becomes in consequence of a collision, in terms of the II’s, viz. IJ, —A,R, then reduces to * (A,B, I, == A,Bell. 55 eee —,b;77; —DNgbytt,— : A,B, +A,?Bo+ ao +776; + Nobo + soc ’s where Aj, Ag...Ay, Ao... correspond to the Ij, hy...4, 15... ete. as defined in the general case (. ¢. p. 262) = amd B,; By..:b;... correspond to By, Boy...by... ; except that the momentoids are not (in general) true (Lagrangian) com- ponents of momentum (a circumstance which dees not atiect the present argument concerned only with collisions and the attendant impulses). Forming the mean square for components of momentum (the mean obtained by supposing ~ the velocities to acquire all possible values while the co- erdinates and points of impact remain the same), and equating (II, | to [11,2], and remembering that the mean products (iL, 1,] vanish, we obtain —AN2A.7( 8 Bolles 1.7, Tle oo 4A,?A37(B, Bs [11,7] — B;{ I? ] W,'=1, 2A, —AA PA? (By; pT?) —b,? [71° | es mere. Ly 2|— —b,° [727] E- ee) Now it is eee (by If 1) that ‘the sought frequency-curve is of the form Const. exp — (©,1?+ C.1y? +... + ey? + com? + ...). Put Cie Bs 10) eee ce and the last written equation becomes — 2A)?A.?(B,/K,— B,/Ky) —2A)?A;3°(B,/K,— B,/K,) ... = DAP NAB Kaen) — =O Since this equality holds good for all the values which the A’s and X’s can assume as the points of contact vary (the co-ordinates remaining the same), it follows that K,=Ko =k Ke Probabilities to the Movement of Gras-Molecules. 245 Thus the frequency-function is of the form (8) Const. exp —XT; where T is the kinetic energy (7), and X is deter- mined by the condition that the energy of the system ne any assigned values of the co-ordinates is constant (/. c¢ p. 263}. ‘This reasoning may at once be extended to encounters in the case of motion in one dimension since the relevant equations (3) are in that case the same for collisions and encounters. In the case of disks on a plane the conservation of energy and that of momentum give only three equations for the four resultant velocities us Wralieg a fourth condition is presented by assigning ‘the ‘angle 20 through which the direction of the relative velocity is turned by a collision or encounter. ‘There can therefore be always arranged a fictitious collision which has the same effect in changing the velocities as any given encounter. Replacing each eneounter by the equivalent collision, we can transfer to encounters the conclusion which has just been proved for collisions. An equivalent collision is not so simply obtained when there ure several degrees of freedom. Let us pass to this case by supposing one set of disks to have its centre of mass different from the centre of shape. The angle between the line joining those points and a fixed axis, say @, defines the phase of an encounter. We may now conceive a second disk of the other (symmetrical) type to be in con- tact with the first at any point on the periphery thereof. But we thus obtain pony four equations to determine the five quantities U’, wu’, V’, v’ and ¢’ the changed velocity of rotation. To obits another equation let us artificially create a second point of impact by affixing a sort of buffer or tentacle (of negligible mass) to the first disk and arranging that there shall be simultaneous impacts at two points. Generalizing the formule given for Pa at one point (1. c. p. 262), we shall now obtain II,’, H,!...7,' in terms of TI,, Hg,...a,... and two sets of the sont cents -called A, 2, each set depending on the position of a point of contact. We have thus with the three conditions given by the con- servation of energy and that of momentum two additional variables, whereby to secure an impact equivalent to the encounter. It would.not, however, be an exact equivalent. For the angle @ would in general be changed by an en- counter, but not by the impact. But since the values of occur oe uniform distribution, the frequency-distribution of the actual system would 1S. the same as that of the artificial system. The reasoning may be extended to the general case where there are several “ momentoids.” eeu dP roksan sy X. Edgeworth on the Application of 3. Correction of the normal formula.—There is reason to believe that the conditions for the genesis of the normal law are not perfectly fulfilled by a molecular chaos. With "regard to collisions, the elasticity can hardly be regarded as perfect (Cp. Burbury, Science Progress, vol. ii. $7). In the case of encounters, the beginning and the end of the encounter being defined arbitrarily, it is to be apprehended that there may be some slight interaction between the molecules. Both in the case of collisions and encounters the ultimate velocities have elements in common (I. ¢. p. 251). Ii the independence of: the elements aggregated and the linearity of the aggregation is not perfect, the function proper to represent the frequency of the aggregate would no longer be the simple normal law, but the corrected form described by the present writer as the generalized law of error (Cam- bridge Philosophical Transactions, 1904; Journal of the Royal Statistical Society, 1906). The first correction, involving odd powers of the variables, may presumably be neglected. The formula given by second correction may be written in the simple case of motion in one dimension Fx {14e(4—2MU0?+2W’U4) + 8G — 2mu? + 2m?u' ) +y1—2M07)(1—2mu7)t, do where F is the normal function (1), «, 8, y are small co- efficients proportional to the excess of the respective mean powers [U*] [ut] [U?u?] over what each would be if the normal law were perfectly satisfied (Camb. Phil. Trans. loc. cit. p. 118, 1904). | The above expression, which may be written asa function of U multiplied by a function of u, if y=0 («8 being negligible), need not express correlation between the velocities. What slight interdependence there may. have been among the elements—the long series of velocities from which the U and win (1) resulted—may have disappeared in the compound (Camb. Phil-Trans. p26); 1904): 4. Correlation.—The correction above made may, how- ever, co-exist with (normal) correlation. Our first argument is not inconsistent with Dr. Burbury’s contention that there may be some interdependence between the velocities of con- tiguous molecules (Science Progress, vol. i. § 33, 1894 ; Phil. Trans. 1886 : “Dense Gases, § 13; ‘ Kinetic Theory of Gases,’ p. 11; et passim). But there are two cases of inter- dependence which should be excluded from the field to which (2) applies, namely (a) between molecuies in an initial distribution, or on their way to the ultimate normal Probabilities to the Movement of Gas-Molecules. 247 distribution ; (6) between molecules engaged in an encounter (referred to below, III. 3). 5. Controverted points.—Our first argument has a bearing on some vexed questions. Thus with respect to ‘“ time- averages” it would hardly occur to one impressed with Laplace’ s theory of error to question Lord Ray leigh’s state- ment that “for a single particle the time-averages of (wv and U*) are equal, provided the averages be taken over a sufficient length of time” (Phil. Mag. ser. 5, voi. xlix. (1900) p. 108). Again as to the reversibility of the motion, consider Galton’s mechanical illustration of the law of error. Shot is poured in through an aperture at the top of the apparatus and comes out at the bottom after repeated collisions with inter- posed obstacles in the form of the normal curve (‘ Heredity and Genius,’ p. 63; Yule, ‘Theory of Statistics,’ p. 298). Need we trouble ourselves with the thought that if the shots were dropped into the apparatus in numbers corresponding at each point to the normal distribution (say by turning the mechanism upside down) they might come out, after re- passing the obstacles, in exactly the same arrangement as that in which they first entered. Recognition of the leading law of Probability disposes the disciple of Laplace to accept the warning of De Morgan, repeated by Tait with reference to the Kinetic Theory of Gases (Transactions Roy. Soc. Edinburgh, vol. xxxiil. p. 256): “No primary considera- tions connected with the subject of probability can or ought to be received if they depend upon the results of a complicated mathematical analysis.” Il. Tae “H” THErorem. The second argument is based on the use of *‘ H,” defined as the integral between extreme limits of flog f, where / is the songht frequency-function of the velocities (or the same multiplied by, or with the addition of, a constant, l. c. p. 253). The argument is to be distinguished from those in which “HH” is employed together ‘with the premiss that the frequency-distributions of the colliding molecules are inde- pendent, e. g. by Boltzmann (‘ Gastheorie,’ I. 5) and by Jeans in his first appeal to the “H” theorem (‘ Dynamical Theory of Gases,’ 3rd ed., Art. 15- et seg.). Without that assumption it is argued that the sought function f is that which makes “ H” a minimum subject to the constancy of the energy (and momenta) of the system. 1. Character of the argument.—The method is akin to the use of Probabilities for the determination of constants per- taining to functions of a given form when the data consist 248 Prof. F. Y. Edgeworth on the Application of of percentiles (l. c. p. 254 and references there given). Both problems involve the following fundamental principle. Suppose a sample numbering N is extracted at random from an indefinitely large medley of things, divided into classes such as (sets of) balls numbered 1, 2, ... m and mixed in proportions vy: V2... Ym—12 Vm, Where Sv=N. © Let. F(2,, 9... %m3 V1, Vo... Vm) be the probability sihat from a given set of v’s there will result a specified set of n’s (and accordingly F(14, vo ... vm; 1, M2 .--Mm)—the variables and constants changing places—the probability that a given set of n’s will have resulted from a specified set of v's). Then if the extraction is subjected to a condition which the ns must satisfy (sets not satisfying that condition being rejected), the most probable set of n’s, corresponding to a given set of v’s, is found by making F (1, no, ...3 v1, ve .--) a maximum subject to the imposed conditions (with a like proposition relating to v’s inferred from n’s). Fer instance let N=30 ; and vy, =v,=v3=10. And let there be imposed on the n’s the condition ny=2n,. Wehave then to maximize the expression (1/3)°° 30 !/ny! ng! ng !+A(n— 2n1) + u(r, + 72+ n3— N), where X and w are indeterminate multipliers. The required set is found with the use of a Table to be ny =7, n= 14, n3=9. Let 1, mg-... be all large. Then by Stirling’s theorem, log F(m, ng ... v1; vg ...) becomes approximately Const. —n, log ny— nz log ng... +7, logy, +nyglogv,+... (10) The variable part of this expression may be called H as being in eodem genere with the “H”’ above defined. (In the inverse problem of percentiles where the v’s are to be deter- mined subject to the condition that they are given functions of certain sought quantities, it is presumed that the differ- ences (v—n) are small; whereby Tavlor’s principle becomes available, and H reduces to the Pearsonian xy”; l. c. p. 254.) Let the given v’s be all equal, the latter terms of (10), n log v1 + ng log vg..., thus becoming absorbed in the constant part (since {n=N). Let the n’s be-represented by columns standing on equal bases, numbered 1, 2, 3... 7, r+1,.. on the right of a central point, and —1, —2, —3... —7, —(r+1)... on the left. Let there be imposed the condition > Mb?(n, + n_,) =R? (Ra given large number) ; the summation extending over all possible values of n. We have thus to minimize Sn, log n+ En_, log n_, 7 + X(2Mr?(n,+n_,)—R?) +4 (2(n,-+n-») —N). Probabilities to the Movement of Gas-Molecules. 249 Whence log n,=log n_,=Const.—AM?r”. Pa 0 7A | AS AAU; R7(AU)?=N2E, and there results for the distribution of ie PoveCorist.exp —AMU "20. (12) - where A is to be determined by treating AU as a differential, multiplying (12) by U? and integrating with respect to U between +0 and —o, and equating the result to N2E (the outside constant securing that }n=N). By parity if U and ware selected by arandom sortition from an indefinite number of values demarcated respectively by the small finite differences AU and Aw, subject to the condition >> (MU? +4 mu?) AU Au=constant, the most probable distribution of U and w—the one which in a prolonged series of trials will be most frequently attained, and around which the sets of N will hover—is AU Au Const. exp —A(MU?+4+ mu?) . . (11) When differentials are substituted for finite differences this may be described as the function which minimizes “ H,”’ subject to the imposed condition (expressed as an integral). 2. Legitimacy of the argument—May we regard the velocities in a molecular medley as analogously determined by a sortition subject to the conditions imposed by Dynamics? The high authority of Professor Jeans may be appealed to in favour of this view (‘ Dynamical Theory of Gases,’ 3rd ed. § 54 et passim). He confirms too the connected proposition that the distribution of velocities in the medley tends to and hovers about some altimate torm. ‘Tait has thus expressed the presumption in favour of this proposition : ‘* Kveryone who considers the subject ....must come to the conclusion that continual collisions among our set of elastic spheres will ....produce a state of things in which the percentage of the whole which have at any moment any distinctive pro- perty must (after many collisions) tend towards a definite numerical value; from which it will ever afterwards markedly depart”—with certain reservations (Trans. Roy. Soc. Edinb. vol. xxxili. p. 67). Professor Jeans confirms this presumption when he shows, with the aid of Liouville’s theorem, that certain incidents adverse to the presumption are not to be apprehended (Hucyclopedia Britannica, ed. 11, Art. Molecule, p. 658; Phil. Mag. ser. 6, vol. vi. (1903) p. 722). 290 Prot. F. Y. Edgeworth on the Application of 3. Analogy of the law-of-error in general statistics—The legitimacy of the argument is confirmed by its use outside molecular dynamics to prove that the normal error-function is the ultimate form approached by the continued super- position of independently fluctuating statistics ; subject to the condition that the. mean square of the compound is constant (/. c. p.256; and ep. p. 269). - The analogy throws light on the implicated question whether there be an ultimate form and its connexion with Liouville’s theorem. The existence of such a form is presumable on grounds of common sense. But hesitation may be felt when we deal: with an unfamiliar case. Suppose MU to be the amount of money possessed by a person of one class, and mu by one of another class; and that on a deal between them U becomes U’ and u, u’, in virtue of the conditions MU#+mut=MU"4*4+ mu? and MU +mu=MU' 4+ mu’ (M and m constants). Let there occur an immense number of such transactions ; while the sum-total of MU + mu, and likewise of MU4+4 mut, in the society remains constant. May we apply the second argument and conclude that the money will be ultimately distributed among the individuals according to a law of frequency of which the logarithm is Constant —r (MU*4+ mut) + w(MU + mu) ? The answer is supplied by Liouville’s theorem. Ifa stable state is attained, the frequency-distribution for U and uw must be the same as for U! and wu’, the values into which U and u pass by a transaction of the kind supposed. Now the frequency-distribution of U' and w’ (given functions of U and w) is obtainable from that of U and u, viz. (hypothetically) AU Au Const. exp —A(MU?# + mut) + wCMU 4+ mu), _ by substituting in the integral part of the above expression for U and w their values in terms of U’ and wu’, and for the differential factor AU Au, AU! Au'( dU du aU du ) aac du’ a): The first substitution leaves the integral value unaltered. Accordingly it is necessary to stability that the bracketed factor of AU’Au’ in (13) should be identically equal to unity : that is, that the theorem proved by Liouville and his followers for a conservative dynamical system should hold good, ~ To apply this criterion to the proposed problem; let us (13) Probabilities to the Movement of Gas-Molecules. 251 first try to find linear functions for U and uw in terms of U' and uw’, say U=AU’+Buw’ and u=aU'+bu’. The Liouville condition gives one equation for the coefficients, viz. Ab—aB=1. Two more equations are given by the identity MU +mu=MU'+ mw’; and five more by the identity MU! + mt= MU" + mut. Though these eight equations are not all independent, it will not be pessible to find values of the variables which satisfy them all; even admitting imaginary roots. Tf the value of U, and lkewrise u, in terms of U’ and w’ is other than linear, it appears necessary to increase the number of equations, with the result that a solution is unattainable. It is therefore significant that when the conditions im- posed are the conservation of energy and of momentum, the condition (13) is fulfilled by the expressions for U and wu in terms of U' and w’ in the simple case of one dimension (1); and likewise in more complicated collisions and encounters. 4. Dense gases.—The analogy of general statistics suggests a warning against the danger of supposing that the premiss of independence (above II. 1) ignored by the second argument is otiose. Consider a normal surface formed by the super- position of n two-dimensioned elements with laws of frequency G&=dilbi,m), So=po(E, m)...- - - (14) Let G&,, 1%), (c&,, oM,)... be successive concurrent values of the pamables pertaining ‘to d, ; and let a= £1 + s+ see = eae Y =o + M+ tee +r sn 3 the &’s and the 7’s and accordingly the aggregate w’s and y’s being measured from their respective average values (Camb. Phil. ‘Trans. p- 116, 1904). Given that the mean value of Mez?+my? is constant, we might conclude by the second argument that the frequency-distribution of w and y was of the form Const. exp —A(Mw?+ my’). But if the values of € and 7» in each element are not independent, the distri- bution of the #’s and y’s will not be of that form, but of one like (2) above, containing the product of the variables in the index. Yet the argument is not fallacious ; it gives the right answer relating to the data, which is all that can be expected from Probabilities (Cp. Keynes, ‘ Probability,’ ch. 1, et passim). The answer would be true on average in the long run of different instances if positive and negative correlation were equally probable (a supposition perhaps not relevant here). It is true of any particular case on the supposition that we no longer tabulate each y against the with which that yi is formed concurrently, but having dispersed 252 Prof. F. Y. Edgeworth on the Application of the y’s take an and any y at random. These considerations may assist in interpreting the proposition that ‘the law of distribution of velocities....remains the same right up to the extreme limiting case in which the spheres are packed so tightly in the containing vessel that they cannot move ” (Jeans, ‘ Dynamical Theory of Gases,’ $57, 3rd ed.). The law of distribution does not hold true of contiguous molecules in the same sense as when the independence of the velocities is postulated. The proposition may be equally true, but it is not equally informing as that which is based on the postulate of independence. Ill. On tHe Lines of MAXWELL. The third argument is based on the incidents of collisions and encounters without the aid of “H.” Itis nowattempted to prove by this argument that the normal distribution is necessary (as well as sufficient; l.c. p. 256). The attempt is discouraged by high authorities (Boltzmann, ‘ Gastheorie,’ i. § 5; Watson, ‘ Kinetic Theory of Gases,’ § 14); but it is countenanced by Maxwell when he argues that the normal distribution of velocities is uot only ‘‘a possible form,” but ““the only form” (“‘ Dynamical Theory of Gases,”’ ‘ Scientific Papers,’ vol. ii. p. 45). The proof which he gives in that context is indeed very different from that offered here ; which is, rather, akin to the reasoning in Maxwell’s paper “On the final state of a system of molecules” (loc. cit. (dn DONO 1. Proof that the normal distribution is necessary to stability.—Let us begin with the simple case of disks moving in a plane (as above, I. 1): two sets of disks of mass M aud m respectively, and radius at first supposed the same for both sets, say R. Let N be the number of disks of each type in a unit of area; the unit being taken so that N is | large. The total area occupied by disks is 2NwR*. Call the ratio of this to the unit of area p, supposed a small fraction, say 1/1000 or less. On a view of the medley at any instant the number of disks which may be expected in anassigned area Ais pAz/4. Ifa small area, say a multiple much less than a thousand of R?, includes a disk of one sect, the frequency with which (at the moment of inspection within a unit area) it will include a disk of the other set is of the order Np. Likewise the probability that a disk taken at random should have in’ close proximity two disks (of assigned sets) is of the order Np’. These propositions are Probabilities to the Movement of Gas-Molecules. 253 easily extended to the case of radii and numbers which are not the same, only of the same order of magnitude. Each set of disks is divided into classes defined by rates of velocity. Thus one class consists of disks with velocities respectively between U and U+AU, V and V+AV. The distribution of these velocities is represented by the frequency-function F(U, V) ; meaning that in a unit area the number of disks of the class specified is NF(U, V)AUAV. The function f(u, v) is similarly related to the other set. It is required to determine the functions F and / so that the distribution of velocities may be stable. Observing the unit area, note all the couples of disks which are in such close proximity that within the short time + (measured forward or backward from the present instant) they either will come into collision, or have come into collision. The species of couples so defined presents two sections according as the partners are consilient or dissilient, say positive and negative seetions. Now let each species of couple be divided into varieties (J. c. p. 260) defined by the mutual orientation of the disks which are just coming into, or from, collision. Saya variety consists of those couples for which the point of impact on one of the pair is between s and s+As, s being the length of an arc measured from a fixed point (rotation being ignored). The content (the number of instances in a unit area) of each variety thus defined will not be the same; but whatever it is it may be assumed to remain constant ; on the hypothesis of a random distribution irrespective of position in space. Let us designate the positive (consilient) section of a variety (U, V; u,v)s and the corresponding negative (u,v; U, V)s. Let U’, V' and w’,v' be the velocities which result from the collision corresponding to the positive section of the specified variety. Then the new velocities will belong to the section (u',v’; U’', V')s: the negative section of a variety which may be termed the reciprocal of the former. Observing the medley during the short time 7, we may expect that the whole initial content of the positive section (U,V; u,v)s will have passed into the negative section of the reciprocal variety. Less than the whole content could not have passed out of the specified section unless one of the. disks included therein were before collision with its partner knocked away by a third disk. But this could only happen through the very improbable double event of a third disk being initially in the neighbourhood and also in such a position and with such velocities as to hit one of the partners within the time rt. More than the whole initial content 254 ~— Prof. F. Y. Edgeworth on the Application of ; could not have passed from the section (U, V ; wu, v)s, unless during 7 a new positive couple of that variety had been created. But this could only happen through a very improbable event; not only must there be initially (within the unit area) a disk belonging to the same class as one of the specified variety—say of mass M with velocities U, V— with two disks in the neighbourhood, one of them of the m-° type, but also the position and velocities of those second and third disks must be such that they shall collide with each other during the time 7 and one of them of the m type should acquire the velocities u,v and the orientation with respect to the first disk which the specified variety connotes. ' By parity of reasoning the whole initial content of the reciprocal section (u,v; U, V)s will have passed out of that section during the time 7; and the final content of the reciprocal section will consis: of the disks comprised initially in the section (U,V; u,v)s. But on the hypothesis of stability the final content of the reciprocal is equal to its initial content. ‘hus initially and constantly the content of (U,V; u,v)s is equal to that of (u',v’; U',V)s. Let the relative velocity of the partners in a positive section be w ; and accordingly in the reciprocal negative, —w. Let the breadth of a section be e=Ascos 0, where @ is the angle made by the line of relative velocity with the line joining the centres at the moment of impact. Then, if at first the distributions I and / are supposed independent, we have, equating the content of the section to that of its reciprocal, wrahU AV AuwAv FE (U, V) f(u,v) =wra dU’ AV! Au’ Ad! F(U',V') fw’, 0’). The differential factors, the product of small finite dif- ferences on the. two sides, being equal, by elementary dynamics (/. c. pp. 207-267), we obtain the equation F(USY) faoeaPaus V") fase e subject to the condition M(U"?+ V) + m(w?? 4-0?) = M(U? + V?) -+ m(u? +e’). ‘The well-known solution of this functional equation is F(U, V)=Const. exp —AM(U? + a f(u,v) =Const. exp —Am(u? + v”) 3 where A is a constant to be determined from the given mean energy of the system. But if the independence of F and fis not assumed in the Probabilities to the Movement of Gas-Molecules. 255 premisses, for F(U,V) /(u,v) substitute y(U, V, u,v) ; and, putting x for log , solve the functional equation v(U, V,u,v)—x(U', V’,u’, v') +2r>{M(U?+ V*) + m(w? + v’) | —M(U?+ V”?) —m(u? + 0”) } =0. Differentiating twice with regard to each of the variables U, V, u,v, we have (5h) —-”AM= (+) (5%) =—2\M= se du? dV?)? \de 2 2 Also i == 0); Be he dU dV _dudv Whence y= Const. — A{ M(U?+ V?) + m(w?+v°)} +a linear function of the variables, which disappears, if, as here generally, the mass-centre of the system is assumed to be at rest ; or otherwise can be adapted to uniform motion of the mass-centre (/. c. pp. 253, 257). Next suppose that in addition to the motion of the masses there are internal movements designated by generalized co- ordinates. As above, the species and varieties of couples may be defined by the velocities of the mass-centres and the place of the point of impact on the contour (of one of the molecules). But that point does not now in general move in a right line parallel to the mass-centre; not like the point of a lance at a tournament, but, rather, like a point on the edge of a sabre which the dragoon whirls as he charges. Collisions therefore will now be divided not only according to the velocities, the classes of the colliding molecules, but also according to the values of their generalized co-ordinates, their genera (l.c. p. 263). IE now stability is defined as steady distribution of velocities for each particular set of values assumed by the co-ordinates, then we may reason as before that each variety of each class of each genus must remain equal in content to its reciprocal ; and therefore that, the velocities must be distributed according to a normal law of error of which the variables are (squares and products of) velocities and the coefficients are functions of the co-ordinates. But if the steady distribution of the classes in euch genus is not zmplied in the definition of stability, it may be deduced as before (J. c. p. 263) from the definition of stability as steady distribution of classes on an average of all the distri- butions in the medley. When we pass to encounters the argument is not materially affected by the circumstances that the change of velocities is 256 Prof. F’. Y. Edgeworth on the Application of not now instantaneous, and that the reciprocal couple consists of different genera as well as classes. But it can now no longer be taken for granted that the differential factor of the frequency-content relating to any species of couple—of the form AQ, AQ»... AUAV AQ, AQ»... Aq Ags... AuAvAdG Ad. (J. c. p. 267)—(where U, V,u,v are velocities of the mass- | centres and the other symbols relate to generalized co- ordinates) retains its value unchanged after the change which is signified by affecting each of the above symbols with a dash. To secure the equality of the differential factors before and after an encounter, recourse must be had to the theorem of Liouville. In general, in order that the distribution of velocities should be stable, it 1s necessary that it should obey the normal law of frequency in two forms: one form (proper to each genus) in which the coefhcients of (the squares and products of) the velocities are functions of the cv-ordinates, and another form (pertaining to the medley as a whole) in which the coefficients do not involve co-ordinates. 2. Proof that the normal distribution 1s suficient.—To show that this distribution is sufficient, as well as necessary, recourse must be had to @ priort probability (U. ¢. p. 257): the presumptions which attach to the hypothesis of random distribution. Of this kind is the generally admitted pro- position that the frequency of a class is not altered by changing the signs of all the components of velocity of momentum (/. c. p. 266). Thus in the simple case first ‘considered where the section (U, V ; wu, v)s passed by collision into (w', v'; U', V’)s; the latter section is equal in content to the section designated by the same velocities with signs reversed, in our notation (—U', —V’; —w!, —v')s; the capital letters now preceding as the section is positive (con- silient). By our third argument (and simple dynamics) the last written section passes by collision into (—u, —v; —U, —V)s; which is @ prior equal to (US Vea as Thus the content of the section first considered is the same at the end of the interval 7 as it was at the beginning. ‘The contents of this section and of every other section thus remain constantly the same if the normal law of frequency holds good. This proof may be transferred to the general case of several degrees of freedom more directly than appeared before (J. c. p. 269). 3. Correlation.—There is of course interdependence between molecules engaged in an encounter. Thus in the Probabilities to the Movement of Gas-Molecules. 257 simplified case of two dimensions, if U and wu be the velocities prior to encounter let U;,,w, designate the velocities just after they have begun to influence each other sensibly ; let U2, uw. ; Us, ug... be the pairs at later stages ; stages being measured by equal intervals of distance or, better, of time. Given the law of repulsion, we can (theoretically) calculate U; and u; the velocities at any time (say before that of least distance between the mass-centres) in terms of U, u,and t ; and thence U, w in terms of U;,, %, and ¢t. To obtain the correlation between U; and «% from the frequency-function of U and wu, viz. Const. exp —A(MU?+ mv”), it is proper to substitute for U and w their values in terms of U; and w; (we need not trouble about the differential factors, since, by Liouville’s theorem dUzdu;=dU;' du). There would result a complicated expression showing interdependence between the velocities at any assigned stage of the encounter, but not normal correlation. But if the averages of the velocities over the period of approach (or regression) are respectively ,U and ,u, then between those variables there will be normal correlation (of the form (2)). (Cp. above (14) and refer- ence). Correlationin this sense may be expected throughout that small part of the field which is occupied by molecules in encounter. 4. Relations between the arguments.—The third argument proves in one respect more, in another less, than the first. The third argument proves that perfect stability cannot exist unless the velocities of the mass-centres are distributed according to the normal iaw of frequency in its simplest form without terms implying correlation between those velocities (of the kind shown in (2)). The first argument cannot prove this ; but it proves that normal distribution will be set up approximately at least in the medley. Whereas the conditional conclusion of the third argument does not by itself confer the power of prediction. Could data such as those which form the premisses of the third argument be supposed, and yet stability not result? Certainly a non- conservative system may be imagined, such that, in addition to the conservation of Momentum MU+ mu (in the case of one dimension), there should be secured the conservation— not of energy, but—of the quantity MU*+mu*. It might prima facie be argued as above that the system could not be perfectly stable unless the velocities are distributed according to the law of frequency Const.exp —(MU*+ mut). But there is not fulfilled the Liouville condition required by the third as well as the second argument (II. 3). The second method has an advantage over the others in the Phil. Mag. 8. 6. Vol. 43. No. 254. Feb. 1922. Ss 258 Mr. A. 8. Percival: Method of readiness with which it is adapted to the case of external forces. We have only to put as the imposed condition the constancy—not of the kinetic energy as above throughout, but—of the total energy (Cp. Jeans, Phil. Mag. v. p. 617, 1903). Of course the acceleration must not be of such magnitude as to mask the random character of the molecular motion (Cp. Watson, ‘ Kinetic Theory of Gases,’ p.35). The second argument, however, is perhaps not so incontrovertible but that it may be indebted to the others for some corrcbo- ration. The three arguments are mutually complementary. XXVI. Method of Tracing Caustic Curves. By A.S8. Perctvar, M.A., MB. Camb.* A CAUSTIC CURVE is the locus of all the primary focal lines formed by the intersection of two contiguous reflected or refracted rays. Many mathematicians have devoted their energies to the problem of discovering the general equation of a Caustic Curve, but without success as far as | know except in a few special eases. The method usually given in the books is first to find an expression for the refracted wave-front, which is a Cartesian oval, and then to find the evolute of this curve. This is a most tedious and laborious proceeding, but in this paper I submit a simple method which will enable one to trace the Caustic due to either reflexion or refraction at a single spherical surface with ease, expedition, and accuracy. As corresponding points can be marked off after refraction at another surface, the Caustic formed by a lens ean be plotted out ina reasonable time. Now different lenses show differently shaped Causties, and it will be found that the general rules for the size and position of the Least Circle of Aberration are by no means true when a lens is used with its full aperture. Refraction at a Single Spherical Surface. Let S be the source of light ; consider the ray SP making an angle @ with the axis SCA, incident at P and refracted as PR. Let RP produced meet the axis at T. From C drop the perpendiculars CM and CN on the incident and refracted rays respectively. * Communicated by Professor J. W. Nicholson, F.R.S. . bo Or to) Tracing Caustic Curves. Let CSP = 86; 1 8C=a, CP=*7, CPM=g¢, Ni 2 eos a 5, CPN=¢q’, CN=p= 7 sin 6’, CTP 4, CM=pp=rsin ¢. Hie 1 O20 aio) ) = 1523, 0. d. ¢'. n. ae Upalee ‘oe | p- ' 0 0 0 5 55325 | 105325] 0 0 | {10° 10’ 55" | —45° |— 27° 39’ 50""| 4-4284 | 35420 | 7-9704 |—2°3214 |4-9766 ! | | | | (i.) From the figure it is seen that p=0+46-—¢’. (The magnitudes are considered as simple geometrical quantities without any vectorial sense of direction.) Differentiate (1.) with regard to wf. dd dd dd! / — — Le: (1.) BN aii, bday ae But sin a= FE, sin gat, sing! =. ag “jp AG a pe ee an =a remit? Ger 7 OP a pet dp y ! = when p'= pe from (1'.) a7 = Eee 0+ “sec — = see Been esti) |, (ai) (i) or p Bh ~ pr sec 0+ jua sec p—asec di S 2 260 Mr.-A. 8. Percival : Method of Differentiate (a) again with regard to yy: me = sec? 7 sin ee Zi “sec” sin “t = rey sin I sect g MPMP + Hsect pHP MP” a ech p'LE ,a¢g' a Yh r Yad fc alll Bef [utr? sect 8+ 42a? sect — a sec 6]. htf (iii. ) p=ptp" =p(1 - = Be E _ per’ sec’ 0+ pa? sec® 6 —a* s ae (ur sec 0+ wasec 6—asec g')? 1 Now if C be regarded as the origin of the p, wy equation to the Caustic (which it is unnecessary to find) as fNP is the tangent to the Caustic at the point f, and as /N=p’ and NP=n, the distance /P is given by p’+n where n=r cos @’. Hence any point f can be readily found on the Caustic and the radius of curvature at that point is given by (iii.). On now introducing signs of direction so that the formulee shall hold universally when the symbols carry signs of direction, it will be noted that in the diagram every magni- tude is positive except ¢, @ and p or rsing’ ; consequently in (i.) the signs of @ and ¢’ must be reversed, and so we obtain (1) p=O-G4+ 9". It will be convenient to regard » as measured always in the direction CN, whatever that may be: for p (O’f in fig. 1) the radius of curvature is always parallel to CN, and when p carries the same sign as p the radius is drawn in the same direction as CN, but “when p carries a sign opposite to that of p it is drawn in the direction NC. When the incident light presents a plane wave-front, §=0 and a becomes infinite, but in that case it m sec = + sec Oe Pp a@cos 0 2 7, ’ = / r SO 2 = psec b—sec years MMe 2) and p=pii-F soa Pe ae (je See @ —sec If the Caustic by reflexion at a spherical surface be con- sidered, we have only to substitute —1 for m, and put oS =u: Tracing Caustic Curves. | 261 In the case of a plane wave-front ¢ must be taken as the variable. In the general case when @ is given, sin d= — “sin 0. The formule for all cases can then be tabulated as below. TABLE I. p=rsngd, n=rcos ¢’ ; Peat p REFRACTION. | REFLEXION. | | p=0—o+t+¢'. (y=8 +29. kee ar oe —ar is psec. 0 +a sec ¢—«a sec 9!” & ~_7rsec 8+ 2a sec ¢! * 7 sec® 0+ 2a? sec’ ¢’ | e=P} ~ (rsec 0+2asec g')3 [° ape p73 sec® 8+ a3 sec? ¢—a* sec} 6’ ig { (pr sec 0+ ua sec ¢—a sec ¢’)3 } P + Plane Wave-front. - Plane Wave-front. vp -o- P — psec o—sec o'* os he pH? sec? 9 —sec? | { 2 sec? ¢! ae (usec d—sec@’)® alae (22sec 9’)? J nae Ts | —_?7 . ; '— = —ZN, se fP=in. Some hesitation may be felt in accepting these results without further examination ; but all doubts will be removed on showing that 7 P as here obtained is identical with v, as given in the usual formula :— pcos? dé cos’*d pwcosd'—cosh i ee r Ve pur cos? d' or 0S oS u(ut cos d —cos d) +7 cos? h On referring to fig. 1, SP or wis seen to ber cos 6 +4 cos 0. eel 3 ar eS: pr sec 0+ ua sec db—asec ¢' _ pr’ cos d' sec 6+ war cos ¢' sec h — ursec@ +a sec d—a sec ¢’ pr? cos? b' cos d+ war cos? d’ cos 7 ~ wr cos ¢ cos f’+ ua cos 8 cos d' —acos 9 cos d pur cos? d’ ~~ 1p COS d+ 7 cos? 6—uUcos dav — eS 262 Mr. A.S. Percival: Method of Whenever p=0, there is either a cusp, as at the point F in figs. 1, 2, 4, etc., or there is a stationary point (point Warrét) as Lin fig. 4, where the angle of incidence is the critical angle and total reflexion occurs at all greater angles of incidence. When the path of light is from the dense towards the rare medium ~ must be substituted for w in the given equa- tions, as has been done in the planoconvex lens represented in fig. 4. Heath fig. 1 is a table of all the values required for finding the position of the first focal Jines when the angle of incidence is 0, and 45°. SC or a is assumed to be 20, and r 5 units, while p, as in all cases, is taken to be 1°523 which is the ordinary value of w in American spectacle glass. Reflexion at a Spherical Surface. As a simple example fig. 2 is given, which represents the Fig. 2. 90° 0 So reflexion of a plane wave-front at a spherical surface where r=5. The adjoining table corresponding to four values of ’ is sufficient to trace the entire caustic in three minutes. The Tracing Caustic Curves. 263 advantage of being given is seen, as by the direction of the curve one can tell whether it is necessary to find any more intermediate values of /P. Tt will be found that if n=, and if one reflexion occurs at each point of the circle, there will be four cusps, all within the circle, as shown in fig. 3 where a=1°5 and r=5. The table below the diagram gives all the values required. Fig. 3. 0. ¢'. ee pl FP. p. 0. 0 0 5 — ‘9375 | 40625 | 0 0 . 59° 54'38" | 15° 2'41”| 48286 | —-5734 | 4-2552 | 1-2979 4466 30° To 2T 20") 47697, | 0 4°7697 | 1:5 0 155° | 7°17' 2”) 49597 | 3:0090 | 7:9687 | -6339 |—5:9571 180° | 0 5) 3°75 8°75 0 0 It is advisable in cases of reflexion as in fig. 3 to tabulate the values when w=90°, for the reflected ray PP’ in that case marks the boundary of the caustic in one direction. When w=90°, 8 and ¢ are easily found, for sin d= — “sin 0, and 26'=90°-8, 2 sin? 6=1—cos 2¢’=1—sin 8, 2a? sin? @+7? sin @—7r?=0. The solution of this quadratic gives sin@, and sin¢d= ge =u with G- U i —— = = = SSS SSS SS SSS | ) 264 Mr. A. 8S. Percival : Method of Refraction at a Spherical Surface. Fig. 4 represents a plane wave-front incident on a plano- convex lens of radius 5 and thickness 3. As refraction only occurs at the spherical surface where the light encounters the rare medium, ~ must replace w in the given equations. The critical angle when #=1°523 is 41° 2' 27-6", so this is Fig, 4. Refraction. Plane Wave-front. 7=65, p=1'd23. | i. tO: Qin iM. pp’. | qibe p. 0. 10 i Or ib pee —9:5602 | 0 0 —27° 89’ 50" | —45° 35855] — 7-4308 | —3-8953 | —3:5355 | 24-5613 —41° 9' 98''| 90° |0 0 | 0 = 0 the maximum value of ¢, and the three values for the three angles @' (0, —45°, —90°) given in the table are sufficient to trace the caustic. It will be noticed that at F’’ and at L p=0, F” being a cusp and L being a point @arrét. Least Circle of Aberration. The extreme ray LTKL’ cuts the axis at T and the caustic at K ; D marks the position of the Least Circle and DK or Tracing Caustic Curves. 265 k its radius; F''T is the longitudinal aberration (usually denoted by a) and F'’L’ is the lateral aberration denoted by. i. Now in the books it is said that F’’D may be taken as 3a or 3F"'T, and kas 4l. In this case the distance F’’D of the Least Circle from F” is about aT and & is about $1. This example is sufficient to show that this simple rule cannot be relied upon when a lens is used. with its full aperture. In most cases it will be sufficient to find fP when ’ is midway between 0 and its extreme limit and the value of pat the point f; K, the intersection .of the extreme ray with the small are drawn through /, will enable one to deter- mine the size and positon of the Least Circle with a high degree of accuracy. In the diagram H "Ar epresents the second Principal Plane cutting the extreme ray at H, and the angle H’/TL or DIK is or $'—¢, i.e. —48° 57! 32, Indeed the angles DTF and H” TH are geometrically equal to >, though they are measured in the reverse direction. Clearly from the diagram Oa on! T= FTN Teh A cot HTH =f" —y cot Wh, where y=H"H. . or hh — PR” tan a tan vp. (a), hor: DK =TDtan DIK =TD tan w. Now (2) and (3) give the correct numerical result which is all that is required for J and #, for they are essentially signless ; in (1), however, it is necessary to know whether a is positive or negative. A moment’s consideration will show that (1) is universally true when the symbols carry sions of direction, for as H"H is negative and H’’TH is positive, every term in the equation carries the same (nega- tive) sign. In order then to determine a and / it is only necessary to trace the extreme ray, but to find the size and position of the Least Circle it is necessary to trace that part of the caustic which the extreme ray cuts. The caustic in fig. 4 is extraordinarily large as it even extends beyond the Principal Plane, although the aperture of the lens is relatively small. A much smailer caustic is formed by the same lens if its position is reversed as in fig. 8, where the semiaperture of the lens is far greater. fr, however, the source of light be at or near the first prin- cipal focus of the planoconvex, so that the incident light is a widely divergent cone, this position of the lens with its plane surface facing the incident light gives far the smaller caustic. ih i | (PAM) | 266 Mr. A.S. Percival: Method of Before dealing with such a case, or with that of the plane wave-front in fig. 8, it will be necessary to consider refraction at a plane surface. Refraction at a Plane Surface. Let OP be ihe plane surface and OT the initial line norma] to it, on which is situated the source S (fig. 5). Let SP incident at P be refracted as PR, where 6=LPS or OSP, and d’ =LPN=OTP=PON, and let ON or p be perpendicular to PT. Then if SO=b, p=OPcosd’=btangcos g’ and n or NP=OPsin ¢'=b tan ¢ sin @’. As ¢’ now replaces , p’ is obtained by differentiating p with regard to ¢’. p=b (cos g' sec? oe —tan ¢ sin $’). But as sin d=psin ¢’, =p sec ¢ cos ¢', so p'=b(psec* ¢ cos? ¢’—ian ¢ sin ¢’), d and, 9 = EE cos? d’ sec® $ tan @ e —Zusec® bcos ¢' sin d | —sin d’ sec? soe —tan ¢ cos é | dd =6(3u? cos? ¢' sec* d tan 6 —3 cos d! sec? db tan f : —tan ¢ cos qd’), p=pt+p"=btan ¢ cos d' (1+ 3p’ cos? g' sec’ b —3 sec? 6—1) = 3b tan ¢ cos ¢! sec® d(p? sec? d cos? d' —1), or 3p sec* d (yu? sec? p cos? d' —1). As before {NP is the tangent to the caustic at the point f, and fP=f/N+NP=p'+n, and as all the magnitudes in the diagram (b, ¢, ¢') are of the same sign, the formule tabu- Jated below are universally correct when signs of direction are introduced. TVasire al, Refraction at a plane surface. p=btan ¢cos o’. J P=bp sec® ¢ cos? 9’. n=b tan ¢ sing’. | p=5p sec? o(u? sec? ¢ cos? g’ —1). Tracing Caustic Curves. 267 In fig. 5 it will be noticed that as before when p and p carry the same sign C7 is measured in the same direction as ON. When ¢=45° pis more than 56, so C’ is beyond the limits of the diagram in the direction fC. Fig. 5. Refraction at a plane surface. SO=b=4: p=1°523. | ¢ q’. pie D. p- | pias 2) at aN Bs .0 0 6092 0 0 | | 45° | 27°39'50"' 13-5165 3°5427 | 56:0971 | Caustic formed by a Lens. In all cases it is necessary to know the axial thickness (¢), the curvature of each surface, and the distance of the radiant point either from the proximal surface or from its centre of curvature. Formule must then be obtained to determine in terms of the data the length of the path (P,P) of any ray within the lens, and the angle of incidence at each surface. We will consider first plano lenses. A (1) The plane surface facing the incident light. Fig. 6 represents a radiant point S on the axis of a plano- convex lens. It will be necessary to find the values of fP, and p,; (due to the refraction at the plane surface) for the axial and the extreme rays and one or two intermediate rays 268 Mr. A. 8. Percival: Method of by the previous method. To avoid confusion in the diagram 7P, has been made greater than its real length for this angle of incidence. Produce fP, to P, on the curved surface of the lens, draw P,L perpendicular to the axis SCO, and let CM and ON (or ,) meet fTP, at right angles ; join fC and CP,. | ' The point 7 is the radiant point for the second surface, fC is the part of the initial line that is represented by SC or a in fig. 1, and CP, f or ¢, is the angle of incidence, and CfP, or 6 is the angle which fP, makes with the néw Panitial lime. Asin’ fig, SO=- 6, ON=p,, OSP,=¢,, and Ont — py. From fig. 6 it is readily seen that (i.) CM=rsin ¢,=TC sin dy’ =(TO—CO) sin dy’ - =(b tan d; cot d)’—r+¢) sin Gy’. r sin d,=b tan ¢; cos di +(t—7) sin dy’ or py++ (t—7) sin dy’. (i.) P,P, cos ¢,’ = OL=t—r vers (¢;' +). Chit rsin do JM fP1 +P P.—71 cos dy" (iv.) eC Or A», =/M sec Goe= (7 P.—?r COS do) sec Oo In the diagram all the magnitudes are positive except do, so to make the formule of universal application when direction-signs are added it is only necessary to change the sign of ¢, as has been done in the table. It has been shown in fig. 4 with what extreme simplicity the caustic can be traced when the light presents a plane wave-front with normal incidence. If the plane wave-front be incident at an angle a on the plane surface and refracted at an angle a’, the initial line for the spherical surface will make an angle 7 (quis) tan'7> — Tracing Caustic Curves. 269 e' with the axis of the lens. The caustic will appear as if rotated about C through the angle «’ from the axis of the | lens. If the emergent: light be received on a small vertical screen, the flare of light seen on it is of course the coma. The difficult subject of coma can be simply introduced to the student by getting him to trace the caustic of such a lens due to a plane wave-front incident at some angle a. With the help of the table below fig. 4 it can be done in three or four minutes. On then drawing a vertical line through the cusp, parallel to the plane surface of the lens, he will find that the rays, which are symmetrical above and below the axis of the lens, when produced beyond the caustic will intersect on this vertical line below the cusp if « be positive. This vertical line presents an edge view of the intersecting comatie circles that produce the flare below the cusp. A (2) The curved surface facing the inerdent light. The construction of fig. 7 requires no explanation ; § is the radiant point, SC=a, sin ¢,=—-~ sin 0, as before, and if fe 7 P, is found by Table I. As the second surface is plane JO represents SO in fig. 5 or 6 in Table II.; since fis the radiant point for the second surface, its angle of incidence is @,=CTP, (or Of P.) = 0,— 94,4 oy. Nest It is required to find P,P, and /O in terms of the known quantities. From the diagram it is obvious that P,P, cos 65 =t—rvers (¢,—49,), and purced—7 Ps cos dy. In the figure all the magnitudes are positive except, 7, so the sign of 7 must be changed in the above to give the general formule in Table III. (V is not shown.) a 270 Mr. A. S. Percival : Method of TasLeE IIT. (Plano lenses.) (1) First surface plane; as in fig.5. SO=5b; OSP|=d;; OTP: = py’. (i.) p,=b tan ¢, cos ¢,'. Plane Wave-front. cos? 6," Incidence Normal. Gi.) fP, =k”. cos? ¢, ur v=r cos ¢'-++———_______, sec P — mw sec b Second surface. p=rsin g’ {1- as p? sec? gh ; 4 sec O—p sec ¢’) (1) sin g¢,= — (7 —¢sin 9,'—P)). | QE t—r vers (¢,' a2) cos ¢,' ae sings OD eS canes oF Re | (4) a, or fC=(fP,—7 cos ¢,) sec 9. Woes re MG? (LE) WY Be oriv=7, cos: oy Ait ae Pym 6 7 (EN) p=rsin ¢,| 1— (7 sec 0,)" +(a, sec 2) — (Ma, Sec J : a a sec 0, +a, sec he Ha, sec we Dah (2) Tis uae cee : asin PE Fe ane . ca om = oe ch,’ a : a, (i.) sin ¢; = — — sin 0. Ce cee pr sec O+ pa see o— asec g! A Geaoud wunbate: Plane Wave-front. Incidence Normal, CD) i= th Slo, = Ghar Ga! First Surface. t-+r7 vers (¢,—9,) : LP cos? ¢,’ D) as aE Se aes (4) P,Po= COS 9, me oS Pus [- COS @,'— C08 @, * (3) Po— Py sm oO, COs. iE Second Surface. cos? ¢, (1) ~2=$1' — >: TE aot I hae cose (2) P.p.— UE vers py a ae c0sg,) / COL) os. St { eee at | ; (3) P= P, sin ¢, cos ¢,. cos’, | (wcos ¢2) a a cos? 0! DR) p= 22s | eee Ee RU COs py)? : Tracing Caustic Curves. 271 On substituting fO for 6 in Table II. we get ON or po=f Ps sin dz cos go’, Cos? ho” ? wos? dy” aL EBPs cos? dy’ 1 ~ cos* dy Lu? cos? be WES or v= FP; It should be noted that in tracing the caustic of a lens all the angles are required, for the first surface only fP; and sometimes p,, but for the second surface VP, and p are necessary. It will be noticed that the usual form for finding the first focal line (/'P,) has been given for a plane wave-front as this is speedier for logarithmic calculation ; when a is finite and given, the method now suggested is the speedier, for 2 troublesome preliminary determination of wu in terms of a, 7, and ¢;, must be made before the usual method can be applied. As an example of class A (2) the caustic is drawn of a plane waye-front incident upon the convex surface of the same planoconvex illustrated in fig. 4. The caustic in fig. 8 in full lines is due to this lens when stopped down to be of the same aperture as the lens in fig. 4,7.e., when $,;=41° 2/ 28”; the focal length FH" = —9-5602 in each. Itis immediately apparent what an advantage is gained by letting the curv ed surface face an incident plane wave-front. Indeed. if the aperture be increased to admit rays whose incident angle is 60° or even more, the caustic is still much smaller than that shown in fig. A, Ali the lines which it is necessary to draw are indicated on the diagram ; a set square will indicate the perpendicular (p2) from 0 on an emergent ray, and C'v is drawn parallel to po, its direction and lenoth being given by the as and figures under p in the explanatory table beneath fig. B. The Bispherical Lens. The construction of fig. 9 (p. 273) requires little explan- ation: C,P, is the radius “of the first surface, C,P, that of the eecond' + Bivky ee PO = ,, C;P,T= dy’, C,PoT=¢s, C,fN=@,, C,MM' and C.N are each at right angles to PE EM. wa 272 Mr. A. 8. Percival: Vethod of Fig. 8. Refraction of a plane wayve-front by a planoconvex lens. t 3 r= —5, t=3. H,O= 7 = {593 =! 9698. ; | : Q- D, dle: bo. ®. G Pee 0 0 — 14-5602 | 0 0 3 30° 19° 9°56") —11-8685 /—10° 50' 4”) —169.38! "9422.37.04 41° 2 98" | 25° 32’ 20") —10:0001 |—15°30' 8") —24° 1713”) —1-8380 60° $4039) 17”) | — 68447 |= 25° 20) 4371) 409 400) aaa bie as Po: 4 v. oO. eb GO2 a0 —7 5904 0 —9-4961 | 1°7103 | —5:9337 — 3:1366 —8 1621 | 1:9925 | —4-8151 — 38-9438 —6:-2915 | 2:0421 | —2-9081 — 52245 The geometry of fig. 9 shows that (.) sin C,PS= "sin CSP, or sin g)= “sin 6 5 (ii.) C.N = M’'C, —MC,=(C,C, sin C,C,M'—O,P; sin 0, PM, andas (C,C,M’=C,TP,;=6,—¢,4+ ¢1, 7 SIN Go=(7,+t—7.) sin (0; —o,+ $;')—7 sin dy’ ; Gii.) P,P, sin C,TP,=C,P, sin LC,P,—C,P, sin SC,P, : or P,P. sin (6;—¢,+¢1')=7 sin (62 +6,—¢)+4 oy) amit hil sin (0,;—¢,) 5 Tracing Caustic Curves. 273 C,N 2 SiN hy ME) pane oN fF Pit PP 7, cos en (v.j JU, or a,=C,N cosec 6,. Fig. 9. In the diagram all the magnitudes are positive except a, (or SC,) and ¢, so the signs of these symbols must be changed in order to make the above formule universally true when the symbols cary direction signs. For a plane wave-front at normal incidence 0, and a, do not appear. These any oes have been made in Table IV. In fig. 10 an example is given of a biconvex lens of the same power as that of the previous planoconvex ; in this case T= —1r,=9°4552 and, as can be readily ne the second principal plane passes through the point H" where H’B=1:0416. From the table of values below the diagram it is seen that when ¢,=0 and a or fC, (t.e. fP,—71,) is —33°9892, v or F’B=—85186, and therefore FH” = —9°5602 as in the case of the planoconvex i in fig. 4 and in fig. 9. “The symmetrical biconvex gives the smallest caustie when the lens is midway between the object and its real image. Fig. LO represents the case when parallel rays are incident ; if provided with a diaphragm to reduce its aperture to that of the planoconvex in fig. 4 tl caustic is fairly small, as is shown by the fuil lines. Even when used with its full aperture the caustic does not quite touch the surface of the lens. Phil. Mag. 8. 6. Vol. 43. No. 204. Feb. 1922. a 274 Mr. A. S. Percival : Method of Tasie TV. (Bispherical Lens.) First Surface. ‘ ; Qa, . Qi.) sing, =— a sin @,. 1 a4? il. P, =r, cos 6,/ + —— oo Tag ia pr, sec 0, +a, sec 9, —a, see 9,' Second Surface. (1) sin ¢,= {(7,+¢—7,) sin ($, —9)/— 8 )-+7; sin 9"), Ps sin (¢,— Qi ate Oram ae sin (d,- hy; Bye PP (2) ite sin (¢,—¢,'—9,) Bi tan = wee sin Go ee 7, cos o,—f P, (4) fC, ora,=—7, sin @, cosec 8. (5) Po= Py SiN Oy. Ma,7 2 hh s yn 5 1 7, sec 0, +a, SCC P, —-Pd, SEC O, . (EE) VP oriv—cacosips __ (Py see G,)8-+(d, SCC bo)> — (Ma, 8eE y')® \ ( 4 10 l (I1.) P=Po | (7, sec 9, +a, sec o, — pa, sec ,')° PuaneE Wave-Front: Incrpence Normau: 6, and a, do not appear. ry : 2 i : 4 1, COS* Gia ty BA NCOsOs pee 5 or mu —_ ‘Cis pe sec b, — BEC [LCOS @, —cOs ¢, : 1 : ; CD) sin@. — a f(r +t -7r,)sin (¢,—¢,')+7, sin gi}. 2 (2) P,P,= 2 sin (? —91'+$2)—7) §1N g . aN, sin ($; — $1’) The remaining formule remain unaltered. Tracing Caustic Curves. Oi Fig, 10. Refraction of a plane wave-front by a biconvex lens, _ r= —9°4552; r,=9°4552 5 f=35 p=1°523. | | ae aaa aan i- pi: FE: | 2: | Pr P,P,. Oy |:.0 —27-534 0 IE, 6 | 3 | sem 9°47) 2") — 261429 | 180507197 |x 29927 28" || 2:49007 | | 302 | 19° 9/56”) —22-4396. | —40° 9! dai’ | —79°12'45” | -5098 | ie 0. (pe v, fase 0. — 24-534 0 — 33°9892 | —8 5186 | 0 0 —93°7222 | — 5° 20' 20” | —32°8131 | —6-4323 | 46499 | —16:0049 —21-9298 | —11° 48’ 52’ | —29°7865 | — -2485 | 92881 | — 83388 The virtual caustic in fig. 11 is interesting. To avoid confusion in the diagram only the lines which havea relation to the extreme ray SP, are indicated ; ASP,;=6, or 45°, C,P,S=¢,, C,P;T’=¢;', and P,P, is the path of the refracted ray through the lens, and of course the angle AT'P, =, or 6,—¢,+¢,'. It will be noted that if P,T’ be produced it will touch the end of the caustic in dotted lines of which the lower limb only has been drawn ; this is the caustic formed by the first refraction. For the second refraction C,P,T" is eZ 276 Method of Tracing Caustic Curves. dy, and C,P,T is d,’. The line PT meets the final caustic at V. Fig. 11. Virtual Caustic formed by a meniscus where SA=4, and 7,=109; == 05 OC, 35 f= ogee ,. Ge FPL. |). 95, ee ea ——_—_<§— Sue | 0 53464 | 0 NG | 39° | 21°30’ 37” 13°55" 50’) 61856 |—15° 37'38" —24° 13/17" 45° 131914’ 5! 19°54! 21", 7-3463 —24° 50! 29'|_ 39° 4631” a aed | | KP. Pest hes oF | Cae ote ee i Ps 3 | 83464 | 0 33464 | 12-8411] 0 Ot re 39735 | 84590 20°17’ 10" 38848 147693) 2.0513 | 66197 1-2527 | 85985 |27°20'52"| 45721 | 255068, 31989 | 3-4034_ —— Now although this caustic 1s very large, as Q is the con- jugate focus of 8, the longitudinal aberration is very small being only QT, while the lateral aberration is the short vertical line above Q, and the radius of the least circle of aberration is indicated by the minute vertical line between Qand T. If an eye, or a camera of very small aperture, were placed behind the lens in the direction P,R and were focussed for the point V, the radiant point S would be repre- sented by a focal line at right angles to the plane of the paper, but if placed in the direction AB and if focussed for the point Q, S would appear as a minute circle. It has been previously pointed out that in most cases it is advisable to determine the angle yy or BTP,. From the figure it is seen that BOW, = Bie + Co Pere AT'P, + BA or Irs ar de — Wry ac do. Forced Convection of Heat from Pair of heated Wires. 277 But : Wi =0,—¢6,4+ dy’. Wro= 91 — $i + $1 + b.— $2’. In the figure ¢, and ¢@,’ only are negative, so the general formula when the symbols carry direction signs becomes W.=9—b1+ $1 — dot Gy. When the incident light is a plane wave-front at normal incidence, the formula is simply Wo= —$,+ $i —h2t $y’ “ Westward,” Tankerville Terrace, Neweastle-upon-Tyne. XXVIII. The Forced Convection of Heat from a Pair of fine heated Wires. By J. S. G. THomas, D.Sc. (Lond.), B.Se. (Wales), AR.CS., A.LC., Senior Physicist, South Metropolitan Gas Company, London *. Introduction. N a recent communication F, attention was directed to certain characteristics of the forced convection of heat ' from a pair of heated wires immersed in aslow stream of air, and constituting a bifilar anemometer of the directional type. It was remarked { that the disposition of the wires could be such that the temperature of the second wire of the pair was unaffected by the stream. In the course of an investigation of the velocity of the air-stream at successive cross sections in a flow tube, employing the hot-wire anemometer, the importance of this consideration became very evident, and the present communication details some of the characteristics of the forced convection of heat from a fine heated wire due to the passage of a stream of air moving with low velocity and heated by prior passage over a similar wire immersed in the stream, The dependence of this thermal effect upon the distance apart of the wires is, more especially, the subject of the present communication. * Communicated by the Author. T Phil. Mag. vol. xl. pp. 640-665 (1920). J Ibid. p. 652. 278 Dr. J. 8. G. Thomas on the Forced Convection of — EKaperimental. Details of the mode of insertion of the heated wires in the flow tube are given in fig. 1. The flow tube was made in two halves. ‘he wires A were of pure platinum supplied by Messrs. Johnson & Matthey, and their length was as nearly as possible equal to the diameter of the flow tube. They were attached to pairs of copper leads B, B of 0°8 mm. diameter as previously described. The following device was finally adopted as the most suitable for maintaining the wires taut during the course of the experiments. The straight copper leads passed through holes affording a nice sliding fit in blocks of ebonite C fixed to the flow tube as shown. One pair of leads passed in similar manner through holes whose respective contours were about three-quarters of the circum- ference of a circle drilled at the edge of brass pieces D carried by supports G, H attached to the ebonite blocks C as shown. Similar holes through which the individual leads passed were provided in the brass pieces D, and Ds, and these leads were securely clamped by means of the screws H, the brass pieces being slotted for this purpose. The wires were maintained taut by means of the weak springs F pressing against the brass supports D,, which slid freely along the surface of the supports G, H. The tension in the wires was adjusted by means of the screws J. In general, the variation of resistance of the wires due to tension was less than 0-1 per cent. of their resistance. The pair of leads shown in the upper part of the diagram was connected with screw terminals K by means of leads of loosely stranded copper wires. The distance apart of the heated wires could be adjusted to any desired value by the insertions of annular bloeks L, accurately drilled to the diameter of the flow tube Heat from a Pair of fine heated Wires. 279 between the ebonite blocks C. Care was taken that the continuity of the inner surface of the flow tube was nowhere disturbed by the insertion of the blocks, which were held rigidly in place against the blocks C by means of nuts screwing on the ends of three brass rods passing longi- tudinally through the whole length of the blocks. The distance apart of the wires was determined by means of a micrometer microscope reading direct to 0°001 cm. For this purpose, the wires could be viewed through the aper- tures M, which were normally closed by plugs whose ends conformed accurately with the inner contour of the flow tube. The tube was inserted in the flow system by means of spigot unions as shown. ‘The wires were disposed in a horizontal plane in the flow tube, which was arranged hori- zontally. The distance apart of the wires having been ascertained, the current in the wires was adjusted by means of a rheostat, so that for the respective distances the wires were, in the absence of any impressed stream, maintained at a constant temperature, and subsequently the current in the flow experiments was adjusted to such appropriate value and maintained constant to within 0°001 amp. The experiments were directed mainly to ascertaining the change in resistance produced in the second wire of the pair by the stream of air, and for this purpose the ends of this wire were connected to a Rayleigh potentiometer, constituted of two P.O. boxes, in which the total resistance was throughout adjusted to 10,000 ohms. The working current in the potentiometer was derived from a Leclanché cell, and the readings were reduced to volts by calibration from time to time against a Weston cell certified by the N. P. Laboratory. The flow of dry air was established as detailed in previous papers, and the mean equivalent velocities of flow were throughout reduced by calculation to 0° C. and 760 mm. pressure. Results and Discussion. Internal Mimnercr of Flow Tube External Diameter of Flow Tube ......... eo 2 Gin: Mean Diameter of Platinum Wires Temperature Coefficient of Platinum {SUES bs ak a SNS a a 0:003750 Eng ObsMeoawcerGiviair ..°......:. 30). 0:2468 ohm. Ey OF 2NGeimerourlatt he 0-2420 ohm. Resistance of 2nd Wire, heated in absence of impressed Stream ............... 06931 £:0011 ohm. Mean temperature of 2nd Wire, heated in absence of impressed Stream 280 Dr. J.8. G. Thomas on the Forced Convection of The pressure in the flow tube did not exceed 0:1 inch of water during the course of the experiments, and at this pressure it was ascertained that there was no appreciable leakage in the flow system. In fig. 2 the changes of resistance of the second wire 4 Pe ri Ka Res | < 3 | a eA ; : Piel ae BEY cap) 7 Aes Sd S eye ota Ly ‘ : 6G val o | VE S ee 5 mw Oe Sey cs SEN ss if. O Ks, Zail aii a Qe 6 -2] tJ ied (o) } 5 2 (8) +3 Fig. 2. 7231 Fo-ai8 60090}4 ©0:189 > SECOND WIRE ALONE HEATED. -4 cr A) “66 l 2 3 4 7 VELOCITY (CMS PER SEC, VOLUMES REDUCED TO O'C ann 760 mm) of the pair from its initial value in the absence of flow are plotted as ordinates against the values of the velocity of the impressed stream as abscissee for various values of the dis- tances apart of the wires. For.the sake of clearness, only about one-quarter of the experimental points are indicated in the several curves. The calibration curve obtained when the second wire alone was heated is likewise shown in the diagram. For purposes of discussion the various curves are best grouped as indicated by the brackets in the table of distances apart of the wires given in the figure. Considering the first group corresponding to distances apart of the wires Heat from a Pair of fine heated Wires. 281 equal to 6°28 cm., 5°31 em., 4°32 cm., 3°31 cm., and 2°31 cm., - indicated by the curves A, B, C, D, and E respectively, it is seen that this group is characterized by a gradual increase in the maximum temperature attained by the second wire of the pair as the distance apart decreases. The initial portions of the curves A, B, and C approximate to the form of the initial portion of the curve L, obtained when the second wire alone was heated. Moreover, the initial cooling effect experienced by the second wire when the velocity of the impressed stream is small is reduced as the distance apart is diminished until it is replaced by an initial heating eftect in the case of the curves D and EK. This is in accordance with anticipations based on the consideration of the resultant thermal effect experienced by the second wire as being con- ditioned by two opposing tendencies, viz. the direct cooling effect due to the impressed stream, and heating effect arising from the forced convection of heat by the stream from the first to the second wire*. It will be noticed that corre- sponding to the higher values of the impressed velocity of the stream, portions ot the curves A, B, C, D, and [6 are, within the limit of experimental error, common to the several members of this group. Moreover, as the distances apart of the wires become less, the value of the impressed velocity at which the respective curves unite is reduced. This result is readily deduced from a graphical consideration of the. direction of the resultant stream due to free and forced convection from the wires as previously explained +. The second group of curves is constituted of curves E, F, G, and H, the curve E being included as affording a transition from the first to the second group. In this group, the maximum temperature attained by the second wire diminishes as the distance apart of the wires diminishes. ‘The respective values of the maximum increase of resistance characteristic of the various curves are plotted and lettered appropriately to the respective curves in tig. 4, which is discussed later. t will be noticed that, as shown by the curve H, when the distance apart of the wires is 0-814 em., for no value of the velocity of the impressed stream does the second wire become heated above its initial temperature. The charac- teristic feature of this group of curves arises owing to the respective distances apart of the wires in the cases in question being such that, in the absence of an impressed flow, the temperature of each wire is determined in part not only by the heated air accumulated in the upper portion of the cross * Proc. Phys. Soe. vol. xxxii. Part iii. p. 203 (1920). + Phil. Mag. vol. xli. p. 247 (1921). 282 Dr. J. 8. G. Thomas on the Forced Convection of section of the tube and originating in the wire in question, but also in part by that similarly accumulated due to the other wire of the pair. This accumulation of a compara- tively stagnant layer of heated air extending over a com- paratively large distance in the upper region of the flow tube, is an additional factor complicating the application of the hot- wire anemometer to the investigation of the velocities (and more especially of low velocities) of gas streams in narrow channels. Thus, as shown by the initial portions of the curves H, I, and M, for small values of the impressed velocity, ‘hs second wire may be more cooled when the first wire is heated than when the wire is not heated. In the case of curves K—H, the initial value of the resistance of the second wire being determined while the wire is subjected to the heating effect just referred to, and as, moreover, this heating effect due to the accumulated layer of hot air clearly increases as the wires approach one another, it is to be anticipated that with the reduction or removal of this effect by the incidence of the impressed stream of air, the maximum increase of resistance of the second wire will be reduced as the distance apart of the wires is decreased. A further feature of the curves EH to H (and the same may be said of all the remaining curves except L) is that, as the distance apart of the wires decreases, the resistance of the second wire corresponding to a definite large value of the impressed velocity of the stream increases. This arises owing to the diminished loss of heat occurring in the forced convection stream during the time of passage from one wire to the other as the distance apart of the wires is diminished. The third group of curves H, I, J, K is characterized by gradual/increase in the maximum temperature of the second wire as the distance apart of the wires diminishes. In the. case of the curve K, corresponding to a distance apart of the wires equal to 0: 189 cm., the maximum increase of resistance of the second wire occurring in the present series of experiments was attained. The degree of symmetry about a vertical axis through the apex shown by this curve indicates the elimination or compensation of the disturbing influences referred to above, and points to a disposition of the wires at a distance apart equal te about 0°2 cm. as most suitable for the construction of a bifilar or directional hot- wire anemometer. The gradual increase in the maximum temperature attained by the second wire as the distance apart of the wires is decreased, contrasted with the gradual decrease in the maximum in the case of the second group of curves (H—-H) discussed above, is to be explained as follows :— Heat from a Pair of fine heated Wires. 283 In the case of all dispositions of the wires, one contem- plates the temperature of the accumulated hot layer of air, due to any one wire, as being greatest vertically above the wire, and diminishing right and left of the wire. Now, if the wires be considered as so widely separated that the accu- mulated layers due to the respective wires just overlap, it is clear that the effect of an impressed stream of low velocity would be to displace away from above the second wire the hot central portion of the one accumulated layer of hot air, and to bring up the comparatively colder border region of that due to the first wire. The net effect, on this account only, would be a considerable fall in the temperature of the second wire. If, however, the wires were so close together that considerable overlapping of the respective accumulated layers of hot air occurs, then any similar displacement of the central hot region of the accumulated hot strata above the second wire is accompanied by the convection thereto of the central hot region of the strata originating in the first wire. The consequent fall of temperature of the second wire is therefore less in this latter case than in the former. Considering the forced convection from the wires, it is readily seen that the net rise of temperature must therefore be greater in the latter than in the former case. The effect of the displacement of the comparatively stagnant accumu- lation of hot air above the wires therefore tends to be reduced as the wires approach one another, and consequently the temperature attained by the second wire due to an impressed stream of small velocity tends to be conditioned to an in- creasing degree by the convection of heat from the two wires alone. The last group of curves is constituted of K and L. The curve L, like the curve K, possesses considerable symmetry. With closer approach of the wires to one another than that corresponding to K, the maximum increase of temperature of the second wire diminishes. That an effect of this nature must occur js clear from the consideration that when the distance apart of the wires is zero, i.e. the wires are coinci- dent, no heating effect due to the impressed stream can be experienced by either wire, the calibration curve then assuming the form M. In the case of distances apart now being considered, these are so small, that due to an impressed stream the approach towards the second ‘wire of the con- vection current arising from the first wire is likewise small. The consequent increase of temperature of the second wire due to such approach will therefore be small. Moreover, as the magnitude of the approach in question is proportional to 284 Dr. J.S. G. Thomas on the Forced Convection of the distance apart of the wires, the thermal effect on the second wire due to this cause will clear ly diminish as the wires are brought closer together. Briefly it can be stated that with the closer approach of the wires at these distances, the tem- perature of the second wire becomes less dependent upon the inclination of the convection current from the first wire to the impressed stream. As the matter referred to above is ae some importance in the design of practical types of thermometeric hot-wire anemometers * for the measurement of the flow of gases in pipes, some further experiments were made with a view to ascertaining how the increase of temperature indicated by the second wire through which a current of 0°01 amp. only was maintained depended upon the distance of this thermo- inetric wire from the heated wire in which a constant current equal to 1°3 amp. was maintained. ‘lhe resistance of the thermometric wire was determined by means of a Callender & Griffiths bridge, and the bridge balance could be accurately adjusted to (‘1 mm, The results obtained are shown in fig. 3. To prevent confusion, a few only of the experimental points are given. An enlarged diagram of the form of the various curves in the neighbourhood of the origin is given in fig. 38 A,in which the effect of the accumu- lated strata of heated air to which reference has been made is clearly seen from the form of the curves F, G, H, and I. Although no actual cooling of the thermometric wire is shown by the curves B, D, H, the effect of the heated strata is not entirely absent, as is door by the forms of the curves in the region of impressed velocities corresponding to 3-4 cm. per second. Certain features of the curves J, K, and Lin fig. 2 are clearly reproduced in the corresponding curves J. K, and L in fig. 3. Attention may in particular be directed to the result that, as shown by the curves K and L, the effect of an impressed stream moving with considerable velocity may be to cool the thermometric wire below its temperature in the absence of flow and in the presence of the heated wire. The curves A’ and A” correspond to greater distances of separation of the wires than any shown in fig, 2. Clearly, for all dispositions of the wires, an appropriate velocity of the impressed stream exists such that the temperature of the thermometric wire attains its maximum value due to an impressed stream of air passing in succession over the heated wire and the thermometric * See e.g. C. C. Thomas, Journ. Franklin Inst. 1911, pp. 411-460 ; Trans. American Soc. Mech. Eng. 1909, p. 655. See also Phil. Mag. vol. xli, p. 258, Bri te leat from a Pair of fine heated Wires. 285 wire. It will be observed that the value of the impressed velocity at which the respective maxima are attained becomes less as the distance between the wires diminishes. Below such value of the impressed velocity, the behaviour of the thermometer type of hot-wire anemometer is conditioned largely by the relative disposition of the wires both with regard to one another and to the walls of the channel in which the flow occurs. This point is of no little consequence 80 70 (10°® om) uw 2) Lb oS 3 CHANGE oF RESISTANCE DUE TO IMPRESSED VELOCITY DISTANCE BETWEEN WiRES( \ 4151) D 182 " Flor { © 131 g O531f ao-si4 4331 9 saa Oe an UO. G9 VY 2-31). +0:400 0-090} ©-0-189 8 roe 16 Ton 32 36 VELOCITY (CMS.PER SEC, VOLUMES REDUCED TO O°C ann 760 mm) in the technical application of this type of hot-wire anemo- meter, as the velocities occurring in gas mains are frequently below those to which reference has just been made. Further consideration of this matter is deferred, however, until the completion of an investigation at present in progress, using three platinum wire grids inserted in the flow tube, the central one functioning as a heating grid between two thermometric grids. ; The values of the maximum increase of resistance of the second wire due to the impressed stream, derived from figs. 2 se 286 Forced Convection of Heat from Pair of Heated Wires. and 3, are plotted in fig. 4 as ordinates against the corre- sponding separation of the wires as abscissee. In general, it is to be anticipated that such maxima will be less in the case when both wires are heated than when the first wire alone is heated, as in the latter case no appreciable cooling effect of the second wire as a direct convection effect due to the im- pressed stream is present to reduce the heating effect arising from convection of heat from the first wire to the second BOTH WIRES HEATED MAXIMUM INCREASE or RESISTANCE (10° OHM) — O D 4 8 8 DISTANCE BETWEEN WIRES (CMS). wire. The results shown are in agreement with this antici- pation, except when the wires are separated by distances of from2to4cm. For such distances the maximum attained when both wires are heated is greater than the corresponding maximum when one wire alone is heated. In these cases, the distances apart of the wires are such that, owing to the impressed stream, the resultant convection from the first wire results in the region about the second wire becoming heated above that ruling in the absence of flow; thereby the velocity of the free convection current arising from the second wire Occurrence of Enhanced Lines in the Are. 287 is reduced, with consequent rise of temperature of the second wire. Such an effect would be absent when the second wire conveys only a very small current. From what has been already said, it is clear that the distances of the wires apart involved in this consideration are somewhat greater than those considered in the discussion of the curves E to H (pp. 281-2). The experimental work detailed herein was carried out in the Physical Laboratory of the South Metropolitan Gas Company, and the author desires to express his gratitude to Dr. Charles Carpenter, C.B.H., M.I.C.E., for the provision of the facilities necessary for the prosecution of the work. Physical Laboratory, South Metropolitan Gas Co., 709 Old Kent Road, 8.E. Aug. 15, 1921. XXVIII. Observations and Experiments on the Occurrence of Spark Lines (Enhanced Lines) in the Arc.—Part lI. Lead and Tin. By G. A. HemsatecH and A. DE GRAMONT *. { Plates III.-V.] ConTENTS. . Introduction. . A convenient arrangement for obtaining the arc spectra of metals having low melting-points., . Spectroscopic methods employed. . Preliminary observations. . Effect of a drop of liquid placed between the electrodes. . Effects of various gases. . Are in liquid air. . Nomenclature employed in this research. . Results of observations on the are spectra of Lead and Tin. . Explanation of Plates. . Summary of results. . Concluding remarks. pe PHP LODEMIUMIOD —— OOCON OTB oD Wr CArAtGr to = bore §1. Introduction. ii the course of his classical series of laboratory experi- ments Sir Norman Lockyer observed that certain spectrum lines which were absent from the flame and relatively feeble in the arc, became bright relative to other lines when a powerful condenser spark was used as the source of light. The name “ enhanced lines” was given to this particular type of lght radiations, and their relative strengthening on passing from the flame to the are and * Communicated by the Authors. 288 Mr. G. A. Hemsalech and the Comte de Gramont on ° spark was attributed to increase of temperature. It appears, however, that Sir Norman Lockyer originally used the term “ temperature ” to include the possible effects of electrical variations*. ‘This fact seems to have been forgotten subse- quently, bor the word temperature has since been used solely in its strictly thermal sense both by astronomers and spectro- scopists, with the result that the enhanced lines (also called spark lines or high temperature lines) play now a prominent role in the temperature classification of the stars. One such enhanced line—namely 14481 of magnesium—has even been strongly recommended for use as a stellar thermometer. The first note of warning against the assumption that changes of temperature are responsible for the emission of enhanced lines seems to have been struck by Messrs. Liveing and Dewar +, who observed the spark lines of magnesium in an are passing between thick rods of the metal in air, nitrogen, hydrogen, and other gases. The con- clusions arrived at by these spectroscopists are best restated in their own words: ‘‘Our observations, however, render doubtful the correctness cf the received opinion that the temperature of the spark discharge is much higher than that Ob Uhe are.) 8 ee ‘‘ Heat, however, is not the only form of energy which may give rise to vibrations, and it is probable that the energy “of the electric discharge as well as that due to chemical change, may directly impart to the matter affected vibrations which are more intense than the temperature alone would produce.” The appearance of spark lines in the are in the presence of hydrogen was again observed by Professor Crew ft and by Messrs. Fowler aud Payne §. Messrs. Hartmann and Eberhard, who obtained spark lines in an are burning under water, attributed their emission under these conditions to the hydrogen released by electrolysis around the electr In a further series of experiments Hartmann varied the current strength in an are passing between magnesium poles in air, and obtained the astonishing result that the enhanced line r 4481 becomes the stronger as the current strength is lessened 4]. Similarly he obtained the spark lines of Zn, Bi, and Pb, which do not show in the carbon are, in an arc * Nature, vol. lxxi. p. 400 (1905). + Liveing and Dewar, Proceed. Roy. Soe. vol. xliv. pp. 241-2 (1888), { Henry “Orew, Astrophysical Journal, vol. xii. p. 167 (1900). § Fowler and Payne, Proceed. Roy. Soe. vol, xxii. p. 253 (1903). || Hartmann and Eberhard, Astrophysical Journal, vol. xvii. p. 2 (1903). 4 Hartmann, Astrophysical Journal, vol. xvul. p. 270 (1903). the Occurrence of Enhanced Lines in the Arc. 289 passing between metal poles using a feeble current with small vapour production. i on the contrary, the amount of vapour was increased by heating one of the poles, the spark lines would disappear. Asa general result of all his observa- tions Professor Hartmann arrives at the conclusion that the spark lines do not correspond to a thermal radiation, but rather to electro-luminescence. By carefully exploring the various regions of electric ares burning steadily between metal electrodes, Messrs. Fabry and Buisson * and Professor Duffield + independently discovered the emission of spark lines in the immediate vicinity of the poles. Messrs. Fabry and Buisson attribute the emission of spark lines under these conditions to the high drop of potential which exists near the poles. According to their view the ions would attain very:-high speeds near “the poles, and their collisions would give rise to the emission of light radiations which could not be produced with low velocity ions. ‘lemperature would in this-case not intervene at all. On the other hand, temperature by itself, if suffi- ciently high, might give similar ionic velocities in the absence of all electric actions. But spark lines have been observed also in flames under certain conditions. Thus Messrs. Hemsalech and de Watte- ville have shown that certain enhanced lines of iron are emitted by the inner cone of the air-coal gas flamet. As most of these lines were not seen in high temperature flames, it seemed most plausible to connect their appearance in the air-coal gas cone with the special chemical actions prevailing therein. In discussing these results one of us directed attention to the existence of several different types of enhanced lines and, in a communication to the International Union for Cooperation in Solar Research, proposed the following provisional classification of onbemestl lines § :— lst type.—Lines which are enhanced on passing from flames of low to those of high temperature. Hxamples : 13934 and X 3968 of calcium. air- goal gas flame But absent from the mantles of the ye et flames. Only traces of a few of them have been observed in the hottest region of the oxy-acetylene flame. * Fabry and Buisson, Journal de Physique, vol. ix. p. 929 (1910). + G. W. Duffield, Astrophysical Journal, vol. xxvii. p. 260 (1908). t Hemsalech and de Watteville, Comptes Rendus de ! Académie des Sciences, t. cxlvi. p. 1389 (1908). § Hemsalech, Transactions Int. Union Solar Research, vol. iv. p. 145 (1914). Phil. Mag. 8. 6. Vol. 43. No. 254. Feb. 1922. U 990 Mr. G. A. Hemsalech and the Comte de Gramont on In the are they are generally confined to the neighbour- hood of the poles. Examples: the iron lines XX3872 and 3936, and a number of “ polar lines ” observed by Professor Duffield. 3rd type.—These lines are absent from flames and from the tube furnace. In the arc they appear as relatively feeble lines but are appreciably enhanced in the spark. Examples: the iron lines AX4924 and 5018, and the cadmium line A. 4416. Ath type.—The characteristic spark lines. Hxamples: ) 4481 of magnesium and AA 4912 and 4924 of zine. More recently one of us observed that certain spark lines which are absent from an are passing between a carbon rod and a metal electrode would show up well when the are is taken between two big pieces of metal*. Further, with the aid of visual observations of the spectrum he noticed that whilst certain spark lines remained visible throughout the duration of the are and could thus be photograpl: ed along with the arc lines, others flashed up only momentarily at the beginning when the are is struck. He thus established the existence of two types of spark lines emitted by the arc, namely : 1. Lines which are permanently emitted. 2, Lines which appear only momentarily at the anode on striking the arc. Experiments made with plate furnaces raised to over 3000 ° C.—thus approaching the temperature of an are but in which the acting electric fields were relatively feeble— had failed to reveal the least traces of spark lines in the case of magnesium yt. It was therefore thought that possibly a high degree of ionization, such as oben in an electric furnace, would prevent the excitation of spark lines by opposing the establishment of high potential gradients. For it seemed to us that the electrical conductivity of the metal vapours at the moment of striking the arc, as also of those in an are which passes between voluminous lumps of metal possessing a big thermal capacity and a large cooling surface, would be appreciably lower than that of the well protected vapoursina high temperature furnace. Stimulated by these considerations we decided to make a series of experiments with the object of ascertaining the relative roles played by thermal, chemical, and electric actions in the emission of spark lines by an electric arc. Accordingly, our observations bear more specially upon the effect of cooling * A. de Gramont, Comptes Rendus de MES ae ces Sciences, clxx. p. 31 (1920). + Hemsalech, Philosophical Magazine, vol. xl. p. "308 (1920). the Occurrence of Enhanced Lines in the Are. 291 and heating the are vapours or the electrodes, and upon that of varying the nature of the medium in which the arc is established. The present paper deals more particularly with the are spectra of lead and tin. § 2. A convenient arrangement for obtaining the are spectra of metals having low melting-points. The usual way of obtaining the are spectrum of a metal is to place small lumps of it into the hollowed out positive crater of the lower carbon rod. The inconveniences attending this method are well known ; the temperature of the crater is so high that metals of low boiling-point are almost immediately thrust out again. The remedy generally applied is to use some compound of the element in place of the metal. It is obviously impossible under these conditions to carefully examine the spectrum radiations emitted by the vapour in the immediate vicinity of the metal, and it is probably for this reason that the emission of spark lines by the are has passed unnoticed for so long. For the purpose of the present research it was essential to be able to explore the region around the metal from: which the are was made to pass, and the following simple method was found to fulfil these require- ments. The piece of metal is laid upon a plate of copper | Fig. 1. Carbon rod. Copper plate. Metal. Method ot obtaining are spectra of volatile metals, about 5 by 10 cm. in area and not less than one milli- metre thick. This plate remains fixed in position and is connected to the positive conductor of the electric supply circuit (fig. 1). A pointed carbon rod, which can be U 2 aad 292 Mr. G. A. Hemsalech and the Comte de Gramont on raised and lowered by means of some mechanical device, is placed vertically above the lump of metal and commu- nicates with the negative end of the cireuit. The are is started by lowering the carbon rod and. bringing it into contact with the piece of metal. As soon as the are is struck the metal melts and forms a globular mass from which the are continues to burn quite steadily. ‘The réle — of the copper plate is, naturally, to dissipate the heat generated by the arcing, and thus to prevent the rapid boiling away of the metal. With this device good continuous ares have been obtained with the following metals: Sb, Bi, Mn, Na, Sn, Cd, Zim, Pb) Tl) Al, Mo vandaiin, § 38. Spectroscopic methods employed. Nearly all our observations were made photographically. In order to cover the whole region of the spectrum between X 2150 and 2» 7000 the following three instruments were made use of : 1. Quartz spectrograph provided with one 60° Cornu- prism for the region from X 2150 to A 3200. . Uviol glass spectrograph by Hilger with two 60° prisms for the region > 3170 to > 5600. . Glass spectrograph covering the region > 36U0 to X% 7000 and consisting of three 45° prisms by Steinheil. Lockyer’s method of projecting an image of the source upon the slit was used throughout, and the focussing and adjustment of the electrodes upon the sit were done in the manner previously described by one of us *. bo Oo §4. Preliminary observations. An are was established between metallic lead and a carbon cathode with the arrangement described in §2. Both visual and photographic observations failed to reveal any trace of spark lines even in the immediate vicinity of the molten metal. During these experiments our attention was attracted by the strong heating of the carbon cathode, the tip of which became white hot almost from the moment of starting the arc. We concluded from this observation that large numbers of the electrons expelled by the incandescent carbon were shot through the are gap and, by reason of their intense ionizing action, were lowering the resistance of the medium through which the are was passing. Accordingly, in a * Hemsalech, Philosophical Magazine, vol. xl. p. 316 (1920). the Occurrence of Enhanced Lines in:the Are. 293 subsequent experiment the carbon was replaced by a rod of graphite. Under otherwise similar conditions as before, the tip of the graphite cathode did not become white hot during arcing and, perhaps as a consequence of this deficiency, the arc burned most unsteadily and was difficult to keep going for any length of time. We concluded that, since evidently fewer electrons were being expelled from the graphite than from the white-hot carbon, the resistance of the medium through which the arcing current passed was greater in the ease of the graphite arc than in that of the carbon arc. Consequently, it would necessitate the application of a stronger electric field in order to drive the electric current through a graphite arc and to maintain the latter stable. Now, with a graphite cathode the spark lines of lead became visible in the are! ‘This result seemed to us to mean that an increase of the resistance of the medium, or otherwise stated a Jowering of the degree of ionization prevailing therein, is one of the conditions for the appearance of spark lines in the are. In order to test this inference the carbon cathode, which in the first experiment had prevented the emission of enhanced lines, was again mounted in the apparatus, and the resistance of the medium between the carbon tip and the lead anode increased by gently blowing through it with the mouth. This very simple modus operandi produced the looked for effect, for the spark lines of lead were brought out quite plainly ! . § 5. Lifect of a drop of liquid placed between the electrodes. Previous experimenters had obtained the spark lines in the are by burning the latter under water ; but, unless the liquid be constantly renewed, it soon becomes opaque with colloidal matter, and detailed observations are thereby seriously impeded. We therefore looked for some more convenient method of attaining the same object, and finally adopted the following very simple process of changing the nature of the medium through which the are was passed, at all events during the early stages of the discharge. The two electrodes, one of which may be a carbon or graphite rod, are so arranged that the lower one (generally the metal and either positive or negative) remains fixed, whereas the upper one can be rapidly moved up and down by means of a vertical rack motion. Both are well rounded off at the ends and present a smooth surface. After the current has been put on the electrodes are first brought to within about two or three millimetres from one another, and by means of a small é ond 301 ‘Mr. G. A: Hemealech and the Comtede Gramont on camel's hair brush a drop of some transparent liquid, such as distilled water, paraffin oil, glycerine, turpentine, aicohol, etc., 1s placed in the small gap between them, where it is held in position by capillary forces (fig. 2a). The electrodes are now brought into contact, and then separated again by swiftiy raising the upper electrode to a distance of several centimetres from the stationary one so as to avoid the establishment of a stable arc (fig. 26). We estimate that in our experiments the velocity of the electrode as it moves upwards is at least one metre per second, so that the luminous Fig, 2. Movable poles y Path of arc flash —_ « aa @& Ge o> Liquid drop Ne Fixed poles a. by } Method of producing are in liquid media. phenomena ‘observed along the path of the are follow each other in rapid succession. The relative position of any luminous effect along the gap will therefore provide us with an indication as to which part of the discharge it corresponds. The luminous phenomenon obtained in the way described is composed of two distinct phases. At the instant the electrodes separate a discharge begins to pass between them through the thin liquid film. When the distance between the electrodes has grown to from about 0:1 to 0°5 millimetre, varying according to the nature of the liquid, the jirst phase of the phenomenon comes to an end with the destruction of the Occurrence of Enhanced Lines in the Are. 2995 the liquid film. It is during this phase—namely, within the liquid film—that the spark lines are brought out and attain a relative development comparable to that observed in condenser discharges. After the destruction of the film a short-lived are flash is formed between the fixed and the upward moving electrode, resulting in the establishment of the second phase. During this phase both flame and are lines are well developed, whereas the spark lines show only feebly, and sometimes stop quite abruptly in the middle of the are flash, as though some sudden change in the structure of the radiating centre had taken place. According to our estimates the first phase during which the spark lines are strongly emitted lasts from about 0:0001 to 0-0005 second ; the second phase may last up to 0°007 second or even longer. The description of the observed phenomena just given applies to various kinds of liquid films, both insulators and conductors. When the liquid is an electrolyte the electrodes need not be metals. Thus the spark line > 4481 of mag- ~uesium is brought out in a film of magnesium sulphate solution placed between graphite electrodes. In addition to the spectrum of the metal vapour the hydrogen lines \ 4862 and 46563 are emitted with all these liquids. Their behaviour is very significant—namely, they are considerably broadened at the beginning of the first phase and grow narrower as the distance between the electrodes increases. § 6. Hfects of various gases. Since our principal object was to ascertain more precisely at which stage of development of the arc phenomenon various types of lines are emitted, it was essential to draw the are out by rapidly displacing one of the electrodes in the manner described in the preceding paragraph. In order to accomplish this in a small enclosed space filled with the gas under examination the following method was employed (fig. 3). A cylindrical glass vessel of two inches diameter and five inches height was closed at the bottom with a stopper which carried the fixed electrode. The top of the glass vessel was covered with an elastic rubber membrane through which passed the movable pole. The latter could be brought into contact with and withdrawn vertically from the fixed electrode by means of a mechanical device. The gases employed—namely, hydrogen, nitrogen, oxygen, and coal gas—were passed through this vessel in a constant stream at a little more than atmospheric pressure. 3 Without entering into details it will suffice here to state 5 ate 296 Mr. G. A. Hemsalech and the Comte de Gramont on that practically in every case the spark lines were brought out strongest during the early stages of the are when the distance between the electrodes was still very small. Both hydrogen and coal gas gave them more prominently than the Fig. 3. Cork flanges Rubber membrane Movable electrode Fixed electrode Yllluw!'- Gork stopper Method of obtaining arc in gases. other gases. The red and blue lines of hydrogen (H, and Hg) were brought out along with the spark lines when the arc flash was taken in the first named two gases, and a significant feature of all these lines is their remarkable broadening, especially during the very early stages of the discharge. In this respect they resemble their appearance in condenser discharges. Although the spark lines are like- wise brought out in oxygen, they are not so prominent relatively to the are lines as in hydrogen. In nitrogen the spark lines of lead are fairly marked at the initial stage, but those of magnesium show only feebly. With the last the Occurrence of Enhanced Lines in the Are. 297 named metal as electrodes in this gas a stable arc becomes established. This fact of course implies a state of high con- ductivity of the vapours—a condition which, as our previous results had shown, is adverse to the emission of spark lines. §7. Arc in liquid air. Our preliminary experiments had taught us that spark lines are brought out when the electrodes, especially the cathode, remain relatively cool during arcing, as for example when a graphite rod is used facing a metal pole. It seemed therefore desirable to make some observations on the spectrum of the are with the electrodes cooled down to the temperature of liquid air—namely, to about —190° C. It was obviously not possible to apply the method of the liquid drop to this case ; therefore the electrodes were mounted inside a double- walled,glass vessel and completely immersed in liquid air. No stable are was obtained ; only short are flashes, each one of which was accompanied by the evolution of numerous air bubbles caused by the boiling of the liquid around the path of the discharge. In the spectrum of these luminous flashes the spark lines or so-called high temperature lines were strongly developed, and generally remained visible till the end of the flash. The flame lines, on the other hand, were relatively feeble. The arc lines showed a normal develop- ment as in an ordinary are. Besides confirming. our previous conclusions with regard to the effect of cooling of the vapour on the appearance of the spark tines in the are, ‘hese results further emphasize the fact that the presence of hydrogen is not essential for their emission. § 8. Nomenclature employed in this research. In connexion with the various modes of obtaining the are spectra of metal vapours as detailed in §§ 2-7, the following denominations will henceforth be adopted in order to secure precision and avoid repeated explanations. 1. Carbon or graphite arc.—Arc formed between a metal and a carbon or graphite rod. 2. Direct arc.*—Arc passing between two metal electrodes. 3. Ordinary are—A carbon, graphite, or direct are burning steadily and continuously. Observations, either visual or photographic, are begun only after the arc has been started and are stopped before the are is extinguished. * A. de Gramont, Joc. cit. 298 Mr. G. A. Hemsalech and the Comte de Gramont on 4, Blown arce.—Air is blown through the are, either with the mouth or otherwise, from the one ae of striking it until its extinction. 5. Water arc, glycerine are, etc.—Ares through liquid films according to nature of liquid employed. $9. fResults of observations. For taking the spectrographic records the image of the are was So adjusted on the slit as to allow of per fect differ eptiation between the luminous radiations emitted at various staves of the phenomenon. [or this purpose the images of the pole tips were made to coincide with the slit in such a manner that the motion of the image corresponding to the tip of the upper pole took place along the slit. Spectro- grams obtained in this way permit of distinguishing at a glance between the radiations emitted by the discharge through the liquid film and those given out a little later during the arcing stage. Since a single are flash did not always suffice to adequately impress the photographic plate a number of flashes, generally trom 7 to 30, were given in succession ; a fresh drop of liquid was placed in position between the electrodes before each flash. It was found that the images of successive flashes were fairly accurately superposed upon each other. ‘This was indicated by the sharply defined borders of the narrow spectrum near the stationary pole © corresponding to the liquid film phase. In fact the definition of the spectra obtained in this way was generally such as to admit of the measurement both of the thickness of the-liquid film and of the total length of the are flash. The electric current used for producing the are was derived partly from a public supply service at 110 volts (Paris) and partly from a battery of accumulators at 80 volts (Manchester). The current strength varied generally between 5 and 10 amperes. On account of the important rdle played by enhanced lines in astrophysical problems, special care was given to the investigation of that region of the spectrum which is most generally used by astronomers—namely, the region between » 3600 and 7000 ; and it is mainly also for this reason that in addition to an estimation of the relative intensities of the lines their characters in either phase have been carefully noted. In order to facilitate the tabulation of these results the following numbers and signs have been made use of for expressing relative intensities and characters. the Occurrence of Enhanced Lines in the Are. 299 a. Relative intensities of lines :— Feeble lines: 4, 0, 00, and 000, this last one being at the limit of perception. Lines of moderate strength: 1 to 5. Strong lines : 6 to 10. Extra strong lines: 12 to 50. b. Character of lines :— —— = no line has been observed. e = a long line, one which passes practically unaffected trom one electrode to the other through the first and second phases. d = a discontinuous line; one which is emitted during the first phase and falls off rapidly or abruptly at the beginning of the second phase and disappears (abruptly or gradually) before the end of the second phase. dd = a short line; observed only during the first phase. id 7 aes tA A line which is not specially marked i eae Gna s or n is tolerably well defined. nn = broad and hazy. b" = winged or broadened towards red. Be ie ni » violet. b'Y = symmetrically winged or broadened. r = reversed. I = first phase. JJ = second phase. enh= enhanced line. These signs are placed behind the numbers indicating the relative intensities of the lines. I. Are spectra of Lead. a. Quartz spectrograph 2150 to 13200. Spark lines which are absent from the ordinary carbon-metal arc, but are brought out with a drop of water placed between the electrodes :— AX 2204 Very feeble in direct arc, but strong in water are and spark. 2698 Well seen. ~950 Strong. 3018 Strong. ee el. ee i _——. Fi ~y 300 Mr. G. A. Hémsalech and the Comte de Gramont on b. Uviol spectrograph % 3170 to 5000. AA 8221) 3240 | 3263 | Characteristic arc lines which disappear in water are. 4341 4603 : 3787 Spark line; absent from carbon arc, feeble in direct are and appreciably enhanced in water are. c. Reversals. AA 2237 2247 2394 | Reversed both in direct are and water arc; not 2614 reversed in spark. 2833 Reversed in water arc, but not in direct are or spark. d. Glass spectrograph A 3600-27000. Winco Ordinary are eres in air between Arc between Lead poles in =e Carbon cathode |-— A --—_~* ——-—_--—_ eA. hes eee Nitrogen. Oxygen. Coal gas. Taquid air. | I | | 3683°5 10 crbtY 10 erbr¥ 4 cr Scr Na nGIer . 3740-0 10c¢ 10c¢ 6c,rat poles 5 ebtyY Ses | 37865 ——- —- 00 dd nn enh, 4019°6 10c¢ Se 10¢ 10c¢ 10 cs 4057°8 | 20 erbtY 25ecrbtY | 30 crbtY 50 erb?® | S30 erbe® 4062:2 | 4c 5¢ 4c 8e 5¢s | | 4168°0 | 8e¢ 6 es 8e hic 8 cs | | 4242-6 Odd nn an 2 dn O0ddn | enh. are . 2nnin I ae 15 btVin I 5in L | | te | { inindt | 2) 10 pam { Vain EP pae. | "| 4340-6 | 4¢ 2en Ocn sen 2e aaa (2nnin i) £5b¥in I | { 15 bv im F | , Hea Ms {genin Il |13\-inI1| (10. im dt) OC eee 5000°4 | 12¢ 8e 8cs 8e 8 cs a | BESO) <= Oddnn | Oddun |{ one | 000da | enh. | 5201°4 | 10c 4¢s 3e¢ | 4¢es | romp. | Dat 3inI 10 b'Y in I aoaile® ice { 0Onin1I 103 in II { eet ue enh. iis taake' IuninI | /2ninI & D* amerk aeteoes mi { 000ninIL| | 00n in IL ' 1 in TE {ee eee | 5608-9 py 10in I 10in I 15 b'’ in I 6inI h | ee | 3in II Ham iE. || Lin 1D ¢ 4) (eerie cee 58764 | -—— 2 dd mun > = enh. - 6002:0 4¢ 6c¢ 4 cn 3 cbt 5es | | 6011 2e 2cn 2 enbt a 2¢en | | 6059°9 | Qe len 0 enb? UME) as | 6109 $c 00c¢ 000 cn 000 ¢ | | 6234 | ic Oc O0c a O0ec | | eso ae (Qinl 5inI SinI |flinI | oon | ‘(00in lin 11 ||) lana { 000 “ogbe 228 : Wave- lengths in ee 3683°5 37400 40196 . 4057°8 4062°2 41680 42429°6 4245°2 4340°6 4386°9 5005-4 5043-0 5201-4 5372°5 55448 5608-9 5876-4 6002-0 6011 6059-9 6109 6234 6660-0 6790 the Occurrence of Enhanced Lines in the Are. Ordinary are Water are Blown are | in air between | between between Carbon cathode} Carbon cathode | Graphite anode and and and Lead anode. Lead anode. Lead cathode. | | 10 erbtY¥ | 9) ae b'Y¥ in I 5 er, btYin I | 5 brvin I ae | 3 in 1 Ge 10in I | Bae | { Ae TE we | 20crb'Y | 380cr,btYinI 30 er, b'¥ in I | 4e¢ 4¢e 5e S8inl : oe (din I reer Oinl ree 000 in II toda (8inI 6in lL | 3in II ane 4¢ O00 enn 2 en | Sin l 6inI iat 3 in IL { sin II | ¢ 8inl | — ea Be 2nn in [ ae 2) eae EOD ae AT 1ddnn fom i EE A et i 4e | 3ddn 2ddn — | I2ddn 1ddn — 8d 8dn | 00 dd a 4c¢ 4 cbt 4¢ 2¢ ien 2¢ | OQen len = Ocn le | en | 4d 4ddn —_ | idd 00 dd | a. (Juartz spectrograph. AA 2531 Strong in direct arc 2558 z 33 39 2632 Fairly strong in direct arc 3284 Absent from direct are po apis ae 4586, b. Reversals. be) 33 23 mi » ., ; well marked in water arc. Il. Are spectra of Tin, ) enh, enh. enh. enh. enh. enh, enh. enh, enh. enh. 301 eee absent from water arc. } very strong in water arc. The following lines are distinctly reversed in the direct are but not in the water are nor in spark: AA 2200, 2210, 2246, 2269, 2287, 2317, 2335, 2355, 2422, 2430, 2707 and 302 Mr. G. A. Hemsalech and the Comte de Gramont on ) \ 2840. The line X 2484 is only very feebly reversed in the direct are and, like the other lines, not reversed in water are and spark. c. Glass spectrograph. Wave. Oey arc Water are lengths etween between : Graphite anode | Graphite anode in : rau ; and and Tin cathode. } Tin cathode. Ist phase only. 38010 15¢s 10 bry 3907°2 000 n enh. 4330°1 00 n enh. 4524-7 10 es 10s 4585'6 if enh. 46182 dd 3 enh. 5100°4 000 enh. 5224°5 O00 enh. 5332'6 lddn 10n enh. 5562°5 2ddn 15 enh. 5589-0 2ddn 10 enh. 5631°7 8 6s Oy 57992 2 dd 15 enh, 6149-6 6 cs On 6452°8 5 dd 20 enh, 6760 1 enh. 6844°3 1 dd 4. enh. : § 10. Heplanation of Plates. The reproductions are made from enlargements of the original negatives. Plate III. gives portions of the ultra- violet regions of the spectra of Pb and Sn. a was obtained by photographing in juxtaposition on the same plate the spectra of the ordinary arc, water are, and capacity spark between Pb electrodes. The spark lines \ 2950 and 2 3018, which are absent from the ordinary arc, are brought out: in the water arc. Noteworthy is the strong reversal of A 2833 in the water arc; this line is reversed only feebly in the ordinary arc and not at all in the spark. Photograph 6 shows the spectra of the, ordinary and water arcs of tin (metal electrodes). The most remarkable feature is the prominence in the water are spectrum of the two spark lines WX 3283 and 73352, which are absent from the spectrum of the ordinary arc. On Plate IV. are reproduced portions of the visible region the Occurrence of Enhanced Lines in the Are. 303 of the spectra of lead. a, b, and c were obtained with a carbon cathode (top) and Pb anode. a is the spectrum of the ordinary are in air, showing the usual development under these conditions. The narrow strip of continuous spectrum along the upper edge is caused by the white-hot, tip of the carbon cathode. There are no traces of spark lines in this spectrum. 6 was obtained with the same elec- trodes as a, but a current of air was blown through the are from the moment of striking the same. The strip of con- tinuous spectrum seen near the anode is produced during the first phase of the discharge ; it is also at this stage that the spark lines 1.4245 and 74387 are most strongly de- veloped. Are as well as spark lines appear symmetrically widened or winged in the first phase. During the second phase the spark lines are relatively feeble, but they remain visible till the end of the arcing ; air is of course passing through the are all the time and keeps its ionization at a low value. Spectrum ¢ was obtained with a drop of distilled water placed between a carbon cathode (top) and a Pb anode. In the first phase the spark lines 14245 and ® 4387 are relatively stronger than the arc lines 14020 and 24168. During the second phase the spark lines die out sooner than either flame or are lines. Noteworthy is also the relatively ereat reduction in intensity of the flame line 14058. The © band at 44216 shows faintly throughout the second phase on the original negative. Spectra d and e were obtained with Pb electrodes in oxygen and coal-gas respectively. In both spectra the spark lines are strongest at the beginning of the are flash when the poles were still near together ; but they are likewise, though less. intensely, emitted throughout the second phase. In coal-gas they are particularly strong and are only surpassed in intensity by the flame line r 4058. Plate V. gives the red region of the are spectra of tin as obtained under various conditions. Spectrum a is that of an ordinary are between a carbon cathode (top) and an Sn anode; no trace of a spark line is to be seen. 6 is the spec- trum of an ordiaary are between a graphite anode (top) and an Sn cathode: a number of spark lines are revealed in the vicinity of the metal cathode. These lines are relatively strengthened in the blown graphite are as shown in c. With the water are the spectrum consists predominantly of spark lines (spectrum d). A feature of this spectrum is the great intensity of the hydrogen line \ 6563. ree 304 Mr. G. A. Hemsalech and the Comte de Gramont on § 11. Summary of Results. 1. A convenient method has been established of obtaining the arc spectra of volatile metals. § 2. 2. Hixperiments are described which seem to indicate that, spark lines are brought out in the arc when the degree of ionization is lowered, as for example, by blowing air through the are. §4. 3. A very simple method has been developed ae studying the effect of various liquid media on the character of the are spectrum of a metal. This method consists in placing a drop of the liquid between the poles before striking the are. §0d. 4, the spark lines or so-called high temperature lines dis- appear from the are when the cathode is formed by white- hot carbon. ‘They are brought out prominently, on the other hand, when the electrodes are cooled down to about —190° C. by immersion in liquid air. §¢$ 4 and 7. 5. A detailed account has been given of the results of our ‘observations on the relative behaviour of different types of lead and tin lines in the are under various discharge condi- tions. §9§ 9 and 10. $12. Concluding Remarks. It has been shown in the course of this research that the simple process of placing a drop of liquid between the elec- trodes of an are provides a very convenient method of obtaining the spectrum of a metal vapour in which the spark lines are prominent and easily recognized as such. In this respect the liquid film are resembles a capacity spark which likewise brings into prominence lines of this type. But our new arc method presents a decided advantage over the em- poyment of capacity-sparks inasmnch as no air lines are brought out with the exception of the not very obnoxious lines of hydrogen. As is well known, the spectrum of a eapacity-spark is infested by hundreds of air lines, which in many cases render the observation of the metal lines very difficult, and although the introduction of self-induction into the discharge circuit suppresses all the air lines, it also similarly affects the spark lines. Thus it seems to us that by further developing the new are method, an effective means should be obtained not only of exciting spark lines in metal vapours, but also of finding ont more about their Bonn id the Occurrence of Enhanced Lines in the Are. 305 origin and meaning. From the results of our present ex- periments we may already safely derive the following fundamental facts:— —- 1. The presence of hydrogen is not essential for the ex- citation of spark lines. 2. Spark lines attain a high degree of development only in a medium which offers a comparatively high resistance to the flow of electricity. a When hydrogen is present in the medium, either in the free state or in combination, its spectrum re- sembles that which is only hecrcd with high tension condenser discharges—namely, its lines are symmetrically broadened as though under the in- fluence of a strong electric field (Stark effect). With the help of these facts we will now attempt to derive some notion as to the conditions which underlie the emission of spark lines (4th type enhanced lines) both in spark and are. As is well known this particular type of lines is most strongly developed in a powerful capacity- spark passing between cold metal. electrodes in air at atmospheric pressure. If one or both electrodes are raised to a high temperature the discharge becomes less violent, and the spark lines are considerably reduced in relative intensity *. The heat given off by the electrodes raises the conductivity of the air-gap between them, and the condenser discharge takes place at a much lower potential gradient than through cool, and therefore less conducting air. Again, in the self-induction spark the whole of the Sondeneee discharge current passes through air and metal vapour which have been ionized already by the initial discharge (pilot spark). The spark lines in this case are either absent or only feebly developed and confined to the immediate vicinity of the poles. Thus the results with spark discharges teach us that the spark lines are brought out when the discharge passes through a medium which to begin with is only feebly ionized, SO oe a high potential ¢ medion is required in order to Foret the electric current through it. As for the arc, it has been shown in this paper that spark lines are not itted in the carbon are when the cathode electrode is white-hot. On the other hand, when a graphite pole is used which does not heat to the same extent, spark lines become visible near * C. C. Schenck, Astrophysical Journal, vol. xiv. p. 180 (1901). Phil. Mag. S. 6. Vol. 43. No. 254. Feb. 1922. X 306 Occurrence of Enhanced Lines in the Are. the metal pole. Again, if air is blown through the conduct- ing vapours of the arc and their resistance thereby increased, spark lines are brought out prominently even with a carbon cathode. But the emission of spark lines by the are is strongest when the formation of a proper arc, and, therefore, a rapid ionization of the medium, is prevented or, at all events, retarded by means of a liquid which fills the are-gap. The flashing up of spark lines at the moment of striking an are between cool metal poles in air may be accounted for by the scarcity of conducting vapours at this early stage ; for | as soon as conducting vapours are formed in abundance re- sulting in the establishment of a steady are with rise in temperature and inerease in electrical conductivity (partly, perhaps, as a result of chemical reactions between the metal vapour and the constituent gases of the surrounding atmo- sphere), the spark-lines either disappear from the spectrum or show only as traces in the vicinity of the poles. The greater prominence of the spark lines when the are is taken in hydrogen, may be due perhaps to the relatively small amount of chemical reaction which takes place between the metal vapour and this gas. In short, the whole of our experimental evidence points to the fact that spark iines are emitted when electric currents 2re passed through media (vapours or gases) which possess a low degree of ionization. Since such a pro- cess involves the application of powerful electric forces giving rise to the establishment of high potential gradients in the medium concerned, we are finally led to conclude that the emission of spark lines is connected with the eaistence of strong electric felds. A simpie consideration shows indeed that for small pole distances the electric field within the arc- gap can be very great even witha small potential difference. Thus, for a pole distance of 0°05 mm. the intensity of the electric field with an applied potential of 110 voits is equal volts voye2,000 As we have shown in the course of this research spark lines are strongly emitted by the are only whilst the distance between the electrodes is precisely very small. | : Manchester and Paris, May 1921. P8074 XXIX. The Dependence of the Intensity of the Fluorescence of Dyes upon the Wave-Length of the Exciting Light. Bu y S. I. Vavmov, Lecturer in Physics at the University of Moscow * ee connexion between the energy of exciting light and _ the intensity of fluorescence can be expressed on the ground of a precisely established experimental law ¢ as [ia P se Than AX ee ee eu G1) where fi is the energy of fluorescence corresponding to the interval of wave-lengths of fluorescence dA,, F the integral energy of fluorescence, I, energy of exciting light, « the bank of the light absorbed by the fluorescent dye in a finite smali interval Ax. The coefficient pee ne LAN gives the energy of fluorescence per unit of absorbed energy; we shall call it “‘ specific fluorescence.” ‘The value of « is generally a complicated function of X. The first stage of the problem which arises here is the dependence of « upon X within an isolated absorption-band. An attempt at an experimental solution of this problem was made by Nichols and Merritt ft. These autnors found that the specific fluores- cence of Hosin and Resorufin increases towards long wave- lengths, for Resorufin by 2°7 times (at the change of wave-length of exciting lght from 520 wy to 600 wu) and by 1°63 times for Hosin (480 wu to 560 uy). The authors have stated a generalization of these results § without giving a theoretical explanation. We suppose that the question of the dependence of « upon wave-length inside an isolated absorption-band is closely related to the problem of the nature of these bands in Caan and solid bodies, and has equally considerable significance for the theory of dispersion and absorption as well as for its bearing upon the theory of fluorescence. The fundamental law (1) shows the secondary character of fluorescence, which is regulated by the value of the energy accumulated in the given molecular resonator. In * Communicated by the Author. + Cf. J. Hattwich, Wiener Ber. Bd. 122. Abt. 11 a (November 1913). t E. L. Nichols and E. Merritt, Phys. Rev. I. xxxi. pp. 376, 881 (1910). § E. Merritt, Phys. Rev. II. v. p. 328 (1915). a 308 Mr. 8. I. Vavilov on Dependence of the Intensity of ° the most probable supposition that such secondary processes as fluorescence and photochemical effect are conditioned exclusively by the intrinsic energy of the resonator and its constant mechanism, the value of « must be constant for the whole region of absorption: 7. e., for a physically simple absorption-band. Such a conclusion is maintained by the old resonance theory of fluorescence of Lommel * and the modern theories which consider this phenomenon as a tertiary process of lighting which accompanies this or that process of dissociation of molecules ft. A somewhat different result is given by the theory of Einstein, which will be mentioned later on. From our standpoint, the inconstancy of « inside an ab- sorption-band is an indication of its physical complexity. Such a band is a result of a superposition of several bands belonging to resonators of different types: 2. e., with a different x. The classical theory of dispersion and absorption in all its modifications is forced to apply broadly the sign & in the explanation of absorption- and dispersion-bands of liquid and solid bodies , which allows us to introduce any number of new empirical constants. The liberty in this operation is practically unlimited. It is essential that none of the expe- rimental curves can be considered from this standpoint as a physically simple one. In all these cases we can, therefore, expect a variability of «. Asa criterion can be used also another secondary process accompanying the absorption of light—the photochemical effect, as was shown by Lasareff §. Therefore the experimental result of Nichols and Merritt is an argument in favour of the classical theory of dispersion | absorption. Another supposition about the nature of broad absorption- bands was proposed by Kravec|| and qualitatively by Webster @. Their supposition consists in that occasional influence of fields of surrounding moving molecules (or parts of the same molecule) can in some way modify the frequency of a resonator towards both sides. The molecules will be distributed along the frequencies following the law * EH. Lommel, Wied. Ann. 111. p. 251, § 19 (1878). + E. Merritt, loc. cet. fe: Ciatek Kay ser, Handbuch der Spectroscopie, Bd. iv. p. 457 fi. B. J. an der Plaats, Ann. d. Phys. xlvii. p. 429 (1915). i iE. Lasaretf, Ann. d. Phys. xxiv. p. 661 (1907). || T. R. Kravec, ‘The Absorption of Light in Solutions of Dyes,’ Moscow, 1912, p. 106. D. L. W ebster, Phys. Rey. II. iv. p. 177 (1914). Fluorescence of Dyes on Wave-Length of Exciting Light. 309 o£ probability. The experimental curve of absorption is therefore a probability curve, enclosing the family of theo- retical curves with a variable parameter—its frequency. For an explanation of the experimental curves from the standpoint of this hypothesis it was unfortunately necessary to suppose at least two types of resonators. Therefvre in this theory the absorption-band of dyes is also a complex one. Many experimental facts and theoretical consequences necessitate a fundamental revision of the classical theory of dispersion and absorption. This is required by a consequent quantum theory, by a complete vagueness of the problem of the nature of damping constant, ete. Still we consider that the question of the physical simplicity or complexity of an absorption-band can .be solved independently of this or other modification of the theory of absorption. The way of solution is already indicated; it is an experimental deter- mination of coefficients characterizing the secondary pro- cesses of absorption inside the considered band. Hinstein’s theory of the simplest photochemical reactions * leads to the result that the coefficient of the velocity of reaction must be inversely proportional to the frequency of the active light. Considering in accordance with modern theories the fluorescence as a production of light accompanying the simple reaction of dissociation, we can hypothetically apply this conclusion to fluorescence. Therefore we can expect the following dependence of « upon X inside a simple absorption- band: ON where a isaconstant. For a complex band, « must be a totally different function of X. Thus we can interpret the experimental results of com- putation of « on the following lines :-— (a) If PAO ey ch te 0) where gis a more or less complicated function of A, the absorption-band is a physically complex one. -(6) If ee ee Te ie al A) the band is a simple one and the theory of Einstein is true. te), Hf fa ealiotemenener care reice ore. .8, CO) the band is a simple one and the theory of Einstein is not true. * A. Einstein, Journ: d, Phys. V. iii. p. 277 (1918). 310 Mr. 8. I. Vavilov on Dependence of the Intensity of EXPERIMENTS. As has been already mentioned, Nichols and Merritt found that the specific fluorescence of Resorufin and Hosin increases towards long wave-lengths. The scope of the present investigation is to state how general is this result. [tis of some interest to note that when this increase is a | general rule we can give a simple explanation to the law ot Stokes. From this standpoint, the fluorescence is excited chiefly or exclusively by the absorption in a small band situated towards long wave-lengths relatively to the resulting maximum of a complex band. The appearance of the curve for Resorufin suggests that the band is complex, while in addition to the principal inaximum, there are three secondary maxima from the side of the short waves*. The band of absorption of Hosin investigated by Nichols and Merritt differs considerably from that of Hosin studied by us (fig. 3). Unfortunately, among six Hosins at our disposal we did not find a dye very closely resembling the Hosin of Nichols and Merritt. These authors excited fluorescence by a Nernst glower which took the place of the slit of a large spectrometer. The narrow regions in the spectrum thus formed were used in exciting the solution studied. The intensity of fluorescence so excited was evidently very feeble; this can explain the very con- siderable deviations of computed points, especially in the case of Kosin. The diminution of errors of observation had a considerable significance for us when proposing to test the equation of Einstein (4), where the systematical deviations do not exceed 15-19 percent. in the conditions of the experiment. There- fore it was necessary to increase the intensity of fluorescence and to avoid the errors in determining the wave-lengths of the exciting light which arealmost inevitable ina prismatical resolving of light, especially in the yellow-red part of the spectrum. ‘Therefore we applied, instead of monochromatic light, the light transmitted through the light-filters quanti- tively measured. . Let us presume that the energy of the exciting source of light in the interval of wave-lengths ) ....4X+aA shall be Ih. The measurement of intensity of fluorescence is made in that place in the vessel with the fluorescing solution where light has already passed the layer of thickness d. The coefficient of absorption of solutions studied in the given * KE. Nichols and E. Merritt, doe. cit. Fluorescence of Dyes on Wave-Length of Exciting Light. 311 interval of waves is ac, where c is concentration. Following (1), the intensity of fluorescence in the named point of vessel will be Ky LS o WC e ered, Let further a light-filter be placed between the source of light and the vessel transmitting the part of light f(A). In this case the integral intensity of fluorescence will be ex- pressed by A2 F=\ CC IORUN) - COmene | SONE® | oto KO) Al where 24, Ay are the practical limits of the disappearance of the function Ty T(r) SUC. C1 7e (these limits depend evidently upon the applied light-filter) and «, is the specific fluorescence (2). . In the experiments of Nichols and Merritt and in the theory of Einstein, «, is an increasing function of X. In both cases we can apply to (6) the theorem of the middle value of a definite integral—+. ¢., we have ! ; F c= eh ee er) i Taf (hk). wee 72%. dX Ai where «’ is the middle value of «, corresponding to 0’, lying between A, and A;. When «) isa linear function (as follows from the theory of Hinstein and also from Nichols and Merritt’s experimental! results for Hosin and also ina long interval for Resorufin), and when the subintegral function in (7) is symmetrical relatively to ), a aia then it is easy to prove that «’ corresponds to A, through which passes the ordinate halving the area Ao o=| MGA) ECC pining yiee fone (8) a ; In cases when the subintegral function is only approxi- mately symmetrical (with which we are chiefly concerned), x’ corresponds only approximately to the halving ordinate of area d. The formula (7) can be applied evidently also in 312 Mr. 8. I. Vavilov on Dependence of the Intensity of the case where x, has a maximum inside of the band of absorption if the limits X,—), are sufficiently narrow. The values of Ih, f(A), ve, d were measured, e~ was calculated. The area ® was determined by graphical inte- gration by means of an Amsler’s planimeter. The error of each separate measurement necessary for the calculation of @ is the usual spectro-photometrical one. The method of lght-filters simplifying observations unfortunately complicates calculations exceedingly. The installation of apparatus is givenin fig. 1. Theimage Fig. 1: / —) ! | eae { ! { ! | { { at lf end | H | Neat oS | | i ! t | I j 1 ! ! ei Ug leet eat Nome} esl {eka ee | eee I. fl i el {oI feast eam | ' 1 Vig 1 Cd Pp wes wee ww eee ee ee a wm of one of the wires of Philips’s lamp (4 W., 200 candles, 106 volt) is projected by a large condenser /,; through a vessel W with water upon the aperture of the Zeiss aplanatic condenser K,. Through the lens J, passes a practically parallel bundle of light of a considerable intensity, and through the light-filter A enters a plane-parallel vessel F containing the fluorescing solution. Through the aperture d, in the screen the light of fluorescence is projected by means of another aplanatic condenser K,, the lens /3, and the auxiliary prism upon the slit of a spectro-photometer of Konig and Martens, B. The passage of rays from the same lamp L entering in the second slit of B is seen in fig. 1. Fluorescence of Dyes on Wave-Length of Exciting Light. 313 The voltage can be regulated by means of a rheosiat R. A trial has shown,that the change of voltage from 100 to 110 does not, influence perceptibly the relation of inten- sities measured by the spectro-photometer. The vessel F by means of slides not represented in the figure can be moved with a micrometrical screw along the screen. In sucha way d can be regulated. The same installation served for measurement of the absorption spectra of the dyes studied, light-filéers, and for determining the distribution of energy of the exciting light. For the measurement of absorption of light-filters, the widths of the slits of the spectro-photometer were taken as 0°1—0°2. mm. (measurements of intensity of fluorescence were made with open slits, 7. e. the spectro-photo- meter served rather as a photometer). Instead of vessel F a prism of total reflexion was put and the aperture d, was closed by a ground glass. Thus the light passing light-filter A enters the spectro-photometer. The second measurement was made without a light-filter. Hence the coefficient of absorp- tion was determined as usual. The absorption of solutions of dyes was measured in different ways: (1) by means cf a vessel with Schultz’s body; (2) by an immediate installation of an absorption vessel in the passage of the parallel light- bundle between J, and 8, ; (3) for determining the slopes of the curves of absorption the measurements were made in vessels of considerable thickness (6-8 em.). The Distribution of Energy. A Hefner’s candle was used as a standard. Its distri- bution ofenerg’y in the visible spectrum has been carefully measured by Angstrém *, and can be expressed quite precisely through the formula of a grey radiation : 1:85 BS OOLGO NO em 2 at 559) (A in microns). The formula was frequently tested. The measurements of Nichols & Merritt and Coblentz+t show that the perceptible deviations begin only in the red region of the spectrum, approximately from 605 wu: i. e., behind the limits of interval studied by us. We can evaluate the exactness of (9) according to the difference of the integral radiation of the Hefner’s candle, measured frequently, and * K, Angstrom, Phys. Rev., I. xvii. p. 302 (1903). + Cf. C. L. Nichols and E. Merritt, ‘Studies in Luminescence,’ p. 178 (Washington, 1912). 314 Mr.8. 1. Vavilov on Dependence of the Intensity of the integral radiation calculated by means of the law of Planck : 207d ib OE ch p) ée «at — J] with constants changed on the basis of formula (9). The measured value of eye integral radiation is Del) Welk —5 gr. cal, sec. em.2? © I= the calculated value Dey Uc sec. cm.” The deviations lie, according to the above, in the red and infra-red part of the spectrum. For the determination of distribution of energy of our source, the prism of total reflexion was put instead of the vessel F, the aperture d, was closed by a ground glass, and a comparison was made between the intensity of the light illuminating the ground glass with that of the light entering the right slit of the spectro-photometer. Afterwards the prism was removed, and by means of an achromatic lens the image of the flame of a Hefner’s candle was projected upon the ground glass in such a way that the part of the flame 1°5 em. above the wick was projected on the aperture d, *. The curve of relations of measured intensities of source and candle, multiplied by (9), gives the distribution of energy of our source of light. The curve determined in such a way satisfies sufficiently the formula of grey radiation: 2°67 Ths Q NTE se At = 1a elle onan In Table I. are given the values of I, measured with those calculated by (10), wherein @ is taken as 4°710. 10°. TABLE I, r T) meas. Ta eale. 0-410 0:125 0°153 0:435 0°258 0:235 0°455 ~ 0380 : 0:345 0-473 0:480 0-489 0-490 0-660 0-645 0:515 ; 0°880 0-905 0548 1:325 1°350 0°565 1:°588 1:560 0-580 1:870 1-910 0°590 2-080 2-050 * A. Becker, Ann. d. Phys. xxviii. p. 1029 (1909). Fluorescence of Dyes on Wave-Length of Exciting Light. 315 Measurements were repeated several times during our work, giving always the same result. We used light-filters, partly coloured glasses, partly coloured films put between two glass plates. Some light- filters were prepared by ‘as in collodion and gelatine films on glass. The curves of absorption of light-filters were measured at least twice. Dyes. The fluorescing dyes were selected so that their bands of absorption covered the whole region accessible to visual investigation from 400 pu to 600 pu. Most careful measure- ments were made with water solutions of the following dyes :-— Fluorescein (Ferrein)+ KOH 9-7. 1OgS and 4°8577 a - i - \ , . =i ss Kosin 8 extra C. (Bayer) ... 1°2.107° 4. Rhodamin B extra (Bayer)... 1:4.1078*. em. Fig. 2. Tan aaee | aoa tH | 1,3 aca 2s Be ray Rede ula Ea SCE AE OR Bee a ew Mee ee Ht le ae EMH eis AL0um HHO “60 500 Kosin S gives a rather weak fluorescence ; still it was the only one whose band of absorption followed sufficiently that of the Eosin studied by Nichols and Merritt. In fig. 2 are 316 Mr. S$. I. Vavilov on Dependence of the Intensity of given the curves of absorption of all three dyes. The or- dinates of Fluorescein correspond in the figure to 0°163 a and for two other dyes wc. The curves are perfectly mono- typical, differing only in the different situation of maximum and the absolute value of wc. This can be proved by a superposition of all three curves. In fig. 3 the curve of Fig. 3. TTP o—Btucreszein. | Bee. | wbsetel CEN gs —\( a ane : 4 ee a aa ea 2 eu Re # | f seaa trae es | ECHL ACE 7 ei ae ALO pans 440 460 $0 500 Kosin is removed to a distance of 45 wy: t.e., the interval separating the maximum of Fluorescein and Hosin. The curve of Rhodamin is removed to a distance of 65 wy. The - ordinates of maxima of all curves are reduced to the same value. On the same figure are marked the points corre- sponding to Hosin of Nichols and Merritt, maximum removed to a distance of 38 up. The established regularity allows of several conclusions being drawn :— 1. All dyes studied by us have the same mechanisin of absorption in the visible part of absorption. 2. The coefficient of absorption is sufficiently approximated by ac=f(X—Np), which is in contradiction with the classical theory for the simple band and likewise for a complex one. 3. The invariability of the form of the curves by removing their maxima to a distance of 65 wy, serves as a criterion of the optical cleanness of our preparations. 4. The same fact serves, of course, as an indirect indication of physical simplicity of the bands of absorption studied. Fluorescence of Dyes on Wave-Length of Exciting Light. 317 The intensity of fluorescence for every dye is measured with 8 to 10 light-filters with a constant thickness of absorbing layer d. For “Fluorescein and Eosin two measurements were made with different d. Every measurement was repeated at least three times on different days. For ascertaining the amount of diffused light in the mea- sured light of fluorescence, the solution of dye in the vessel I was replaced by pure water. In all cases the quantity of the diffused light was not more than | per cent. : 7. e¢., could be disregarded. Results. We abstain from the reproduction of lengthy tables with elements necessary for graphical calculations of ® on the basis of (8). The values of I,, xc, e-7*¢, f(A) were taken from experimental curves for every 5 wy. As an example of the subintegral function so obtained we include fig. 4, representing nine curves obtained for Hosin with different light-filters. Beside every curve is given the value of ®, obtained by a graphical integration in the relative units, the observed value of intensity of fluorescence, also in relative units, and the relation K=h- As shown in the figure, the curves are approximately symmetrical. According to the above mentioued, we can con- sider that « so computed | corresponds more or less to the wave- length through which passes the halving ordinate of area ®. On every curve this ordinate is marked by a dotted line. The Tables II., I1I., IV. contain the following data :— (1) lght-filter “used, (2) the value of X through which passes the halving onplinee: (3) F, (4) d, (5) «. The tables give also d and concentration. The tables correspond to the most careful measurements. TABLE II. Fluorescein. c=3'7.1076 255 ; d=0°80. r. Ine fe Ks arte fo 446 pp 2-940 61:0 4°81 . 10-2 figs ee ee 448 0:0579 1S 4°70 i a ear oe 449 0°920 19°3 4°77 LO: ae 453 3°70 7-50 4°93 UR eae. 482 14-60 296 4°88 le dete 506 3°62 7°45 4°86 ho eee 510 4-90 97-0 5:05 Disenn | Ole 0:218 4°55 4°80 318 Mr. 8. L. Vavilov on Dependence of the Intensity of From Table II. it is seen that « is constant in the limits of errors of measurement of dand F. The greatest deviation reaches 7 per cent. There is no perceptible increase or Fig. 4, * T 1 HT = BS i \ { : 1 1 ia ! | ag | . | Ces r 10 1| Sho 5 oO! : i 4 t --—-+ 00 | @p: 30 lol CES peabel Ze | v Te Asta C0 SSSA NM SPACE Ais ; at or oe { 0! a a i LN | } ve ! Life pt —-4- ye — el ai | hl ei ype 60 | ABO L Soo | S44 , | ai ary RE | Jarra |_| iG on@ent vavion 4,1040°S5 em, d= 0,83 cm decrease of x. bootie to Einstein’s equation (4), it must be a linear increase to 16 per cent. within the interval 446 to 919 wp. Fluorescence of Dyes on Wave-Length of Exciting Light. 319_ PARLE EET. Hosin § extra. c=1:20.10752%,; d=0°83. Ne F, Q. Ke ce 473 pe 0-0845 9:3 9-05 . 10-3 es. 475 0-247 27-7 - $91 Vie. 477 0-160 18-1 8-80 HO ae 487 0-470 52:0 3-02 Ae 495 0-615 68:9 8-97 Sa 528 6:42 725 8-87 (a 534 3:00 330 9-09 PGE... 542, 0-570 6°55 8-69 ay. 547 1-48 167 8°85 TABLE LV. Rhodamin B extra. c=2°4. 10-* =; a Wica Ne 1} . K. ale 478 pp 0-364 10-6 3:43. 10-2 10 4... 492 8:25 23:9 3-46 ES 513 1:67 48°5 3-45 Oy we gai4 11-4 318 3:58 ieee. 547 29°i 809 3-60 er. 552 15°8 447 3:53 Baek 559 12°6 339 3-7 2 579 1:05 © 29:2 3:60 Table III. gives the same result for Hosin. The deviation of « reaches 5 per cent.; according to Hinstein, there must exist the increase of 16 per cent. For Rhodamin there is a small increase of « with wave-length, which lies still inside the limit of errors of experiment. ‘The greatest deviation is 8 per cent. This systematic deviation can be possibly explained by the existence of some discordance between the Angstrém equa- tion (9) and the true distribution of energy in the spectrum of the Hefner’s candle. According to Hinstein, the value of x must increase in this case by 21 per cent. In other less careful measurements of « the deviations amounted to 10 to 14 per cent. Conclusions. (1) Within the limits of errors of observation the specific fluorescence of our dyes is independent of > within their band of absorption. 320 Prof. W. B. Morton and Mr. L. J. Close on Hertz’s (2) According to all above-mentioned, this result is equi- valent to the conclusion that the curves of absorption of our dyes are physically simple ones. (3) The theory of Kinstein is not confirmed, but the devia- tions required by this theory are so small that they exceed only a little the experimental errors, and we are unable to make a definite statement. (4) The intensity pf fluorescence radiated by a definite molecular resonator depends only upon the value of the absorbed energy and upon the mechanism of the resonator. _In the case of an excitation by a white light we can there- fore write Xa Parl Ly ac. eT?" dks ee ee AL ; (5) The result of Nichols and Merritt shows probably only the physical complexity of bands of dyes studied by these authors. ; This work was carried out in the Physics Department of the Scientific Institution of Moscow, to the Director of which, Prof. Dr. P. P. Lasareff, [am much indebted for much valuable help and for his interest during the course of the work. XXX. Notes on Hertzs Theory of the Contact of Elastic Bodies. By W.B. Morton, M.A., and L. J. Crosz, .A., Queen's University, Belfast”. 1. Introduction. rBX\HE elastic problem of the deformations and _ stresses which arise when two bodies, having continuous cur- vature, are pressed together was solved by Hertzf in a classical memoir. “He showed that the area of contact is, in general, an ellipse, and that the displacement in the neigh- bourhood of the contact could be expressed by means of potential functions belonging to a certain distribution of surface-density on this ellipse : viz., that reached as the iimit of a uniform solid ellipsoid. In view of the complicated nature of the mathematics, Hertz contented himself with establishing some general conclusions and with drawing a diagram of the lines of principal stress which was partly conjectural and which was afterwards found to be erroneous * Communicated by the Authors. + Hertz, Miscellaneous Papers (Engl. trans.) p, 146. See Love, ‘Elasticity’ (3rd ed.), p. 191. ; Pry eae = ocr Theory of the Contact of Elastic Bodies. 321 in some of its features. The subject was further developed by Huber* for the contact of two spherical surfaces, in which case the potentials can be expressed in finite forms. For this simple case the question of the correct forms of the lines of principal stress was settled by Fuchst+, who obtained the result by carrying out a laborious process of arithmetical integration. — The present work had its origin in an endeavour to apply the method of expansion in zonal harmonies in order to obtain numerical values for the stresses, etc., in Huber’s ease of axial symmetry. The formulz of Huber contain an elliptic co-ordinate, the parameter of the spheroid through the point which is confocal with the circle of contact. This is inconvenient for purposes of calculation. On the other hand, a good deal of labour was found to be needed in evaluating the series to which we were led, so it is doubtful - whether one method has an advantage over the other in this respect. It does not, therefore, seem worth while to reproduce the details of our analysis ; but some of the results obtained, with regard to the magnitudes and directions of the principal stresses, may’ be of interest as adding a little to what is already known. We give graphs for the stresses along lines running from the centre of contact in directions making angles 0, 30°, 60°, 90° with the normal. A further note is added on the limiting forms assumed by the lines of principal stress at a distance from the contact. It is shown that the characteristic features of these curves, as discovered by Fuchs, can be simply deduced from the solution given by Boussinesq of the problem of a body acted upon by a pressure concentrated at a point on an otherwise free plane boundary. 2. Outline of Method. Let the axis of z be the inward normal to one of the bodies at the centre of the circle of contact. Suppose a distribution of matter on this circle equivalent to an oblate spheroid of vanishing axis, the whole mass of the distribution being equal to P, the normal force with which the bodies are pressed together. This makes the surface-density at distance r equal to 3P(a—r?)*/2 7a’, where a is the radius of the circle. Let ¢ be the ordinary inverse potential of this distribution and y the logarithmic * Huber, Ann. d. Phys. vol. xiv. p. 153 (1904). + Fuchs, Phys. Zeitschr. vol. xiv. p. 1282 (19138). Phil. Mag. 8. 6. Vol. 43. No. 254. Feb. 1922. if 322 Prof. W. B. Morton and Mr. L. J. Close on Hertz’s potential Jlog(e+7) dm. Hertz showed that 4c x (dis- placement) is the resultant of the two vectors (1) —slope{2¢+ (1—2c)y}, (2) (1—o)dq, parallel to the normal, where w= rigidity, c= Poisson’s ratio. - From this the strain-components can be obtained in any chosen system of co-ordinates, e.g. polars (r @ ¢) or cylindri- cals (a zd). It was found most convenient to use a rather unorthodox combination of these, the cylindrical components being expanded in polar series. The advantage of the (a z @) system lies in the relation to the fixed direction 3, and in particular the loci in the body at which they vanish and change sign. As the result of some further calculation, these loci are found to have approxi- mately the forms shown on fig. 5. There is a region, surrounding the origin and based on a circle very slightly smaller than the circle of contact, within which all three principal stresses are pressures. The boundary of this region is marked p,>=0. Springing from the top of this is the locus p3=0, above which both p, and ps are negative ; below it, and outside the former region, there is a hoop- pressure, while the smaller principal stress in the axial plane is a tension. The main stress p; is everywhere a pressure. As the area of contact shrinks to a point, the locus p,=0 Theory of the Contact of Elastic Bodies. 327 disappears and p3=0 becomes a cone, one of whose genera- tors is shown by the broken line. This is Boussinesq’s case * of pressureatapoint. Inside the cone there is hoop-pressure, outside hoop-tension. The angle of the cone is found to Fig. 5. rm 2 ‘s be independent of the elastic constants and to satisfy the equation cos’? 0+cos 0=1, | giving cos0=3(V75—1), O=51° 50’. In this limiting case, p, is everywhere negative. 5. Lines of Principal Stress. The other question suggested is the limiting value approached by the angle between p; and the radius vector as 7 increases in any direction 6. To find this it is sufficient to use the results of the simple case just referred to, in which the area of contact is regarded asa point, and @=P/r. On working out the strain-components in polar co-ordinates the inclination of the principal stress p; to the radius vector is found to be given by the equation tan 2a = 2(1—2c) sin @ cos 6/{ (5 —4c) cos? 6 +3 cos @—(1—2c)}, * Love, ‘ Elasticity,’ p. 189. 328 = Hertz’s Theory of the Contact of Elastic Bodies. As @ increases from 0 to 90°, y also runs through the same range but the variation of wis marked by a sudden transition from a slow and steady increase over most of the range to a very rapid increase near the end. For example, with o=%5 we find wr =10° 98/ for §=80°, abe = 90° for 6=904 This means that p, keeps near the direction of r as the normal is departed from, being inclined to r on the side of the normal ; but just before the radius vector reaches the boundary there is a rapid tilting of the direction of py, bringing it perpendicular to r, z.e. to the boundary. Along with this, of course, the magnitude of p; becomes zero. For the purpose of obtaining the forms of the lines of principal stress, it is convenient to tabulate (@—w), the inclination of p, to the fixed direction of the normal. This is found to have a maximum value at about 02=79°. The lines of principal stress are thus two families of similar curves, intersecting at right angles and having points of. inflexion along the radii 9=+79°. Since the direction of the tangent is known for each point of the plane, an approxi- mation to the forms of the curves can be found*. The result is shown on fig. 6. The forms are similar to those given by Fuchs f for the outer curves in the more general case. ‘The broken line drawn to the intersection of the two curves is the locus of inflexions. A remark may be added about the displacements in this case. In the account of Boussinesq’s solution given by Love t it is mentioned that the particles move towards or from the line of action of the apptied force according as they lie outside or inside a cone whose angle is given by cos? 6+cos 0=1—2c. It may not have been noticed that the differential equation of the “lines of displacement” can be integrated in finite form. Using polars, the differential equation is dr/r= —d0(1+ cos 8) {4(1—c) cos 0—(1—2¢) }/ . sin 0{(3—4e) cos 0+2(1—o)}, with the integral —logr=A log (1—cos @)+ B log (cos 0+ C) + const., * @’Ocagne, Caleul Graphique et Nomographie, p. 155. + See Love, loc. cit. p. 196. ft Loe. crt. p. 190. Natural Convective Cooling of Wires. 329 where A=(8—20)/(5— 8c), B=(11— 260 +4 160°) /((5—6c)(3—40), : C=2(1—c)/(8—4c). Fig. 6. A curve of this family is shown on fig. 6, marked “ displt.” The point nearest the axis lies on the cone above referred to, as shown by the upper broken line. Its angle in the present case (c=4) is about 74°8°. XXXI. Natural Convective Cooling of Wires. ip AnH. avis, asc.” [From the National Physical Laboratory. ] (1) Introduction. > ie object of this paper is to investigate the apparent discrepancy between published data for the natural convective cooling of wires and the hydrodynamical theory of the phenomenon. ‘Convective cooling” is taken to refer to the total heat * Communicated by the Author. 330 - Mr. A. H. Davis on Natural transfer from a hot body by the medium of a fluid moving past the surface. Such cooling is said to be “ natural” or ‘free ” when the fluid is still except for the streams set up by the heat from the hot body itself. The theory involves the dilatability of the fluid, the hydro- dynamical equations of motion, the Fourier equations of heat-flow, and the appropriate boundary conditions. From this point of view Boussinesq* has studied both natural and forced convection for similar bodies immersed in an infinite inviscid fluid. His formule may also be derived from the; principle of similitude by simple consideration of the variables involved, and may be extended to viscous fluids by the same means. The following formula is obtained for the heat loss by natural convection from similar bodies similarly immersed in viscous fluids. Al/kO=F(eglab[k?)flcv[k), . . - . (Cy) where j h=heat-loss per second per unit area of the body, k=thermal conductivity of the fluid, c=capacity for heat of the fluid per unit volume, @=temperature excess of the body, a=coefficient of density reduction of the fluid per degree rise of. temperature, g=acceleration due to gravity, /=linear dimensions of the body. For a given kind of gas cv/k appears to be constant as required by the kinetic theory, and the formula becomes simpler h=(kO/)E(egad/l2).. . . . . (2) The formula involves an assumption that a and g always occur as a product, that is, that the expansion of the fluid is negligible, except in so far as it alters the weeght of unit volume, and thus supplies the necessary driving force for the convection currents. This restriction may impose limits to the temperature excess for which the formula is applicable for a given series of bodies. If only the temperature and size of the model are varied, the gaseous constants (c,a,and £) and gravity (g) remaining the same, the formula becomes A=(@DEOP). . 3 ae, * Boussinesq, Comptes Rendus, exxxii. p. 1382 (1901). Convective Cooling of Wires. a3l In a previous paper * where this formula was studied, it was shown that in general, both for forced and for free convection, the hydrodynamic formule are in very promising agreement with published data, except in the case of free convection from hot thin wires. The present paper investi- gates this apparent disagreement. 4 e “a e (2) Convection loss from wires. Hxperimenters have almost invariably conducted wire experiments at temperatures much higher than those used with larger bodies, and this suggests two possible sources of the apparent failure of the theoretical formula. Firstly, it may be that in this extreme case it is no longer legitimate to assume that “a” and “g” occur only as a product, the mere volume changes of the air now having sensible eftect. Secondly, it may be now necessary to allow for the change of the conductivity and specific heat of the fluid due to the temperature rise caused by the hot wire. The present paper shows the remarkable improvement which follows an attempt to allow for this second effect, so that even for thin wires the equation appears to be substantially correct. The formuia may be put in a more convenient form for our purpose. In (1) “A” refers to unit area, and if the body is a long cylinder, “h” is obviously unaffected by the length. It readily follows that for long cylinders (diameter dj the heat loss H per unit length per degree temperature elevation is given by H/k=F(egd8a0/k?). . . 2. . (4) Writing v for k/c (since cv/k is constant for a given kind of gas) and regarding “a” T and “g” as constant, we have Hic EP @Hryes . 2s 2B) Consequently, if data for the natural convective cooling of long cylinders be plotted on a graph with H/k as ordinate and @d*/y? as abscissa, the result should be a single curve independent of the size of the cylinders, and also of their temperature excess, if appropriate allowance be made for the variation of & and v with temperature. Further, from the method of its derivation, formula (5) would appear to hold for all fluids for which cv/k is the same. For gases the * Davis, Phil. Mag. xl. p. 692 (1920). + The convection currents depend on the density change relative to tne density of the cold fluid, so that “a” must be regarded as applicable to the cold gas, and independent of the temperature of the wire. BaL Mr. A. H. Davis on Natural value of this variable appears to depend on the number of atoms in the molecule, and has thé same value for the diatomic gases oxygen, hydrogen, and nitrogen as it has for air, so that these gases should give convection results in agr eement with the curve for air. (3) Method of examination of data. Published data have been examined in the light of the formula, and to obtain a convenient tabular summary the following o general procedure has been adopted. The published data have been converted into c.g.s. units, corrected for radiation if necessary, and plotted on a graph from which values have been read off for even values of temperature excess. This gives corresponding values of Od? and H. To obtain 6d?/v? and H/k it was necessary to decide on the values of £ and v appropriate to the particular experiment. For a hot wire in air at room temperature (15° C.) the appropriate values are neither those for air at the temperature of the hot wire nor those for air at 15° C. As a first approximation the values actually taken were those for air at the mean of these two temperatures. In cases where the pressure was not atmoshane allowance has been made as before, and in addition it has been assumed that, as indicated by the kinetic theory of gases, both the conductivity & and the viscosity 7 are independent of pressure, and consequently. the kinematical viscosity v has been taken as inversely proportional to the pressure of the as. In following this general procedure certain auxiliary constants have been used, as explained in the following paragraphs :— (a) Radiation correction.—Data for heat losses are often published in the form of a total loss, radiation being included. Langmuir *, who used platinum wires, published his data with the radiation eliminated, and in this paper results . obtained by other experimenters for platinum surfaces have been corrected, using his values for the emissivity for radiation. For steam pipes, total heat losses given by Petavel have been corrected by the amount attributed by him to radiation, an amount equivalent to 04x10°™ x (T*—1,*) t calories per sq. cm. per sec., where T and T, * Langmuir, Phys. Rey. xxxiv. p. 401 (1912). } 5x10~'°(T*—T,*) B.T.U. per square foot per hour. Convective Cooling of Wires. 333. refer respectively to the absolute temperatures of the body and its surroundings. (b) Convection constants of gases.—Below are given data to express the variation of & and v with temperature. The viscosity (n) of a gasis given by Sutherland’s formula and the density by p=po (273/T), where T is the absolute temperature : from these the kinematical viscosity v (=7/p) has been derived. The conductivity (k) has been calculated from the viscosity (n) and the specific heat at constant volume (c,), using the formula k/cyn=constant, as given by the kinetic theory of gases. Hucken* has shown that the constant has the value 2-50 for monatomic gases and 1°90 for the diatomic gases hydrogen, oxygen, nitrogen, and also for air. The specific heats at high temperature have been obtained from the data given by M. Pier t, who has given formule which, for the above gases, may be written Cy=A(1+0-0002T), where A depends on the nature of the gas. Langmuir{ adopted this method of calculating conductivity when dealing with heat convection from the point of view of conduction through a film of gas adherent to the hot body. Thus, for the gases considered, Wiener se od hs SCO (emit OS) eee: Ga Ran cen mea ar where nas PERO MMeS 5) eo eee (8) PREV STI 2000/00 31.) a a 2) P= 2 Lapp pemnibe sty Fe duc ih oh(L.O) and where the constants have the values given in the following Table :— TABLE I. (¢.g.s. units.) oak e010 ee, Cia ae pe ae Pee Ua:Or |) 1a 0161 | 1-293 | Hydrogen ......... ao Ta S/O he 0:09 | Oxygen: ots eli 127 0145.) | 143 * See Tables by Kaye and Laby, or Fisher, Phys. Rev. xxiv. p. 385 (1907); xxix. p. 146 (1909). * Eucken, Phys. Zeit. xii. p. 1101 (1911). + Pier, Z. f. Elektrochem. xv. p. 536 (1909) ; xvi. p. 899 (1910). { Langmuir, loc. ert. Byaye Mr. A. H. Davis on Natural Calculation from these data yields the values for k and p given in Table II. TABLE JI, Air. Hydrogen. Oxygen. oG, | | MOze Bnew: 101. k. v. 104. &. v. | Ot Rivers ees SS ee ee Ora Oar a ecOals 37 + 09 0:58 “14 17 | 058 | O15 4-1 1-1 0°61 15 108) | M072. F023 47 16 0:75 “24 2007 10:86. 1 Oj85 57 24 | 0-91 36 | 300 | 101 | 0-48 65 3°3 1:00 50 | 400 114 | 0-62 74 4-2 1:20 66 | 500 | 1:27 0:78 8:2 5:3 1:33 83. | 1000 | 1:89 aye | aes. | a4 |) ea gen ees | (4) Summary of collected data. Following the procedure explained, Tables III. and IV. have been compiled summarizing a wide range of data for convection losses from cylinders. In the experiments of Table III. the wires were used mainly in free air, or in enclosures so large that the free air figure would evidently be obtained. It appears, however, that heat losses from wires in even a small enclosure (a few centimetres diameter) are not different from those obtained in an unlimited medium. In Table III. the results by Ayrton and Kilgour were obtained using water-cooled enclosures 5°08 em. internal diameter, and they agree excellently with data for similar wires in practically free air, as given by Langmuir and by Kennelly. Table IV. refers to an enclosure of 2°06 cm. diameter, and this is still fairly large compared with the diameter of the wire, and the results for air at one atmo- sphere agree satisfactorily with the curve derived from data for free air already considered. Results given in the tables are plotted in fig. 1. The sources from which the summary was compiled are briefly indicated below, the distinguishing letters corre- sponding with those used in the tables :— (A) Langmuir * conducted experiments practically in free * Langmuir, loc. crt. Convective Cooling of Wires. 305 TaBueE III. Natural Convection from Long Cylinders in Air. (Logarithmic values of @d°/y? and H/é are given.) Temperature Excess (° C.). | 7 Dijana |) ade 100°. 200°. 600°. | 1600°. Source. pce. | | aa | | eae | | ea | fade | joa a | bopeet lair sp? k Petal ad 1) A 7 Mal bat v2 k | ae '0:0081 | 5:55 014) 5:90 |0-21| 5-98 032) .. |... | .. | eae, | 000404) ... |... | ... | --- | #82 (0-25) 422 0°20] 5-74 \o-24 ee 00051 | 4:20 0-22| 4:55 [0°31| 463 0-31) ... | ... Bee deer 000601) ... |... | .. | - | 302 |0-29| 492/026 4-44 0-29 ee 00074 | #69 199 304 024) S11 |o-2 | | Bie 4:.: | 0-0114 | 359 0:29) 3:67 |0-29) ... | . ES ae | 001262) ... |... | .-. | -- | 380 |0:32| 370030] 3-23 /0-29 ht 00152 | 3:62/0:30| 3-97 0-34| Fos [036]... |... | pas bei | 0°02508| ... | ...| ... | +. | 270 0-44] 2600-41 | 312 /0-34 [eee ooss5 |... |...) Fos (053) 116 joss). |... PAu eh 0°051 | fetal 162 0°55) 1:52 0°52) 1-05 0-49 Dole 00691 1-94 0:38 0:02 058 | | Bae 01106 | | 0-48 (0°63 | | | cee ae 0:51 255 |L-11| 262 |1-17) era eae (Cee 5:08 5:55 |1°88| 5°62 |1-93 Sy | ee: 30-48 7-98 [261/795 267 | | |D*......{ O-0114 300 0:26 307 |0-23 | | ipD** .... 0-0114 2-20 0°35 227 Deoan ae HUTTE aE | BISSAU. 0:0691 | IES on ODO ee OroUn mer) Coe | so D ** 0:0691 0°55 0°67 | 062 0°67 Pee | | | | | * Air at 3 atmosphere pressure. ** Air at 2’atmospheres pressure. air with fine platinum wires at temperature excesses from 200° C. to 1600° C. He carefully corrected for radiation. His results may be represented toa certain degree of approxi- mation by the formula H « d¥/°6"*4, which does not agree with (3), where & and v are regarded as constant; the disagree- ment is brought out even more clearly if @d? is plotted against H. 336 Mr. A. H. Davis on Natural Taste fy, Convection Loss from Wire in Various Gases, at various pressures. Wire diam. 0°1106 cm. Temperature Excess (° C.), 'g q |Pressure 100°. 300°, 900°. | ource, as. Atm. pd? | HH -| @d* | sen eae ae pe k y? EN om ee oe ae Air .......] O01 | 248 | 040 | 259 | 0-45 | 2:34 | 053 10 | 0-48 | 0°63 | 0:59 | 0°64 | 0:34 | 0-67 10 | 248 | 0-95 | 2:59 | 0:97 | 2:34 | 0-99 | 100 | 448 | 147 | 459 | 144 | 434 | 739 | | siaeey eta Hydrogen| 1 | 482 | 034 | 4-91 | 0:34 | 4-69 | 0-38 10 | 2:82 | 038 | 2:91 | 0-40 | 2-69 | 0-44 10 | 0:82 | 0:60 | 0-91 | 068 | 0-69 | 062 100 2-82 | 1:00 | 2-91 | 1:03 | 2-69 | 0-99 Cathie: Oxygen ...| O71 | 2:47 | 0:40 | 2:56 | 0:44 | 2:30 | 0°57 | 1:0 | 0:47 | 0°66 | 0:56 | 0:66. | 0:30 | 0-67 10 | 247 | 0-93 | 256 | 0-95 | 2:30 | 0:96 100 | 447 | 147 | 4:56 | 1-45 | 4:30 | 1-45 In fig. 1 all the Langmuir values of Table III. are plotted. (B) Ayrton and Kilgour * worked with fine wires at more moderate temperature excesses. Table III. contains values for all their wires, but for convenience fig. 1 only shows values for the largest and the smallest. It should be mentioned that one wire (0°:0074 cm. diam.) gives results at low temperature excess in poor agreement with the line in fig. 1. However, study of their curves shows that the behaviour of this particular wire is exceptional. (C) Petavel tf has given data for the total heat loss from the oxidized iron surface of steam pipes of various diameters up to 1 foot, a diameter which is ten thousand times the * Ayrton & Kilgour, Phil. Trans. A, clxxxiii. p. 371 (1892). + Petavel, Proc. Manchester Assoc. Hngineers, 1915-16. ae OH Convective Cooling of Wires. 337 diameter of Ayrton and Kilgour’s smallest wire. Values with radiation eliminated are plotted in fig. 1. (D) Kennelly * investigated the heat losses from wires in very large enclosures at various pressures, thus introducing another experimental variable. Fig, 1. OO Bit Hee OQaAIGOW > 5 4) ad?/" Sees eae Spee es ee. wee 0 5 uw Langmuir + has corrected these for radiation, and his corrected values have been used in compiling Table III. The points, if plotted on fig. 1, are in excellent agreement, but to preserve a clear figure only two points are shown, these corresponding to the extremities of the range. (E, F, & G) Petavel t, who worked over a wide range of pressure (1/10 to 100 atm.) and up to 1100° C. excess, has given data by means of which the performance of different * Kennelly, Trans. A.LLE.E. xxviii. (1) p. 363 (1909). + Langmuir, Trans. A.i.E.E. xxxi. (1) p. 1228 (1912). { Petavel, Phil. Trans. A, excvii. p. 229 (1901). Phil. Mag. 8. 6. Vol. 43. No. 254. feb. 1922. Z ee ‘ 338 Natural Convective Cooling of Wires. gases may becompared. He studied the cooling of 01106 cm. wire, using a cylindrical water-cooled enclosure. Values have been plotted on the graph of fig. | instead of on a separate graph. The radiation correction is not large and in its calculation the enclosure temperature has been taken as 300° absolute. EH, F, and G refer respectively to air, hydrogen, and oxygen. The point for the case of 1/10 atmo- sphere and 900° C. temperature excess is seen to lie rather high for each of the gases considered. (5) Conclusion. The curve of fig. 1 shows that very diverse data give points which, in general, all lie on one line independent of the size and temperature excess of the object, and of the nature and pressure of the gas involved. Ittherefore appears that the hydrodynamical theory of convection represented by our formule is substantially satisfactory even in the case of hot thin wires, provided allowance* is made for the temperature change of the properties of the fluid, an allow- ance brought into prominence by the high temperatures to which wires are usually heated in convection experiments. High temperatures are also used in experiments on jorced convection f:om wires, and it is desirable to notice that in this case approximate agreement with the appropriate formula is found even without allowing for the temperature variations of the properties of the fluid, but that agreement is improved when such allowance is made fF. Langmuir developed a theory of convection in which bodies are considered to be surrounded by a film of gas through which the heat is transmitted by pure conduction, and with certain assumptions, and allowing for the tempe- rature change of the conductivity of the gas, he obtained results in agreement with experiment for free convection. However, he concludes that “for forced convection the film theory does not seem to apply ” ¢. It may be counted a superiority of the hydrodynamical theory that the treatment which reconciles theory with axperiment for free convection is satisfactory for forced convection also. Besides testing the hydrodynamical theory, fig. 1 effects a graphical correlation of diverse convection data for various gases and gas pressures over a wide range of wire diameter * Only approximate allowance is made in this paper. t Davis, Phil. Mae. xli. p. 899 (1921). if Langmuir, Trans. Am. Electrochem. Soe. xxiii. p. 829 (19138). er ey eee eee ha “a Frequency of the Electrons in the Neon Atom. 339 and temperature excess. The tendency of the curve appears to be, at its upper extremity, towards the form lee OG ee, | ELLY. and at its other end towards | ee —CONstaMt 8 to yyy sd dhe Le) Hquation (11) would imply that for large bodies the heat loss per unit area is independent of the size, and (12) that for very fine wires the heat loss per unit length is independent of the diameter. Kquation (12) would also imply that the convective cooling of wires, very fine and very slightly heated, is a measure of the conductivity of the fluid, at any rate for fluids of such similar molecular constitution that cv[k is the same. May 1921. XXXII. The Frequency of the Hlectrons in the Neon Atom. By Laurence St. C. BrouGHatt *. * attempts have been made to arrange the electrons in the atoms of the inert gases, but it has been found that no stable arrangements can be obtained if only electro- static forces are present. We are thus obliged to search for some other force, and the most reasonable assumption is that the electrons are in motion in circular orbits and so have an acceleration towards their axis of rotation. The main objection to this theory is that the electron since it rotates should radiate energy, and the diameter of its orbit should gradually diminish. Hxperimental evidence shows, however, that this is not the case, and so if we are to allow the electrons to rotate then we must assume tbat they radiate no energy. ‘This assumption made by Bohr has met with great success in the case of hydrogen, where he assumes that the electrons radiate energy only when dropping suddenly into a different orbit. In consequence of this we have assumed throughout this paper that no energy is radiated by the electrons in the neon atom under normal conditions. The number of electrons in the neon atom is ten, and Langmuir has shown that the most satisfactory arrangement of these ten electrons is to place eight of them at the corners of a cube and to place the other two within the cube. He * Communicated by the Author. 7-9 a 340 Mr. L. St. C. Broughall on the Mrequency is led to this arrangement by a consideration of the valency of the elements, and also his hypothesis shows that the pro- perties of neon should be similar to those of helium and argon. There is also considerable evidence that the atom or molecule of neon is spherical, and the diameter of the atom has been calculated by two methods. It has been found by means of the viscosity of the gas, and secondly, W. L. Bragg has given a value for “d” the diameter of the atom of on alan consideration of the diameters of the atoms of elements which have atomic numbers near that of neon. Representing the diameter found by viscosity measure- 99 ment by ‘a” we obtain the following values for neon : o=2°35 x 107°, d=1:30%105%-em. Nowthenaloethameag sd is very much less than the value for ‘“‘o”’ and it is supposed that when two molecules collide they do not come into contact. W. L. Bragg considers'*that his value for the atomic diameter is the distance between the outside electrons. iia consequence of this we shall use the value d=1-30 x 10-8 em. for the real diameter of the neon atom. In order to obtain a spherical atom by the rotation of eight electrons at the corners of a cube there are several possible axes of rotation, but the simplest case is to assume that the atom rotates about three perpendicular axes XN’, YY', ZZ’. YY’ is determined by the mid! pomits of jihe surfaces of any opposite pair of sides of the cube. In a similar manner XX’ and ZZ" are determined by taking different pairs of sides. Using these axes of rotation the atom will have a radius equal to one half the diagonal of the cube, and this will be represented by “c.” Further, for simplicity the lengths of the edges and of the surface diagonals will be represented by “27” and “‘2s” respec- tively, since the representation of ‘f/’ and “s” as functions of “c’’ tends to make the work obscure. We e see that these axes of rotation give all the electrons in the outer shell or octet similar velocities, and it is for this. reason of svmmetry that these axes have been adopted in preference to all others. The question now arises as to what will be the positions of the other twoelectrons. We desire if possible to make the forces acting on the other electrons the same for all of them, _and the only possible positions for the electrons e, and ey is to place them on one of the axes XX" YY, on GA. ii is quite immaterial which axis we take, put ihe electrons are assumed to be upon the axis XX! thr oughout this paper, and =) are at a distance ‘“‘7”’ cms. from the nucleus, which latter is of the Electrons in the Neon Atom. 341 at the centre of the atom. Since these two inner electrons are upon one of the axes of rotation, it follows that they only have two velocities, but this is sufficient to make the shell mapped out by these electrons spherical in shape. Fig. 1, Y’ The next point that arises is with regard to the laws of force holding between two electrons in the atom, and as there is no experimental evidence to the contrary we imagine that the inverse square law holds good. We are now in a position to calculate the forces“acting on any electron in the outer shell. We shall resolve the forces along the three perpendicular lines eg, e,¢., and e,e;. Since the electron is to-be in equilibrium under the forces present, it follows that the total force acting along any one of the three perpendicular lines must be zero, and so we obtain three equations. Let us first take the forces acting along the line ¢,¢s, and == further let the positive direction be eg. The charge on the electron is “‘e.” Charge of nucleus =EH=10e. Mass of electrons when moving with a small velocity =m. Angular \ 342 Mr. L. St. C. Broughall on the Frequency velocity about YY’=o,. Angular velocity about XX'’=o3. Angular velocity about ZZ'=@,. Using this notation we have :— a Z : (1) Force due to eg= — a (2) Force due to eg=force due to = — ee. a a (3) Force due to e,=— Te 2 \ (4) Force due to nucleus= + —— 2 l ee a (5) Force due to eyo (ee Vetbias Sea ae ee (CDs ds 2 —l 6\ Force due + Ea Se ce ee (6) Force due to eg = + C48 pees ec.) = [(r—l)?+57]* (7) Forces due to e;, e3, and e,=0. (8) Force due to centrifugal force owing to rotation about axis YY’= —a,’s “m= —o,’ml, (9) Centrifugal force due to rotation about axis ZZ’ = —W,’s ém= — "ml. Since the sum of these forces must be zero, we obtain the equation SOC aL iter aes e’(r+ 1) (.) ‘dee. ALP am 28° [r+l)2+s?]2 e*(r—1) RC If we resolve along the line e,e, and equate the total force to zero we obtain the equation oe. e3 eed e7l i) garmin: Sp eapm peut eT: : Zz +E a + mo,7l +m@3"l- [(r— 1)? + as of the Electrons in the Neon Atom Similarly resolving along the line ee, we have 343 B07 Ley e7l see) ee e?] 7 gem N48] + me@,7l + mws;*/ SyPirdcting equations (II.) from (III.) we have — mo2l—mol=0; and therefore @), —Wo- Replacing ‘ w,” by “ a, in equation (1.) we obtain S9e ls. Chae e?(r +1) rv) r= apt gat [ (rt 1)? +5? ]3 uae Care) [autos and * 7,” 3] +2mo,7l. Another equation is required before we can find @,, a3, This equation can be found by finding the force acting on electron “ é” the total force to zero along the line ee¢;,, and equating Performing this operation we have sae arieey Loy ica De lock : (V.) —> THis +58 * Te pe ee aa ae may be expressed in the form 7 397 Ae*l(r +1) 4e*l(r —1) i a tla al ee (VI) 2ma"l= 4p® r[ (rtl)?+s7]2 rl (r—l)?4+s?]3° Substituting this value for 2mw,7/ in (IV.) and dividing throughout by “ e?”’ we obtain the equation > (VIL) 34 —p- r+l 4] daa 0 eran? Pe Aly av - (eaarey + PUT aS as This equation contains no unknown quantity except “ Now c=6°9 x 107° (VITE.) 1-04 510" — r+3°75 x 107° ~ [(r 43°75 x 10-9)? + 282 x 10-*]3 : 1 Oe i r—3°75 x 10-9 { r ~ [@—3'75 x 10-9)? + 28-2 x 10-* ]# ’ ate ° =o ee, eee : 7”? ot j2=3"(5 % 1072 and) s=o751 K 10-9 Petting these values for s, 1, and c, we obtain the equation 344 Frequency of Electrons in the Neon Atom. Solving this equation in “‘7” we find that it is satiéfied when r=6°1 x 107° em. From cae (VI.) we ‘may ane an ae for @, : oo A(r+l) (r—l) J Ce Salva ae Kal a ee = Ay? [ (rl)? +5715 re 24 5215 Putting e=4:774x 10°" E.8.U. and m=8: 8 x 107% orm we find that o,=6°28 x 10" radians/sec. ee equation (II.) from equation (I.) we find that | oe 2m Lol es 2 2 (X.) es : Ge ep (+e ES =ml(@, Os is Substituting in this equation for e, 7, /, s,m, and w,, we find that 3 = 4:58 x 1016 radians/sec. From this we can now find the frequency about the axes ZZ' and YY' (n,) and about the axis XX! (ns). Also we may calculate the instantaneous linear velocities of the electrons in the outer or inner shells. Taking the case of the outer shell we determine the linear velocity due to rotation about ZZ’ or YY' (v,) and due to rotation about »:0:¢ (v3). Tabulating the results we find that :— Frequency of the electrons about XX! =n =~ ap IO poe Frequency of the electrons about YY’ and ZZ’ =n 100 ao: Angular velocity of the electrons about xoG = 0;,—4°58 x 10"rad_/sec. , Angular velocity of the electrons about YY’ and ZZ! = @,= 6°28 x 10!° rad./sec. Instantaneous linear velocity of the outer electrons about XX'’=v,== 2°98 x 108 cms./sec. Instantaneous linear velocity of the outer electrons about YY’ and ZZ’ =v, =4°08 x 10° ems./sec. Instantaneous linear velocity of the inner electrons about YY' and ZZ'’=3°83 x 10° cms./sec. The value of ‘v” will be seen to be small compared with the velocity of light, in consequence of which it follows that we have committed no appreciable error in not correcting for the variation of mass with velocity according to the equation m,=m(1—v/c?)? where “c” is the velocity of light. October 15, 1921 =e es XXXII. An Attempt to determine whether a Minimum Time is necessary to eweite the Human Retina. By J. H. J. Rooms. Se. De 1g a letter to ‘Nature,’ published on April 7th, 1921, Sir Oliver Lodge suggested that possibly there might be a certain minimum time necessary for a beam of light to liberate an electron from a photo-electrie surface. Thus, assuming that the area of wave-front which can contribute energy to a very small resonator, like the electron, is of the order of 2/7, he showed that for ordinary sunlight the surface would have to be illuminated for about ;;, sec. before an electron could be liberated. It seems, however, improbable that such a comparatively large interval of time should be necessary to excite an electron, especially as photographs, in which the action of the light on the sensitive film is probably due to the liberation of such electrons, can be taken in ordinary sunlight with exposures shorter than this. _ Asa matter of fact, at the time that Sir Oliver Lodge published the above suggestion in ‘ Nature,’ Dr. Joly and I had already undertaken some experiments with the view to seeing if any such effect held for the human retina. The general principle of the method is described by Dr. Joly in a letter to ‘Nature’ for April 14th, 1921, but perhaps a recapitulation of it is desirable. The general idea of the method was to cause a parallel beam of light to revolve very rapidly by reflecting it from the surface of a rotating mirror. The revolving beam was then viewed through a narrow slit placed at a considerable distance from the mirror; and it is obvious that if the slit and the light-beam can both be kept narrow, then the resulting flash, seen by the eye, may be made very short indeed by running the mirror at a high speed; and also by making the distance between the mirror and the slit large. In order to get as high a speed of rotation of the mirror as possible, it was mounted on the top of the vertical spindle of a Legendre Centrifuge. This centrifuge was capable of speeds up to about 8000 r.p.m., as was determined with a revolution counter and stop-watch. The mirror itself was a small speculum metal one, of about 4 cms. diameter, manufactured and ground optically plane by Sir Howard Grubb. During the course of the experiments this mirror * Communicated by Prof. J. Joly, F.R.S. 346 Dr. J. H. J. Poole on the Minimum was re-polished by Hilger Ltd., as it was found that there was rather an undue amount of irregular reflexion taking place at the surface of the mirror. To avoid, as far as possible, any stray light from the mounting of the mirror or from its back reaching the eye, they were both carefully blackened by means of a small turpentine flame, which gave a very good deposit of lamp-black. The chief difficulty experienced was with the optical arrangements. It is plain that it is impossible to realize the ideal of a perfectly parallel beam owing to the finite size of the light-radiant. The best we can do is to use as small a radiant as possible, and also as long a focus lens as we can without reducing the illumination too much. In the preli- minary experiments a slit was used as the radiant, the light from a projection type of electric glow lamp being concen- trated on it with a condensing lens. Between this slit and the mirror a long focus lens was placed, and the distanee of the lens and the slit was so adjusted that the image of the latter was formed on the second slit behind which the eye was placed. The lens is naturally placed as close to the revolving mirror as possible, as the image formed will be magnified in the ratio of its distances from the two slits, while the speed of the image will be proportional to the distance of the mirror from what we might perhaps call the eye-slit ; thus the distance between the lens and the mirror only tends to enlarge the image without increasing the speed, and should hence be kept as small as is convenient. ' In the first trials this lens was of about 2 metres focal length, which was hardly sufficient. These experiments were conducted in an ordinary room, and it was found impossible to keep it sufficiently dark for the eye of the observer to become truly dark-adapted. The distance between the mirror and the eye-slit could only be made about 15 metres, a distance which it was thought could be increased with advantage. On beth these accounts it was decided to remove the apparatus to the cellars of this building. ~ These cellars were particularly suitable for the purpose, as a distance of 30 metres could easily be obtained between the mirror and the eye-slit, and no trouble was experienced in keeping them nearly perfectly dark. Some modifications were also introduced into the arrangements as a result of our previous trials. Thus we had already found that even at the maximum speed of the centrifuge the light was plainly visible in the mirror. This showed that a flash of the order of about 10~° sec. or rather less was visible if repeated sufficiently often. To be able to deal with one flash ot this duration, Time necessary to excite the Human Retina. 347 however, it is necessary to use some-method of only illumi- nating the mirror at intervals for less than a complete revolution, thus ensuring that only one flash reached the eye for each light interval. This end was brought about by interposing a revolving disk, with a sector of suitable size cut out, between the lens and the radiant. The dimensions ef the sector removed from the disk were arranged so that when the disk was running at about 120 r.p.m. the length of flash should be about z, sec., which would be less than one revolution of the centrifuge. The frequency of the flashes seen by the observer in this case was about two per second. ‘To drive the disk, it was mounted directly on the shaft of a variable speed repulsion motor. As a matter of fact, in practice the disk was driven at a slightly greater speed than 120 r.p.m. This increased the frequency ot the fiashes slightly, but diminished the chance of any light from two successive revolutions of the mirror reaching the eye. The optical arrangements were also improved. New lenses of various focal lengths were obtained from Adam Hilger & Co. Finally a lens of about 4 metres focal length was found to answer most satisfactorily. If a longer focal length than this is used, the question of obtaining sufticient illumination in the beam becomes rather troublesome. Different forms of radiant were also tried. The plan of using a slit as radiant was abandoned as suflicient light could not be obtained from it. After trying various otler light sources, a small galvanometer electric lamp supplied by the Cambridge & Paul Scientific Instrument Co. was found to work most satisfactorily. This lamp had a single loop filament, and by turning this loop parallel to the direction of the light-beam, a very fair approximation to a line source of light could be obtained. The method of observing the flashes was as follows. One observer took up his position at the eye-slit, which was fitted with a small tube so that the direction of vision might be preserved constant in the dark. ‘The other observer then started the centrifuge and ran it up to the required speed. In the meanwhile the revolving sector shutter was kept permanently open, so that what the first observer perceived was a steady light in the mirror. When he had satisfied himself that this light was visible, i.e. that the direction of the light-beam was correct, he gave a sigual, and the sector shutter was started. This shutter was not, however, run continuously, but it was stopped in the open position occa- sionally, so that the observer might satisfy himself that the light-beam was reaching his eye correctly. This precaution 348 Minimum Time necessary to Excite the Human Retina. ig very necessary, owing to the fact that between the very feeble flashes that are visible when the revolving shutter is running, there is a great tendency for the direction of the eye to alter, as in the dark intervals there is Tepe to fix its position very exactly. It is also very necessary that the dark lie eee of the eye should be nearly perfect. For this purpose the observer should be at least twenty minutes in the dark before attempting to make any trials. -The difference in seasitivity between ordinary daylight vision and fully dark-adapted vision is very large. Selig Hecht, in a paper published in the ‘General Journal of Physiology’ for May 1920, states that the dark-adapted eye is easily from 5,000 to 10,000 times more sensitive. To attain the maximum sensitivity it is necessary to be at least 45 minutes in the dark, but after 25 minutes the increase in sensitivity is very slow, hence probably the period of 20 minutes adopted in these experi- ments was sufficient to ensure that the eye would be nearly fully dark-adapted. As regards the results obtained with this apparatus, it was found that a flash of 2 x 1077 sec. was still visible. A flash of duration 8 x 107° sec. was, however, found to be invisible. This flash was, however, found to be visible if viewed directly without a slit,in which case the time of flash would be about 2°4 x 107‘ sec., thus confirming the previous result. The energy in each of these flashes could be approximately calculated from the candle-power of the radiant and the various dimensions of the apparatus, and it was found that, neglecting any loss by reflexion, etc., the energy in the visible flash was about 4 x 10°" erg, w rhile that in the invisible flash was about 8 x 10° erg. As to whether the invisibility of the shorter flash is due to its smaller duration alone or simply to the decrease in the energy available, is uncertain. It seems, however, that the latter hypothesis is more probable as the intensity of the longer flash was excessively feeble. The problem of getting more light into a flash of this nature is not very easy, as the amount of light which can be obtained depends only on the brightness of the radiant, and it is not very easy to get a suitable one which would be brighter than an electric light filament. Anare lamp might have given more light, but would not have been so convenient as a slit, and a condensing lens would have been required. Probably the increase in brightness would also not have been very large. Iveagh Geological Laboratory, Sept. 1921. bepege 7 XXXIV. The Analysis of Sound Waves by the Cochlea. By H. KE. Roar. (From the Department of Physiology, London Hospital Medical College.) * NHE variations of air pressure which produce tlie sensation of sound are conveyed to the /fenestra ovalis through the foot plate of the stapes. The mechanics of this conveyance is well described by Wrightson, who points out that the liquid in the internal ear is practically incompressible, therefore movements of the stapes can take place only by mass movements of liquid and by yielding of the Nentbrane closing the fenestra rotunda f. Mass movement of the liquid can take place in one of two ways. «Liquid may pass up the scala vestibuli through the helicotrema and down the scala tympani, or the scala media may be pushed towards the scala tympani. The resistance to these movements is in the former case the inertia of the mass of liquid to be moved, and the friction of the liquid against the walls of its containing tube, and in the Pee the tension of the basilar membrane (Reissner’s membrane is usually represented as being flaccid). The relations of these two movements are shown in figs. 1 and 2. Fig. 1 represents the cochlea uncoiled, and arrews indicate ie direction of movement of liquid if it were to take place through the helicotrema. Fig. 2 represents a cross section through one part of the coiled tube, and an arrow indicates movement of liquid when the basilar membrane is pushed towards the scala tympani. The relation of movement of liquid along the scale to deformation of the basilar membrane is difficult to assess because the viscosity of the liquid and the tension of the basilar membrane are unknown, but the following factors are in operation. The resistance fp movement is high i in such small tubes as it is inversely proportional to the fourth power of the radius in circular tubes, and the shape of the scalw is such that there is a relatively larger surface to area than if they were circular tubes. Such ‘high resistance will damp any movement, so that the movements of the liquid will be “ dead beat,” such as are those associated with the recognition of tone i. The width of the basilar membrane is oreater the further up the cochlea it is measured, therefore it will be * Communicated by the Author. + Sir Thomas W rightson, ‘The Analytical Mechanism of the Internal Ear,’ Macmillan & Co. (1918). { H. Hartridge, J. Physiol. vol. liv. Proce. p. vii (1920). 350 Mr. H. EB. Roaf on the Analysis of deformed by less force, and at the same time there isa larger area for any pressure to act upon it. Thus the distal end of the cochlea will be moved by a lower pressure than will affect the proximal end. Diagram to show variations in area of the cochlear tubes and basilar membrane. _The cochlea is represented as uncoiled. From Sir Arthur Keith, appendix to ‘The Analytical Mechanism of the Internal Ear.’ Diagram to show cross secticn of cochlear tube, the connexion of the scala vestibuli and scala tympani with the stapes and fenestra rotunda respectively are indicated by a break in the wail closed by a dotted line. From Sir Arthur Keith, appendix to ‘The Analytical Mechanism of the Internal Ear.’ : S.V.=scala vestibuli; 8.T.=scala tympani; S8.=stapes; R.= fenestra rotunda; H.=helicotrema; B.=basilar membrane. 5B’.=surface view of basilar membrane. The mechanical conditions on which movement of the organ of Corti depend are practically unknown. If the basilar membrane consists merely of stretched strings and the rods of Corti behave as hinged structures, the force. & Sound Waves by the Cochlea. 351 required to produce a definite movement in either direction will be inversely proportional to the length of fibre extended, 2. é., Young’s modulus, or stretching force per unit area extension per unit length for circular fibres Sd iiayel If, on the other hand, the organ of Corti (including the rods of Corti and the tectorial membrane) acts as a rigid structure, the problem is similar to that of the bending ofa bar, and the force required to produce the same amount of movement will be inversely proportional to the cube of the length of the structure, 7. e., Young’s modulus=4FL*/bd*/. The mathematical treatment of this problem is not possible because we do not know the nature of the unions nor the physical properties of tie structures concerned, but it is possible that the variation in ease of deformation may be inuch greater than is to be expected from the relative widths of the various portions of the basilar membrane. ‘The problem is further complicated by the fact that the pectinate portion of the basilar membrane» must also be stretched, even if the organ of Corti acts as a rigid bar. Thus we see that rapid variations in pressure will not be able to set a long column of liquid in motion, but the fenestra rotunda being at atmospheric pressure there will be greater differences of pressure across the lower end of the basilar membrane, with the result that the proximal end of the basilar membrane will be deformed, thus stimulating the hair cells at the proximal end of the cochlea. Slower variations in pressure will cause movement along the scale without producing sufficient difference in pressure to deform the proximal end of the basilar membrane. Some- where along the basilar membrane the lower difference in pressure w iil be sufficient to deform the basilar membrane, with the result that the hair ¢ells in that region will be stimulated. Accordingly, it is evident that the impedance due to the mass and friction of the perilymph will tend to produce greater differences of pressure at the narrower end of the basilar membrane with rapid changes, whilst slower changes will cause lesser ee so that the basilar membrane will be moved at a wider part. Irregularities in the area of the scale will cause the impedance to vary, so that certain ranges of frequencies will be more accurately analysed than others. It must be pointed out that other factors, such as the mass 392 Mr. H. E. Roaf on the Analysis of of the organ of Corti, will aid the analysis of the pressure variations. | The factor that is analysed is acceleration, which is pro- portional to the second differential coefficient of pressure in relation to time. ‘The pressure-time relation is a sine curve, Zin be = —sin X. . | It must be remembered that a tone depends not on a single rise of pressure but on a succession of pressures, so that continuous movement occurs only at the part of the basilar membrane where the inertia and friction produce sufficient pressure to deform the basilar membrane. The hypothesis of M. Meyer * does not treat the subject from the same physical standpoint as that described herein, because his hypothesis is based on the amplitude of the wave and not its frequency, nor do other hypotheses such as those of H. ter Kuilet and of Hurst ¢ explain analysis on the_ same basis. It is assumed that the number of nerve-fibres and end organs are sufficient to account for the various frequencies distinguishable by the ear. The mechanical conditions described above do not depend upon the presence of an organ of Corti, but the problem of tone analysis in birds requires further’ investigation. The exact anatomical arrangement of the cochlea and the range of tone analysis must be compared. In order to show that a mechanism such as that described can analyse complex pressure waves, several models have been made of which the simplest for purposes of demonstra- tion is that shown In fig. 3. A glass U tube was made with a narrowed portion in the connecting limb to damp the oscillations, and several smaller tubes were blown on one limb of the U. The open ends of the side tubes are closed by glass bulbs of different sizes to represent the different elasticities of the parts of the basilar membrane. The apparatus is filled with water and a rubber bulb is fastened to the U tube at A: pressure variations are made by squeezing A. | With such a large scale apparatus it is not to be expected that it will respond to such rapid pressure variations as most sound waves, but it will respond to slower variations. For therefore * M. Meyer, Arch. f. d. ges. Physiol. vol. Ixxvii. p. 346 (1899) and vol. Ixxxi. p. 61 (1900). + E. ter Kuile, Arch. f. d. ges. Physiol. vol. Ixxix. p. 146 (1900). { ©. H. Hurst, Trans. L’pool. Biol. Soc. vol. ix. p. 821 (1895). Sound Waves by the Cochlea. 395 instance, if one squeezes A gently and slowly there is a move- ment of liquid in the far limb of the U and also rapid oscillations in the tube nearest to A. These rapid oscilla- tions are due to irregularities in the muscular contraction, Fig. 3. Diagram of apparatus for the analysis of pressure waves. A =rubber bulb for producing pressure variations, and with a proper apparatus it should be possible to show the individual contractions of which the complete movement is composed. In fact a similar apparatus is being used to analyse various sorts of tremors in patients. Models more like the cochlea have been made, but they are not so convenient for purposes of demonstration. Phil. Mag. 8. 6. Vol. 43. No. 254. Feb. 1922. J 354 Mr. F. M. Lidstone on the Summary. The factors of inertia of fluid and friction of perilymph against the walls of the scale, with the variation in elastic tension of the basilar membrane and organ of Corti, must have considerable influence in analysing complex pressure variations, and such factors are capable by themselves of analysing coarser pressure variations such as the irregular contractions constituting a slow muscular movement. This process of analysis would satisfactorily account for summation and difference tones. XXXV. Notes on the Measurement of Absolute Viscosity. By Frank M. Lipstone*, T appears to have generally escaped notice, that in any form of absolute viscometer where a variable head obtains, it is not strictly accurate to use the mean head when computing the viscosity. When the difference between the initial and final heads is at all large as compared with the mean head, the error introduced becomes far from negligible. If we take the general formula of Poiseuille, iy Me nV mee Oc Here A stands for the head, V for the volume, and ¢ for the time of efflux. K represents the remainder of the constants, with which we are not at present concerned. By taking all the heads, from the initial to the final, summing the times, and integrating the expression we get nV Fda 2 Keo” where H is the initial and F the final head. This gives us in place of the original equation oR; F) Vi 7) og. T * Communicated by the Author. a Measurement of Absolute Viscosity. 300 The factor oe HF rapidly approaches the mean head 3(H+F) as the initial and final heads become large as compared with their difference ; and as long as this fact is not lost sight of, there is no reason why the more cumbersome calculation should be necessary. The subjoined table shows at a glance the extent of the error involved. | Ratio of mean Ratio of Ratio of head to differ- initial to mean head to Percent. error. ence in heads. final head. corrected head, H+F) H xH+F) H-—F - y loge 7 2n+1 2n +1 2n 2 ant m log, oy 100( log, a = 1) 2 e0) C3) e 7) = g 137383 37°33 # 5 1:2071 20:71 4 32 1:1369 13°69 1 3 1:0986 9:86 13 2 10397 3:97 2 a 170186 1:86 is a 1/0094 0-94. + 2 1:0052 0-52 10 22 1:0008 0-08 20 2x 10002 0:02 30 61 100015 0-015 The formule which have been advanced to correct for the kinetic energy of the liquid are almost as numerous as the factors they contain. It might, therefore, be not out of place to work it out once again from first principles. Tf W is taken as the total work done in the system, and Wx as that expended on kinetic energy, then we can write V n =n (found) x cu ) The total work W is equal to V.d.g.4(H+F), where V is the volume which passes any fixed point in the system - d the density ; H the initial, and F the final head. The allowance to be made for the work expended on kinetic energy is rather more elusive than it appears at first sight. 2A 2 356 On the Measurement of Absolute Viscosity. Let AYY,D represent a section of the capillary and ABCD the volume which passes in unit time. Then if AY is made equal to 2. AB the paraboloid of revolution AOD (which is equal in volume to ABCD) represents the actual flow of the liquid which passes in unit time. Then if OX=hA and radius OY=*, we have: apparent kinetic energy of volume AYY;D is equal to mrdh’ — ne Actual kinetic energy of volume AYY,D at any instant of time in the tube is equal to mT ee rrdh? a yr? — y?)?dy = ie (2) 0 Kinetic energy of volume equal to AYY,D which leaves . the tube, is twice the kinetic energy possessed by the volume AOD, namely, awd ard h? Ox al y (7? —y?) dy = 7 5 tears (3) The value of (8) is exactly double that of (1). Hence if a liquid flows through a eapillary with a mean velocity of v, the work done in giving it its kinetic energy is 2x mv’ if m is the mass which has left the tube. If then we take this value and integrate between the initial and final heads, H and F respectively, we get for the total work expended on kinetic energy, na Ni AV dv? Hog Wr= cron ada shee (1 (A+)? and HE Wr ay 8v? {1 — (H+FY } ee ; 39(H + I) s i are e = — - — On the Phenomenon of the ‘Radiant Spectrum.” — 357 The amended equation of Poiseuille thus becomes ( Sy? “(1 HF ) _ wrigdt (H— F) | iil): i H+ Fy) 8Va here | 39(H + F) | He ) NA The apparent mean velocity v being equal to er it Tt becomes more convenient for purposes of calculation to write it thus : . vi amr gdt( A a) Vd( HH? — F%) 8Va loge = d7at(H + F)3 loo H * S° # XXXVI. On the Phenomenon of the “ Radiant Spectrum”’ observed by Sir David Brewster. By C. V. Raman, WA, Palht Professor of Physics in the Calcutta University *. N a paper on “ The Scatter ing of Light in the Refractive Media of the Eye,” published in the Philosophical Magazine for November 1919 {p. 568), I discussed the explanation of the luminous effects observed when a small brilliant source of light is viewed directly by the eye against a dark background, and especially of the marked difference between the cases in which the source emits white light and highly monochromatic light respectively. In both cases the source appears to be surrounded by a diffraction-halo ; but the structure of the halo is markedly different in appearance. In the former case, the source appears to shoot out streamers of light radiating from it in all directions, these streamers showing marked colour, and in fact appearing as elongated spectra in the outer parts of the halo. With the mono- chromatic light-source, on the other hand, the radiant structure of the halo is not observed, and we have instead surrounding the light-source a halo showing dark and bright rings and exhibiting a finely mottled or granular appearance. It was pointed out in the paper that these effects are pre- cisely what might: be expected on the hypothesis that the halo seen surrounding the sourve is due to the diffraction of light by a large number of particles of constant size—pre- sumably the corneal corpuscles—present in the refractive media of the eye. The radiant structure of the halo in * Communicated by the Author. Read before the Royal Society of Edinburgh, Noy. 7, 1921. 3098 On the Phenomenon of the “Radiant Spectrum.” white light and its granular structure in highly mono- chromatic light is, on this view, due to the field of light diffracted by individual particles varying arbitrarily in intensity from point to point as the result of the mutual interference of the effects of the large number of such particles. A closely analogous structure of the luminous field may be observed in diffraction-haloes obtained in other ways, e.g. with the aid of a glass plate dusted with lyco- podium powder through which a small distant source of light 1s viewed. “The facts mentioned above provide a very simpie explan- ation of a remarkable observation made long ago by Sir David Brewster, and communicated to the Royal Society of Edinburgh (Proceedings, vi. p. 147%; see also Phil. Mag. September 1867), which has not up to now been satisfactorily accounted for, and to which my attention has been recently drawn by Dr. G. G. Knott while I was on a visit to Edi burgh. Brewster noticed that when a spectrum of a small brilliant source of white light is formed, either by a prism or by diffraction, and viewed directly by the eye, a patch of light is seen lying in the continuation of the spectrum well beyond its violet end and exhibiting streamers radiating from its centre. That this is a diffraction-eftect is shown by the fact that a similar and even more striking effect may be observed in the diffraction-halo due to a glass plate dusted with lycopodium held together with a 60° glass prism before the eye, when a small distant source of white light is viewed through the combination. The prism disperses the image of the source into a spectrum. It also disperses the diffraction- halo, and sinee the diffraction-rings are of different size for the different wave-len othsand are shifted to different extents owing to the dispersive power of the prism, the achromatic centre of the halo is shifted laterally to a considerable extent, its new position generally lying at a point much removed ea the violet end of the spectrum of the source itself. The elongated spectra which form the radiating streamers are rotated through various angles by the dispersion of the prism, being drawn out laterally on one side and shut up or drawn together on the other side, and they then appear to diverge froin the shifted position of the achromatic centre of the halo, which, as remarked above, now lies well beyond the violet end of the spectrum of the source. The analogy between this effect and Brewster’s phenomenon is so striking that there can be no doubt that the latter is essentially of the same nature, the diffraction in this case being due to the structures within the eye itself. * See also a brief note by Tait, Proc. R. 8. E. vi. p. 167. fe 3591) XXXVII. On the Form of the Temperature Wave spreading by Conduction from Point and Spherical Sources ; with a suggested application to the Problem of Spark Ignition. By i. Tayuor Jonss, D.Sc., Professor of Physics in the University Colleye of North Wales, J. D. Moraan, B.Sce., and R. V. WHEELER, D).Sc., Professor of Fuel Technology in the Unaversity of Shefield*. T is natural to assume that the power of igniting inflam- mable gaseous mixtures possessed by an electric spark depends essentially upon its ability to impart heat to the gaseous mixture, and that in this respect it behaves similarly to any other source of heat. Most, if not all, investigators of the phenomena of the ignition of gases by electric sparks have entered upon their work with this assumption. An obvious inference is that the heat just necessary to ignite a given mixture under standard conditions should bea constant quantity, and that the heat contents of all sparks that are Just capable of igniting such a mixture, no matter how those sparks are produced, should be the same. It has, however, been definitely established by experiment that the thermal energy of a spark that is just capable of igniting a given mixture varies considerably with the electrical conditions under which the spark is produced. This has led to the view that electric-spark ignition is not mainly, if at all,a thermal process; that it is, in fact, mainly - due to ionization. Inasmuch as the rate of reaction between oxygen and inflammable gas is increasedif the mixture is ionized, and the attainment of the ignition temperature of a gaseous mixture is mainly dependent upon this rate of reaction, it can readily be understood that means of ignition that perform the dual function of ionizing a mixture and imparting heat to it (either successively or concurrently) may be more effective than those which only impart heat, if such there be. The common means of ignition of gaseous mixtures— heated wires or surfaces, or jets of flame—are powerful sources of ionization, though they are usually supposed to cause ignition by virtue of the heat they supply. The debateable question would therefore appear to be: Are electric sparks so much more potent sources of ionization than, say, heated surfaces as to warrant the process of ignition of gaseous mixtures by the former being labelled “ionic ” in contradistinction to a “thermal” process of ignition by the latter? Or, assuming that there are marked differences * Communicated by the Authors, 360 Prof. Taylor Jones, Mr. Morgan, and Prof. Wheeler on between the ionizing powers possessed by various igniting agents (or by various types of electric. sparks), do these differences effect changes in the rationale of ignition of a given series of inflammable mixtures ? The object of the present paper is to show that, given gaseous mixtures or series of mixtures of constant com- position, a variation in the energy of the electric spark required to ignite them, dependent on the character of the spark, is to be anticipated on thermal considerations alone. — The ignition of a gaseous mixture depends primarily on the attainment ‘of a sufficiently rapid rate of reaction within a sufficient mass, or, expressed on a thermal. basis, on the heating of a sufficient volume to a sufficient temperature. If whilst the requisite volume of a given mixture was being raised to the required temperature it could be isolated, so that no heat could escape to the surrounding gas, then (presuming that no complicating effect were introduced by the mode of imparting the heat) it might reasonably be expected that the quantity of heat required to ignite that mixture would be constant. An unavoidable condition in practice, however, is that whilst that portion of the mixture adjacent to the source of ignition is receiving heat, it is also communicating heat to the surrounding gas. The distri- bution of temperature at any instant throughout the volume of the mixture will vary with the character of. the source, even though the total quantity of heat delivered by each source may be the same. A starting-point in an examination of the manner in which thermal energy is conveyed to a gaseous mixture by an electric spark, is given by an inquiry into the manner in which the distribution of temperature varies with time when a source of heat corresponding in general character with a spark is introduced into a gas. Llectric sparks can be divided into two main classes: (i.) those of exceedingly short duration (such as single capacity sparks), and (11.) those of relatively long duration (such as inductance sparks). It will be advantageous, therefore, to consider the thermal distri- bution in a gas afforded by hypothetical sources of heat of {i.) Instantaneous, and (1i1.) continued character. In order, also, to obtain some idea of the effect of the volume of an electric spark on its igniting power, or “ incendivity,”’ point and spherical sources of heat will be considered. The subject is here treated mainly as a problem in thermal conduction in a uniform medium, and, to avoid complication, the medium chosen is air. The numerical results obtained may not represent at all closely the manner in which heat = Form of Temperature Wave spreading by Conduction. 361 actually spreads from a spark in aninflammable mixture. In a full treatment of the problem, the effect of the pressure- wave emanating from the spark, and the effects of convection, radiation, conduction through the electrodes, and variable conductivity of the medium must be taken into account. Above all, the fact that during the process of ignition chemical combination is proceeding causes the temperature wave to be more elevated than it would be as a result of a purely physical transmission of heat. The general effect of the heat added to the system by chemical ‘action would be to intensify those differences that are shown to arise from purely physical causes between one type of source of heat and another. A temperature wave-form in which a con- siderable volume of gas is raised by conduction of heat to the temperature required for active combination to take place, would be accompanied by a greater supply of “‘chemical heat” than one in which the region of high temperature is more restricted ; the chemical heat would theretore more greatly enhance the effect of the source in the former than in the latter instance. Our immediate purpose being to inquire what influence, if any, the manner of supply of heat to the medium has upon the wave-form, we will suppose that the total quantity of heat (Q) supplied i is constant. This quantity we will assume to be supplied to the medium either at a point or throughout a space symmetrically surrounding a point which is taken as origin. Under these circumstances the ges ature, @, at any point in the. medium ata distance, 7, from the origin, and at a time, ¢, after heating begins, can be deduced from the well-known equation : | tO (dh i dd kr Gee g + 2k —iy Aine 5 > E - 5 (1) which expresses the fact that the excess of heat flowing into any elementary concentric spherical shel] through the inner surface, over the heat flowing out to regions beyond, is equal to the heat stored during the same time in the element. The coefficient, k, is the thermometric conductivity of the medium—that is to say, its thermal conductivity divided by its thermal capacity, c, per unit volume. We will consider the supply of a constant quantity of heat, @, under four different conditions : (1) Instantaneously at the origin. (2) At the origin at a uniform rate during the time T. (3) Instantaneously over a spherical surface of radius a. (4) Instantaneously throughout a spherical volume of radius a. 362 Prof. Taylor Jones, Mr. Morgan, and Prof. Wheeler on The following numerical values, appropriate to air as the medium, will be assumed throughout: k=0°5, c=0-00014, both in ¢.g.s. units. The total heat supplied, Q, will be taken as 0'001 calorie, and the original temperature of the medium will be Aeanmedieomse 0. I. Instantaneous Point Bounce’ The solution of equation (1) for this case was given by Fourier * in the form Men 7 Se(arkty? ° ° e ° ° ° (2) Values of @, the temperature in degrees centigrade for various values of > and t, calculated from equation (2), are given in Table I. TABLE. Temperature at vt sec. 0 ‘05 ‘075 ‘J 15 | -2em. POA 0.) em 0 0.4 30 001 | 14350 | 4111 862 97 Ol sO 002 | 5074 | 2716 | 1244 | 416 20 2 003 | 2762 | 1820 | 1os1 | 521 65 | 35 004 | 1794 | 1318 |-7888 | 514 |.108 |. 12 - 005 | 1283 999 | ¥31 | 472 135042) 9B 006. 8 2076 798 \eerl | 494 150° | 35 At any distance, 7, from the origin the temperature of the air reaches the highest value it can attain there after an interval of time, t=r?/6k. Thus, at a lstenee of 0°05 em. the maximum temperature is reached after 0°00083 sec., and at a distance of 01 cm. it is reached after 0°:00333 “el at this latter distance the temperature of the air never wees higher than about 526°. If an inflammable mixture of the same thermal properties is assumed to be substituted for the air, and the ignition temperature of this mixture is assumed to be 700°, then the greatest volume of the mixture that can be simultaneously raised (by conduction of heat only) to a temperature not less than this ignition temperature, is approximately that of a * M. Fourier, Théorie de la Chaleur, § 385. Form of Temperature Wave spreading by Conduction. 363 sphere 0-091 cm. in radius, or about 3:16mm.? This volume may be regarded as a measure of the ‘“‘incendivity ” of the instantaneous point source of heat (with respect only to its ability to disseminate heat by conduction), for com- parison ‘with the other -hypothetical sources shortly to be discussed. The time at which the temperature of the gasat a distance 0-091 cm. from the origin reaches its maximum (namely, 700°) is about 0:00275 sec. The form of the temperature wave at this time is shown in fig. 1, curve A, in which the abscissee are values of r, and the ordinates temperatures. Big. 1. A. fpstontenegus Fort Source. €= -OO27S sec. B. Continued Point Source 7=:O08 sec. €= O06 Sec. C. Instantaneous Spherical Surface Source. t=O0005 sec. BD. /rstanteneous SpehericH Volume Source. t= O02 sec. ‘3000 |} 2000 | The errnaion of temperature due to different quantities of heat supplied instantaneously at the origin is easily calculated from Table I., the temperature at any place ana time being proportional to @): Il. Continued Point Source. The solution of equation (1) for the case in which heat is supplied at the origin at a uniform rate g per second for a time interval T is obtained by integration from equation (2). When the time ¢, for which the temperature is calculated, is less than T’, the solution is Vy ete pA kana if. Se( wht)?” 364 Prof. Taylor Jones, Mr. Morgan,'and Prof. Wheeler on which with the substitution <=7/2 / kt becomes ; lee) ot og da 2V kt When ¢ is greater than T, the solution takes the form : pm . 4eé-T) Ser" J [kt—T) | 2Veb—T) == ee ede. 9 Ay Mio laters ae (4) In Table II. are given numerical values of @ calculated from equations (3) and (4) for a total quantity of heat of 0-001 calorie supplied uniformly during 0:005 sec.—that is to say, By ak =-, =~ Cal./see. q= ip = 5 cal./ Tasxe II. Temperature at ~ sec. |- Sar SRE ae Om 05: S07 zi jal vem 001 | B18 fd iy Bs eee 002 00 1197 (| 288 1) 58a et 003 © 1641) | 517 «| ba aS 004. | 1952 | 714258 ees | 005 20 2181) 876") | 38 a wae Wasdo6 3301 || “1841 | | 956) | Aaa ew | 007 | 1880 | 1803 | 838 | 470 | 109 | | It will be seen from Table II. that after the heat supply ceases, the temperature at points near the origin (for example, at a distance 7=0'075 cm.) rises a little before falling in consequence of the diffusion of the heat throughout the Form of Temperature Wave spreading by Conduction. 365 medium. Again, assuming an inflammable mixture with an ignition temperature of 700° to be substituted for the air, this temperature is approximately the maximum temperature attained at a distance r=0°0855 cm., and it is reached after an interval of time ¢=0-006 sec. The temperature wave at this time is shown in fig. 1, curve B. Thus the greatest volume which is raised to at least 700° (by conduction alone) is that of a sphere of radius 0°0855 cm., or 2°62 mm.* This volume accordingly represents the incendivity of the con- tinued point source of heat. The incendivity of a point source of total heat 0-001 calorie, uniformly in action over 0-005 sec., is thus about 17 per cent. less than that of an instantaneous point source of the same total heat. Similar calculations for other values of the time of heat supply T showed that as T is increased, tie surface repre- senting the 700° limit continues to shrink, and that as T is diminished, the temperature distribution continually ap- proaches that given by an instantaneous point source (T=0) of the same total heat. The incendivity of a source of small rate of heat supply may therefore be very much iess than that of an instantaneous source of the same total heat. The general conclusions to be drawn from a comparison of Tables I. and II. are: (1) if an inflammable mixture is to be ignited by a given quantity of heat supplied at a point, the more quickly it is supplied the better ; and (2) the total quantity of heat supplied by a given source is no criterion as to its incendivity unless the rate of supply also is specified. Ill. Instantaneous Spherical Surface Source. The distribution of temperature when the heat is supplied instantaneously and uniformly over a spherical surface has — been stated by Lord Kelvin *. In this instance, if a is the radius of the spherical surface source, the temperature at any point distance 7 from the centre is given by —(r—a)2 —(r+a)2 e 4kt wig 4kt = iW il A eee alae Q Scars” (kt)? (9) Some values of @ calculated from equation (5) with a taken as 0°l cm. and gq as 0:001 calorie, are given in Table ITI. * Ene. Brit. 9th ed. Art. ‘‘ Heat,” Appendix. The expression given by Kelvin should, however, be divided by zx. 366 Prof. Taylor Jones, Mr. Morgan, and Prof. Wheeler on TasuF III. 7 Cm. t sec. : nT; ] - = 0 05 1 125 15 0 0 0 i 0 0 0005 | 18 | 166 | 1015 | 436 55 001 OF A411 718 |° 420 +] qBy7 002 | 416 | 540 507 |) 347, felon 003 | 522 527 414 | 299 | 182 004. | 514 | 482 356 | 265 | 175 As might be expected, the temperature in the neighbour- hood of the source falls to small values much more rapidly with the spherical surface source of heat than with the instantaneous point source (Table I.). Thus with the spherical source the temperature, after an interval of time t=0:002 sec., is at no point in the medium as high as 700° C.. It can be shown that at a time t=0:0005 sec., the volume of the spherical shell bounded by the surfaces r=0-075 em. and r=0°115 cm. is at or above 700° C, (see fig. 1, curve ©). This volume is 4°605 mm.°*, which is considerably greater than the greatest volume raised simultaneously to or above 700° by the instantaneovs point source, and is 1°76 times as great as the maximum obtained with the continued point source (Table II... Arguing solely on the distribution of heat by conduction, it is therefore to be expected that the effectiveness of a source of ignition will be improved by spreading it over a surface rather than by concentrating it into a small space. It is also a fair conclusion that a number of simultaneous sparks arranged close together in parallel would be more effective than a single spark of the same length and the same total heat content. IV. Instantaneous Spherical Volume Source. We arrive at the solution for this case by integration from equation (5). ‘Thus, if qg is the heat generated instanta- neously ‘per unit volume of a spherical source of radius a, so Form of Temperature Wave spreading by Conduction. 367 + that ©) = 3 TOG, a my a hear) ke eee x dx 4 a = oe } Qer(akt) "Jo r+a T= 2V kt 2V kt 3Q 2V kt ; Was -e *dy— e "dy iF 32 Acar” 0 0 | ‘ 4\l/2 —(r—a)2 —(r+a)? ‘a 3) (kt) \ ike, \. EPA CR) 32 ° Aca*a™! ? The temperatures given in Table IV. are calculated from equation (6) with a=0:05 cm. and Q=0:001 calorie. The initial temperature of the source is 13640° C. TaBLE IV, 7 cm. | t sec. “| 0 ae nel 0 | 125 15 | cp ce ae Tr aacye : 0 | 13640 | 13640 | 0 0 0 ‘O01 | 7207 | 3388 | 284 | 38 2-5 002 |. 3530 | 2174 | 498 | 160 40 | 003 | 2162 | 1518 | 526 | 236 88 | 004 | 1493 | 1130 | 494 264 | 124 005 | 1105 879 | 447 268 143 Comparison of this table with Table I. shows that, at the times given in the tables, the temperature at short distances from the centre, due to the spherical volume source, is lower, and at great distances is higher, than that due to the instan- taneous point source. The greatest volume raised to or above 700° by the spherical volume source is nearly the same as that raised by the instantaneous point source— namely, the volume of a sphere 0:091 cm. in radius. With the spherical volume source this is effected after a time t=0°002 sec. The form of the temperature wave at this time is shown in fig. 1, curve D. It appears, therefore, for the particular values assumed in 368 Form of Temperature Wave spreading by Conduction. the above calculation that there is no advantage to be gained, from the point of view of the distribution of heat by con- duction, in enlarging the source of heat from a point to a uniformly supplied spherical volume 1 mm. in diameter ; and that both are inferior to a spherical surface source of 2 mm. diameter. It is clear, however, that much more effective distributions of temperature can be obtained by further increasing the size of the source of heat. The best possible source of ignition is obviously such a volume of the inflammable mixture as can be raised instantaneously by the given quantity of heat Q to its ignition temperature ; any further spreading out of the source, either in space or in time, can only resuit in a diminution of the volume that is simultaneously raised to the ignition temperature, and no improvement can be attained by altering the shape of the volume initially heated. With Q=0:001 calorie, and an assumed ignition temperature of 700°, the volume of the most effective source for ignition is 10-2 mm3 (that is to say, the volume of a sphere of radius 071345 em.), a figure which represents the maximum incendivity according to the scale of measurement defined. The results of the above, calculation show that, if a source of ignition be regarded solely as a source of heat, the effec- tiveness of a given quantity of heat in raising a sufficient volume of an inflammable mixture to a given temperature (by conduction alone) depends essentially upon the manner in which that heat is communicated to the mixture. Differences in the heat contents of the least sparks of different types capable of igniting a given gas mixture are therefore to be anticipated. To quote a single example, it has been found that the energy necessary to ignite a certain coal-gas mixture by a capacity spark was 00025 joule, and by an inductance spark ‘0006 joule. The ratio of these figures is of the order of magnitude which would be antici- pated from a purely thermal theory of ignition (¢f. discussion of Tables I. and IL. ). Itwill-be apparent from the foregoing that the observed differences in spark energy required for ignition of a given gas are not in themselves sufficient to warrant the assumption shat ignition is due to ionization, as such differences are consistent with a purely thermal theory. [369°] XXXVIII. The Specific Heats of Ammonia, Sulphur dioxide, and Carbon dioxide. By Prof. J. R. Partineton and Mir. J. Cant *, EE recent years the theory of specific heats has been prominent, but the subject is still very incomplete. In the case of solids, the work of Planck, Einstein, Nernst, Lindemann, and Debye, based on the Quantum Theory, has resulted in notable advances. In the case of gases, however, the treatment has prineipally been based on the theory of Kquipartition of Energy, which is now known to be of very restricted validity, and fails altogether in the case of solid - bodies at lowtemperatures. The application of the Quantum Theory to gases has been most incomplete, and the results are far from satisfactory. The determinations described in the present communication were carried out with the object of obtaining reasonably accurate data which might be used in theoretical discussion ; the paper deals only with the experimental results, and does not enter into questions of theory, which are deferred until more data are available. The method employed was that of Kundt, as modified by U. Behn and H. Geiger, the method of filling the gas-tubes being that described by Partington (Phys. Zeit. xv. p. 601, 1914). A tube of suitable dimensions is filled with the gas under investigation, and contains also a small amount of some light powder unaffected by the gas. The tube is sealed at both ends, which are as far as possible symmetrical, and is clamped in the middle. When the tube is set in vibration by stroking, it is usually found that the dust figures are very irregular, if anv appear at all. Such a tube is chosen that its mass is a little too small to vibrate in resonance with the coniained gas. To produce resonance, circular rings cut from thin sheet-lead are cemented on the two ends of the tube symmetrically. The hole in the middle of each ring is to admit the projection at each end, or at one end, of the tube where it’ was sealed off. One end of this tube is then brought inside another, slightly wider, tube, open at both ends. ‘This. tube is fitted with a cork piston and contains a small quantity of the same powder as the sealed tube. By varying the position of the piston the position for resonance in the air inside the open tube is found, and when the gas tube is set in vibration sharp dust figures are formed in both tubes. The wave-lengths of * Communicated by the Authors, _ Phil. Mag. 8. 6. Vol. 43. No. 254, Feb. 1922. 2B : 370 Prof. Partington and Mr. Cant on the Specific Heats sound of the same frequency can then be determined in the gas and in air. In the experiments described below a gas tube was used which was about 125 cm. long and with an internal diameter of 4 cm. There was, therefore, no necessity to apply the correction for diameter which is required for narrower tubes (Kundt, Pogg. Ann. cxxvil. p. 497, 1866; cxxxv. p. 247, 1868 ; Kirchhoff, aiid. cxxxiv. p. 177, 160%); Eiiesem Ann. Phys. xxiv. p. 401, 1907). The tube had practically the same internal diameter throughout, was tree from irregularities in the walls, and was straight. It was cleaned with nitric acid, washed and dried. One end was sealed off and blown as nearly hemispherical as possible. ‘To the other end a short tube with a ground connexion, J (fig. 1), was Fig. 1. Bay : en | Nad 8 7 | , SS t A sealed, and the remaining part blown hemispherical as before. The tube was then washed out witha mixture of concentrated sulphuric acid and sodium dichromate, then with dilute nitric acid, and finally several times with hot distilled water quite free from grease. The water was then allowed to drain out by inverting the tube. Complete cleaning is most essential. The powder used was silica. This was prepared by pouring water-glass diluted with its own volume of water into con- centrated hydrochloric acid with constant stirring. The jelly formed was then heated on a water-bath for some time, and the acid decanted off. ‘lhe residue was extracted and washed several times with concentrated, then with dilute hydrochloric acid, and finally with hot distilled water until the liquid was of Ammonia, Sulphur dioaide, and Carbon diomde. 371 free from iron and chlorides. ‘The silica was dried on the water-bath and afterwards strongly heated in a silica crucible. The perfectly white mass was finely ground in an agate mortar, and the finest powder obtained by shaking through satin. 1 304"'4-abs. ; dy|d, = 4400/2898. Thus 6=1:0110; «=cp/e,=1°303; C,—C,=2°087 ; Ce ibis. cal, =O? wz. cal. This result is in good agreement with the value found by Partington (loc. cit.) by the method of adibatic expansion : viz., 1°302 at 17°. The values found by other experimenters are shown in the following table. { | 380 © Specific Heats of Ammonia, Sulphur diowde, ete. Tase VII. ne Carbon dioxide. 1 atm. | | a eoy d/2 aie | | ig. (Cp en Cy ae | | | ° ° ¢ te . Author, 72°C | in em,, |) inemieiys | ? le g.cal. 'g. cal.'g. et | corr. ies | | | | | | | [= een | gaan a | | Thibaut ... 15 7-404 9518 |1-288 1-0116 1303) 2039 | 8-77 673 : | { 4 Schéler ... 20 7397 9°50 1-290 1-0110 1:303/ 2-037 876 6-72 Schweikert, 0 wCO,= wair= 1-283 10139 1°301| 2-048 | 8-85 | 6-80 | | 257°6 m. | 331°9 m. | egaeseei per sec. per sec. | | | | | Cant and | ee | Mig Partington. 20 , 6378 -. 8195 1-280 1:0110 (1303 | 2037 | 8°76 | 6°72 | | i | Summary of Results—The ratio of the specific heats « =c,/¢y ammonia, sulphur dioxide, and carbon dioxide have been determined by a method depending on the relative velocities of sound in the gases and in air. The specific heats were calculated. Berthelot’s equation of state was used in all calculations.. ‘he following results were obtained at 1 atm. pressure. TaBLe VIII; Ammonia at Sulphur dioxide~ Carbon dioxide 14°°5 C, atl 3e2'C. ce | be ZOr es a 1 i IR aes eens |i Se oe rr OPS GSR oar 1:308 1:290 1:303 Gia A esasosoae 8:77 g. cal. 947 ¢. cal, | Sule eneale Gry, ssash0u30090903 6°70 g. cal. 734 ¢. cal. | Geiemealn The authors wish to express their thanks to the Department of Scientific and Industrial Research for a maintenance grant to one of them (H.J.C.), and to the authorities of East London College for a grant which contributed to de- fraying the expense of the work. The experiments described were submitted as part of a Thesis for the degree of M.Se. of London University by H. J. Cant. East London College, University of London. a Ro 38la XXXIX. The Absorption of the K X-rays of Silver in Gases and Gaseous Mixtures. By P. W. BursipcGe, M.Sc. (V.Z.), B.A. (Cantab.), 1851 Exhibition Scholar, Trinity College, Cambridge *. = Te. published results of experiments on the ionization of mixtures of gases f show discrepancies from the results expected on the accepted theory that the ionization is due to corpuscles ejected by the Rontgen rays from the atoms of the mixture. It seemed worth while, in view of the failure of other co ee to test directly tie assump- tion that the absorption of the energy necessary to eject these corpuscles was independent of the association of the gases in ae mixture—z. e., that this “ionization absorption ’ 2s an atomic phenomenon and therefore in a gaseous mixture strictly additive. From the well-known results of the direct test with solids and liquids, this assumption for gases seemed quite ds and received support from the work of H. Moore t, who has calculated absorption coefficients on this basis, but, since the accuracy of Moore’s comparison was necessarily very limited, it was Judged advisable to make a few direct experiments. During the course of these, some measurements of absorption coefficients were also obtained. EXPERIMENTAL METHODS. (1) X-Rays. As a source of strong homogeneous radiation an attempt was made to utilize the direct 1 rays from a palladium anti- cathode, but the nature of the radiation was too dependent on the hardness of the bulb, and the method was therefore abandoned and secondary silver characteristic rays (K type) used instead. These were excited in the usual manner, and the best arrangement for the greatest homogeneity was determined by obtaining a series of absor ption curves in aluminium. Using a bulb with a platinum anticathode, it was found best to have an alternative spark-gap of 2°7 cm. (between points), a thin plate of silver (99-9 per cent., ‘075 mm.), and a filtering screen of silver foil (-026 mm.). The bulb was placed in a large lead-lined box directly above the radiator, which was again in a small lead box from which * Communicated by Prof. Sir. J. J. Thomson, O.M., leisy + Barkla and Simons, Phil. Mag. Feb. 1912; Barkla and Philpot, Phil. Mag. June 1918. + H. Moore, Proc. Phys. Soc. June 1915; also E. A. Owen, Proc. Roy. Soe., xciv. A, 1918 Boe ee Sa al aS SEs aoe 382 Mr. P. W. Burbidge on the Absorption of the a lead tube allowed the silver rays to pass out to the absorp- tion chamber (fig. 2). Direct radiation from the bulb fell on the silver plate only, and scattered radiation from surrounding walls was reduced to a minimum. Fig, 1.—Absorption-Curve of X-Rays, Cms. of Al Absorption curves of the rays were determined by placing sheets of aluminium betweem the absorption chamber and the large lead box. The curve for the rays used (fig. 1) showed considerable homogeneity to exist. he radiation appeared homogeneous till 95 per cent. was absorbed, the effective coefficient of absorption in aluminium, ja), being 6°01. Assuming the mean for silver K rays in aluminium to be 6°6 (corresponding to Barkla’s figure 2°5 for p/p, the mass absorption coefficient), the curve can be analysed into two straight lines corresponding to radiations with w=6°6 K X-rays of Silver in Gases and Gaseous Mixtures. 383 and initial intensity 96 per cent., and »=2°34 and initial intensity 4 per cent.* (2) Measuring Apparatus (see fig. 2). To determine the absorption produced by various gases in the absorption chamber, a ratio method was used in which the intensity of the main beam after passing through the chamber was compared with the intensity of a beam of similar rays passing always unaltered into a standard ioniza- tion vessel. Fig. 2 (to scale). J—main Ionization chamber; S—standard ; A—Absorption chamber ; F—-silver filter ; R—silver radiator ; B—X-ray bulb. The absorption chamber was a stout aluminium cylinder, 50°8 em. long, 12 cm. in diam., with each end closed with two circular aluminium plates, 4 mm. thick, holding between them a thin aluminium sheet t+ (1 mm.). These two plates for each end were drilled with the maximum number of holes (8 mm. diam.) so that the X-rays could pass through the thin sheet in the “ windows” thus formed. Three long rods ran the length of the cylinder and bolted the ends against the cylinder, and all the joins were cemented with a plastic wax made with resin, beeswax, and a little turpentine. * The analysis was made from the absorption curve (log I/thickness of Al), generally graphically, but also by the analytical method given by Sir J. J. Thomson, Phil. Mag. Dec. 1915. + Celluloid and vaselined parchment were both attacked by SO,. d84 Mr. P. W, Burbidge on the Absorption of the Thus arranged the vessel withstood the strains due either to evacuation or to an internal pressure of 2 atmospheres, with- out developing leaks. The main ionization vessel was a short brass cylinder placed so that the rays entered through the aluminium potential plate and fell on the brass electrode connected to the measuring electroscope. The front aluminium plate was ‘4 mm. thick, to absorb any soft L radiation from the lead used in screening. eae The standard ionization vessel was a long brass cylinder with parallel aluminium plates (14 <4 em.) between which the beam of rays passed. Both ionization vessels. were sealed with air inside. The difference in structure between the two vessels resulted in surprisingly good detection of changes in the homogeneity of the X-rays, this being shown by the variation in the ratio of the ionizations in the two vessels. The electroscopes used were of the Wilson tilted type, and for greater constancy of readings were enclosed in wooden boxes to minimize convection currents in the leaf-chamber, and had earthing keys consisting of weighted sharp knitting- needles pressing into brass or copper. These keys were both connected to a potentiometer, the middle of which was earthed, and were arranged to lft simultaneously, so as to isolate both measuring systems at once. Corrections were made for natural leak and for insulation leaks. (3) Gases. The gases used were air, sulphur dioxide, hydrogen, carbon dioxide, sulphuretted hydrogen, oxygen, and methyl iodide. In all cases but the last they were passed slowly through two calcium-chloride tubes (15 x 1:5 cm.). The absorption chamber was evacuated with a water-pump, filled with dried gas, and the operation repeated once at least, generally twice. The pressure of gas in a U-tube mercury manometer gave the proportion present. No rubber connexions were used in standing contact with the gases. The sulphur dioxide was obtained from the commercial liquid ; the carbon dioxide from marble and pure bhydro- chloric acid solution, acid spray being absorbed in a tube of glass wool moistened with sodium-carbonate solution ; the methyl iodide was evaporated into the evacuated chamber trom liquid in an attached air-free bulb, the liquid boiled at 41°-9 and the pressure of vapour in the chamber kept constant after the tube connecting with the liquid was closed. nat ‘ ‘ 4 w : 5 5 ! K X-rays of Silver in Gases and Gaseous Miatures. 385 The absorption chamber in this experiment was evacuated with an oil-pump to <1 mm. Hg; the stopcocks were lubricated with a brominated grease from paraffin wax and vaseline, to avoid leaks from action of the iodide vapour. } Ii. Resvts. (1) Absorption Coefficients. If I, is the measure of the intensity of the rays entering the ionization vessel (as recorded by the observed potential on the electroscope) for a pressure of gas p; mm. of mercury in the absorption chamber, I, is the corresponding measure for p, mm. pressure, # is the linear absorption coefficient of the gas at atmospheric pressure P, d the length in cm. of the absorbing path, an HEP) then “eae @ B i e@., _ 23x 760, login Li[ls ea Ce pS) The quantity d is the measure of the average length traversed in the chamber—in this experiment the difference between this and the axial length (50°8 cm.) was negligible. Inserting the value for d, we have . logyo I, —logyo I, = 34:4 i (P2— 1) This coefficient represents the total loss suffered in the particular apparatus. Some of the radiation scattered in a forward direction is collected by the ionization chamber, and thus reduces the absorption as measured. In the lack of definite data for scattering by various elements at different wave-lengths, a rough estimate of the amount so collected was made, assuming the mass scattering coefficient to be °2, using Crowther’s* distribution figure and taking a mean value for the solid angle of rays collected. Adding this correction, a value for the total absorption is obtained. A further correction is necessary for the lack of homo- geneity of the rays used. As will be seen from figure 1, over the range of absorption used, the rays appear homo- geneous but with a lower absorption coefficient in aluminium * J, A. Crowther, Proc, Roy. Soc. lxxxvi. A, 1912. Phil. Mag. S. 6. Vol. 43. No. 254. Feb. 1922. 2¢ A TEES, On a Ee a cee * Sgn eh 386 Mr. P. W. Burbidge on the Absorption of the than silver rays. From the relation | 7/0 — ane (where 7/p is the mass ionization absorption, A is the wave- length, A is a constant for a particular material, and wis a universal constant, approx. 3), then T, /pi= A,X" T9 ‘1 po= Aon? T1/P) a AA’, T>/p2= Aad’, whence T/T 7 9 as Barkla and Collier * found. Hence the correction can be made for the different wave- lengths, since 7, for the gas is measured with the rays used, tT, for aluminium is also measured (6°14), and 7! for alumi- nium for silver K rays (6°62) is known. The corrected values are tabulated :— Press ye Range : (corr. for ae (corr. for lie ees 0. tio. | p/p (mm. Hg). observed. scattering). (u—"2p). Rays). (7+ 2p). ee ee wee eee a Oe a ae eed a 1450 "00059 00068 00042 | :00045 | -00071 | -00129 | ey Ges 1450 "00091 00105 ‘00065 | 00076 | :00110 | -00198 9) | 205) 1440 "0069 “0072 ‘00662 | 00715 | 00773 | -00288 2°50 | 2°70 130 ‘0905 "1060 1047 114 "1153 ‘00688 |179 | 181 In fe. 3 are shown the present experimental results together with those of Barkla and Collier (oe. cit.) and of H. A. Owen te the graph representing the variation of log 7/p with log X. The figures given by Owen in his paper are for total absor ption, as he used a narrow beam ; those of Barkla and Collier are given as total absorption, but since probably they used a fairly wide beam there would be some correction necessary. However, for this purpose the value of the mass scattering coefficient, 69, has been subtracted from both sets of values. Siegbahn}{ has plotted similarly some of Barkla * Barkla and Collier, Phil. Mag. June 1912. + E. A. Owen, Proc. Roy. Soc. Ixxxvi. A, 1912. { Siegbahn. Phys. Zevtschr. 1914; Siegbahn gives these values, by mistake, as Owen’s. kK X-rays of Silver in Gases and Gaseous Mixtures. 387 and Collier’s values for the total absorption coefficient. The wave-length chosen is for the Ka doublet. Fig. 3. CH,! aanDs eee | ee eae g 4 i OwenkoS0:,COL sAir Beark/s & Collien- * CH31, 503, Air 0K Author- 000,,Air, ete. eee O -2 Log A [A.U.for kx) / Ag ~2 From the slopes of the lines the constant 2, of the equation t/o=AX*, has the values 2°60 for methyl iodide, 2°80 for sulphur dioxide, and 3°13 for air and carbon dioxide. (2) Absorption in Mixtures. Three mixtures were used—air and sulphur dioxide, carbon dioxide and sulphur dioxide, methyl iodide and air—and, in addition, the absorption in the mixture sulphur dioxide and hydrogen (equal volumes) was compared with that in the mixture, sulphuretted hydrogen and oxygen (equal volumes) ; each of these latter mixtures has the same number of atoms of each element but the chemical combination is different. In the case of any mixture the absorption, if additive, will be in accord with the relation —(f,Pi +H 2P2)a/P t= eee 1.@., (My pit Mop2) =34'4 (logioli —logiols) t -_~ bo 388 The Absorption of K X-rays of Silver in Gases. © | (a) For the first two mixtures, w, and pw. were measured, p, and ps, observed, and the value of the left-hand side of the equation was compared with that of the right :— | Geese. tae, Pi + HaP2 sbee ae | Seon] HR | ome | ie ae cl fo | (998 | eee ore) 7 | oso | sor | mon |, 88 bee (6) In the experiment with methyl iodide and air, the absorption due to the additional air was determined. Gases. Pressures, he ats (deter- ail : mined above). Cle ghare meer aL) Bra, er | 138 00066 06059 CHL...) Ot oo ite) See 00065 00059 The agreement is not so close as before, possibly because the decrease of ionization due to extra absorption by the air was difficult to measure. Owing to the chemical activity of the iodide, it could not be mixed with denser gases like sulphur dioxide which would give greater change in ionization. (c) For the chemical combination test, the sulphuretted hydrogen (from iron sulphide and hydrochloric acid) was found, on analysis over caustic soda, to contain 7°5 per cent. of hydrogen. This was allowed for in the two mixtures, so that in each case there was a small surplus of hydrogen. The ionizations resulting after absorption in each mixture were determined in quick succession. Mixture. lTonization. 700 mm. SH,+700 O+0d7H...... 66°1 Excess of H estimated by analysis. 700 mm. SO,-+700 H+57H...... 66°5 Excess of H added. On the Absorption of Narrow X-ray Beams. 389 The hydrogen was generated from zine and sulphuric acid and was passed through moist glass wool to eliminate acid spray. The oxygen was 99 per cent. pure from a commercial cylinder. Some time after the experiment had been completed, a paper by Simons * was noticed in which he has recorded the same result for this particular comparison. | The results of these four direct tests show that the assump- tion of additive absorption in gaseous mixtures and compounds is justified. Summary. (L) The total absorption coetficient of silver K radiation has been measured in air, carbon dioxide, sulphur dioxide, methyl iodide. (2) Using this silver K radiation, it has been shown by direct test on four mixtures that. the absorption in gases. is atomic (as in the case of solids and liquids) and therefore additive in mixtures and compounds. My thanks are due to Professor Sir J. J. Thomson for suggesting this research and for his interest in the experiments. XL. Note on the Absorption of Narrow X-ray Beams. By feo. Bbursmen, M.Sc .(N.Z.),. B.A. (Cantab.). 1351 Hehibition Feescarch Scholar, Trinity College, Cambridge +. HE experimental data on the scattering of X-rays at small angles is somewhat meagre and the following rough experiments were done, at the suggestion of Protessor Sir Ernest Rutherford, to ascertain whether at very small angles there was any very large scattering effect which may have escaped observation. The principle of the method is that used by Baier ford and Nuttall { in determining the scattering of ie ied absorption of the rays in some medium is measured (1) with a ~vide-angle beam, (2) with a very narrow beam. In the case of X-rays the ideal arrangement for the former case would be to have the radiating point at the centre of a sphere and to measure the absorption in a surrounding spherical shell. If the coefficient.so determined were then subtracted * Simons, Univ. of Durham Phil. Soc. Proc., 1912-13. a Cnn by Sir E. Rutherford, F. KS ¢ Rutherford & Nuttall, Phil. Mag. vol. xxvi., “Oct. 1913. 390 Nir ee Burbidge on the Absorption from the greater value obtained by passing a very thin pencil of the same radiation through the same medium, the dif- ference would approximate to the coefficient ef scattering in the medium ™*. Such an ideal comparison is experimentally difficult, but itis possible to compare the absorption of a wide- angle bean with that of a very narrow one, and so detect any very large small-angle scattering. Haperimental Arrangements. The source of X-rays was a palladium bulb run under constant conditions (7. e., speed of mercury break, currents in primary and secondary of the coil, alternative spark-ga [1 cm. between 1 cm. spheres, corresponding to 27,000 volts |) to give the maximum of palladium radiation, which was then filtered through silver foil to transmit mainly the palladium Kalinef. This gave a fairly homogeneous primary radiation (see figure) the absorption of which in sulphur dioxide was measured by a ratio method, with reference to an unaltered beam of rays in the manner described in the previous paper f. For the absorption chamber, a wide brass évlindee was used, 29°7 em. long by 11 cm. in diam. , lined with aluminium 1 mm. thick. In experiments with the wide- angle beam the full aperture of this vessel was used, the ends being of perforated aluminium as described in the previous paper. The narrow beams were obtained by using two sets of horizontal slits, in 3 mm. lead, placed one at each end of the absorption chamber and independent of it ; one set had 4 mm. slits, the other °6 mm. The chamber had special ends, each fitted with a continuous narrow aluminium window ; these special windows and the independence of the slits from, the absorp--. tion chamber were necessary to avoid errors from mechanical displacements consequent on pressure changes in the chamber. Two separate ionization vessels were used corresponding to the sections of the X-ray beams. Any characteristic radiation (Aluminium K or Lead L) excited in the passage of the palladium rays was reduced to a negligible amount by the time the interiors of the ionization vessels were reached. Owing to the small ionization from the narrow beams, sulphur dioxide was used in the ionization vessel with he 4 mm. beam, and methyl] iodide vapour mixed with air in the * The actual total scattering coefficient would be obtained if a correction could be made in the spherical case for the average increase of path of the scattered rays. + Hull, Phys. Rev., Series 2, vol. x., 1917. t See p. 383. | of Narrow X-ray Beams. 391 case of the'6 mm. The methyl iodide had always an effect on the insulation even when the gas mixture was well dried*. The plane angle of rays entering the ionization chamber Fig. 1.—Log Absorption Curves of X-rays. ae Los | ~ f fe) "2 4 Crs. oF Au in the wide-angle beam was 13°. The rays fell also on the sides of the absorption chamber, but a check experiment, in which the beam was limited by a lead screen so that it Just * Amber, sealing-wax, and rubber are attacked by the iodide; the insulation of quartz was impaired ; quartz coated with sulphur answered fairly well but was affected—it recovered if SO. was passed through. 392 On the Absorption of Narrow X-ray Beams. filled the solid angle subtended by the ionization chamber, ‘showed that there was at most a 4 per cent. difference (due probably to the increase in path of rays scattered from the sides). For the 4 mm. slits the vertical plane angle of the beam was ‘5°, and for the °6 mm., ‘07°. Results. In the table below the results obtained are tabulated, the absorption coefficient being reduced to atmospheric pressure. The comparison between the various beams was made in two stages—the absorption of the wide beam being compared with that of the 4 mm. beam, and this latter then with the ‘6 mm. beam. The coefficients for the two narrow beams were almost identical, a small increase of 1 per cent. being shown, on the average, for the finer beam. | Plane Angle Slit. pile or | eee ele Rays. ie (SU ci:) NViderer ee Be ‘0072 2°50 NTMI ceacon “Apso ‘OOT7 2°69 (5) imine hs aee "07° 0078 2-71 The use of a gas-filled bulb and the consequent variation of the quality of the X-rays necessarily limited the accuracy of the experiments, and all that can be said is that there does not appear to be any large small-angle scattering of the rays, a result confirming and extending to smaller angles the direct method used by Crowther * My thanks are due to Professor Sir Ernest Rutherford for the suggestion of this research and for the interest he has shown during its prosecution. * J. A. Crowther, Proc. Camb. Phil. Soce., voi. xvi., 1911. ae [esa al XLI. A Note on Beta Rays and Atomic Number. By J. L. Guasson, M.A., D.Se.* ss A LL workers on the passage of fast @ rays through matter have noted that the uncorrected absorption coefficient appears to be a periodic property of the weight of the atom. A résumé of the present state of our knowledge of this subject is to be found on pages 227 to 234 of Ruther- ford’s‘ Radioactive Substances,’ where the results of Crowther and of Schmidt are fully stated and discussed. Crowther’s curve connecting w/D with atomic weight seems to show that there are two effects at work, one of which increases steadily throughont the series of the elements while the other is a periodic function of the atomic weight. The investigation of Schmidt ne dS Pihysmcxilte pot ev lI0t Phys. Zert: x: poze. 1909; Phys. Zeit: xi. p. 262, 1910) has disentangled two Bifecis. An outline of Schmidt’s theory i is given by Ruther- ford together with a complete table of his resuits. Schmidt finds that the passage of 6 rays through matter can be use- fully considered in terms of two coefficients, the absorption coefficient a, and the scattering coefficient @ ; and he reaches the conclusion that the absorption coefficient « is directly proportional to the density (D), and inversely proportional to the cube root of the atomic weight (A), whereas the scat- tering coefficient @ is directly proportional to the product of A and D. Schmidt finds that a=C,DA7? and B=C,AD, and considers that C, and Cs are universal constants of matter for a given radiation. Professor Rutherford interprets these results to mean that « is proportional to the cross section of the atom and £8 to the volume of the atom. Schmidt believed that his results proved that there was no periodic effect either in @ or in #, and that his results were at variance with those of Crowther. . In view of recent tendencies it appears desirable to ae Schmidt’s results in terms of atomic number, and this is done in the following table. It is found that several obvious discrepancies disappear and the regularities become more obvious. (Guided by analogy with the a ray case, it seems likely that the results will be easier to interpret if instead of measuring the coefficients in reciprocal centimetres as Schmidt does, we use the atomic absorption and scattering coefficients defined by the relations aA - BA Ca aa and Vea * Communicated by Prof. Sir E. Rutherford, F.R.S. 394 Dr. J. L. Glasson on The values of the atomic coefficients are shown in the last two columns of Table I., and their values plotted logarith- mically against atomic numbers are shown in fig. 1. : TABLE I. Sede At. Wt.| Density | At. No. Schmidt’s values. Atomie coefficients. ymbol. | ~", D | ‘ ; ; a B a co Me | ofa Ne Tian io 6°30 5716 89 81 NMS 27/211 2°65 18 8°65 9°63 89 99 Fe fer DY) 78 26 24:0 55°9 172 400 Con vane) cer) ie ie 250°7 60:0 178 417 Na aon 89 28 26°1 70°5 172 465 Cu ...| 63°6 8:93 29 27:0 70:0 192 496 Zn ...| 65°4 719 | ~~ 30 21°6 56°9 196 518 Dawe Wate) | itis 46 | 300 160 268 | 1432 Ag ...) 108 | 105 47 26-0 144 968 | 1480 Syaity calmer WAU) 73 50 15°6 100 260 16380 Pee ton eis 73 | 410 468 372 | 4240 Nose LOT 19:3 TOT | 362 480 368 4910 bias: 220 11-4 82 19-3 266 : 350 4820 IB ao 5 Ln 20s Tia 8s 163 254 348 5400 3. THE ATOMIC ABSORPTION COEFFICIENT.—tLhe figures in the penultimate column of the table do not increase regularly as one proceeds to elements of higher atomic number, but tend to group themselves about four different values, being approximately constant for elements in the same period of the periodic table. This is shown graphically in fig. 1, curve A. Unfortunately the elements used by Schmidt are not well enough distributed to give a complete idea of the shape of the curve ; by analogy with other periodic curves we may expect the missing elements to show intermediate maxima. But it is a remarkable fact that the values for these four groups are almost exactly in the ratio 1:2:3:4. This may be merely coincidence, but in any case it seems to be a consequence of Schmidt’s work, that the atomic absorp- tion coefficient is dependent chiefly on the arrangement of the electrons in the atom and not so much on their total number. In view of the importance this result would possess if it could be definitely established, it is highly desirable that Schmidt’s results should be confirmed and extended, if possible by an experimental method which does not involve so many theoretical assumptions. It is obviously possible to draw a straight line through the points of curve A which fits fairly well, and this is in 109 4, 4-0 | Beta Rays and Atomic Number. 395 effect what Schmidt did. He was thus led to deny the existence of any periodicity in the absorption coefficient. But when exhibited as they are here it seems most likely that the curve is really periodic, and that Schmidt’s con- clusions were only rendered possible by the fact that he itieg i 2:C } TE : . - 2°25 log Ne accidentally confined his work to elements which were all grouped very closely about thé centres cf the periods. I think the work of Schmidt therefore does not establish the conclusions which he drew from it, and that his results really confirm the existence of the periodicity which Crowther had earlier detected. 4. THE ATOMIC SCATTERING CORFFICIENT.—The figures in the last column of the table increase regularly as one 109 o. 396 | On the Einstein Spectral Shift. proceeds to elements of higher atomic number. The relation- ship is exhibited graphically in curve B of fig. 1, in which the values of log 6 are plotted against log N. The points fall accurately on a straight line, whose equation is log b= 2°09 log N—-44 and therefore b—-36 N2, ‘The constant of proportionality is of no significance, being merely determined by our arbitrary choice of the centimetre as the unit of length. Provided one does not extrapolate over too great a range of atomic numbers, Schmidt’s results agree with the ~ idea that the atomic scattering coefficient is proportional to the square of the atomic number. This result is predicted by Sir Ernest Rutherford’s theory of nuclear scattering, and has been verified for @ rays by observations made in a very direct manner by Dr. Crowther. The agreement of Schmidt’s results with these direct obser- vations seems to be prima facie evidence of the substantial accuracy of his theory and experiments. Cavendish Laboratory, Cambridge. Nov. 38rd, 1921. XLII. Reply to Sir Oliver Lodge’s paper on the Einstein Spectral Shift. i Phil. Mag. June 1921. ] IR OLIVER LODGE’S criticism of my paper on the HKinstein Spectral Line effect appears to be sound in so far as my argument is stated more briefly and more roughly than would be justifiable if the work were given as a theorem in Pure Mathematics. Strictly speaking, if ds=0, the limit of = does not exist, and there is no differential coefficient 5 In the problem under consideration, however, there is the underlying physical assumption that the path corresponding to an initial velocity c is the limit, as v tends to c, of the path corresponding to an initial velocity v. This assumption appears to be supported by the transformations of the Special Theory of Relativity and it is difficult to imagine any alternative. Equations (5) and (6) of my paper lead to du ke? eS ee eee | where & and h depend on the initial velocity v but not on the coordinates. | | | : The Fundamental Principles of Scientific Inquiry. 397 As v tends toc, k tendstoinfinity. yw? is finite, and, except in the case of a beam passing through the centre of the sun, du dd finite, and the above equation reduces to is finite. It follows that the ratio (4/h) must remain eae, 7) uaa University of Queensland, H. J. PRIESTLEY. Brisbane, August 5th, 1921. XLII. The Fundamental Principles of Scientific Inquiry. To the Editors of the Philosophical Magazine. GENTLEMEN,— ()* p- 373 of the September number of the ‘ Philosophical Magazine’ Drs. Wrinch and Jeffreys accuse me of having stated in my ‘ Physics’ that ‘‘ only those processes, logical and experimental, are admissible in physics, which are universally agreed to be valid.” On pp. 21-22 of that book I intended expressly to disclaim that interpretation of my views, and used against it much the same arguments that Drs. Wrinch and Jeffreys have used. The passage concludes thus :—“ The view that I wish to assertis.... that science starts by selecting for its consideration those judgements alone concerning which absolutely universal agreement can be obtained, rejecting unhesitatingly any concerning which there can be any doubt or personal opinion ; that at every stage of the reasoning to which it submits these judgements a personal element is introduced, and with it the possibility of error and of difference of opinion ; and that its final results, those that represent its greatest achievements and possess the greatest intellectual value, are almost as individual and as personal as the greatest seliee ements of art.” Drs. Wrinch and Jeffreys probably think that nonsense ; but it is not the nonsense that they have imputed to me. Yours faithfully, Noy. 6th, 1921. NormMAN R. CAMPBELL. 398 Notices respecting New Books. GENTLEMEN,— wat | R. CAMPBELL mentions on p. 29 of his book three classes of judgments which he believes satisfy his criterion: they include judgments of betweenness in time and in space and judgments of number. We would say that the making of each of these judgments isa logical or an experimental process. We were considering only funda- mental judgments, and regret that in criticising the criterion we did not make it clear that he apphed it only to those classes of judgments which he treats as fundamental; we apologise if we have misrepresented him on any important matter. We suggest that in oar paper the words “as fundamental” should be inserted after “ admissible.” We think Dr. Campbell must have misunderstood us, for the only argument we used that has any counterpart on pp. 21-22 of his book was directed against a view that we explicitly stated he didnot hold. The argument we directed against the doctrine of universal consent was based on the obvious facts that the fundamental judgments depending on experience are Judgments of sensation, each of which is absolutely private to one individual, and that the judgments of others can be utilised only as constituent elements in individual judgments. We did not consider whether judg- ments about the opinion of others are ever used in science, or even whether they can be correct; we were concerned only to point out that they are not fundamental, and require much further analysis before they can be used in a theory of scientific or other knowledge. This argument stands without alteration. Yours faithfully, D. M. Wrinca, 192] December 22. HAROLD JEFFREYS. - XLIV. Notices respecting New Books. Meteorology: an Introductory Treatise. By A. HE. M. Guppxs, O.B.E., M.A., D.Sc., Lecturer on Philosophy in the University of Aberdeen. [Pp. xx+390, with 20 plates, 103 figures, and 11 charts.| (london, Blackie and Son. 1921. Price 21s. net.) ITHIN recent years the development of meteorology has been so rapid and it has increased in practical importance to such an extent, that this concise and up-to-date introductory treatise is very welcome. In writing a volume of this nature, it is always somewhat difficult to decide what to include and what to omit, and no two authors would adopt exactly the same point of view. Allowing for such personal differences, the author appears: to have succeeded in including all that is of importance for the beginner, and in avoiding the development of one part of the subject at the expense of another. The chapters on atmospheric Notices respecting New Books. 399 electricity, optics, and acoustics are particularly welcome; much of the matter contained in them is usually omitted from textbooks. The author has wisely emphasized the broad prin- ciples of the subject and kept them to the forefront ; mathematics has been avoided as far as possible, so that the book can be used by students with little mathematical training. After mastering it, the student should havea clear understanding of the basic principles and be well equipped for the study of more advanced treatises. The volume is well illustrated by plates and line diagrams; the excellent photographs by Mr. G. A. Clarke, of Aberdeen, illustra- ting the various types of cloud formation may be specially mentioned. It has a very complete index, and is well printed. Relativity, the Electron Theory, and Gravitation. By HK. CuNNING- HAM, M.A. Second Edition. Pp. 148. (Longmans & Co. 1921.) Lecons Hlémentaires sur la Gravitation @aprés la Théorie d’ Einstean. By E.M. Lemeray. Pp. 93. (Paris: Gauthier-Villars et C*. 212) : Amone the books on Relativity which, despite all the difficulties in cost of production of which we hear so much from the publishing trade, are appearing like “leaves in Vallombrosa,” these two deserve very special notice. What, distinguishes Mr. Cunningham is his clear straight- forward forcible exposition. We feel that not a word is wasted nor a difficulty slurred over. It breaks no new ground and offers no first-hand criticism, but this makes it especially valuable to the student. M. Lémeray’s book is also a manual for the student. It is an account of the equations which Hinstein uses set forth in the clearest and most precise form and without any extraneous explanations. The Mechamsm of Iife. By James Jounstonn, D.Sc. Pp. ix+ 247. (Edward Arnold & Co. 1921.) : THE special interest of this book is the author’s attempt to bring biology into general line with the recent development of mathe- matics and physics. It might seem a far cry from the principle of relativity to the theory of the élan vital, yet there is nothing extravagant, or in any sense unscientific, in the association which Prof. Johnstone finds between them. The argument is clear and sustained. It begins with an account of the mechanism of the organism and the scheme of its psycho-physical behaviour, drawn largely from Prof. Sherrington’s work. It compares the new mechanistic concept with the old Cartesian mechanism, and shows the change which has been brought about by the new physico- chemical concepts and the modern science of energetics. The author then criticises neo-vitalism, particularly the psychoids and entelechies of Driesch, “‘ vestal virgins dedicated to God and barren,” and expounds his own “ vital” concept that, in living processes the increase of entropy is retarded. 400 Notices respecting New Books. The Emission of Electricity from Hot Bodies. By Prof. O. W. RicnaRrpson, F.R.S. Second Edition; pp. viii+320, (Mono- graphs on Physics Series. London: Longmans, Green & Co., 1921.) Price 16s. net. THE second edition of Prof. Richardson’s book has been brought up to date, in the subjects with which it deals, by the extension of certain, sections and the rewriting of others. There is now included an account of von Laue’s treatment of the problem of the equilibrium of electron atmospheres in cases in which the volume density effects are not negligible; fuller explanation is given of the deviation, at low voltage, from the three-halves power law of thermionic current. Noteworthy also is the description of the author’s recent experiments on the emission of electrons under the influence of chemical action. No reference is made to the application of the thermionic current to the rectification and amplification of alternating currents, a short account of which would surely have been of value, if only to help students to realize how purely scientific researches in this subject have made possible the recent strides in the development of radio-telegraphy and radio-telephony ; again, many properties of thermionic emission, such as the cooling effect, are now being studied by means of the triodes used in radio. telegraphy. The book remains a store-house of ideas for those who are seeking problems for research, especially in the chapters con- cerning the emission of positive ions, the emission of ions by heated salts, and ionization and chemical action, subjects which as yet are only in the early stages of development. Bibliotheca Chemico-Mathematica: Catalogue of works in many tongues on Exact and Applied Science, compiled and annotated by H. Z. and H. C. 8. Two volumes, pp. 964. london: Henry Sotheran & Co. 1921. THIS unique work consists primarily of a list of many books on -pure and applied science, including mathematics. It was begunin 1906, and was expected to reach about three hundred pages. It developed, however, in the hands of the compilers gradually from a catalogue in the ordinary sense into a storehouse of information about almost all the great works on science. It is indeed a fascinating work, for there are many biographical and historical references both in the descriptions of the various books and in the notes which are appended to many of the entries. The illustrations are exceedingly delightful. They have all been obtained from the actual books by a photegraphic process, and include, besides facsimiles of illustrations and portraits, textual passages from works of historical importance, many of them produced for the first time. We notice specially an illustration from Boyle’s New Experiments Physico-Mechanical (1660), showing his air-pump ; and portraits of Torricelli (1715), John Napier (1616), and Thomas Young (1855). Hemsatecu & pp GRamont. Phil. Mag. Ser. 6, Vol. 48, Pl. ITI. Ultra-violet Spectra of Lead and Tin. x < Spark sines. DK HN ~ Ss no) SS > eo a aaS RS V8Ss 3 SSS | ‘oF source. nN . N N NEN N 2 o [ 3 x Capacity ee S i ee spark. po @ eS Ordinary arc. a. Arc and Spark Spectra of Lead. N NS) ) ™ Spark lines. b. Arc Spectra of Tin. Heusatecn & pr Gramoyt. Phil. Mag. Ser. 6, Vol. 43, Pl. IV. Arc Spectra of Lead. % s cy) ee ~ ~ = Sign and Nature AQ w 05 8 Nature of of Source. YY - * vs 2f/Ectrodes. eee 2. Ordinary are. Ee eC/ectroge. Direction oF motion or uv, 9 & MH s 99 ~ yy NY % t Nac vs yy > ¥ Spark tines. qemsanecn & pr Gramont. Phil. Mag. Ser. 6, Vol. 43, Pl. V. Arc Spectra of Tin. Sy 197 and Nature of Neture e/ectrodes. oF source. —-5350 7/ 45632 Sn are sine, ice Ordinary ere. b. Ordinary ere. ee Blow Si ad ah S7 spark ses. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. AER AES fis. | he MAR 15 4099 a Ag? 2. tw Sa @ Ds Ns parent oo [SIXTH SER, MARCH * XLV. The Theory of the Intrinsic Field of a Magnet and the relation of its Magnetic to its characteristic Klectric and Thermal Properties. By J. R. Asawortu, D.Sc.* INDEX TO PARAGRAPHS. The Intrinsic Field of a Magnet—Discussion of the subject. . Resistivity under an alternating field. Thermo-electric Power under an alternating field. Specific heat under an alternating field. Corresponding States. Theory of a combined magnetic and molecular intrinsic field. 7. How it accords with the facts of magnetism. &. How it accords with the facts of specific heat. 9. How it accords with the facts of resistivity. 10. Recapitulation. 1. The Intrinsic Field of a Mugnet.—The theory of the magnetism of a ferro-magnetic material, which regards the intensity of magnetization as analagous to the density Ore a fluid and treats it as'a function of both field strength and temperature, has been developed in some detail in several papers published in this Magazine ft, and an equation has been used for magnetism which is the counterpart of van der Waals’s equation of state for afluid. In this theory it is necessary to introduce an intrinsic field, just as an intrinsic > OR go 10 * Communicated by the Author. + Ashworth, Phil. Mag. vol. xxvil. p. 357; vol. xxx, p. 711; vol. xxxii. p. 334. Phil. Mag. 8. 6. Vol. 43. No. 255. March 1922. 2D 402 Dr. J. R. Ashworth on the Theory of pressure has to be introduced in the theory of fluids ; and this intrinsic field may be evaluated either trom the known facts of intensity of magnetization (1) as a funetion of the temperature (‘J’) or as a function of the field strength (H), just as the intrinsic pressure in a fluid may be derived from the relation ot the density to the temperature or the density to the pressure. In the theory of fluids the constant of the intrinsic pressure (van der Waals’s a) has consistent values by whichever method the calculation is carried out, but in the theory of magnetism the constant of the hee field has two values which differ enor mously according to the way in which they are derived. For exa umple, if the. constant of the nti field is derived from I=/(T) it gives to this field at the highest intensity a value about 10’ in iron, and similar values in nickel and cobalt, but if it is derived from I=¢(H) it gives a value of the order of a few units. We have then to choose between two magnitudes for the intrinsic field of entirely different orders, and the first step will be to find grounds for excluding one or the other. The value of the intrinsic field derived from the equation to the critical tentperature in spite of its enormous magnitude has been in genera! accepted, probably because such a large intrinsic field explains successfully the behaviour of intensity of magnetization under changes of temperature, and also because it may be made to account for the. rise in specific heat which the ferro-magnetic metals exhibit when the tem- perature is raised up to the critical temperature. But while the effects of temperature on ferro-magnetism are satisfied by the introduction inte the equation of an immense intrinsic field, the demonstration that this field is eutirely a magnetic field is w anting, and it is allowable to suppose that it is in part, perhaps in greater part, a field of force of some other kind than magnetic. At the critical temperature the sudden loss of magnetism in ferro-magnetic substances although the most prominent is not the only effect, and the modification which takes piace in other properties, such as electrical resistivity, thermo- electric power, and specific heat, is evidence of 2 change in ferro-magnetic substances which is probably akin to a change of state. If we treat the intrinsic field as the independent variable and the other properties as dependent upon it, then, assuming this field is entirely magnetic and of enormous magnitude, we ought to find some ‘large effects in the de- pendent properties when this magnetic intrinsic field is varied or suppressed. There is an interesting change in the magnetic critical rr | »" - the Intrinsic Field of a Magnet. 403 temperature, especially prominent in nickel, which can be brought about by the application of an alternating magnetic field, which may be used as a test of the magnitude of the magnetic intrinsic field. If a nickel wire is subjected to an alternating field, then its ferro-magnetism is lost at a tempe- rature 50° to 100° C. lower than its normal critical tempera- ture. Curves illustrating this are shown in the upper diagram * ofsfig. 1, which is taken from a former paper published in AC - £20 opin on j / Kohkes SFO: 400. > 4 Ss = oe Tera peraifvee |. abs. | o he. 200 £2 4co a0 = ’ ln hystere tac 7 dnc Waaren\| WZ | } Or SOM eRerS strengths of alternating | Currents | T Vid i erie s by hy AC WEEE piencid > Ne x > \ Senne zN ES ; \ | H Tempekature 17, Gebsolysre Deg j iv f2 wo 2pe a For | HERMO~ELECTKRrIC FOWEF of IRON ane IYICKEL ‘= /VICKEL. considerable divergence which is perhaps due to a change in iron, the molecule of which appears to undergo a subdivision at or near the critical temperature. There is, on the whole, evidence that corresponding states approximately hold for iron and nickel in relation to thermo- electric power. 6. Theory of an Intrinsic Magnetic and Molecular Field.— Returning now to the problem of the intrinsic field, the results of the experiments which have just been described lead to the conclusion that the magnetic intrinsic field has not the enormous magnitude which has been assigned to it, but on the contrary has probably a very small value. Nevertheless, the effects of temperature on ferro-magnetism demand an the Intrinsic Field of a Magnet. 413. intrinsic field of some kind of very large magnitude ; and we are led, therefore, to think of the intrinsic field as the - combination of two fields, one a true magnetic field and the other a field of force, not magnetic, arising from molecular forces. These two fields may be distinguished as the magnetic intrinsic field and the molecular intrinsic field. The extreme ease with which an external magnetic field can turn the magnetic molecules shows that the molecular intrinsic field does not exercise any constraint on the orientations of the molecular magnets, the only rotational constraint to which they are subject being the magnetic intrinsic field *. On the other hand, the molecular intrinsic field controls displacements and translatory movements of the molecules, while the magnetic intrinsic field is not primarily concerned with movements of this kind. Thermal action produces to and tro displacements which, increase as the temperature rises, and at the critical temperature the molecular intrinsic field becomes very small, as witnessed by the fact that elastic forces then. very greatly diminish. But, although thermal action does not directly produce rotational vibrations of the molecules, it is a property peculiar to a ferro-magnetic substance that rotational vibrations can be communicated to the molecules from translatory movements in virtue of the mutual magnetic: forces existing between them. If, for example, we consider a pair of magnetic molecules with their axes in alignment, a to and fro displacement of one, at right angles to this axial line, would produce a forced vibration of the other around its centre, and the magnetic moment of the pair would diminish. Thus it is by the intermediate agency of mutual magnetic forces that thermal action affects the intensity of magneti-. zation. As the to and fro displacements of the molecules become larger and larger with rise of temperature, ultimately losing their oscillatory character at the critical temperatyre, so the amplitude of the rotational vibrations becomes propor-. tionally wider and wider, and ultimately rotations are developed at or above the critical temperature, and, in consequence, ferro-magnetic properties disappear. Thus, at the critical temperature, both the molecular and the magnetic intrinsic fields become negligible or very small, and the concurrent loss of elastic and magnetic properties at this point finds a ready explanation. Briefly, the view here taken of the structure and behaviour * It is not vital to the view here taken to decide whether the magnetic axis of the molecule is fixed within it, or whether it is free to. be orientated within the molecule. thus AlA Dr. J. R. Ashworth on the Theory of of a ferro-magnetic substance is that it consists of an assemblage of molecules held in definite positions by molecular forces and subject to a molecular intrinsic field; . that these molecules are magnets subject to a magnetic intrinsic field ; that the molecular field imposes no rotational constraints on the molecular magnets; that a magnetic field does not appreciably affect displacements and trans- latory movements; that the translatory movements due to thermal action can develop corresponding rotational movements in the molecules in virtue of the mutual magnetic forces existing between them, but that rotational movements cannot produce translations ; lastly, that the molecular and magnetic intrinsic fields become small and, in effect, vanish at the critical temperature. 7. We now apply these views to the ferro-magnetic equation, which, written in terms of susceptibility, is (H+aP)(p-p)=Ro, . sa tnt H being the applied field, I the intensity of magnetization, I, its maximum value, T the absolute temperature, and R’ the reciprocal of Curie’s constant and such that I,?R’=R the gas constant. Instead of the large intrinsic field a’I? of the order 107 -gausses, we substitute for it a small magnetic field b1’, where b is a small constant, and add aterm A’ to the left side which is a function of the molecular field and the temperature, A'+ (H+ 01y(7 7 J=RIT.. +, AG ‘Consider first the effect of the temperature varying while the applied field remains constant; then, since thermal agitation induces rotational vibrations, which are a copy of the molecular movements, both may be treated as the same function of the temperature, and the equation may be put licens oats a Wi ee ! (H+ O28) (7 p)=RT—a'= 2 RT. Seat) where » to a close approximation 1s equal to a'/b and is numerically of the order 10’. From this we get i —R/ : CO n ive R Bas alg ae fi ee ‘(aoe per) ee the Intrinsic field of a ALlagnet. A415 and the first term in the denominator instead of being negligible, as in the former treatment of the ferro-magnetic | equation*, is of importance. By thus recognizing the influence of the applied field the correspondence between the curve deduced from the equation and the curve constructed from experimental data is found to be improved. The equation to the critical temperature (T.) remains the same as before——namely, 8 6] 8 ala Leas ede 2 Bad tw eRe La ©) Secondly, let the temperature be constant and the applied field be variable ; then, A’ being constant in the absence of external constraints, the right side of equation (3) is constant, and the consequences formerly derived from the ferro-magnetic equation are the same t+ except that the small factor b replaces the large factor a’, with the result that appropriate numerical values of I and H are now obtained. For example, the relation of I to H is given by the equation ee eb acl Eloy le) a= (i-p)ite C25) - © which shows that hysteresis is in evidence, since the second term in the denominator, involving the small quantity 6, is comparable with the first term E: If 6 were to diminish to a negligible quantity, as it may be made to do under certain conditions, then the equation represents the anhysteretic cunye of [= G.(H). | Again, the equation to the critical field (H,) is ORR Ii oc rae Dane S and this correctly gives the critical field as about the order of a tenth of a unit, instead of the order of 10° formerly calculated with a’ instead of 6. Lastly, when both the molecular and the magnetic intrinsic fields become small enough to be negligible then the ferrc- magnetic equation reduces to st Slay ns un '(8) * Ashworth, Phil. Mag. vol. xxx. p. 711. + Ashworth, Phil. Mag. vol. xxxili. p. 334. i ik 416 Dr. J. R. Ashworth on the Theory of the paramagnetic equation, which applies above the critical temper ature. / ~ It remains to show that the abnormal changes of specific heat and of electric resistivity with temperature, which are characteristics of the ferro-magnetic metals, are to be expected if this view of the intrinsic field is adopted. 8. Specific Heat.—Since the magnetic molecules are almost. entirely free from rotational constraints except for a small magnetic intrinsic field, whilst the constraints under which thermal agitation takes place due to the large molecular field are very great, the energy at ordinary temperatures is almost entirely concentrated in the three degrees of freedom associated with translatory movements; but as the temperature rises, the molecular field becomes weaker, and, when the critical temperature is approached, the energy becomes mainly kinetic, and, in virtue of the magnetic forces, it is shared with the additional two degrees of freedom associated with magnetic vibrations and rotations. The specific heat thus increases up to the critical temperature. But just above the critical temperature, experiments show that there is a sudden drop in the specific heat to lower vaiues. This is accounted for by the fact that just at or above the critical temperature, when rotations of the molecules are becoming established, the mutual magnetic forces become very small, and the energy of thermal translatory movements is not as freely commu- nicated to and shared with rotational movements; thermal agitation for higher temperatures will again be mainly confined to translations and the specific heat will fall to a lower value. The experiments of Pionchon * and of Weiss & Beck t on the change of specific heat with. rise of temperature in ferro- -magnetie substances are of great interest in this connexion, and show how far these views are supported by facts. The following table of specific heats is constructed from their observations :— At At Crit. 17°C. Temp. Ratio. Rise. Fall. Ratio. ‘Magnetite ...... 0165 0275 3/5 0110 0055 2/1 Nikole deci: 0098 0153 38/47 0055 0097 2/4 Those a eae aed 0101 0314 3/93 0218 0112? 2/1 Coualeys ic we 0108 01922 3/542? 00842 — es * Pionchon, Ann. Chim. Phys. 6 sér. xi. p. 23. + Weiss & Beck, J. de Phys. sér. 4, vii. p. 249. the Intrinsic Field of a Magnet. 417 The critical temperature of cobalt is about 1100° C., and at this high temperature Pionchon remarks that investigations on specific heats become very difficult and the results must be accepted with some reserve ; there are no observations for cobalt in sufficient detail to allow the sudden fall of specific heat above the critical temperature to be traced exactly. Magnetite gives precisely the required ratio of 3 to 5 for the specific heats at 17° C. and at its critical temperature, but nickel and cobalt have not so exactly this ratio. If, however, the specific heats of nickel and cobalt are com- pared at corresponding temperatures, the ratios of their specific heats are more nearly alike, and if the chosen corresponding temperature at the lower point is that of iron at 17° C., they are more nearly 3 to 5. Tron, which has nearly the same specific heat as nickel at 17° C., has an abnormally high specific heat at the critical temperature, which is very nearly the double of that of nickel, and the ratio of the specific heats, namely 3 to 9°3, is thus double the ratio of 3 to 4°7 for nickel. This may be explained by assuming, as before, that the molecule of iron is subdivided at the critical temperature and, if into two parts, the number of degrees of freedom would be doubled and the | specific heat would be doubled. There are two interesting facts which the table brings out, which can be no more than mentioned here: first, the rise or the fall of the specific heat is in each case almost an exact multiple of 0°027, the smallest number ; and, secondly, the abrupt fall above the critical temperature is always the half of the rise. 9. Hlectrical Resistivity.—If resistivity is expressed in terms of the heat which is emitted from a conductor, it is evident that a change in the specific heat must produce a change in the resistivity. If s is the specific heat, p the resistivity, and ¢ the temperature, then the relation may be put, dp a ae where & is a constant. As long as s is constant, the resistivity will vary linearly with the temperature, but if s increases, the slope of the line, giving the relation of p to T, will also increase as the diagram shows (fig. 3). Now, the specific heat of nickel changes, as we have seen, from air temperature to the Phil. Mag. 8. 6. Vol. 43. No. 255. March 1922. 2 418 Dr. J. R. Ashworth on the Theory of critical temperature in the approximate ratio of three to five, and we should therefore expect the slope of the line to change in the same ratio. From the reduced curve we get the following results :— Nickel.—Beginning at air temperature ....... ¥ H = MEOR nn . dp At the critical temperature .......,.... a4 er 1°63 Ratio’. 2 ee 3/5 This ratio is approximately the ratio of the specific heats at these temperatures. | Jron.— Beginning at air temperature in s One ee ; dp . At the critical tenfperature....). he ia 2°15 Ratio... 0:50 ae 3/9 And, again, this ratio is nearly the ratio of the specific heats. There seems little doubt that the large and increasing change of resistivity up to the critical temperature is due to the increase of the specific heat; and as this in its turn depends on the mutual magnetic forces between the molecules, we have an explanation of why ferro-magnetic substances exhibit an abnormally large change of resistivity with rise of temperature. ; 10. Recapitulating, we find from the ferro-magnetic equation, based as it is on a kinetic theory of magnetism, that a magnet possesses an intrinsic field, and that this field must be in magnitude immense if energy effects are taken into account, but only very small if the effects of an external applied field are considered. In short, when the intensity of magnetization is treated as a function of the temperature the intrinsic field must be very large, and when treated as a function of the applied field the intrinsic field must be very small—results which are not consistent with one another. Further, we find from the effects of an alternating field upon ferro-magnetism—more definitely its ability to lower the critical temperature and at the same time its inability to alter the temperature at which electrical resistivity, thermo- electric power, and specific heat change abruptly—that the the Intrinsic Field of a Magnet. 419 intrinsic field cannot be regarded as simply and solely a magnetic field of immense maguitude. This and other aifficulties disappear if the intrinsic field is treated as the combination of two fields, one a molecular field, and the other a true magnetic field; the former in magnitude very large and the latter relatively small. It is then assumed that thermal action operates directly on the molecular field by giving rise to translatory movements of the molecules, and that it only indirectly affects the magnetic field by producing rotational vibrations through the mutual magnetic forces of the molecular magnets, the rotational move- ments so developed being a transcription of the translational movements ; on the other hand, orientations of the molecular magnets which result from the application of an external magnetic field cannot appreciably influence translations or displacements of the molecules. This view of a two-fold field explains why the critical temperature is lowered by an alternating magnetic field, while the temperature at which the specific heat changes sharply is but slightly altered; for an alternating field exciting rotational vibrations in the magnetic molecules is highly effective in reducing magnetic intensity by increasing the amplitude of the vibrations, and therefore a lower temperature than the critical temperature is sufficient to bring about the magnetic critical condition ; but, on the other hand, an alter nating field adds little, or a negligible amount, to the energy of thermal agitation, and SO the “specific heat is almost unaffected. This view also gives a reasonable account of the relations of mechanical and magnetic forces, and shows why elastic and magnetic qualities tend to disappear at the same high temperatue e. Further, it removes a serious defect in the former statement of the kinetic theory of magnetism and now allows consistent values to be calculated from the ferro-magnetic equation, whether the intensity of magnetization is treated as a function of the temperature or a function of the applied field. And, lastly, it gives an explanation of the abnormal rise of specific heat, and the abnormally large temperature coefficient of resistiv ity found in the ferro- magnetic metals. July 30th, 1921. 2H 2 i eae a XLVI. The Relation between the Projective and the Metrical Scales, and its bearing on the Theory of Parallels. By Lupwik SILBERstTHIN, Ph. D.* - 1. F HT us recall briefly the structure of the projective scale as given by v. Staudt. OUT being a given segment of a straight line, let us affix to the points O and U of this segment the numbers, labels, or, as we will call them, the Staudtian indices 0 and 1, respectively. The fourth harmonic to 0, 1, 7, conjugate to 0, whose construction is shewn in fig. 1, will then have the index 2. Continuing this. Fig. 1. graphical process we shall have the points with indices 3, 4, etc. With equal ease points with any fractional indices can be constructed, and irrational indices are obtained by limit considerations. We need not give here a detailed description of this pro- jective scale construction t.- Suffice it to say that, the points Q, 1, and 7 being chosen, the position of a point of the segment OUT having any real positive index n is uniquely determined. The point 7 itself will have the index n=x, To points following upon one another from O, via U, to T correspond indices n increasing from 0 to o. Beyond O and beyond 7’ we have negative indices, so that the supple- - ment of the segment represents analytically but one domain, —x suG=@=a Thus, after easy reductions, and replacing 0 by //R, be UG i ly aes cotp rl coth + (n—1) cot Ty Se ery 7) * Notice that, according to a famous theorem due to Schur, every projective space is a space of constant curvature, and vice versa. ¥ Or better, a limited portion of it. the Projective and the Metrical Scales. 423 This is the required relation between the scale of Staudtians (x) and that of metrical ‘divisions’ (/) setup upon a straight line in any space of constant curvature. Tor real finite R the formula is ready for the Riemannian straight line ; the Enclidean case corresponds to R=«: and the Lobatchev- skyan or hyperbolic one to R?<0. Leaving out the first case, of no interest for our purpose, let us consider the consequences of (2) for a Euclidean, and then for a hyperbolic segment. The latter will offer some interesting points connected with Lobatchevsky’s parallels. 3. Euclidean segment.—Puttng R=x, that is, making the ratios of /, 4,1, to & tend to zero and dividing by &, we have at once l Sih - {i+e@-1t ch is aot eA OD and, conversely, i i » fh em, oe oO These equations enable us to write down the Staudtian index corresponding to any metrical scale-division, and vice versa. If l,, l,:are rational numbers, to every rational / corresponds, by (4), a rational index », and the corresponding point can therefore be constructed by a finite number of straight-edge operations. Of particular interest is the value of n corresponding to = (“point at infinity’). This is by (4) l E has We een oe Geen Bylot AC) ly and the same index corresponds to /=—«. We assume that J, >1,, so that the point corresponding to l —n= ~~ —1 i lies within the segment OUT. Constructing its negative as shown in fig. 2, we find, through any point Y, the parallel to the base line. In particular, 1 Jo = 21), so that n=1 is the metrical mid-point of OT, we have for the parallel (i. e. for /= +o ) n=—l, whence the known construction of the Euclidean parallel to a bisected segment, as a sub-case of the construction repre- sented by (5). en Fer Ry ee ere es AQ4 Dr. L. Silberstein on the Relation between It will be well to restate this result :—The knowledge of the mid-point J/ of a segment AB enables us to draw through any point Y the parallel to AB (fig. 4), and vice versa, the knowledge of a parallel to AB enables us to find the point (by means of a straight-edge alone). Fig. 4. We can express this also by saying that with every _ Euclidean segment is associated a certain characteristic point, its mid-point, the unique image* of both the (coalescent) points = +0 andl=—x of the straight line AB. We shull see that in the space of Lobatchevsky there is on every straight segment: a pair of distinct points which take over the réle of the mid-point. The latter is thus a double point, the coalescence of the pair. 4. Lobatchevskyan segment.—Let the curvature of the contemplated hyperbolic space be KA=—1/R?, and let us take R, areal length, as unit length. Then, our previous OUT being a straight segment in this space, we have to put in equation (2)7= /—1linstead of R. Thus the relation between the metrical and the Staudtian scales becomes ‘otl=~[Coth+(n—1) Cot i], . - - ©) where Cot stands for the hyperbolic cotangent. Without dwelling any further upon the general relation (6), let us specialize it by taking for the Staudtian n=1 the metrical mid-point J/ of the segment. Thus, if 2d be the total length cf the segment, with # as unit, let us put i=), (2h hen Fob l= : | Cot r+ (n—1) Cot 20], for any /. Solve for n and, aiming at the Lobatchevskyan * Or fourth harmonic of 4, B, and the point at infinity, conjugate to the latter. the Projective and the Metrieal Scales. 425 parallels, put =o. Then, since l'an «© =1, Tan 24— Tan 2r a . Sp cyte ae he 7 ) Fi tam (‘Tan 2A4—1) - (7) Similarly, for /=—o, Tan 2x~—Tanzr i | — Eat ae UBher at yes | (7 ) Tan A (‘Tan 2X 4+ 1) These two different indices correspond to the two “ points at infinity ” (or pencil centres) of the Lobatchevskyan straight line. 2 ' i eek Remembering that Tan v= -..——,, the last formule are x s : got a? easily transformed into y iy, SRDS EO a Pa eae a Fe) i = em giving the simple relation | ARE ey cite ee hy (OD Now, consider the two points P and P’ (fig. 5), whose Staudtian indices are the negatives of those in (8), Ver ANN ammo sm dee. a hO!) Fie, 5. A pA eT oar} — >: ts) yay oO In order to find their distances from the origin, /= OP, and J'=OP"’, substitute the indices (10) into (6), with 1 =2l,=2X, as before. Then the result will be ipeG pere L ek —] a aa PT ee 5 The metrical: distance OM being X, let us find the length P'M=nr—l’. Remembering that | Tan (A—U') = (Tan d— Tan l’) : (1—Tand .Tan 1’), we shall find, after easy reductions, Tan (A—0')=Tan? a. And, in quite the same way, for /@P=1—n, Tan (—X)=Tan? v, (11) 436 Dr. L. Silberstein on the Relation between so that P and P’ have a symmetrical position within the seoment, as might have been expected. Thus if the semi-length of the gap * P'P or each of the equal distances MP', MP, be denoted by 7, we have the surprisingly simple result Marra —" Van? The connexion of these two characteristic points P, P’ - with the Lobatchevskyan (arrowed) parallels through any point Y is shown in fig. 6, which, after what was said before, searcely calls for further explanations. Fig. 6. To resume :— Every Lobatchevskyan straight segment has intrinsically associated with ita pair of characteristic points P and P', symmetrically situated within it, whose mutual distance 2n is given by am = Tan? = 12 7 tan? (12) and is thus a function only of the total length 2r of the segment. It may be interesting to notice that for segments which are small fractions of & we have, approximately, n/R=(A/R)’, or PP’: OTOL: 2R, . o> ee ele that is to say, the “gap” is to the whole segment as this is to twice the radius of curvature of the contemplated space. * This portion of the whole segment OUT deserves such a name because if the end-point .X of a vector X=OX falls within it, there is. no negative vector —X with O as origin. Cf. Proj. Vector Algebra. the Projective and the Metrical Scales. 427 Thus if we wish to characterize palpably a particular Lobatchevskyan space, we have only to draw a segment and mark upon it the “ gap” or its extremities P and P’. If we do the same thing for a pencil of equal segments from J/, we have, within the circle of radius A, a concentric circle of radius 9:; and similarly within a sphere of radius » a con- centric sphere of radius » determined by (12). This little sphere is an image of the inaccessible locus of ‘ points at infinity,’ which is thus seen also to bea real quadric “at infinity.” It is certainly agreeable to have its natural image or correlate near at hand. li & becomes infinite, or better, if A/R becomes sinaller and smaller, the gap becomes small of the second order, until the two points coalesce into the Huclidean characteristic point, the ordinary mid-point of the segment. 5. Verification of formula (12).—Since some readers may find the above method based on Staudt’s scale and especially our deduction of (2), not wholly convincing, it has seemed well to deduce formula (12) directly by the aid of Lobat- chevskyan trigonometry or the equivalent analytic geometry, and thus to verify it *. This can be done mest easily in Weierstrass coordinates. ae ie PMP A In fact, let the mid-point M of the given segment AB be the origin, /B and the perpendicular WY (fig. 7) the axes. Then, if V be any point of the plane and a, 6, 7 its shortest * Another way of deducing (2) and (12), based on Cayley’s definition of distance by the logarithm of a cross-ratio, may be left to the care of the reader. 4.98 Dr. L. Silberstein on the Relation between distances from the axes MY, MB, and from the origin, respectively, the Weierstrass codrdinates of NW are (with geal), £=OSma,) y= Sin), 2 Wosae satisfying the condition # + y?7—2?=—1. All that is required for our purpose is to remember that the equation of a straight line passing through two points #7, y, 2, and Lay Yo, &2 1S A lige fy 2, | BOL 2 re Bo. Yo 25 and that the angle e contained between any two straight lines ax + by +cz= 0, aa + b'y +¢'z=0 (where a?+0?—c’=1, etc.) is given by cose= aa’ + bb' cc. Se ee It will be convenient to denote the Weierstrass codrdinates x,y, 2 of any point WV of the plane by \,, No, V3 respéctively. Since the position of the required point-pair P, P’ is independent of the choice of the point Y, through which the two parallels are drawn, we may take Y on the ordinate axis, in a distance fh above he origin M@. Thus Yo =0), 0 — sinh) Vee Cosme The auxiliary line through the end-point B of the segment being arbitrary let us make it perpendicular to AB. In order to find the point P we have to draw through Y the right-hand parallel YZ (fig. 7), 7. e. the join of Y with the point ty. 2—o , 0, co. ies equation is, by ae aNG +yV3—40) the constant is 1°13; inorganic salts gave a mean constant of 2°0 ; organic compounds, 2°2-3°0. It is known that the boiling-point is a ‘‘ corresponding temper- ature’ in the theory of corresponding states (see Partington, ‘Thermodynamics,’ p. 234) ; if T, and T;, are the critical temperature and melting-point of a substance, then, according to F. W. Clarke (Amer. Chem. J. xviii. p. bie, 1896), T./T,=2, so that fusion is also a “corresponding state.” E. Mathias (Le pornt critique, p.59, 1904), on the assumption of the law of the rectilinear diameter, also deduces that T,, is a corresponding temperature, and the same assumption is made by V. Kourbatuw (J. Chim. Phys. vi. p. 339, 1908). P. Walden (Zeitschr. Elektrochem. xiv. p. 713, 1908) assumes that the relation M/T=const. holds for the fusion of non- associated substances, the value of the constant for 35 organic substances being 13°5 (12°1 to 14:2). For other substances the ‘‘degree of association,’ wv, is assumed to be given by i 5 [a Benzophenone, betol, and anthracene, for example, are assumed to be normal ; formic and acetic acids, phenol, and benzene have association factors less than 2. E. Baud (Comptes rendus, 152, p. 1480, 1911) proposes the equation L=k(v,—vs), where v, and v, are the specific volumes of the liquid and solid at the melting-point. Accord- ing to G. Tammann (Krystallisieren und Schmelzen, p. 41, 1903), the constant & is the absolute melting point Tp. J. Narbutt (Zetschr. Hlektrochem. xxv. pp. 51, 57, 1919) finds large discrepancies in the application of Tammann’s formula; Walden’s rule gives fairly satisfactory results if 444 Mrs. K. Stratton and Prof. J. R. Partington on different constants are used for ortho-, meta-, and para- compounds. The value of & in Baud’s formula, for 1 gram molecule, Narbutt gives as 272°5. Narbuttattempts to apply the Nernst-Lindemann formula, the frequency being calcu- lated by Koref’s rule (see Partington, ‘Thermodynamics,’ pp- 927, 530), but with unsatisfactory results. H. G. Wayling (Phil. Mag. xxxvil. p. 495, 1919) proposes the rule nL/T,=N ; in this “equation Noneineee of atoms in molecule, n is the oneleenlae number,” 7. e. the sum of the atomic numbers of the atoms in the compound (H. 8. Allen, Trans. Chem. Soe. exiii. p. 389, 1918). In the case of salts containing water of crystallization, and in the few organic compounds considered (except formic acid and chloroform), this rule breaks down. The most attractive theoretical investigation we have met with is that of K. Honda (Sci. Rep. Tohuku Imp. Univ. vil. p. 123, 1918), which is based on the quantum theory. The atoms or molecules in the solid are allocated an amount of / ed energy c=lv| (em —1) =i, ‘(er =i), or re | ( ef 1) per gram molecule. The constants h and & are Planck’s constant and Boltzmann’s constant, respectively, 6=h/k is an absolute constant =4°863 x 1074, and R=Npok is the gas constant per gram molecule. Rotational energy in the solid is assumed to be small, except at very high temperatures. The molecules set free on liquefaction are assumed to retain the energy referred to, but to acquire also rotational energy corresponding with 2 or 3 degrees of freedom, the energy per degree of freedom being $e. The total energy acquired on fusion will then be , per gram-molecule, Noe=E or 3H, according as the liquid molecules have an axis of symmetry or not. Thus ML= tn, where n=2 or 3. The value of H is calenlated by Hinstein’s formula (given above) with a value of v, the frequency, given by Lindemann’s equation : ie MV y= 3-08 x 10% This formula is applied both to elements and compounds, and it is noteworthy that although n is 2 in the case of many elements, the value 3 is also required for some elements (e.g. zinc) usually assumed to be monatomic. Honda obtains a fairly satisfactory agreement for a number of elements and compounds, although in some of the latter Latent Heats of Fusion. 44.5 values of n much higher than 3 have to be assumed ({e. g, 6 for KF, 7 for PbCl). It is to be expected on theor etical grounds that the energy content of molecules at the fairly high temperatures of fusion would be more satisfactorily given “by the formula of Debye than that of Hinstein (see Partington, : Thermodynamics,’ s ae Jeans, ‘Dynamical Theory of Gases,’ 2nd edit. 422). On Debye’ s theory the energy content E corre- edit with three degrees of freedom is given per gram molecule by The values of H/T for given values of @y/T are tabulated by Nernst (Grundlagen des neuen Warmesatzes, p. 206, 1918), and greatly facilitate the calculations. The values of E/T taken from these tables may be regarded as giving the values of 3H,/T, where Eris the energy per degree of freedom for rotation. A comparison of Hd2T with ML/2T (observed) will then give the value of n, the number of degrees of freedom of rotation it is necessary to assume for the liquid molecule. The following example will illustrate the method of calculation :— Benzophenone.—_M=182. T,,=321. d (density) =1:098. oy torn. Lindemann’ s formula=0°7447 x 10”. meeye ao 2... Svpl, =" 1t23. Value of 3E,/T,, corresponding with 8y/T=from Tables. =5°709. Observed ML/T,, =12-28=nk,/2T,, = 12223: *, n=12°9. This, it will be observed, is much higher than any value recorded by Honda. This example also serves to bring out the complete failure of Wayling’s rule : ‘“‘ molecular number ” of benzophenone=96. .. nL/T,=6°5,, whereas the number of atoms in the molecule is 24. In the table are given the values of H,/2T,, calculated by Debye’s formula, with the values of ML/T,, from the experi- mental results of various investigators. For convenience in applying other quantum formule the values of vy from Lindemann’s formulaare alsoincluded. Itwill be seen that for many elements and simple compounds the modified theory of Honda which has been proposed is fairly satisfactory, although in some cases the interpretation of values of n greater than 3 is not yet clear. It is possible that the large values of n represent intramolecular vibrations ; at least it seems to be 4A6 Latent Heats of Fusion. that complicated molecules, in which the possibility of intra- molecular vibration is greater (as is known from the values of y=C,/C, in the case of gases and vapours), have high values of n. It is proposed “to return to this question ina further communication. TABLE I. | | | Ey | ML | y | Substance. nee on rT. iu, By. a rae : 2. ieliydirogen | e..n 0 hi 16, |'14 | 2:669 | 129°8 1) 00247) Tae eG NERO sh atiaenagt csc 28 192 62°5 | 1-402 | 68-17/0°645 | 3:07 | 48 W)ilorimes yaeeeeee | 70°} 814 | -169:5) 1-578) 76:72) 0: e3an e790 ian MVexcuny ice. 224.55: 1200 | 550 | 2343; 1361 | 61:19) 0°89 | 2:34. 2-6 | Bromine) a) sae | 159°8 | 1293917265" 7-| 1-069) | a-Ga'0:38 4°879| 5 | @Cueas 7 eee Maen | 254 | 1487 386°5 | 1:0238 | 49°75) 0°944 | 384 | 4 InBOtassiime!).22. | 39 ] bat 1. 836°5)| 2:58.) 1230 Ors Zou iain le) MSodimmie e230... | 23 | O26) Side x) 2:92 azar ts 1-68 | 1:9 | Wead..:.........0....1207 *| T1I8" "600 | 1:99.) 96:8 O93 ieeGr a2) | Sodium hydroxide. 40 | 1610 | 633 | 4-601 | 2238 | 0:868 | 2-543] 2:9 _ Potassium fluoride.) 58 we 1133 4-737 | 230°3) | O92.) SoZ a 26.6 i Silver bromide .../187° | 2370 | 703 | 1:92 93°4 |0945 | 1:89 | 2:0 | Lead chloride ...... '178 | 5150 764 11:40 | 6d°1 |0°955 | 6-67) 7-0 | Barium chloride.. | 208°38 | 5581 |1232 |1°900 | 92°59] 0-96 4471) 4:7 | Acetic acid ®. ......). 60°03) 2641 | 289°7) 1-757 | 85-43) 0°888 |10-91 |12°3 lPienol......:........, 94 |2735 | 313 4 1261 1) 6-3) O23 Nag omen yBenzene '........42.. 78 | 2340 | 2785 1:306 | 63:5 | 0-909 } 85 9-4 Benzophenone...... 1182 | 8950 | 321 | 0°7447| 36-20)0:9515)12-28 19-9 | Naphthalene ...... 1128-1 | 4483°5) 853 | 1-064 | 51°71) 0-941 [12°70 |138°5 Summary. The latent heats of fusion of benzophenone (21°70 g. cal. per g¢); phenol (29-06 g. eal. per g.);° and’ monochme co) sulphur (8°85 v. cal. per g ay 1ave been determined by electrical heating at the melting-point, A theory of fusion from the point of view of the quantum theory, with the use of Debye’s formula for the energy, has been suggested. In some cases ‘the existence of intramolecular vibrations, as well as rotations, ‘is shown to be probable in the case of liquid molecules. The authors desire to express their thanks to the Chemical ‘Society for a grant which largely covered the expense of the ‘investigation. East London College, University of London. XLIX. On the Convection of Light (Fizeau Effect) in Moving Gases. By C.V. Raman, M.A., Hon. D.Sc., Palit Pro- fessor of Physics in the Calcutta University, and NIHAL Karan SeErui, D.Sc.* (Plates VI. & VII. ] 1. Introduction. T was early in the last century that Arago tried his famous experiment with the prism to detect whether the ether in the interior of a material body and the light- waves travelling inside it are carried along with that body in its motion, and it was to explain the negative result of this experiment that Fresnel propounded his well-known hypothesis that the eether outside a moving body remains stationary while that inside it drifts along with it, though with a diminished velocity. He deduced a law according to which this diminished velocity is given by the relation . il u = (1—-,)u, ; \ i where w is the velocity of the body and p its refractive index. This result also explained why the aberration of the fixed stars was found by Airy and Hoek to be independeit of the nature of the substance filling the telescope tube. And on account of its fundamental importance in the theory of Optics, some of the most eminent physicists have devoted considerable energy to verifying the different aspects of this law. More recently it has gained additional importance on account of the fact that this law follows as a matter of course from Hinstein’s remarkable principle of the relativity of space and time 7, and its experimental verification is now looked upon as one of the proofs of the correctness of )is theorem of the addition of velocities aud consequently of the special Principle of Relativity. We are, in the present note, not concerned with the first part of Fresnel’s law, which demands a fixity of the sether outside a moving body and which found support in the experiments of Sir O. Lodge t, who failed to discover a drift of zther in the neighbourhood of moving matter, even in the narrow space between two revolving disks or a crevice in a massive sphere. The apparent disagreement of these * Communicated by the Authors, t+ See Cunningham’s ‘ Theory of Relativity,’ p. 61. { Phil. Trans, A, 1893, p. 727. A448 Prof. C. V. Raman and Dr. N. K. Sethi on the with the experiments of Michelson and Morley *, which seem to contradict this fixity of the external ether, has been explained away by FitzGerald and Lorentz as being due to an inevitable change in the dimensions of the apparatus on account of its motion with the earth, an explanation very considerably simplified and made almost self-evident by the advent of the Relativity Principle fF. But so far as the second part is concerned, the only evidence of a positive character in favour of this law of ether drift was furnished by the celebrated experiments of Fizeau, in which a delicate interference method ‘was em- ployed to detect and measure a change in the velocity of light on account of a velocity of about 7 metres per second of a column of water through which it was made to travel. This experiment was repeated by Michelson and Morley ¢ in 1886 with improved apparatus, and it demonstrated the surprisingly good agreement with theory. But the appear-. ance of the elaborate electron theory of Lorentz made this agreement much less brilliant; for he showed that in dis- persive media the convection coefficient was not 1— —,, but 1— S _ aoe This necessitated a more careful repetition pro pee of the experiment which has recently been accomplished by Zeeman §, and the result is decidedly in favour of the new theory. The subject has been further followed up by him, and he has succeeded in overcoming the enormous experi- mental difficuities, and has actually determined for glass and quartz not only the Fresnel coefficient, but also the Lorentz correction for dispersion. In view of this recent work and the accuracy which has been attained in the measurements, it is hardly necessary to refer to the work of Sir J. J. Thomson ||, which led to the result that an electromagnetic wave inside a moving body should drift with half the velocity of that body. In support of this, it was argued that the substances fer which Fresnel’s law was actually verified in a positive manner happened to 1 ‘ A be such that 1— -, was in their case very nearly equal ! to 4. But even before the work of Zeeman, it was quite evident from the negative results of the experiments with * Plil. Mae. 1887, p. 449. + Cunningham, ‘ Relativity and Electron Theory,’ p. 34. { Amer. Journ. of Science, xxx. p. 377 (1886). § K. Akad. Amst. Proc. xvil. p. 445 (1914); xvii. p. 898 (1915) ; xxii. po. 462 & 512 (1920). || Phil. Mag. April 18&0. ee Convection of Light in Moving Gases. 449 moving air conducted by Michelson and Morley that Thomson’s value for the convection coefficient could not be correct. While, therefore, all the available evidence is strongly in favour of Fresnel’s value or, rather, Lorentz’s corrected value, at least in the case of solids and liquids, it is highly desirable that it shonld be confirmed in the case of gases also. A special grant secured from the Caleutta University through the kind offices of Sir Asutosh Mookerjee has enabled the writers to undertake this research, and it is proposed to give here a preliminary account of the progress of the work and of the difficulties encountered. But before we do so it might be of interest to deduce Fresnel’s law for gases from slightly different considerations. The late Lord Ravleigh* in discussing the scattering of light by small patticles and in referring the blue colour of the sky to the molecules of air, deduced an expression for the refractive index of the gas in terms of the molecular constants.’ Following the same line of argument, we may also deduce on Doppler’s Principle an expression for the altered refractive index when the air molecules are set in motion with « constant velocity v in a definite direction. On this principle, the incident light of wave-length X is received by the molecules as light of wave-length ft where 0 is the velocity of light in the medium without these scattering molecules. Thus, in the notation of Lord Rayleigh, the expression for the vibration scattered from the molecule in a direction making an angle @ with that of the primary vibration is D'—D oT 1 i Q Qa a —~—.. ~5{ 1—- } sin 8 cos —({ bt—r NP b r And considering the particles which occupy a thin stratum dx perpendicular to the primary ray 2, the resultant, at a point on the incident ray, of all the secondary “ennai which issue from this stratum is oe by oT »\? 2 n dr | ra eae a ( 1— *) cos = (nr) Dorr dr bik - 2 Bog yl D ~.5=(1- +) sin { Leia Fay r). * Se. Papers, vol. iv. p. 395 Phil. Mag. 8. 6. Vol. 43. No. 255. March 1922. 2G 450 Prof. C. V. Raman and Dr. N. K. Sethi on the ; 2 This, combined with the primary wave cos 2) : . 2 h 9! 1 will give cos —— t—xvx—o ), where : co DD 2v d J =nT dx op ase) If uw’ be the effective refractive index of the medium as modified by the moving molecules, that of the medium without the particles being taken as unity, 6'=(p'—I)de. ee. 2v —l=nT— aD (1- +). But it has been shown by Lord Rayleigh that if w be the refractive index with the molecules when they do not have the velocity »v, And therefore —D YT) 3 ; 2v ps ~1=(¥-)(1— ae ; Bikes v 1@., f—h=2(u—I) >, which, in terms of the corresponding velocities of light V and V’, reduces to pb eee 2(4—1) as In the case when p does not much differ from 1, as in air, this becomes V’—V=v. ; 1 =v{1— ‘) ‘ [oa 2. Experimental Methods. It was proposed to employ the same optical arrangements © as had been used by Fizeau and which constitute in effect a compensated Rayleigh Interferometer, and which are eminently suited for this work. Michelson’s arrangement might have been better, but owing to the great length of aia p+1=2] Convection of Light in Moving Gases. 451 path necessary, the attempt to observe the fringes with it did not prove to be a success. A large refracting telescope with a 7 inch object-glass and «a focal length of over 7 feet was available from the observatory of the Indian Association for the Cultivation of Science, and was made use of in the work. The plan of the optical arrangement is shown in fig. 1, from Fie, 1. Which it is clear that if / be the length of each of the tubes B and CD, the shift of the fringes observed when a © Current of air is sent round with a velocity v in the direction °f the arrows will be 4108 rV of the distance between two consecutive maxima or minima, where @ is the convection coefficient and V is the velocity of light in the medium. If, however, the current is first allowed to flow as in the figure and then suddenly reversed, the total shift will be double this or equal to ae It is evident, therefore, that what is required in order to increase the shift and render it measurable is that both J and v should be increased. Consequently we proposed to utilize all the space available to us and make the tubes about 200 feet in length. For driving the current it was proposed to utilize a Pressure Blower driven by a 3-h.p. gas engine (fig. 3, Pl. VJ.). With this we expected a velocity of air of over 50 metres per sec. in the tubes. The amount of shift of the fringes which we could therefore expect became 1 9 ‘ = teh SEE em 8 x 200 x 30 x 5000 x (1- area) mone xo LOM ; or about one-tenth of a fringe-width, which should be easily ineasurable with sufficient accuracy. In Michelson and 26 2 452 ~—s- Prof. C. V. Raman and Dr. N. K: Sethi on the Morley’s experiment the size of the apparatus was very much smaller and the velocity of air not more than about 25 metres per sec., so that the observed shift was, as stated by them, certainly less than +)5 of a fringe and probably less than =) >. The actual setting up of the apparatus was taken in hand about July 1920, and the site chosen was the compound of the Indian Association. One of the out-houses served to house the optical parts at the observing end, and a small hut was erected at the other end to receive the reflecting arrange- ment L,R (fig. 1). The big telescope body with the object- glass was mounted on solid brick-work pillars erected for the purpose, and adjustments had to be provided for both vertical and horizontal movements. Some of these are shown in fig. 3 (Pl. VI.). A fine spectrometer slit was mounted at the eyepiece end of the telescope, and the thin plate-glass, P, was also mounted in the same tube. A micro- scope was used to observe the fringes. For the other end, another shorter telescope (4 inch objective) was similarly mounted on two pillars, and the eyepiece was in this case replaced by one of the front-silvered Michelson inter- ferometer mirrors, so mounted as to allow of accurate adjustment of its plane. But the real difficulty lay in the fixing of the tubes. Galvanized-iron pipes of 14 in. internal bore were selected, and each piece was carefully straightened. Brick pillars were erected at intervals of about 10 ft., and their tops carefully aligned by means of a theodolite so that the deviations of the tp line from the mean did not exceed qo of an inch for any individual pillar. This having been accomplished, the pipes were assembled and placed on top of the pillars, so that ultimately there was a pair of tubes quite straight from end to end and separated from each other by a constant distance throughout. There was no appreciable sag of the pipe between.the pillars. This seems to have been avoided automatically by having such a von- siderable continuous length. This was finally tested by placing a small hole near one end and illuminating it by a strong beam of light. Observing at the other end, the whole of the aperture should be uniformly illuminated if there was no bend anywhere. Unfortunately this was very difficult to secure, for the slightest cutting-off of the light anywhere cast a tremendously large shadow at the great distance where it was being observed, and it was not at all easy to locate where the fault lay. The width of the diffrac- tion fringes at the edge of the shadow, however, gave a rough and ready indication of the approximate position of the fault, Convection of Light in Moving Gases. 453 and it was with the help of these that finally the major part of the aperture was freed from all obstacles. A parallel beam of light now sent down any one of the tubes passed unobstructed and filied almost the whole aperture at the other end. The ends of the tubes were closed by windows of about 7 in. diameter covered by thick interferometer plates secured between leather washers very much after. the manner employed by Zeeman *, and the current of air was led into the pipe by two side tubes inclined at about 60° to the axis of the pipe. The end pieces for carrying the windows and these inlet tubes were cast in brass, and can be seen in fig. 6 Gey VII.) This method of leading the current of air into and out of the tubes is calculated to cause the least disturbance in the path of the light. The manner of connecting the two tubes with each other at one end and with the arrangement of cocks to facilitate the reversing the direction of flow of air at the other, is shown in figs. 5 and 6 (PI. VIJ.). This connexion is by means of lead pipes, and the reversing arrangement consists of four cocks connected by means of a handle, so that at any time two alternate cocks are open and the other two simul- taneously closed. The central vertical pipe in the photograph leads to the blower, which was used rather as an exhauster, so that the air of the atmosphere entered one of the open cocks of the reversing arrangement, passed through one pipe to the other end, and returned through the other pipe and thence into the blower. ‘This method was followed in order to avoid the temperature changes which would have certainly occurred within the tubes if the compressed air from the blower had been led into them. The temperature of the whole of the air in the tubes was thus kept equal to that of the external atmosphere. With all these arrangements complete in January 1921, the two telescopes were carefully adjusted so that a fine slit sent two parallel beams down each tube, and the reflecting mirror at the other end sent tiem back still str ictly parallel. It is evident that the slightest want of adjustment in the direction or the para lelism of the beams was enough to stop all light trom passing through the tubes, and even the smallest angle between the two surfaces of each plate closing the ends of the tubes was inadmissible. But when these adjustments were carefully carried out, the fringes obtained were excellent and surprisingly steady, though the light = LOC, Cit, A454 Convection of Light in Moving Gases. itself ‘“‘ boiled” very badly, except in the cool mornings. The heat of the sun during the day caused variations in the temperature of the air inside the pipes, and the convection currents deflected the light in all sorts of ways. This was to some extent uvoided by covering the entire length of the pipes by bamboo screens, but the most satisfactory results were obtained only in the pormnes or an hour or two atter the sun had gone down. To test whether or not the paths traversed by the two interfering beams were identical, the method sug ogested by Micheison was employed. A plane- -parallel plate of glass was inserted in front of one of the ends of the tube, and the effect of its rotation on the fringes was observed in the microscope. With the final adjustments no displacement of the fringes could be observed by this means, and there seemed to be no reason to doubt that the paths were really identical and not merely parallet. The current of air was now turned on, but 1t was at once apparent that the air was not flowing at a constant rate, but was being driven through the tubes in puffs: and although nothing happened to the fringes which, whenever visible, appeared in the same position as when no current of air was flowing, yet the spots of light themselves “boiled” very badly indeed. An attempt was made to overcome this difficulty by inserting a reservoir of air to steady the motion, and, after some preliminary trials with a brickwork and cement reservoir which developed a leak and proved unsatis- factory, a large weoden box, 4 ft.x 4 ft. x 6 ft., covered over with galvanized-i iron sheets was installed, and inserted in the line of flow of the air between the pipe-line and the blower (see fig. 3, Pl. VI.). This successfully checked the oscil- lations in the speed of the current of air, and the spot of light in the microscope containing the interference fringes: could be steadily seen, even with the current of air running. To measure the velocity of the air-current, a pair of Pitot tubes with water-manometer were inserted in the channel between the pipe-line and steadying reservoir. 3. Results. When the arrangements including the steadying reservoir were complete in April 1921, it was found that the engine and blower were unequal to the task of drawing the air through the system at the originally estimated velocity of 50 metres. per second, and that a speed of only about 20 metres per second could be attained. The hot weather bs” peel ~ 2 oe : * Modifications of Hydrogen and Nitrogen by « Rays. 455 which had then commenced also made the temperature con- ditions in the tubes very unfavourable for systematic work. Nevertheless, on two evenings when observations were made after a smart shower following an April nor’wester, the fringes were seen very steadily, and appeared to show a slight but unmistakeable shift on reversing the direction of the air- current. On the second occasion an attempt was made to estimate the magnitude of the shift by setting a cross wire on the fringes and comparing the shift observed on reversing the air-current with that produced by flexure of the micro- scope tube by a known small load. The shift was estimated to be about 54 part of a fringe, which was of the right order of magnitude and in the direction indicated by theory. Subseguent attempts to confirm these ovservations and measurements under less favourable conditions proved un- fruitful, as the fringes then showed a distinct rotation as a whole when the air-current was reversed. The rotation proved to be a very troublesome and disturbing factor, and before the cause of it could be ascertained and removed, the work had te be suspended, owing to the departure of one of us for Kurope. While, therefore, the results so far obtained cannot be regarded as entirely conclusive, they hold out a distinct promise of success when the work is resumed under more favourable conditions, particularly if a more powerful blower with steady electric drive can be obtained and installed. L. Active Modifications of Hydrogen and Nitrogen produced by «a Rays. By F.H. Newman, M.Se., PhD., Head ef the Physics Department, University College, Exeter *. 1. INTRODUCTION. T has been shown recently + that nitrogen and hydrogen are both absorbed in the electric discharge tube, in the presence of various elements, when an electric. discharge passes through the gases. This effect is due, in part, to chemical action, and is greatest when the element under investigation is deposited on the cathode of the tube. The chemical action appears to be produced by the gases assuming active modifications when an electric discharge is = Communicated by the Author. be, + Newman, Proce. Phys. Soc. xxxiii. part ii. (1921). 456 Dr. F. H. Newman on Active Modifications of passed through them. Strutt * found that nitrogen drawn from a discharge tube had active properties, and concluded from his experiments that the modified form was atomic in composition. Wendt+ has sho. n that a modified form of hydrogen, probably consisting of H3, is produced when an electric discharge passes through hydrogen at low pressures. In both cases the active form is not due to the ions produced. The radiations from radioactive bodies, and especially the a rays, produce marked chemical, effects in many substances. In general, the chemical actions produced resemble those due to the silent electric discharge. In some cases complex molecules are dissociated, in others more complex mole- cules are built up. Thus ozone is produced from oxygen. Carbon dioxide is transformed into carbon, carbon monoxide, and oxygen. Carbon monoxide is decomposed with the appearance of solid carbon and oxygen. Ammonia is changed into nitrogen and hydrogen, and in all cases the chemical action is proportional to the amount of radio- active body present, showing that the transformation of each atom of the radioactive body produces a definite chemical effect. It would be expected that the effect of the rays from any radioactive substance, more particularly the a rays, would be similar to the electric discharge at low pressure. It should be possible to produce the active forms of hydrogen and nitrogen by the « rays.- The object of the present work was to “activate” these gases in this manner, and show by the formation of chemical compounds that the modified form of the gas is more active, chemically, than the ordinary gas. 2. Description of Apparatus. The apparatus used is shown in fig. 1. The gas was pre- pared, and stored in a very pure condition in a reservoir from which it could enter D. The latter was a known volume (0°106 e¢.c.) enclosed between two taps. A was the experimental bulb containing the radioactive substance— an electrolytic deposit of polonium on the plate G. The support of the plate was sealed through a glass stopper. The polonium could thus be removed while the bulb A was exeing cleaned after each experiment. The element used was introduced by the side tube B. The whole of the apparatus was evacuated by a Toepler pump. F and EH were * Proc. Roy. Soc. lxxxw,. (Goi); t Nat. Acad. Sci. Proc. v. (1919). Hydrogen and Nitrogen produced by « Rays. 457 entrance and exit tubes respectively for moist air, used in the detection of chemical compounds produced. Asa rule, the active gases will not react with any element unless the Fig. 1. surface of the latter is clean and free from occluded gas. In all experiments a thin film of the substance being tested was deposited over the surface of A by distillation in vacuum. The volume of A was about 50 c.c. 3. Huperiments with Nitrogen. The nitrogen used was prepared by warming a solution of sodium nitrite and ammonium chloride. The gas was then passed over sodium-potassium alloy, calcium chloride, and phospkorus pentoxide to the reservoir. ‘The whole of the _ apparatus was thoroughly washed out with nitrogen. The gas was then pumped out, and the element at the bottom of A heated to produce the pure deposit on the interior of A, care being taken that there was no film formed on G. Again the apparatus was evacuated, and the gases liberated by the melting of the element were removed. The tap H being closed, the oil manometer C was calibrated by admitting known volumes of gas from D, the pressure before and after the gas was admitted being noted. The arays then acted on the gas in A, producing active nitrogen which combined chemically with the element present. As a result the gas gradually disappeared, this absorption being shown by a gradual decrease in the pressure indicated by C. From the initial and final readings of C the amount of gas absorbed could be calculated. This was repeated, using different, pressures of gas. The polonium was never more than 6 em,,_ away from any part of A, and the range of the a rays was always greater than this distance, at the pressures used. The latter were measured by a mercury gauge, not shown in the diagram. es er 458 Dr. F. H. Newman on Active Modifications of The substances selected were those which are fairly volatile in vacuum. The temperature of the experimental bulb A was maintained at —40° C. by immersion in a freezing | mixture. Table I. shows that the rate of absorption in- creased as the pressure of the gas increased. The chemical TArie 1, Volume of nitrogen (at 760 mm. pressure) absorbed in 30 mins. | | Pressure. Pressure Pressure Substance. 3mm. He. 96 mm. Hg. 3804 mm. Hg. OGHMM ees eee 0-04 c.e. 0:07 c.c. 0:24 c.c. Potassium yee 0:05 0-07 0:27 Sodium- Potassium MUON Meer cht Woo 0:09 O11 0°36 Suliphin ye 0-11 0-14 0-48 Phosphorus ...... 0-12 0-16 O51 MOGI eS Pe es 0:09 OnE 0°46 PESO Gut ete 0-01. 0-08 0-21 Magnesium .. ... 0-03 0:06 0-19 MMiereuty jee. 22 se. 0-04 0-08 0-21 effect due to the a rays, which is not necessarily equivalent to the ionization and probably exceeds the latter, appears to: be a function only of the velocity of the @ rays and ot the number of collisions with the gas molecules. As the pres- sure decreases, the number of molecules present decreases, and as a result the amount of chemical action decreases. After absorption had occurred no gas was re-liberated when the bulb A was heated to 100° C. The chemical compounds produced must be fairly stable. The disappearance of the gas continued for many hours, but the rate of the absorption decreased with time, and finally ceased altogether. If the substanee was re-heated, so that a fresh surface was prepared, the absorption re-commenced. The chemical compound pro- duced at the surface of the substance protects the solid from further action, and accounts for this fatigue effect ; unless the surface under examination was very clean there was no absorption at all. When about 3 ce. of the gas had been absorbed by the sodium-potassium alloy, moist air was drawn through the bulb A from F to E, and was then passed through Nessler’s solution. The presence of ammonia in the stream of air showed that the nitrides of sodium and potassium had been produced. A null experiment indicated that this ammonia was not present as an impurity in the air. The product obtained with magnesium was boiled with caustic potash, aud the formation of a nitride was shown by the Hydrogen and Nitrogen produced by « Rays. 459 ammonia test. Negative results were obtained with sulphur, iodine, and phosphorus when tested for nitrides. The absorption cannot be due to occlusion, otherwise the gas would have been re-liberated on heating. The effect must be due, in some cases at any rate, to chemical action. 4. Heperiments with Hydrogen. The gas was prepared by the Heo es of barium hydrate. After passing over sodium-potassium alloy to remove any oxygen, and then over phosphorus pentoxide, it was stored in a reservoir. Experiments were conducted in the same manner ‘as those witl: nitrogen, but it was found that the gas was absorbed only by sulphur, phosphorus, and iodine. The results obtained are shown in Table IJ. The buib was main~— tained at —40° C. TasBueE If. Values of hivdroeee ae 760 mm. pressure) apeenee in 30 mins. Element. | Pressure Pressure | Pressure 5mm. Hg. | 101 mm. Hg. | 299 mm. Hg. Sulphur « .2..:.... 003" cie: 0-13 c.c. 0°46 c.c. Phosphorus ...... 0°10 0-16 0:49 crite ae ees 0°08 0°12 0°43 The rate of absorption decreased with time, and on heating the bulb to 100° C. most of the hydrogen which had been absorbed was re-liberated. This behaviour is different to that in the case of uitrogen. This indicates that with hydrogen the effect is either due to occlusion, or if it is due to Chemical action, the resulting compounds ona be unstable. To show that the disappearance of the gas was due to cheniical action, the apparatus shown in fig. 2 was utilized. Two strips of platinum foil, about 8 em. “long and fitting close to the interior of the glass tube, were introduced at B. The strips were connected to the terminals of a 600-volt water battery. Any ions present in the gas were removed as they passed through B. Sodium- -potassium alloy was prepared in D, and, “after the whole apparatus had been exhausted, the alloy was run into C. As a result, the alloy in G had a bright, clean surface. Hydrogen was fen drawn com) through the apparatus, and was subject to the action of the a rays. When examined by a microscope, the surface at C was seen to be covered with a white crystalline substance, which afterwards changed into a blush-grey coloured film as 460 Dr. F. H. Newman on Active Modifications of more hydrogen was drawn through. The experiment was repeated without the polonium present, and in this case the surface of the alloy remained quite clear. The Slm produced is due to chemical action of the active hydrogen on the alloy. The white crystalline compound appears to be a mixture of the hydrides of sodium and potassium, while the bluish-grey coloured product afterwards formed is probably a solution _of these hydrides in the alloy. For the investigation of sulphur, a film of this element was deposited over the interior surface of C by distillation in vacuum. D contained some filter-paper soaked in lead- acetate solution, together with a little of the solution. After hydrogen, subject to the action of the « rays, had been drawn over the sulphur for about 30 minutes, the filter-paper in D became blackened. ‘This indicated the presence of hydrogen sulphide in the stream of gas, and it had been produced by the action of the active hydrogen on the sulphur. A similar result has been found by Wendt and Landauer*. ‘This chemical action is not due directly to the ions present in the gas, for they are all removed by the electrostatic field before coming into contact with the sulphur. 5. Other Heperiments. Some radium emanation of strength 57 mg., enclosed in a glass tube, was substituted for the polonium. The thickness of the glass—about 0°5 mm.—absorbed the a rays but trans- mitted the other rays. After several hours of test there was practically no absorption of hydrogen or nitrogen by any element: this indicates that the a rays are the effective ones in the chemical actions observed above. Ultra-violet ght was employed instead of the radio- active compound. A quartz-mercury lamp was used as the source of illumination, and the experimental bulb A was of * Amer. Chem. Soc. Journ. xlii. (1920). Hydrogen and Nitrogen produced by « Rays. 46} uartz. There was, however, no sign of any activation in either gas. This negative result is explained either by the ultra-violet light not. producing active gas, or by the quartz not being sufficiently transparent to those rays which pro- duce the active modification. The amount of gas absorbed was independent of the thickness of the element deposited on the glass surface: this indicates that the chemical action occurs at the surface. The rate of absorption was increased by using a larger surface area of polonium, and was decreased when the. temperature of the gas was raised. The activation of the gases thus appears to become greater as the temperature is. lowered. 6. Discussion of Results. Langmuir™* found that hydrogen and nitrogen at low pressures disappeared in the presence of an incandescent tungsten filament, and he has utilized this fact in the elimi- nation of the last traces of gas in valves, etc. The nitrogen forms a nitride with the tungsten. He accounted for the disappearance of the hydrogen by assuming that the mole- cules of the gas on impact with the hot filament are disso- ciated into atoms, and these atoms, at very low pressures, have a clear run to the walls of the vessel, and condense on them. This disappearance is marked at low temperatures, On heating the tube, Langmuir found that part of the gas was recovered. his re-liberated gas he supposes is due to the re-combination of the atoms driven off from the walls of the vessel by the heat. At the pressures used in the present work, it is unlikely that the active gas consists of atoms. It is probably dis- sociated at first into atoms by the action of the a rays, and the atoms then unite with neutral molecules to form tri- atomic molecules. The existence of these triatomic forms of nitrogen and hydrogen has been shown by Wendt, in the papers previously quoted, by passing electric discharges through these gases at low pressures. The function of the ions produced by the @ rays does not appear to assist a chemical action between the gas and the element which does not otherwise take place, for the com- bination occurs after the ions have been removed. More probably the activity of the gas is due to an atomizing effect of the a rays on the gas, the active product being monatomic * Amer. Chem. Soc. Journ. xxxv. (1913); & xxxvii. (1915). 462 Modifications of Hydrogen and Nitrogen by « Rays. ‘gas. Some of the atoms will be formed at the surface of the element, and will react chemically. Others will be pro- duced in the interior of the gas, and will form triatomic molecules by collision with the neutral molecules. The primary action of the 2 rays appears to be the liberation of -atoms of high activity. In addition, the rays produce 6 rays by their motion through gases, and these 6 rays may also be effective in producing the active modifications. The chemical -actions occurring are probably determined by the heat neces- ‘sary to effect ne decomposition of the molecules, and not by the heat required for the final transformations and resulting products. That the a rays produce active modifications of hydrogen and nitrogen, which in turn are able to react chemically. with certain elements, strengthens the view that the disappearance of these gases in an electric discharge ‘tube, in the presence of various elements, is also due mainly to the formation of the active modifications of the gases by the electric discharge. 7. Summary. 1. « rays from polonium were allowed to act on nitrogen -at different pressures in the presence of various elements. ‘Some of the gas was absorbed. 2. The elements tested were sodium, potassium, sulphur, “phosphorus, iodine, magnesium, arsenic, mercury, together with the alloy of sodium and: potassium. Similar experi- ments with hydrogen gave absorption with sulphur, phos- phorus, and iodine. 3. The absorption of the gas was shown to be due to chemical action resulting in the production of nitrides and vhydrides. 4. The chemical activity of the gas is due to an active modification produced by the a rays. The active form, probably consisting of neutral atoms and triatomic molecules, ‘is not due directly to the presence of ions. 5. The a rays are the only ones effective from radioactive .bodies. [ 463 ] LI. The Analysis of Microsetsmograms. By J.B. Dan, M.A. * 1. FN his monograph on Modern Seismology, Walker has called attention to the inter esting records traced by seismographs when undisturbed by earthg uakes. Miesaccismie motion falls into two classes, of which one is due to the motion of wind and is wholly irregular, while the other prevails even in calm weather and at all times of the year, although it is more marked in winter than in summer, Tt is this latter type which is discussed in the present paper. The general characteristics of the records of this seismic motion are smooth sinusoidal curves of fairly constant period, but with an amplitude which rises and falls at somewhat irregular intervals whose average length is about 1 minute. The period of the oscillations ranges from 4 to 8 seconds, the longer periods and larger average amplitudes occurring in the winter months. Although the main features of the motion are obvious, their exact specification is by no means easy, and analysis discloses the simultaneous existence of oscillations of different periods. eoere also appear to be discontinuities in phase and period. . The method of analysis employed is that published by a se a few years ago t, and the following summary will explain the notation.» It is assumed that a function y of t can be expressed in the form Cae casi, (Ont tole i. LS eh) when ¢9, ny On, &n are constants to be determined from the observational data. Vt is further assumed that the values of y are known for a sufficient number of. eddistant values of t, the values corre- sponding to ¢=0, 1, 2, ... n being denoted by Oi ORO nat ps As a first step co is eliminated by forming differences which are denoted by a, so that 4 dy=—Yr41—Yr- ° > 5 : “ b (2) Next an operator E is defined by the relation Ha,= 4741+ 4-1, SMe e EV 3) and in a similar manner E?a,, K?a, are derived. The number of times the operation E is to be performed is equal to the number of independent periods. In practice * Communicated by the Author. + “The Resolution of a Compound Periodic Function into Simple Petiodie Functions,” Monthly Notices, R.A.S., May 1914. AG64 Mr. J. B. Dale on the this number can be found only by trial. Supposing, for example, that three periodic terms are involved, then it must be possible to find constants p,, ps2, p3 such that the relation Ha, + pHa, + pola, +p3a,-=0 . . . {A4) is identically satisfied for all values of r. In any actual problem, on account of errors in the data it is impossible to satisfy this relation accurately ; but if the residuals obtained on substituting successive values of a, in the expression are always small and exhibit no systematic run, we may regard the assumption of three independent periodic terms as justified. In the problem considered in the present paper it is found that the assumption of two periods satisfies this condition, and the relation to be satisfied 1s Wa +p,Ha,+ poa,=0. . 2 . 2 4’) The values of »,, pe are found by solving two of the equations. With the values of the p’s thus found, the equations e+ Die + poet P3s— 0 rt or +912 + ps = 0, 52 i eee as the case may be, are formed and solved. Considering (5), if the roots are 21, 2, 23, then the three speeds 6, 6, 03; of the periodic terms are given by 6,=cos 112,, @:=cos '52,, 0; Cespasc ero Finally, to obtain the amplitudes and arguments corre- sponding to 6;, we form the series of quantities Pot = Wa, (29+ 23) Har+ eo23dp 2. CeCe CT) Them. ce, sin (7 -+a)=(P),.1—P ere €, COS (Oyr+ 1) = Comes + Payer C A (8') where : = 1/2 sin 0; (21—2.)(2;— 23), S, = 1/(2—23) (41-2) (21— 2) Py - - (9) In like manner the arguments and amplitudes of the other periodic terms are obtained. 3. The curve here submitted to analysis is part of that taken at Pulkowa, Sept. 18, 1910, and reproduced as Plate 7 A in Walker’s book. By means of tracing-paper ruled in millimetre squares, readings of the ordinates were taken at intervals of 1 mm. Analysis of Microsersmograms. A695 The time-scale was such that 2°8 cm. corresponds to 1 minute ; hence 1 mm. corresponds to 2°14 seconds. The ordinates are tabulated in units of 0°-l mm. The breadth of the trace being about 0°3 mm., it was not possible to read the values accurately in the neighbourhood of acute intersections of the curye with the ordinate lines, and so our readings may be 0-3 mm. or more in error. 4. The main steps in the determination of the periods of the oscillations prevailing during 1 minute are set out in Table I. It was found by triai that all the equations (4) could be well satisfied by the assumption of only two periods. TABLE J. { beh) |) @) (3) (4) (d) (6) (7) (8) | t y a a.) Ba. R y' Rie | | 0-0 él +21 | ol mete? 7 a 52 pete iT) th ee (238) P|) 45 4 ge 28 +32 | —42 | +61 + 4 29 =) Ae GO == oy +34, -|. 48 Bie) 60 ey 5 39 4 |) 56 + 9 — | 30 +3 Reesor 19” | S27 tee7 | te | sg 0) 8 mot 31, le £38 BON lirresy 2,1 i659 2 ee rit fir P13 ft 6 99.) meta | 890 | ap I on eg | go oni 1:0 61 —34 | -—35 —37 | 0 a9 +1 1 27 a a ( am Ai L 28 249 Pee eS 7) |. a ee 10: || Orr | 43 0 3 59 — 29 +27 —20 | + 5 59 —1 | Pr ae 10) Ml tor ieee (eS ae! 99 0 & | 40 eon | nao BATA eek Tl 40 4] 6 63 —40 +47 NU) emer 65 +1 Bie | 484 | 80) | pe BEG 99 ag Beene; | 20 | —a6) | eo Ge as af Houg 7 25) +49 5A 6 65 ao ine 20 17 +29 — 28 +23 — 4 18 —3 eta 46 +22 6 +41 229) +3 Piee |) 6s 5 68) B87 4 8 |! 67 =i | Peas 13 aA 265 +81 Se salt Laat hea © 60 — 7 | +12 —27 -- 84 59 +2 | MMe | ao ets | See ge | a 6 18 +42 —5dD +73 1 20 —3 eh GO i 20a E88 s bh Beet a | Et 0 8 40 — 4 — 4 40 Sy) 36 +16 36 30 52 | nD, | | Jolumn (1) gives the abscissee, the common difference being 1 mm. or 2°14 seconds. Column (2) gives the corre- sponding ordinates y. Column (3) contains the differences (a), and columns (4) and (5) the values of Ha and Eva. Phil. Mag. 8. 6. Vol. 43. No. 255. March 1922. 2 H 466 Mr. 1p B. Dale on the Equations of the type E’a+p,Ha+p,a=0 were formed and divided into two groups, the equations in each group being added to form a single equation. From the two resultant equations the values of p,; and py were found to be p == 2°4290, Do= 14188.) ee: Hence y= — OTA, 2g — LAO ae ee) and Gee O5— Laon : E - (12) In order to test how far the assumption of the existence of only two independent periodic terms was justified, the values of R=H’a+p,Ha+p.a were calculated with the above values of p; and py and are placed in column (6). It is seen that with one or two exceptions the residuals are small and present no systematic run. Hence it is coneluded that the deviations from zero are entirely due to errors in reading the values of y, and would be removed by appropriate corrections. 5. The determination of corrections is really an indeter- minate problem, for the number of ordinates exceeds the number of equations. Expressed in terms of the y’s, each equation H?a,+ p,Ha,+ poa,=0 becomes rat QYrs2+ QYrs1— Jor — DYr-1—Yr-2=0, (18) where qg;=p,—1 and gg=2+po—pi. . . - = - CA) The observed values, however, differ from the true values by small quantities v, and hence these deviations from the true values satisfy the equation Urea t+ QUrt2 + Glr41— Qotr— Qilr-1— r-2=R,. (19) To render the problem of finding appropriate. values of the vs determinate, we add the condition that the sum of their squares must be a minimum. Since the labour involved in obtaining an exact solution of the system of simultaneous equations arising from the expression of the problem in this form would be extremely oreat, it was decided to proceed by a method of successive approximation. _ It is clear that any particular v will appear in six equations of the tvpe (15), and these together with the minimum con- ditions will give a value for v. This method was tried, but. was not found to lead to any more satisfactory results than those obtained by the simple method of obtaining six separate Analysis of Microseismograms. 467 corrections to each v by solving equation (15) alone subject — to the minimum condition. Taking then (15) subject to U'r413 55 Vr pat Wed +v7,.+ Unt + U'p—2 being a minimum, we obtain Uri3= —Ur_2=A, ncar Be tea | Urt1= —lr=Qor, r (16) where A=R,/20. 4+ 9.7 +92”). 3 On repeating with the v’s substituted for the corresponding ys the original series of operations, a new series of resi- duals was obtained whose average value was double the original R’s. The values of y were therefore corrected by subtracting from them half the total values of the wv’s as given by the above formule. These corrected values of y are given in column (7) and the resulting residuals in column (8). Tt is seen that all large residuals ‘have been removed, and this has been done without altering any y by more ‘than wo units, an amount well within the probable limits of error in reading the curve. Seeing that the values of g; and q,. are about unity, it is clear that an alteration of a y by a single unit may give rise to a unit residual ; hence a complete removal of the residuals is difficult: but the fact that of the residuals only two are equal to 4, six are equal to 3, and the remainder are 1 or 0, shows that all the equations of conditions are now well satisfied. 6. We now proceed to determine the amplitudes and ar guments of the component terms. With the corrected values 2 y we form iresh values of a and Ka (Table II.). This table also contains the values of P!.=Ha,— Ley: (col. 4). Column (5) contains 8, (P4,-1—P1,) where 8, =1/(2—z,) X (2; —2.) =9°71, and column (6) ©C,(P1,_, + Pt.) where C,= 1/2 sin 0,(2;—<) =1-20. Incolumn (7) the corresponding arguments 0,t+«,=¢, are tabulated, and in column (8) the amplitudes ¢. Column (10) contains the corresponding arguments dy» for the terms involving @,, and column (11) the values of Cy. These values are calculated with S,= 1/(2 — 29) (22— 21) = —*6L and ¢=1/2 sin 09(z2—2,) = — 1°53. The last column gives the values of ¢, which are obtained by subtracting from 7’, the sum of the values of 8;(P1,_;— P},) and S.(P?,-1— P?,). The constancy of the values so obtained affords a check on the accuracy of the work and a further justification of the assumptions made. 2 2 ee ee ae eee ; 468 Meee sale oni TABLE If. | l Oe ey 1 -@) (4) (5) | (6) |(7) /(8) | (9): | 0)| (D2) ay | Ay Ay ada age a y shai Mea i Ay ay a4 t a Ea. iPS, oe | c ob, Cj oom po G Co ek oe sts Rok | < - 2} =16 | 4-24) 4 1) — 29> = 24-1230) 4 ee iGae on miaeaeee 3 | 4311 243) + 2-|) —.0-7 | + 3:6 | 348) 3 ae Ue aie A) 97593236) = 38 |b 8°6 | — 12) 108) 4 eee Oia aye ec Flos hl 6 + 1 | =] 2:9 1-2-4) 23018 Gain oom 6 | 421 | 95 | +5 |} — 29 | 4 7-2 |338|, 9s sane 7 |) 30 | +34 | —10 | 410-7 | — 6:0: | 119 | 12) “2G oe 8 e138 18) 4+ 6 [71-4 |= '4-8 "| 247 | 12 PSB Oe 9) 417-7) —18 |) +7) — O74 +15°6 | 358) 16) = OG se eee 10) 2231-) 4-32 | —13 | 414-2. | — 97-2 V7 elGs SaaIcS ail Oraeees 125 |) 15 | + 7 | 142 | — 7-2 | 248 6s ee Oe ees 2) 516) —15 | 4+ 8) — OF | +180 | 358) 18 | a0 Geis Rees 3-20 | +27 | —16 | 4171 | = 9:6 | 119) 2072 Meer ait ees A) ig 5) +11] 17 960 | 251) Tei e oe nee eeu h | 95 | -82) 4+ 4) 4+ 50 | +180 | 16) 1S | ison Res 6 | 48 48 | —14 } 419-8 | —12:0 1198) 18 220s 5 eos 7 | 2238 93 | +10 | —17-1 | = 48 | 254) 48 2943s 20or oats 8 | +90 | —241 + 5)4 36 | +180 | It) 19) ee @ | —47 | +49 | —19 | 417-1 | —16:8 | 134) 247) er oe et ees 90) 4299 | 97 | +15 | —249 | — 4:8 | 259) 25) = Oe Ions 1 | 20 | —27 ) + 2) + 930) 4204 | 251122), Seb oiaead eee 956 | +68 | —18 | +107.) —13:2 | 141) 17 eee ee ees 312148 | —61 | + 9| —156 | — 48 | 253) 1G |= 1G 2a on log ers Alama e E Abie abe tt) is | A eNO 4/19) +140 1| 16 | 44 5h | — 34. | 4-36 |) —13.| 4142) 67-20 | 107 | 160) Se ee ees 6 | 441 |'—55 | + 5 | —12°8 | — 96 | 233) 16) SOM e2ei ieee GP ee Ih Sy a ie mya ome | | +144 |354|) 15 | +196 | 99) 20 | 43 Sa 4 = iil | eins Bay 48 |1111 14} —15:8 | 232) 20 | 43 7. The results here tabulated show numerous irregularities, but this is not surprising when we consider that tke argu- ments and amplitudes ultimately depend upon the values of P!. and P2,, all.of which are less than 20 in absolute magnitude, and of which the majority are less than 10. Assuming the values of y' to be now correct within half a unit, and this is certainly not true in every case, there is a possible error of one unit in a and of two units in Ha. Hence P1, which is equal to Har—z,a, may be over three units in error, and P?, which is equal to Ha—z,a, may be in error by nearly the same amount. Bearing this in mind, the consistency of the values obtained may be regarded as quite satisfactory. | An examination of the values of ¢, shows.a regular Analysis of Microsecsmograms. 469 increase, in good agreement with the value for 0, obtained trom the equation for the periods, namely 119°°2. A least square solution from the values obtained for ¢, increases the value of 6, by a degree, making it 120°2. This value indicates an oscillation whose period is three of the units of t or 64 seconds. But the values of c; show a rise from a value which is nearly zero to one of 25 followed by a fall. On plotting these values against the time, it is seen that they can be well represented by the expresion 22 sin (6° —12°) ; and since the unit of time is 2°14 seconds, this indicates a complete fluctuation of amplitude in 2:14 minutes. - The phase when t=O of the angle d, may be taken without sensible error to be zero; hence the complete expression for the first periodic term becomes 22 sin(6o— 12) sin L205 26.0 ysis She CLT) 8. The values of ¢, exhibit much greater irregularities than those of ¢,, particularly in the neighbourhood of ¢=10, where’ the amplitudes are nearly zero. If, however, we assume that there is a change of phase at this point of 180°, we can remove the most serious discrepancies. The mean of the first 9 values of ¢, gives 323° as the argument at t=6, and the mean of the last 9 values gives 44° as the argument at ¢=24. Increasing 323° by 180° we obtain for the in- erement of $, per unit of t, 134°°5, which is in good agree- ment with the value 136°°5 previously obtained. Plotting the values of cy against ¢, taking those before t=11 to be negative on account of the change of phase by 180°, we find that the points are satisfactorily represented by 20 sin (5°t—70°}, and the complete expression for the second periodic term is AQ (5° é— 70°) sin (W34°-"5 t+ 96°). . .., (8) It is to be noted that no great degree of accuracy is claimed for the coefficients 6° and 5° of ¢ in.the factors sin (6°¢—12°) and sin (5°t—70°), the data analysed being too restricted in extent for a closer determination, but they are probably correct to within 10 per cent. 9. An obvious interpretation of the results here obtained is that the microseismic motion was due to two groups of waves of periods of 64 and 52 seconds respectively. The group velocities were slightly different, but were of such an order that on the average each group attained a maximum within an interval of two minutes. Since the maximum amplitudes were nearly equal, the combination of the two 470 Analysis of Microseismograms. sets of waves gave rise to maxima at average intervals of one minute, and this feature is clearly shown on the records. 10. Another interpretation, however, is that the analysis employed has effected a separation of the observed movement into more than two simple harmonic components, for we may write 22 sin (6° é—12°) sin 120°°2¢ + 20 sin (5° t—70°) sin (134°5 t+ 56°) in the form 11 cos (114°°2 ¢— 12°) —11 cos (126°2 ¢—12°), +10 cos (129°°5 t— 126°) —10 cos (139°'5 ¢—14°). In view of the uncertainty attaching to the values found for the speeds, the terms 10 cos (129°5 £—126°) and 11 cos (1262 t—12°) are hardly distinguishable one from the other, and there is a possibility that the number of component harmonies is only three, with speeds about 114°, 127°, and 139°. The smallness of the amplitudes makes a closer determi- nation of the angles impossible. It is therefore desirable to see whether other portions of the seismogram confirm the result just obtained. The portion’ of the trace analysed formed part of the fifth line of Plate 7 A, and portions of the traces in the first, second,.third, seventh, and eighth lines were measured and treated in the same manner. As arule, the equations of conditions were well satisfied. by the assumption of the existence of two periodic terms, but in two cases somewhat smailer residuals were obtained by the assumption that three terms were involved. The values of the speed angles thus found are here tabu- lated. ier) ist solution ey..ccceneeee 1299-5, 114°] 2nd) SOLUtIOM a) <2 3x eee eenee IS 1°-6; 119°°8 Line 2. Assume two periods ...... 1340-2, 116°°4 Assume three periods ...... lesbo) 1220) 82°°2. Line 3. Assume two periods ...... 148°°7, 125° Assume three periods ...... 149°°5, 120° 2, 45°°5. Line 5. Assume two periods ...... 134°°5, 120°°2 Since, however, the amplitudes vary, it is more correct to say that there are three periods whose speed values are 139°, 127°, 114°. TBST me 0A Me? aa ge ee 1Z9°°6; LEZOY2 Lave (SPR ee een ene Mre Sek eect ie. . ee 140°-9, 110°:0° Interferometer Method of determining Phase Difference. 471 The results agree sufficiently well to establish the exist- ence of at least two periedic terms, but the differences between the various solutions are so large that itis not easy to say whether three or even a greater number may not be involved. The somewhat unsatisfactory nature of this conclusion is solely due to the circumstance that the measured ordinates may be in error by an amount of the order of one-tenth of the maximum amplitude of swing, and as analyses of arti- ficially constructed functions show, small changes in the values of the ordinates make unexpectedly large eeions in the values of speed and other constants. Until means are found for obtaining a higher degree of magnification of the movement, and, if possible, a diminution in the breadth of the trace, it will be difficult to obtain more definite results. University of London, King’s College. Aug. 1921. LIf. An Interferometer Method of tage the Phase Difference resulting from Metallic Reflevion. By H. P. WaRAN, ee Government Scholar of the University of Madras * [Plate VIIL.] HILE investigating the possibility of a parallel plate interferometer, by floating one transparent liquid as water over another like mercury; the question arose as to the difficulties that would arise from the difference in the character of the reflexions from the two surfaces. In the parallel plate of glass both the surfaces are identical and the reflexions take place under identical conditions. But in the present case nearly total reflexion at the top water-air surface and simple reflexion at the bottom mercury surface are utilised to get multiple reflexions from the parallel plate of liquid. ‘To see if this metallic reflexion at the bottom surface was producing any disturbing factor prejudicial to the success of this type of interferometer, a control experiment was devised with a glass Lummer plate which led to a very interesting observation. The glass Lummer plate (made by Hilger) was mounted on a specially made ebonite mount, resembling very much the * Communicated by Prof. A. W. Porter, F.R.S. 472 Mr.H.P.Waran: Interferometer Method of determining usual brass mount, but provided with a trough arrangement as shown in fig. 1 in part section, by which clean mercury could be introduced below the plate and made to cover its bottom surface. This combination being of identical size with the usual mount of the parallel plate could be fitted on the Hilger interference spectroscope between the collimator and the constant deviation prism, asin fig. a (PI. VIII.). Fig. 1. Z| Z Z Z Zi Z| aA Zi 2 % SANA The slit was illuminated by a mercury vapour lamp, and the resulting interference pattern displayed an interesting change as shown by fig. b (Pl. VIII.). For the first appearance it was as if an extra system of bands had crept in between the normal system of bands. That it was not due to a simple reflected system was evident from the absence of constant spacing between the two sets of bands. An idea that it might be due to a thin layer of air between the glass and the mercury led to a repetition of the experiment by silvering the bottom surface of the plate chemically. The same type of band system observed made it conclusive that it was being brought about as an effect of the metallic reflexion at the bottom surface. Considering the effect of such a metallic reflexion, we know from the optical properties of metals that the emergent light will be elliptically polarized. On this consideration, the extra system of bands could be explained away directly as due toa simple phase difference effect brought about by the metallic reflexion. Insucha case the effect of polarizing the incident light pught to be to make one of these two systems of bands disappear as shown in fig. 2, according as the incident light is polarized in the plane of incidence or perpendicular to it. A Nicol polarizer introduced into the path of the incident light brings about this change as expected, confirming the validity of our explanation. Phase Difference resulting from Metallic Reflexion, 473 Considering the accuracy with which interferometer measurements could be conducted, it occurred to the writer that this method may be used for experimentally determining the phase difference resulting from metallic reflexion ii ereater accuracy, since it is free from the defects of other ectiiods which suffer * from the unavoidable impurities of the polished metal surfaces exposed to air. Before going into the theory of the method, a word may Ke said about ihe ‘experimental arrangements adopted to secure ce) the necessarv photographs of the fringes for measurement. Fig. 2.—Showing the types of changes resulting from metallic reflexion in polarized and unpolarized light. Light not polarised Light polarised One of the Revers At 45 to | Perpendicular iene plane normal the plane to the plane band system Tie uaLe ey of or reflection. incidence. Hncidence greidentcs The glass Lummer plate chemically silvered on the bottom surface was taken, and half the silvering along its length removed by gentle rubbing with dilute nitric acid, and the glass portion thus showing cleaned and dried carefully. It was then mounted in its usual stand on the Hilger inter- ference spectroscope. Light from a mercury vapour lamp was focussed on the slit by a pair of collimating lenses, between which was interposed a 2in. Nicol polarizer as illus- trated in fig. ¢ (Pl. VIII.). The polarizer was set to polarize the incident light in the plane of incidence, and thus cut off the other component set of bands resulting from the metallic reflexion and corresponding to the case of light polarized perpendicular to the plane of incidence. Under tliese circumstances, the collimated beam in passing through this half metal and half air plate was split into two halves, namely, the one that underwent metallic reflexion and the other that did not, and the fringes due to these two were obtained consecutively on the same plate, but with a natural * Drude’s ‘ Theory of Optics,’ p. 366. 474 Mr. H.P.Waran: Interferometer Method of determining displacement, vertically between the two sets that represented the phase difference. The fringes due to the latter acted as the reference marks from which to measure the distance to the bands corresponding to the former, and a slight lateral shift given to the latter system cf bands by rotating the wave-length drum a little served to identify the two in practice, as shown in fig. d (Pl. VIII.). For mereury, which was the next metal studied, the arrangement had to be slightly modified, and the specially made ebonite trough described above and shown in fig. 1 and Pl. VIII. fig. e, had to be used. A modification had to be intro- duced in it in the form of a narrow partition to divide the mercury chamber into two longitudinal halves, so that, as with the silver, we might get half the plate covered with mercury, leaving the other half free as shown in part section fio. 1. We may proceed now to consider the theory of the method and the process employed for the actual evaluation of the phase difference from a measurement of the displace- ment of the metal fringes from the normal system. Let ¢t=the thickness of the parallel plate and w=the refractive index for the wave-length 2X used, which may be calculated out from the given optical constants of the plate, using Cauchy’s formule. Then the path difference between two adjacent beams emerging out of the plate because of the multiple reflexions is given by the relation Zut cosr=nr, lt) rl) where r is the angle of incidence for these rays within the late. , If we count this as the path difference for the starting band of the Lummer system of fringes, we have a bright band at every position corresponding to an increase in the path difference by X. Thus for the pth band we have | Qut cos (r—Or,)=(n+p)rA. 2 . 2 . Cy Subtracting (1) from (2), we have, since 67» is small, Qué sin 7 OP = pr. a re Further, we have the relations sini=p sin 7, and SIN 2)= SIN Tp, | sin 2—$1n tp= yu (Sin 7—Sin 7) PAP eae =p.2.c0s 2yin? i D) 2 Phase Difference resulting from Metallic Reflexion. 479 But : : " a 7 ede i MOT; 7: (ieee : ast ‘ @ phe ics Pere ee epee 1B a. rapprox. and = sin 5 yal: Therefore Sim Sine COSsT OF) fee. A) fr Bat rom (3) | ot i ae a 2é sin r substituting in (4) for 67, we get | Lal's ea Cos 7pr sin?—sin ip = ECON ae 2ut sin r ae pr B26 tani Expanding 7-2 pr = 2 cos ’ sin P— ft" — Ky Deeded dy Detainee nN where gee ie Sean Rn Let t—ip=0,, sothat 1,=1—0, and Bebit pete Ls ey and oti a Opi. Op Mog te ea We cet Sent p Kp - See Ea CO aah Wa aes) 2 O» Now @, can be found by observation of the distance of the pth band from the zero band and dividing it by the focal length of the camera lens. Substitating in equation (9) gives us the value of 2, since & is readily evaluated from the known constants. By taking every value of p from l upwards, the corresponding value of 7 is deduced and the mean value of 2 found out. Values of 7 for the successive bands can then be found out by calculation from the relation Sing, = sim, Kh ps. 2 YB) After this preliminary consideration of the elementary theory applicable to the present case, we proceed as follows to apply it to the case of metallic reflexion. When simple multiple reflexion takes place in a parallel plate of glass bounded by air, we have the simple relation 2pt cosr = nn. 476 Interferometer Method of determining Phase Difference. But, for the case of light reflected from one of the surfaces covered by metal and polarized in the plane of incidence, this becomes : nie, 2ut cos(r+6r) + 5 el (7) when we account for the displaced band as being due to a change of phase on metallic reflexion. When the band shifts to its new position as a result of the phase difference, the change in the angle of incidence corresponds to the term 6r and the phase difference of @ is taken into consideration by id the term 5—A. 21 Expanding (7), Qut cos 7 cos or— 2yt sin r sin or + — AN=NA, 0 i.e. 2putcosr—2pt sin’ or + gp MSNA, 0 1. @. nr — 2pt sin r or - an == i); Oh 2 pt om ror — Sah and therefore | 27 ne F 0 = ~~ 4xé sini fhe Ack Reape 3) But sin?=psin ’. Differentiating cost bi=pcos 7 or. cos 102 Therefore or = pcos? Substituting for 67 in (38) nr Cosi 02 TOT : é= x (2pt sin aaa = an t tan 7 cos 7 62 ay r = K' costo 2 er) where K'= i tan 7. In practice, with the constant K’ evaluated from the known constants of the plate, to evaluate @ the phase difference, the normal (glass-air) system of bands is measured out, and getting by subtraction the distance of each from 0 and Stopping Power and Atomic Number. ATT dividing by the focal length ot the camera, the values of 6,, and fe aretrom the mean 2, 2,, and cos ty far each value of p are evaluated. In each case 62, is given by the measured displacement on the plate of the metallic fringe from the normal system divided by the focal length of “the camera lens. The mean of the values of cos7, Se for each of the values of p is the value for cosi5i employed in the final substitution in (9) to give the value of the phase difference. In practice, the method works out quite satisfactorily, and the preliminary study in the case of silver and mercury has yielded quite concordant results. Because of freedom from surface impurities in the present method, and considering that the effect of surface impurities is to lower the values, the slightly higher values obtained by this method are easily explained. This view has also in corroboration the fact of the observed difference in the case of mercury being much smaller than with silver, which is more liable to surface contamination than mercury. Further, the method has also the advantage of being applicable to all the other metals which can be coated on to the Interferometer plate by cathodic deposition or otherwise. A detailed evaluation of this phase difference for all the metals is in progress. In conclusion, I beg to express my indebtedness to Prof. R. Ll. Jones, M. im Professor of Physics, Presidency College, Madras, for his kind and sympathetic help during the progress. this investigation at Madras under his direction. University College, London. LIT. Stopping Power and Atomic Number. By J. L. Guasson, M_A., D.Sc.* 1. BT has recently been established that certain properties. of the atom are more simply and accurately expressed in terms of atomic number than in terms of atomic weight. The following is an attempt to find out if this is also true of the atomic property known as stopping power for @ rays. According to the work of Sir William Bragg, it was shown that the stopping power of an atom is proportional to the square root of the weight of the atom. Professor Bragg’s. own figures which form the basis of the work are given in. * Communicated by Prof. Sir E. Rutherford, F.R.S. "oo 478 Dr. J. L. Glasson on columns III. and IV. of Table I. The value for argon is that obtained by Adams (Phys. Rev. vol. xxiv. p. 108, 1907). The constancy of the figures in column IV. is a measure of the range and of the success of the rule. TABLE I. (i Hs UVP Vi) Nad Stopping | g *| Power. | MV w. DA |, 240 Ae | 214 Sars 246 ‘04-| 259 1:00 264 1705. 4 262 1:49 288 1:76 312 leytor un 299 1°83 290 229 ale Oy, PAs eS 246 |. 809 2°60 | 291 3-28 | 315 3:56 | 326 3-44 |} 307 Re 414 | 997 LO We) 3x0) 497 | 298 2. In fig. 1 the logarithms of the atomic stopping power are plotted against the logarithms of the atomic number. The observed values lie nearly on a straight line whose slope is almost exactly two-thirds. In fig. 1 the line is drawn so as to pass through the air or standard point and have the slope two-thirds. It wiil be seen that it fits the observations very well. The relation between o and N is of the form o=kN?*, where k =-262 and is the atomic stopping power of hvdrozen. We may say therefore that “ The stopping power of an atom is proportional to its atomic number raised to the power two-thirds.” 3. It is perhaps desirable to inquire how well this rule fits the observed values and to compare its success with Stopping Power Stopping Power and Atomic Number. 479 Brage’s rule. ‘The limitations in one respect are the same for both. ‘They do not apply when large velocity variations are permitted in the « raysemployed. Nevertheless Bragg’s tule is extremely useful over a considerable range of speeds, and there is no reason to believe that the new rule should be less applicable over the same range. The degree of numerical agreement is shown by the figures in columns V., VI., and VII. of the Table. The constancy of the numbers in column VII. is the measure of the success of the rule. The average deviation from the mean is Fig. 1, Atomic Number. ) per cent. for column VII. as against 83 per cent. for column IV. The maximum deviation is 11 per cent. as against 25 per cent. for the square root rule. The case of the helium-hydrogen ratio is interesting. According to the square root law, remembering that hydrogen is di- _ atomic, the stopping powers of the two gases should be equal. All the observations prior to 1914 had made the ranges of « particles nearly the same for the two gases, that in helium being slightly less than that in hydroven. In 1914 Taylor found (Phil. Mag. vol. xxvi. p. 402, 1914) that if special precautions were taken to purify the helium, then: the range was actually 5 per cent. greater in helium 480 Stopping Power and Atomic Number. than in hydrogen. This is a remarkable result when we consider that the density of helium is twice that of hydrogen. Nevertheless it is successfully predicted by the atomic number rule, but not by the atomic weight rule. 4, There is a recent paper by von Traubenberg (Zeztschrift fiir Physik, vol. ii. p. 268, 1920) dealing with the range of a particles, in the first part of which ‘he describes a new method of determining the range of »% particles in solids, which is apparently simple and accurate. But the values of the stopping power deduced from these ranges depend on _ the value given to the range of the @ particle in air, and no uniform value has been employed by different experimenters. Until therefore this new method has established itself in the confidence of experimenters in this field and, more important perhaps, until the meaning of the word range is more clearly defined by them, I have thought it best not to endeavour to incorporate von “Traubenberg’ s results into Brage’s table. But a study of them leads to ‘three conclusions :-— (1) The two-thirds power rule fits von Traubenberg’s results as accurately as it does Bragg’s. , (2) The stopping powers of platinum, gold, and lead are distinetly greater than those deter: ened by Brage. Thus the slight deviations which these three elements show in fig. 1 are reduced to about half their present amount. (3) The new elements lithium, magnesium, and calcium are made available. I have inserted these in fig. 1 and they fit well. — In the second part of his paper and in another paper (Phys. Zeit. p. 588, 1920) von Traubenberg examines the agreement of his results with Brage’s rule and, finding the well-known discrepancies in fhe region of low atomic weights, tests certain new rules which give, he claims, better agreement than Bragg’s. The first two of ieee express the stopping power as a complicated function of both atomic weight and atomic number. One is that o is proportional tv “ANS, The other is that o is proportional to At Ns. These fit slightly better than Brage’s in the revion of hich atomic ‘weights, but distinctly worse in the region ee low acho weights as von Traubenberg himself shows. Neither of ahean predicts the hydrogen- helium ratio to which attention has been drawn. A third rule, that the stopping power is proportional to the square root of the atomic number, fails to express the sesillis either for high or for low aie? weights. Dielectric Constants of Esters at Low Temperatures. 481 It is important to notice that for the test of a power law the first nine elements are as important as the other seventy- three taken together. In the upper part of the table the rate of variation of stopping power and of atomic number is not nearly so rapid, and many rules could be devised which would fit the results over a limited range of the periodic table. The logarithmic plotting which is used in the figure brings out this fact, whereas the method of direct plotting which von Traubenberg uses disguises it. It seems evident, therefore, that while von Traubenberg’s experimental results are a valuable addition to our know- ledge, his rules fail either in simplicity or in generality or in both. 5. In view of the complexity of the phenomena asso- ciated with the passage of a rays through matter, it is rather surprising that a simple relation can be found between stopping power and atomic number. The index number suggests that the stopping power of an atom is related to the number of electrons in the atom in much the same way as the cross section of a sphere is related to its volume. A theory of the passage of @ rays through matter has been worked out by Bohr (Phil. Mag. vol. xxx. p- 581. 1915), and it has been shown by him to express the results for light atoms. The data necessary for applying it to heavy atoms are unfortunately not available. The two- thirds power rule is therefore propounded as a simple working rule until such time as a general theory of the phenomena is available. Cavendish Laboratory, Cambridge, October 11, 1921. LIV. The Dielectric Constants of some of the Esters at Low Temperatures. By li. C. Jackson, M.Se.* HE dielectric constants of the members of homologous series of organic compounds, when measured at ordinary temperatures, are generally found to change from member to member in a manner that can be correlated with the structure of the compounds. This is exemplified in the case of the two series uf esters (formates and acetates) for which the values of the dielectric constants as determined by Drude by his well-known method, using high-frequency oscillations, are given below. It will be seen that the values * Communicated by the Author. + Zeit. Phys. Chem, xxiii. p. 267 (1897). Phil. Mag. Ser. 6. Vol. 43. No. 255. March 1922. 21 482 Mr. L. C. Jackson on the Melectric Constants change fairly regularly from member to member of each series, and that the values for the corresponding members of the two series are distinctly different. Methyl formate 8°87 (19°) | Methyl] acetate 7:03 (20°) Ethyl formate 8°27 (19°) Ethyl acetate 5°85 (20°) n-Propyl formate 7°72 (19°) n-Propyl acetate 5°65 (19°) n-Buty! formate — n-Butyl acetate 5:00 (19°) The figures in brackets are the teniperatures at which the determinations were made. : The object of the present work was to determine the values of the dielectric constants of the above series of esters at the low temperatures obtainable with liquid air so as to find the actual values of the dielectric constants and compare them with those obtaining at ordinary room temperatures, and to find whether the relations hoiding at ordinary tem- peratures between the various members of the series and between corresponding members of the series of formates and acetates hold also at the lower temperatures. The results obtained, for which great accuracy is not claimed and which are intended as preliminary values, show that apparently the acid radicle contributes but little, as compared with the rest of the molecule, to the value of the dielectric constant at the temperature of liquid air, since the constants for the corresponding formates and acetates are found to be approximately equal. ‘The values for the various members of either series do not apparently exhibit the same relationships to each other as hold at ordinary temperatures. Method of Haperiment. The principle underlying the method used in the present work may be stated briefly as follows :—An electric cireuit containing a capacity C and an inductance L will oscillate with a frequency determined by the relation T=297,/LG, where T is the time of oscillation. If, now, a second condenser is connected in parallel with the first one, and the latter is so reduced that the frequency of oscillation of the circuit is the same as in the absence of the second condenser, first when the latter has air (or, more correctly, a vacuum) as its dielectric, and secondly some other substance, the ratio of the amounts by which the first condenser had to be reduced in the two cases will give the of some of the Listers at Low Temperatures. 483 dielectric constant of the substance under test for the parti- cular conditions of temperature and frequency of oscillation obtaining at the time of the experiment. The actual arrangement by which the above-described method was applied to the investigation of the dielectric constants of the esters is shown diagramatically in fig. 1. It consists of two simple ‘ valve” circuits placed side by side, so that, as in the heterodyne method of reception, a beat note is heard in the telephone placed in one cf the circuits. In parallel with the variable condenser ©, is placed the experimental condenser Cy. the change of the capacity of which with change of dielectric is to be determined. ‘The cireult 2 remains untouched throughout the experiment, and serves aS a means whereby the frequency of oscillation of circuit 1 can be brought to some standard value whenever Buea: = Nei required. The method actually adopted for this purpose was to adjust C, so that the beat note in the telephone always just fell to zero. The accuracy with which the scale of the variable condenser could be read determined the accuracy with which it was necessary to adjust the beat note to the standard value, it being found that the method of ad- ‘justine to zero gave all the accuracy of setting required, so that the more elaborate and accurate method of comparing the beat note with a standard fork was not used. The experimental condenser was a brass cylindrical one, consisting of four coaxial tubes connected together alternately and fastened at one end into a fibre separator. The diameter and length of the outer tube were 2°06 cms. and 12°36 cms. respectively, this size being such that the condenser would just slip into an ordinary boiling-tube. The whole of the two circuits, with the exception of C, and the telephones T, was enclosed in an earthed tin-plate re ———-_ A484 Mr. L. C. Jackson on the Dielectric Constants case to prevent any change in the frequency of oscillation of the two circuits being caused by external disturbances or the movements of the observer. This precaution was - quite necessary for accurate work on account of the well-known delicacy of the arrangement. It was also necessary to keep the whole apparatus dry throughout. _ The procedure of the experiments was as follows :—The - variable condenser ©, was adjusted until the beat note fell to zero when the experimental condenser was removed from the outer end of the leads. The condenser ©, with air as dielectric was then attached and the process repeated. ‘The air was then replaced by the substance to be tested, and the whole, which was contained in an ordinary boiling- ane was placed in a bath of liquid air. The test substance solidified (the substances used in the work being all liquid at room temperatures), and was allowed to remain in the liquid air until a constant state was obtained, and the condenser Cy adjusted as before. The shifts of the condenser C, gave first the capacity of C, with air as the dielectric, and secondly with the test substance, the ratio of the two. shifts (after being suitably corrected) being the value of the dielectric constant of the test substance at the temperature of the liquid-air bath and for the particular frequency of the oscillations used. The temperatures were measured throughout by means of a platinum resistance thermometer, and the frequency of the oscillations with the aid of a wave-meter. Care was taken to make the measurements of the capacity of Cy with air and with the test substance as dielectric under as nearly as possible identical conditions (except as to tempe- rature—see below), and the effect of the leads was eliminated by keeping them permanently attached to ©, so that the difference of any two readings on CU, did not contain any contribution from them. The scale of the variable condenser having been previously calibrated, a correction may be applied tothe observed values for the known unevenness of the scale. The fact that the capacity of C, with air as dielectric was measured at ordinary temperatures and with the test substance at the temperature of the liquid-air bath, necessitated a correction for the temperature change of the dimensions of the condenser. Quite recently » number of methods for the application of “valve” circuits to the measurement of capacities have been described by several investigators : arrangements similar to the above have been proposed by J. Scott Taggart (‘ Hlec- trician,’ Ixxxil. pp. 466-467, Apr. 18, 1919} and J. Herwee of some of the Esters at Low Temperatures. 485 (Deutsch. Phys. Ges. Ver. xxi. pp. 572-577, Sept. 30, 1919), and more complicated methods involving the use of more than two valve circuits have been described by Pungs and Preuner (Phys. Zeit. xx. pp. 543-545, Dec. 1, 1919) and Falekeuberg (Ann. der Phys. |xi. 2, pp. 167-172, Jan. 15, 1920). The latter worker has applied his method to the determination. of the dependence on pressure of the dielectric constants of water, ethyl alcohol, methyl alcohol, and acetone. Purification of the Materials. The substances used in the investigation were methyl, ethyl], n-propyl, n-butyl formates and the corresponding acetates. The esters were obtained from the firm of Poulenc Fréres of Paris, and before use were fractionated, washed, very carefully dried, and re-distilled. It was found, however, that after the most elaborate precautions had been taken to eliminate water, acids, cr other possible impurities, the methyl, ethyl, and, to a less extent, n-propyl esters still possessed a conductivity, while liquid at ordinary temper- atures, which, though very small, was sufficient to so reduce the intensity of the sound in the telephones as to make the method inapplicable, quite apart from any change in the value of the effective dielectric constant due to the finite conductivity of the dielectric. (Had it been possible to obtain readings of C,, the true dielectric constant could have been calculated in terms of the known resistance of the dielectric and the constants of the circuit: cf. R. T. Lattey, Phil. Mag., June 1921. Possibly some arrangement by which the sound in the telephones could be considerably amplified would enable this to be done.) In the case of the butyl esters it was found that a much less rigorous method of purification than had failed with the other esters would suffice to reduce the conductivity, even when the substances were liquid to a value small enough to make the method applicable. The facts that the conductivity of the esters remained after very rigorous purification, and that this residual small con- ductivity decreased from member to member as one proceeded up the series, would seem to show that this conductivity was a property of the esters themselves. The conductivities of the esters when solid at liquid-air temperatures were so small as to make the present method quite applicable. In general, the method provides a very simple and direct means of measuring the dielectric constants of substances of sufficiently small natural conductivity. 486 Mr. L. C. Jackson on ihe Dielectric Constants. Heperimental Results. The results obtained for the dielectric constants of esters at the temperature of liquid air are exhibited in tabular form below. The values have been corrected for the in- equalities of the condenser scale and for the temperature change of the dimensions of the experimental condenser. TABLE I. Dielectric Constant Dielectric Constant Substance. Temperature [T. at temperature T. at ordinary Obs. Mean. Obs. Mean. temperature. 76°°38 A 2°54 Methyl 78°°0 A ayes BOE ale fy ate 399-8 A 78°69 A a5 2°56 8:87 OAM 2615 83°°2 A 2°38 Ethyl 82°70 A A 2Z-o1 5: ae pete A goon Sea 9-48 2°40 8:27 Ce Die 81°70 A 2°36 n-Propyl TIOO A ve 2°40 pb formate. 990.6 A EO a 9-44 ae Be: (Moos I 2°36 820-0 A 2-45 Dee «70 Ae eeaa alan Wore ONEa a Tie OeN 2°42 Weak 2°63 NGonG) A 2°63 ee |) 79°:0A Wook 260 258 7-03 T7°0 A Wao 78°0 A 2°53 Ethyl 81°°:0 A Ey 2°45 Beataeal W003 A TAS alsy IS. 9-515 2°48 5°85 82°°8 A 2°41 i 82°83 A 2°49 n-Fropy! 802-9 A 9°42 565 acetate. TTeOA 9-44 81°0 A 2°34 Pec OR 2°38 ae T7°-6 A i es - 5-00 2°44 acetate. 732-4 A of some of the Esters at Low Temperatures. 487 The frequency: of the oscillations used was throughout of the order of 4°7 x 10° per sec. Tt will be seen from the above tables that the values of the dielectric constants of these esters when measured at the temperature of liquid air are throughout considerably less than the values obtained at ordinary temperatures. ‘These low values are characteristic of the solid state, for the general course of the change of the dielectric constant with decrease of temperature is first an almost linear rise of the value of the constant as the temperature falls until the melting- point is reached, then a sudden fall during solidification to a value less than that obtained at ordinary temperatures, then a very slow rise as the temperature is further decreased. This phenomenon is illustrated by the case of n-butyl acetate, for which the results obtained for the dielectric constant throughout the range of temperature 292° A—80° A are given below. That this is probably a general phenomenon is shown by the work of several investigators, as, for example, that of Abegg and Seitz on p-azoxyanisole (Zeitsch. Phys. Chem. xxix. p. 491, 1899) and various alcohols (ad. xxix. p- 242, 1899). This almost linear rise in the value of the _ dielectric constant with decrease in temperature is in agree- ment with the theory of the temperature variation of this constant put forward by Debye (Phys. Zeit. xii. pp. 97- 100, Feb. 1, 1912) on the basis of the assumption of the presence of “dipoles” in the liquid dielectric. Returning to the tables, it will be seen that, though the values of the dielectric constants at ordinary temperatures of the corresponding members of the series of formates and acetates are distinctly different, they become approximately equal at low temperatures. This would seem to show that at these low temperatures the acid radicle contributes very little to the value of the dielectric constant. An inspection of the table will also show that the ratio of the dielectric constant of any member of either series to that of the next higher member does not remain throughout the series the same at low temperatures as it is at ordinary temperatures. In Table II. are given the results obtained for the case of n-butyl acetate for the range of temperature 292° A-80° A. The temperatures were obtained by surrounding the vessel holding the experimental condenser by air (292° A), ice (273° A), freezing mixture (264°5 A-258°7 A), solid CO, and ether (200°5 A), and liquid air (77°6A) Readings were taken at each of these temperatures, the results in the table being mean values. Readings were also obtained when a eer brelectrve Constant, , mama ewem wm we | eee Melting Pont, 458 Dielectric Constants of Esters at Low Temperatures. the n-butyl acetate was Just beginning to solidfy and when it was solid just below the melting-point. The melting-point was afterwards determined, the result obtained” being —77°9C., TaB.e IT. n-Butyl acetate. Yemperature. Dielectric Constant. PRION 5:05 280°°5 A 5°10 273°°5 A 5:25 264°°5 A GRO) } 258° 7 A 5625 200°°5 A 6°85 195°°1 A just melting. 6°965 195°-1 A just solid. 2°395 T7°6A 2°41 Fig. 2. 100 200 300 Absolute Temperatures. The results are plotted graphically in fig. 2. It will be seen that the dielectric constant arises from 5°95 at 292° A to 6°965 at 195°°1 A, the melting-point, in a manner such that the relatien of the dielectric constant to the temperature is almost exactly a linear one. Then there is a sudden fall in the value during solidification to 2°395, and finally a very A Study of Franklin’s Experiment on Leyden Jar. 489 small rise to 2°41 at 80° A. The present arrangement did not permit of an accurate determination as to whether there was an actual discontinuity in the value of the dielectric constant at the melting-point, or whether the change took place ina small range of temperature. The former alternative is probable, and the change is so represented in the graph (fig. 2). Summary. A method is described for the determination of the di- electric constants of liquid or solid substances by the aid of triode valve circuits. The results obtained by this method for the dielectric constants of methyl], ethyl, n-propyl, and n-butyl formates, and the corresponding acetates at the temperature of liquid air, are then given and compared with the values obtained by other investigators at ordinary tem- peratures. It is shown that at the low temperatures the acid radicle apparently contributes but little to the value of the dielectric constant. Results showing the dependence on temperature of the dielectric constant of n-butyl acetate throughout the range of temperature 292° A-80° A are also given. It is found that the dielectric constant of this sub- stance increases linearly with decrease of temperature down to the melting-point, at which a sudden fall in the value occurs, followed by a very gradual rise as the temperature is further lowered. The author desires to express his indebtedness to Prof. H. H. Barton, F.R.S., for the facilities afforded to him in his laboratory, to Mr.C. F. Ward, B.Sc., for his kind assistance in the rigorous purification of the materials used, and to the Department of Scientific and Industrial Research for a grant by the aid of which the above work was carried out. Physical Laboratory, The University, Leiden (Holland). LV. A Study of Franklin’s Experiment on the Leyden Jar with Movable Coatings. By G. lL. ADDENBROOKE, M.L.E.E.* OQ understand the actions in dielectrics it is very desirable to have a clear idea of the principles which underlie electrostatic actions. For the most part these can be found in text-books, but there is one respect in which they are all defective, and that in connexion with one very * Communicated by the Author. 490 Mr. G. L. Addenbrooke: A Study of Franklin’s important action which bears largely on a class of problems which has much practical importance. This is known as the “ Franklin Experiment” on the Leyden Jar with movable coatings. The experiment in its simplest form is briefly thus :—The Jar is charged, the inner coating is lifted out by an insulated hook, and this inner coating is then touched against the outer coating and put back. It is found that when taken out the inner coating carries scarcely any charge, and that when it is put back a full or nearly full discharge can be got from the reconstituted jar. The experiment is sometimes varied by placing the charged jar on an insulating stand and taking away by insulated tongs both the covers, leaving the glass jar exposed. Both covers are found to be pre actically unchar ged, they are then touched against each other. The whole jar and coatings are next reconstituted, when a full discharge can be obtained. This has been taken to mean that when a dielectric between two electrodes is charged, if the electrodes are taken away while still charged, the energy of the charge is left stored in the dielectric, and remains there antl the electrodes are restored and short-circuited, or the charge dies down by some indefinite form of leakage. For a long time I could not reconcile this experiment and its general interpretation with the views I had been led to form of electrical actions in dielectrics in other respects. I therefore searched through a number of works for the pur- pose of finding any variations there might be in the methods of performing the experiment, or in the explanation of it, when I noticed that there seemed to be no mention of it in araday’ s or Maxwell’s work. It is not mentioned in Gordon’s ‘ Electricity and } Magnetism,’ nor in Sin; Wid. Thomson’s Text-book, though itis deseribed in Poynting and Thomson’s ‘ Electricity and Magnetism’ of 1914. This was significant, though it seemed almost equally significant that if the experiment: was omitted for any reason, no reasons should be given for so doing. At last, to clear up the matter there seemed no other course but to investigate it further oneself. In considering how to do this the first point which struck me was that a glass jar was invariably mentioned as the dielectric. The question arose what would happen if another dielectric were substituted. Finally, solid paraffin was selected as a suitable substitute on account of its high insulating properties, and because its Experiment on the Leyden Jar with Movable Coatings. 491 surface is not so hygroscopic as glass. After some trouble I succeeded in casting a thin but perfect jar of parafiin fitting the same metal coatings asa glass jar I had. | The usual experiment was tried with this, but instead of charging the jar from an electrical machine, the jar was connected to a gold-leaf electroscope,and charged trom an electrophorus. By this method one does not need such high potentials, there is less loss from leakage, and it 1s possible to see what is going on and make appr oximate quantitative measurements. Tn this way the jar was charged to 500-600 volts, and the deflexion on the electroscope noted. The inner coating of the jar was then lifted out by an insulating handle, ee ched to the outer coating and replaced. It was then re- connected to the electroscope. Contrary to the case with the glass Jar there was no appreciable charge remaining in the jar ; in ee repeated experiment showed that on lifting out the inner coating and earthing it, or connecting it to the outer coating, the dielectric was completely discharged in the act, and only the barest trace of charge remained when the inner coating was put back. Also the charge evidently came away on the inner metallic coating when it was lifted. out, as could be plainly seen on the electroscope. In fact, the action was exactly the reverse to the account. of it usually given. On the other hand, the actions as now found seemed conipletely fulaeeord) with the Faraday-Maxwell theory, and with ae n electrostatic laws. Briefly they showed, that the action on the dielectric is inductive, that if the inner charged electrode were brought outside, the lines of force going to the outer electrode no longer went through the dielectric, which ac- cordingly lapsed to its normal and neutral state. If when the jar with paraffin dielectric is charged it is placed on an insulated stand and both coatings are taken off by insulated handles, both coatings will be found to have strong and equal charges of opposite signs. If these two coatings are then touched together, and the jar is reconsti- tuted, it will be found to have no charge as before. Having thus clearly shown that there is a difference in the results of the experiment when glass is used and a better dielectric such as paraffin, I determined to vary the glass experiment as follows :—The glass jar was first thoroughly warmed, It was then put ina “large dry cupboard, and Kept dry for a couple of days by means of plenty of calcium chloride. This dry cupboard is so arranged that, by means of oil | silk sleeves passing through holesin one of the sides, and wearing 492 = Study of Franklin's Haperiment on Leyden Jar. rubber gloves, one can manipulate inside without introducing moisture The Franklin experiment with. glass dielectric was then repeated with the dried glass in a thoroughly dried atmo- sphere. The effects observed were now. no longer the same as betore, but were the same as described with the paratiin Jar. That is, after charging and taking out the inner coating and touching it to the outer coating and replacing, there was only a slight charge remaining, due doubtless to absorption. For all practical purposes the characteristic actions as de- scribed in the text-books were wanting, and the results come in line with the Faraday- Maxwell theory. It is clear from other experiments I have made (see Physical Society's Proceedings, 1912) that, in the case of the glass jar, condensed moisture on its surface is under. all ordinary circumstances sufficient to form a semi-conducting film, of high resistance, but sufficiently conducting for the charge on the slednadies to escape to it when one is removed, or before, especially as when the experiment is tried alee conditions as above, the electroscope shows that if the electrodes are separated the same action takes place as when the cover is removed from an electrophorus, that is as the electrodes are separated their difference of potential rapidly increases. This is more the case as the dielectric constant of glass is so high that the slope of the potential is chiefly concentrated on al is very great across any small air space between the glass and the ‘electrodes. A small motion of the electrodes already charged to a fairly high potential, therefore, raises this potential at least two or three times so that Frets: is a strong tendency for the charge to flow to the glass surface, even if there is only momentary contact at two or three points. I have related my work on this experiment at some length because it became more and more clear to me that to arrive at any clear understanding of the actions of electric fields on dielectrics, the correct interpretation of the experiments Is essential. It has always appeared to me that this experiment was about the most striking and most fundamental in all the realm of electrostatics. It seemed so convincing that for many years I, and I know numbers of others, have accepted it and still do without question, as ordinarily ‘interpr eted. It is such a striking experiment that when I began to doubt its interpretation it appeared to me that to leave it. without rational explanation left a fundamental point in A New Model of Ferromagnetic Induction. 493 an undecided state, as there was no other experiment which gave an equally convincing demonstration of the real actions. In drawing attention to this matter I trust it may not lead to teachers abandoning the experiment, as it is now omitted in some cases. As reconstructed, besides demonstrating the main theoretical point, it brings into prominence the other actions, and coordinates these actions with the behaviour of the electrophorus and other electrostatic facts, while it also draws attention to the question of leakage and surface effects which are very indefinitely understood at present but which are being found increasingly important. Instead of being tried only with a glass jar I would suggest that the experiment be tried also with a high-class ebonite jar, which would be less fragile than parafiin, as I think it would succeed if the ebonite was kept in the dark in a closed glass jar with calcium chloride up to the moment of use. It is difficult to speak definitely of glass, its surface state varies so, but I think in most cases if the experiment were tried as usual, and then the glass jar were heated to 100° C. and kept over calcium chloride for 20 minutes till it was cold, if the experiment was then quickly made in the open air, it would be found to fail from the Franklin point of view, although it succeeded well in the first trial. LVI. A New Model of Ferromagnetic Induction. By Sir J. AtFreD Ewine, K.C.B., P.RS., Principal of the Uni- versity of Edinburgh *. | HAVE lately reconsidered, in the light of what*is now known about atomic structure, the theory of induced magnetism in iron and other ferromag netic substances which I put forward more than thirty years ago, and have come to see that it needs substantial amendment. A new model of the process of ferromagnetic induction has to take the place of the model then suggested. The new model is the subject of a recent communication to the Royal Society (Proc. Roy. Soc. Feb. 1, 1922), but a brief account may be offered to the Philosophical Magazine, in which the model of 1890 was described on its first introduction fF. The revised theory and the new model retain this funda- mental feature, that there is in every ferromagnetic atom a Weber element possessing magnetic moment and capable of. * Communicated by the Author. + Phil. Mag., 5th series, vol. xxx. p. 205 (Sept. 1890). A494 Sir J. A. Ewing on a New being turned into alignment by a sufficiently strong external field, and that the control of the Weber elements—which resists their turning and gives rise to the phenomena of hysteresis—is due entirely to magnetic forces. When they are caused to turn there is first a small stable deflexion, then a breaking away from the position of stability and a falling over, through an unstable phase, into a new position of stability. In this essential characteristic the new model resembles the old one. But in the old model the only magnetic forces that contributed to the control of any one Weber element were the forces between it and other Weber elements. In the new model a great part of the magnetic control is due to forces between the Weber element and other portions of the same atom. The Weber element—the thing that turns—is probably only a small part of the atom*. Before describing the new model it is necessary to point out in what respect the old one fails. The old model consisted entirely of pivoted magnets which represented the Weber elements in the atoms of a erystal, say of iron. Their mutual magnetic forces made them form rows. When a weak external field was applied, the magnets in any row that lay more or less transverse to it underwent a small stable deflexion, corresponding to the initial stage of magnet- ization which the late Lord Rayleigh showed to be quasi- elastic. When the field was sufficiently increased the rows broke up and other rows were formed in a more favourable orientation. ‘This corresponds to the “steep ” stage in the magnetizing curve. Now it is known that in ordinary iron harely one per cent. of the whole magnetism of saturation is acquired in the quasi-elastic stage before the effects of hysteresis set in. ‘To conform to this condition, the magnets of the model must have only a very narrow range of stable deflexion, and consequently they have to be set very near together, rich the result. that am thetel@nodeltn mutual control Tyeganne excessive. A calculation of the force required to break up rows of pivoted magnets, of atomic dimensions, when set near enough together to satisty the above condition, shoved it to be many thousands of times greater than the force which is actually required, in iron, to reach the steep part of the curve. The original model, therefore failed quantitatively, and a model had te be sought for which would exhibit similar qualitative features but would provide a far weaker comttel, while still allowing no more than a very narrow range of stable deflexion. This is arrived at as follows. * Cf, A. H. Compton and O. Rognley, Phys. Rev. xvi. p. 464 (1920). Model of Ferromagnetic Induction. 495 Let W, fig. 1, be a pivoted magnet controlled by two fixed magnets A and B, which point in opposite directions. Assume, in the first instance, that the attracting pole of A acts on W somewhat more strongly than the repelling pole of B, so that there is a feeble control due to this differ- ence. Assume further that the clearance between W and the adjacent poles of A and B is very small. Let an external field H be applied. Then as H is increased W will deflect stably through a small angle, however nearly balanced are the forces of control exerted by A and B. When the deflexion exceeds a certain smail angle, W becomes unstable. Fig. 1. 72 W aN If there is a second pair of fixed magnets placed as at C and D, W will tend to assume a stable relation to them, but a further increase of H will make it break away from their control also into a position nearly parallel to the field. To make such a model completely imitate the process of ferro- magnetic induction we have only to think of it as extended into three dimensions, letting W be pivoted with two degrees of rotational freedom, and surrounding it by say four pairs of fixed controlling magnets, situated on the four diagonal axes of a cube. The sketch, drawn for a diagonal plane, shows two such pairs. It is obvious that if a substance were made up of atoms on this model all the familiar character- istics of ferromagnetic induction would be reproduced in it when a field was applied, removed, reversed and so on, or when the piece was made to rotate in a constant field. Since the strength of the control may be reduced to any desired extent, a quantitative agreement with the results of experiment becomes possible. | pt nal eee ts agen yh ey eta ee ee em oe met pairs of elliptical coils and supposing W free to turn about any axis in its own plane. Here a single electron orbit suffices to constitute the Weber element. It is interesting to find that all the main characteristics of ferro-magnetism are capable of representation by means of models which embody the same ideas in two widely different forms, based respectively on the two conceptions of atomic structure which physicists and chemists are endeavouring to reconcile. 502 A New Model of Ferromagnetic Induction. Though in these models the turning part has, for the sake of simplicity, been represented by a single coil or a single magnet, it will be obvious that in developing the theory here outlined the Weber element of a ferromagnetic atom may be regarded as possibly comprising a complex group of electron orbits, capable of turning as a whole within the outer system, or even more than one such group. Reverting to the model of fig. 1, where there is an octet of fixed magnets, namely the four that are shown in the sketch and other four in a plane perpendicular to that of the paper, the Weber element might itself have eight magnetic poles, placed at the corners of a cube and capable of turning as a rigid system about the centre. Such a model is conveniently made by using, for the part that turns, a cubic steel boss with projecting magnets screwed into its eight corners in the direction of the diagonals of the cube, and magnetized so that four of the projecting poles have one polarity and the four opposite poles have “the other polarity. In fig. 6 the eight fixed magnets are held in a skeleton cube of brass rods, and the octet which forms the Weber element. is, for the sake of clearness, removed from its place on the pivot and shown separately. When the model is arranged in this form it is practicable to reproduce conditions of © extreme magnetic wolotropy. Thus we may advance one pair of opposite fixed magnets (in the outer shell) until Amplitude of Vibrations of Double Frequency. 508 they touch one pair of opposite magnetic poles in the turning part which represents the Weber element, with the result that one of its axes becomes fixed and rotation can consequently occur only about that axis. The magnetic properties of the model system then resemble those of pyrrhotite, a crystal of which, as Weiss * showed, will take up magnetic induction readily. in one plane, but not in the direction perpendicular to that plane. ‘The model may also be adjusted to exhibit differences of magnetic quality along different axes in the plane of magnetization. Such differences were in fact observed in pyrrhotite by Weiss. He found that when a crystal was turned about its non-magnetic axis in a fixed field there were abrupt magnetic changes at intervals of 60°. Thisis just what a study of the model would lead one to expect, for the projecting magnets in the turning element lie in planes 60° apart round any one of its magnets taken as axis. Periodic variations along axes inclined at 120° to one another in the plane of easy “magnetization are consistent with cubic symmetry on the part of the iron atom: they follow directly from the assumed grouping of magnet poles at the corners of a cube. And they will occur at intervals of 60° if we ascribe the hexagonal structure of the crystal to tw inning in successive layers. If Hull’s view of the structure of the iron atom be correct, it seems not improbable that the Weber element includes not only the duplet of electrons which he places near the nucleus, but also the innermost octet, the members of which are somewhat further away from the nucleus, leaving the other and more distant octets to constitute what I have called the fixed elements. LVII. On the Amplitude of Vibrations ted Ge For ces of Double Frequency. By N. ©. Krisunatyar, I.A., Lecturer in Physics, University College, Rangoon t. ORD RAYLEIGH ¢ was the first to discuss the theory of the vibration maintained by an influence whose frequency was double that of the vibration maintained. A well-known example of such maintenance is the longi- tudinal form of Melde’s experiment. His differential equation took the form y + kot (v? — 2a sin 2pt)y =0, h Ae a being small quantities. * Weiss, Jou. de Phys. iv. p. 469 (1905). ai Communicated by the Author. t ‘ Theory of Sound,’ vol. i. pp. 81-85. 504 Mr. N.C. Krishnaiyar on Amplitude of Vibrations Assuming | y=A,sin pt-+ B, cos pt + Az sin 3pt + B; cos 3pt + As sin dpt+ B; cos 5pt+ ...- | and equating to zero the coefficients of sinpt and cos pt in the relation obtained when this value of y was substituted in the differential equation, he showed that A; and B; were of the order « and therefore negligible reiatively to A, and B,. Hence as a first approximation, Ay(n? —p)— (kp+ a)B,=0, By(n? —p*) + (kp = a) Ay = QO. Therefore | and = (v? —p?)? = a — I? p?. According to this theory, the phase tan71B,/A,; is con- stant and independent of the amplitude maintained, and the amplitude V A,?+B,? is indeterminate. But it is known * experimentally that the phase of the vibration is not inde- pendent of the amplitude, and a study of the maintained amplitudes brings out a number of results which the above — theory does not indicate. : In the case of fork maintenance the vibration of the wire reacts on the vibration of the fork and alters the amplitude of the latter, and so a difficulty arises in interpreting the results observed. The author f found that when an alter- nating current of 50 cycles, and therefore of 100 heat cycles, was passed through a wire whose length and tension were adjusted to a frequency of 50, the wire on account of the periodic thermal expansion vibrated with a large amplitude. The effect of the earth’s magnetic field was found to be negligible, the experiment succeeding equally well in all positions of the wire including the position when the wire was along the direction of the earth’s magnetic force. The periodic thermal expansion due to the alternating current heating being independent of the vibration of the wire, there is no reaction between the vibrating system and the * C. V. Raman, Physical Review, December 1912, pp. 451-453; N. C. Kvishnaiyar, Physical Review, December 1919, pp. 494-496: R. N.Ghosh, Proc. Indian Assoc. Cultivation of Science, vol. vi, pp. 75 & 84(1920). t+ N. C. Krishnaiyar, ‘On the maintenance of vibrations of wires by electric heating,” Physical Review, n.s. vol. vi. no. 6. AINPUML AE SS maintained by Forces of Double Frequency. D095 maintaining influence. There is no mechanical force acting at a node or any other point of the wire. Hence a strict comparison of the observed amplitudes with theory is possible It is the aim of this paper to develop the theory so as to explain the facts of observation regarding the amplitude maintained. The details of the experiment are given in the paper cited above, and it will suffice to summarize below the main results. «is the constant factor of the periodic maintaining influence and in the experiment, for the same wire, depends upon the current through the wire and can be kept constant for an indefinite length of time or can he altered at will. mis the natural frequency of tke vibrating wire, and can be altered by altering the tension or the length of the wire, It was found that the maintenance started when the tension was slightly greater than that necessary to make n=p, i.e. when p?—n’? was negative. Maintenance ‘continued as tension diminished _so as to a ae and n less than p, i.e.as p’—n? became zero and as p*—n? increased positively. pt as (1) When the observed amplitudes are plotted * against the differences of the squares of the forced and the natural frequencies, there is no ‘“‘ peak” or maximum resonance as in the case of the ordinary forced oscillations, but only a continual increase in the amplitude from one end of the range ynere P *—n? is negative to the other end of the range where p? —n? is positive. (2) The amplitude increases to a maximum value after * Figure | of the author’s paper cited above. 906 Mr. N.C. Krishnaiyar on Amplitude of Vibrations which any diminution of tension brings about —— collapse of maintenance. (3) The amplitude graph though parabolic in parts is ei continually so; the are at one end produced parabolieally does not coincide with the arc at the other end but is nearly parallel. (4) The maximum amplitude maintained increases wpe 22 increases in value. (5) There is a lower limit to the value of « for maintenance to happen. (6) There is an upper limit to the value of the excess tension, 7.e. an upper limit to the magnitude of p?—7? when n>p if mainéenance should happen. Beyond this, there is no vibration of the wire at all. The projection of the wire by a magic-lantern on a distant screen remains a sharp image. | (7) This upper limit to excess tension increases with increase of «. In other words, when @ is larger, main- tenance starts with a larger excess tension. Most of the observations detailed above have been con- firmed by R. N. Ghosh * by an entirely different method of maintenance, viz. by means of an electric motor vibrator originally designed by J. A. Fleming and subseqte ey, improved by C. “V. Raman fo In the theory developed below, a term fy? is introduced in the part of the equation representing tension to denote the variation proved { experimentally to occur in free oscil- lations of sensible amplitude. This variation is due to the second order differences in length between the equilibrium position and the displaced position of the wire and is propor- tional to the square of the displacement. The frictional force is put down as the sum of two terms, one proportional to the velocity and the other proportional to the square of the velocity. Ordinarily, as Sir G. G. Stokes has shown, the frictional force is proportional to the velocity when the velo- city is small, and proportional to the square of the velocity when the velocity is relatively large. This apparent change of the law of friction with increase of velocity can be ex- plained by the assumption that the frictional force always consists of two terms, the first term proportional to the velocity and the second term proportional to the square of the velocity, the coefficient of velocity being relatively larger * Proc. Indian Assoc. 1920, vol. vi. + Physical Review, Nov ennlbes 1919. + C. V. Raman, “ Photographs of vibration curves,” Phil. Ae. May 1911: Physical Review, December 1912. maintained by Forces of Double frequency. D07 than the coefficient of the square of the velocity. When the velocity is small the first term will be the dominating term, and as the velocity increases, the second term will increase more rapidly than the first and will become ulti- mately the dominating quantity. Froude * has shown that the forced oscillations a a ship rolling among waves are due to a periodic forcing cause damped by resistance varying with both the first and the second powers of the angular velocity. Routh f has shown analytically that the period of a pendulum in a very rare medium resisting partly as the velocity and partly as the square of the velocity, is constant throughout the motion and independent of the are. Parker Van Zandt + has shown experimentally that the free vibra- tions of a system with a resisting torque due to both first and second power damping are “isochronous. This law of frictional force when introduced into the differential equa- tion brings out all the above facts of experiment in double frequency maintenance. The differential equation may be written as yt kytky +(v?—2asin 2pt+ By*)y=0. The sign + is introduced in the third term to indicate that that part of the frictional force changes sign as velocity changes sign. Assume as before, y= A, sin pt + B, cos pt + A; sin 3pt + Bs cos 3pt +A; sin dpt+ B; cos 5pt+ ...., and consider in the limit all coefficients other than Ay and B, to be negligible. Collect the coefficients of sin pt and cos pt with the aid of the following relations : y=A,sin pt +B, cos pt+.... y =p(A, cos pt— B, sin pt)+ .... y sin 2pt=$(A, cos pt + By sin pt)+ . y®=3A,(A;?+ B,”) sin pt + 3B (A+ BY) cos pt+ . The above are easily obtained by differentiation and trigono- metric transformation. 4? = pA,’ + B,”) cos? ( pt + tan-! B,/.Aj). * Sir Phillip Watt’s article on Shipbuilding in the Encyclopedia Britannica. Tt Routh’s ‘ Advanced Rigid Dynamics,’ Art. 364. { J. Parker Van Zandt, Physical Review, November 1917. 508 Mr. NAC. Krishnaiyar on Amplitude of Vibrations Expanding the even function cos? (pt + tan~ By/A;) by Fourier’s Series into a series of cosines of multiples of (pt+tan~*B,/A,) between the limits — and = the term containing cos ( pt+tan7!B,/A,) is obtained as *(A,?+ B,’) cos (pt +tan- 1B,/ Ay); and it can be written as 8 Te. oe ea OF 8 Tar) Tey) ea Re 3 PAA / AP a Biz COS pt —s 3) By / A? -- B,? sin pt. Kquating the collected coefficients of sin pt oe cos pt separately to zero, we obtain Ayj{n?—p? + $8(AP? + B,’)5 = B; { a+ | pk! - bs 2h \/ A 248 "| } t = 30? ‘ t and Bj’ —p? + $B(Ar’ + B,’)} ; 8 qe a | ae [pe £5 pk /AF+ Be | } Therefore 9 : ! § JA 2, oD : $72 — pp? + 28(A,? + B,?) b2 92 7/2 | & + 3 pk VAP+B, | : If A stands for the amplitude of the maintained vibration, / A? + Be=A. | 2 {R8A°—(p*—n8) Pap? [Wt go pha | . (1) The amplitude A is not symmetrical with respect to p’—n*. So there will be no “ peak” or maximum resonance. The last term on the right involving the squares and product of the small quantities & and 4’ will be ot small importance in the change of the value of A. So the relation between A and p?—n? will be nearly parabolic. 2 (2) Since the right-hand side «?--p’ Le: = pha] is equal to a square and therefore cannot be negative, the maintained by Forces of Double Frequency. 509 jar magnitude ot A cannot be greater than apr PE) Increase of p’—n? above the value required to give this maximum value of A renders A imaginary. Hence a collapse of Ln se occurs. (3) Substitute for 5, sh ates —pk’) the symbol A,, ea maximum maintained pea Then . yee ‘Roane | 384?—(p'—n") | =a" — Ph +(a—pl) 5 | At the excess-tension end, A/A, is very small and if neglected the graph pede to the form 3BA? = Je — pk? + (p?—n’); and at the defect-tension end, A/A,, reduces to unity and the graph to the form 26R2 = p*— n?- The two limiting forms differ only by a constant. This shows that the shape of the graph at the two extremes is. similar. OTT 4 A, = (e— pk’ ( ) LOG 8p k ( aed ). Therefore A,, Increases if « increases in value. (5) The least value of «necessary for maintenance isthat which makes the expression for the maximum maintained amplitude just equal to zero. Since A,.= 5 2p ZG — pk’), the lower limit of the value of «is pk’. Or since a?—p?[k' + 8/3 pkA]? is equal to a square, the least value of « is p(k'+8/3q pkA), and sine there is no maintenance, A=O and the least value of «is pk’. (6) At the excess-tension end, he? — p*k!? + (p? gee is equal to #@A? and p’?—n? is negative. ‘So p’—n* cannot be arithmetically greater than ~2?—pk’?, This sets an upper limit to the value of excess tension. (7) This upper limit being Wa?—»?k'? increases with increase in the value of a. If the frictional term involving the square of the velocity 510 Prof. C. V. Raman and Mr. V. S. Tamma on a be omitted, 7.e.if k be zero, the amplitude graph will be- come a strict parabola. There will be no upper limit to the value of the amplitude, and therefore the collapse of main- tenance will not be explained. If the frictional term in- volving the first power of the velocity be omitted, i. e. if k’ be zero, there will be no explanation for the fact that there is a lower limit to the value of « below which there is no main- tenance. Facts (5) and (7) will be left unexplained. With the two frictional terms all the facts are indicated. University College, Rangoon. May 10, 1921. LVIII. On a New Optical Property of Biaxial Crystals. By C. V. Raman, M.A., Palit Professor of Physics im the Calcutta University, and V.S. Tamma, M.Sc., Lecturer in Physics, Meerut College, India”. 1. Introduction. FACT of singular interest and importance in the optics ai of crystalline media which appears hitherto to have been overlooked is that a plate of a biaxial crystal bounded by parallel faces is capable of focussing divergent rays pro- ceeding from a distant light-source and forming real images of it. Very simple apparatus will suffice to observe this phenomenon. The incandescent filament of a tiny 2-volt lamp or an illuminated pin-hole serves as a suitable source of light. At some distance from it is placed a crystal of aragonite cut and polished with parallel faces at right angles to the bisectrix of the acute angle between the optic axes. On suitably orienting the crystal and examining the pencil of light which has passed through it, a real erect unpolarized image of the luminous filament. may easily be picked up and traced continuously away from the crystal for a considerable distance. There is no difficulty in receiving and observing the image directly on a plate of ground g class if desired, The image is sharp and bright and pr actically achromatic if the object and the place of observation are both within a few centimetres of the crystal, one on each side. As either the object or the place of observation is drawn away trom the crystal the image spreads out into a spectrum. Using monochromatic light, however, it can be seen that the image remains quite well defined and sufficiently bright to be * Communicated by Prof. A. W. Porter, F RS. 1 ate cra i lal New Optical Property of Biamal Crystals. D11 traced for a distance of some 30 or 40 em. from the crystal. The effects are specially beautiful if observed in the light of a mercury vapour lamp. Separate images of the pinhole corresponding to the yellow, green, anil violet radiations of the mercury vapour may be seen, and images corre-. sponding to the fainter components of the line spectrum may also be observed by cutting out the superfluous light with suitable colour filters. A simple slit with a plate of aragonite crystal and an eye-lens to observe the image thus functions as a little spectroscope, which has quite a marked dispersion in the region of shorter wave-lengths. It should be remarked that these optical images formed by an aragonite plate differ from those formed by an ordinary converging lens in several respects. The images in the present case are real, erect, and of wnit magnification irre- spective of the distance of either object or image from the erystal. Further, the imige is continuous, that is, it may be observed anywhere in the prolongation of a certain line for a considerable distance from the crystal and not merely at a single point as in the case of the images formed by a lens. Also the images appear sharply defined in a field of diffuse light, showing that only part of the ener ey passing through the crystal is ~ brought to a focus. The object being fixed, the image moves when the orientation of the crystal is altered, but not when the plate is moved in its own plane. The focussing property, in other words, appears to be related to a fixed direction within the crystal. In order that the image may be within the field of observation it is necessary, in fact, that the bundle of light-rays should pass threugh the crystal roughly in this fixed direction, which appears to be that of either axis of single ray velocity in the crystal. 2. Explanation of the Phenomenon. The fact last mentioned suggests a mode of approach to a theoretical explanation of the phenomenon. As already mentioned, the object may be placed at a distance from the crystal, but for simplicity we shail first consider the case in which the object is a point source of light placed just on one ot its surfaces. The torm of the wave-fronts in the disturb- ance diverging within the crystal is the Fresnel surface of two sheets which, as is well known, has a singular point in the direction of either axis of single-ray velocity. If we resolve the wave-front into the groip of plane waves of which it is the envelope, we see at once that the singularity is really the crossing point of an infinite number of plane waves the pie oe, lee 512. ~=Prof. 0. V. Raman and Mr. V. 8. Tamma on a normals to which generate the surface of a cone, and hence the intensity of the disturbance at the singular point must be very great In comparison with that at other points on the it wave. On emergence from the crystal the singularity per- im sists, and, since in its immediate neighbourhood the waye- q front is approximately symmetrical in shape about the axis | of the cone of normals, it can easily be seen that the ad- wi vancing front would retain the same general configuration exhibiting a singularity or concentration of luminosity along its course, as indicated in fig. 1. We have thus in effect a continuous image of the source along a line. When the source instead of being placed on the surface is removed to a distance from the crystal, the waves which diverge from it are in the first instance spherical, but on entry into the crystal these divide at once into two sheets, the points of intersections.of which must be in the nature of foci or concentrations of luminosity in the: wave-front. On emergence of the waves from the crystal the same effect is propagated outwards, giving a cortinuous ai focus or image of the source in much the same way as in the | ease illustrated in fig. 1. When the luminous source is of finite dimensions we have an image corresponding to each Wt point of it, and it is easy to see that we should have a com- 4 plete picture built up which would be of the same dimensions . as the source and similarly oriented. | The spectral dispersion of the image is also easily under- | | ae ES a re Oe aan ee ee ae 4 : j stood. For, the direction in which the singularity travels within the crystal being inclined to the normal to the plate, the direction in which the singularity travels on emergence New Optical Property of Biaxial Crystals. 913 from the crystal depends on the refractive index and hence must make a greater angle with the normal to the plate for the shorter wave-lengths. This is exactly what is actually observed. In an Be quantitative discussion the fact that the direction of the axis of single-ray velocity in the crystal varies with the wave-length must also be taken into acta A fuller discussion of this point and of the resolving power of the crystalline plate regarded as a spectroscope in itself would be of interest. This would obviously require a deter- mination of the distribution of luminosity in and around the point of singularity in the wave-front. It is hoped at an early opportunity to carry out a detailed investigation on these points. Summary and Conclusion. (a) The paper describes a new optical effect observed with a biaxial crystalline plate, viz., that such a plate though bounded by parallel faces, is capable of forming on one side of it when suitably oriented a real erect image of a source of light such as a luminous filament placed at some distance from it on the other side. (b) The characteristics of this image-forming property are set out in detail, a noteworthy feature e being the spectr al resolution of the image which occurs when either the object or the image is at a considerable distance from the Sal (c) A theoretical explanation of the phenomenon is sug- gested, viz., that the singular point in the Fresnel wave- surface isa locus of maximum intensity, and, owing to the fact that the wave-form on emergence ieon the crystal retains 1ts general character, we get a continuous concentra- tion of luminosity alone aaliee any point of which is in effect con) an optical image of the source. The investigation described in this paper was carried out in part in the senior author’s laboratory at Calcutta, and in part at University College, London, during the visit to England of the senior author, who wishes to express his cordia!| thanks to Prof. A. W. Porter, F.R.S., for his hospitality and kind interest in providing the necessary facilities for the work. Phil. Mag. 8. 6. Vol. 43. No. 255. March 1922. 21 LIX. Some Problems of the Mass-Spectrograph. By WW. Asron, D.S8e., F.R.S., and Rh. El. Foye eViein § 1. Lntroduction. N the account of the principle of the Mass-spectrograph which appeared in this journal +, and in subsequent papers describing the constructional details and the results obtained by its means tf, attention was called to certain points, e.g. the linearity of the mass-scale and the possibility of improved focussing, which at that time had not been fully investigated. The following paper contains a mathematical analysis of these points and suggests the directions in which development is most likely to take place. It also supplies an answer to a criticism recently expressed as to the efficacy _ of the apparatus for the analysis of positive rays. § 2. The distribution of the mass-spectrum over the photographic plate. An inspection of actual photographs shows that the various lines of the mass-spectrum are distributed along the plate in such a way that the distance of any image from the fiducial spot is very nearly a linear function of the mass m. It was suggested § that the unexpected linearity might be due to some special feature in the geometry of the apparatus. We have now worked out the distribution exactly, and can account for the linear mass-scale. We can also show that the actual observed positions of the lines agree closely over the whole scale with the positions calculated on simple assumptions. This is satisfactory in that it shows that the paths are not seriously distorted by stray fields. In fig. 1, let O be the centre of the magnetic field, assumed cmcoreiiand circular, of radius d. Let Z be the virtual focus from which the rays diverge after ee the electric feld, and F their focus on the plate which lies along ZF. Let R be the radius of the circular path (centre C) of rays of mass m in the magnetic field, and let p be the length of ON, the perpendicular from O on ZF. The angle ¢ is the angle at C and FZ0=26. The angles @ and ¢ are the angles through which the rays are bent by the electric and * Communicated by the Authors. + Phil. Mag. ser. 6, vol. xxxviii. p. 707 (1919). } Lbid. vol. xxxix. pp. 449, 611; vol. xl. p. 622; vol. xlii. pp. 140, 436. § Ibid. vol. xxxix. p,. 454. ee On some Problems of the Muass-Spectrogrg¢ph. o1d magnetic fields respectively. Then tan $p=d/R, and we know that, as the energy is constant, R varies as ,/m. We may therefore write tana (MIGHT) yn ee LEY where mp (a constant) can be interpreted as that mass which under the conditions of the experiment is bent through one right angle in the magnetic field. Hquation (1) expresses in an exact and convenient form the same facts as equations (1) and (2) of the first paper already cited. It is only Reouch mo that the actual abies of the fields affect the mass-scale on the plate. ronan; It appears from fig. 1 that 1+tan @ tan 20 | mg ole = 28 tan “S 20 ° 2 OE, tp _ 2,/ (mm) tan? in m— Mo Dut tan d= by (1). Hence NF ‘m—m—2¥V (mmo) tan 20 x art 7 2/tmmy) —(m—my) tan 20° ~ = A) N is of course a fixed point, which in the actual apparatus is approximately 5-4 cm. behind the fiducial spot. When mo and the geometrical constants p and @ are known, equation (2) enables us to calculate the position of the image on the plate for any desired mass. ed ie: 516 Dr. F. W. Aston and Mr. R. H. Fowler on $3. The linearity of the mass-scale. It was observed that in the most important part of the plate the mass-scale was nearly linear—more precisely that NF was proportional to m over a wide range. Hquation (2) enables us to explain this, and in fact to prove that such linearity must always occur near 6=40, which agrees exactly with experience. or if we write Cie we have Ni pep yeah oe came, mimy 27{2z2— (7 Si \tan2e ts An approximately linear scale of the observed nature will occur where “ (~P\=0, Ae d SEI) dm \m/no dz\m/mo On differentiation and simplification we find that d (2) _ Aztan 20 — 1) {(82*—1) tan 20 + 22? = dz\m|[my )} 2{2z2—(2*—1) tan 20}? which vanishes when 1/z=tan 20, 7. e. when tan $¢=tan 20. Thus the mass-scale will be approximately linear near 6=46. Actual numerical calculation by (2) shows that the approxi- mation to linearity should be (as was observed) very close. For the actual apparatus. @= 4, radian, tan 20=0°168. Values of ] IP in arbitrary units are given in the following mm table :— MM. aie mo. LEWD M/[Mo mM 44 139 32 138 42 138 ‘ 30 13 40 138 28 139 38 137 26 140 36 137 24 14] 34 137), aoe 143 The value of m/m, atietoednte to ¢=46 is given by / myjm—tan? +6=—tan? 26—0°02 72. i] Mg Ou b ee ee mons i | ‘ 8 some Problems of the Mass-Spectrograph. ity § 4. Practical use of the mass-scale. Let us denote the distance of an observed image from the fiducial spot by D. Then D and NF differ by some constant k—about 5:4.m. in the existing spectrograph. Equation (2) eee that in all cases the relation between D and m has the orm Meigen ee Ya ia) Key) where f is a function in which all the coefficients, p, k, and tan 20 are geometrical constants; the tields affect m, and mo alone. In the actual apparatus, which was not quite rigid, p, k, and 26 probably altered slightly each time the apparatus was assembled, but otherwise were strictly constant, The only assumption made in the actual analysis of the plates was the following :—Jf D, and D, are the distances from the fiducial spot of any two points on the plate and m, and my the corresponding masses, for given values of Dy and D, the ratio my/my will be the same in every photograph taken with the same setting up of the apparatus. This is a direct Consequence of (3). For so long as the apparatus be not subjected to a new set of stresses, the function f will remain unaltered. In any one photograph we have D,;=f(m/m), D, =f (m2/mo), and in any other (with different fields) D,=f'm,'/m'), D, =f (me'/m ). But if mo! =am, it follows at once that we must have m,'=amy,, mM, =amz,, and therefore in all cases Leh Magee yo Hh) bee tA) which is the hypothesis mentioned. The argument is more general than equation (2), as an equation of the form (3) must hold even when actual fields are considered instead of the idealized fields of § 2. The practical accuracy of the theoretical relations (3) and (4) was amply verified by the fact that the construction of a consistent calibration curve—that is, the evaluation of the function f—was possible for all photo- graphs taken with one setting up of the apparatus. Owing to the fact that f is very nearly linear, the calibration curve results in applying only asmall correction to the observed D to make it proportional to m. The certainty of the results is therefore greatly increased. 518 Dr. F. W. Aston and Mr. R. H. Fowler on § 5. A comparison of the calculated mass-scale with an actual photograph. The existing spectrograph was designed so that k=5-4 em. approximately and tan 20=0°168. The values actually realized may be slightly different. “ The designed value of p is less certain, but is about 2°3 cms. In order to compare a calculated and observed mass-scale we have assumed the above values for £ and tan 20. Equation (2) now takes the form D+5-4 | Pp where the function f is exactly known. We do not know my directly or the exact value of p, and therefore mp and p may be regarded as disposable constants. In order to deter- mine them, values of logy(D+5°4) were tabulated for observed values of D, for which the values of m are known, and also values of logig f(m/mp) for selected values of m/mp. The difference of these logarithms, log,) p, should be constant and mj) must and ean be selected so as to obtain this constancy. In the photograph analysed it happens that, thus determined, mp=1. The corresponding value of p was 2°388, in agreement with the designed value. Assuming the values of mp and p, and the above values of k and tan 20, values of D were calculated for a series of values of m, and are shown by the continuous curve of fig. 2. Observed values of D for the actual lines on the plate are shown by circles. The agreement is excellent, in view of the fact that the assumed values of k and tan 20 are not reliable to the required degree of exactness. F (mmo) 5 § 6. A discussion of some recent criticisms. In this connexion some recent criticisms by Sir J. J. Thom- son* call for comment. He there discusses the focussing effect of the electric and mugnetic fields deflecting in opposite directions, and assumes an ideal arrangement practically identical with the existing instrument. He points out that the emergent rays for each value of ¢/m must have a caustic, but that when (as bere) rays of constant kinetic energy are selected only certain portions of the caustic will be touched by the existing rays, and the photo- graphic plate must be placed so that it passes through the * Proc. Roy. Soc. A, vol. xcix. p. 93 (1921). Mass expressed on the Oxygen scale. e some Problems of the Mass-Spectrograph. 519 existing element of each caustic. He gives an equation (p. 94) for the position of the plate. This position had been already established by another argument in the first paper cited, so that to this order of accuracy Thomson’s discussion agrees with ours. He then goes on to give a formula for the distance of the image along the plate as a function of e/m, and points out that this distance will not be a linear function of the mass. Fig. 2. Distances from the Fiduce/ spot tm cms. —— > The order of approximation, however, which he uses is, we think, inadequate for the purpose: the full analysis given in the preceding sections is needed to obtain that true relation. We have there seen that the observed distributions are a good fit with the theory, and that the deviations from a linear relation are very small. Such deviations as occur are simply allowed for by the use of a calibration curve in reducing the observations, so that criticism of the results based on this non-linearity cannot be regarded as of great weight. 520 Dr. F. W. Aston and Mr. R. H. Fowler on § 7. The resolving power. Hquation (2) enables us to discuss accurately the resolving power of the apparatus, but we begin with some preliminary considerations. In fig. 1 we denote ZO and OF Py banda respectively, and the centre of the slit system by 8, where SZ=c. We consider the spread of an otherwise homo- geneous beam of rays which pass through the slits at slightly different angles distributed over a range de. This spread. arises of course from the width w of the slits and we have ot 20/Ts i) where T is the distance between them. The beam may be taken as diverging from S. We are not here concerned with the focussing for different velocities, which may be assumed perfect. The total path of the rays is of=length SZ-+ZO+OF or a+b-+e very nearly, and therefore the linear spread of the beam at F is (a+6+c)ée at right angles to OF. This can be converted when desired into an expression for the width of the image on the plate. At the moment we regard it as equivalent to a spread in @ equal to (a+b+c)é6a/a. Now the instrument will resolve beams of different masses if the change in ¢@ for change of mass is greater than the geometrical spread, and the greater ¢ for a given mass and given spread the greater the resolving power. Thus we may t take a (a+ b+c¢)da. as arough measure of the resolving power of the instrument. Now the relation between a and 0 is (loc. cit.) bla=(— 20)/28, and ‘we find on eliminating the variable a that the resolving power is measured by a ey ee ee 1 gD Tt is our object to make (6) as large as possible. To do this we can keep da and c/b as small as possible, but there are strict limits to what can be done in thisdirection. The only other method is to increase ¢@ without increasing the denomi- nator, which can only be done by making ¢@ and @ both some Problems of the Mass-Spectrograph. D21 large together. It is important to note that larger resolving power can only be obtained by increase of ¢ if @ is increased as well. These arguments are too rough to give an exact value for the resolving power of the actual apparatus. This we can obtain by considering linear displacements along the plate. The width of the image on the plate is. (a+b + c)dacosec (bd — 26), Consider the region near @6=40, a=b in the actual appa- ratus, wherea+6+c=26cm.andT=10 ecm. Thus d¢=w/d and the width of the image (in em.) is 31-4. By direct calculation with formula (2), with p= 2°388 cm., we find that there is an interval of 0-797 cm. between the corresponding parts of the images for m/m)=34, 36. This corresponds to an interval of 0°139 cm. for a change of 1 per cent. inm. The actual width of the slits used was about 1/25 mm., so that the theoretical width of the image in this region is 0°126 cm. This agrees very well with the actual image widths, showing that there was little (if any) increase in width due to inexact focussing, and further that theoretically the apparatus should be able to resolve lines corresponding to masses differing by just less than 1 per cent. In actual practice a slightly greater resolving power was obtained, probably by using the ends of the images which correspond to a narrower part of the slits. i These considerations show that the theory and practice of this form of mass-spectrograph are in very satisfactory agreement, and present no anomalous and disturbing dis- cordances. § 8. Possible improvements in focussing and in the position of the photographic plate. The focussing achieved by the existing apparatus is only “first order focussing”; it is not exact, but the rays all touch a caustic and the plate is placed to pass through the points of contact of the rays actually existing. Asa result, many of the rays must have a very oblique impact on the plate, which is unsatisfactory for a variety of reasons and should be avoided if possible. It is natural therefore to try so to modify the apparatus that both the focussing and the position of the plate may be improved. The only feature which is really at our disposal is the shape of the magnetic field. It ought to be possible so to modify the shape that 522 Dr. F. W. Aston and Mr. R. H. Fowler on either or both of the objects might be achieved. It turns out that it is unlikely that any serious improvement is feasible in the position of the plate, but that improvements should be possible insthe focussing. The arguments by which one can obtain these results are as follows. Consider, first, the problem of trying to achieve focussing at a (roughly) constant distance from the centre of the magnetic field. The equations governing the deflexions are (loc. cit.) vO Comstn, vb/L=const., where L is the length of the path in the magnetic field and is now no longer assumed constant. On differentiating we have Gh) ay db du dh Of ees og Oa and on eliminating v, Qdp dO 2dU Se We shall, for example, achieve focussing at a constant dis- tance a from the magnetic field (fig. 1) if we make db=2d0 for all values of @ by proper adjustment of dL. That is to say, we must have by (7) dp db _ dl 5 o- ——— (8) for 0 is a constant 4, the same for all rays. Hquation (8) can be integrated and we find that, putting ¢o=40), L _ 9-9 L= 20-8 Po This gives the necessary length of path L as a function of ¢, and thus determines the shape of the trazling edge of the _magnetic field. It is clear that it is the shape otf the trailing edge only which is relevant here, for the rays for various values of m all enter at the same point on the leading edge. So far all is satisfactory, but the result will only be of practical value if the curvature of the trailing edge is not too large compared with the size of the field. If the edge is very sharply curved, the stray field will be large and may be expected to spoil the effect which we desire to produce. The calculation of this radius of curvature p is straightforward. some Problems of the Mass-Spectrograph. 523 If we take as axes of coordinates the tangent and normal to the rays at their point of entry into the magnetic field, the coordinates of their point of emergence may be shown to be Lsin ¢ ne L(1 —cos ¢) ae db » ¥y Cw) b} and therefore by (9) ee Gs rice “=> singe 9% , y= — (l—cos dje™ 40 The values of p can now be determined for general values of ¢ by the usual formula. The expression is complicated and for our purpose it is sufficient to consider 6=qgy and assume that @, is moderately small. The leading term in the expression for p is then —t1o¢,”. The minus sign denotes that the concave side of the trailing edge is outwards. In the existing instrument ¢y)=40)=3. To achieve the desired result we should require a concave trailing edge in this neighbourhood of radius 34], where Ly) may be taken to be the diameter of the magnetic tield. This is far too severe to be of practical use. For d)>=40,=32, the radius of curvature would have to be Lo, which is perhaps just practicable if Ly is Jarge. Thus a serious improvement in the position of the plate is barely feasible. § 9. Second order focussing. In order to obtain the conditions for more exact focussing we must start by examining the form of the beam for given m after deflexion in the electric field. It is easily shown that, when the electric field is a uniform field of intensity X acting over a length J, the rays of various velocities all diverge exactly from a virtual focus Z at the centre of the field on the line of entry. For the path of the particle, charge e, mass m, in the assumed field is a parabola, whose equations referred to axes Owy are (fig. 3): 2=Vv COS at, y=vsin «t—4 Xe??, Xe y=ea tan a— ae sec? a wv’. hed L 524 Dr. F. W. Aston and Mr. R. H. Fowler on The emergent ray starts from the point (1 ltan «— ete a ) | 20 at the slope tan a — ao Sec? a v Fig. 3. Tts equation can therefore be put in the form 2 peace sec? “= (tan He ieee *) (@ wee or y—a tan a=— eeENS: eS, AD) Ol AS Pek) which passes through the i (30, 41 tan Ale 1.0. L, ror all values of v. To tind the conditions for more accurate focussing after passing the magnetic field we may therefore think of the rays incident on the magnetic field as diverging exactly from a point distant 6’ from its leading edge, where 0’ is less than the b of fig. 1 by d, or $h, the radius of the field. We shall assume, to simplify the discussion, that the leading edge is plane and at right angles to the median ray, which is deflected through an Pangle Oy in the electric field. The path of a typical - ray is shown in fig. 4, in which A is the point of entry and B the point ‘of emergence. After reduction the exact equation of the emergent ray, referred to axes Oxy, can be putin the form y cos (@—a)—a# sin (6—a) + Bu(1—cos ¢) +b' tan acos(@b—a)=0, . (11) where B is the constant L/(v@) and « and v are related by some Problems of the Mass-Spectrograph. 525: the equation, valid when @) is small, for the electric deflexion (0,+ «)v?=const. The focus of the “emergent beam is the point where the line (11) touches its caustic (7.e. the envelope of the family (11) when the parameter « is varied). This point can be determined (by the usual rule) by differentiating with respect to a, and determining dd/da and dv/d« from the deflexion equations. This leads to a position of the focus agreeing when @¢ is small with the position already deter- mined (loc. cit.) by simpler arguments. In order to obtain Fig. 4. second-order focussing the coordinates of this point on the caustic must also satisfy the equation obtained by differen- tiating (11) twice with respect to a. The exact condition thus obtained is an equation for d’L/da? and is somewhat complicated. In order to appre- ciate its meaning, we shali assume that dL/dz=0 and retain only the lowest powers of @. It then reduces to the relation i ae ee L 46,2" b'+ L(i—¢/40,) ° Near 6=46), the important neighbourhood, and for A= j)5 we have L/L somewhat greater than 36—for 6)=4 some- what greater than 9. If we work out the radius of curvature p of the trailing edge of the field we find, with dL/de=0 526 Dr. F. W. Aston and Mr. R. H. Fowler on and retaining only the lowest powers of d, __ {+ L(1—4/46,)}? (be! _ _462{0' + Ld —$/40)}5 a L(o'-+ £1) We observe that this radius of curvature can be kept reasonably large by making 0’ fairly large compared with L. For example, if b’=2L, 6=46, the radius required is $*6,"L. Even when 6)= 5, as in the present apparatus, this is 4 L, and if O>=4 it is }$L, which is by no means unmanageably small. It is therefore possible that the focussing can be improved by this means, although it is not so likely to be possible to improve the position of the plate. The reason why this improvement is possible without the use of an unreasonably small radius of curvature on the trailing edge of the field is that, by making 0’ large, we can scatter the rays of different velocity well before they impinge on the leading edge of the field. Thus the necessary difference of path length can be achieved on paths which are not too close together—that is, by a trailing edge of reasonable curvature. In actual practice the curvature would be applied to the leading edge, since this arrangement possesses all the advan- tages, from a structural point of view. In the region of o=46),, where the paths are symmetrical about the magnetic field, it is theoretically immaterial to which edge the curva- ture is applied. § 10. Considerations of the dimensions of the apparatus. In designing physical instruments of precision, it is of importance to study the absolute scale of dimensions likely to yield the best results. Now, pressures being equal, the shorter the path of the beam of positive rays between the cathode and the photographic plate the less likely it is to lose intensity and sharp outline through collisions. Also the smaller the area of the plate affected by a beam of given intensity the easier will be its detection and the more accurate its measurement, for we are very far at present from the limits of accuracy of the measurement of position on the plate determined by the fineness of the grain of the emulsion. Hence both these considerations point to an instrument of the smallest practicable dimensions. some Problems of the Mass-Spectrograph. D20 The electric field offers no restrictions to such a develop- ment, for, once having decided the value of the deflexion 0, and the potential at our disposal to produce it, the only quantity fixed is the ratio of the length of the plates to their distance apart. Their absolute dimensions are only limited by considerations of convenience of construction. The magnetic field, on the other hand, unfortunately dictates an inferior practical limit to the size of the instru- ment in an unequivocal manner. Wehave from the original exact equation for the motion of the charged particle in the magnetic field of intensity H, es mv m SEE yee me be vy: where V is the potential through which the particle has fallen in the discharge tube. If we express V in volts and m on the ordinary chemical scale (O=16) we get approxi- mately HU -144(mV):. Now H cannot very well be greater than 17,000 gauss for large pole-pieces. Actually the highest value used so far in this work is 15,000 gauss. Taking the values of the existing apparatus peed 3 radian and B15, 000, and allowing V to range from 20,000 to 50,000 males we find that pahile for the hydrogen atom (m=1) the length of the field required ranges from 0°45 to 1-0 em., for mercury (m=200) it must range {rom 6°4 to 14:1 cm. (The actual length is 8 cm.) We see therefore that.while it is possible to design a mass-spectrograph of precision on a small scale to inves- tigate elements of the lightness of hydrogen or helium, an apparatus capable of resolving the isotopes of the heavy elements must of necessity be on a considerable scale. In- crease in the scale of the apparatus brings the necessity for extremely low pressure and other technical difficulties in its train, so that it appears probable that really great increase in resolving power, as in the case of X-rays, will have to come ultimately from increase in the intensity of the beam of rays, enabling extremely narrow slits to be employed. Summary. Some points raised by the performance and further design of the mass-spectrograph are discussed. The linearity of the mass-scale and the resolving power of SS a a a re : 7 a Pe \ an "| 528 | Mr. E. H. Synge on a Definition of the existing apparatus are explained. The agreement between theory and experiment is shown to be good. The questions of improvement in position of the plate and second-order focussing are discussed. The latter is shown to be feasible. Lines on which future improvements are possible are suggested. Trinity College, Cambridge. November 1921. LX. A Definition of Simultaneity and the ther. By i. H. Syneu*. ~ PROPOSE to show that a definition of absolute simul- taneity is involved in the generalized coordinate frames assumed in relativity. Jonsidering two events E, EH’, located at the points P, P’, in an arbitrary frame F, and joining P, P’ by any line Lin F, we may assign, in an infinite number of ways, another frame F’, which, at every instant, at every point of L, has a finite component of relative velocity normal to L. The points of L trace out a surface S in F’, and if Q be the point inS occupied by P ata specific instant, and if M be any line in S passing through Q, then the point R in L, at which M and L instantaneously intersect as L crosses M, will move along Las L moves in 8, and in the immediate neighbourhood of P this motion of R along J, will, according to the direction of M in 8 at Q, be either iawards to, or outwards from, P. The possible directions of M at Q are divisible into two groups, corresponding to these two different senses of the inaction of R, and it is clear, on grounds of continuity, that the two groups in question are separated by a limiting direction of discontinuity, for which the motion of R along L at P, and obviously also of R along M at Q, would be indeterminate in sense, and of infinite velocity. We shall define this limiting direction as instantaneously codirectional with L at P, and, considering the entire length of L, it appears without difficulty, in the supposed circum- stances cf motion, that through each position instantaneously occupied by P in 8, a single ‘line in 8 can be drawn, which at all points of its length is instantaneously codirectional with L as the latter crosses it. For, supposing, on the contrary, that two such lines passed through a point Q! jim Sjiand * Communicated by the Author. Simultaneity and the Aither. 529 considering a small triangle in 8, whose sides are formed by two lines M’, M"”, which pass through Q’, diverging slightly from the supposed two lines of instantaneous codirectionality, and whose base is the track of a specific point P’” of L, we see that the point R’ of inter- section of L, Gish with M’, and then with M", travels along L to Q', and then along Lin the reverse (eee to meet the point P”. But, by’ taking the lines M’, M”, suffi- ciently near to the critical’ directions the velocity of R' can be increased beyond any assigned limit, while that of P” travelling through 8 remains finite. We should therefore have two points proceeding over finite distances, the one at a finite, the other at an infinite velocity, and reaching the same point together, a thing which no question of time units can render conceivable. It will be noticed that the velocity of R along IL is in no respect limited by, or in any way dependent on, or connected with, the velocity of light, which does not enter into the question at all. Some additional considerations are perhaps called for here as being necessary to justify the attribution of a definite sense of motion along L to the phase point k, where the velocity exceeds that of light. For this purpose we may for simplicity suppose that the surface § is the plane of the paper, and conceive an apparatus in which two very fine split needles N, and Ny, are pivoted about points O, and O,, O, and O, being out of and above the surface 8, and situated near the tangent to M through Q, and on opposite sides of M with respect to @, where & is no longer taken as the terminal point of M. Ny, and Ny, are free to move in such a way that each always touches M ata point, and they are set initially with their extremities passing through 8, verv close to Q, the needles making very small angles with M. where they touch the latter, ani each touching M at the same side of Q as its own pivoting point. N, and N, are now caused to move so that, as L slides over 8, the om where each touches M continually coincides with R. In this motion of L. one needle will be struck by L on the lower edge and raised, while the other needle will be struck on the upper edge and depr essed, and the direction of motion of R along L can be defined according to the particular needle which is struck on its upper edge. In this apparatus QO, and O, can, except for the limiting direction, always be chosen so near to the tangent to M at Q, that no portion-of the needles need have a velocity as great as light, although the velocity of R at P may be many ‘times oreater Piul. Mag. 8. 6. Vol. 43. No. 255. March 1922. 2 M 530 A Defimtion of Simultaneity and the ther. than this. It is obvious that we are justified in speaking quantitatively of the velocity of the phase point R along L, precisely as if we were speaking of a material point or light wave, while the argument employed to demonstrate the uniqueness of the line of instantaneous co-directionality becomes quite definite when considered in connexion with such apparatus. . We shall call the system of these lines of instantaneous codirectionality, corresponding to the different positions occupied in 8 by P, the system (LE’) and we shall define the events E and E’ as simultaneous if both occur in the transit of L across the same line of (LE’). To prove that this definition is independent of F, F’, and L, we first infer from the definition of instantaneous co- directionality, that if two lines cross one another, as, for example, L and M, and have at all points of this transit finite normal components of relative velocity, then if they are instantaneously codirectional at any point, the line which is the locus of the events of their transit in any third frame, will, at that point, be codirectional with both lines. Considering the locus in a third frame of the transit of L across a member of (LE’) the definition of simul- taneity is seen to be independent of fF’, the third frame being interchangeable with E’, where the condition of a finite normal velocity relative to Lis observed. It is also obvious that F’ is interchangeable with F, M taking the place of L. We now join P, P’ by any second line A in F, and select i’ in such a way that the condition of a finite normal com- ponent of relative velocity is satisfied with regard to the closed line L+”. There will then, through Q, be a unique closed line M+, in EF", which is instantaneously co-direc- tional at all points with L+A, as the latter crosses it, and it is clear that the same criterion of simultaneity is satisfied by the passage of L+2 across M+ yp, as by that of L across M or of 2 across w, L and ® beine thus interchangeable and the definition’s independence of the particular line L being established. It is clear, further, that if two events are simultaneous with a third, according to this definition, they are also simultaneous with one another, since the line joining their locations in I’ may be chosen to pass through the third. The definition is thus absolute, and can be extended throughout the whole universe, while it does not depend in any way upon the Huclidean character of space. The ether also is that privileged frame of space for which, where On Ionization by Cumulative Action. 531 matter is not in question, light simultaneities coincide with these absolute simultaneities. | The doctrine of relativity, as such, therefore breaks down, and the mathematical transformations it introduces must, if they are valid, be susceptible of an interpretation in terms of these absolute data; while with regard to hyperbolic Space-Time, in which the instantaneous co-directionality of two moving lines is not an absolute property, but dependent on particular frames of reference, it appears that intrinsic contradictions are involved. Note.—In the above discussion it might have conformed more strictly to the criteria of relativity to have stated that the condition satisfied by EF’ is that, in a finite region and time considered, each point of the surface traced out by L in EF’ is occupied by one point only of Land for only one instant. The remark should perhaps also be made that the argu- ment for the indifference of the coordinate frames can be made definite by the introduction of the apparatus described xbove, in which case we assume, as not open to question, that for great velocities of R the same sense of motion of R along L would be indicated whether O,, O, were fixed in forin F. LXI. Remarks on Ionization by Cumulative Action. By K. T. Compron, Professor of Physics, Princeton University *. A RECENT, very interesting paper by Professor F. Horton and Miss A. C. Davies} brings convincing proof of the importance of radiation in the production of ionization in certain cases in which ionization by direct impact is impossible or improbable. These cases include the formation of arcs in metallic vapours and in helium at voltages less than their minimum ionizing potentials. Ex- periments now being conducted in this Laboratory by Mr. Duffendack, to test the possibility of the production of similar low voltage arcs in multiatomic gases, have indicated that it is impossible to cause ares to strike in such gases at any voltages less than their minimum ionizing potentials, even when stimulated by the most intense thermionic currents which can be produced between heavy incandescent tungsten strip electrodes heated by currents through water- cooled leads. Hxperiments on low voltage arcs in atomic = Communicated by the Author. 7 Phil. Mag. xl. p. 746 (1921), 2M 2 — ee 532 Prof. K. T. Compton on hydrogen and atomic iodine, produced by dissociation within an incandescent tungsten tube, are now in progress. | The two characteristics of monatomic gases, which probably account for the relative ease of production of low voltage ares in them, are elasticity of electron impacts and ability of any atom to absorb the resonance radiation from neighbouring atoms. The former characteristic results in the gaining of sufficient energy by every electron to produce either radiation or ionization at an impact, and also in greatly increasing the number of impacts made by an electron in its path between the electrodes. The latter characteristic permits the radiant energy, liberated by each electron impact which results in radiation, to be passed on from atom to atom, and thus multiplies the fradtion of atomne) whieh ienenam the “abnormal” or partially ionized condition. In multiatomic gases the undissociated molecules do not appear to be capable of absorbing the radiation which is produced by electron impacts and which is believed to be characteristic of the dissociated atoms instead of the molecules. In a previous paper * the writer has summarized some evidence that ionization in low voltage arcs cannot be due to effects of single impacts, but must be the cumulative effect of two or more impacts. Additional evidence may be suggested along the following lines. Consider parallel electrodes distant d apart in a gas at. pressure p millimetres and with a difference of potential V, giving an average electric intensity X=V/d. Let V; and V, be the minimum ionizing and radiating potentials of the gas and let N be the average number of collisions made by an electron per centimetre path. Its mean free path /=1/N. If the potential difference V is made equal to or slightly greater than V;, what is the probability P that an electron may gain a speed sufficient to ionize a normal atom by impact ? In a monatomic gas every electron will attain a speed corresponding to V, despite possible collisions. In passing from the point of potential V, to that of potential V,, it is liable to expend its energy in the production of radiation at any impact. If one of these intervening impacts happens. to be elastic, the electron is deflected through such an increased path that additional impacts become probable, each with an additional chance for loss of energy by production of radiation. Taking account of this and of the average direction of motion of an electron in the region between V,. and V;, it 1s found that the probability of ‘getting through * Phys. Rev. xv. p. 476 (1920). Lonization by Cumulative Action. 533 this region without loss of energy by radiation is approxi- mately : V 9pN(V;-V,») Ee 2 ; which must at least be of the right order of magnitude. This represents the fraction of the bombarding electrons which attain sufficient speed to ionize a uormal atom by a single impact, and is independent of the initial kinetic energy of emission of the electrons from the cathode, provided this does not exceed V,,. In Table I. are given values of P calculated for various pressures in mercury vapour and helium, assuming the electrodes to be 1 cm. apart. For mercury: V,;=10°4, eo N= 75; for helium ;, V;=25°2,; V,=20°4, N=S'5. These gases represent extreme cases of small and large values of P, respectively, under given conditions. Papin. p (mm.). P Hg. | P He. 0-01 0-59 0:98 0-1 3-1 (10)73 0-79 1:0 7-8 (10) 7° | 0-09 eR!) 50: (10)— 76 Meola 10:0 GENO wi) 27) OAL? At the pressures, between 2 mm. and 10 mm., at which the most intense low voltage ares are obtained, it is evident that any contribution to this ionization due to single impacts against normal atoms is entirely negligible. The same conclusion must be drawn from experimental evidence. In a recent paper* I have shown that the ionization curve in helium shows no discontinuity at the ionizing potential 25°2 volts for pressures exceeding three or four millimetres. Further experiments on arcs in helium + have also indicated that the eritical voltage for the helium are is the minimum radiating potential, near 20-4 volts, rather than the ionizing potential. It appears, therefore, that while ionization by single * Phil. Mag. xl. p. 553 (1920). + K. T. Compton, E. G. Lilly, & P. S. Olmstead, Phys. Rev. xvi. p- 282 (1920); K. T. Compton & E.G. Lilly, Astrophys. Jour. li. p. 1 (1920). 534 3 Prof. K. T. Compton on : impact may be of primary importance in producing ionization in rarified gases, as in Geissler tubes, ionization by cumulative action is of preponderating importance in ares in monatomic gases and vapours. As to the nature of this cumulative action, there are four possibilities which have been suggested. ‘The majority of atoms which are in an abnormal or partially ionized state may be in this state either as a result of a previous impact or by absorption of resonance radiation produced by electron impacts against neighbouring atoms. Then the second stage of complete ionization may be brought about, either by an electron impact against the abnormal atom or by its photo- electric ionization by additional radiation. As the result of a tentative formulation of the theory of ionization by direct successive impacts * and experiments on helium ¢, the writer came to the conclusion that the actual amount of ionization is much larger than can reasonably be accounted for in this way. It can be shown, for example, that if the observed results are to be explained simply by successive impacts, it would be necessary to take the time constant of damping of resonance radiation to be about a million times greater than any value of this constant which has been found tor any substance by direct experiment. Hence, it was concluded that the energy of resonance - radiation from neighbouring atoms must contribute to the total energy required for ionization. If impurities are present in helium, they will be photo- electrically ionized by the helium resonance radiation. Goucher { and others who have obtained experimental evi- dence of ionization in helium below the ionizing potential have attributed it to such photoelectric ionization of impurities. Professor Horton and Miss Dayies § have criticized the writer's proof of the existence of true ionization of helium below the ionizing potentials as inconclusive on two grounds: (1) the probable photoelectric ionization of neon or other impurity, and (2) uncertainty regarding the significance of a measured ratio R, on whose values the conclusions were based. The following evidence, however, proves these criticisms to be unfounded :— (1) Although earlier experiments were made with helium, in which a trace of neon was spectroscopically deteciable, later experiments in which similar lonization was obtained * Phil. Mae. loc. cit. + Phils Magy voc xen: i Proc mhys-Soe, xxxir.p. 13 (1920): § Loc. cit. Tonization by Cumulative Action. 535 in absolutely pure helium are quoted in the writer’s original paper. The degree of purity may be judged by the fact that, in this helium, arc currents as large as an ampere were obtained at the minimum radiating voltage without observing any spectral line of an impurity. Every line in the visible line spectrum of helium (excepting the enhanced lines) was observed and, in addition, more than two hundred lines of the band spectrum. It is impossible that such a degree of ionization and such an intense helium spectrum could be attributed to ionization of impurities which were present in too small quantity to be detected in the spectrum. (2) The criticism of Horton and Miss Davies to the effect that the writer’s measurements of R (which is the ratio of the electrometer deflexions observed with the foil and gauze ends, respectively, of a hollow cylinder presented toward the filument) do not give an exact estimate of the proportion of the observed effect due to ionization or radiation separately, is well taken. The uncertainty is pointed out as due to the presence of a photoelectric emission from the sides and back of the cylinder; as well as from its front, due to resonance radiation passed on from atom to atom through the gas. Sach photoelectric emission from the sides was neglected in the writer’s calculations. Yet the observed variation in R is entirely too large to be accounted for in this way. For instance, at 25 mm. pressure, it would require a photoelectric emission from the sides of the cylinder twelve times iarger than that from the foil end to account in this way for the observed value of R. But the area of the sides is only four times that of the face, and they are much less favourably placed as regards the electric field and their accessibility to the radiation. For these reasons the writer stated that the absolute values of R are of little importance, but that the variation of R pointed conclusively to ionization. For these reasons the writer cannot admit that Professor Horton and Miss Davies have been the first to definitely prove the existence of ionization between the minimum radiating and ionizing voltages in helium, or the essential part play ed by resonance radiation in producing this 1oniza- tion. But the work of Horton and Miss Davies strikingly proves that radiation may play an even more important role in cumulative ionization en ‘had previously been suggested, being active not only in maintaining atoms in an abnormal state, but also in photoelectrically ionizing them when in this condition. It is reasonable to suppose that cumulative ionization becomes an increasingly important factor in ionization as 536 On Ionization by Cumulative Action. conditions become more favourable to cumulative action and less favourable to direct ionization by single impacts. We should expect, therefore, that cumulative action would be particularly important in the temperature ionization of gases, and therefore that the part played by radiation in promoting temperature ionization would be very important. An illustration of the possible importance of such a concept of temperature ionization is seen in the opportunity which it affords of explaining certain failures of the exceed- ingly interesting and important theory of Lonization in the Solar Chromosphere, which has recently been proposed by Dr. Megh Nad Saha*. He applies Nernst’s equation of the reaction isobar to the calculation of the percentages of ‘ionization of different elements, by considering ionization to be a dissociation with heat of dissociation measured by the ionizing potential, and by assuming the element to be in an enclosure at the temperature T of the Sun’s chromosphere. This is equivalent to assuming that the element is subjected to, molecular bombardments and black body radiation characteristic of the temperature T. This léads to the con- clusion that the degree of ionization of an element at any’ pressure depends only on its ionization potential and the temperature T. The theory thus developed accounts beauti- fully for numerous characteristics of the solar spectrum. It has been pointed out by Professor Russell, however, that there are several instances in which Saha’s theory seems inadequate tT. For instance, the ionization potentials of barium and sodium are practically equal, vet barium is apparently completely ionized in the chromosphere, whereas sodium is not, as evidenced by the absence of all except the enhanced lines of barium, which are strong, while the sodium D lines prove an abundance of un-ionized sodium. This may be accounted for if the influence of radiation in promoting ionization be considered, as will be pointed out in a Note in the Astrophysical Journal by Professor Russell and the writer. Whereas the energy of molecular bombard- ments is characteristic of the temperature T, the spectral distribution of radiant energy is not, since it is present as an outward flux, from the hotter interior through the selectively absorbing photosphere. Various factors, such as abundance, atomic weight, and chemical affinity, may determine the extent to which any element in the chromosphere is shielded from radiation of its own resonance type by the “ blanketing ” * Phil. Mag. xl. pp. 472, 809 (1920). + Astrophys. Jour., im print. way se ‘Ai t —- Resistance of Electrolytes at High Frequencies. — 537 layers of this element in the photosphere. The Fraunhofer spectrum proves this “blanketing” effect. Sodium, in the chromosphere, is subject to radiation which is deficient in those particular wave-lengths which could put its atoms into an abnormal state, whereas the effective wave-lengths for barium are present. The effect of those types of radiation present in the Sun’s chromosphere is equivalent, therefore, to a lower effective temperature or to a higher effective ionization potential for sodium than for barium. Thus, Saha’s treatment of the chromosphere as a black body is but an approximation to the actual conditions, and variations from the results of his treatment may be expected because of the deficiencies in particularly effective types of radiation as shown by the Fraunhofer spectrum. Palmer Physical Laboratory, Princeton University, U.S.A. LXIT. On the Resistance of Electrolytes at High Frequencies. By Joun J. Dowie, W.A., F.Inst.P., and KATHARINE _M. Preston”. in lai primary object of the experiments described in . this paper was to ascertain whether the resistance of electrolytes is the same at high frequencies as for direct currents. The problem was attacked as far back as 1889 - by Professor (now Sir) J.J. Thomson f. He used a Hertz oscillating circuit, and showed that at frequencies of the order of 10° ~ there was good reason to believe that the ohmic - resistances of electrolytes were unchanged. As he pointed out, there isan element of difficulty in reconciling these facts with the ionic theory ot electrolytic conduction. The problem, therefore, seemed worth further investigation, particularly in view of the fact that we had devised very accurate methods for measuring resistances at high fre- quencies. § 2. These methods are developments of principles made use of by the senior author in another connexion f. The earlier experiments were carried out with a view to testing the possibilities of the methods contemplated, and it will, therefore, be convenient to describe the experiments “Communieated by Prof. J. Joly, F.R.S. + Proc. Roy. Soc. vol. xlv., January 1889. t “A direct-reading Ultramicrometer,’ Proc. Roy. Dublin Soe. xvi. p. 185, March 1921; also Brit. Assoc., Edinburgh Meeting, 1921. 938 Mr. J. J. Dowling and Miss K. M. Preston on the more or less in the order they were made. The time at our disposal for the work did not, unfortunately, permit of the observations being pushed to the highest possible accuracy, but it will be evident that the new methods lend themselves to measurements of considerable precision. § 3. After several trials of various possible oscillation- circuits, and with different values of battery voltages, etc., the following was adopted as most likely to prove satisfactory for the present purpose. It was made up of existing apparatus, and could, no doubt, be much improved. The coil AB (fig. 1) was of 170 turns bare (22) copper Figat, wire, wound on a 10 cm. square section frame 60 cms. long. ~C was a graduated sector condenser (100-1200 pu farad), An Kdiswan (HS,) valve was employed with a °6 ampere filament current. An anode battery of 20 volts was found suitable. A potential balancing arrangement, inserted in the anode circuit, enables a sensitive galvanometer to be employed to detect small variations of the anode current. The resistance to be measured is inserted in the oscillation- circuit at RK, where it has no direct-current component passing through it. | §4. The potential balancing device, just referred to, is represented by HT (fig. 1), and the theory of its action is important. It consists of a few cells (EH), in series with a rather large resistance (T), the whole being in parallel with the galvanometer branch (GS). Let g be the resistance of the (shunted) galvanometer (GS). | If 1 is any value of the anode current, a current Resistance of Electrolytes at High Frequencies. 539 pee 10) passes through the galvanometer branch where I,)= 1: Thus for values of I in the neighbourhood of I,, a galvanometer of high sensitivity can be used to measure slight deviations of I from the “ Standard value ” I). § 5. The application of this apparatus to the measurement of resistance will be clear from a consideration of the Fig. 2. 70 = ¥3). 30 Paul” galvanometer (shunt 4 p.u.farads. 200 400 following experiments. By a series of preliminary adjust- ments of the coils AB, the apparatus (fig. 1) was made to yield an “anode-current ... capacity” curve of the form shown in fig. 2. An electrolytic resistance * was then introduced at R and the observations made with the electrodes at various distances apart. An examination of the resulting family of curves (fig. 2; A, B, C, D, E corresponding to 5, 4, 3, 2, 1 ems. of * A straight tube of CuSO, (Normal) with adjustable copper disk- electrodes. 540 Mr. J.J. Dowling and Miss K. M. Preston on the electrolyte) shows that by confining our attention to a fixed value of the capacity (say, 150 wp f.) the corresponding ordinates decrease by amounts nearly proportional to the increments of the resistance. § 6. Observations were now carried out on these lines, both with electrolytic resistances and with the practically non- inductive resistances afforded by lengths of No. 40 s.w.g. Manganin wire wound in small helices. The results of such observations are shown in fig. 3. Fig. 3. 320} 240} ¥300). ay) S Galvanometer (spunt Van ee 5 Curvel; | div.= 20 cms. Manganin. ves ==) 4 Electrolyte (3 cm. tube). oy ANUS 9 = 5H » 9 lcm. » A comparison of the same resistances was then made by es the Kohlrausch method, using a metre bridge. The following i" table shows the results. et Kohlrausch method :— iY 100 ems. wire has same resistance as 12°5 ems. of CuSO, sol. in 3 em. tube. ae 1:0 95 «e yy a9 37 Resistance of Electrolytes at High Frequencies. 541 From fig. 3 :— Galvanometer deflexion dueto 100 ems. wire = 110 divisions. bs sh » 12:5 ,, CuSO, in3cm. tube= 106 2 9 39 be] 10 3) 33 1 br] = 1138 ) The discrepancy between the galvanometer readings is not as small as one would desire, but the Kohlrausch bridge arrangement available was rather unsatisfactory. Probably a considerable error is attribatable to this. § 7. Since the curves in fig. 3 do not approximate, even along their middle parts, to straight lines, a trial of another circuit was made—namely, that shown in fig. 4. The Fig. 4. resistance R to be measured is here introduced in series with a coil Y. This coil consisted of about 40 turns of No. 22 copper wound on a frame 25 cms. square. It was found best to place it coaxial with the coil AB and closeto X. The oscillations of current in BCX then induce an oscillation in the circuit RY, and the amount of power contributed by the valve circuit for this purpose is a function of the resistance R. Figs. 5 and 6 show the variation of the anode current in the valve circuit as a function of R*. The curve (fig. 6) consists of two branches, that on the left (lower resistances) being almost perfectly straight for values of R up to about 400 ohms. This circuit is suitable for the measurement of large as well as small resistances; but we shall confine ourselves tv the latter at present. §8. The modus operandi is as follows. Using suitable resistances of the fine Manganin wire, observations of the * The abscisse in fig. 6 are lengths of a capillary tube of electrolyte: * 1 em. corresponds to 923 ohms, 542 Mr. J.J. Dowling and Miss K. M. Preston on the galvanometer deflexion (shunted .},) were made. A curve so obtained is shown in fig.5 *. By a method of substitution the resistance of an electrolyte can be easily obtained ; but, to attain the highest accuracy, the following procedure is necessary. '§ 9. Having introduced the electrolytic resistance in the circuit RY, the resistance T is changed until the galvanometer is at zero. The galvanometer shunt S may then be reduced to unity and the galvanometer reading taken. Having restored the shunt S to a high value the electrolyte is removed, and, by means of a suitable set of non-inductive resistances, Fig. 5. aS O = — our a 2) Ga!vanometer. Ohms. 82) eo v7 40 Fe, 60 2 a resistance of nearly the same value is introduced at R in place of the electrolyte. On again reducing the galyanometer shunt a second reading is obtained. The difference between the two galvanometer readings is proportional to that between the electrolyte and the known resistance, in virtue of the linear relation shown in fig. 6. For calibration it is simply necessary to alter the known resistance by a small amount and to observe the effect on the galvanometer. Great precision is obtainable in resistance determinations by this method. A consideration of fig. 5 shows this. The * The curve starts from zero because of the previous adjustment of the potential balancing device TE. Galvanometer sensitivity =10~° amp. per diy. Resistance of Electrolytes at High Frequencies. 5438 observations were taken with the galvanometer shunted 300 times. The slope of the curve (left) is nearly 10 galvanometer divisions per ohm. In the balance method, the galvanometer being unshunted, ome galvanometer division would clearly correspond to 3)9X jy or 8x107* ohm. In other words, in the measurement of a resistance of about 60 ohms an accuracy of 5 parts per million might be reached. For higher resistances the error would be iess, but the method fails, of course, when the crest. of the curve is reached. § 10. Fig. 6 indicates that, when the resistance R (fig. 4) is increased sufficiently, the anode current in the valve-cireuit commences to decrease regularly. Although the relation NT aa between the anode current and R is not linear, it is, of course, clear that high resistances can be measured in much the same way as that described in § 9 for the smaller — ones. This method was therefore employed to investigate the high frequency resistances of glycerine solutions of copper sulphate. Gilmour *, who investigated these solutions, found that the specific resistance (for D. ‘G) varied irregularly with the concentration over a certain range ; Jones and others have described similar effects in other glycerine solutions. It is, of course, of some interest to determine whether these ir regularities reproduce themselves at high frequencies. * Phil. Mag., March 1921. Ame ee eG a ae 8001-02 400}: 544 Resistance of Electrolytes at High Frequencies. §11. These solutions were of such high resistance that it was necessary to use a container of what would be rather an unsuitable design for high frequency work—particularly as the solutions were such poor conductors. The electrolyte was contained in a shallow copper vessel with a flat bottom (8 cms. diameter), which formed one electrode. The other electrode (copper disk 6 cms. diam.) was suitably supported from an accurately fitting wooden cover, so that it was 5 cm. above the bottom electrode*. A galvanometer circuit, as described by Gilmour (loc. cté.) was used for the determina- tion of the direct current resistance. To 74 2 Reciprocal of | galv. deflection. Ohms. Concentration. ‘Gihe | eT Tie ee (03 Having prepared a series of solutions covering the range of concentrations where anomalies occur, the container just described was filled in turn with each of them and observa- tions taken, first with it inserted in the D.C. circuit and immediately afterwards in the H.F. circuit. Curves I. and II. (fig. 7) show the results so obtained. In Curve I., which refers to the D.C. experiment, the ordinates are proportional to the resistance plus a constant. § 12. The two curves closely resemble one another, but are * The vessel had appreciable capacity due to the size and proximity of the electrodes. A Graphical Synthesis of the Linear Vector Function, 545 not quite identical, the H.F. Curve II. showing a much more pronounced “kink.” It is unfortunate that the time at our disposal was insufficient to enable this problem to be subjected to a more careful examination. It is doubtful whether this discrepancy is due to an inherent defect in the methods of observation or whether there is actually here a difference between the H.F. and D.C. resistances. We hope to pursue the inquiry at another time. Summary ° Methods of utilising H.F. oscillation valve-circuits for the the measurement of resistance have been devised, the principles involved being extensions of the senior author’s ‘ altra- micrometer’ circuits. ‘These methods are capable of great precision and are suitable both for metallic and electrolytic resistances. Preliminary tests indicate that ordinary electrolytes offer the same resistance to high™* as well as to low frequency currents. Observations on glycerine solutions, which show anomalous “ resistance-concentration ” variations for D.C., are somewhat inconclusive, but seem to show this to be more pronounced under H.F. conditions. . It is hoped to continue the work, especially at still higher frequencies and in connexion with the anomalous effects in glycerine solutions. | University College, Dublin. yee, 25, 1921". LXIII. A Graphical Synthesis of the Linear Vector Function. By Prof. PRepERICK SLATE fF. REVAILING use of the polygon-graph employs the same scale for the resuitant, and throughout any group of its components. But the resultant can be ex- pressed more comprehensively and still simply, when an individual scale-factor is allotted to each constituent vector. This idea is found to open an approach to linear vector * The frequency used was about three-quarters of a million. Even at this frequency the “skin effect ” is negligible in such poor conductors. + Communicated by the Author. Phil. Mag. 8. 6. Vol. 43. No. 255. March 1922. 2N 546 Prof. F. Slate on a Graphical Synthesis functions that is in some respects supplementary to the traditional methods and more direct *. Let (O) be a base-point for three vectors (Vy, Ve, V3). Suppose them physically homogeneous (compoundable) and mutually perpendicular. Their union to a resultant (V) drawn from (QO) is shown fund: unentally: in terms of umtt- vectors and their tensors by Vv= Viv, + Vov2t V3V3. Mer tis ly) The unit-vectors (¥,, Vo, V3) in this order of their indices may fit either cycle—right-handed or left-handed. Tensors cannot be negative under a primary ruling that every vector shall be rated positive in its own “forward” direction. This agrees, moreover, with algebraic usage regarding the radius vector of polar coordinates, and the perpendicular from the origin to any plane. Negative tensors are attendant later upon. the algebra of assembling vector elements parallel to the same line, after giving them a unit-vector in common. The reversed (neutralizing) vector (—V) can be described equally well both ways: (= V)v=(=Vi)ii (= Vas tener Vv'=Vy,v, + Veve’ 4+- Vavs’ ; ty Waal) Viv =v. =v =v =O Note that the cycle (circulation-rule for axial vectors), with the same index-order, must exchange between right-handed and left-handed at passage from (Vj, Va, Vs) to (Vy’, Vo’, Vs’) or vice versa. Represent (Vy, Vo, V3) graphically by the arbitrary lengths (OA=a,v,, OB=y,v., OC=c1v3) asso- ciated with the scale-factors (a, 6, c) sueh that V= (ax,)V, + (by) Ve + (€21) V3 § @) and consequently, —V=(aa,)vy' + (by) Vo + (¢21) V3’. As a standard, all six factors of (vj, Vo, v;) will be positive ; but consistent algebra covers the usuai variations. Let (V) cut the plane of (ABC) at (R), the foot of the normal from * The generalized construction belongs to the older developments for centre of inertia. Its essential thought is carried back to Leibnitz by Minchin: ‘Statics,’ vol. i. p. 18 (1884). Itis given bare recognition in recent books, but is scarcely made of any important service. See Greaves. ‘ Klementary Statics,’ p. 19 (1886); Love, ‘Theoretical Mechanics,’ p. 15 (LSS@): Its control over ine factors adapts it well to “contraction hypotheses,” adding a resource there. ee of the Linear Vector Function. 547 (O) to that plane be (N), and (S) the centre of figure of the triangle. With notation OR=r, ON=n, NR=u, OS=s, -SR=k, . (8a) the quoted proposition leads to yp aren tusetk, BaP CAR eas @) where on clearing of fractions the last members revert to using the composition with one scale-factor (a+b-+¢) ; a vec- torial addition is visualized without distortion in the graph. To locate the intersection (R), take (D) in (AB) satisfying (AD /DB=b/a) ; the line (CD) contains (R), the ratio of seoments being (DR/RC=c/(a+b). Hence with given intercepts (a1, y;, 2,) the direction of (V) is fixed by two ratios ; the magnitudes of their terms do not enter. Planes parallel to (ABC) preserve these ratios, and only magnify its graph by changing the sum (a+6+c). This segregation is a working convenience. Next particularize on making (OA, OB, OC) the unit- vectors themselves. Since here (#,=y,;=2,=1), the scale- factors (a, b, c) become numerically the tensors (V,, Vo, V3). It is common practice that the latter shall include every dimensional adjustment. Distinguish the special values of (3a) as (r’, n’, wu’, s', k’), and remark that (n’=s', uw’ =k’) because (N, 8) now coincide. Choose (w'=w'w,’) at right- angies with (u’) in the present plane (ABC); (n’, w’) are perpendicular, and another correct general form for (V) follows, provided that the tensor (w’) is given proper magni- tude and dimensions : V=(V,4 Vet V3)(n’ +0) =(Vi + Vo+ V3) [n’+(0'x w’)). (5) With the indicated order of factors (n’, w’), accept the separation in the second member as conforming more closely to physical data. Then (w’) is a merely mathematical auxiliary, whose sense in its line must be governed by either one convention or the other concerning circulation ; (w') must be reversed at transition between right-handed eycle and left-handed. But should the third member register the primary record of physics through its variables und its form, the alternatives regarding assumed circulation clearly reverse (u’). As though, without changing the circu- lation-rule, we came to deal with a new vector by applying DIN? aa ad s CP Ne eh 548 Prof. #. Slate on a Graphical Synthesis the same scale-factor to (+u’), v"=(V,+ Vo+ V3) [n’ —(n' x w') | =(V,+ V.+ V3)@'—v), (6) convertible into (V) by imposing the new convention. The last member involves a different) interséction with the same (ABC), marked by vl eens, aes See Vit V.+ V3 This conical symmetry of (1, r'’) relative to an axis (n’) secures equal tensors (V, V'"’). Such opposed behaviour of linear and axial vectors towards change of cycle is inherent in separable assignments of axes and of their cycle-order. To illustrate how the above discrimination affects mixed conditions, subdivide (w’) into a mathematical auxiliary (w’’) and a physical vector (w). Let (L’, A’) express the physical facts about linear and axial contributions to (V), additive under its rule ; so that V=L’/+ A/=(V,+4+ Vo+ V3)[n’+ (' x w") | +[(VitVo+V3)(@7’xw)]. . (8) A change to the other rule, on whatever grounds executed, is to be offset in (L’) through reversal of (w’’) as a detail of mathematical routine ; the determinate physical element (w) can only persist in direction, making thus the last term negative. Then having Vel—a'; V-2W=V; WV4V) =n) ee (9) (V, V’) are mutually convertible on the basis explained for (V,V"). Their sum is purely linear, and their difference purely axial. Whatever is superficially ambiguous in really occurring cases must be removed by exploration of phe- nomena, in order to disentangle (w’’, w). Adopt (v,/3, V2/3, v3/3) for coordinates of (N), and resolve (w’) of equation (5) as shown by fa’ 0): ae W! = )V, + WeVn + Wag. 2 ee LO) The suppositions include perpendicularity of (n’, w’), which implies one negative tensor or more in the last equation. , The group (Vy, V2, V3) and corresponding components of (V"’), written in parallel expansion on either background of the Linear Vector Function. 549 outlined under equations (5, 6, 8), appear as Vi =3(Vi4 Vet V3) [Vy + (ves — v3) Vi | 5 =2(V, + V.+ V3) [¥) — (vgs —- V3) V; | 3 qi) etc., etc. The insertion of the unit-tensors (v, vs, v3) preserves the formal type ready for algebraic evaluation. These pre- liminaries have broadly suggestive aspects: the conversions of (n’) into (V, V’, V’’) are after the pattern of stretch- operators and quaternions, for instance. But equations (5) to (9) also place a patent emphasis upon taking due © account of axial combinations with the operand-vector, if (V, V’, V'') become generally representative of linear vector functions and of relations among them. This is for the present our chief concern. “Return accordingly to equation (4) and repeat the above sequence, modified only as that more general form demands. Begin with the arbitrary partitions, ‘which may finally be algebraic sums, @=A+A.3; b=d,4+b.3; cH=ceyteg; . . (12) and express (V) to match their terms: V= [Cav + iy) V2 + (e121) V3] — | (aoX,)Vy + (bsY/1) Vo + (Ca¥ )v3 | = Q, + Q.. (13) The components (Q, @,) are given by means of the same intercepts; their graphs utilize the same plane (ABO), though scale-factors are varied. Make equation (4) model and put Q,=(4, +0) +6) = (4 +0,4+¢)(a4+u)=(q 4+ 6+) (8 +k, ‘ (14) ) Q.= (a, =e by + C,)¥o= (do+ b, -+ Gy) (n+1,) — (dy + bs + Cy) (s aa k, ; The arbitrary character of (Q,, Q) enters their geometry through the elements (U,, U2, Kk), k,), since (n,s} belong to both. We can treat (r, 1), rg) as making réal intersections with the plane (ABC), whose equation their extremities satisfy, which yields useful corollaries. The negra o- 100 of (Q,;, Q,), by applying to the second members the general construction detailed specially for equations (3, 4), is “simple to verify. The third members reproduce the scheme of Wl 550 Prof. F. Slate on a Graphical Synthesis equation (5), and justify as general expressions Q,=(a4+6, + ¢}) [n+(nx w) | | Q.= (a, + by + C2) [n+ (n x w.) | aie! where (w,, w.) successively replace (w’). Obviously, the ideas underlying equations (8, 9) continue into this phase, which enables us to indicate compactly a conversion of (n) into any complementary constituents of (V). And when a mean vector (w) has been determined from (15) (at+b+c)W=(a,+ 61+) Wy + (dg + 62+ ¢2)We; U=nXw; : (16) (Q,, 2.) can be summed into V=(a+b+c)[n+(maxw)]. . . . (17) But an attempt at similar general conversion of (s) into (Q,, Q, V) halts at the possible obliquity to (s) of (ky, k,, (k)). The situation that involves mean vectors (8, w) to this extent dislocated, remains for analysis after taking out three particular cases. First, the mean vector (w) may prove to vanish; (V, n) become colinear. Or (s) can be perpendicular to (k), though oblique to each separate part. Or again, (s) can be perpendicular to both (k,, k,), yet not coincident with (n). If for the then colinear sum the relation among tensors be added: hy (a, +b, + ¢;) + ko(ag tbo +e,)=hk(at+b+c); and k=sxq, (18) with permissibly oblique factors, the type of equation (17) is retained by V=0,+0,=(a+b6+c)[s+(sxq)], . . (9) whose peculiar lesson is more fully read later. Under natural guidance of several aspects in the foregoing results, reserve one component (say (Q,)) as stated for equa- tion (13), and transform (Q,) for general discussion through the equivalents ’ (9%) Vi = (Y1V2 + 21V3) xX (Y,V_ + Z1V3) = (4 ZL, — 21 Yi)vy 3 (botf )V2= (21V3 + @V1) X (ZoV3 + Xov1) = (4 X2— 21 Zp) Ve 3 > (20) (¢921)V3= (avy + Vo) x (X3v, + Y3Vo) = (2 Y3—Yy1-X3) V3. The vector products employ first factors that are distances from the lines of (vj, v2, V3), and that combine significantly of the Linear Vector Function. HOE as projections into the radius vector linking with equa- tion (4), XV} + y1Vo + 21V3= 38. oF tte nparens (21) This diagonal from (QO) of the parallelopiped (a;y,21), deter- mined by given axis-intercepts, takes prominent place among the data. The second factor in each vector product is seen to specify legitimately a vector parallel to a coordinate plane, 2, 4, Xs, X;, Y;) being adjusted positive magnitudes. Because (Q,) is arbitrary, the third members must be inde- pendent ; which bars their general union into one vector product of which (3s) is a factor. Observe that the specifi- cations parallel to the coordinate aves in this group amount in the aggregate to (X,+ X3)v, +(Y1+ Y3)¥ot(Z,+ Zojvz=A. . (22) Further, since every such summation by planes effectively takes each component by axes twice, the last requirement can be met as a total with cS + 4,+Z Beste, es (23) assigning components to axes. On the face of it, this sub- stitutes an average, and effaces some particulars of equation (20). The latter can be summed into these terms : Q,=[ 3s x A]—[(1Z.— <1 Y3)v1 + (21X3—2Z,)Vo+ (a; Yi y Xe)V3| 5. (24) an important result that we put into the condensed notation, Q.=M+Q,'; Q@+4+@/=M; Q@=M+Q,"; Q,'4+0,"=0; (25) where the double sign has been made to allow (M, Q,) as alternates for the resultant diagonal. Evidently (Q,’) admits a new sum of vector products : Q,' = [(yi:Vot+ 21V3) x (Y3v. + Zvs) | + [(G1v3+ @,¥1) X (Z,v3 + X3v;1) | + [| (@v,+y:V,) x (Xov,+ Yive)]. . - (26) The first factors throughout (Q,, Q') are identical, and the average fixed by equation (23) reappears; though in general the distributions differ, since the second factors of (Qs, Q,') contain pairs that are complementary within (A). Con- tinuing to coordinate the graphs through the plane (ABC), 502 Prof. F. Slate on a Graphical Synthesis other scale-factors applied to the same (a, 4, 21) give Qo" = (ag! vy) Vy + (bey) Vo + (C521) V3 = (ag! + be! + 05!) 19" 5 27) M=Q, — Q," = (a,—ay!)vyvy + (bo — ba!) yo + (Co — Ce!) 21V3- Thus the vanishing of (M) depends upon the equality in pairs of tensors for the components of (Q,, @,’). Also (M) serves as a measured consequence of unequal scale-factors there, on reduction to the same (ABC), through whatever circumstances such differences come into question. The indications of reversal by equations (2) have pruma facie validity. Yet on proceeding to the algebraic equiva- lents for (—V) in equations (3), an issue of fact will never- theless be raised: Whether phenomena harmonize with making freely exchangeable, as related to the first forms, the specific factorings in —V=a(— 2V,) + O( Vy) + (As), ; (28) ~V=(—a)(w1v1) + (—8)(yrv») + (—O)(er¥8)- Where physical properties sanction the first equation, they legitimize at once unaltered scale-factors, 1o be partners of what the reversed vector (—3s) stands for mere widely. This holds for coefficients that belong to a line, and not to one direction in it; moment of inertia and light-speed in crystals are instances. The other equation refers more at first-hand to opposed states like stretch and squeeze. The proved precision of either equality Jeads to decisive infer- ences in other directions. The consideration of such matters, for which equations, (13, 14, 25) may offer a starting-point, bears upon classifying linear vector functions, especially as regards conjugate pairs. The operand of the function that here builds up (V) was indeed defined particularly by equa- tion (21). However, with (#, y;, <;) of equations (3) selected at will, and adopting the superpositions proved for principal axes, the procedure enlarges itself at this stage into treating functions of ,any vector, whose ecmponents (@V1, Y1V2, 21V3) will then be chosen to represent. Primarily by using for the operand some one scale-factor (planes parallel to (ABC)), though the algebra may be shaped to break down that restriction. The developments are adaptable to whichever (3s) the working basis best yields, bearing in mind about two differing scales, that either may be norm for the other’s distortion. It is true that equations (14) are perhaps simpler for (n), another standard parameter of a plane. Still the initiative at (s) or (n) will always be weighted in favour of expressing physics; a mathematical gain is less fruitful. A thought that governs, too, the parallel] sequences of the Linear Vector Function. D093 within ellipsoidal geometry that have the same graphical origin, and that have long been exploited. The derivation of (M) has led to an invariant form, that makes (Q@,—4Q,’’) necessarily perpendicular to the mean vector (s), the second factor in the vector product being their common value of (A). Otherwise the invariance is to be delimited in the light of equations (14, 27). Therefore every combination whose net sum is expressible by (M) must at least be self-cancelling in the line of (s); the factor (A) preserves for (M) a margin of flexibility proper to all vector products. Observe how well all these features align with the specialized equation (19). This mode of statement adapts (M) to embody the dominant difference between a pair of “conjugate vectors,’ according to standard usage of that term. For if we define a companion to (V) by . V'=V—M=@,+Q,”, chine eR od. e's (29) (Q,) has been replaced with (Q,''). Then bring out further the mutually supplementary relation of (V’, V) to (M) in the symmetry of V—3M=Q,+ (Q)" 43M); V’+4M=Q,+ (Q—3M); (30) and compare with the usual established forms. By reference to equations (13, 20, 24), the nine coefficients in each set of the rectangular components for (V’, V) can be picked out, and their interchanges recognized at sight, which render this pair conjugate, (Q@;) being common to both. Proceed to note that 2(V+V')=O0,4+5(04+4"); 3°V—-V')=3M; (81) showing complete formal resemblance to equations (9) through the obvious identifications with (L', A’) in the latter. The application is quite direct, if inquiry has isolated an intrinsic axial constituent, as in electro- magnetism; this treatment then acquires clear physical status immediately. But though equations (20) are less vital, if they merely prepare for deviations from equa- tions (28) which simulate that experimental trend towards composite vectors, the artifice does not lose its value. It creates a widely comprehensive scheme*. What correlates * The range of the plan can be broadened usefully to cover a “@omplex”’ of non-homogeneous vectors, like forces and couples in a forcive impressed upon any system as related to the centre of inertia. Fusion into some analogue of screw-motion would follow easily. The track pursued has several contacts with Poinsot’s force-transfer, which likewise turns upon shift of vectors to secure lever-arm. Our moment- vectors. (Qs, Qo, M} drawn at (O) presuppose (A) “off centre.” The conditions underlying equations (84) below are not altogether remote from those of a couple. dd4 Prof. I. Slate on a Graphical Synthesis both typical situations rather naturally under one point of view is the essential nexus made explicit for equations (2) : Reversal of axes with retention of their identity (so to speak), makes abandonment of the original cycle inevitable. It is plain from what precedes how that mode of reversal would be carried through to any valid equivalent of ail a V) ar [ (aya) vy’ ali (by) Vo" + (¢124 )v3" | + [ (221) ¥1! + (bey) Vo + (6021) V3], . (82) intentionally emphatic of original direction as given by equation (1). Similarly, lay down (v,’, v2’, vs’) as original umt-vectors for (+V’), and throw, all necessary changes of scale-factor into the second term by compensations in the other term. Reversal now substitutes (v,, Vo, V3) and gives —(+V')= [(ay)vi t+ ry) V2 + (421) Vs | + [ (ay'at1) V1 + (b2'y1) Vo + (¢2'21) V3]. « (33) Reference to equations (27) makes it plain that addition of equations (32, 33) hag for result the negative of (M). The terms of the comparison inverted lead to (M) itself ; the thought is the same and the end reached. Finally, it is important for the context to remark how (M) can change sign, the order of its factors being untouched. First by reversal of either factor separately ; or secondly, by reversal of both factors, and transition to the other circulation-rule as well; or thirdly, by change of rule alone. Other bearings of the last possibility, on reversal of (Q», @,'’), are evident. The device in equations (20) is prompted by the an- nounced purpose that prefaces equations (12). But at this point its consequences are available to round out the method and also add some particulars about (M). Whenever (Q,) is parallel to (ABC) by choice or necessity, (r,) loses definite- ness as first defined; every such case, however, is still tractable by means of a difference of two vectors drawn from (O). Let these be (8,, 8), on which (ABC) makes intercepts (8), 8); the sum-diagonal ($,+8,) cuts that plane at (p;), while the difference-diagonal taken in the sense that gives its parallel at (O) a real intercept (p,) 1s (S:—§,). Consistently, within the generalized construction, Sj =78;; S,= 823 8, +8,=(m+%2)p:; S:—So=(%1— 22) po. (34) of the Linear Vector Function. 550 So long as complete coincidence of (8), S,) is excluded, the last expression cannot vanish. It is indeterminately satis- fied for (mj=72.; pe=«), which at once justifies the proposition and adds its necessary condition. It is to inspection true that a plane parallel to any difference- diagonal cuts off equally proportional segments of both components, and of their half-sum. In illustration, inter- pret what comes of combining equations (14, 25) and dividing by (ag+6,+¢g) : = | Q, Q," " aad —p= ay roe la 4 Ag +bytey GED, eae dy + bg + C5 Pp (s+ kz) (s 7 Ug ) (39) The vector (p) will be both perpendicular to (s) and parallel to (ABC); (Q,, Q,’") neutralize each other’s components in (s). If (u,’’) completes a ee gt decomposition of (k,) in (ABC), it will lie in (SN) of (3a); but (p, uw”) may be oblique. Of course the ea tromber sis) lid throughout two groups of vectors, subject to a proviso of common difference between pairs. And again there is brought forward a reminder of equation (18). Return to equations (27, 29, 30), whose parts (Qs, Q,'’) are less special than those supposed in equation (35). When (a9 + by + C2), (dq +b +¢5') are unequal, a common divisor does not “reduce (Q,, Q,'’) to the plane (ABC)”; some analysis of the latter relations is needed. Let the projec- tions of (12, 12’) have tensors (#2, ys, 22), (#2, Yo» 2%): Equating equivalents for each of (Q, Q,’), record the consequences (as + bo —- Cy) &g = Ao#, 5 (az + b! + Cp) ia = Ao @;, ste. 5 yh, , a I! al Me is, Ao + bs + C9 = Ago oes boo pe Co<9 = =r =e a 36 Ny Che! 2 be! ali iy Ay! ko bo" ys C5 Zo ; (36) Ayal! + boys! + Co29)! Er dal ty + dg! Yo + CQlzZ <2 whose second and last members connect interchange of scale-factors with the ratio (m/m,"’). Because change of cycle cannot neglect it, that crossed pairing of intercepts (T2, ro’) and Seales icron: (ms, m,'') should be examined in some ‘detail, and its bearing upon conjugate relation extracted. To this end, begin with two general vectors (S,, 8,) drawn 956 Graphical Synthesis of the Linear Vector Function. from (O), and let (R,, R,) be given by their inversion; so that in notation like equation (34), Ry ==N5S) 5 R, = 1485 5 4(R, am R.) == np £(R, —R,) = (m— 71) py! The mean scale-factor (n) is common to (§,;+8§.), (Ry + Re) sa but (p,, p;') differ in general. Omitting steps of plain reduction, the working out shows an Cu £(S, +8,) +4(R, + Ry) =$(8, + Ry) + $(S, + Ro) =n(Si +82) 5 ) 3($,—S») + 3(R, —Ry)=3(8; + Ry) —5(S2 + Re) ms n($1—S») 3 3(S; + 8) — 3(Ri + Ry) = 3(S8;—R,) + 3(S2— Ry) | : = M6, 5) 3(8, —8,) —3(R, — Ry) = 3(81 —B1) — 3(82— Re) | _ ore al (38) It is almost intuitive that the mean factor (nx) should corre- spond to the mean vectors in the way that these equations confirm. The second and third are interpretable at sight in their last members as expressing colinear vectors parallel to the plane (ABC). Denote these respectively by (2D,, 2D,) ;. it is clear that a changed sign for (ng) interchanges them. In terms thus chosen, it follows that S;(—D,+D,)=R,+ (D,+D,); S.+(D, —D,)=R,—(D, ~ D,); (S, —R, — 2D.) + (8, —R,—2D,) =0; (So+R,+2D,) — (8, + R,—2D,)=0. (39) Tf next the extra condition be attached that (s,;—s,) shall be perpendicular to (s), the symmetries of equations (39), in comparison with those of equations (29, 30, 31), make corre- spondence permissible of (M), either with (2D,) or with (2D,), according to the actual occurrence of the positive sign for the scale-factors. In this scheme, equation (35) is easy to locate as a special combination ; and equation (19) finds its place. The last of equations (39) bring to light the well- known “symmetrical average function.” The important part played in all these developments by the mean factor (n) must be explicitly recognized. That is intimately associated Contact Potential and Thermionic Emission. Oe with its necessary (automatic) appearance in (np;, np;'); it is one main connectine link between the original vectors and those obtained by inversion (crossed pairing). The perpen- dicularity of (IM, s), requiring equal orthogonal projections upon the line of (s) for the vectors (Q:, Q’’), considered by itself demands no more than equality of scale-factors applied to the same seginent of that line. The consideration that here precedes makes the selection (ms) natural. The material of this particular discussion traverses familiar ground, so far as the final facts about conjugate vectors are concerned. But the line of attack there formulated is capable of deducing novel results elsewhere. University of California. LXIV. Contact Difference of Potential and Thérmionic Emis- sion. By O. W. Ricwarpson, F.R.S., Wheatstone Pro- fessor of Physics, and B.S. Ropertson, ML.ELE.; Lecturer in Llectrical Engineering, University of London, King’s College *. 7 ; has been pointed out by one of us f that the following relation should subsist between the contact potential difference V between two surfaces at the absolute temperature. T and their thermionic electron saturation currents 2, i, per unit area at the same temperature — 7 le ON (1) @ ay In this equation, & is Boltzmann’s constant and e the elec- tronic charge. Observations which contirm the validity of this equation, approximately at any rate, have been made by us in the course of some experiments with a thoriated tungsten filament. The filament was of tungsten containing 1 per cent. of thorium and was kindly supplied to us by the Tungsten Manufacturing Co. Ltd., 231 Strand, London. It was 3:0 cm. long and 0-100 mm. in diameter, and was mounted axially in a cylin- diical glass tube and surrounded by a coaxal copper anode. The general arrangements for evacuation and for controlling the temperature of the filament, etc., were similar to those deseribed in our paper on the effect of gases on the contact difference of potential between metalst. After baking out. * Communicated by the Authors. + O. W. Richardson, Phil. Mag. vol. xxiii. p. 265 (1912); and ‘ Emis- sion of Electricity from Hot Bodies,’ p. 41. Second Edition (1916), t Phil. Mag. vol. xliii. p. 162 (1922). 558 Prof. Richardson and Mr. Robertson on Contact the tube, exhausting with the mercury-vapour pump, etc., and heating the filament for some hours, it was found to have a fairly stable characteristic curve in the neighbourhood of that to the left-hand side of fig.1. Later on, the emission was found to have risen to an enormously higher value, and was ue I. : Sat fi BCC a i Tees i a | CJ a a 420 & ° ~~ IGAUVA! a TER | | | 12 8 a= o “b+ 4 8 42 16 2 ry right outside the scale of the available measuring instruments with the same temperature as indicated by the resistance of the filament. The emission was now lowered by reducing the filament temperature to a value comparable with that ruling before the change occurred, and the characteristic was now found to be stable in the neighbourhood of the right- hand curve in fig.1. This displacement of the characteristic Difference of Potential and Thermionic Emission. 559 corresponds to a change of 0°71 volt in the potential difference between cathode and anode, and is in such a direction as to indicate that the filament had suddenly become relatively more electropositive to that extent. There seems no reason to doubt that most of this chanze in the contact potential océurred at the heated filament. Assuming that the whole change is at the filament, these observations furnish the material for an approximate test of equation (1). The test is only approximate in any event because it was not possible to ascertain the temperature of the filament with accuracy. When its inverse resistance was 0°521 ohm7™?, corresponding to the left-hand curve in fig. 1, its temperature was estimated with a Cambridge & Paul Optical Pyrometer at 1583° K. Another estimate ean be obtained if we assume that it was giving the pure tungsten emission at this stage. Using K. K. Smith’s tungsten data we calculate T=1450? K; and using Langmuir’s T=1507° K for this limit, the average of these three esti- mates is 1480° K. The emission for a resistance of (°521)7' after the rise was too large for us to measure with our instruments, but was obtained by extrapolation. It was found that the currents at various values of the inverse resistance were as follows :— po 2706 X10") amp.)...... NO Zee 21-0 45 Oi-be “Gore B18 te ee HO S230) G53" | F962" 2218 9-508 T=) sn) re ie 64) 740), 726.704 +698 The relation between the Jogarithm of this current and R71 is found to be linear, and trom the plot it appears that the value of logit. for 1/R=°521 is 5°96. The value of logit, for 1/R="521 was 2°62. Substituting the known values of & and e and the experimental value of T, viz. 1480° K, and inserting the factors for volts and natural logarithms, it appears that these data give V=0-96 volt as compared with the measured value 0°71 volt. This is probably as substantial an agreement as could be expected, considering the uncertainty of some of the data. We are engaged in further experiments which it is believed will eliminate these elements of uncertainty. The difficulties, however, are not inconsiderable, and it is clear that some time must elapse before further.results of value are obtained. A reference to these experiments was made by one of us in his Presidential Address to Section A of the British Association at the Edinburgh meeting, 9th September 1921. PM 560°] LXV. On the Deformation of the “ Rings and Brushes” as observed through a Spath Hemitrope. By B. N. CuuckEerBUTTI, M.Sc., Assistant Professor of Physics, Calcutta University * hd [Plate IX.] 1. Introduction. NRYSTALS are frequently found which are obviously of / a composite character, that is, are composed of more than a single individual crystal of the same substance and in which the parts belonging to different individuals are united in a definite and regular manner, the peculiar mode of union being characteristic of the substance. The true nature of the composite structure is often betrayed by the presence of re-entrant angles, forming notches (as in Diamond crystals), arrow-head shapes (as in Gypsum), knee-shapes (as in Cassiterite), cruciform (as in Staurolite), or heart- shapes (as in Fluorspar) ; but not infrequently the two or more individuals are so intimately blended, that the ap- oe at first sight is that of a single individual crystal. The law defining the manner in which the composition occurs is aa stated as follows :— The two individuals are first supposed to be arranged parallel to each other, with one face of each in mutual con- tact, and then in order to produce the twin, one individual is supposed to be rotated to 180°, upon a ‘plane, which is called the twin-plane, and about the normal to the plane as axis, which is called the twin-aais. But it is necessary to point ont here, that in the actual process of the forma- tion of the twins, no such two different crystals are brought into contact and then one turned through 180°, as stated above. The orientation takes place amongst the erystalline molecules of their own accord during the process of for- mation. Hence, the twin-plane is not necessarily always the plane of contact. In general, the plane of twinning may be any actual or possible face, except obviously a plane of symmetry of the crystal. The twinning may be either of Primary or of Secondary origin. The former refers to the case when the formation takes place of itself without the intervention of some external agency during the process; but when the twin- ning takes place subsequently to the original formation of * Communicated hy the Author “ Rings and Brushes” in a Spath Hemitrope. 561 the crystal or crystalline mass, by means of external pres- sure, it is called secondary. Closely related to the cleavage- direction in their connexion with the cohesion of the molecules of a crystal, are the Gliding planes (Gr. Gleit- flachen) or directions parallel :o which a slipping of the molecules may take place under the application of an external force. ‘This molecular slipping may be attended by a rotation through 180° of the molecules and the resulting twin formed. | According to the mineralogists, Iceland-spar belongs to a class of crystals which always produce twinning lamelle, that is, one twinning plane in the case of this crystal is always followed by another at a very short distance so that a thin layer of similarly oriented molecules is formed. These are found abundantly in nature and, moreover, the pressure upon the cleavage-iragment of Iceland-spar results in the formation of a number of thin laminz in twinning position to the parent mass. Secondary twinning lamelle are often observed in natural cleavage masses of calcite. The explanation of the mechanism of formation of twin erystals is rather difficult. Lord Kelvyin*, however, has touched upon the problem. He has given a very lucid exposition of the whole process of crystal-building from a solution. In the particular class of crystals producing twins, he considers the constituent molecules symmetrical on the two sides of a plane passing through itself and also on the two sides of a plane perpendicular to this plane, that is to say, his crystalline molecules are egg-shaped. A real erystal which is growing by addition to a face would give layer after layer regularly. But, if by some change in internal circttmstances, the molecules that would go to the formation of a layer are all oriented to 180° with respect to the molecules that formed the previous layer, a twinning plane is formed, and if the remaining layers form in the same way as the last mentioned one, then we shall get a erystal having two different parts separated by a twin-plane between, If, again, the process of orientation continues only to some layers and then, due to the re-establishment of the initial conditions, the layers form as at the start, we shall get two portions to the two sides with similarly oriented molecules, enclosing a thin layer in which the molecules are turned through 180°. The case will resemble that of Iceland- spar. But the main difficulty in the explanation is to under- stand what is that change in the circumstances that causes the erystalline molecules to turn through 180° in the process of * Baltimore Lectures, p. 629, Art. 37-39. Phil. Mag. 8. 6. Vol. 43. No. 255. March 1922, 20 562 Prof. Chuckerbutti on Deformation of “ Rings erystal formation, and how is it in the case of Iceland-spar that the intial conditions are re-established after a thin layer is formed under the changed conditions. The difficulty is still greater when we come to consider the case of repeated twinning as in the case of potassium chlorate crystals, where it is found that a very large number of twinning layers may be formed with a most surprisingly regular periodicity and constancy of thickness. Thus there is a good deal in regard to the mechanism of formation of these twinning layers that is as yet but imperfectly understood. 2D. Optical behaviour of a Spath Hemitrope. On looking at a source of light through a twin crystal of Iceland-spar, generally it is found that three images of the source are formed, the central one of which is always sta- tionary. If the source of light is an incandescent electric lamp, then the images are beautifully coloured, the nature of the colour changing with the orientation of the plane of the erystal to the incident beam. Itis also observed that the two outside images are polarized in perpendicular planes. On rotating the crystal, the two outside images are found to rotate about the central one, and in the course of a revolu- tion there are positions in which both of them disappear, the remaining one becoming the most brilliant for the time. The author has also found that when a beam of light is allowed to fall upon a crystal cut perpendicular to the axis and also polished in such a way that the lines, in which the twinning lamina cuts the planes of the main crystal, are on the surface, some interesting diffraction effects are observed. Fringes are noticed in the region of the transmitted light which show a remarkable asymmetry. The edges of the lamina also appear to be luminous. The phenomena appear te belong to the class of laminar diffraction effects, but there are certain features regarding them of which the explanation is not clear. The author is at present engaged in a fuller study of these effects, and the investigation men- tioned above will probably prove of considerable interest in relation to the determination of the optical nature of the twinning layer. On examining the crystal with convergent plane polarized light and between crossed nicols, the ordinary “rings and brushes” are found to be distorted, the amount and the nature of the distortion changing with the orientation of the crystal. However, in some cases, the distortion more or less resembles that produced by the superposition of a and Brushes” observed through a Spath Henutrope. 563 quarter-wave mica-plate upon a erystal of Iceland-spar cut perpendicular to the axis, so that the plane containing the optic axes of the mica-plate makes an angle of 45° with the vibration planes of the crossed nicols. The explanation of the phenomena of refraction has been given in text-books on Optics*. There, the formation of the three images and the polarization are all explained. The disappearance of the outside images in certain positions, as stated before, is connected with ie orientation of the optic axis of the intervening layer to that of the main crystal. The problem of the form of the rings and brushes through a twin crystal has been treated mathematically, but the case has always been considered where a twinning plane separates two crystals. The problem of the two similar crystal wedges separated by a thin twinned lamina inserted between them in an inclined position appears not to have been solved, and an investigation in this direction forms the subject-matter of the present paper. 3. Explanation of the Photographs. The crystal that was employed in the following experi- . ments was cut perpendicular to the axis, and the portions free from the twinning layer showed the ordinary ‘“‘rings and brushes” peculiar to erystals cut in such a manner. The angle which the twinning plane makes with the faces of the crystal is determined by observing the marginal outline of the plane as seen on looking through the ae of the erystal. The Hemitrope, as examined with tourmaline tongs, shows very beautiful distortions and changes in the usuai circular ring-system. It is difficult to photograph the successive stages completely, and it is found that the changes repeat themselves four times during a turn of the crystal through 277, when the tourmalines are crossed,—the position for which most of the observations were made. The rings lose their circular nature altogether, and they will be found to be elliptic as if pressed up and down or sideways. Moreover, the change due to the introduction of the thin twin plane is most prominent near the central part of the system. These changes at the centre mixing up with the crosses make the central portion look rather cumbrous. Figs. 1 to 4 (Pl. IX.) represent the changes for a rota- tion of the crystal through 7/2, when the analyser has its * Mascart, Trazté d’ Optique, vol, ii. p. 192. 202 564 Prof. Chuckerbutti on Deformation of * Rings plane of polarization vertical and the polarizer horizontal. In fig. 1 the crystal is placed so that the twinning layer extends vertically up and down, inclined in such a manner that its sections with the faces of the crystai are vertical straight lines. Light is always allowed to fall normally upon the first face of the crystal, as otherwise the three images of which mention has been made before will make the system more complicated. Figs. 2, 3, 4 are obtained by rotating the crystal to the left in its own plane till the rota- tion amounts to 7/2, when the system corresponds with fig. 1 but rotated through 90°. As we proceed further beyond 7/2, the figures 2, 3, 4 come in the reverse order, fig. 4 being followed by figs. 3 and 2 and at m7 we get back to fig. 1. On rotating further, figs. 2, 3, 4 come in turn, similarly as in the first quadrant, and the figure at 37/2 corresponds with the one at 7/2. The order of succession of the system along the last quadrant corresponds with that of the second quad- rant. Vigs. 5 to 8 were obtained by rotating the tourmalines in some position other than the crossed and at the same time rotating the crystal also. In general, the systems are rather cumbrous in these cases and the four cases are selected out, as showing the changes rather systematically. Figs. 7 and 8 have the peculiarity that they are complementary in nature to figs. 4 and 1 respectively, although they were obtained when the tourmaline planes were not parallel but inclined at an angle of about 45°. 4. Physical Theory. In general, if 6 be the path retardation of two rays due to their passage through a uniaxal crystal the equation to the generating curve to the isochromatic surface is { (o”— Me ly? + 0° j= Aug O°(a? +’), where the optic axis lies along the axis of X. In case of a regular crystal of thickness T, the section of the isochromatic surface by a plane ec=T gives the equation tor the rings, and the points giving the same path-retardation are equidistant from the centre of the system. Now the interposition of the thin layer which again has its optic axis inclined to the optic axis of the main crystal, causes different path-retardation upon the interfering rays, so that the points of equal path-retardation are not equidistant from the centre, and as a result we should expect distorted curves. and Brushes” observed through a Spath Hemitrope. 565 Let lE+mn+n,6=1, lE--mn+n6=1, be the equations of the two waves in unit time after passing through a point o of the surface of the erystal. | [, and n, are the reciprocals of the intercepts made on the normal (axis €) to the plate through the point o by the refracted waves in unit time after passing through o.| If @ be the azimuth of the plane of incidence with respect to that of €f and v be the velocity of ight in air, then sinz. cos @ sinz.sin 0 MS v v , where z=angle of incidence. Suppose, that the plane €€ contains the greatest axis oz _of the ellipsoid of polarization, the plate being on the posi- tive side of €, then if ox, oy, oz be the axes of the ellipsoid and zof=y, we get by iransformation from the axes of optical symmetry to the new axes, x=E&cos ¢ cos y—7 sin 6+ € cos g sin x, y=Esin d cos y—n cos $+ € sin G sin x, e=—E&siny+€cos y. The equation to the wave surface, referred to the axes of optical symmetry 1s ana b?y? . C22? G = 0-7 1G @e6-) ao" = C" From the condition that the plane /E+mn+n€=1 should touch the wave surface in the new system of coordinates, we find Ve—a'sin?i (a?—c) sin x cos x cos O sini a a” cos? y +c" sin? y — — q 7, uf VW (a2 cos?y + sin? y) (v? — ce? sin?) —c(a? —c?) sin’ . cos? . sin”, a” cos’ y +c? sin? v where 6=vT(ny—7). Hence, for a crystal cut perpendicular to the axis (y=0), 0 ) = 4 Nv? —¢? sin? 1 — r/v?—-a? sin? 7| / a. ie . i | | a ea er = itl 6 ye ee 566 “ Rings and Brushes” in a Spath Hemitrope. Let T, ¢ be the thicknesses of the main crystal, and of the thin twin layer respectively, and 6, and 6, be the corre- sponding path retardations. In the present case, if we assume that the twin layer is also cut perpendicular to the optie axis, then from the angle (30°) at which its plane cuts the faces of the crystal, we must have y=60° for the thin layer. So that the expression for the retardation in terms of the refractive indices takes the form, 2 2/2 ==G? 2 <é ae ve Sin? . cos 8 a / et —3pe? —4 pry sin? 2 cos? 6 Me 2 pio Mo” + Bpe’) For the ae which fall normally upon the face of the crystal (2. e., the rays which make an angle of 30° with the ial to tue fami layer), 6o=t {13186 +0°05 cos 0 + :0028 cos? O} and 0, =T 9 {Mo Sof: Moreover, cl = L1L0 | = eee aa Ko fe where i,;= angle of incidence (very small). The thickness of the twin layer as seen under a high power microscope and estimated with the help of a micro- meter eyepiece is about 2X. So that 6,= 2A(1°3186 +. 0°05 cos 8+ 0°0028 cos? 6). Now as the azimuth angle increases from 0° to 90° the value of 6, diminishes. On the other hand, 6, goes on increasing as the angle of incidence is increased, and is constant for the same angle of incidence. Tf r be the radius- Teotor of the interference minima or maxima, then we may put 7 = k( 6, + Soy) where k isaconstant. Hence we can expect the elliptic form of the curves, the radius-vector for an azimuth otf 90° being the least. Moreover, at a distance trom the centre where 2 is great, 6, will be very great in comparison to 62, which latter may then be safeiy neglected, the radius-vector in those cases being given by 7?=k6,. Coupled Vibrations by means of a Double Pendulum. 567 Another important feature of this thin twin plane is the asymmetric laminary diffraction* pattern to be observed near the two edges of the twin plane, with a beam of light incident normally upon the surface of the crystal. This problem, however, is under investigation. The experimental work described in this paper was carried out in the Palit Laboratory of Physics and the writer wishes to express his thanks to Prof. C. V. Raman for kindly providing all facilities for work. University College of Science, Calcutta. 24th August, 1921. LXVI. Coupled Vibrations by means of a Double Pendulum. By A. L. Narayan, M.A.T [Plates X. & XI. ] ae the production of high-frequency oscillations .re- uired in radio-telegraphy, one of the most important ican in the oscillation circuit is the oscillation-trans- former, which is simply a magnetically-coupled oscillating circuit possessing a pr ‘imary and a secondary each containing inductance and capacity in series. In order to elucidate and interpret the theory of these electric circuits and illustrate how, when oscillations are excited in one circuit, these two circuits act and react on each other so that the result is that oscillations of two different periods are set up in both circuits, one greater thau and the other Jess than the natural period of either circuit taken separately, mechanical systems were described by Prof. Thomas Lyle, Phil. Mag. vol. xxv. pp. 567— aeecepril 1913) + Prok. aon and Miss Browning, Phil. Rien vol. xxxiv.. ps 246 (1917), vol. xxxv. ip. 62 (1918), and vol. xxxvi. p. 36 of the same year; and Mr. Jackson, Phil. Mag. March and September 1920. In this paper the author describes an entirely different kind of mechanical system, which gives results very similar to those cbtained by Prof. Barton and Miss Browning, and Mr. Jackson, in the papers referred to above. At the same time, the whole arrangement is more neat and much simpler than the very elaborate one adopted by Mr. Jackson, and the method of photographing the vibrations in the present case is essentially different from that of the previous authors. The coupled system treated in these pages consists, as shown in the two diagrams (PI. X.), of two rigid pendulums A and B, Fob. N. Ghosh Etoeinoy0c.A4 yok xcevi., (1919). + Communicated by the Author. - 568 Prof. A. L. Narayan on Coupled Vibrations of which A consists of a steel rod furnished with a heavy bob which can be screwed at different points so as to vary the moment of inertia of the system, and it can turn freely about a horizontal axis by means of a knife-edge which ean be fixed at different points—thus varying the point of suspension also. Sunilarly the pendulum B consists of a steel red and a heavy bob. and it is suspended from A by means of a highly polished steel knife-edge resting in a V-groove of asmall steel bracket which can be screwed at different points along the length of A. Thus the degree of coupling can be varied at will. The paper includes twenty photographic traces of the motion of the pendulums under various conditions. The method adopted in this case for photographing the vibrations is wholly different from that adopted by Prof. Barton and Miss Browning and Mr. Jackson in their various experiments on coupled oscillations. Hach of the pendulums A and B carries (as can be clearly seen from the second photograph) a small galvanometer mirror of about 1 m. radius, with its plane perpendicular to that of the vibration of the pen- dulum, and the mirror can be turned (by means of a simple mechanism) about a horizontal and a vertical axis, for final - adjustment to get the two reflected spots exactly one above the other on the slit of the camera. When the pendulum vibrates, the mirror mounted on it rotates about the axis of rotation of the pendulum. taser 42°6 1601 "452 BO)! A) ict 49°5 1757 "D5 1 ee ert ae 52°8 1:860 6 Figs. 1 to 20 (PI. XI.) are reproductions of the photo- graphs obtained with the above apparatus under various conditions ; figs. 1 to 10 represent the oscillations of the DT4 Coupled Vibrations by means of a Double Pendulum. system when the upper bob is displaced and the lower bob free, and figs. 10 to 20 represent those of the system when the lower bob is displaced and upper bob free. The figures clearly show the effect of progressively increasing the coupling ; the first figures, in which the coupling is loose, show the phenomenon of beats. Thus the coupled system treated eee presents an exact mechanical analogy to the case of twe mutually interacting electrical circuits ; and at the same time it is much simpler, © and therefore a more convenient model by which the pro- perties of the electrical case can be demonstrated to a class or anaudience. Toa large audience it may be much better demonstrated by ailewing the reflected spots of lght to suffer another reflexion from a mirror rotating or oscillating about a vertical axis and then projecting the spots either on a screen or on a wall of the room. With a modified form of apparatus, experiments have been carried out to illustrate the theory of three mutually inter- acting oscillating circuits, and the mathematical theory of the mechanical system ‘triple pendulum ” is developed ; and these results will be submitted for publication, shortly, in another paper. Abstract. The author, following Frof. Lyle, Prof. Barton and Miss Browning, and Mr. Jackson, describes an arrangement of mechanical systems as an éxact analogy of two magnetically coupled circuits. The mathematical theory of the system is developed; and the paper is accompanied by 20 photographic reproductions of the simultaneous traces of the two pendulums, the method adopted for photographing the vibrations being entirely different from that adopted by the previcus writers. At the same time, on account of its simplicity of arrange- ment it is a very convenient model by which the properties of electrical circuits can be easily demonstrated to a large class to satisfy the non-mathematical student, who is cften mystified by the perfect silence of the high frequency circuits. Department of Physics, Maharajah’s College, Vizianagaram City, India. 25th April, 1921, Pre L LXVII. Mechanical Illustration of three Magnetically Coupled Oscillating Circuits. By A. L. Narayan, UA.* [Plate XII.} Synopsise HE paper is a continuation of the experiments made by the. author on Coupled Circuits, and gives a short account ot three mutually interacting pendulums which he used, in order to illustrate the theory of three magnetically coupled circuits. The mathematical theory of the mechanical system is developed. The apparatus is a modified form of the one used by the author in his previous paper on Coupled Circuits. The paper is illustrated by ten photographic reproductions of the vibrations of the three pendulums under various con- ditions of starting, the method adopted for photographing the vibrations being essentially the same as that adopted by the author in his previous experiments, and_ therefore entirely different from that adopted by Prof. Barton and Miss Browning. The whole arrangement serves as a very convenient model for demonstrating the action of three electrical circuits to a large class. In the Philosophical Magazine for November 1920, Prof. Barton and Miss Browning describe a model of triple pendulum as‘a mechanical analogue of three mutually interacting electrical circuits. In this paper an entirely different kind of mechanical system, which the author used in order to elucidate and interpret the theory of three magnetically coupled oscillating circuits, is described, and the mathematical theory of the mechanical system is developed. The coupled system in this case is a modified form of one used by the author in his experiments to study variably coupled vibrations of two interacting circuits which illustrates the theory of the oscillation transformer. It consists (as shown in photograph fig. IJ., Pl. XII.) of three rigid pendulums, A, B, and ©, of which A. can turn freely about a horizontal axis by means of a steel knife-edge resting on a bracket. The pendulum B can be suspended fron A by means of a steel knife-edge resting on a V-bracket, which can be screwed at different points along * Communicated by the Author. Se = se Lae 5 PIER I ITE I TT a ae —— a ‘ aa pa! ce a > SS oes a = — Se = SSS SS See ——————— == — sa es += SS SSS a ee — = TP Oia wh eee! er ee 576 Prof. A. L. Narayan on a Mechanical Illustration the Jengtb of A so as to rotate about a parallel axis O’. Similarly, C is attached to A. Thus the degree of coupling between the various circuits can be varied at will. Tt is worth noting that, at first, the three pendulums were connected as shown in photograph fig. I.; but subsequently as it was found that owing to want of symmetry in the load ihe pendulum A began to tilt slightly to one side as the coupling was increased, the arrangement shown in photograph fig. IT. was adopted throughout. And it may be noted that there is essentially no difference between these two arrange- ments, as can be seen from the fact that in either case we get a cubic in n?, thereby showing that the resultant motion in each case is composed of three superposed simple harmonic motions. | The paper includes ten photographic traces of the motion of the pendulums under various conditions, the method adopted for photographing these vibrations being exactly the same as that adopted by the author in his previous experiments. For a detailed description of this method, the reader is referred to his previous paper on coupled oscillations *. Equations of motion and coupling of the mechanical system. The annexed figure represents a projection of the three pendulums on a vertical plane, perpendicular to the axes of rotation. * Supra, p. 567. of three Magnetically Coupled Oscillating Circuits. 577 Let: my, m:, and m3 the masses of the three pendulums ; G,, Ge, and G, the centres of gravity of the three pendulums ; K, the radius of gyration of the pendulum A about the axis of rotation, and K, and K, the radii of gyration of B and C about their centres of gravity ; hy, ho, and hs; the distances between the centres of gravity and the axes of rotation ; a, b the distances of the points of suspension of B and (© from that of A ; and let the pendulums.be inclined to the vertical at 0, ¢, and w respectively. _ In the case of a conservative dynamical system like this, where there are no extranecus forces, by Lagrange’ s equations we have for the equation of motion of three pendulums : pooh ole Y He: a iT pendujum gpn0T. OE. oW B ‘dt od’ OU a 0d’ : 5 by) ° ol «ole OV A di. "Sue! PMP —— a ° ° ° ‘ Je Now the kinetic energy of the pendulum A about the axis Pola. eno of rotation is ym 2(F) ; Cc pila ayy) that of B relative to its centre of gravity is bmaK?( TP and hence the kinetic energy of B relatively to A is im,K,? (F ) +4» {a = at a ar (4 +) } dé dd aie ee Fe Similarly, kinetic energy of C relatively to A is dé =1m;K,? (ty 4 +4ms tat, (G) +0 (Sy y d: yy oo 2bhs (3) ° (+) 4 cos p— 8 ° Phil. Mag. §. 6. Vol. 43. No. 255, March 1922. ee we cos $— ooh. niiaeton A Le Narayan on a Mechamcal Illustration Therefore, if ‘T represents the total kinetic energy of the system, , 2 2 : 2 oT = = milk "(nd ye | + “a t ed oe ep 2(ey van iy oe = and V the potential energy of the system (7. e, work done) = (mh + mga + mszb) g cos 8 + maghs cos P+ msghsz cos +K (a constant). .. Equations of motion for small oscillations are :— Ge (m1 Ky? + moa? + m3b”) - + mashes + rable He = (mhytmoa+msb)g0, . (1) dO * (my KK") a meotths ——~ Alp ram = moghod, fa ° acamtons : 5 (2) Bie a? ad?é mK.” “a + msbhs ro m3 ghsyr.: -. 2. a a These equations show the analogy that exists between the mechanical case in question and the case of coupled electrical cireuits, the equations of which we may write down as follows :—— ae d*g d?q Ie as =M—F+u A d*q dq a i, e+e =M,7 + Meee eee (5) Ga, Os ne d? iis Pio 2 Mf 4+ My — T(E) where L’s, q’s, and M’s have their usual significance. * The equations of motion of the first arrangement, namely that shown in photograph fig. I., will be of the form: ab+b,o+teay = d,9 9 a0+bhptow = do, aotbhptew = dw. The essential difference between these equations and those of the actual arrangement adopted finally being that the method of attachment of B and C to A in the second case reduces the coefficients c. and 43 to Zero, of three Magnetically Coupled Oscillation Circuits. 579 By analogy from electrical practice, we can write down the coefficients of coupling as: | ; meea7he? Dt) Wye ote Wess ms3"b7hs? = A and © m,K,?. mK? , between A and B, where =m Kk? = m,K,?+ mea? + m;6?, and so on. To solve the equations (1), (2), and (3), assume tentatively 9=Acos (nt+e), 6=Beos(nt+e), and w=Ccos (né+e). Substituting these values in (1), (2), and (3), and eliminating 0, b, and yr, we have Alam Ky? xn? +2 myhyg} + Bm ahgn? 1 Umsbhn = OL. (0) Amgahgn? + Bom, Ka?n? + mghz) =O. 4 . (8) and Amsbhen? +C(m,K,?n?+mgh;) =0. . -. (9) From this we get the determinantal equation : (2 Dm KYe+gSm,h,) : n?me2alho : n?msbhz . | n?moah, (n?megK4? + moghg) : 0 = 0. | n2msbhs : 0 : (n?m3K3? + mgghs) (10) On simplification this gives us a cubic in n? of the form ayn® + bynt+ en? +d, = 0, having three real positive roots. Assuming that m,, m2, and nz are the three roots of this cubic equation, the most general solution would be 6 = Aj cos (mt+€) + As cos (not + €:) + Az cos (mgt + €), h = Bi cos (mt + €1) + Be cos (ngt + €) + Bz cos (nat + €3), and Ww = C1 cos (mt +1) + Ce cos (net + €2) + C3 cos (nzt + €3), where 71, 2, and nz; depend upon the constitution of the system, and A, B, ©, and e are arbitrary quantities and enable us to satisfy any prescribed initial conditions. Thus the three pendulums act and react upon one another, thereby Za 2 580 Prof. F. D. Murnaghan on the Dejlexion of a creating in each a motion which is made up of three super- posed S.H.M.’s of different periods. Thus, in accordance with the general principle, there are as many normal modes of vibration as there are degrees of freedom in the system. Photographic reproductions of the vibrations of the three pendulums under different conditions are given in Pl. XII. figs. 1-10. These curves show clearly that there are more than two simple harmonic vibrations in each. With a modified form of apparatus, experiments are being carried on to illustrate the theory of combinational tones and their objective existence, showing thereby that when two simple harmonic forces of small bui, finite amplitude and of frequencies N, and Ne are imposed on a vibrating system, there are generated in the system simple harmonic vibrations of frequencies 2N,, 2N2, Ni+Nos, and Ni—Ne besides N, and Ne, the results of which will be published in a separate paper. Department of Physics, Maharajah’s College, Vizianagaram City India. LXVII. The Defleaion of a Ray of Light in the Solar Gravitational Field. By Francis D. Murnacuan, .A., Ph.D., Assoc. Prof. of Applied Math., Johns Hopkins University *. ee... the most important experimental test of the General Relativity Theory is that dealing with the deflexion of a light-ray in its passage near the Sun. It is essential, therefore, that the mathematical discussion of the amount of deflexion to be expected should be clear, and should avoid all unnecessary assumptions. The treat- ment given in Kddington’s classical and excellent Report + does not seem to meet these. requirements, since it is somewhat indirect and appeals to theorems of optics which are proven in the text-books for Euclidean space alone ; whilst it is the essence of. the Relativity Theory that the space surrounding the sun is non-Hnclidean. It is hoped that the following treatment will appeal to the reader as at once direct and not unnecessarily difficult. * Communieated by the Author. + The Physical Society of London. ‘ Report on the Relativity Theory of Gravitation,’ 2nd ed. (1920). } “ ee i ho, ee Se! Ray of Light in the Solar Gravitational Field, 581 §1. The minimal Geodesics or Light-Rays of a Gravitational Space: the Fermat-Huygens Principle of Least Time. The metrical geometry of the Space-Time continuum being characterized by (ds)? = Ymn air, lin, (m, n dummy or umbral symbols), it is postulated that along a ray of light (ds)?=0. Further, by (ds) is understood the positive square root of the expression for (ds)?, so that the arc-length integral [= \ds has an extreme or minimum value (zero) when extended over a light-ray. For this reason, these lines of zero length are known as minimal geodesics. However, the usual method of the Calculus of Variations for determining the differential equations of the geodesics is not immediately applicable. In this method we express the co-ordinates of a point in terms of two independent parameters 7 and « : i, = eee) (P= Lis. 4). The value of « being assigned, we have on varying + a particular curve C,, and when the integral I(a) extended over this curve, e=( let us ‘say, has an extreme value, él ( which is defined as = =(<) da) is zero,—granting a that I(a) has a Taylor development near «=0 which requires the existence of the derivative (S.) _. However, in the a/a=0 particular problem confronting us, I(a) = \/ San lin eae (m, n umbral), where &m = &y(T, a) and primes denote differentiations with sae to 7, and =\{32 5 (Yn Lm: Xn N/A Onn bm Lo) Lm, Ln 4 aT is not defined at p26 on account of the zero factor Ce intn Ua os occurring in the denominator *. This * This is the reason for Eddington’s remark (Report, p. 55) that “ the notion of a geodesic fails for motion with the speed of light.” “This statement seems unfortunate, since the curves are uniformly referred to by geometers as “ minimal geodesics.” 982 Prof. F. D. Murnaghan on the Deflexion of a difficulty may be overcome* by considering, instead of the are integral, the integral ie) = (aya = t Com Un, ie Ne No matter what the parameter 7 is, this has an extreme value (zero) when extended over a light-ray, since the quadratic form (ds)? is supposed one-signed. (In the discussion of the mnon-minimal geodesics which give the path of a material particle it was convenient + to take for 7 the are length along the extremal; but this cannot be done here, since this length is no longer a variable but a constant—zero.) Writing for convenience Gmn€m kn =F, we find the limits of the integral being independent of «+. On integrating the second term of the integrand by parts, after interchanging the order of differentiation with respect to « and 7, we find | > +S Gea renee (rv umbral), oF OL» where the symbols 1, 2 refer to the ends of the curve and ba, = (9 de. If the ends are first kept fixed, 7 a=0 we get the familiar Huler-Lagrangian equations 02; dt Ove 4 * Cf. Weyl, H., Raum, Zeit, Materie, Dritte Auflage (Berlin, 1919), 9. 210. ; Attention is directed to the fact that the integral l(#) is not, properly speaking, a line integral attached to the curve C(#). Its value depends not merely on the curve C(«) alone, but on the particular parameter r chosen to specify points on it. ; + Eddington’s Report, p. 48. { This does not imply that the end-points of all the curves Ca coincide. Although + takes the same values at these end-points, the functions 2,~=2,(T, a) do not necessarily do this, since there is a second variable « on which they depend. — Ray of Light in the Solar Gravitational Field. 583 and then allowing the ends. to vary, we get the additional end condition oO ; ae One ia If a particular x, does not appear in F, the corresponding == COnsivas 0 Ox, and so, if the’ other co-ordinates of the end-points are kept al Huler- Lagrangian equation gives at once S is not zero. This may be conveniently written jy dx,=0. This 2 : fixed, we find Mohs) provided the constant is the explanation of Eddington’s statement * that a light-ray “is determined. by stationary values of \ dt instead of \ds.” In the special case of the solar gravitational field PF. (67)?= (ds)?= (1— 2m/r) (dt)? —(1—2m/r)-*(dr)? | — 7? (d0)?—1? sin? 6 (dp)? ol 7 does not involve ¢, and a Come #0 as ¢ is not constant along the light-ray ; so that ON ai 0, which is known as a Fermat or Huygens’s Principle of Least Time. The space co-ordinates 7, 0, @ of the end-points remain fixed : it may be observed that since @ does not occur in F, we have equally well S\dp=0 along the world line of a light pulse, the co-ordinates 7, @ of the end-points being fixed, as also the time of departure and arrival of the pulse. We shall not make any use of this Fermat Least-Time Principle, but think it worth while to point out its exact significance. § 2. Determination of the Elliptic Integral giving the Displacement. A first integral of the Huler-Lagrangian equations is immediately found. EF being homogeneous of degree 2 * Report, p. 55. The writer does not see how this “ end-condition ” can of itseif replace the Euler-Lavrange differential equations. If, for example, we are investigating the shortest distance from a point to a curve in the ordinary Huclidean plane, the Euler-Lagrangian equations tell us that the curve must be linear (using Cartesian co-ordinates), whilst the end-condition says it must be at right angles to the original curve: quite a different kind of statement altogether. rs 584 Prof. F. D. Murnaghan on the Dejflewion of a in the z,', we have (by Huler’s Theorem on homogeneous functions, or directly) = 2F (rv umbral). oly ad 2) Ea) 02, dt OL, my by 2, , and ae y as an umbral symbol, we find or tae or ak On: vy We ¢ (oF ) hs peel ay ae Oe or ae = 0, verifying the fact that I is constant along the extremals— the constant in the case of the minimal geodesics being zero. The differential equations are On multiplying, then, the equations fo (5. dine) tlt! =. 2G) Q Von ¢ / aa 20 pn Ba + 2—— tm Ip! van eo n umbral), i / / or Dr Ln. tens te| Brediay 4 — since {Orn , OFrm ! _ 9 8% rn y ++ i Um & t pal = Tiel Ln . fo) Ab nL re) v nN ae Xm On multiplying by g’? and using r as an umbral symbol, this gives F he Up + {mn, Pp} Lm Un = 0. Writing, for convenience, B= g(t’)? +yilr!)? + 92(0)? + 93($!)?, we have, on account of the absence of ¢ and @ from F, the integrals — = const. ; 3d == CONMsts or gt = const:; 936 — const. which give, on substituting the values of g, and gs, (1—2m/r)t’ = C; sin? 6.6 =h, where C and / are constants. The equation fur @ is found, by writing p=2 in 1] I [fies = Lp AM per, Ga Oe } to be : Be G?0')—7? sin 0 cos 0(¢')?7 = 0" | * Cf, Eddington’s Report, Equation (30.12), p. 49. Ray of Light in the Solar Gravitational Field. 585 on eliminating the parameter 7 by means of 7? sin? @.¢’=/, Css gopaula d dr r* sin’? @ dd’ ) = sin 8 cos @, we find without difficulty, since d?0 dp dd which simplifies on writing <=cot 8 to —2 cot 6( Henee, if initially z=0, —=0, z=0. Choosing accordingly a 10 our co-ordinates so that initially @=7/2 : @=7/2, and our integrals simplify to ae . rp =h; (L—2m/r)t' = C, to which we add F=0, or (1—2m/r) (t')? — (1—2m/r) "(7 P — 7° (G')? = 0. =0, we have On writing moe and again eliminating 7, we find : Pe a + u? ae: ee do i h? or dus? C2 u i = 2mue—wv? +73: Here m is the gravitational mass of the sun, which has the dimensions of any one of the co-ordinates * (the velocity of light tends to unity as r->#, so that a time-unit of 1 cm. =4.10-'!° second). Taking, with Eddington, the kilo- metre as the unit of length, m=1:47; the smallest vaiue that r can have for the light-pulse is equal to the radius of the sun, or 697,000. Hence we obtain a Ast approximation to the light-r a neglecting the term in u® in comparison with that in u?, and ad immediately Ch u= 7 sin(p—¢o), where ¢o is a constant of integration. (This would be the ae of a straight line if the space were Euclidean, * If the gravitational constant of Newtonian mechanics is regarded as a mere number, mass has the dimensions L?T7?= L if L and T ‘have the same dinfensions. 986 Prof, F, D. Murnaghan on the Deflexion of a the perpendicular distance from the origin being A/C). Hence is is the largest value of wu; so that it is a small quantity of the order +.10~°. We shall denote this. small quantity by «, and for the next approximation shall keep in any expression only the lowest powers of @ Pe 8 in i. The discriminant of the literal cubic equation’ Ay @ +4, 27 +a,u+az3 = 0 is Ay? dy” + 18ay ay Ay Ag— 4a ag? — 40,3 ag — 2T a7 a5". Hor the cubic in question 2mur—u?+ a? = 0; this is 4a?(1— 27m?«?), which is positive since « is small. Hence the oe roots of the cubic are real; when 2=0 they are 0, 0, 5 In and writing as jirst approximations when “” a is small, in e +ksa, we find from the symmetric ay 2m functions of the roots ae ky +hy tks = 0; (hy +h) 5 — = 03 Kika = er ; that = Cee k= 0, ee and we may write the three roots as a * W=—-a235 US=+as w= 2m The next approximation is found in the same way or te direct substitution in the cubic to be Uy = —atme*s; u=atma?; u=— —2me’*. 2m ales Now the cubic (5 = 2mu®—u? +? cannot be negative in our physical problem, whilst also uw starts out very small and is always positive. As the ray of light is traced out, wu starts at zero (let us say) and increases to ug, at witich value ae is zero and uw has a maximum value corre- sponding to the perihelion of the light-path. The variable w must then retrace its values (it cannot lie between uz and us, 1 * It is just as convenient to substitute the trial solutions Smt oe etc... directly in the cubic, and thus determine the f. me Ray of Light in the Solar Gravitational Field. 587 since values between these roots make the cubic negative), the sign of the radical changing as wu passes through the value w,. Accordingly as u varies from wu to us, the angle d traces out the angle to the perihelion froma remote distance. The excess of twice this angle over 7 gives the amount of deflexion of the ray. Solving for dd, we have as the required deflexion Cu D = 2 ; du ==) Tike ae J Imre — wu? + 02 §3. Evaluation of the Elliptic Integral for the Dejlexion. It is convenient to make a simple linear change of variable under the sign of integration. We write u=a+6z and determine the coefficients a and 6 of the transformation so that to the roots wand uz of the cubic will correspond values 0 and 1 of 2 respectively. The values are a=w,; b=u,—u,, and then the third root ws; goes over into 2 where hk? = (ug—w,)/(ugs—u,). The cubic 2mu?—u?+ a, f transforms into 2mb?z(1—z) (-*) so that Seca |" dz oii an / 2mb) op / 2(1—z) (1 — 22) This simplifies considerably on writing z= sin? @, when, in fact, He Ak ie dé r/ 2mb~ egen/) Sy Se sin? 7 ; 5 9 Ug—U Qe welts Now # = *—* = ——— (to a first approximation) Ug—Uy h : 2m is a small quantity of the same order of magnitude as «a. Hence we can expand (1—k?sin®@)-? in a rapidly con- vergent series, and a mere integration of the initial terms will .cive a very good approximation to D. The multiplier of the integral in D is Ak/\/2mb = by ag : from 2m(uzg—v1) the expressions for kand 6. From the values given for wg and wu, this is 4(1+2ma)-?=4(1—me«). The lower limit eee U; of the integral is os) sin714/4(1—mz), where Uy — Uy 988 Deflexion of a Ray of Light in Solar Gravitational Field. here it will be observed that it is necessary to use the second approximation for w,, since it is to be divided by uy—ug— a small quantity of the first order. This may be votsles sin? 7(1—""), and expanding in(7 +e) by the Taylor theorem, we see that the lower limit is 7 Using the well-known result that \ sin? 0d0 = 4(0—sin @ cos 0), we find D+a = 4£(1—72) E +ih?(8—sin @cos@).. Pies: : ies where, to a first approximation, k*=4ma. In the term multiplied by 4? it is sufficient to use for the lower limit 50 Vass the rough approximation ~, and we have 4 4. D+7 = 4( (1—me) a + +ma(T +2) | = (l—ma)[ar+ma(4+7)] = 7w+4mact.... Hence the defiexion D is 4ma, where « is to a first approximation the largest value uw, of u: that is, the reciprocal of a is the smallest value of the distance r from the attracting centre to the light-ray. The closeness of the approximation is easily seen by taking the second approxi- mation u=«-+me?, giving, on solving this quadratic for a, 2ma = —-14(14+4mu,)? = 2mu,—2m?u,?, whence Be aa m a = Uys— MU = R , 3 il Ug = being the sun’s radius. ae RAs, i Hence writing «=. gives us an accuracy which allows R an error of about 1 in 5.10°, since m=1-47 and R=697,000. eee A Am : : On substituting these values in D=-— and converting radian on) R a measure into seconds of arc, we recover Hinstein’s prediction of a deflexion of about 1'°73 for a ray which Just grazes the sun. Johns Hopkins University, Baltimore, July 11th, 1921. L 589 J LXIX. The Hifect of Variable Specific Heat on the Discharge of Gases through Orifices or Nozzles. By Wiuuiam J. Waker, Ph.D)., University College, Dundee *. Soumary. The following paper raises the question whether or not it is desirable to account for abnormal orifice or nozzle discharges by the consideration of changes in the yalue of y, the index in the equation pvY=constant, for adiabatic changes of state. This appears, generally, to have been the custom hitherto, but, since the actual adiabatic - equation under linear variable specific heat conditions is pve =constant, the analysts in this paper has been carried out on the latter basis for the purpose of determining, as nearly as possible, what effect such specific heat variation has on discharges. An exact solution does not appear to be derivable, but the method of analysis adopted here may be carried to any degree of accuracy required. The method of analysis is somewhat similar to that adopted in a previous paper dealing with another effect of variable specific heat. The result obtained in the present paper brings out promi- nently the fact that the error in computing discharges, by the usual constant specific heat theory, increases as the density of the medium in the reservoir is diminished. This fact appears to have been neglected in previous considerations ot the subject. It is pointed out, also, that, by means of the dis- charge formule -obtained, the method of orifice discharge may be used as a reliable and convenient one for the deter- mination of specific heat variations with temperature. In the flow of gases through nozzles and orifices the analysis generally applied to any interpretation of test results is that based on the constancy of the specific heat of the gas concerned. The formulz derived under such an assumption are, for the velocity of the gas, co ey ; ee 2 eS \ wo=| 2g Pvi{l—2 Y } | feet persec., . (1) and for the quantity discharged, Ap 2 ae — eras (aa hs) 1. lb, per sec...” (2) * Communicated by the Author. 590 Dr. W. J. Walker on the Effect of where p is the pressure in the reservoir in lb. per sq. ft. abs. v, is the specific volume of the gas in the reservoir in cubic feet per Jb., p _ pressure at discharge py pressure In reservoir * f= The suffix c in u, and Q, refers to the coustancy. of specific heat as assumed in the derivation of these formule. In applying the foregoing equations (1) and (2) to experl- mental results it is customary to account, by one means at least, for departures from theoretical discharges by a suitable variation of y. Such a procedure can scarcely be called a satisfactory one, and it is proposed in what follows to deduce a rational formula (assuming, at first, linear variable specific heat conditions) which will require no such adjustment. This can then be extended to include quadratic variable specific heat conditions and so on for any variation of specific heat with temperature of the form K=a+6T+4+cI?+dT?+... which may be specified. In the adiabatic expansion of a gas, the velocity at any cross-section of the stream is given by 7, = 20K, | al feet per sec. 5) iy eee) it assuming a reservoir of infinite capacity, K, being the specific heat at constant pressure, in ft. lb. per lb. per degree centi- grade, T being temperature in degrees centigrade absolute Let Ka Sl: K,=B+ Sie K, being the specific heat at constant volume. Also, let n= = and Ne= . | S w=2g| A{T,—T} 4 5 {te | m c Xu Se Opp Pit PP) | Leet } ae) Now, during adiabatic changes of state poet = constant. Le ak eS Variable Specific Heat on Gas Discharges. 591 1 Uv pa. Pi m Xx Bi < = (BY {2+ any f to a first appromimation 1 : | ( 5 eh (9) Equation (4) may therefore be written 9 m—1 m— 1 : 1 =2g Mo pmatn ™ —p ® xt} aS API ms Gs mod a : ee 2mkR (> : we ) m—1 1 mM m—1 m—Il x ) ie. p ae, == 2 eeu, i , mu — i aos Ap) alae SEE eins op) P Aa) ( m—1 m— | )t 41 rAd } , m —P ™ + P1 l mR }’ mu—1 m— ‘) where p= at m ee m 19) m |. a m—1 m—1 2g pi™ ee m = ne ) CO pay felis t ee Pie (145) ne m—1 m—1 =F mpm) a 71 a7) 7 m—1 1 wee Me v1 ) to a first +5 R = m }) { approximation m a m—1 m—l1 =% 1, ™m a nu — m } Im—1? Py ~ ] x Pp m al m—1 m—I1 {i+ pun (a * —P mm) t m— iL ay m—1 "ope {ine bf 14S (10) j m—-1 d ese 41 = ae ee ) i to a first approximation, Ji9% Z a ME ae a. @. u=ur 15 ee )\ PARTE Te?’ (GB) 592 a of Variable Specific Heat on Gas hihi ur ite Be ye ee a . yee (5). and v?= am m+1 , oe 2 Pl m — m oe 1 am 2 {;-oh( 2m approximation, De m—1 Ma Glee Oe ie —om)\. » tidioy dy et ana If, however, K,=A+S8T4+9’T? and K,=B+S8T+98’'T? g ith A’ = —, wi B then the formule (6) and (7) become respectively ; m—1} NT?\ tl—a m usted 1+( 00+ 9 )l Tes iP Se) 4 m—1 GN a ee, O= Q.4 1-3 ( 3(an+ {2 ™ be 4m It will be observed from formule (6) to (9) that, when § (or what is the same thing, 2) is positive, the velocity wu is greater under variable than under constant specific heat conditions, while Q, the quantity discharged, is less. The same applies, of course, to S’ and 2’. If ni dad stim nega- tive, then the velocity w is less and the discharge Q is greater than under constant specific heat conditions. ‘The foregoing analysis suggests another method of specific heat determination, besides those already in use. By the provision of a suitably short convergent nozzle of invariable coefficient of contraction under all conditions, the application of equations (6) to (9) to the discharges obtained would enable the values of X and 2! to be derived. .This method has, at least, decided advantages over most others, particularly in the simplicity both of the experimental work and ealcu- lations involved. It is to be observed, of course, that the same qualification must be applied to these variable specific heat equations (6) to (9) as are applied to the usual constant specific heat formule. That is, they must be used as they stand, only for cases of flow in which the discharge pressure is above or equal to the critical pressure, 7. e., when the stream velocity at The Motion of Electrons in Argon. 593 discharge is less than or equal to that of the velocity of sound through the gas under the discharge conditions which prevail. For discharge pressures below this value, the assumption (which is found to accord closely with experimental fact) is made that the maximum discharge value holds for all values of a. Other points worth noting are :— 1. That the percentage error involved in the computation of discharges is three times the percentage error in velocity computations on a constant specific heat basis. 2. That the discharge error increases as x decreases while the velocity error also increases. 3. That the error in both velocity and discharge compu- tations on a constant specific heat basis increases as the temperature of the gas in the reservoir is increased. In other words, for a given reservoir pressure p the errors increase as the density of the gas in the reservoir is diminished. The conclusions reached here with regard to the effect of variable specific heat on gas discharges are somewhat like those reached in a previous paper* by the writer on ‘‘ The Effect of Variable Specific Heat on Thermodynamic Cycle Efficiencies.”” The equations deduced in that paper were of analogous form to those given here for orifice discharges. The applicability of the former, to, and their value in, practi- eal internal combustion work has since been fully demon- strated, and it is hoped that equations (6) to (9) may prove of equal value. LXX. The Motion of Electronsin Argon. By J.S.TowNsEnD, M.A., F.RS., Wykeham Professor of Physics, Oxford, and V. A. Battey, M.A., Queen’s College, Oxford t. 1. JN the December number of the Philosophical Magazine we gave an account of the motion of electrons in nitrogen, hydrogen, and oxygen, and it was shown that the loss of energy of an electron in colliding with a molecule, and the mean free path of the electron, may be found from the experimental determinations of the velocity of the electron in the direction of the electric force and its velocity of agitation. We give in this paper the results of similar * Phil. Mag. Sept. 1917. + Communicated by the Authors. Phil. Mag. 8.5. Vol. 43. No. 255. March 1922. 2Q x Se ee ae. 594 Prof. J. 8. Townsend and Mr. V. A. Bailey on investigations with argon, which are of particular interest, as they show that the molecules of this gas are very different from those of the other gases. Thus when an electron moving with a velocity of the order of 10% centimetres per second collides with a molecule of argon it loses about one ten-thousandth part of its energy, but when it collides with molecules of the other gases it loses more than one per cent. of its energy. Also the mean free path of an electron moving with these velocities in argon is about ten times as long as its mean free path in the other gases at the same pressure. 2. The experiments with argon were made with the apparatus described in the previous paper. Some prelimi- nary investigations were made with impure argon which had been dried for several days in a vessel containing phosphorus pentoxide. ‘The gas was supplied in a cylinder by the British Oxygen Company, and was said to contain 88 per cent. of argon, 10°5 per cent. of nitrogen, and 1°5 per cent. of oxygen. It was found that the lateral diver- gence of a stream of electrons in the impure argon was | remarkably large, and for a given electric force X and gas pressure » the velocity in the direction of the electric force and the velocity of agitation of the electrons were greater than in nitrogen or hydrogen. The gas was then purified by Rayleigh’s method. It was admitted to a pressure of about 50 centimetres into a flask containing a solution of caustic potash, and oxygen was admitted in excess of the amount required to combine with the nitrogen. Two platinum electrodes were sealed in the flask and a discharge was passed through the gas for several hours. An approximate estimate of the rate of combination of the oxygen and nitrogen was made by observing the change of pressure of the gas, and when the proportion of nitrogen was reduced to about one per cent. of the argon the gas was passed over hot copper in long quartz tubes to remove all the oxygen, and into a vessel containing phosphorus pentoxide, where it was dried for several days before measurements were made of its electrical properties. As it was important that no air should leak into the gas all the apparatus, including .the quartz tubes, had been exhausted by a mercury pump down to one-hundredth of a millimetre, and tested for leaks by a McLeod gauge. Also before using the copper it was heated for several hours in order to expel the occluded gases. This first purification did not have an appreciable effect on the velocity in the direction of the electric force, but the divergence of the stream of electrons became wider than in the more impure argon. a the Motion of Electrons in Argon. O95 These results indicated that quantities of nitrogen of the order of one per cent. of the argon would have large effects on the velocities. The gas was therefore purified a second time, the process being continued until the estimated quantity of nitrogen remaining in the gas was very much less than one-thousandth fo) of the argon. ‘Ihe excess of oxygen was, in this case, can) removed by passing the gas over hot copper several times. The second purification had the effect of reducing the velocity in the direction of the electric force and of making a further increase in the lateral divergence of the stream of ions. 3. The following examples of the measurements made with argon show how the velocities of the electrons are affected by reducing the proportion of nitrogen. With the gas Ae pressure of 20 millimetres and an electric force Z of 17 volts per centimetre the velocities W in the direction of the electric force were 22 x 10° cms. per second in impure argon, 22 x 10° cms. per second in argon containing about one per cent. of nitrogen, and 9°3 x 10° ems. per second in argon containing less than a tenth of one per cent. of nitrogen. With the same force and pressure the quantities & representing the factor by which the mean energy of agitation (mu?/2) of an electron exceeds the inean energy of agitation of a molecule of a gas at 15°C. were 48°8, 73, and 155 in the three cases respectively. For the ratio Z/p=17/20 the velocities W in nitrogen and in hydrogen are 7:6 x 10° and 11:1 x 10° ems. per second and the values of k are 19 and 7:9 in the two gases respectively. Owing to the very large divergence of the streams of electrons in argon it was impossible to make accurate measurements of the values of W and & corr esponding to the lower pressures which were used in the experiments with nitrogen and hydrogen. In the experiments with _the purest argon W and & were measured with pressures from 20 to 120 millimetres and with electric forces from 12 to 34 volts per centimetre, so that the experiments are over a range of comparatively small values of the ratio Z/p. A new apparatus is being made which is suitable for cases where the streams of electrons are very divergent, and we hope to extend these experiments with a view to measuring the velocities of electrons in argon corresponding to a wider range of the ratio Z/p. 4, The velocities W, obtained with the argon which had been purified twice, are shown in terms of the ratio (Z/p) by a curve in fig. 1. The velocities W in nitrogen and hydrogen are also given by curves for comparison. yO? 596 Prof. J. 8. Townsend and Mr. V. A. Bailey on It will be seen that no great difference between the three gases is to be found from the measurements of the velocity W in the direction of the electric force alone. Fig. 1. RESBa ee ea Z HEE pe pL oN ee | mee oe ee The values of & for the pure argon, for nitrogen, and for hydrogen in terms of the ratio Z/p are given by the curves in fig. 2, which show a great difference between the three gases. ). The velocity of agitation w of an electron, its mean free path /, and the proportion % of the energy of an electron lost in a collision with a molecule of argon, are given in Table I. The numbers for W, J, and X are obtained from the formulee Ts Sen! se wie Zz r = 2°46 x gil U the numerical factor in the last formula being an approxi- mate estimate as explained in the previous paper *. * Phil. Mag. Dec. 1921, p. 889, RCE eae BPS abe eee Ss axe a hi i Ne eee iad bees edhe Ke siecle. a 5 livetea Hydrogen | ro gen O 0-5 ; Si 2 2-5 Zs P TABLE I, Argon. Zz : a k W x 10-8. uwx10—‘: lp x 100. Ax 104, 5) 275 == 19-1 — — 2:0 2317. — Wr — — 15 200 aa 16°3 eo a8 1:0 162 — 146 — — 8 147 9°15 139 pats 1:07 ‘6 129 8°65 ial 13°38 1:08 ‘4 109 8:05 12-0 16:8 1:09 3 96 7°65 IES; 20°2 1:12 = 50 i fal | 10°3 25°8 1:18 bd | 5D 6°05 85 36'3 1°24 598 Prof. J. 8. Townsend and Mr. V. A. Bailey on 6. Since the effect of a collision of an electron with a molecule depends on the velocity of the electron, it is of interest to compare the effects of collisions in argon, nitro- _ gen, and hydrogen when the electrons are moving with the same velocity u. From the determinations that have been made the effects obtained in the three gases may be com- pared when the velocity of agitation is between 8°5 x 10’ and 14x 10’ cms. per sec. The values of Z/p required to maintain the velocity of agitation u in each gas are given in Table II. with the corresponding values of the velocities W. TABLE II. Ki Argon. Nitrogen. Hydrogen. ey Z/p. Ww LOT: Bp Wi Sc lO me een —wx10-?. 14 "82 9:15 74 215 O08 299 12 “40 8°05 50°5 172 32°4 128 10 19 7:05 32 124 19:2 69 85 ‘1G 6:05 16 72 13:0 49 From the numbers given in Table IJ., the mean free paths J and 'the proportions of the energy of an electron lost in a collinson % may be calculated by the preceding formule. These are given in Table III. Isic NOG, Argon. Nitrogen. Hydrogen. a5 . , UxNO~ ipx100. XXx104 Ipx100. Ax 104 1 xO MeNeaIOe 14 ~10°8 1:05 2°85 580 4°35 815 12 16:8 1:09 2-86 905 Bee 280 10 26°9 1:19 DOT 380 2°52 Li 8:5 36°35 eye 2°68 IAT 2°25 82 These numbers show that the mean free path J of an electron is about ten times as long in argon as in hydrogen or nitrogen at the same pressure when the velocity is 108 cms. per second. This result is remarkable, especially as the radii o of the molecules as deduced from the viscosity of the gas are not much different, the values of o x 10° being 1:80 cms. in argon, 1°88 cms. in nitrogen, and 1°34 cms. in hydrogen. Also there is a large increase in the free path 1 in argon as the velocity wu of the electron is reduced. A similar effect was obtained in nitrogen, oxygen, and hydrogen with lower velocities. Thus in nitrogen at one millimetre the Motion of Electrons in Argon. 599 pressure / increases from 0°266 mm. to 0°455 mm. when the velocity is reduced from 8°85 x 10’ to 3°15 x 10’ cm. per see. The values of X for argon are very much less than for the other gases. They can only be considered as approxi- mate values, owing to the large effect of impurities, one part of nitrogen to 10° of argon being sufficient to increase the value of X for argon by 3 per cent. The values of A» for nitrogen and hydrogen diminish with the velocity u, and in nitrogen when u=4 x 10‘ cms. per sec. X is 5°35 x 10~4, which is still about five times as great as the value for argon with the velocity u=J0*. 7. If the argon were absolutely pure, and if there were no loss of energy when an electron collides with a molecule of argon except the small loss corresponding to the momen- tum imparted to the molecule regarded as a perfectly elastic body, the value of X would be about a quarter of the value found experimentally. The question therefore arises as to whether the amount of nitrogen remaining in the argon after the second purification would be sufficient to increase the value of X from about 0°33 x 10~* to the observed value b2ec10=4. ; The following calculations show what amount of nitrogen would be necessary to produce this effect. If the argon contained one molecule of nitrogen to 4000 molecules of argon, an electron would make one collision with a molecule of nitrogen for 400 collisions with molecules of argon, the velocity of the electron being about 10% cms. per second. If EK be the energy of the electron, the average loss of energy in each collision with a molecule of nitrogen is 3°8X 107~*H, and in each collision with a molecule of argon the loss would be about 3x 10~*E, if the argon be supposed to be perfectly elastic. Thus the average loss of energy in a collision would be | 0:3 x 10-4 +3°8 x 10-2 E/400, and the value of > would be 1°25 x 1074, which is near the observed value. From the observations made on the contraction of the gas in the process of purification by sparking, we concluded that the proportion of nitrogen to argon remaining in the gas was much less than one part in 4000, but we cannot be certain that other impurities were present in a smaller proportion, so that notwithstanding the experimental results it may be possible that no part of the energy of the electron is transferred to the molecule in the form of internal energy. ol ee 600 Dr. G. B. Jeffery on the Identical We expect, however, to be able to obtain more definite evidence on this point, by extending the investigations over a larger range of the velocities of agitation w. The pro- portion of the energy of an electron lost in a collision with a molecule of nitrogen diminishes rapidly with the velocity of the electron, so that the value of »% for electrons moving a in argon would be much less affected by a small proportion 4 of nitrogen, if the determinations were made with velocities i of agitation of about 4 x 10’ cms. per second instead of the Soloei tes given in Table ITT. Nore Since this paper was written we have noticed that Mayer* and Ramsauert have determined the penetration ef electrons in gases, and obtained estimates of the free paths of the electrons or cross sections of the molecules. They found that the free path in argon was about 15 times the free path in nitrogen or hydrogen. They also found that the free paths in argon were much increased when the velocity of the electrons was reduced from 6°3 x 10‘ to 5°2 x 10’ cms. per sec. Ramsauer conciuded that there was no noticeable change in the free paths of electrons in the other gases for similar changes in the velocity. The latter result is not in agreement with our determinations. LXXI. The Identical Relations in Einstein’s Theory. By G. B. Jerrery, W.A., D).Se., Fellow of University College, London tf. N Hinstein’s general theory of relativity it is assumed that the gravitational field is completely defined by the coeflicients on in the expression for the interval in the four- dimensional sontinemaa, oye ie Os = 7,0, OL, ee GL) where, with the usual convention, the repetition of a suffix | in the same term implies summation for values dens coy iy that suffix. It is further assumed that these coefficients satisfy the * A. F. Mayer, Ann. d. Phys. vol. lxiv. p. 451 (1921). Tt C. Ramsauer, Ann. d. Phys. vol. Ixiv. p. 518 (ee 1 Communicated by the Author. Relations in Hinstein’s Theory. 601 tensor equation G 39,0 = —87reI,,, ots 7-2 where T : is a tensor expressing the density and motion of matter, « is a ioe and G,,.=— iw, a} + {ua, B}{vB, a} . . AO: ‘ eee 2.0m, log V— —y— i 2} =o log Vv —% gn (3) in which g is the determinant of the g,,, g*” is the co- -factor of g_, divided by g, the Christoffel sailoell are defined by O9up , Ore OFuv = 1y28( <2 #6 ue AS Te ee (4 and Gis the scalar g'"G, ,. Further, it is found that the principles of the conservation of mass, energy, and momentum in their generalized form are expr essed by the contr: acted covariant derivative equation DV We te gee ahaa ea (9) From (2) and (5) the following relations are deduced : 0G ; pape OS. Ge Se ee eat (6) in which the contracted covariant derivatives are defined by Cee on em Ct LG) where Gi.=9"?Gup. The four relations corresponding to w=1, 2, 5, 4 in (6), if they are true at all, must be identities derivable from (3). Written out in full théy are extremely complicated, and _ Prof. Eddington’s remark in his ‘ Space, Time, and Gravi- tation ’—that he doubts “ whether anyone has performed the laborious task of verifying these identities by straightforward algebra ’’—must have come as a challenge to many students of the subject. However, he himself has given an algebraic proof, which is not at all laborious, in the French edition of his work*. In the meantime we had obtained a proof which is perhaps a little shorter-than Prof. Eddington’s and may be easier to those English readers whose knowledge of the subject is based mainly upon his “ Report.’’ In view of * Espace, Temps et Gravitation, Complément Mathématique, p. 89. 602. Dr. G. B. Jeffery on the Identical the fundamental importance of these identities in the theory of Relativity, we venture to put forward this proof. It is known that a transformation of coordinates may be found in an infinite number of ways so that at one definite point of the continuum the y,, and their first differential coeflicients have prescribed values. Select a definite point of the continuum which we will call the origin, and if necessary make a transformation of coordinates, so that the first differential coefficients of the y,, 1 vanish at the origin. ‘The differential coefficients of ie second and higher orders will then in general not vanish at the origin. “We will first show that in the new coordinates the relations (6) are identically satisfied at the origin. In the case when the first differential coefficients of the 9,» Vanish, the Christoffel symbols also vanish at the origin, and we Have from (7) (en _ oO @ al fe) (g vel ) roan vere: i = prOGur ax, Substituting from (3) and omitting terms which obviously vanish on differentiation, we have ce Av aB 0 + ( Site a OIus one 2 J 04,02,\ 02, ) Of, ieee on | Av a Ryne Tg ae Vad The second and third terms in the bracket cancel on summation, and we have veges) Ay AY a8 0° Op Av on om 02, 02,0Xq vy 5H 08. es V-g. (8) Again OG 0 ae 02, a Oa Cr G,,) moh WS wees 2 OL Bee fo) OX Ou; (9) Av fo) 79 Dadaon, 8° 9 ag The first and second terms are identical on summation : neat Oe (ee Ome a) Relations in Einstein’s Theory. 603 the third is Dnv pay Av «af fe) Yas ic 02,0204 ay 02,02, Av oO a a a a ie 7) > om ag 5 . |: aus lee ON: ‘My poy = 2 (=). aia Now V =|) ate Vv cy? ay 2__9¢¢, cos 6 Ae, fe db Yaad, r/1- —k? sin 4 2 ae | = 2k + 1) log a ea + yo yok e+ _ Lee — 92 te e f j 9 4 ee - @ 5 where = nee (c os Cy)? and eee a (assumed to be small). C1 a Co * Communicated by the Author. : Mutual Induction between two Circular Currents. 605 From (1) we get peel OV | a) ! Urea Wea Neale eS 1—k’' ok 5h! we have My _ ki +1 Aue oe ap where A is the constant of integration. Rejecting 4’? and higher terms in the expression for V, eo) Cs Ate A 4\ #41 a, A+ = 2m re oly 82 | (los aay 28 87rc,” 4 ! ~atereai ee orf lee (143k!) dk 4 a k' a mewsee } + 87 |e log me a a log > 4 acalt To determine the constant, we observe that the quantity independent of c, on the right-hand side is 87. Hence, in view of the actual value of Mo, we get A=8r. Accordingly Be at a Taye Sle 8t ea a5 =1] +87 [ log G++... el It will be seen that if we put c.—c,= a, and reject powers about 2”, the first two terms reduce to 47} cy log = —2], which agrees with Maxwell’s result. : 3. In the case of two parallel wires at a small distance apart, w=0, and we have Manta, 3(7) aed 23) Ge +3 all 1— wry | du’ + ue wr wi) 2 da ma 24 C OE a aks =M,—47’ Cy = aa a EnGakcs LR Mee ite A} ace + |; since Gan =(), iad 0 ec a 60G) Mr. Bernard Cavanagh on ? Moreover, dp! = Fe 1 where r is the shortest distance between the wires. The expression for « can accordingly be expressed in ie following symbolical form :— We have [| ak dp P,,(w) = (p+ : qt) Pale 2h die bel otPrlp) _ ] Newey dp Hence if GAP | = eh : AG en ane Gib (F) BY; Moc eta (=) UD ay Be” oP 7 C9 since (P.) c= OUT ane: LXXIII. Molecular Thermodynanucs. I. By BERNARD A. M. Cavanacu, B.A., Balliol College, Oxford *. I. THz GENERAL CONDENSED PHASE. 7 EXHE treatment of the thermodynamics of solutions given by Planck ft for the simple limiting case of extreme dilution appears to be capable of extension to the general case. Planck himself, indeed, suggested { the expansion of the specific Total- ‘Energy and Volume of the solution, in integral powers of the various concentrations. But for _ general applicability this assumption of integral powers would appear to be quite unnecessarily arbitrary and narrow. We have, in general, for the Total-Energy and Volume per unit quantity of solution, functions F, and F, (say) of the various concentrations, which will involve parameters dependent on Temperature end Pressure. Now it would seem that the minimum assumption required * Communicated by Dr. J. W. Nicholson, F.R.S. + ‘Thermodynamics’ (Trans. Oge’) 1917, Chap. Vv. t Ibid. p. 225. Molecular Thermodynamies. 607 to make possible an extension of Planck’s method to the general case, is that F, and F, shall permit of expansion in the forms By =Duzfe (creas » - + +) (1) ey (€469¢3 eeciae zi) where 7, /o,... are functions independent of temperature and pressure, while wy, %..-, Vj, vg... are parameters dependent only on temperature and pressure. Some of these parameters may of course be zero, since we use the same series of functions, 7; f..., for both F, — and F,. This minimum assumption is clearly a very wide one which will probably always be, at least very approximately, applicable. Now as there can be little doubt that the linear form treated by Planck is the limiting form at the most extreme dilution, we may conveniently separate the simple linear terms from the rest which we shall call the “General terms” : Fy = Duye, + Suz f2 (eye. state i) B= Sve) + Sve’ fr (c¢o hee ) .lhe general terms must be of degree higher than the first IN ¢¢)...., in order that the linear may be the limiting form. For general treatment we may best express the concentra- tions as molecular ratios referred to the total number, 7; or Xn, of molecules, as denominator For special applications, as we shall see later, other “ con- centrations” may be preferable. Taking the mean~ molecular weight in grams as the unit quantity of solution, and writing U and V for the Total- Energy and Volume of the whole solution, U os uyC; + dat, fir( ce se .)s t V . *y == 2M Ue f( Creo, «+ -)s é that is U = yy + date feeyeg . . . .) \ V=dryuy + ngdvz fr(CyCg . . . «) 608 Mr. Bernard Cavanagh on Now for a change of total Entropy while all the n’s remain unchanged, fea dU oe % n, es rie et pie es. cs = yn ds, =P Needy (dss VAG nineaate * . . su hi aNe, 2 oF Ans (3) Integrating, S = 2y(s,+ &,) + ndlse + he oles - 2) 2 The integration constants are determined by conceiving the system, by an ideal process of raising the temperature and diminishing the pressure, to pass into the perfect gaseous state, the number of the various molecules remaining unchanged. It is not necessary that this should be practicable. The ideal process of “rushing” the system over the intervening unstable states, to exist, at least momentarily in the perfect gaseous state, all without appreciable change in the numbers of the various molecular species, is conceivable. It is not intrinsically impossible, or incompatible with our more limited experience, and that is all that is required. For then, however momentarily, the entropy of the system will be expressible both by the integrated expression (4) and by the known expression for the entropy of a perfect gaseous mixture: S=37,[5,+4—R log ey], which gives us. j ky =1,—R log a, ie hee = > S25 the latter s,/ corresponding to the ideal limiting state so that ky’ (ete.) like J, (ete.) and unlike hy Sie are independent of concentrations. Returning now to our solution under ordinary conditions, S=n,(s,+4—R log ¢,) + 2B(se' + he!) frlerce - - . -), and so p=(S— oo Ve chee log 6) + nediyo + kr’) ner er as ays where wv is the thermodynamic potential (of the dimensions of Entropy) used by Planck, which will be called simply *‘ potential ’ throughout this paper. | Molecular Thermodynamics. 609 We mav write, more briefly, v= > (¢,—R log e) + rBdbii flees... B yanaareg (2) where every ¢’ as well as every ¢ is of the form ps; =(const.)— = oa +| Tae a eee 6) so that 3 a Os s Op Ds Ty ) (i) CE Ceo UNG or Sia epee a mp2? : y z * (8) where Q=U+PV = 2 Quy + pry) + m4Z(us! + prr')fr(ereg « « .) a fc(Ciln segs 9. os oe oe. |.» (9) The formal convenience of this use * of Q will appear clearly below. Now from (5) BY =, -R log 4 34,'[ fit 26 30,94], or writing, for convenience, Of: ) (f.-Ee1 = a oe aN, (10) we obtain oY = ,—Rlog Ct dbz’ [e.+ 22], te Ce) while from (9) o2 = 95 Te B +27 eta 0 and from (3) oT a1, +3e,'| F.+ aE He ha GS) We may now consider the general system of any number of components in any number of phases from the “ molecular” point of view. Any infinitesimal change in the molecular constitution of the system can be represented by 26n;,, where 6n;, represents an increment of the number of molecules of the species (1) in * Planck writes Q for what will be 6Q in this paper. Phil. Mag. 8.6. Vol. 43. No. 255. March 1922. 2K 610 Mr. Bernard Cavanagh on the phase (a). Clearly this covers chemical reactions as well as transferences between phases etc. The total Mass increase of the system will be 6M = 2m, On, : > rn es) The total increase of Potential will be | pe ow . Ou = ee 671, é . : . 5 oy The total Heat Absorption of the isothermal isopiestic change: 5Q=3( 2" Jom, ee and the total Dilatation of the system : ye =(2* JB audi se a ET) It will be seen, of course, that equations (7-17) lead clearly and directly to the general mee. relations sy OW) = m (6Q), 5 ae) 9. (Sq) = 2 OV). Finally we have the conditions :—- For conservation of the mass of the system, sM—0,. . | ie For equilibrium (with regard to a particular virtual change) Sb=0'.: 2 Il. MotecuLak THERMODYNAMICS, VALUE AND GENERAL APPLICATION. Strictly general thermodynamics is independent of atomic, molecular, or other hypotheses. : The term ‘‘ Molecular Thermodynamics ” is suggested and used here to signify a slightly less general treatment of thermodynamics which rests upon the molecular theory of matter. In the case of a low pressure gaseous phase the advantage of such treatment was early (though not perhaps explicitly) recognized. Planck * first showed how it could be extended to condensed phases in the simple limiting case where one single * Loc. cit. chan. v. Molecular T hermodynamics. 611 molecular species vastly preponderates in each phase *, the ease in fact of very dilute solutions in non-associated, unmixed solvents fF. In section I. above, the ‘‘molecular” treatmentis extended to the perfectly general system of phases of any composition or state of ‘aggregation, subject only to the very broad assumption (1). That the “molecular” treatment when possible is essentially a step towards simplicity can scarcely be doubted, but ‘apart from this, it appears to the author oP) that the great importance and value of what is here called. “* Molecular thermodynamics” lies in the fact, apparently not recognized so far, that by expressing the Total-Hnergy in terms of the concentrations of the various molecular species (instead of, as in general thermodynamics, in terms of the concentrations of the components) we relegate all the chemical energy entirely to the linear terms. The “ general terms” must be considered to represent only potential ¢ energy of physical intermolecular forces. This suggests at onee ‘the possible use of Dynamical theory, which is, at least theoretically, applicable to the calculation of such energy. In section Vea striking and important example of the realization of this possibility will be given. As regards the general application of molecular thermo- dynamics, the determination of dw, 6Q, OV, for various isothermal isopiestic changes and of the relations of these quantities to the molecular concentrations in the various phases, to the temperature, and to the pressure, enables us, proceeding from the simpler to the more complex cases, to draw conclusions as to the form of the general terms in the above expressions for U, V, Q, andy. Itis clear that the complete solution of the pr oblem would give us one general form for these “general terms” which would be applicable to any phase of any molecular composition, at any temperature and pressure. It would then be a simple matter to describe completely any system of phases given the temperature, pressure, and total masses of the components. We can proceed at first only with relatively simple special cases. Qn the one hand, the approach to the limiting linear * This is not the case, for instance, in dilute aqueous or alcoholic solutions. + The remarkable “gas-analogy” discovered by van’t Hoff (and so often mis-used by others) was thus for the first time elucidated,—at least in the case of non-associated simple solvents. t Conceivably also, some speciai Kinetic Energies (such as kinetic energy of rotation) which might depend upon specific intermolecular effects to some extent. 2R 2 612 Mr. Bernard Cavanagh on form in ‘‘ dilute solutions” enables us to learn something of the molecular condition of the solutes, the important first step of all. On the other hand, we have the possibility of using dynamical theory as pointed out above and eh alee in section In the general case, owing to the complexity of the dynamical problem, one may expect that deductions from dynamical theory will be suggestive aids to empirical methods — rather than explicit solutions in themselves. Til. A Usgrun MopiricaTIoN AND SOME EXAMPLES OF APPLICATION. Though unsuitable for general treatment it is very con- venient in many practical cases to use concentrations referred to one molecular species called the “solvent.” When such a “‘solvent’’ is present in very large excess we have a “dilute solution ” of the kind considered by Planck. If m) be the number of molecules of this “solvent,” one may express the concentrations as Ny es | Ronea Cy ee = No” ny N Il | | | | i— e It is also a formal convenience to introduce the gas-constant R into the “ general terms,” writing U= anu, + Ray, frie 22 ee) “Unit quantity of the solution” containing one gram- molecule of the solvent. cy being unity, the general terms now depend only on ()Cg +--+ - V being similarly modified it is easily seen that vr now takes the form C I p= Sn,($.—B log =a) +RnSdbe'feleree -- »), N s or, since ¢) is unity, and using suffix s for solutes, / = ap =n pot Rlog (1+2c1)]+ 2s (% Fae 1+ 2c + Rrodobzfz(cies .-.))2 (28) Naturally ae is now formally distinguishable from ow et a Os wlan general terms in both, however, are some- Ny , what shorter and more easily obtained. ov =;—R | log cs—log (1+ 3e)— 34.25) (24) Ons 4 ay Molecular Thermodynamics. 613 while oY = goth | log (1430) +24," Lim Sa 22] f. (25) And similarly 0Q ) Of aii =q,+Rqz aa C ° - - > ° (26) ee os and again of 0,4 REx! OF fam. OMe BY ae, +RSe/ [A Ba8Z].. . . 29) A few illustrations of application may now be given. We deal generally with two phases (whether naturally or artificially separated) and we may expect the information to be most clear and direct when we keep one phase (the second or (a) phase) as simple as possible. We may place the experimental data under two heads :-— (1) Concerning the dependence of ov tions—Solvent-Separation-Data. M% Here we need only consider the common and preferable case, where the second or (a) phase consists of pure ‘‘solvent”’ though not necessarily in the same molecular condition as in the first phase. The modifications required when this is not so will be sufficiently obvious and need not be given in these illustrations. (a) Cryoscopic, Kbullioscopic, Vapour Pressure, and Freezing ° Pressure * Data. Five important quantities characteristic of the solvent are first accurately determined, viz. :— Gi.) The two Latent Heats (L calories per gram) for the two phase-changes of the solvent. aan sea sauce for the change (61% + 67,) in the limit when Xc, is zero (pare solvent)—or since by (20), 5M=m, dng +o, dn, =0 oe we get pes ee a being an absorption of heat on passing into the second phase. * This method has not so far been used. Its possibilities are being investigated. PN ee filmed me geagee oS pa A eee eee ee ee i ee vr) en ee 614 Mr. Bernard Cavanagh on Thus by (8) 2 Oona Po) C30) es nS Po). aa (ii.) The three Specific volumes of the three phases of the pure solvent Egy Pop Mo By (7) | nas =) } 3] a= ee ago ot And similarly | | (— a = oe ! 207 men i" oa = (He — 2) = 9 2 oe 0) oy 4989) mo, Mo Op Tg, eee | Now for the virtual change (6%, +6,) as we have seen Lye —O6no, 5 ON == mo, ° Mo 5 And by (21) we have for equilibrium between solution in the first phase and pure “ solvent”’ in the second ape. & {log (1+ 3a)+3¢.'[f-Ea2Z]}=—0 (83) M9 3 OC Mo, Mo —at temperature T and pressure D- Making 2c; vanish we get pure “solvent” in both phases, and the condition for equilibrium at temperature Ty and pressure po reduces to Poa al Po 51) yy, ERD Clearly therefore (T—T)) and tae being small *, Esai 0 t Po ek {( To)am + (p - poe, | i = =~ jog (1+2e)+2¢, [A= — 2c S| * When they are not small the right-hand side of {34) has to be treated as a differential and integrated by means of known equations expressing the physical behaviour of the pure) = solvers ” Similarly with (39). ’ Molecular Thermodynamics. 615 —at temperature Tand pressure p. Thatis, by (31) and (83) = \ log (1+ 3c) + =¢,' aa } , = ey ee where i cals. per gm. is absorbed in passing into phase («). (<= )c.c./am, is dilatation on passing into phase (4)— P0aq Po: and ¢y', etc., correspond to temperature T and pressure p. In Cryoscopic and Ebullioscopic data, Ap is zero. a In Vapour pressure and Freezing pressure data, AT is zero. (b) Osmotic Pressure data. Here the phases are artificially . . "ad bs d separated, and maintained at different pressures, the “ solvent being in the same molecular condition in both. Since mp, and mo are the same dN,= — SN: For equilibrium of the two phases under different pressures as indicated by the suffixes p and po, That is = { log (1+ 2e1)+ 2x’ [f- 2S] }= a Cy Cy Gaby eh 0) = le Ap _ vo Ap oy m, T P att = —m> ig OS! 9* Can AR Re as sr do at (35) where P (or Ap) is (a small) “osmotic pressure.” Now from the form of the expansion of log (1+) it is clear that log (1+ 3c) approximates to ec; when, and only when, 4>¢, is negligible compared with unity. When in addition the general terms are negligible we get P 0) ees a (aa 2 ‘ 616 Mr. Bernard Cavanagh on which is easily seen to be the “ Perfect Gas Law ” of van’t. Hoff, as obtained of course in Planck’s simple treatment, but only, as already pointed out, for a simple non-associated “‘solvent ”-such as is considered in this section. ov Ons on Concentrations— (2) Concerning the dependence of Solute-Separation- Data. For each solute-molecular species we have from (24) oY 5. —R] log ¢;—log (1+ 21) — 3g S5). We have two main types of data to deal with : (a) Partition data which includes solulility data; (b) E.M.F. data in the special case of Electrolytes. (a2) For partition data to be of much use the condition and behaviour of the solute considered must be simple (or accurately known) in the second phase. Thus it may be ee in the second phase | as in solubility data, or it may form a “perfect solute” in the second phase, Dee os ow Ons, ee Oe 1+ a as for a constituent of a mixture of perfect gases. This is the case to a close approximation in the gaseous phase, in dilute solutions of non-electrolytes, and in non- conducting dilute solutions of electrolytes. Then for the equilibrium partition, ; g i Ofs t $.,—Rlo aa R{ log S- = 3p 2 2.é., at constant temperature and Loe yz! —— Ofs = loo ee leo + Const. OCs = eae, rae Thus a Oss Ad tog 54 eS — log —2— Sa be =A2¢.'5 =Sg/A52 a Scns So that Files vs = 2 only be obtained after ical to of, Zero Pencentration unless fie etc. are already known. , Cs | se Molecular Thermodynamics. 617 If, however, 4 oh etc. can be reproduced (whether known or not) at different temperatures and pressures, then since by (8) fe) Oe Of: = ‘A= — pAteces 5 4 5 ul Ay Yb, Oes Yn A Acs 9 ( ) and by (7) 2 SAE =o Jae, ges 1 BS) we can thus determine g,'A a and dv, and by extrapolation to zero concentration &q,' = and >», oe Soe Of oh SE ets. el ae etc. or ae grounds we thus obtuin q;’ v,’ dy’ etc. at once. Generally, however, it will be possible to determine them by empirical methods given an adequate set of data as to the variation ot etc. be determinable on _ theoretical > S Px 5 ae —s concentrations, temperature, and pressure. In Pe ai, Methods of course log —— vanishes. lq The modification when the molecular weight of the solute is not the same in the two phases is obvious (cp. (33) above). (b) H.M.F. Data in the special case of Electrolytes.—This gives essentially the same information as_ the partition data, but is more widely and easily applicable. A virtual transference of the solute is effected by means of electrodes, and for this RSM Sene et is (2) sin eh > (39) where de units of electricity are involved in the transference of the 6m, gm.-mols. of electrolyte or (say) vén, equivalents of electrolyte, so that CE MV OMiapas oo. otro aye CLO) F units being one faraday. Thus if A signify the (finite) difference between two phases, (39) and (40) give ov —TA = a Bm BSe = yas rae). (48) 618 Mr. Bernard Cavanagh on | E 1 eas | to Bs qv Se Se ae leak c c (42) re) sate grok) We x(a) ~wAGn! ee é OV 1 5p T= 5 Se oe Now when we consider that, when the solvent is the same in both phases, fo x AST =—Ra | log ¢;—log (1+ 2¢) ~Xg,1 22 : (45) a ee OW 1 Ofe : os == RAYv, Aa 5 6 5 s ‘4 . A - 5 ° (47) it immediately becomes clear that the measurement of H/T and its temperature and pressure coefficients for concen- tration cells in which the molecular concentrations are known gives us precisely the same information as the partition data already discussed. But (within its obvious limitations) the E.M.F. method is much more easily and widely applicable. IV. PotyMrErizep and Mixrep SoLvEnts. Hitherto, like Planck, we have assumed a complete knowledge of the molecular constitution of the solution. In the case of an absolutely non-associated solvent such as a paraffin this will be readily obtainable, but many of our - best solvents consist of two or three different kinds of molecules in proportions of which we have no more than a rough idea. Concentritions referred to the ‘“ number of molecules of solvent ’ as denominator, are obviously of no practical use to us in these cases, for indeed our definition of the “solvent” as a particular molecular species has here to be altered to ‘‘qa particular group of molecular species, incompletely specified.” We have, therefore, to consider carefully how far both Planck’s treatment of extreme dilutions and the above treatment of the general case can be made practieally applic- able—by suitable modification—in the face of this deficiency of ‘data. Molecular Thermodynamics. 61% Now in “ Solvent-Separation-data,” cases we are now considering, not one molecular species. but all those constituting the “solvent,” and in the propor- tions in which they occur. This applies also to the use of miwed solvents—of which the molecular constitution may or may not be. known. We shall now obtain a general modification of the method of the preceding sections, suitable for these special cases, and which will show how far we can go with limited knowledge of the molecular constitution-of the solvent. We begin by referring the molecular concentrations to tbe mass M, of the solvent, which, incidentally, brings us into closer touch with experimental practice. n , Cor ie ihe ML My = 2NyMoy; where mm, ™m,-.... are the molecular vw eights (in grams) of the various molecular species constituting the ‘“ solvent,” and * OM) 0M, =m a ON, at Ong, =Mpp, etc. The mean molecular weight eo or may be written io. =) iQ) At first, let us assume that the molecular constitution of the solvent is independent of the concentrations c, ¢,... of the solutes. Then ¢o, ¢o,.. .!can only depend on tempera- ture and pressure, are not differentiable with respect to Cy... , and, in the general terms, can therefore be relegated entirely to the parameters w' uy’... and v,' v9’ .... Taking now as “unit quantity of solution,” the amount containing one gram of solvent, we have U=2no vo, + Sri + RM Sun! f(e,¢9 Beet ah CAS < V=219,%, + Sritit RMode'f(creg.--.). . (49) Whence clearly, as in preceding cases, ¥=2n( ,—B log 3) + Sm(g.—R log — ete Mo@s fr (Cr Co.-.).. =...) . (20) But Le= Xo, + Lei = = sae (l+m,%¢1), Mo Mo we separate, in the 620 Mr. Bernard Cavanagh on so that y= Mo| Seo, $n, —R2Xco, log moe, + - log (1 +77gSay) | + 3n,[6,—R log mM ¢, + R log (1-4 mBq) | SRM Sid. fr(Ci6g ee). bn ee err We have assumed mp, Cy, Co, --- to be dependent only on temperature and pressure, so that we can include —R log m under the symbols ¢, ¢, etc., and we can write om for the quantity S60, (go, - —R log my v0.) getting ap= My [out = 2 log (1+ TmoSey) | J inl Ry loga—log (1+ 77,¢)) yb] +RM ai ie ep) Thence we have R m ae ov = du + ae (1+ Mody) ae Rid | fe Sa | | an . (53) oy =$,-B J log cs—log (L+moXe1) — 2a 55. = j We have treated im, ¢ ¢,,.---, as he only on temperature and pressure. ; In a solvent which is a mixture of inert non-associated liquids these quantities will clearly be independent even of temperature and pressure. The modification just given will of course apply and be useful here, but might be dispensed with since the exact molecular constitution of such a selvent could cael be determined. In a polymerized solvent mo Co, Co, +--+ depend on the chemical equilibrium between the aaroe ont magleetles of the solvent, which in the general case cannot be assumed to be entirely independent ONC 8 In the general case then, ou di dy ete. in (52) will be differentiable with respect to c ¢,..., and (53) eé seq. will not be valid. We have to consider the order of magnitude of the effect of the variations of cy c,.... upon the values of the quantities ov and ov ; : On, Om In the first place, we note that variations in ¢ ¢,.... & Molecular Thermodynamics. 621 will affect the quantities cy), co, etc. only indirectly and relatively slightly, or in other words will produce only second-order variations in é , Co, ..... In the second place, since & mo, ¢o, is constant (unity), it is clear on examining (51) that the effects upon w of varia- tions of ¢, ¢&,.... are throughout more or less mutually compensatory. In the case of Sy, log mp ¢,, for instance, we note that the mutual compensation will be greatest when co, co... . are of similar order. When, however, this is far from the case, and any of them, co, (say) approaches either of the extremes zero or 1/imp, then, since the relative magnitude of the effect on v of a variation in ¢, may be estimated as ee 0 log cp, [ Co, log Mo Co, | = C0, [log Mo Co + 1] ; we see that this is small whether co, approaches zero or 1/m. Finally, the ‘‘ general terms’? (which may depend some- what on ¢,, o.----) are themselves small when ¢ ¢. are not large, so that altogether it is clear that the effects upon wy of variations in ¢,, Co... are relatively small, that is, second-order effects. That is, variations in ¢, c,.... will produce only third- order variations in 1p. From the general form of this argument we must expect of course that special cases will be producible in which it does not hold, but, in general, we are able to conclude that it is possible by means of the modification given in this section to go a surprisingly long way without complete knowledge of the molecular constitution of the solvent. Naturally‘ the most general case will require complete knowledge of the constitution of the phase for accurate treatment, but even then the grouping of equation (51) may be found convenient, concentrations being referred to the mass of the whole “ solvent,” rather than to the numerical] quantity of one molecular form of the “solvent.” The ov “cS Ae st a derivatives 5M, and an. obtainable from (51) are of course obvious. Returning to the equations (53) we see that they depend on m, in respect of the term log (1+m)2c;). Expanded, this term takes the form MoXC1| L—mpc, + sete «| - 622 Mr. Bernard Cavanagh on As a first approximation we may neglect all but the first term of this series, and in oe clearly we may neglect even the first term in comparison with logc,. Thus we obtain the equations ap = M,(bar+ RE) + 21 (G,— RK log c,) + RM Xbs' fe(41es.--) 1.5 gs ne) and thence SF du tR {2 4+3p/(f-BaSe) py] tg 19) oe =d¢d. —R ( log ts 2pz = ole \ —from which mp is completely eliminated so that no know- ledge of the constitution of the solvent is assumed. In this first approximation we have neglected dim dce, in comparison with unity. Now, for example, taking m) for water as roughly 36, we see that in half-molar solution the error involved in this first approximation is about one per cent. It is clear that at or above tenth-molar concentration we ought to proceed to a further degree of approximation by oe the jirst term and in a the second term of the expansion of log (1+) Xc;). We thus eet the equations ineluding in (96) in which, it is seen, 7%) only enters in the form of small correction terms, so that, until considerable concentrations are reached, a quite approximate idea of 7 will suffice. These equations (56) may profitably be substituted for equations (55) at and above tenth-molar_ concentration. Whether the inclusion of a further term of the expansion of log (L4+™Xe,) would be of any value (except possibly in certain special cases) is very doubtful, particularly when, as in this section, we are considering associated or polymerized solvents, for at really considerable concentrations the third- Molecular Thermodynamics. 623 order effects already discussed will generally enter and invalidate equations (53). Appendix to Section IV. : Non-validity of ** Mass-Action Equations” for Perfect Solutes. It has been common to use equations of the form of (55) with the “ general terms” | Sumitted, or rather the equivalents thereof, as the criteria of a “ per fect solute.” Ii may be suggested that this is incorrect, not being in accord with the obviously rational definition of a perfect solute as a solute which behaves like a constituent of a mixture of perfect gases, in fact, which does not affect the “general terms” but only the linear, and for which = —Rilog c;— log (1+ mc.) |. - ; (07) Now at very low concentrations (57) approximaies closely to OF =9,— PM OMECGE ON Ah hale (OB) and (58), or something equivalent to it, has hitherto generally been the criterion of a perfect solute. The question arises, as an interesting example of what has been said with regard to the “‘ second approximation ” (56), whether cases may not easily arise in which a solute remains “ perfect” in accordance with the true criterion (57), within the error of experiment even at concentrations for which (57) does not, within the experimental error, approximate to equation (58). We have seen that in aqueous solution when %e, is half- molar concentration the value of log (IL+m)Xc,) is of the order of ‘02. Now consider a chemical reaction between solutes which are “‘nerfect”’ in accordance with the true criterion (57). We have from (21) for equilibrium with regard to the virtual change {61s that is no 624 Mr. Bernard Cavanagh on Or if the lowest possible integers v,, v,.... be chosen so’ that Of] ON; DOnaeae. . = Vy 302203. eee we obtain, (for perfect solutes) from (57) ; Yv,{ds— Rf log c, —log (1 + imc) ]} =0, whence Sy, log e— (Sx) log (1+, Eey) = 5 Sod, =log Ko uit). eee) whereas the ordinary ‘“ mass-action equation” has been SA Owe =log K") i.) te ey eee We see therefore that log K'=log K+ (Sv,) log A+ mpc) = eel or ; K’=K (1. + mo2c;) 2s, 2 eee) or as long as mc, is small =K1 4 md¢,20.): oe We thus find that, except when Sv; is zero, the equilibrium constant as hitherto written down should depend linearly upon the total concentration of solutes present (whether taking part in the reaction or not) if the participants in the reaction are behaving as perfect solutes. The pose of the relative variation of K’ is seen to unity. Now supposing for simplicity that the Re evihs relative error in the experimental determination of each of the concentrations c; is uniformly equal to #, then the “ probable error” in K’ is of course oe so the possibility of detecting the variation which we are expecting in K' depends on the magnitude of XV; Modcy Sve] erage Since, as we have seen, for molar aqueous solution mjc, is probably at least -04, we find that evidence of the variation we are considering will, in easily chosen cases, become decisive as soon as the experimental error 1s reduced much below one per cent. (64) Molecular Thermodynamics. 625 It is to be noted that Yc, represents the range of variation of the total concentration of solutes present, not of the concentrations of the reactants, which latter alone, however, are required to behave as perfect solutes. There seems, therefore, good reason to believe that in a few well-chosen cases this effect might be demonstrated beyond doubt, and the quantity mm, found with some approximation in eacn case. Agreement between the several values of m, thus found would justify some confidence in this estumate of the mean molecular weight of the solvent. It seems probable, however, that measurements of the solubility of a moderately soluble gas would be the most hopeful first approach to this investigation,—the theoretical treatment being perfectly obvious from what has gone before. Hxperimental investigation of this question is contem- plated. V. A SpecrAL APPLICATION. THE THEORY OF EXLECTROLYTES. It was pointed out in section II. that the “ molecular” treatment of thermodynamics opens up the way for the use of dynamical theory. The calculation of the general energy terms will be complex, however, in the general case and only rough ideas and indications may be obtainable at first. The idea put forward by Sutherland in 1902, that strong electrolytes are completely ionized, at any rate in dilute solution, is of great interest in this connexion. For in a solution of ions we have forces at work which are amenable to relatively simple dynamical treatment, and which are considerable in dilute solutions where other inter- solute forces are negligible. The credit of first showing that this is so is undoubtedly due to Milner *, and the full value of Milner’s calculation of what he calls the *‘ Viria]l ” of a mixture of ions has perhaps; scarcely been realized. Milner calculated }27¢4(7), the sum of all the electrostatic forces f(7) in a chaos of equal numbers of oppositely charged massive points, each force being multiplied by the distance (7) over which it acts. Calling this the “ virial ” of a mixture of ions he proceeds to apply to the osmotic pressure of a solution of ions the Virial theorem established by Clausius for a gas. He points out + that because the law of force is that of the inverse * Phil. Mag. ser. 6, xxiii. p. 551; xxv. p. 742. toes Cth: SX. Pa (a. Phil. Mag. 8. 6. Vol. 43. No. 255. March 1922. 28 Ful es'ra- Ae tee 626 -Mr. Bernard Cavanagh a square, this “virial”? represents also the potential energy of all the electrostatic forces. Milner then gives a thermodynamic demonstration * supporting his application of the Virial theorem “to the complex phenomenon which the osmotic pressure of electrolyte undoubtedly is.” But this demonstration is open to grave doubts, for it ‘depends on treating the solute thermo- dynamically as an independent system, whose “ external pressure ”’ is the osmotic pressure of the soiution, and also on regarding K, the dielectric constant of the medium, as independent of temper ature. Of course, neither of these can possibly be strictiy permissible, but also one cannot, @ priori, assume even approximate validity as regards the conclusions to which they may lead. Using the general molecular-thermodynamic method developed in this paper, we shall, by means of a simple example, reach the conclusion that this application of the Virial theorem cannot be, in any general sense, valid. The Non-validity of the “ Virial’”? Equation for Osmotic Pressure. Consider a solution of a binary electrolyte, containing ¢ gram-molecules of electrolyte per gram of solvent. Suppose the “virial” per gram-molecule of electrolyte to be given by apy, Ru’, > where w! is pe pends of ¢. Then ¢ being £ iL we have U=Mywy + 2nu—nRu! = Mju + 2nu—RMyu’c!, . 2) (gh, ae GGG) where u is a mean value for the two ions f. Assuming, further, that owing to the smallness of the ex- ternal pressure the general terms in V can be neglected in comparison with that in U, we shall write | V=Myyt2nt, 9)... ey * Loe. cit. xxv. p. 748. + The discussion of the possibility of distinguishing the thermo- dynamic properties of the complementary ions of an electrolyte is reserved for a later paper. Molecular T hermodynamics. 627 and then obviously for dilute solution [ef (55)] we shall obtain w=M)| dut2Re] + 2n(¢— Rlog ce) _RM,¢'e, (68) where pegs u' ‘du’ gow T tho and Ps) | au! or Ce aE i ae du' sm (Pl)=¢ ty =k + (4. ° e ° (70) From (68) we derive oe -dy +2Re+4R¢d'c2 so that for a small osmotic pressure P [¢f. (35) | Rae tia a Re ok é 5 : : “ (71) Now Milner, applying his thermodynamic demonstration, with the form of *‘ virial’? we have assumed would obtain fe Comparing (71) and (72) we see that this would mean A NT a orca ate) i. e., by (70), a Mori Oran) ls deka 2 is( aad Ee. Dy Cho), cs CANO res. VS We Caen) Now Milner’s thermodynamic demonstration will apply to the form (65) if w'/p 9s is independent of temperature, i.e. if: (5) so that rigorous thermodynamics Caen bears out his demonstration in this simplest case, making nearly the same demand in order to arrive at the “virial” equation. But 282 Srd(7) is the implicit assumption that K is independent of the temperature, saying that “if it were not constant, the electrical force between two ions would depend on their kinetic energies as well as on their positions.” There appears to be some mistake here. The electrostatic force between two ions will depend on the effective dielectric constant of the intervening medium. In dilute solution—to which Milner is limited—the large majority of the ions will be separated by many molecules of solvent, so that, on the average, the effective dielectric * Loc. cit. xxv. p. 748. es ee See ee a eh ey ae Molecular Thermodynamics. 629 constant will be that of the pure solvent, corresponding to the statistical temperature. In centinormal aqueous solution of KCl, for instance, there are about 3000 solvent molecules (“ dihydrol ’’) to every molecule of electrolyte. But quite apart from this, it is far from clear how the kinetic energies of particular ions can enter into the question of the electrostatic forces in which those particular ions are concerned. The logical force of the above quotation does not seem at all obvious. Thus Milner’s calculation of SS7rd(v) appears to rest on a stricter basis than he himseif has claimed, though his application of it in the form of the Virial theorem seems altogether doubtful theoretically. The calculation proved to be complex, and the form in which the final approximation was cast RTAP(h) depends on a tabulated function ¢(h). In an inconspicuous footnote *, Milner remarks that ‘It may be shown as an approximation, and when hf is very small” that PU in Sah) ee es” C17) Now in dilute conducting solutions h is small, and owing to the rather striking result to which it will lead, one is very interested to find, on examining the values for 3Ad(h) given by Milner in the later paper, that up to centinormal (or even higher concentrations) we shall not be very far out in applying this approximation (77), which, since h is of the form A’c3, means that >Srd(r) is of the form Act per gram-molecule. . . . . (78) This gives for the general energy term GC ed tek A No CEO) instead of the expression nRuw'c? used in (66), and we shall obviously obtain w= My(dut 2Re) +2n(d—Rlog c)-RMy¢'ci, - (80) whence ce eh SY =2{$-R(loge+ $6'e)!} | * Loe. cit, xxiii. p. 576. oy =dou+ R{2c+43¢'c3} 630 Mr. Bernard Cavanagh on and for solvent-separation data [cf. (35) and (34)] By LA aly ot Vs Ps. Bye = HPs py, py) ee 2+ HA «CD Now in the nomenclature of G. N. Lewis, as will be clear from (42), we could write oY = = 2{(const.) — Rlog ey) een for a completely ionized electrolyte, this being simply the equivalent of Lewis’s-definition of y in this case, and y being what Bronsted calls the “ stoichiometric activity coefficient ” of an ion. Comparing (81) and (83) we find log y=20'e... eee But Lewis and Linhart* have recently shown as a purely empirical fact derived from the most accurate data then available, that this form Bc? is actually the limiting form of log y or uni-univalent strong electrolytes in very dilute C solution | below ian) This seems to be rather striking ; mt confirmation both for Milner’s calculation of Yrd(7) and for the “‘ complete ienization”’ theory. It will ba very interesting to test Milner’s calculation more closely by obtaining © the experimental value for w’ itself. If (and only if), as we have assumed, the general terms in the expression for V can be neglected in comparison with those in U f, then, as we have seen woes (69) | o¢' Cae DP that is, at a given concentration (c) by (82) ! Fae. . eee aa and by (84) : : fe) ' ap logy det aes 2 Be) * Journ. Amer. Chem. Soc. xli. p. 1951 (1919). + This uae requires that ie pressure coefficient of log Y whien multiplied by 7m be negligible in comparison with the temperature coefficient of log y, and of this there can be little doubt. Molecular Thermodynamics. pie IGan so that accurate data, whether so!vent-separation data or solute-separation data, for low concentrations (below ai ut different temperatures would enable us to calculate w’, and compare the experimental value with the theoretical value calculated by Milner. from dynamical theory. Mixtures of Strong Electrolytes. It will be of interest, now, to consider the extension of this theory of electrolytes to the more general case of a mixture of strong electrolytes, particularly as experimental data on this head are forthcoming. Consider a very dilute solution of various univalent ions, in total number Sn, and in total concentration ¥c;. These quantities correspond to 2n and to 2 respectively in the simple case of a single electrolyte. In such a solution we may regard all the ions as point- charges, and (as we have seen, in that case) express the total SErh(r), (of (79), a8 (Soe (bees ke eee BE) We may conveniently write 4>¢q= C;, ° . : . . ° ( 88) the total equivalent concentration, or total concentration of ions of one sign, so that pee ih eterna? (89) And, neglecting again the general volume terms, we have + ; 3 U=Moupt+ Sryuy,—RMyu'C?, . . . (90) Ne Vea te. hi ocev a rea GeO) so that r= Modu + 2RCi) + X21 ($1 — BR log cy) —RMod'C3, (92) and thence oe = out RC,{2 +144'C?} | = =o: —Riloge:+3¢/C;7} that is eee een erie, (94) leer Oe i ake, (95) cena ea 95 ——-+,-- 2 632 Mr. Bernard Cavanagh on This means that the thermodynamic behaviour of all the ions in the same solution of (entirely univalent) ions is the same and depends only on the total equivalent concen- tration C;. Now this is precisely what has been deduced in recent years as a purely empirical rule by Lewis, Loomis, Harned, Ming Chow, and others *. It is a very significant fact that these workers have shown this rule to hold quite closely at concentrations much higher than are contemplated in the above. Thus, probably we shall be close to the truth in regarding >Srd(r) as a function of C;, even at these somewhat higher concentrations where it does not take the simple form Ru'C3. For tentative preliminary treatment of the higher con- centrations, the use of the general form RSu! COM o 6: ss ron may,be suggested, being fairly elastic since mj, ms, ...... can have any positive values, and possessing the advantage of being easy to work with since it gives U= Mou) + Snu,— RM Sy", 5 ae b= Mo(ou +2RC;) + 2n(G1— R log «1) —RM>g,'07™, iS, Seago) aad ‘ ee Sur, = out RO 2 + Sings'Or be Vee oe) i m That is 1=2 + Smidy Cr 4 has a ea ) logy So eres mee i On the basis of Milner’s calculation the first m would be 3, and the rest higher than 4, so that (94) and (95) - would be the limits of (101) and (102), as found by Lewis and Linhart. For bivalent binary electrolytes one would expect the same form of virial with a different limiting parameter B. * Journ. Amer. Chem. Soc, xxxi. p. 385 (1909); xxxviii. p. 1986 (191G)Nexxxvinl. p: 25800 (1916); xxxix. p. 38h (IGig meee p. 632 (1911) ; xii. p. 488 (1920). xxxvii. p. 2460 (1915). ? Molecular Thermodynamics. 633 If the limiting form has really been reached by Lewis and Linhart in the few such cases investigated, this expectation would not appear to have been realized, but the data can scarcely be regarded as very satisfactory. Owing to the electrostatic forces being four times as great in this case, we should expect the limiting form to be attained only at much greater dilution than in the case of univalent binary electrolytes. Now when we consider a mixture of ions of different valencies, the corresponding calculation of {=rg(r) becomes somewhat more complex. Milner’s intermediate functions F.(™), Ga(m), and s(m) become modified and less simple. Lewis has propounded an empirical rule which is equiva- lent to substituting for C; (the total equivalent concentration) of the preceding, a quantity S; called the “ionic strength ” and given by Sea i Pe eet (103) where wz; is the valency of the ion species (s). It is probably not incompatible with a generalized form of Milner’s calculation that this rule should have a certain limited validity as an approximation in practice, but whether it is anything more is rather doubtful. Chenucal reactions in Electrolytic Solutions. In dilute solutions non-electrolytes behave as_ perfect solutes for which OF = $,—Rlog en ae eey soy bee de Cl O44 Not so zons, for which, as we have seen, ov = 5 Mey a 1 £ (NM, h ee =o:—R | log C+ 9 1 C; We supposing only ions of the same valency are present. Now from general theory (equation (21)) in any equi- librinm with regard to the virtual chemical change én; we have Spiele. 4: aa = Ons= 0. When all the participants are solutes obeying (104) this gives, as we have seen (and as Planck showed in the case where the solvent is not “associated” or polymerized), ae: > 634 Mr. Bernard Cavanagh on the mass-action equation in its ordinary form. “When ions participate we get a modified equilibrium-equation. The case of a weak (or ‘“semi-weak ”’) monobasic acid EDN will suffice as an example. All the 6n’s are equal, but while ov fe) NHA the undissociated molecules behaving as a perfect solute ; on the other hand, =dya—Rlog Cra, oe =doy—Rh log Co RS mitt OP) Jel’ 2 >: fo s ae f ym, | oy ~$.—Rbgt, Ry ae Whence we get ; be | gg CH Va _ Ont bal OMA Sin CHa Rh =log K+ 3 (mm, + 1)6)'C, 1) ai which, of course, might have been obtained from the fact that log y=32(m +1) pC", where y is now the activity coefficient as distinct from the ‘stoichiometric activity coefficient”? generally used by G. N. Lewis. For the very meaning of y involves (yO) (yCu) Me: CHa fligh Concentrations of [lectrolytes. The study of really high concentrations of electrolytes is at present necessarily rather devoid of significance from the point of view of this paper. Other inter-solute forces enter into the problem, besides the electrostatic forces which no longer remain readily amenable to dynamical calculation, and finally the solvent must en ae be a polymerized one of little-known * As far as experience goes. Molecular Thermodynamics. 635 constitution, whereby, as indicated in section IV., very great uncertainties are introduced at high concentrations. It seems advisable, therefore, that the study of strong non- conducting solutions and of non-polymerized solvents should precede that of high concentration conducting solutions. SUMMARY. In section I. it is shown that on the basis of a very broad assumption, a general treatment of the thermodynamics of any system of phases, from the point of view of molecular theory, can be developed. This “ molecular thermodynamics ” which is thus slightly less general than strictly ‘‘ general thermodynamics” was really initiated by Planck in his classical treatment of a very dilute solution in a non-associated simple solvent. In section II. the meaning and value of this “ molecular ”’ treatment is discussed, and also the general aim and mode of application. It is shown how this treatment paves the way for the introduction of dynamical theory as an aid to the solution of the general problem. In section III. a slightly modified procedure, useful for many practical cases, is introduced, and the significance of the chief types of experimental data is concisely demon- strated. Section IV. contains a special treatment of the case of polymerized and mixed solvents, and it is shown how far we can proceed in this case without precise knowledge of the molecular constitution of the solvent. Some second-approximation equations are put forward for use at medium concentrations, and in an appendix to this section the importance of a second approximation at such concentrations is illustrated by the non-validity of the ordinary mass-action equation for perfect solutes. In section V. the question of electrolytes is considered from the point of view of molecular thermodynamics. It is shown that Milner’s calculation of the “ virial” of a mixture of ions is of considerable value here, although his application of the Virial theorem is rejected on theoretical grounds. In fact, a striking example of the use of dynamical theory as an aid in the solution of the thermodynamic problem is here forthcoming, and considerable confirmation of the con- clusions reached is found in already (recently) established empirical facts. The question of weak and “ semi-strong” electrolytes is _ briefly treated as an example of the participation of ions in 636 Notices respecting New Books. chemical reactions. Some lines of experimental investiga- tion are indicated in the paper which it is intended to follow up. In conclusion the author wishes to thank Dr. J W. Nicholson and Mr. H. B. Hartley for encouragement and interest in this paper and for helpful advice with regard to preparation for publication. : Balliol College. Nov. 10, 1921. LAXIV. Notices respecting New Books. A Treatise on the Integral Calculus. By JosrpH Epwanrops, M.A., Principal of Queen’s College, London. Vol. I. 907 pages. (Maemillan, 1921; price 50 shillings.) FIXHIS work when complete will challenge competition with Bertrand’s Calcul intégral, in size and scope. And the price will make us all reflect, gone up about fivefold by comparison with the pre-war era. Chapter I leads off in the manner to be anticipated in the modern treatise, with a review of fundamental metaphysical con- ceptions of difficulties the beginner has not yet encountered, and best kept out of his sight as long as possible. The author does not settle down in his stride till Chapter II, where it is a comfort to find that the old-fashioned procedure is tolerated of treating Integration as Anti-Differentiation, as leading most rapidly to the applications to give reality to the subject. Practice is required here as much as in the scale practice of the musician, and this elementary practice is best carried out in the viva voce catechetical instruction of a small class, of two or three ; in this way the beginner acquires confidence in his initial judgment of the first steps, and can be made to apply them at once to some well chosen elementary applications, and state his result in words. The instructor will be inspired by this treatise in his hand, equivalent of the corresponding German volume by Stegemann. Real progress has been made in the foreign language when the ear seizes the whole phrase and does not dwell on the separate word. The musician does not read the separate notes of the score, but plays the whole musical phrase, ignoring a misprint. - So too in the Integral Calculus: the object is to see at a glance the nature of the integral, and the class to which it belongs. A rapid method is described here, p. 179, where the integral is sought as a function of the integrand, as in \sec w dv=ch ‘sec a, called integration without preparation. Ce ee ee ee N e a ee ee Oe et Se Notices respecting New Books. 637 Good instances are provided in Chap. VI, where the integrand is the reciprocal of a+bcosx+csinw,....; and the method is : ; R : WV z a6 lx developed in Chap. VIII in its application to Se dix pat ib be ae : where X, Y are both quadratics; the substitution y= =~ breaks. xX up the integral without preparation into two simple standard forms, circular and hyperbolic. This integral may be cited as the degenerate hyperelliptic integral, where the sextic under the radical breaks up into X°Y. So too when X and Y are both linear, a degenerate elliptic integral is encountered, the cubic under the radical is X*Y, having the repeated factor X. Once started on the Elliptic Integral in Chap. XI, and the author finds it difficult to pull up; we are promised a sequel of complete treatment in vol, Il. The lemniscate and cassinian are treated as applications and with elegance, but had better have been delayed till later on. The pious adhesion here to the old standard forms of Legendre should give way to the more elastic treatment of Weierstrass, as not requiring preparation, taking the irrationality in the integrand as reduced to the square root of a cubic; but disregarding the undimensional abstraction of the p function, to consider the Second Stage functions of Abel and Jacobi, where the cubic is resolved into factors X=4, v—w%,.4%—4,.x—x, in $388, and to discuss the various forms that arise and the double periodicity, as the variable a traverses the regions bounded by the branch points w,, v,, v, As indicated above, with #,>2,>,, the degenerate form is investigated, circular or- hyperbolic, as the middle root x, moves up to z,, or down to «,. The author works the circular and hyperbolic functions together in harness, and so maintains an analogy of assistance in the treatment. Although so innovating as to introduce the elliptic functions, and to employ the Jacobi-Gudermann notation of sn, en, dn, and their inverse, he has stopped short of the modern French abbreviations of sh, ch, th, for the hyperbolic functions, analogous to the old sin, cos, tan, of the circular Trigonometry. Bertrand’s tangsecthyp strikes us to-day as a very cumbrous form of tanh, or th. A picture of the integration of y=a”" is provided in fig. 16, p. 112, where the curve PQ then divides the gnomon in the ratio of one to x; this is evident by elementary geometry when Q is brought close up to P, and so still holds as Q separates again to any distance along the curve. Here is the most evident repre- sentation of the integration, with no apparent flaw in the argument for the young Berkleian to attack, who is repelled by the gritty approximation he meets in the course of a treatment by summation, necessitating the lengthy abstract apologies of 638 Notices respecting New Books. | Chap. I, to meet the objections there suggested, of which the 5 author seems to go in fear. The integration of the circular functions, Yy=S10 @, COs can be shown in a similar picture, equating the differential element of the curve to a corresponding element associated with the circle. No need then for the long-winded summation penultimate j to the integral, with its gritty approximation so indigestible to the careful young Berkleian thinker. The author goes in fear of this young critic, apt to conclude ! that the Calculus is after all only approximately true; hence much of the lengthy explanations in Chapter I, only to strew his path with difficulty and trip him up. But if the young heretic will not be convinced in his heresy, agree with him then in his error, and allow that there is an outstanding error, but that the error is finally proved zero. The young engineer is inclined to these heresies, but Perry knew how to interest him. Perry states the result of a differentiation of sin ma is to ( increase the phase angle ma by a lead 37, and to multiply by m; ; a sin (me+e)= m on (mate+ia i da ae without changing from sine to cosine. Conversely an integration reverses these steps; it subtracts a oe of Ln, and divides by m, sin ls ee (ma + e) Une = B08 5 (me +e— Aq), and no confused thinking is Aes) of any change of sign. Similar results are easily remembered for a combination of circular and exponential or hyperbolic functions, Pee in a change of phase and a multiplier or divisor. ’ d sin ay + BO? (og (mathe) we to . The result of the operation of A _ymA give a lead tan B and amultipher / Be en. ; the HN Bont ae / oii be : 5 dy sin operation solves the differential equation a Bye (ma+e). The differential equation (D.E.) is not included in the scope of I this treatise, and the name even is kept out of sight; go that a third volume looms in the future, to replace the treatise prepared by Bertrand, destroyed in the siege of Paris. But the novice should become accustomed at the outset to ‘the name and the simple operations. To cure his fright, the integration of f(«) should be proposed as the solution of the D.E. ES TT, LE Te Re eS a eS ge Re Ce ls, a ye Fer =57/(79)). and then the interpretation of the arbitrary constant, 59 singular solution, node-, tac-, and cusp-locus, can be illustrated in a graph as it occurs. Fa en an ee TS hm Geological Society. 639 Take as a type the simple case of the D.E. ss 25 The solution of the D.E. of the second order with constant coefficients is implied tacitly in all treatises on Physics, in the 2 ae + ; ; form qe tn e=0 for simple vibration, and the penultimate pendulum oscillation. We put in a plea here for a return to Aristotle’s name of Centre of Gravity, instead of the variants coming in of centroid, centre of mass, of figure, of volume, of inertia. A mental reservation is implied here of the effect of a radiating field of gravity, on a body according as it is non-centrobaric. This is a refinement that cannot be detected in a careful: weighing in a balance, the most accurate operation in a measurement in Physics. Call the areal coordinates of § 460 centrobaric, and extend the idea to any number of particles, to make great use of the idea in a number of dynamical problems. Colonel Hippisiey’s Ephelkustikon curves deserve mention, as described by a jointed tail, leading sometimes to a good illus- tration of the asymptotic circle, encountered in some central orbits. The figures are careful and accurate; and we are spared the too frequent footnote. ‘ There is a copious Table of Contents, but no Index. LXXV. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from vol. xlii. p. 1024.] December 7th, 1921.—-Mr. H. G. Oldham, F.R.S., President, in the Chair. : | : ‘HE following communications were read :— 1. ‘ Jurassic Chronology : [1.—Preliminary Studies. Certain Jurassic Strata near Eype’s Mouth (Dorset): the Junction-Bed of Watton Cliff and Associated Rocks.’ By 8. 8. Buckman, F.G:S. This paper deals with certain Jurassic strata near Eype’s Mouth on the coast of Dorset; but, as it forms part of a series of pre- liminary studies in connexion with Jurassic chronology, certain details connected with other localities are noticed. A general section of the main mass of Watton Clif east of Eype’s Mouth is given. A detailed section is recorded of a remarkable white litho- graphic bed in Watton Cliff, one in the same position as the Junction-Bed, but differing much in faunal and in stratal details. This bed shows faunal inversion, presumably due to redeposition of material from older deposits. “to eer wore 640 Geological Suciety. The dating of this Watton Bed is discussed, after preliminary investigations into the sequence of horizons in the Upper Lias of various areas, in the Junction-Bed, and in the pre-Junction-Bed strata of Thorncombe Beacon. A theory of stratal repetition and coalescence is discussed in regard to the Watton Bed. Its main date is taken to be Yeovilian, Hammatoceras hemera. The white hthographic bed of Burton Bradstock is cited as evidence of stratal repetition, and a theory as to the deposition and partial destruction ot this Burton bed is put forward. The Watton and Burton lithographic beds are cited as evidence of Alpenkalk conditions prevailing in Western Europe during two — well-separated Jurassic dates, yr of them far earlier than the times of Alpenkalk deposits in Central and Hastern Europe. Certain remarks are made upon sections at Milborne Wick and Haselbury (Somerset) in regard to the dating of their deposits, and a table of the succession and distribution of Hammatoceratids is given. A paleontological note describes a new species of Rhyncho- nellid—a species marking a particular deposit at Thorneombe Beacon. Appendix I, by Mr. James F. Jackson, gives the result of his studies of various sections of the Junction-Bed. Appendix IT, by Dr. L. F. Spath, Mr. J. Pringle, Mr. A. Temple- man, and Mr. Buckman, deals with the Upper Liassic sequence, with especial reference to facts disclosed by recent excavations at Barrington (Somerset). 2. ‘ Banded Precipitates of Vivianite in a Saskatchewan Fire- clay.’ By John Stansfield, B.A., F.G.S. A pale-grey Tertiary fire-clay worked for fire-bricks at Clay Bank (Saskatchewan) contains bluish-black spherical, ellipsoidal, and cylindrical patches varying from + inch to an inch and a half in greatest dimension. The central portions of the spheres, and the long axes of the cylinders are deeply coloured, and are usually sur- rounded by a uniformly stained area or by several concentric stained layers of varying, but less intense, tint. The colour is due to an earthy amorphous variety of vivianite, formed presumably by precipitation. The precipitation has been brought about by dilute iron-solutions reacting on solutions of phosphates of organic origin, such solutions being brought together by diffusion through the colloidal matter of the clay. The spacing of the vivianite-bands is irregular, and appears to follow no known law; but similar bands of precipitation may be produced in gelatine by the reaction of certain soluble salts one on the other. A small specimen of plattnerite (lead dioxide), a very rare mineral recently rediscovered at Leadhills (Lanarkshire), was exhibited by W. Campbell Smith, M.C., M.A,, Sec.G.s. Raman & SETHI, Phil. Mag. Ser. 6, Vol. 43, Pl. VI. * servolr. , Blower, and Re ied) = 4 on = ae 2 [ = ¥ o ral om i i ipa li _ a i "8 aD (=| oe a do ee Tie. 2.—Showing 200-foot pip co Raman & SETHI. Phil. Mag. Ser. 6, Vol. 43, Pl. VII. I'1e. 4.—Showing observation end. Fic. 6.—Reversing Gear. ia. 6.—Showing further end of pipe-line. Waray. Phil. Mag. Ser. 6, Vol. 43. Pl. VITT. (b) (d) CHUCKERBUTTT. Phil. Mag. Ser. 6, Vol. 43, Pl. IX. TT i tal arrangement. xperlmen Photograph showing the e NARAYAN ES OS er Ne ae eee 2 Ss ee a a ns ee Phil. Mag. Ser. 6, Vol. 43. Pl. X. Photograph showing the optical arrangements, Phil. Mag. Ser."6, Vol, 43. Pl. XI. Ine ae lL ATAYAVAT ti AL ay WWW WW VAY / f y as | . Mag. Ser. 6, Vol. 43, Pl. XII. ARAMA \ b e INNA peat All "y ‘é yy"? * ae | THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND LAL OF SCIENCE. ‘e ; Ren A aN j ; is LAND \ APR 2 0 [SIXTH SERIES.] AP BAL A922. LXXVI. Note onthe Theory of Radiation. By C.G. Darwin, Fellow and Lecturer of Christ's College, Cambridge”. te: HE theory of optics and the thermodynamic theory of radiation both deal with the electromagnetic changes of free space, and yet their methods of treatment are so widely different that hardly any point of contact is to be found in their fundamental definitions and procedure. In optics the primary description is in terms of the electric and magnetic forces, and an arbitrary field is specified by giving those forces at all times and places. On the other hand, in thermodynamics the analysis is entirely different, since it is in terms of radiant energy, passing through every point in every direction and sorted out according to its frequency. Now, though the spectroscope makes it fairly clear what is meant by “the energy in a certain range of frequencies,” yet a rigorous definition is lacking, and even the conception of streams of radiation going simultaneously in all directions does not lend itself at once to the ordinary energy formule of electromagnetic theory. It appeared to me, tnerefore, that it might be useful to place on record the formal connexion between the optical and the thermodynamic descriptions of an arbitrary field of radiation ; its outline must have been present in the minds of most writers on radiation, but I have not seen it in any work on the subject. * Communicated by the Author. Phil. Mag. 8. 6. Vol. 43. No. 256. April 1922. Zee 642 | Mr. C. G. Darwin on the It should be said at once that it seems improbable that any practical use could be made of the relations, or even that an example of them could be given. Tor in an example it is only possible to specify the optical quantities by virtue of some regularity in their character, and this very regularity makes them unsuited for thermodynamic expression. 2. The only process by which the two types of quantity can be related consists in expressing as integrals the whole flow of energy across an area in terms of both, and then identifying the elements of the two integrands. In no other way can a rigorous definition be given for the thermo- dynamic quantities. The modus operandi consists in the repeated application of Fourier integrals. In the present note no pretence is made to mathematical rigour in the deductions, but there can, I think, be no doubt of their validity. The most important step in the process is based on a theorem, due originally to Stokes*, and used for the present purpose by Rayleigh t. Stokes and Rayleigh both proved this theorem by introducing an exponential factor to help convergence. Here I have not used one, as it would probably be as hard to justify its introduction as to prove rigorously the result without it. ‘To save confusion with a more celebrated theorem, and because Stokes’s work dealt with a rather different aspect, I shall call it Rayleigh’s theorem. It will be convenient to exhibit it here first, so as to make its repeated use in the later work easier to follow. T/2 Let a — 2f Z(t) cos 27vt dt, ~T/2 a (2'1) v2 ( Z(t) sin Qarvt dt, J-T2 where Z is a function of t, which is restricted in such a way that ¢, and yr, are continuous, and vanish for large values of v according to some appropriate rule of convergence (the sequel suggests that the necessary condition is d,, th =0(v 2), e>0), while they vanish for v=0. T is so large that for yp=0(1/T), ¢, and wy, are negligible. By the principle of the inversion of Fourier integrals, if Z,= | ($, cos 2apt-+p, sin 2rvt)dv, . . (2:2) 0 then Z,=Z in the interval +T/2 and vanishes outside it. * Stokes, Edinb. Trans, xx. p. 817 (1852), 7 Rayleigh, Phil. Mag. xxvii. p. 460 (1889). ee ae a ee a ai ey ; - tas ns ) Theory of Radiation, 643 Let Z? be the mean of the square of Z taken over the range == 1/2. ‘Then T/2 ae Ran we dy'(d, cos 2arvt +, sin 27) x (p,, cos 27rv't+h,, sin 27v't). With the restrictions on ¢, and yf, it is permissible to invert the order of integration, and to take that for ¢ first. Then gil ‘a a 4 sin 7(v'—v)T Basi) ol a} 6.409) aor +b. — tah) emt. Following the usual reasoning for Fourier integrals, with the conditions imposed on T the important part of this is the first term, and for it v! can be changed into » in the first factor. Performing the integration for v’, we then have Fao CEES | eee ere which is the theorem. There is one point in this result that deserves mention, and that is the presence of the factor 1/T. For at first sight this suggests that if the time considered were long enough, the average would tend to vanish. ‘This is not so, hechtise the magnitudes of @, andy, will vary with T by (2:1). Still if Z were quite arbitrary, Z? would naturally depend on the exact value of T. The utility of (2°3) depends on its appli- cation to rather more specialized types of function. For use in radiation theory we must attribute to Z the property which the radiation, in fact, has—that Z? has a value independent of the exact value of T: and @, and yx, will then be proportional to.) But it we not seem possible to make T disappear from the expression (2°3). 3. The radiation field is specified in the optical manner by means of the six components of electric and magnetic force. ‘To derive from these the thermodynamic description it is necessary first to express the optical quantities as a set of plane waves going in all directions; then the Poynting vector must be calculated from these, and must be integrated Zen 2 644 Mr. C. G. Darwin on the so as to give the total flow of energy through a given aperture in a given time. By successive applications of Rayleigh’s theorem the result can he brought to an integral of the same form as the corresponding thermodynamic expression, and the Oe is then given by equating the two integrands. Let the optical field be given be X, Y¥, 4; o&, 63, wih are arbitrary functions of «, y, <, t, subj ae to hee eee the electromagnetic equations and to suitable conditions as to continuity and convergence. We shall evaluate the total senda crossing the yz plane between y=+p/2 and = + 4/2 during the time between t=+T/2. Here p and 4 are to be so lar ge that they include many wave-lengths of the types of waves that occur, and the same must be true for T and the periods. On the other hand, they must be sufficiently small to allow for the gross variations of radiation in the different parts of space. These limitations are like those which are imposed in the kinetic theory of gases. More precisely the functions X, Y, Z; «, 8, y are such that the averages to be constructed shall be independent of p, q, and T. It should be said that this independence will not be apparent formally, but will occur in the same way as in (253). 4. We proceed to consider how the radiation is to be analysed into plane waves. A typical plane wave is described by the expression S= 91° COST + Js SIN Ti, where ; tT=2rv{t—(asinOcosd+ysindsing@+zcos@)/ch, (4:1) while S and g§ represent in turn each of the components of electric and magnetic force according to the rule *=Ucos@cos¢6—Vsng g*=—Usind—V cos 6 cos ¢) =Ucos@sind+Veosd g=Ucosd6—VecosOsmg >». 4——JU sin? g¥=V sind ( Here U corresponds to the wave polarized with electric force in a plane through the < axis and V to that in the perpendicular direction. The arbitrary field can be represented by taking four quantities U,, i oe V.as functions of vy, #, @ and expressing any of the forces ae 0 Aloe as Ton Si) dig ( sin 6d0 do(gi§ cos T+ go* sin T). e 0 e 0 " (4:2) Theory of Radiation. 645 As we are to consider the flow of energy through the aperture in the ye plane, we put «=0 in this expression, thus changing 7 into T) = 2rv{t—(ysin @sin @d+ ++ + Px, but is independent of the space variations of these quantities ; (i1.) F depends not only on o, but on Oc Oc Oc Ox’ oy’ 0z where o is ascalar quantity defining the state of aggregation ; (iv.) Fis a continuous function of the n+8 independent Walables O, Gx, Gy, Oz, OY, pi, Po -~ +n: Ur, Dy, D;z, where Oc Oc Oc On OY; OF D. are the components of the electric displacement ; (v.) all the independent variables o, ox, oy, 72, 9, 21, Po, +++ Pay De, D,, Dz are continuous functions of the rectangular co-ordinates xv, y, z; (vi.) F is sensibly independent of Ox, Oy, Fz in all the points outside of the transition layer ; and (vul.) the elementary work done per unit volume on the electric field when the densities are kept constant is E,dD,+E,dD, + E.dD., , and not on the higher space variations of a, Or, Cy, Gz Stand for respectively, and D,, D,, where Ez, H,, Ez are the components of the electric force. The quantity o above stated requires some explanation. The functional form of F differs according as we take the first fluid or the second. For the first fluid we put c=o’",a * Communicated by Prof. A. W. Porter, F.R.S. 650 Mr. Shizuwo Sano on the T hermodynamical constant, for every point in the fluid for all values of , P15 Po, +++ Pn, Dz, Dy, Dz. For every point of the second fluid, is assigned another constant value of o, say a”. I consider every point in the transition layer on a line normal to the layer to have different values of c varying from oa’ too”. It was hinted in my former paper on the equilibrium of fluids in an electromagnetic field * that a theory explaining the existence of surface tension can be formed by assuming that contains oz, oy, oz. According to my theory, every point along the orthogonal line in the transition layer is in a different state of aggregation and is in stable equilibrium, both mechanically and chemically, although I shall not enter into any investigation of the conditions of stability. Though I assume that the fluids vary continuously from one phase to the other, the eonception differs entirely from that contained in van der Waals’s equation of condition. In the paper referred to, 1 proved that a conditions of chemical equilibrium are o a= GiVetV == Ue ct 2, OR Sass n|, Snore (1) where We | W, are electric and gravitational potentials respectively, and q; the quantity of electricity associated with the unit mass of the 7-th constituent, and C; is independent of v,y,2. Inobtaining (1) it was assumed that the gravitational field was uniform, but it can be easily seen that the proof is more general. In (1) the effect of mutual gravitational attraction of different portions of the fluids under consider- ation is neglected. It was also proved that if there is a reaction equation of the form Sy =0, 0 1 where Si represents a molecule of the i-th constituent and v’s are integers, then the equation of chemical equilibrium cor responding to the reaction (2) is Sram =(0), 2. ce m; being the molecular weight of the i-th constituent. Proceeding in a similar way as in my paper above men- tioned, the equations of mechanical equilibrium are easily * Proc. Math.-Phys. Soc. Tokyo, ii. p, 365 (1905) ; Physik. Zeitschr. vi. p. 566 (1905). Theory of Surface Tension. 651 found to be ~ eed ven es 3 OF \ =2 {- F4 “6. > pi St HyD, + Bed. }) ae: peels at —H,D,—B,Ds | | aes Be +. & a # 9.8F up,-u.p,}, +35 Y=. ee J where X, Y, Z are the components of the impressed force per unit volume. Now, suppose that the layer of transition is not placed under an external electric field, so that we may take the component electric force parallel to the surface c=const., which. passes through the point under consideration, to vanish. When the sere layer is not horizontal, there will be a component of the electric force parallel to the layer, due to the effect of gravity, but this is so small that for all practical purposes it may be neglected. The electric displacement outside the layer is, by the same reasoning, taken to be zero. Let us consider a line orthogonal to the series of surfaces «= const. and take two points P’ and P” on this line, P’ being in the first fluid and P” in the second, both being very near to the transition layer ; and let ds be the element of the line. Then it immediately follows from (4) that the surface tension is = (on ( z felon where os denotes the magnitude, of the vector (ox, cy, oz), and E and D are the electric force and electric displacement respectively, and the integral is to be taken from P’ to P”. By dint of these pare ene the position of P’ along the curve does not alter the value of the right-hand side of (5) so long as P’ lies outside the transition layer. Now suppose that the axis of # is vertical, and that the surfaces ¢=const. are horizontal planes and that the variables are independent of y and <. In this case we may write (9) in the form a oF 1=((c. felons (9) (6) a 652 Mr. Shizuwo Sano on the Thermodynamical It pe denotes the electric charge per unit volume, then C= oD pe= 2 gi, >. a so that (6) may be written T= { ( or 95 —Wepe) de, er since D vanishes at P’ and P”’. The phenomena of pyro-electricity teach us that E does not vanish at the same time as D in crystals having polar axes. It seems to me that it is quite natural to suppose that at any point P within the transition layer the fluid may be considered just like a crystal having a polar axis, normal to the series of surfaces o=const., 2. e. the electric force g, contains oy and does not vanish when D=0O. If thisis really the case, both ox i : and V., contain terms depending Cz upon the distribution of electricity in the layer of transition, so that it is impossible to conceive that — \ Wrepe dx would be the term arising from the double sheet of electricity in the layer. If we assume that F is independent of oz, then (6) would become T= — { ED dz, where E vanishes simultaneously with D. Since we expect that E and D have the same sign at P, this value of T would become negative, 7. e. the mechanical equilibrium would be labile. Hence if we assume that I is independent of cy, then the fluid cannot exist in two phases in contact. Discussing this from another point of view :— In the following, I assume that the temperature is given, and neglect the effect of gravity altogether. Let the quantities referring to the point P’ be accented once, and those corresponding to P” twice. Now, assuming that F does not contain o;, then from ; ‘Ly. (ye 2) we get = a oe De gt wk) ccliee le EF -F'+3p/S% = TS we wl Theory of Surface Tension. 803 First, suppose that there is only one independent con- stituent, and that all the molecules are electrically neutral. Let there be n—1 kinds of molecules S,, 83, ...S,, all of which are formed by the polymerization of 8;. In this case there are n—1 equations of the type (10), so that there are 2n equations (Clo); and a detormun ie 2n unknown quantities p> Dee Pn 5: Pili Pols © ae eee Hence the constants C in the equations F iy - = = =0 5. l= He he . n|, (12) oF * ue op: SS seal ic 5 5 5 : (13) become determinate. Since we have supposed that all the ° oD e molecules are electrically neutral, Si =0, from which we have D=0, since by our supposition D vanishes at P’ and P”. Hence F contains only » unknown quantities p;, 02, . . . px for any value of o between o’ and o”. But these quantities would have to satisfy n-+ 1 equations (12) and (13), which is impossible. Next, take the case of the system composed of h inde- pendent constituents, the molecules of which are all supposed to be electrically neutral. In this case there are n—h inde- pendent reaction equations of the form (2), to which cor respond n—h equations of the type ClO Since we can give any assigned values to po'/pi', ps3 ‘Tex sn 7. ‘/Py between proper limits, 2 2n unknown quantities p;', ae Me Bis Pin wPorwuee Pn are determinate corresponding to any set of values of p2'/p1',- - - pr |p, , so that Cy, Cy, . . . Cy41 become determinate. Just as in the preceding case in which there is only one independent constituent, the 1 densities p,, ps, ..~. px would have to satisfy n+1 equations (12) and (13) in the interior of the transition layer. Thus, if we suppose that there are only neutral molecules in the fluids, and also assume that Fis independent of oz, then it follows that it is impossible to consider that there would be a transition layer in equilibrium between two fluids. Next, suppose that F is again independent of ow, and that there are electrolytically dissociated molecules. First, take the case in which the two fluids consist of only one independent constituent, so called. In this case the molecules §,, 85, ... Sn are connected by n—2 independent > 654 Mr. Shizuwo Sano on the Thermodynamical aos esos and the 2n+1 unknown quantities ilo JONG WOU ora er aroha Pr’ ¥. aa are ice at In+1 nea quantities Die Poe ok. Pn! Pine wv "—W,! are determined by 2n+ 1: equations (9), "(10), iy. and Zoid =0,; Vero: =0. |. es Hence, if we write F : as +q(V.—WVe)=Gi, 2=1, 2 » 10 | Se te) F ie —H+ 3p; = =Grai ot 0 eel then the constants G are determined quantities. Solving the n+1 equations (15) and (16), the n+1 quantities ~(V.—WVe'), pi, po ---pnand also the right-hand side of the equation oD Ow become functions of D and o only. Differentiating V,'—W, with respect to # and making use of the equation oF =Sgpi 3 47 ie and then eliminating D, we get an equation in the form do d’ce > dx’ de> 0, from which we deduce a solution of the form o==3(2 4A, B), wc. Cee (19) where A and B are constants of integration. During the process of reaching the final solution of o as a function of x, we have differentiated V. —‘V, with respect to xz, so that we ‘cannot expect (19) to satisfy the condition that We —Y, vanishes at P', unless the arbitary constants in (19) are so chosen as to satisfy the condition. All the solutions of (17) satisfy the condition Qo" =0 at =o’, but not D=0 at the same point. Hence, in order that equation (19) may be our solution, it must satisly the three conditions that y.—WV,' and D aah at P’ and o becomes o’ at the same Theory of Surface Tension. 659 point. But (19) contains only two arbitrary constants, so that it cannot satisfy all these three conditions. Therefore in this case the transition layer cannot exist. Tt will be remarked that in this case the transition layer is just like a fluid in which there are two independent con- stituents, since n kinds of molecules are connected with each other by only n~ 2 independent reaction equations. In the case in which there are more independent constitu- _ ents, the reasoning quite similar to the above holds good, and we see that the neglecting of terms containing go, in the expression of F' leads to impossible conclusions. Hence, it seems to me that when we are considering the equilibrium of fluids in the transition layer we must suppose that F depends upon or so that the existence of surface tension is itself closely connected to the fact that the system can exist in two phases in contact. Now suppose that F depends upon oz. At first let there he neither cation nor anion in the fluids. In this case our equations are op Sp; Cn =L2---mb. - - . (20 pre actgn | hs —FtosS= +3055 —Oun - - QL) WE Ope eet gee tartan (2 where the n+1 constants © are to be ane as deter- minate. A glance at equations (20), (21), (22) shows that _the equation ‘determining oasa Fatdtion of a is a differential equation of the first or der, the constant of integration being determined by the fact that oo" at B. It is more general to suppose that a does not vanish at P when D=0, and that Ye eo = = dx does not vanish. When V,'—W," does not vanish, the layer of transition may be looked upon asa double sheet of free electricity, but not of real electricity. Finally, let us suppose that F contains cz and that there are electrolytic ions in the fluids. In this case our equations are (15), (17), (18), and (21), and the equation for determining o as a ae of « is a differential equation of the third or der, 656 Mr. Shizuwo Sano on the Thermodynamical so that o can satisfy the three boundary conditions that V.—wW,. and D vanish at P’ and o becomes o' at the same point. Atthe same time 94, po,... pn, D become definite functions of 2, and the distance between the two points at which oe becomes sensibly equal to zero gives the thickness of the ae layer. It appears to me at present that the case last miéntigned contains the most natural assumptions as to the conception of F, i. e. (i.) the fluids are electrolytically dissociated, (i1.) F . gy Oo we depends upon Saree: Oe okey le when o becomes either o’ or co’, although the Ow’ Oy’ OZ > absence of electrolytic ions does not prevent the existence of the transition layer. In will be remarked that Mes aae to this theory the difference of electric potentials between P’ and P” is a definite quantity provided: that the physical and chemical conditions at P’ are given, and that the condition of the maximum surface tension is quite different from the condition that the difference ef electric potentials between P’ and P"’ vanishes, so that it is not probable that these two things always occur at the same time. I shall conclude this paper by giving other expressions of surface tension and by describing the expressions for some quantities relating to 1, always neglecting the effect of gravity. If we write (1) and (4) in the form (ii.) F’ is sensibly independent of os +07 Ve Nn, [2=1, 2,..-)m]5 fe mre (23) E —F+oy o + pi See 3) Ao ee where p’ denotes the pressure at P’, then (8) becomes 1=n T= =(Fdx+p'\de—>Ci\ pide. . ee) In the case of only one independent constituent, so called, (25) can be written / ft = ra +p'(de _ 5a? dx ! ! v] é = ((-2” a ) pdx, halt (26) Theory of Surface Tension. where p;’ is the density of an electrically neutral constituent at P’, and p and p’ the densities at P and P! respectively. In (23), O; stands for | | | iw SF 7 +qiVe ? so that C; contains a term of the form M@+ N, where M and N are arbitrary constants. F contains a term equal to (M@+N)o. so that the right-hand member of (25) is independent of the arbitrary constants M and N. _ If it were possible to choose the constants M and N so that p' \de— Cif pide could be neglected with respect to T for all the possible values of the temperature and densities at P’, then (25) would simplify into NBT OW ky a) | ee) which, I consider, is not a proper expression of [. Equation (27) has apparently the same form as commonly given, but it is different in these respects: F contains 0¢/O2, and is made up of two terms, the first being equal to f, EdD and the second being independent of D but a function of n+3 variables oa, oe 0, Pi, Po; - ++ Pa, Where during the integration these n+3 variables are to be kept constant. Next let us find the relation between the temperature coefficient of surface tension and the entropy per unit area q | : ST SES er Se RR al 7) the integral being extended from P’ to P”. In the case of one independent constituent, so called, the temperature coefficient of T is definite ; but ee fnere are more independent constituents, there is an infinite number of the temperature coefficients according to the different modes of change. Let us take any definite mode of change and denote the corresponding variations by using the symboi d. By (23), (24), (25), and (28) we have at once EEE! 2 Re an OY Gea tus hy | ae ~ 5g | pede ena 4S) Phil. Mag. 8. 6. Vol: 43. No. 256. April 1922. 2U 658 Thermodynamical Theory of Surface Tension. Now take the case of only one independent constituent, so called. Here (29) becomes aT dp' {° OF!) ¢ 10 +H= Wo da— (Soy) Jae - . (30) Take the same case and let a unit area of the capillary layer be formed from the first liquid under a constant temperature and in a reversible manner, and let W be the work done on the mass spreading over the unit area of the layer, and Q the quantity of the heat absorbed by the same mass during the change, and U its intrinsic energy. Then it can easily be shown that w=t+2{ (— —p' dz, ne dT 1 dp’ =o - : ; FV (p—plde, . (82) U=w'(pde+t—0 + ae! see 5) (le p')da, (33) where w’ is the intrinsic energy per unit mass of the fluid at P’, As may be expected, equations (31), (32), and (33) have the same form compared with the case in which the electric field in the layer of transition is not taken into account. Now consider the case in which there is neither cation nor anion. When the layer is placed in an electric field normal to it, the surface tension T depends upon the electric displacement D’ at P’. By using (6) and observing that the right-hand sides of (20) and (21) are even functions of D’, it is immediately seen that when D" is infinitely small and the temperature is kept constant during the change, which shows that the surface tension is maximum or minimum for, D'=0, although, perhaps, V.'’—V-.' does not vanish. Lueso | LXXVISI. A Significant Hexception to the Principle of Selection. By Paut D. Foorr, Ph.D., F. L. Mouwuer, Ph.D., and W. F. Mreacsrs, Ph.D.* [Plate XIIL] CCORDING to the Rubinowicz f principle of selection, the azimuthal quantum number, in any transition of an electron between two stationary orbits, resulting in radiation, may change by only +1, 0, or —1. Now, in the enhanced lines X= 4656 and 3203 emitted by the ionized helium atom, there appear to be certain experi- mentally observed components which on the basis of the above principle should not exist. Rubinowicz states that the fact that these extra lines have been observed, for which the change in the azimuthal quantum number is 2, obviously tf may be attributed to the presence of the electrostatic field exciting the radiation. While from a consideration of the application of the quantum theory to the Stark effect, and the resulting necessary introduction of a third quantizing process in the mathematical analysis of the lines, Rubinowicz’s explanation is consistent with possible fact, it is, of course, by no means a necessary deduction. On the aspherical nucleus theory of Silberstein §, for example, there should be many more components than would arise with the radially symmetrical (2. e. spherical) nucleus assumed in the Rubinowicz theory, and in the more restricted theories of Bohr and Sommerfeld. A most interesting extension of the quantum theory to the derivation of general spectral series relations, for atoms with several electrons, his been made by Sommerfeld ||, in which he associates definite azimuthal quantum numbers with the series terms: for example, nz=1 for the ms terms, 2 for the mp terms, 3 for the md terms, and 4 for the md terms. Here, again, exceptions to the principle of selection have been noted and have been attributed to the presence of the exciting electrostatic field, an explanation entirely satisfac- tory from the mathematical standpoint, if one admits that the lines would not be present in the absence of the field. * Communicated by the Authors, Published by permission of the Director, Bureau of Standards. t Rubinowicz, Phys. Zeit. xix. pp. 441 and 465,1918. An application of the principle of conservation of moment of momentum to the inter- action of ether and matter, in addition to the postulates originally proposed by Bohr. { Loe. cit. p. 466, “ handgreiflich.” § Silberstein, Proc. Roy. Soc. xeviil. p. 1, 1920. || Summarized in several chapters of ‘ Atombau.’ 2U 2 eee ee _ ae se a T= ee —— a 660 Drs. Foote, Mohler, and Meggers on a Significant In all the discussions so far published in regard to the failure of the principle of selection, the field referred to has been that applied between the two electrodes of a vacuum tube. The ‘ breaking through ” of the principle with helium is a trifle more pronounced in a high voltage capacity dis- charge than in a lower voltage direct current discharge, supposedly because of the oreater field in the former case. It is readily possible, however, to eliminate the effect of such a field in a suitably designed ‘discharge tube. One form of such a tube is illustrated by fig. 1. This consists of a central electrode closely surrounded by a cylindrical net which is similarly enclosed by a cylinder of much larger diameter. The net and outer cylinder are in metallic Radiation Fig. 1.—Schematic representation of a discharge tube in which the character of the radiation is not Gieonee by the exciting electrostatic field. contact, and the field is applied between the central electrode and net. Thus in the region between the net and cylinder, where the character of the radiation is observed, there exists no applied field. By use of an induction coil or a high voltage transformer very high intensity of radiation is excited, probably sufficient for fine structure analysis. It is desirable for certain types of work to replace the central electrode by a Wehnelt cathode or heated tungsten filament, in which case the applied field may be reduced to the order of a few volts, to the ionization potential if are spectra are studied. Currents of 25 amperes and more at 20 volts are readily obtained with hydrogen. The radiation fr om such a tube containing potassium vapour is shown in‘tig. 2 (Pi XITE). The total applied ‘field was hut 7 volts, cl the spectroscope was sighted on the light emitted in the force-free space between the outer electr odes. With a current of 70 milliamperes leaving the filament, the Huception to the Principle of Selection. 661 ordinary arc lines (principle, 1st subordinate, 2nd sub- ordinate, Bergmann series) are excited, as shown by the upper spectrogram. On increasing the current to 1000 milliamperes, all other conditions remaining identical, the pair \ 4641 and A 4642, entirely absent at 70 milliamperes, now become one of the strongest pairs in the spectrum, as shown in the lower half of fig. 2 2 (PI. XIII.). These lines have the spectral notation 1s—3d ‘and are accordingly, if Sommer- feld’s interpretation be accepted, the result of an interorbital transition in which the azimuthal quantum number changes by two units. Similar results were obtained with sodium. The fact that the linesarise when the ewciting field is only 7 volts, and in a location well shielded electrostatically from even this small field, makes it extremely doubtful if this and the other above mentioned exceptions to the principle of selection may be attributed to the incipient Stark effect of the applied electrostatic field. The effect in general appears closely related to the magnitude of the current, and might ther efore he of a magnetic reir: However, the magnetic effect of a projected beam of electrons should not be large within the beam itself, from which the radiation is observed. A possible explanation may lie in the influence of the electric field of neighbouring ions and electrons, the concentration of which may increase with the current. On the other hand, the explanation may involve a reconsideration of the ee wher eby single azimuthal quantum numbers have been assigned to each of the s, p, d, and b terms. Summary. A new form of discharge tube is described in which the applied electrostatic field can exert no influence upon the radiation. Using this tube, the lines 1s—3d were observed to be oneot the moan. intense pairs in the potassium spectrum, with a current of 1000 milliampers es, while at 70 milliamper es their presence was not detected. The excitation of these lines, for which the change in azimuthal quantum number according to Sommerfeld is two units, is contrary to Rubinowicz’s principle of selection as extended by Sommerfeld. It cannot be attributed to the Stark effect of the applied exciting potential. This fact suggests that the extra components in the fine structure of the helium lines likewise may not be ealled out by the applied field. In conclusion, the writers desire to express their apprecia- tion for the suggestions made by Dr. L Silberstein, who very kindly read the manuscript. Bureau of Standards, Noy. 8, 1921. LXXIX. A Modified Ronn of Double Slit Spectrophotometer. By A. L. Narayan, M.A.* [Plate XIV. [* different kinds of spectrographic work which have been carried on in the Physical Laboratory of the College, the need for a good spectrophotometer in the visible and ultra-violet regions is keenly felt for the exact location of the absorption centres and for the measurement of their intensity. One of the spectrophotometers adapted to the purpose is the one designed by Vierordt. In this, the colli- mator is furnished with a double slit in front of which the absorption cell is placed in such a position that the dividing line between the two layers of different thicknesses or the dividing line between the liquid and air coincides with the junction of the two slits. And the slit widths are adjusted until equal illumination is obtained in both the spectra. But the difficulty in the way of regulating the brightness of the spectrum by altering the width of these slits is that it seriously interferes with the purity of the spectrum. A modified form of double slit spectrophotometer has been devised, which is free from the defects of Vierordt’s type. At the same time, it possesses many of the advantages of the sector photometer. It consists essentially of an electromagnetically maintained pendulum the period of vibration of which is °5 second nearly, so that it executes 120 oscillations per minute. ‘The bob of the pendulum is a heavy frame of brass containing two rectangular apertures one above the other, there being a well-defined line of demarcation between the two. Two photographs are given, of which No. 1 shows the photometer and its parts, and No. 2 the photometer in position before the spectrograph. The widths of both the slits can be adjusted by independent micrometer screws. The whole pendulum is mounted in front of the collimator slit of the spectrograph such that the line of separation between the two slits almost bisects the collimator slit. And the bob of the pendulum can be raised or lowered by a small amount by working a nut by means of which the whole frame constituting the bob is fixed to one end of the pendulum rod. Since the period of the pendulum is *5 second, the whole * Communicated by the Author Lol Bi A Oy te ee ed i oF ig toy Wt, Tension of Soap Solutions for Different Concentrations. 663 length of the pendulum will not be more than 25 to 30 cm., so that the whole arrangement is very compact. Further, by the electromagnet arrangement, the oscillations will have a constant amplitude, and the amplitude also, if necessary, ean be varied within certain limits. Tt is also possible to produce very small diminution of intensity unlike the case of the sector photometer. It is free from the complications of a motor. The slit-widths are adjusted so as to get equal illumination in both the spectra. If I’ and I” are the intensities of the incident and absorbed beams, we have Ani ee a rv P=Ve ‘ where & is extinction coefficient, and d the thickness of the layer, so that tk" log e= log (1 Ft) = log (b!/b""), where 6' and 0" are slit-widths. Therefore, by knowing 6’ and b", & the extinction coetiicient can be determined. Department of Physics, Maharajah’s College, Vizianagram, 10th November, 1921. LXXX. Surface Tension of Soap Solutions for Different Concentrations. By A. L. Narayan, VW.A., and G. ele nin BAS [Plate XV.] dhe question of surface tension of soap solutions has from time to time attracted the attention of several scientists, of whom the late Lord Rayleigh was one, who in a series of investigations drew several conclusions of abiding interest. In his paper, “On the tension of recently formed liquid surfaces” (Proc. Roy. Soe. xlvii. pp. 281-287, 1890), Lord Rayleigh, following Marangani, attributed the capability of extension of a soap film into large and tolerably durable laminz to the “‘ superficial viscosity’ due to the presence upon the body of the liquid of a coating or pellicle composed * Communicated by the Authors. 664 Messrs, Narayan and Subrahmanyam on Surface of matter whose inherent capillary force is less than that of the mass. In the case of soap, the formation of the peilicle is attributed to the action of atmospheric carbon dioxide liberating the fatty acid from its combination with alkali. This pellicle, Rayleigh remarks, is more akin to a solid than to a liquid. Since the formation of a coating would be a matter of time, Rayleigh confines himself to this particular aspect of the question, viz., how the surface tension of a soap solution (oleate of soda) changes with the freshness of the surface formed. From observations upon a jet of liquid issuing from an elliptical aperture, he concludes that the addition of a little oleate of soda does not change the suriace tension if the surface be fresh (Proc. Roy. Soc. xlviii. pp. 127- 140, 1890), proving thereby beyond all doubt that the change in the tension is due to external agency. The observations of Dupré (Théorte Mécanique dela Chaleur, Paris, 1869, p. 377), and the theory of Prof. W. Gibbs (Connecticut Acad. Trans. vol. il. pt. 2, 1877-78) lend strong colour to Rayleigh’s co:clusions. A strong argument in favour of Marangani’s theory is afforded by his observations (Pogg. Ann. vol. exliil. p. 342, 1871) that within very wide limits the superficial tension of soap solutions as determined by capillary tubes is almost independent of strength. In a recent issue of the Journal of Physical Chemistry, vol. xxiv. pp. 617-629 (1920), in the course of a long article on the surface tension of soap solutions, and their emulsifying power, Messrs. M. G. White and J. W. Marden described a series of experiments with the capillary tube, with the result that the surface tension of soap solutions decreases with increase of concentrations, from 49 to 20 and 40 to 21 dynes per centimetre, and that the emulsifying power is closely connected with the surface tension. As there is wide discrepancy between the results obtained by Lord Rayleigh and others on the one hand and the above authors on the other, the present investigation is taken up with a view to test the accuracy of the results, in which a long series of determinations of the surface tension of soap solutions at different concentrations is made by different methods—strictest possible precautions being observed im maintaining the temperature uniform, and in measuring the various quantities. Taking advantage of the bubble-forming property of a soap solution, the authors have thought it desirable te a Tension of Soap Solutions for Different Concentrations. 665 measure the surface tension by blowing bubbles, and finding the excess of pressure within the spherical bubbles so formed. This method, besides being more simple and elegant: in itself, is free from all assumptions of an uncertain nature, and is one which can very conveniently be applied for an accurate determination of surface tension of soap solutions at different concentrations. The experiment therefore essentially con- sists in finding the excess of pressure within a spherical bubble, and as this is found to be very small, extreme care is taken to measure it as accurately as possible. This has been effected by a differential micro-manometer, the essential part of which—the bed or the inclined plane, the inclination of which with the horizontal can be varied by an accurately cut micrometer screw—was made by C.8.1. Co., England, and designed to measure angles correct to a micro-radian. The manometer is an almost straiglit tube of 4 mm. diameter, with its arms making an angle of nearly 180 degrees and having a suitable length of xylene (density 0°850 grm. per e.c. at 28° centigrade) for the manometric liquid, which has been chosen for its extreme mobility and relative involatility. The tube is mounted on a triangular bed of steel resting on a hole-groove-and-point support and hinged about the first two. The platform can be tilted by working a screw having four threads to a mm. and its drum-head divided into 240 equal divisions. It is found by the authors that the whole arrangement far surpasses all other methods of measuring pressure differences both in delicacy and quickness of action. The manometer is connected to one end of a T-piece, the other ends of which are joined to a compressed air-chamber and a glass tube of 5 mm. diameter respectively. The tube is enclosed in a glass case with sliding doors, and is fed with the minimum quantity of the soap solution. By setting up communication with the compressed air-chamber, a bubble of suitable size is blown. On account of the difference in pressure the manometric column is displaced laterally and the position of the meniscus of the inner end is marked by focussi:@ a comparator, arranged in frout of the manometer. When the bubble is off the manometric column resumes its original position and the screw-head is now turned until, as judged by the comparator, the meniscus takes up the same position as when the bubble is on. The readings of the drum before and after the adjustment give the angle of tilt, I, of the foot-board. If is the length andd the density of the manometric liquid, it is evident that the difference in pressure is d.g./sin I. 666 Messrs. Narayan and Subrahmanyam on Surface To measure the diameter of the bubble a magnified image of it is projected on a graduated paper placed at a distance. The bubble is illumined with an electric glow-lamp arranged on one side of the glass box, and a good definition of the periphery is secured by making the side of the glass box next to the bulb of a ground glass plate. Some screens are also interposed between the lamp and the glass box, and a preliminary observation with a delicate thermometer indicated no temperature changes within the enclosed space. Without disturbing the position of either the lens or the screen, a translucent scale is heid horizontally in the position previously occupied by the bubble, and each of the equal divisions of the graduated paper is calibrated. This avoids the necessity for measuring u and v and the consequent error in calculating the magnification. This, of course, is done at the conclusion of the experiment. To make the description clear, a photographic reproduction of the arrangement of the apparatus is given herewith (Pl. XV,). TABLE I. Surface Tension of freshly prepared Sodium Oleate Solution. (By the Bubble method.) , Length of the manometric column... 9°10 em. Density of Xylene (Temp. 28° 1 C.)... 0°850 grm. per c.c. } | No. of rotations- Diameter Concentration. |of the micrometer| (to be multiplie (Dynes per cm.) by 3/5°35). a ae | a Surface Tension. | screw. Saturated | soiaian ql) 87 3°35 em. 23:50 | 0-667 of __,, 10 108 3°66, 93:53 | 0:445 ‘ ll 27 Bay) 23-71 0:297 i! 12 25 SU 23:23 0-198 is 10 57 3:60 ,, 93:19 0-132 i ll 82 BRAN 23-93 0-088 i hy Be | 344 |. 23°57 0-059 i 12 66 B10 23-4] | 0-039 u 12 58 312 ,, 23-50 Length of the manometric column: 9:00 cm. Other data precisely the same. , 0-:0909 5 ll 88 3°38 cm. | 23°33 | 0-:0625 - 11 199 3°26 ,, 23°46 | 00476 a 12 130 307 =, 23°42 | 00385 i 12 179 2 OSie: 23°11 | 003822 si No bubbles could be blown. | ie een Tension of Soap Solutions for Different Concentrations. 667 TABLE II. Surface Tension of Castyl Soap Solution used for Water Analysis. (By the Bubble method.) Length of the Manometric column...... 9°40 c.m ; No. of rotations Diameter ls face T Concentration. |of the micrometer} (to be multiplied | ae ran ay screw. by 5/5°70). Ea Oe Saturated solution. 13 121 300 cm. 23°66 peo) bof! ,, 13 185 2°93. ,, | 23°61 0°33 53 13 00 | 309). 5; | 23°52 Length of the manometric column: 9°75 em. | | | Diameter | | (to be multiplied | | bby 8/5734). | 0°25 SI 12 189 | 2°86 cm. | 23-70 O17 tient 12 69 296 ,, 23-57 0-090 Se 12, 237 280. ;, 23°57 | 0062 sel) | id 87 238 ,, 23°69 | 0-048 Bt | 14 71 2°55) ,, | 23°62 | 0-038 et | 14 236 | Pare Ti Me 23°98 | 0-032 sy hl 15 46 2°40 ,, | 23°62 | 0028 | 16 45 ‘7 S14 24-23 | 0-024 STINET 16 45 | vs) ae 24:23 For further dilutions no bubbles could be formed. To compare the values thus obtained with those by the eapillarity method, the surface tension for the same concen- trations has been determined by the capillarity method. Most of the tubes. to be ordinarily met with have been discarded for want of a perfectly uniferm and circular bore. For this purpose, the stem of a broken thermometer is found to answer excellently well, notwithstanding the elliptical nature of the bore. If 2a and 2b are the major and minor axes, the area of the section of the bore is evidently zab, and the perimeter 4 (* (a? sin? 6 —b? cos? b):dd J0 =4a ager. cos” f)2d¢, 0 668 Messrs. Narayan and Subrahmanyam on Surface where of course e is the eccentricity defined by eat" 7 2C, , (4-1) AC, =4a 54 lade 22 eee a4 94 MENARD 4 604 9, 3 74 which in the present case, as e is smaller than /, is a rapidly converging series and can be calculated to the. desired degree of approximation. Some readings have, however, been taken with a few capillary tubes having very nearly a circular bore. In the case of such tubes, correction for the weight of the liquid above the bottom of the meniscus in the capillary tube is applied aceording to Hagen & Desain’s formula ar 3 a’ ae ye ? where a? is the capillary constant, hy and fh are observed and corrected heights. The height of the liquid column and the diameter of the tube are measured in all these cases by a comparator (reading to hundredths of a mm.), for which we are indebted to the C.8.1. Co., England. The following observations (Tables IIT. to V.) are the mean of a number of readings, which, from considerations of space, we could not give in detail. Before each set of measurements the tubes are thoroughly cleaned by a hot mixture of nitric and chromic acids and washed well in plenty of running tap-water, and finally dried in a current of warm air. (In preparing the solutions, excess of soap is at first taken, and is dissolved in water. ‘The solution is then allowed to settle, and the clear liquid decanted into a beaker. It is then filtered twice. One hundred c.e. of this filtered solution is successively diluted with 106, 200, 300, 400,....c.c. of water. Hence the concentrations are expressed in terms of the saturated solution.) Té will be seen that the surface tension in all these cases never differs much from 23°5 dynes per cm. (Temp. 29° 1 C.), and is about the same «as Lord Riyleigh’s results, whose values range between 25°3 and 24°6 dynes per cm. (Temp. too: On the tension of water surfaces, clean and con- taminated, investigated by the method of ripples” ee Mag. OG.) 0) 0h 386-400, November 1890). ° h=hy+ Tension of Soap Solutions for Different Concentrations. 669 TABLE III, Sodium Oleate Solution: with capillary tube with an elliptical bore. 2a = 0:0255. 2b =0-0178 cm. Concentration. Height (corrected). ae i gta Saturated solution, 4-658 em. 22°57 05 of 4-707 ,, | 22-70 | 0°33 es 474... 22 88 0°25 ye ATSS oe, 23°00 0°17 % ALOT 23°63 0-090 * 4-908 _,, ASSET | 0°062 ie 4:999 ,, 23°93 0-048 ve D0SO.. 24°38 0-038 us Bay oy. 24-83 0032 a D048) ,, | 24°23 0-028 = 5000 _,, 24:00 TABLE LV. Castyl Soap Solution: by a capillary tube with an elliptical bore. Surface Tension. | | pane com | Height (corrected). (Dyes nesen) | Saturated Se sae solution. 5-038 em. 24°27 | Doan. is, 4949 ,. 23°85 0°33 Ties 4-894 ,, 23°59 0:25 4 4906 _,, 23°64 0-17 5s oh 4874 ,, 23:49 | 0-090 as 4873 ,, 23-49 0-062 \. 4-964 ,, 23°92 | 0-048 : | 4-934 ,, 23°78 | 0-038 %5 | 4-923 - ,, 23°73 | 0-032 x | Ze SHE Ty 23° 59 | y TAREE: Castyl Soap Solution : by different capillary tubes with cir cular bore. Diameter of Bore. | Corrected Height. Surface Tension. | (cm.) (cm.) (Dynes per cm.) | = = SS es 029 3°398 24°10 0:020 4-796 23°49 | 670 Messrs. Narayan and Subrahmanyam on Surface | In order to examine how the surface tension of soap solutions varies with dilution at great dilutions, we have thought it desirable to institute another series of experiments with soap solutions. As no bubbles could be blown at these great dilutions, 2. e. below 0°22 gm. per 100 c.c. of solution, in all these experiments measurements of surface tension are made by the capillary tube method with the tubes used in the former expcriments. TABLE VI. Castyl Soap Solution: Surface Tension for different concentrations (at great dilutions). Coneentration—Gms. | Surface Tension. | per 100 c.c. of soiution. Height (corrected). (Dy nes per cm. ) | 0-20 544 cm. 27°10 | 0175 DBO) be 27°70 | 0150 | 5768, 28-70 | 0-125 6015 ,, 29:90 | 0-100 | 6653, 33°10 | 0-050 | 8-025 ,, 40-09 0:025 | F212). 45-91 00125 | 10525. ,, 52°36 00062 | eg 57:03 00031 | 12-475)... 61-940 0-001 | 13:02" 7 =. 64 71 It will be seen from comparison of the observations recorded here with those already recorded, that the surface tension of soap solution begins to increase with dilution only at great dilutions, up to 0°23 erm. per 100 ¢.c. of solution, the surface tension being pr actically constant, and then rapidly rising with dilution. The results of the experiments are graphically represented on the diagram, where concentrations in grams per 100 c.c. of solution are plotted along the X-axis, and the surface tension values along the Y-axis. The curve, which may well be styled the isothermal of Castyl soap solution at 28° C., is an hyperbola. Further experiments are in progress— (i.) to construct the isothermals of different soap solutions ; (ai.) to construct the isothermals of one and the same soap (Castyl soap) for different temperatures from 30° to 1007 C, Tension of Soap Solutions for Different Concentrations. 671 Oo on o O aynes per ov. 10) oO SOV?” iS (6)! Surrace t0/7 W 6) ) 2) 25 i O 0-025 0-05 O10 O25 O15 O75 0-20 Concentration: orms. per 100 c.c. of Solution. Conclusions. (1) These experiments place beyond all doubt the fact that the surface tension of soap solutions is the same for all concentrations between very wide limits. (2) They further lend weight to the reliability of the bubble method for studying the surface tension of soap solutions although the apparatus is seemingly more complex than that for the capillarity method. (3) The close agreement between the results obtained by. the soap-bubble method and those obtained by the capillarity method further shows that the angle of contact in this case is 0, as has also been conclusively established by the experi- ments of Bosanquet and Hartley (Phil. Mag. vol. xlii. Sept. 1921) and Richards and Caver (Journal American Chemical Society, 1921, p. 827). Department of Physics, Maharajah’s College, Vizianagram, 15th November, 1921. LXXXI. On the Kinetic Theory of Solids (Metals) and the Partition of Thermal Energy.—Part I. ‘By B. M. Sey, Dacca College, Dacca, Bengal*. Preface. | ia Kinetic Theory ef Gases has been developed in great detail, but hardly any progress has been made with that of the solids. In the absence of any such well- developed theory, attempts are sometimes made to apply the results of the Kinetic Theory of Gases to the case of solids, as, for example, in the theory of electric conduction. The results can hardly be regarded as satisfactory. One funda- mental assumption of the Gas- theory is that the volume of the molecules is negligible in comparison with that of the gas itself. Clearly, this does not hold good even approxi- mately for solids. Lately Nernst and others have approached the subject from the standpoint of Planck’s Quantum Theory. In the present paper an attempt has been made to develop the theory for metals on the basis of classical mechanics. Tt is known that for 1° rise of temperature, every atom of any metal absorbs 4x 107 erg in round numbers. On the principle of equi-partition of energy, every atom having three degrees of freedom of translation, this is explained by the supposition that the translational motion absorbs one-half of this quantity of energy [4mv?= a0, where a2=2°02 x 107°], and the other half is absorbed by the increase of potential energy of vibration, whose mean value is the same as that of the kinetic energy. But there are difficulties in the way, as the specific heat increases gradually, whereas the degrees of freedom can increase only by leaps+. It may be pointed out that Born{ and others have developed the theory of crystal lattices, which gives the twenty-one constants required by the mathematical theory of elasticity. Between two neighbouring molecules there must be a force of attraction and another of repulsion which keep the molecules at their proper distance. Born assumed the form a/r+/r” for the mutual potential energy, the first term representing the force of attraction and the second repulsion. But this does * Communicated by the Author. + MacLewis, ‘Physical Chemistry,’ vol. i1. p. 29. t Born, Dz ynamike der Kristalg Me Aufbau der Materie, where full references are oven. On the Kinetic Theory of Solids. 673 not easily explain the heat expansion of the body. If heat be a form of motion, it is clear that the motion of the atoms must have something to do with the force of repulsion. I have, therefore, assumed that the repulsion is of the nature of an impulse due to what are called impacts in the Kinetic Theory of Gases. The problem thus becomes a dynamical one and the mathematics much more difficult. Again, in order to allow impacts, it is necessary to assume that the mutual distances of molecules are not large enough for free vibration. For simplicity I have assumed that the amplitude is small in comparison with the atomic dimen- sions. The results obtained completely bear out with the assumption. It may be pointed out that the phraseology used is that of the Kinetic Theory of Gases. Use has, therefore, been made of terms such as impacts, radius of atoms, which in the light of modern researches on the structure of matter will have to be accepted in a generalized sense. The numerical values have been taken from Kaye and Laby. 1. Some numerical values. Let N be the number of molecules in a metal supposed monatomic per c.cm., and / the average distance between the centres of two adjacent ones. ‘Then N?=1*. The number of molecules of gas per c.cm. at N.T.P. = 2 fox 10%. Let W be the atomic weight, and D the density of the metal. Let w be the weight of one molecule of the standard gas (O=—16). Then D=wt. of 1-c.cm. of the metal=N.W.4w. wt. of 1 c.cm. of oxygen =2°75 x 10 x low=1°429 x 107%. ie aa 23 iS eeu 16 x 10 =the number of atoms in one gram atom of the metal =e€E say. * Strictly speaking this relation holds only for the cubical arrange- ment of the molecules. But for the face-centred cubical arrangement the equation ought to be N2=72. This would increase the valu- of J given in the table by about 11] per cent. But there will be no occasion to use the arithmetical values of 7. All that we need trouble about is the order of magnitude of J. ‘ Phil. Mag. 8. 6. Vol. 43. No. 256. April 1922. 2X 674 Prof. B. M. Sen on the Kinetee Theory of Solids TABLE I. Density | Atomic M. wt. |, coeff. of, ¢ compres- pals. D. | weight. Oy S of mol. expansion. sibility. Cire 8:93 | 63°57 |8'64x10”|2-26 x 10-81:03 x 10-2/16-7 x 10-"| 74x 1057” UN ae 105 |) 107°9° 1597 ©) 4,: 7/250 9 fF TO: |S SS See ier Hen. 786 | 5584 (867 ,, i226 ,, i091) 4) lO 2 ie Gee Ay Nees TA A637 (OCIS 5. e2AGs it OGRE ae ee . Deas... | 1137 | 207-2) (3:38, 13:02) S36). (een 2a ommemmenY BS Au. ...| 19°32 1197°2 6:05 |, (2°55 >. B19) 3) SON adie eee. ZAOO MI OD 23 HOGS tn. Weider et nele ied (ae 8's i) Oe Sch aan T2909 AIST 878. 4. 7298) 7. OSes os pial eae ee 3 ANE as 27 | 271 G16”... \203).,, i944 9.) | (Zoe ee iacnee 2. Arrangement of molecules in an tsotropic body. To find the number of molecules which can be arranged about a central one at a distance / from it and from one another. The problem is the same as the determination of the number of tetrahedrons having all their edges equal to / and having one vertex at the centre which can be fitted ‘nside a sphere of radius /. For every tetrahedron we have an equilateral triangle of side / having its vertices on the sphere. If a, 6, ¢ be the sides of the spherical triangle so formed, a=b=c=60°. *. CosA=t, and) A= 70 .. the spherical excess =30°. The total number of tetrahedrons= TF = 24. For each tetrahedron there are three molecules and each molecule comes in or about five times. The number of 360 On molecules, therefore, is about 14. 3. Rough working model. It is not geometrically possible to have an arrangement of molecules on a sphere so that the spherical are between any two adjacent ones is 60°. But in a rough model form these 14 may be supposed to be arranged as follows :—The centres all lie on a sphere of radius 4. Two may be placed at the two poles which may he arbitrarily chosen. Three meridian circles are drawn making an angle of 60° with and the Partition of Thermal Energy. 675 each other. On each of these, beside the two at the poles, are placed four others, one at each extremity of a pair of diameters making an angle of 60° with the axis. There are thus six molecules on each of these meridian circles, one at each vertex of a regular hexagon inscribed in the circle. This arrangement gives the 14 molecules arranged about the central one. It is symmetrical through the centre and has an axis. 4. The Potential Energy of displacement. Let us consider the motion of a molecule O in the direction AB which we take as the direction of the axis of the model (fig. 1). Let two molecules be placed at A and B and let Jaireayals (e F C, D, H, F be the four other molecules in the same meridian planes co tna OA—OB—OC=AH...=!. Wor simplicity, we shall assume that the central one is the only one vibrating, the others being stationary in their mean positions. Let L be the extreme position of O. Take OL=a, and r=the radius of the molecule. Then lf=a-- or Let f(l) be the potential energy of two molecules at a distance J. Then for a displacement x taken small, along the axis of the model, the potential energy due to A and B is Alta) +f—2) =f) +2°f" (0. The potential energy due to a pair like C, D =f(7;)+f(r2) where 1.=CL, and r,=DL. Now Real ae al, a? el) se ipso 7p neglecting higher powers of 2, Da. fd a 676 Prof. B. M. Sen on the Kinetic Theory of Solids and al dete.—. ° pines Ire TL, an?’ ! Lp2f/! Kr i) (42+ 8 T)F Q)+gef"(), 2 and f(r.) =f) + (—12 +2 “7 + $a7f"'(1). Coe | oe fly) +f) = 2/0) +27 f'O +42? FO. There are six pairs of such molecules. The total potential energy | / =14/(1)+42° {258 +57" : It may be noted that the attractive force is f'(1). And if, as is only natural to assume, the force diminishes with the distance, f"(/) is negative. And if the force varies as any power of the distance, the two terms of the variable part of the potential energy are of the same order of magnitude. It is obvious, moreover, from general considerations that the potential energy for any displacement # must be of the type A+ Bx’+.... If the coefficient of x be not zero, there wili be a permanent force independent of the displacement. The equation of energy, therefore, is MV?+ 2 ee +57" } =iCDMSt.). ot peel) The motion, therefore, is simple harmonic if the coefficient of «? be positive. If the molecule can complete its oscil- lation, the average potential energy would be equal to the averave kinetic energy. We assume that this is not the case, so that the molecule comes into collision with its neighbour as soon as it has described a small fraction of its path *. 5. Order of magnitude of the Kinetic and Potential Energy terms. . In the equation (1) above, the first term MV?=2a0 is of order 10~™, for «= 2°02 x 10716 and @ is the temperature. From known values of the specific heat and the coefficient of linear expansion, it will be proved that /'(l) is of order 107° and the maximum displacement of order 10° °. Now, l being of order 1078, the potential energy term is of order * The reason for this assumption has been given in the preface. and the Partition of Thermal Energy. 677 10-". The error in neglecting it is therefore small. Its effect is felt at the boundary where it gives rise to a powerful ‘surface pressure. We neglect, therefore, the change in the potential energy and take the velocity as constant throughout the motion. Also we neglect the duration of the impact in comparison with the period of vibration. If t, be the time from one extreme position to the other, and ¢, the duration of the impact, ¢, will be neglected in comparison with ¢, *. 6 Heat Relations: To consider the heat relations of a body it is necessary to formulate a relation between the molecular kinetic energy and the temperature of the body. It may be pointed out that the equation IMV?=26 for gases does not follow from dynamical considerations, but is a direct consequence of Boyle and Charles’ law. Asa matter of fact, we may take it as the definition of temper- ature. Fora solid we take Q=the average external kinetic energy of a molecule =3MV’. P=the internal kinetic energy per molecule, z. e. the energy of motion relative to the centre. These are, of course, additive in the ordinary way. If Q be an analytic function of @ vanishing with it, we may take Q=a0+a,07+.... We shall follow the usage of the Gas-theory and suppose that the temperature is determined by the average external kinetic energy of the molecule, taking Q=20. 7. Partition of Energy. When the temperature of a solid is raised by 1° C., the quantity of energy absorbed per unit volumeis DsJ, s being the specific heat and J=4°18x10' ergs. Now Dulong and Petit’s law states that Ws is approximately constant for all elements. Hnergy imparted to every molecule eas Wig N € * Cf, J. J. Thomson, ‘ Corpuscular Theory of Matter,’ p. 93. 678 Prof. B. M. Sen on the Kinetic Theory of Solids For a gas such as oxygen, Cp 24, y=l4; 124 ets | Energy absorbed by each molecule for 1° rise of temper- ature Cy = eal — 3-74 x 10738, whereas the external kinetic energy is increased by a=2:02x 10735, In the Kinetic Theory of Gases this difference is explained by the diatomic nature of the molecules, there being no intermolecular forces*. A portion may aiso be accounted for by the absorption by the electrons. But for solids the matter is complicated by the presence of intermolecular forces against which work has to be done by the heat expansion. For a solid we assume that the total energy imparted is split up into three portions :— (1) a quantity « absorbed in increasing the external kinetic energy Q=4MV?=a0, where 2=2°02 x 107%, and @ the temperature. | (2) a quantity B absorbed in increasing the kinetic energy of rotation relative to the centre. This is presumably absorbed by the electrons in the outer rings. We take P=A+ BO=in mv’, where A represents the store of energy, n the number of electrons taking part in the absorption, m the mass of an electron for velocities small in comparison with that of light, and v the velocity. (3) a quantity 8 absorbed in increasing the potential energy of the molecules, 7. e. in doing work against the inter-molecular forces. a+@B+B= — Taking 4MV?=2@ for Cu at 0=17° C., Wie x Oe Thus for metals at ordinary temperatures V is of order 10*. A table of values is given for solids and gases. It may * Jeans, ‘ Dynamical Theory of Gases,’ art. 270. and the Partition of Thermal Energy. 679 be noted that at first sight the similarity in the values of the total gain of energy for solids and gases may tempt one to apply the diatomic explanation to solids. But the value of N has been calculated on the monatomic basis. If the molecules be taken as diatomic the numbers in the third column will be doubled and the similarity will disappear. Total gain of energy per molecule for 1° C. == x OO OC Una TABLE II. | ; Total gain | Specific: 8 Metals. aes | Sa of energy B+B. B (art. 9). | F (art. 9). | | per mol, =a ae a Can hes rien: | rents. OD G3 1S'8i 10-2" |1:85.x 10-7*| 76 LOA sc10: betes, reer Ss00 | ier 3008 1 eB ietigay mx teAMape Ooi LOT O02 a0. 12 Bey Needed iali[einisy: 0 Sas PPE es: MOS a 15a Aide ee 09 ON Bi Mea ye Rec: ee 5G. Moshe, IGS om See ey ( \jceuue hte ari lies A ee EO ee AOR CID ah COD O)SGay oe | Sn Betas OOM a Te Qe to 11D Tray Gay ail Pik sae. emi GSry |sOS)o),0°) a6. -;, i | Gases.| Cy. Density. pone 0746; ... (1976x1078 | 13 (em 2:40 |0:09X10-33-27 __,, Nieeeees Me OD. a) (S34, Oe. Peer ikctom nator nO! 5! COMP Oni 13-23) 5 C0, Mi atGo O76 4) 96 |, 8. The Amplitude of Molecular Vibration a. We revert to the consideration of a vibrating molecule O surrounded by fourteen others stationary in their mean positions. Its average velocity in any direction may be taken to be V//3. At each collision this forward velocity is reversed. Thus there is a change of momentum at the point 2MV//3 at each collision, or per sec., ee V3 (ty + te) 2(¢t; +t.) being the period of vibration. Neglecting ¢, the duration of the collision, and putting t;=2WV/3a]/V, the 2 me =2 There being N** molecules per sq. cm., the pressure at the surface within the change of momentum becomes ae: 6d mass of the metal is N°? . = per sq. cm. 3a 680 Prof. B. M. Sen on the Kinetic Theory of Solids We now calculate the pressure at the surface due to the molecular forces, the atmospheric pressure being of course negligible. We take into account only seven of the fourteen molecules lying on one side of the equatorial plane. The potential energy in the displaced position, « being the displacement from the mean position, =fUl—2) + 6fl— 32) =7/()—4af"(). Force along the normal =4/" ()=4E, where I is the force between two molecules. Pressure at the surface=4N*". F, ad — =4F da ab or =) i (2) 9. Compressibility. We proceed to find the simplest of the elastic constants, viz. compressibility, on the basis of our hypothesis. The work done against the molecular forces=8 per molecule for 1° rise of temperature. , Now the pressure at the surface=4F . N*”. Work done against the surface pressure=N””*.12F a, where a, is the coefficient of linear expansion. Work done against the inter-molecular forces per unit volume=7F61.N=7Fa,N**, there being fourteen mole- cules arranged about each individual molecule and d6a=6él. N@=19Fa,N*. o. F= ht Meee |) ao) Leal a ie ENN We the Now the surface pressure=4F . N*??= ae : 1 If the pressure be increased by one dyne without change ike : of temperature, it increases by - - th part of itself. 86 aa Uae Ge ol iene ds oy 192, Li AN ies and the Partition of Thermal Energy. 681 " oe hs dTaja bbe Dfer Lad Compressibility C= 4ANBl = ana * Tze? poe kL xyes Penns a0 WON so we oN Taking C= 720 and G— Tine... we get _ Ba te x lO” for Cw Substituting this value of 8, we have for Cu Se <0 and) oa oe ee The value of 8 for the different metals has been calculated and shown in the second table. The values of B are obtained by difference. 10. The calculation of Young’s modulus gives good support to the previous work. Suppose that the body is stretched so that / increases by a. é/=da=x, where l=a+2r. Putting a+e instead of a in art. 8, the half-period of vibration i=2V73(a+2)/V. The change of momentum per sec. is, therefore, MV? MV? ( *) G(a+a) 6a See The force of repulsion due to impacts is diminished by MV? ba? The force of attraction =f/(l+2)—f' (D)+af"O), J’ (L) being of course negative. The total change in the force of attraction per molecule Ftroh, and the total change per unit area on”, {TY — ‘Ot. Now the extension is w/l. eg s modulus is, ait Nf Ga th Ob = 7 {aa +7" } 682 On the Kinetic Theory of Solids. It is known that the value of Young’s modulus for the- common metals is of the order 10”. Taking the case of Cu by way of illustration, the first term=5 x 10” roughly. The experimental value=1°23 x 10”. a a is of order 10”, which makes /"(1) of order 10+. The potential energy term in the equation of energy in art. 4 is therefore of order 1071, 11. Residual Rays. We can make a rough calculation of the frequency of vibration of the molecules at the surface. We know that for a Cu molecule, V is of order 10%, ME Meat eo nek Oe Bey REO ion Neglecting the variations of the force, the frequency p= a which is of order 10”. This agrees with the experimental values *. 12. Conclusion. We have, therefore, arrived at a consistent theory of the solid state. Starting from known values of the specific: heat, compressibility, and the coefficient of linear expansion, we have investigated the amplitude of molecular vibration and the inter-molecular forces. The partition of the gain of thermal energy has been dealt with. We have found that for Cu, for example, (1) 2:02x107'% erg is absorbed by each molecule to increase the energy of translational motion, (2) 1:2x107'% erg is absorbed by the rotational motion, and presumably increases the energy of the electrons, (3) *76x107'6 erg is absorbed in doing work against the inter-molecular forces. The cases of cubical and face-centred cubical crystals have been dealt with in the next paper ; the results obtained. do not differ materially from those obtained here. * McLewis, ‘Physical Chemistry,’ vol. 11. p. 58. Pereser | LXXXII. On the Kinetic Theory of Solids (Metals) and the Partition of Thermal Energy.—Part II. By B. M. Sey,. Dacca College, Dacca, Bengal”. Preface. lb my previous paper on the “ Kinetic Theory of Solids (Metals),” Part I.t, Linvestigated the theory of the solid state with a rough working model of fourteen molecules placed on a sphere about each individual molecule at the centre. The idea was to make the distance between any two adjacent molecules equal to the radius of the sphere (the body being isotropic) so that the spherical distance between them is 60°. This is geometrically impossible, but the model satisfies this condition approximately together with the condition of symmetry. In the present paper, I have restated the theory for the cuhic and face-centred cubic crystals. These are the two arrangements which are common for the solid state. The arguments have been put briefly. Fora more detailed state- ment the reader is referred to the previous paper. 1. Potential energy of displacement. Let us suppose that six molecules are placed at a distance 1 from the central molecule, two on each axis. Let the components of the displacement of the central molecule be &, y, 2, the others being fixed in their mean positions. Then the potential energy for the displacement « (1) due to the molecules on the x-axis = f(l+2)+ f(t—2)=2/0) +2?f"(D, (2) due to the pair of molecules on the y-axis Le | x. 2f('+ 57 )=2t0 + FFU: (3) due to the pair of molecules on the z-axis Qf) + Sf (0). The combined effect of these six molecules for the general displacement 2 = 67() +a? | FN (2) + f".(1) } where @=a?+y?+27 XQ * Communicated by the Author. t+ Suvra, p. 672. 684 Prof. B. M. Sen on the Kinetic Theory of Solids The equation of energy is, therefore, bahia 4MV?+a7 jf ©) +f" (2) } = const. If the molecule were at liberty to swing to its natural amplitude, the mean kinetic energy would be equal to its mean potential energy. We make the assumption that this is not the case, the vibrating molecule coming into collision with the adjacent one before reaching the extreme position of the natural vibration *. 2. Amplitude of molecular vibration. | Ai each collision the forward motion is reversed, so that there is a change of momentum at the point =2MV/WV3 at he Ne ; é each collision, or = per sec., where ¢, is the time from one oly extreme position to the other, and a the maximum displace- ment. Putting t,=2,/3a/V, the change of momentum per I eevee) Oa eda, As there are N molecules per sq. cm., the pressure within the sec. = 40 mass is found to be N8 3, ber sq. cm. Od To calculate the pressure at the surface, we suppose that the molecule on the positive side of the yz plane is missing. The potential energy for a displacement x along the z-axis =1f(l4 57) +f +0 =eothy (ree i (+a {700 oe The first term is a constant. The second gives a constant force F=f' (1) in the direction of the inward normal. The third term gives a force of restitution proportional to the displacement and changing sign with it. Its mean value 1S Zero. The permanent force, therefore, is F along the inward normal, where F is the attraction between two molecules. The pressure at the surface is FN". a8 ag oe ssi) OF a= Buy . This is the equation which is to replace equation (2) in the former paper. * A comparison, of the relative magnitude of the kinetic and the potential energy terms has been given in the previous paper. and the Partition of Thermal Knergy. 685. 3. Compressibility. To find the compressibility, we notice that the work done against the molecular forces = per molecule for 1° rise of temperature. Now the pressure at the surface =FN?*. Therefore the work done against it =3a,FN?° where a, is the coefficient of linear expansion. The work done against the inter-molecular forces per unit volume = 3FSIN =3a, FN??, there being six molecules arranged about each atone molecule, and 6a=01— ol, N@=62, FN". since N/?=1. ~ Gayl? _ 28 _ 208 . ml a= 5H = 2B : Now the surface pressure = NH¥?8= = If the pressure be increased by one oe witheut change of temperature, it increases by ies th part of itself. *. a diminishes by th part of itself. da _ 6a, _ 8 es, (ANGRY pe aad RR en él _ 18%) a C=—37 = ye ° 6 = 362; o ) B= 36a.9 x A or B=6a, — This is the equation which has to be substituted for (4) in the former paper. It will be noticed that this value of 8 increaxes the value of F and diminishes the value of a in the ratio of 24:19. 686 Prof. B. M. Sen on the Kinetic Theory of Solids 4. Face-centred Cubic Crystals. This arrangement consists in having a molecule placed at the eight corners of a cube and one at the centre of each of the six faces. To find the arrangement of the molecules about any individual one, we note that there are two types, viz. those at the corners and those at the centres of the faces. Tor both the types it will be found that the following arrangement gives the requisite number of molecules situated nearest to each individual molecule O. With O as centre describe a sphere of radius 7 and draw a system of three great circles intersecting one another at right angles. Hach great circle is divided into four quad- rants by the other two. At the middle point of each of these twelve quadrants place a molecule. With a series of such models we can build up a body of the face-centred cubic crystal tvpe, the edge of the cube being’,/2/. It may be noted that there are twelve molecules surrounding each individual molecule instead of the fourteen in the isotropic arrangement, and that the spherical distance between any two adjacent molecules not on the same co-ordinate plane is ‘60°. In the isotropic model the distance between any two on the same great circle is 60°. Let us calculate the potential energy of the displacement of the central molecule. Take the planes of the great circles for the co-ordinate planes. Take two straight lines in the xy-plane, the 2, and the y; axes, bisecting the anzles between the x and the » axes. Place four molecules at the extremities of the x, and the y, axes. Put similarly four at the extremities of the y, and z, axes which bisect the angles between the y, ¢ axes in the yz-plane, and also four at the extremities of the <3; and a2; axes which bisect the angles between the z, v axes in the za-plane. Let (wx, y, z) be the most general displacement of the central molecule. It can -also be written in the form (a, ¥1, 2), (@ Yo, 22), (#3) Y> 23)5 so that +yptoaartypt eat yy t+ efaag ty +egae (say). For the four molecules on the 2, y, axes, the potential -energy due to the displacement ) Tay? =f(l+ x1) +f(l—m) + 2f(1+ 5+ ) =u()+a2{ 7 ze ro}. and the Partition of Thermal Energy. 687 ‘That due to the displacement y; Ls =4p() nef OP +7" }. ‘That due to displacement 2 mis(l+ 4) mp) 2.8 2 aapye2{ 0 yy p+e{ AO -7"o}. The total energy due to the four molecules KC) ee Sl Bel Aig aapntay Orpen bred ep }. The totalenergy due to the twelve molecules =12/ (4 aa | 0) + ry +0 {oo he One =12 (1) + 2a f a +f" f. It may be noted that the variable part of the potential energy is just double of the corresponding portion of the potential energy for the cubical system. It vanishes if the force is proportional to the inverse square of the distance. This fact seems to give a special significance to the inverse- square law. Following the same line of argument we get the following equations :— Esha? ao 1 Roeh aly ro) This value of @ is greater than the corresponding value obtained in the case of the isotropic arrangement by about 5 per cent. The value of F is increased and that of a@ diminished by the same amount. June 30, 1921. a [ 688 ] LXXXITI. The Thermometric Anemometer. By J. 8. G. Tuomas, D.Sc., A.R C.Se., A.L.C., Senior Physicist, South Metropolitan Gas Company, London*. Introduction. | eee thermometric type of anemometer due to CC. C. Thomas 7, which has found application in industrial practice, more especially in the measurement of large volumes of gases flowing in mains etc., is based upon the principle that if heat is imparted from a source to a stream of gas so as to raise the temperature of the stream by a constant amount, then the heat energy imparted to the gas to produce such rise of temperature is proportional to the rate of flow of the gas. Assuming constancy of the specific heat of the gas, the supply of energy to the stream to effect a given rise of temperature is independent of the initial temperature of the stream. Such a type of anemometer possesses the desirable characteristic that it measures the mass-flow of gas, and hence its indications may be made to read directly in terms of standard pressure and temperature conditions. An elaborate series of calibrations of this type of anemometer against others of the Pitot and Venturi types has been made by C. C. Thomas t, and over the range of velocities employed, the indications of the three types of instruments were found to be in very close agreement. The calibrations referred to were carried out in a pipe of 24 in. diameter and the lowest velocity corresponded with an hourly flow of 7,000 Ib. of air, 7. €. a mean velocity of about 230 cm. per sec. The straight lines obtained by plotting the rate of revolution of the fan producing the flow, as abscissa against the flow deter- mined from the indications of the Pitot, Venturi,and Electric meters respectively, were extrapolated for low See of the flow and passed through the origin of co-ordinates. Such extrapolation is justifiable in the case of the Pitot and Venturi meters. In the case of the electric meter, however, such extrapolation is clearly unjustifiable, as under no cireum- stances is the whole of the heat supplied to the heating coil imparted to the gas. There are necessarily heat losses due to radiation, conduction, and convection, and such losses * Communicated by the Author. + Journ. Amer. Soc. Mech. Eng., xxxi. pp. 1825-1840 (1909). Journ. Franklin Inst., vol. 172, pp. 411-460 (1911). Proc. Amer. Gas Inst. vii. pp. 340-381 (191 2). t See e.g. Journ. Franklin Inst., loc. cit. p. 433. On the Thermometric Anemometer. 689 become relatively more important the less the velocity of the Stream. It is clear that if the thermometers be symmetri- cally disposed with regard to the heating coil employed, no value of the heat supp'y to the latter would be such as to maintain the desired ditference of temperature between the two thermometers in the absence of flow. If, however, the second thermometer be disposed closer to the heater than the first thermometer, in the absence of flow, the value of such heat supply is perfectly definite and determined principally by radiation and convection losses from the channel in which the heater is placed. The effect of the predominance of such heat losses for low velocities ae to have been overlooked in the papers already referred to*. It will be found that the curves obtained by plotting the rate of flow of gas as abscissee against the rise of temperature due to a constant supply of energy to the heating coil as ordinates, is not asyraptotic to the ordinate axis. Owing to the heat losses referred to, there exists a finite maximum te: Wee aay rise through which the stream is heated by a constant supply of energy. The valne of the velocity of flow at which such maximum rise of temperature occurs is conditioned by a variety of factors, the size of pipe, heat insulation of pipe, ete. For velocities of flow less than this critical value, decreasing velocity is accompanied by a decreasing rise of temperature. Attention has been directed to the ph enc- menon in a previous paper T. Its consideration is of some consequence in technical practice, e.g., in the design of heat interchangers or regenerator furnace- settings for the attain- ment of a maximum temperature in the gases to be heated. In the technical form of thermomenic meter, the ratio of maximum flow to minimum flow for corre«t registration is about 15:1, and by employing a second pair of. “tempera- ture-difference resistances this ratio can be increased to about 60:1. Velocities quite commonly occurring in tec!- nical practice are of the order of a few cms. per sec., and upwards, and it is important to determine the forms of the calibration curves of a low capacity anemometer of the thermometric type for such velocities, and more especially the variation in such calibration curves accompanying a variation of the heat losses due to conduction and radiation. The present paper details some of the results obtained in the course of such an investigation. * See e. g. Journ. Franklin Inst., loc. cit. p. 447. + Phil. Mae. vol. xh. p. 258 (1921). Phil. Mag. 8. 6. Vol. 43. No. 256. April 1922. ine 690 Dr. J. S. G. Thomas on EHeperimental. The flow tube consisted of a brass tube of 2:011 cm. diameter wound with one layer of ashestos cord (diameter of cord about 3 mm.), inserted in the flow system as detailed in previous papers *. The sup: ly of air was derived from a gas- holder of 5 cubic feet capacity connected with the laboratory high-pressure air service, and a steady and calculable flow of dry air in the system was established as ae detailed Ff. The heating element employed is shown in fig. 1 (a and 6). A is a por tion of brass tube 6 mm, in length, exactly siml'ar to the main flow tube. It fitted tightly within the ebonite ring B. A number of thin copper pins, bent at right angles as shown at C in fig. 1 6, were driven securely into holes drilled in the ebonite ring, and slots were cut of an appro- priate width and depth in the brass ring A so that these pins were insulated from the latter. A length of fine platinum wire was soldered in zigzay fashion to “these pins as shown, the minimum amount o/ solder being employed for this pur- ose. The ends of the wire were connected by short copper leads with the terminals IT’. Holes H were drilled at angular * See e.g. Phil. Mag. vol. xli. p. 242 (1921). + Phil. Mag. vol. xxxix. p. 509 (1920). eS ee eee the Thermometric Anemometer. 691 intervals of 120° C. in the ebonite ring B. These served for the insertion of the heating element in the flow svstem. This was effeeted by means of brass rods passing through the holes H and through similar holes drilled in ebonite rings attached to the main flow tube. The whole was securely fixed by means of nuts on the ends of the brass rods *. The heating element was inserted in avertical plane in the horizontal flow system with the several wires about equally inclined to the vertical as shown in fig. 1, so that the free convection currents tended to equalize the temperature distribution across the section of the flow tube. The construction of the two platinum thermo- meters employed resembled that of the heating element: there was in this ease, however, no necessity to employ an inner brass ring, and the pins for the support of the platinum wires were made considerably smaller than in the case of the heating element. The thermometers were inserted in the flow system in a similar manner to that employed for the insertion of the heater. The distances b:tween the respective thermometers and the latter could be adjusted to any desired values by the use of suitable lengths of tube provided with ebonite ends affixed to the tube and suitably bored. For very small distances between heater and thermometer, thin separating disks of ebonite bored to the appropriate diameter were similarly employed. The thermometers were calibrated by the determination of their respective resistances at 0°C. and at atemperature of 50°+-°02° C., on gas scale, corre- sponding to 49°62+0-02° C. on Pt scale (6=1°5), this temperature being maintained thermostatically in a water bath, and determined by a mercury thermometer standardized at the N. P. Laboratory. The thermometers were adjusted as nearly as possible to equality of resistance at 0° C., and they were, in use, connected differentially with a Callendar and Griffiths’ bridge, the scale of which was provide! with a vernier, enabling the balance to be deter- mined to 0°01 cm. The supply of energy to the heating element was determined from a knowledge of the current supplied and the drop of potential occurring across the heater. The current was read from a Siemens & Halske ammeter reading to 0°00] amp. The voitage drop across the heater was determined by means of a Rayleigh pote: tio- meter composed of two P O. Boxes adjusted so that their total resistance was 10,000 ohms throughout. The potentio- meter readinys were calibrated by means of a Weston cell standardized at the N. P. Laboratory. * See Phil. Mag. vol. xliii. p. 279 (1922). ra Oe. 692 Dr. J. g. G. Thomas on The mode of experiment was as follows :—A steady flow of dry air was established in the system for about an hour, and the balance of the Callendar & Griffiths’ bridge ascertained in the absence of any current in the heating element. The point of contact on the bridge wire was then moved to the point corresponding to 2°. difference between the two thermometers, and the value of the current in the heater adjusted until the bridge balance was restored at the point so determined. The necessary displacement at 15° C.was 2°65 em. of bridge wire. The bridge current was throughoul adjasted to 0°010 amp. On account of the relatively large thermal capacity of the flow system, a considerable time—of the order of 2 to 3 hours when the thermometers were widely separated —celapsed before the system attained a steady condition. Owing to slight inequality of the two thermometer resistances at O° C., a slight alteration of balance accompanied a change of atmospheric temperature. Such alteration was taken account of throughout the observations. ‘The potential drop across the heater was determined as already explained. The velocity of flow in the system was determined as detailed in previous papers. The results were plotted in the form of curves in which the abscissee represent velocities of flow, and ordinates the respective supplies of energy to the heater to maintain the second thermometer at a temperature 2° C. above that of the first. ; Results and Discussion. Internal diameter of flow tube............... 20 ena. External diameter of flow tube ............ 2°22,eme Ry of 1st Thermometer (cins. of bridge NYA ee oo ae aca YG a ie 363°40 Temperature caatieeul of Ist ie WNCTOR eye an Ou via hee yeaa aR eee 0°003556 Ry of 2nd Thermometer -(ems. of bridge CRTC) RTS te PD dE Ceueleree aie eM ia iat i 368°75 Temperature coefficient of Znd Thermo- TIMO UOT Peek tun soe: uel t rete aren ey ete ae 0:003552 Shift of balance for 1° C. change of atmo- spheric temperature (cm. of bridge NUD OR sr gil ei cais keene ese ee 0-017 In fig. 2 are given the forms of the calibration curves obtained when the first thermometer was throughout placed at a distance of 30 cm, from the heating element while the the Thermometric Anemometer. 693 distance between the latter and the second thermometer assumed the values 30, 15, 7°5, 6°3, 3°0, 1:1, 0°5, and 0°15 em., as indicated by the respective curves. The initial portions of some of the curves are plotted on an enlarged DISTANCE BETWEEN HEATING COIL AND SECOND THERMOMETER (CMS) © 30, Xi5, + 7:5, 463, ©30,Gtl, VO5,0015 WaTTS SUPPLIED PER 2°C RISE OF TEMPERATURE. 66 scale in fig. 2a. The variation of distance between the heating element and the second thermometer affords a convenient method of studying the effect of a variation of the heat losses due to radiation etc. upon the form of the calibration curves. The broken line B in the figure gives the theoretical form—a straight line passing through the a 694 . Dr. J. S. G. Thomas on origin—of the calibration curve in the absence of any such losses, the whole of the energy supply being utilized to heat the air stream through 2° C.on the gas scale, In all cases within the range of velocities studied, the necessary energy supply to the heating element in order that a difference of 2° C. may be indicated by the thermometers is considerably greater than that theoretically necessary. Owing to the absence of a device for stirring the gas so as to render the temperature of the gas stream uniform over a cross section, an indication of 2° C. difference of temperature in the thermometers does not necessarily correspond exactly with a 2°C. rise of temperature of the stream. ‘The use of any efficient stirring device was precluded by the experimental conditions in the present case and is impossible in many of the technical applications of anemometry, more especially where gas streams carrying dust etc. in suspension are to be measured. The various calibration curves show considerable departure from the straight line B which would be obtained in the absence of any heat losses and if the temperature indicated by the second thermometer accurately represented the mean temperature of the stream. The greatest departure from the linear relation between the energy supply and the velocity is sgen in the case of curve A, corresponding to the greatest distance between heating element and _ second thermometer, as is to be anticipated owing to the heat losses being greatest in this case. The approximaie form of the curve A can be best discussed by reference to the relation given by Callendar* and em- ployed in the determination of the specific heat of gases and vapours by the continuous-flow method. Where the velocities of flow are larger than those concerned in the present series of experiments, and where the heat losses are small, Callendar has shown that W=SQU +1d0-+ —d6, where W is the energy supply to the heating element, S the specific heat of the gas, Q the mass flow, d@ the rise in temperature, hd the portion of heat loss independent of the flow, and kd6/Q the residual portion of the heat loss varying inversely as the flow. Applying the appropriate value of S, and employing v the velocity of flow in place of Q, the relation becomes in the case of a flow of air in the present flow system :—W =0:004147 vd@ + hdd + es senting a series of hyperbole, h and k being variable * See e. g. Phil. Trans., A. 535. p. 390 (1915). . repre- the Thermometric Anemometer. 695 parameters. The asymptotes are given by v=0 and W=0-004147.1d@+hd@. The value v corresponding to the minimum value of W isv=242 vk. It is seen therefore that if the experimental conditions are such that the relation between the energy supply W and the velocity.v is of the hyperbolic type discussed, corresponding to very large values of the velocity, the energy supply to maintain a difference of temperature d@ will be in excess of that represented by the eurve B by an amount Ad@, and practically independent of k. Moreover the thermometric type of anemometer would, under the same conditions, permit the velocity to be uniquely deter- mined from the energy supply, only if such velocity were, in the case of the flow tube used in the present experiments, known to be either lexs than or greater than 242 Wk. Similarly in the general case, when the energy supply is given in the form W=SQdd+hdé+ 5% the minimum value of Q that can be uniquely determined from the value 7 of the energy supply is equal to ae The close approach to parallelism of the curve B and the various calibration curves, more particularly those for small values of the distance apart of the heating element and the second thermometer, for large values of the impressed velocity of the streain, indicates that the actual increase of temperature of the stream was very approximately 2°C. Owing to the increased facility for mixing occurring with slow flows, the same was probably true in the case of low velocities of flow also. The calibration curve A is approximately hyperbolic. It will be noted however that for velocities of from 30 to 60 ems. per sec., the curve is slightly concave to the axis of velocities. This sime feature of slight concavity is seen to be present in all the other curves of the series. This charac- teristic 1s probably attributable to the asymmetrical dispo- sition of the resultant convection current from the heating element with regard to the two thermometers. Consider the case of curve A for which the two thermometers are disposed at equal distances of 30 em. from the heating element. With zero flow, the two thermometers would indicate equal temperatures, and the highest temperature in the flow system would be found at the point immediately above the heating element. When a slow flow is imposed, the region of maximum temperature in the flow system is moved towards the second thermometer. The energy requisite to 636 Dr. J. S. G. Thomas on maintain an indicated difference of 2° C. in the two thermo- meters 1s diminished on this account below the value it would have were the region of maximum temperature not so displaced. This effect increases with increasing velocity of the stream, as with such increase of velocity, the region of maximum temperature in the flow system advances towards the second thermometer. Such increase of effect must, however, attain a limiting value as it is clear that for large values of the impressed velocity cnly a small proportion of the energy supplied is utilized to raise the temperature of the How system. The form of the calibration curve A may therefore be regarded as derived from the theoretical hyperbolic form by drawing it through a series of points the ordinates of which are slightly less than those of points on the hyperbola, the proportional difference of ordinates intially increasing with increase of velocity from zero until a limiting velocity i is reached, and thereafter decreasing. - The remaining calibration curves in figs. 2 and 2a were obtained with the second thermometer placed closer than the first to the heating element. These curves are not asym- ptotic to the axis of energy supply. Clearly, with such dis- positions of the respective thermometers, a finite energy supply would establish a difference in the temperatures indicated by the two thermometers. This effect is clearly seen in the curves C, D, Einfig.2a. The curve E obtained with the second thermometer placed at a distance of onl 1-5 mm. from the heating element is characterized by the fact that the minimum value—if any such occur—of the energy supply necessary to maintain the indicated difference of 2°C. must correspond with a very small velocity of the stream. The lower limit of velocities of gas streams for which the thermometric anemometer may be employed is clearly reduced by reducing the distance between the heating element and the second thermometer. For low vaiues of the velocity, the time lag of response of the thermometric anemometer to a change of velocity is conditioned principally by the distances between the heating element and the thermo- meters. For small time lag, these distances should be as small as possible. The disposition of the second thermometer has been dis- cussed above. There is alsoa most suitable position at which the first thermometer should be placed. Such position is clearly not at too great a distance from the heating element, nor near thereto. In the latter case the temperature indicaled by it would be conditioned largely by that of the heating element and its surroundings. An indicated difference of the Thermometric Anemometer. 697 2° C. between the two thermometers would under these circumstances correspond to « greater rise of temperature than 2° C. in the stream. This effect is shown in fig. 3. O'4 @) 3 1 . DISTANCE BETWEEN FIRST THERMOMETER | AND HEATING COIL | | © 30 cMS xX 0-8 CM. 0-2 | : > } © WATTS SUPPLIED PER 2°C RISE OF TEMPERATURE. 0 4 8 12 ie VELOCITY(CMS. PER SEC., VOLUMES REDUCED TO O°C AND 760 M.M), Both curves were obtained with the second thermometer at, a distance of 0°8 cm. from the heating element. The respective distances of the first thermometer therefrom were 30 cm. (curve A) and 0°8 em. (curve B). Summary. An experimental investigation of the application of the thermometric anemometer to the determination of slow rates of flow of gases is detailed. It is shown that values of such low velocities cannot, in general, be uniquely determined by such type of anemometer. This arises owing to the existence of a minimum value of the energy supply required to heat 698 Mr.G. A. Newgass on a Physical Interpretation of the stream through a definite range of temperature. The dependence of the limiting velocity upon the disposition of the thermometers with regard to the heating element is shown by means of calibration curves obtained for various distances between the respective thermometers and heating element. The author desires to express to Dr. Charles Carpenter, C.B.E., his thanks for the provision of the facilities neces- sary for carrying out the work detailed in this paper. 709 Old Kent Road, S.E. Dec. 29, 1921. LXXXIV. On a Possible Physical Interpretation of Lewis and Adams’ Relationship between h, c, and e. By GERALD A. Neweass, B.A. (Cantab.)*. ONSIDER the hypothetical case of two spheres. carrylng equal but opposite charges e, of no mass other than electromagnetic, of equal radii. Imagine that the two spheres are touching, but that they are prevented from discharging. Ifthe system is set rotating about its centre of mass and an axis at right angles to the line joining the centres of the spheres, there will be a certain angular velo:ity, angular momentum, and rotational energy at which the centripetal force would balance the electrostatic attraction. These values were roughly calculated, making use of the simplifying assumptions that the electromagnetic masses as given by the equation m=e?/(67rc”) were con- centrated at the centres of the spheres, and that any possible effects of relativity etc. due to the high velocities. cancelled out, and it was found that while the angular velocity and the rotational energy of the system varied with the radii of the spheres, the angular momentum was independent of the size of the system and was given by | aul 2/67 © If the electricity is assumed to be distributed uniformly throughout the spheres instead of on the surface, or if, instead of both spheres being equal, one is considerably larger than the other, the only effect is to alter the coefficient. Accordingly, it is suggested that if the values of the moment of inertia and angular momentum were * Communicated by the Author. Lewis and Adams’ Relationship between h, ec, ande. 699 calculated accurately by considering each element of the field of force separately and integrating up, assuming a not improbable distribution of electricity in the spheres, the angular momentum might be found to be given by 8/87 > 16n3.4/ pe ah eae Tae This expression, if the spheres were shrunken or swollen negative or positive electrons, is that calculated by Lewis and Adams for h (Phys. Rev. (2) ii. p. 92 (1914)). And it might also be found that (2) the energy of the system was 1/c? times its mass, (3) that the diameter of the system was c/w, and (4) that the velocity of the periphery was c. In the case that has been considered the coefficients were not unity. Regarding the physical interpretation of the critical angular velocity, there are two possibilities: (1) The radii of the spheres might be fixed; if the angular momentum were less than the critical value there would be a tendency for the spheres to fall into one another, if it were greater they would fly apart ; (2) the criterion might be that the spheres were always in equilibrium, the centripetai force being equal to the electrostatic force, and this would require that the angular momentum should always have the critical value. Accordingly, it is found that the spheres wculd have to decrease in size with increase of energy. It is not quite clear what would cause the spheres to alter their radii in this way, but it is possible that, if there was an inward force which balanced the outward pull on the surface of the tubes of force with an increase in angular velocity, the tubes would lag behind and leave the surface of the spheres farther from the normal, and hence, the effective outward pull being reduced, the spheres would shrink. It is suggested that some such electric doublet, where the distribution was such that the coefficients were correct, is the unit of radiation. Space free from all radiation would be considcred to contain an infinite number of doublets of infinite size. Ifa certain volume of space is fil!ed with homogeneous polarized radiations of frequency , some of these doublets would shrink and rotate with frequency o, angular momentum h, and energy hw. If the radiation were of sufficient density, the whole volume of space would be full of rotating doublets, similar to a train of cog-wheels, the electric force atany point would be varying periodically, and there would be also a magnetic force at right angles to the plane of the rotations of 700 Dr. B. van der Pol on Oscillation Hysteresis in a the doublets also varying periodically. Hach doublet would occupy a volume c/@ cube, and so would give interference effects. It is also seen that if both polarized rays are eliminated by crossed nicols no energy will get through, and it would be impossible to have “ longitudinal radiation ” by itself. A source of radiation of frequency w would consist of a number of rotating doublets. Knergy would be dissipated in two ways:—(1) A doublet losing all its energy, as its periphery is moving with velocity c, would send out a pulse to infinity—if all the doublets were rotating in phase, the disruption of a number at a fairly steady rate would give the ordinary periodic radiation and interference effects ; (2) a doublet owing to a “collision” would leave the cluster and travel as a whole in a straight line from the source until through another collision it gave up all its energy—its velocity would be less than ec, but would probably give effects that would appear instantaneous. Accordingly, it seems that a source could be very weak aud give interference effects, and yet be able to start the photoelectric effect in an inappreciable time. LXXXV. On Oscillation Hysteresis in a Triode Generator with Two Degrees of Freedom. By BALTH. VAN DER Pon sum. See" HEN two oscillatory circuits are linked together by means of a magnetic, electrostatic, or resistance coupling, it is well known that the circuit combination possesses two degrees of freedom. If one of these oscillatory circuits is, moreover, a part of a triode generator, it is natural to ask whether the two modes of vibration can exist simultaneously or, if this is not the case, whether the one or the other mode of vibration will obtain for any particular conditions. Now it is found experimentally that, when the system oscillates in one of the two modes of vibration and the natural frequency of the secondary circuit is varied gra- dually, the system suddenly jumps at a certain point from the first mode of vibration to the other. If afterwards the natural frequency of the secondary is varied in the re- verse direction it is found that the system jumps from the second into the first mode of vibration, but at a point which is not identical with the first one mentioned above, and thus * Communicated by Professor H. A. Lorentz, For.M.R:S. Triode Generator with Two Degrees of Kreedom. 701 _ a kind of oscillation hysteresis is obtained, which, apart from its importance in technical applications, is of interest from a physical point of view. The normal experimental arrangement is shown in fig. 1, where an oscillatory circuit L,C, is shown coupled through Fig. 1. ee it the mutual induction M to the circuit L,C, belonging to a normal tricde generator. We may consider the indications of the thermal ammeters 2, and 22 when (i.) C, is brought from a small value, through the point of resonance (L,C, = L,C,)L, to a large value ; and (ii.) when the value of C, is thereupon gradually de- creased through resonance to the first small value. The relations between 7, and @,? and 7% and @,” thus ob- tained are shown in figs. 2 and 3, where the arrows indicate the paths followed. Further, in these two diagrams the total systemis found to vibrate for conditions represented by EFB in one cf the two modes of vibration, 7. e. with the higher one of the two coupling frequencies, while for con- ditions represented by DUA the system vibrates with the lower coupling frequency. Hence at the points B and A discontinuities occur in the modes of vibration, resulting in a discontinuity both in frequency and amplitudes of the currents. But, further, it is also seen from these graphs that the system has the tendency to go on oscillating as long as 702. Dr. B. van der Pol on Oscillation Hysteresis in a possible in the mode of vibration in which it is already oseil- lating, though the other mode of vibratien is possible for the same parameters. Fig. 2. a? —> we | These phenomena were noticed by the author in February 1920, but it was felt that no sati-factory explanation could be given unless progress was first made in the development of a non-linear theory of sustained oscillations. For it is obvious that, when the problem is treated with linear differential equations, the principle of superposition 1s valid, and in tis case oscillations in the one mode are uninfluenced by oscillations in the other. It is therefore somewhat sur- prising that up to the present, though several theoretical Triode Generator with Two Degrees of Freedom. 703 contributions to the problem have already appeared *, the phenomenon has, as far as we are aware, only been dealt with in a linear theory. The solutions of the differential equations in this case are of the form e+ sin wt, and it depends on the sign of « whether an oscillation will build up or decay. But whether both oscillations will be present simultaneously or whether the one mode of vibration will suddenly be replaced by the other when a parameter of the circuits is gradually varied, and whether a hysteresis loop will be obtained, these questions can not be answered by a linear theory. In order therefore to retain in the analysis the necessary interaction of simultaneous vibrations which determines the stability of the oscillat-ons, non-linear terms which oceur through the curvature of the triode charac- teristics may not be ignored. Before attempting, however, to set out a non-linear theory of the phenomena under consideration, a few remarks may first be made concerning the terminology. The notion of one or two degrees of freedom is used here as an extension of the usual meaning attached to these terms in the ordinary linear treatment of oscillation problems. We are weil aware that, e.g. to speak of a system as having one degree of freedom when more than one stabie oscillation is possible for a given set of parameterst is not altogether satisfactory, but 1t is hoped that from the description of the phenomena the meaning will be sufficiently clear. Further, we shall di-criminate between a “ possible” vibra- tion and a vibration that can actually be realized. With “possible” is here meant a solution representing a stationary oscillation with a constant amplitude. It may, however, be that this oscillation cannot be realized, it being unstable. Finally, it is obvious that for a system such as shown in tig. 1, when the secondary circuit is very loosely coupled to the primary, the reaction of the secondary on the primary will be small. It is found experimentally that under these circumstances an ordinary resonunce curve can be obtained as the secondary circuit. This case will, however, not be considered here and : ¢ shall confine our considerations to cases where the coupling is strong. * J. S. Townsend, Radio Review, i. p. 369 (May 1920). K. Heeeoner, Archie fiir Elektrotechnik, ix. p. 127 (1920). F. H+rnts, Jahrbuch Sur drahtl. Telegraphie, xv. p. 442 (1920). Hl. Vogel und M. Wien, Ann. d. Phys. \xii. p. 649 (1920). H. G Moller, Jahrbuch fiir drahtil. Tele- grapme, xvi. p. 402 (1920). H. Panli, zbed. xvii. p. 322 (1921). W. Rogowski, Archiv fiir Elektrotechnik, x. pp. 1, 15 (1921) See also Moller, Die Elektronenrdhre (Vieweg, 1920). + See, e.g. Appleton and Van der Pol, Phil. Mag. Jan. 1922, 704 Dr. B. van der Pol on Oscillation Hysteresis in a In order to simplify the analysis we shall not treat the actual circuit shown in fig. 1, which, from an experimental point of view, is the simpler, but will replace the magnetic coupling by an equivalent capacity coupling. We shall also replace the series resistance of the oscillatory cireuits by equivalent shunt resistances, and therefore deal with the circuit of fig. 4. In this way we retain all the essentials of the problem while, through simple phase relations, a con- siderable simplification of the analysis is obtained. Fig. 4. e e } : e C In fig. 4 the total equivalent capacity te of the 2 f primary : circuit 1s given by | secondary age Chics oe eo and the square of the coupling coefticient, &’, becomes CiC. B= oa The tree angular frequency te of the | i scogiaee circuit is further @ = : het Pere 1 Triode Generator with Two Degrees of Freedom. 105 ' 5 oa Similarly the damping coefticient ath of the | eaten el circuit is given by 1 fi iia ties: a, /= 1 CRy’ We call 7, the variable part of the anode current and va the variable part of the anode potential. The application of Kirchhoff’s laws to the circuits then leads (with neglect of the grid current) to: d*tq Hee fo ae F ‘ OR re ee dt! + (a, + Qo ) ys Fie + @» +(1—h yay Ay dt + (L= 2) (2a! + copay") OM + (1—K?) e209 i at, a4 di ae 171 12) ta 274 __ 2) Ua =~ ¢ [ge te” UP) Se + ov! eae ay We further notice that ti tla di, aaa tre where v, is the grid potential, so that a constant ratio exists between the variable anode potential and grid potential, namely, Uy M Vy - i, ; Hence, though in general the anode current is a function of both the anode and grid potentials, by means of this constant ratio we are able to express the anode current as a function of the variable anode potential alone. A method of deter- mining experimentally this relation 7,=W(v,) has been previously described*. For conditions for which free oscillations are possible this characteristic has in general a negative slopef for v,=0. It is therefore appropriate to deve.op the function i2=W(v,) as 7 gay BU Pye. SA (a) where the index of v, has been dropped for simplicity as will be done in the further treatment. * Appleton and Van der Pol, Phil. Mag. xlii. p. 201 (1921). + When developed in this way the theory applies equally well to ‘“‘dynatron” circuits. Pist. Mag. 8.5. Vol. 43. No. 256. April 1922." 2.2 706 Dr. B. van der Pol on Oscillation Hysteresis in a It may here be noticed that stable oscillations are only possible when both a’ and y' as defined by (1 a) are positive. No further terms are needed in this series to enable us to account for the hysteresis phenomenon under consideration, though, naturally, in order to obtain a more exact numerical result in all details, further terms may be necessary. We further write ae! Th on TA, = &, 1 Ay = 29, ley Ci =, oy =e Oh) and assume, in agreement with the usual circuit dimensions, that the initial logarithmic increment of the total primary (triode included) and the logarithmic decrement of the secondary are small compared with unity, 2. e. that Uo 2h 20a @1 DW On making these substitutions in (1) we arrive at d*v dt* are = ee +f + wo’ 27— (1 — k? een} Se = d + (1—h?)(ws2ccy— oo2a11) @ + (1 =F) Pay2v dB a? age =— 4 et al —@) ap tor — BY a | (Be + 94). (2) On neglecting small terms in (2) this equation can further be simplified, and we obtain as the fundamental differential equation of our problem d‘y dis es +(1—/? )@,? Wo” U + | - aoe k*) 5} (Be? byt) + (oa) dv +(1— Ce glo) QO. oa Seeteanen anne (3) e Triode Generator with Two Degrees of Freedom. TOT The general solution of (3) seems not to be possible, but : a a since we consider — and — as small compared with unity, a @ We an approximate solution can be obtained. From the -general vature of the problem two modes of vibration may be expected to be possible, and to express this mathematically we are thus led to a trial solution, vV=asino,t +6 sin (@,,t+)), where a and # are certain unknown functions of the time and @, and w,,; are unknown frequencies. 2 is an arbitrary phase constant. As ay a ae ify and w, and w, are of the same order of magnitude, we may expect the possible building up or decay of the amplitudes to occur slowly compared with the oscillations themselves, that is, da dt < wd, d db dt < @5,), Hence the second and higher differential coefficients of a and 6 with respect to time will be neglected. We thus have v =asino,t+bsin (@,t+A), > P =@,0 Cos wif + Asin wy + wy) Cos (yt +2) | +6sin (w@pyt+r), | 6 =—@/asin ot + 20,% cos @t—@,,b sin (@,;¢ +r) | + eb cos (@yt+Ar). - (4) 0 = —@/7d COS @yt — 30772 sin w,t— @,°) cos (@_t +2) | —3e,,°b sin (ot+2), | v =@,'asin of — 40a cos wt + 4b sin (Opt +2) —40,°b cos (wt +2). ) We shall further have to consider the terms involving v? and v®. These non-linear terms obviously suggest the pre- sence of higher harmonics and combination tones, but as the increment and decrement are small, the main effect of the non-linear terms is in their influence on the amplitudes. 2 Le 708 ~=Dr. B. van der Pol on Oscillation Hysteresis in a For under these circumstances the series representing the amplitudes of the harmonics may be expected to converge rapidly so that the influence of the harmonics on the ampli- tude of the fundamental may, as a first approximation, be neglected. We are thus justified in neglecting as a first approximation the presence of these higher harmonics and SUL LAE tones and shall retain, therefore, in the terms with v? and v? only those parts involving the frequencies o1 and w,; We thus see that the term @v? has no influence on the result. In considering, however, v= {asin o,f +6sin (o,¢+r) |, terms of several frequencies occur, such as w,, @;;, 8@7, 3@,,, @,+2@,;, ©; —2o@;,, 20;+@,,, ... etc., but only the terms involving the frequencies w; and @,, will be retained. Hence we have v® = 2a(a?+ 26°) sin wt + 2b(b? + 2a’) sin (@y,t-+r). (5) It may here be noticed that 6 occurs in the coefficient of sin w,¢ and a in.the coefficient of sin (w,¢+)). This funda- mental fact in the non-linear treatment of our problem shows the mutual influence of simultaneous vibrations, and it will further be found that the presence of one oscillation makes it more difficult for the other to develop. When more than three terms are used in the series expansion for @, as is advisable when working on the lower bottom part or higher top part of the 72—v, characteristic, then the presence of one oscillation is, however, occasionally favourable to the develop- ment of anietlier peeillationt ‘Such special cases will, however, not be considered here. | We now proceed to substitute from (4) and (5) in (3) and thus get an equation of the form _ A sin ae B sin (@yf +X) + C cos wt + D cos (yt +0) =0, (94) where A, B, C, and D are functions of the variables w,, da. ab" at Gee These expressions A, B, C, and D contain terms of three orders of magnitude, viz. a, b, but they also contain a, bee first order: ‘a, w*); ‘second order: aw*®a, yw*a*®, wa.... etc. ; : vas third order: ew7a, yo Boe! ete, ; but we only retain the first two orders of magnitude. Triode Generator with Two Degrees of Freedom. 709 In order to satisfy (5a) identically we equate separately to zero the four coefficients A, B, C, D. Thus four equations are obtained for the four variables a, 6, w;, and @,,. They are found to be ot —o,(@,"+0,")+(1—/”)oo.=0, . (6a) | Oy, — @y,7(@;? + 5”) + (i — fk?) o7o.? = 0, (6 b) da dt 2 w7(a, —ay)+at Pye + 267) {w7(1—k?)-—o7} =0, crake ere Ca) pat OO ie 2(@,? + wy’—20,,”) a + {(1 — h?) (01% — wo?) 22+ 07— 20,7)= +{A—F)(@y?a2 — w72)) + @y7"( 41 — aa) } b + Byb(b? + 2a?) {@.2(1 —k?) — 77} =0. (7 6) Equations (6a) and (60) give us the coupling frequencies @, and @,,, while (7a) and (74) enable us to find the pos- sible stationary amplitudes a and } and to determine their stability. Since (6a) and (60) are of the same form it is necessary to define w, and w,, quite definitely. We shall take o> =t(@+o.")+4 (@, + w,”)?—4(1 —k*)o"o.”, | 8) (D @y; =}(o? + @,”) —$ V(@ +0,")? — 41 —F )o7o,”, | where the roots are to be taken positive, so that @y" > Oy, O17, @2”, 2 2 2 9 and Wy < @,", @1", Wy”. The equations (7 a) and (76) can further be written 2 \ = = H,a?( ao? — a? — 26”), | aie oes Mee CAL eid) db? Bll ite ae = H,,0?(b.?—b°— 2a*), | where, with the aid of (6a), (6b), and (8), we have 2 2 2 By=3y—5. ; Yor Or Ore: | WOME Bas Aes aiae (0 710. Dr. B. van der Pol on Oscillation Hysteresis in a and thus H> 0, Hi; > 0. Further, the term a)? introduced in (9), represents the square of the stationary amplitude which would be obtained when an oscillation in the first mode of vibration, 2. e. with a frequency @,, alone was present. ‘Similarly bo is the stationary amplitude which would be attained if the system vibrated only in the second mode of vibration. These ampli- tudes ao? and 69? are obtained directly from (7a) and (76) Pee OG! OUD by putting —-= — =0 and are found to be : Qt at » » 9 \ 2 ay Or ae” @)~ As t OG) ern cane = Rol Say ee RO : zi) te 8 RN ae US en 0 / a ae 2 2 9 A eS ON a Garr SO ee 9 by = 3 PR ee ; 9 ° 9 2 oY Oo Oi. Op aan, However, a and by are not the only “ possible” stationary amplitudes as may be seen from (9) by putting ay CER ii res We thus have in general for the “ possible” stationary amplitudes a, and 6, the two equations s’(dp’ — a — 267) =0, 6.7 (by? — b9 — 2a,?) =0, the four sets of solutions of which are (i. a =; b= (i1.) As” = 3(2by? — a7), -,? = 4 (2ag? — b?), i (yy eae be = 6. | og (iv.) a bP = pe But we shall further have to investigate separately which one of these four stationary solutions (11) will be attained in any given circumstances. Therefore (9) would have to be solved, which is a difficult, if not impossible, matter. However, in order to investigate the stability of each of the four solutions (41) we may consider the effect of a small forced change of amplitude from the stationary value due to some disturbing cause, and investigate the tendency of the Triode Generator with Two Degrees of Freedom. 711 amplitude either to return to or to depart further from its initial stationary value, thus applying the usual method in questions of dynamic stability. In this way we shall find a certain conservatism of the system in that the particular mode of vibration persists (when one parameter is varied gradually) even when conditions have been reached which are not favourable to it and which are such that, were this mode of vibration not actually present, yet the other one would exist. In other words, metastable oscillation con- ditions may arise. Let these small changes of amplitude be represented by the type 6. We thus substitute in (9) a? =a," + 6a,’, b? = b? + db,”, & - e sire 9 and only retain first powers of the small quantities ga,” and 66,2. Hence we have : as ) = By (ao? — 2a,? — 2b,2)8a,2—2Eya.28b,?, | d(66;") ty [in Hir(bo? — 26,? — 2,7) 6b? — 2 Ey7b,76.a,’. j / These linear equations (12) are solved by putting Saz= A’e'*, 002 ==.B' ce", and we obtain as characteristic equation for / i? +k { Ky(2a, a 2b? — dy") Si BEyr( 2a, + 2b? — bo”) } + Hy Ey { (24,7 + 2b,? — ap”) (2a? + 26,7 — bo”) —4a;76,"} = 0. (18) In order that a set of stationary values a, and bs should be stable neither of the two roots & of (13) may be positive as this would show the tendency of the system to depart from the stationary solution in question. A set of stationary amplitudes a, and /, is therefore only stable when Ey( 2a2 + 26,? — ao”) + Hqr(2a,? + 26, — by”) > 0 ) and (2a? + 2b? — ao”) (2a,? + 26,2 — by?) —4a,°b" > 0. { (14) 712 Dr. B. van der Pol on Oscillation Hysteresis in a We shall now proceed to investigate the conditions of stability of our four solutions 1., 11, 11) Iv.) or separately. ie ie az =0, ba): After substitution of these values in (14) we find. as the condition for which both amplitudes remain zero : . — ay’ Hy — by’ Eq; > 0 and ay’by? > 0, or, as Ey and Ey are both positive, ay? < 0 and | By These inequalities are expressions for the fact that only when the circuit conditions (resistances, retroaction, etc.) are such that no oscillations are “possible” at all, can the system be kept in the non-oscillatory state, from which it may be concluded that, when oscillations are “possible ” at all, some form of oscillation (either i1., iii., or iv.) will build up automatically. ii. aoe This represents the case in which both coupling frequencies would be present simultaneously. But the conditions of stability here are from (14) easily found to be | i Eya,? + Ey162 > 0 : ‘ (15 a, b) —a,’b.? > 0. Now for a, and 6, to be possible at all we must obviously have as’ > 0 De > 0, which relations are incompatible with (155). We thus see that the simultaneous occurrence of finite stationary oscilla- tions of both the coupling frequencies represents an unstable condition and can therefore not be realized in practice. Triode Generator with Two Degrees of Freedom. 713 This is in complete agreement with the experimental results. We saw (i.) that when a vibration is possible at all the system will automatically start vibrating in some form. It cannot, however, produce stationary oscillations in both frequencies at the same time (ii.), so that only one of the two coupling frequencies will build up. Which one this will be depends on the circumstances and can be found from a consideration of iT ee az=a?, b?=0, lV; a;°=0, b= by These cases may be conveniently treated together. Before considering, however, in detail the stability of the system when oscillating in one mode of vibration only, we shall first determine the conditions for which such an oscillation is ‘“‘possible” at all, apart from its stability. Moreover, as the peculiar discontinuities, described in the introduc- tion, occur when the natural frequency of the secondary circuit is altered, we shall leave all parameters unaltered except the detuning of the secondary and consider how these possible amplitudes a, and 6, vary as a function of this detuning. For the circuit under consideration this variation of , is brought about by varying L, (fig. 4), which is equivalent to a variation of the secondary capacity in a case where the electrostatic coupling here considered is replaced by its electromagnetic equivalent. Now (9a) can be written a 2 o nt i gy x — fils"), | r « (16) ty =" Sir(@") | ay ao ia Ay Il 2) Is) J where a ee Ces ie aera I Wo DW | NS UE a MOE (10 d : = IE ee Il ; @) : an F11(@2”) os — Or | 9? These functions fj and fi; are the coefficients with which the damping coefficient of the secondary must be multiplied in order to transpose the secondary damping to the primary 714 Dr. B. van der Pol on Oscillation Flysteresis in a circuits. They are represented in fig. 5 for a constant k?=(05, Since in (16) a, a2 and y are independent of @» we can trace in the latter diagram the dependence of a,” a and b,52 on — and @,?. a) For example, if = is represented by OB the amplitude a” D ‘is is by (16) proportional to the vertical distance between the Fig. 5. w? oe: we & line BE andthe curve /;(@,”). Ina similar way b9° is pro- portional to the distance between BE and fi(@.”). Thus when @2=OH < OG, ay? is proportional to CD, but when w,? > OG, a? would be negative and oscillations of the fre- quency @, are impossible. In the same way oscillations of the frequency @,;,; are only possible when o,” > OK. , ; a a If we next consider a larger value of —!, e.g. ~=ON, ao As “ Triode Generator with Two Degrees of Freedom. 715 markedly different possibilities arise. For example, the range of values of ow,” for which a,” is possible is now represented by OL and is seen to extend beyond the resonance position @)°=@,". In the same way the ampli- tude ),” is now possible for all values of Ws" ‘ereater than OA. Hence a region Al for o,’ exists in “which both modes of vibration are separately ‘‘ possible.’ We thus must have recourse to a consideration of the conditions of stability in order to decide which mode of vibration will actually be present for any given conditions. Now. in order that Chama (ay) be stable, we must have according to (14) Hiya? + Eq (2ap? — by”) > O and ag (2a97 — by”) > 0, where the second condition is the more stringent one. Hence for a,” only to be stable, we must have (18 a, b) Be A aTiT Ay > Sho 5 and similarly for 6)? only to be stable we must have bo? = tay. . te : By ahs a These conditions are represented in fig. 5 for “TON. ao The vertical SP is so chosen that SR=RQ and the vertical evecuch that TU=UV. Hence, though we found pre- viously Al as the region where ap? as well as bo? were separately “ possible”, we may now further conclude that the common region where a,” as well as 6,? are separately stable is given by the smaller distance PW only. Which one of the two possible and stable oscillations dy Sin @yt or bysin (wyzt+A) will be attained in the region PW? The answer to this question, which must also include the explanation of the hysteresis effect, is given by (9), and will be seen to depend on the initial conditions. For let us see what happens when @,? is gradually brought from a small value such as represented by OH, through resonance a to a big value represented by OX. (We again assume — to be given by ON.) 2 Fir st; when @,? OH. only the first mode of vibration is possible and stable and we therefore have dg =A’; b= 0 716 Dr. B. van der Pol on Oscillation Hysteresis in a Whether 6? has the tendency to build up when once a; =a) and b?=0, is seen trom (96), which can”? be written d log b? dt = By (bo? — 249° 2), which shows that log 6? or 6? itself will only increase when w>” has been given such a value that This is the ease when or = OW" (lies): We can therefore bring @,? froma value @,? < @,? through resonance (@>7=@,”) Oa sale @°=OW > while all the time the system continues to vibrate in the first mode only. But as soon as w,? has reached the value OW where the square of the amplitude which would obtain if the system vibrated in the second mode only equals twice the square of the amplitude of the vibrations in the first mode actually present (b9’=2a9 or TV=2TU), then the oscillation sud-- denly jumps fon the first mode to the second. 1, the system starts vibrating in the second mode. The initial conditions here are therefore fon et Oe da Do eae Now (9) may be integrated graphically for these initial conditions, and fig. 6 is Shs result for a special case where @.” < w) and therefore a,? >. This figure shows clearly Ve | Triode Generator with Two Degrees of Freedom. 717 how originally both vibrations build up simultaneously but also that the initial rate of increase of 6? is smaller than that of a’. Further, as follows directly from (9), 6? reaches its Fig. 6. ae maximum when 6?=l)?—2a?, and thereupon it goes back asymptotically to zero while a? increases up to a2, its stationary value. Summing up, our results can be described in short with the aid of the schematic fig. 7. Mode I. is stable for Migs 7. @:” < w, (AB), Mode II. for o.2>,? (EF). Mode I. is metastable for w,?>,? but only up to the point C (BC) Mode I. is metastable for w,? > @,? up to the point G (FG). The part CD is unstable for Mode I., and the part GH for Mode II. DK represents the part where Mode I. is . s 718 Dr. B. van der Pol on Oscillation Hysteresis in a imposstble at all, LH the impossible part for Mode II. On closing the primary circuit only stable oscillations are obtained, while the metastable states can only be realize when the system is first in a stable state and thereupon slowly (compared with the damping coefficients) (or adia- batically) brought to the metastable state. Some doubt, however. still exists whether it is exactly the point of resonance which separates the regions where on closing the primary circuit either a9? or 0,7 is finally attained, as some dissymmetry still exists in the formulee (Br # Bh). But a value for w,”? very close to w,? may easily be obtained experimentally for which it is a mere matter of chance whether a or by will finally be obtained. A simple way of demonstrating this fact is by putting a big leaky condenser in series with the grid of the triode circuit. It is well known that with this arrangement the oscillations are periodically quenched in an automatic way, so that regular trains of vibrations are obtained. The group frequency may e.g. be made of the order of one second. When next the secondary circuit is coupled to the primary we can, with a heterodyne arrangement, produce an audible combination tone corresponding to either the one or the other of the two frequencies w; and w,;. In general, each time only one of these two combination tones is obtained, but with w, close to or equal to w,, the combination tone heard every second jumps erratically between the two tones corresponding to @, and @,, respectively, and it isa mere matter of chance which one of the two occurs. Finally, (9a) yields for o;=@, 1 Ap? = = (ay—a 0 ay 1 2) b Ss EM) J) 0 a 2) and thus shows that, as far as our approximations go, the two amplitude carves intersect at the point of resonance. But, moreover, these amplitudes at the resonance point are independent of the coupling coefficient. Fig. 8, which gives a set of observations of the mean square secondary current (in a circuit like that of fig 1) asa function of @,? for dif- ferent coupling coefficients (increasing with the numbers 0, 1, 2, 3, 4, 5), is a confirmation of this theoretical result. For very loose coupling (Curve O) an ore resonance curve is obtained, but for closer coupling (1, 2, 3, 4, 5) the ficure clearly shows that the intersection of the two branches Triode Generator with Two Degrees of Freedom. 719 (1—1), (2—2), ete., occurs practically at resonance and for all curves at the same height. Fig. 8. 3° 29 2 . 1 : f, ¢ Po Bee, 03 4 O ue \ Pl, ! oS 50 : , : A re cee OI’ [\ OO 4o ; O 0 . 5 ~ ; : . 5 3 °° 2 5 ° 2 i 8 2 w 2 In conclusion my sincere thanks are due to Professor H. A. Lorentz for kis kind interest and some valuable sug- gestions. Physical Laboratory, Teyler’s Institute, Haarlem (Holland). O20 ag LAXAXAXVI. Polarization Phenomena in X-ray Bulbs. To the Editors of the Philosophical Magazine. (GENTLEMEN, — R. 8S. RATNER in the January issue of the Philo- sophical Magazine contributed a paper, entitled ‘* Polarization henner in X-ray Bulbs,” in which he describes certain experiments made with vacuum tubes, and puts forward certain explanations. Mr. Ratner, however, appears to have overlooked some previous contributions which have an important bearing on the subject of his investigation, and which in fact would seem to anticipate some of his results and contradict some of his conclusions. I refer particularly to the publications of Winkleman in 1901, of Campbell Swinton later, and my own notes in the Proceedings of the Cambridge Philosophical Society, vols. xvi. and xvii. of 1912. The results of these investigators challenge Mr. Ratner’s opening statement that ‘ Hitherto the sparking potential in well exhausted vacuum tubes has been considered to be the property of the gas in the tube, being determined entirely by the nature and the pressure of the gas.’ It is well known that the disposition of the electrodes within the evacuated envelope exercises a profound effect; one only has to mention the Lodge rectifying valve in this connexion to illustrate the point. In the notes above cited it was peinted out that a probable explanation of the influence of the position of the electrodes on the hardness of the vacuum tube was to be sought in the charge gathering on the walls of the tube surrounding the cathode, and some supporting experimental evidence was adduced. Such an effect would clearly be influenced by the amount of metal sputtered on the walls during the working of the tuhe. In fact, many if not all peculiarities in working such tnhes wc) T hare eo Be met, receiveatany rate qualitative explanation on the lines just mentioned. o The University, Leeds, Yours faithfully, January 28, 1922. | R. WHIDDINGTON. — Se ee LXXXVII. Appiication of the Electron Theory of Chemistry to Solids. By Sir J. J. THomson, O.M., FRS.* N discussions on the structure of solids and crystals atien- tion is usually confined to the distribution of the atoms and little or no consideration given to tliat of the electrons. On the view I discussed in the Phil. Mag. Mar. 1921 the electrons play a very important part in determining the arrangement of the atoms and the properties of tie substance. Kach kind of atom has associated with it a definite number of electrons which form its outer layer when it is in a free state: itis by the rearrangement of these electrons that it is able to hold other atoms, whether of the same or different kinds, in chemical combination. When these atoms aggre- gate and form a solid there will be in each unit volume of the solid a definite number of these electrons, and. the problem is to distribute the electrons so that they will form with the atoms a system in stable equilibrium. We shall begin with the simplest case when the atoms are all of the same kind, 7z.e. when the solid contains only one chemical element. We suppose that the electrons are arranged as a series of cells which fill space and that each cell surrounds an atom; the number of cells is equal to the number of atoms. If the atom is monovalent the number ef electrons is equal to the number of atoms, if divalent to twice that number, if trivalent to thrice that number, and soon. This condition will determine the shape of the cell. If the cells have to be similar and equal and to fill up space without leaving gaps, they must be of a limited number of types. ‘These are as follows :-— (1) Parallelepipeda: if the atoms are of the same kind these may be expected to be cubes. (2) Hexagonal Prisms. (3) Rhombic Dodecahedra. (4) Cubo-octahedra. Let us consider these in order. Parallelepipeda. If there is an electron at each of the eight corners of the parallelepipedon, then since each corner is common to eight parallelepipeda, the number of electrons is equal to the number of cells. Thus this is a possible arrangement for monovalent elements. If all the atoms are of the same kind the parallelepipeda will be cubes and the atoms themselves will be on the simple space lattice formed by their centres. * Communicated by the Author. Phil. Mag. 8. 6. Vol. 43. No. 256. April 1922. 3A —————— —— ee ee a CCC CCC (22 Sir J. J. Thomson on the Application of the If electrons are placed at the centres of each of the six faces of the cubes as well as at the corners, since each middle point is shared by two cubes, each of the six central electrons will count as one half; the number of electrons will be four times the number of cells, so that this is a possible distribution for a tetravalent element. The atoms at the centres of these cells may either form a rectangular space lattice of the simplest kind, or a system built up of different lattices; since a series of layers of these face-centred cubes will still fit, if one layer, for example, a horizontal one, is moved relatively to the layer above or below it parallel to a diagonal of a horizontal face of the cube and through a distance equal to one-half the length of the diagonal. By moving the layers about in this way we can get distributions of the atoms corresponding to the distributions of the centres of the spheres in the different methods of piling shot. If instead of placing electrons at the centres of the faces we place them at the middle points of the twelve edges of the cube, since each middle point is shared by four cubes, these twelve electrons count as three, so that in this arrange- ment, as in the previous one, the number of electrons will be four times the number of cells. This arrangement would be possible for a tetravalent element; we shall see, however, that it is much less stable than the previous one. Another symmetrical arrangement with cubical cells is one where four electrons at the corners of a regular tetrahedron are placed inside the cell. These are to be placed according to the following plan :—Let AB be two points at the opposite ends of a diagonal of one of the faces of the eube, C and D the ends of the diagonal of the parallel face, CD being at right angles to AB. Join O, the centre of the cube, with ABCD; measure from O equal lengths, OP, OQ, OR, OS, along OA, OB, OC, OD; then P, Q, R, 8 will be the corners of a tetrahedron symmetrical with respeét to the cube. If there are electrons at each corner of the cube this arrange- ment will give five electrons per cell. The preceding arrangements are all symmetrical with respect to three axes at right angles to each other and so would correspond to the cubical system in crystallography. If instead of placing electrons at the centres of all the faces of the cubes we place them only on the faces at right angles to the axis of w, we get a distribution which gives two electrons per cell but which is not symmetrical about the three axes wz, y, z, and so could not correspond to a crystal belonging to the cubical system, but to one belonging Electron Theory of Chemistry to Solids. 723 ‘to some uniaxial system. If we put electrons at the centres of the faces at right angles to y as well as those at right angles to #, we get a system with three electrons per cell and again having a uniaxial symmetry. Hexagonal Prisms. When the electrons are at the corners of a hexagonal prism, since each corner is the meeting place of six prisms, the twelve electrons at the corners of the prism will give an average of two electrons per cell. This would be a suitable arrangement for a divalent element. The arrangement of electrons would have a symmetry corresponding to that of the hexagonal system in crystallography. If, in addition, we place electrons at the middle points of the flat ends of the prism, each of these points will be shared by two prisms; this arrangement will give an extra electron per cell, so that there will be on the average three electrons per cell, a possible arrangement for a trivalent element.. If electrons are placed at the middle points of the six side faces there will be three more electrons per cell, so that each cell will account for five electrons if there are no elec- trons at the flat ends and for six if there are, and would give arrangements suitable for pentavalent and hexavalent elements. The atoms accompanying the electrons will also be arranged in hexagonal prisms; each of these prisms will have an atom at the centre of each end. Big. 1. The Rhombie Dodecahedron. (Fig. 1.) There are six corners at which four edges meet and eight at which only three meet ; when the dodecahedra are fitted together so as to fill space each of the four edged corners is the meeting-place of six dodecahedra, and each of the three edged corners of four; thus there will, on the average, be dA 2 724 Sir J. J. Thomson on the Application of the three electrons to each of these cells. This would corre- spond to a trivalent element and would possess cubical symmetry. The disposition of the atoms round the dodeca- - hedron is as follows :— Let us call the corners where four edges meet the octa- hedral points, as these points are at the corners of a regular octahedron, and let us take the axes of the octahedron as the ~ axes of w, y, and 2. Hach of these octahedral points, e. g. P the one whose coordinates are (0, 0,d), has four atoms round it, in a plane through P at right angles to the line joining the point P to the centre O; the lines joining P to these atoms are parallel to the two axes of the octahedron which are at right angles to OP, and the distance of each of these atoms from P is equal to d the distance of P from the centre. In addition to these atoms there is one at the centre of each dodecahedron. It will be seen that this arrangement of the atoms is equivalent to that of a system of face-centred cubes, the centres of the cubes being at the octahedral points of the dodecahedra and the sides of the cubes parallel to the octahedral axes. | The Cubo-octahedron. This has twenty-four corners, and when the cells are packed together so as to fill space each of the corners forms a part of four cells ; there will be thus on the average six electrons to each cell, so that this gives a symmetrical arrangement for the arrangement of the electrons in a hexavalent solid. (Fig. 2.) Fig. 2. The disposition of the atoms may be represented as follows :—Take the centre of one of the cells as origin and the perpendiculars to the three square faces as the axes of coorinates, then the atoms are represented by two lattices ; in (1) the coordinates of a point on the lattice are represented by pd, qd, rd, where p, g, 7 are even positive or negative integers and d is the distance of one of the plane faces from the centre of the cell ; in (2) the coordinates are represented , . } Electron Theory of Chemistry to Solids. 725 by pd, gd, rd, where p, q, 7 are positive or negative integers, two of these must be odd and one even. ‘These two lattices are again equivalent to a system of face-centred cubes. The preceding results may be summed up in the following table, where the first column gives the valency of the atom, the second the possible shapes of the cells formed by its electrons, and the third the nature of the symmetry of the arrangement. Valency. Shape of Cell. Symmetry. ' Cube with electrons at the corners ............ cubical. 2. JE (S50) 12 IG Cif o 0S onan ee ee Pac ee hexagonal. Cubes with electrons at the centres of one set of prameel TACSS) 2 keh 62 halen ia tye bah ab ye «ie a etd ee tetragonal. 3. Hexagonal prisms with an electron at the centre BIB qe Meu, ha ele eo can Megat ainda a Sueho reas hexagonal, Cubes with electrons at the centres of two sets eg AT MOD TACOS: ee a Fea.a clei niacy ole w big be nar dee tetragonal. Fuombie dodecahedron’) 2)... 5.4 peur ss v2. Cubical, 4. IBaeereentred Cube. deca,\h scr eete ee eta ke cle ts cubical. Hexagonal prism with two electrons alung the axis at equal distances from the centre...... hexagonal. 5. Hexagonal prisms with electrons at the centres of the side faces ....... PANY ace, SO aaa ar age hexagonal. Electrons at the corners of cubes with a regular tetrahedron of electrons inside ............ cubical. 6. Hexagonal prisms with electrons at the centres SUE AC Carel is kai apt Las Gch! ated ys bye, we homie uty hexagon, Cubo-octahedrony hiss 6 2 \.cp34 dyes t We dhs eves cubical. Face-centred cube with two electrons inside.... tetragonal, i. Electrons at the corners of a cube with six electrons inside at the corners of a regular He etl Caron ist aprons ty seine Ake as doin oa cubical, Electrons at the corners and centres of side faces of a hexagonal prism with two electrons inside, on the axis, at equal distances from FEES COMBE cima antmens eAh eee hs a oe vie ld oi oe! Boop hexagonal. Though the table is not complete it will be seen that for most of the valencies more than one arrangement of the electrons is possible, indicating that for such elements there might be allotropic modifications with different crystalline forms. Hitherto we have supposed that the cells in which the electrons may be arranged are regular solids, that the parallelepipeda are cubes, the hexagonal prisms regular, and so on; it 1s evident, however, that such an assemblage of regular cells with the atoms at their centres will, if it be 726 Sir J. J. Thomson on the Application of the strained homogeneously in any way without fracture, still remain an assemblage of cells which would enclose an atom in each cell and have the right number of electrons per cell. Thus the cubes might be distorted into parallelepipeda and the cubical symmetry replaced by symmetries represented by the triclinic, monoclinic, rhombic, or tetragonal systems in crystallography. Such a distortion of the cells might be expected to occur when there are many different kinds of atoms in the system. It may be impossible to arrange these so as to give complete cubical symmetry so that bodies with — complicated constitutions would tend to crystallize in the more irregular systems. The view we are discussing is supported by a comparison of the way in which elements of different valencies crystallize with the results indicated by the preceding table. The state- ments as to the forms in which the different elements crystallize are taken from Groth’s Chemische Krystallo- graphie, vol. i. Monovalent Hlements. 7 Very little seems to be known about the crystallization of the alkali metals; copper, silver, gold, which are monovalent, all crystallize in the regular system. Divalent Elements. Beryllium, magnesium, zine, and cadmium all crystallize in the hexagonal system; an observation by Moissan sug-. gests that calcium does so also; nothing is known as to the crystallization of barium and strontium. Trivalent Elements. Very little seems settled as to the crystallization of these elements, except that aluminium crystallizes in the regular system. Quadrivalent Hlements. Carbon, silicon, and lead crystallize in the regular system, tin probably in a tetragonal one. | | Pentavalent Elements. Phosphorus crystallizes in the regular system ; arsenic in two forms—one regular, the other trigonal. Sexavalent [lements. Sulphur crystallizes in several forms, monoclinic and rhombic ; selenium is monoclinic or trigonal; tellurium trigonal. Septavalent Elements. Iodine crystallizes in a rhombic system. Electron Theory of Chemistry to Solids. 127 Stability of the Distribution of Electrons. Hitherto in considering the possible distribution of elec- trons we have only taken into account geometrical con- siderations ; it is, however, of fundamental importance to consider the stability of the distribution, as no distribution is of any use for our purpose unless it is stable. We shall begin with the case of a monovalent element when the electrons are at the corners of cubes and the atoms at the centre. Take the origin of coordinates at one of the electrons and the axes parallel to the sides of a cube; let 2d be the length of a side of the cube. Then the coordinates of the array of electrons are given by the equations id e200. ee — 210. those of the atoms by m—(2p-+ ja, “y—Qq tld, 2=(2r-+ 1), where p, 7,7 may have any positive or negative integral values. If the electron (p, g, 7) has a vertical displacement py, the force due to this displacement tending to increase pp, the displacement of the electron at the origin is : (po 7 Pp, 4, 3) Byars where oes Sl if: pate ko: 3r? ) pr (2d (preter? (p+etryey When the attraction between an atom and an electron separated by a distance r is expressed by if the (p,q,7) atom has a vertical displacement @,,,,,, the force tending to increase pop, the displacement of the electron at the origin is — (Po — per) Cpgrs where Coar= BY Reg? Ry d when Rog = (2p +1)? + (2¢ +1)’ + r+)’. i 728 Sir J. J. Thomson on the Application of the Thus if ™ be the mass of an electron me 2 ) maa = =(po— Peer) Byer — 3 (Po— per) Opa + A) Since the electrons and the atoms are so distributed that i if one of these has coordinates p, g, 7, there are others whose coordinates are 9, P, 1 P79, Ts Pr Qs U2 Po GTP and since for such a group of six 2 1 te 272 )=0 (P+? +77)? (p?+q? +77)? ee N ( LY ee Ay? )=- 2 (p? + ge +r)? (p?+ gq” + 77)? rae (p?+q74+77)??’ 2 Orne 2 Boor =0, Ob — > (P+ etry? xX a x ae Thus equation (1) may be written, do ILD ape — A po —_— » Ppor 1B “+ > Dyqr Chars ; where 2c é i A= 7p Ginga” where p, g, 7 must all be odd. : q If the disturbance is represented by 4 2a ra ea, 20 =C0S— &. COS cos — i xc PO we may put Pr, r= 1” €9? €3 Poy where €, €, €; are the roots, real and imaginary, of the } equations q and the equation for py becomes, 2 “po = — pol A+ 2 €1? €5? €3” Byg,) + wo (Er? €2* €3” Cogr) ‘ (2) mv dt? To find the equation of motion of the atoms we must make a some assumption as to the repulsion between two positive a charges at a distance 7+; we shall suppose that this is expressed by D Cris q oD f(r). | ee Electron Theory of Chemistry to Solids. 729 Then if M is the mass of an atom, Wa qU 7 ‘ M aE ar — Ka, + po= (er? Eo! €5” C> g.0)s where H=A+ > Foo + De? eo? €3” (Bagr + Pyen)s and AG: ny To 3) Fvor= Pg - 7 (: pq Ypqr am Tor where 7,9, is the distance between the atom p, q, rand the atom whose displacement is wp. As the mass of an atom is very large compared with that of an electron, an approximate solution for the motion of the electrons will be got by supposing that the atoms are not displaced ; putting w, equal to zero in equation (2) we get m “Po = — po(A+ dey? €5% €3” Brow), » » + (3) or, if po varies as (1/r), where the summation extends over all the other charges. If we try to calculate S(1/r) by taking first the electrons and then the positive particles we get divergent series which are unmanageable ; if, however, we regard the solid as built up of cubes with a unit positive charge at the centre and 1/8 of a unit negative charge at each of the eight corners, and, taking this as our unit, calculate the potential energy of the electron with reference to each of the cells, we get a series which rapidly converges and which can be calculated without great difficulty. Suppose O is the electron for which we wish to calculate =(1/r). O will be the meeting-place of eight cells, remove Fig, 3. these, leaving a cubical cavity around O. Then ¥(1/7r) will be the sum of terms due to the cells outside the cavity, plus the part due to the charges we have removed by taking away the eight cells. These charges were —1/8 at each of the eight corners, ABCDEFGH, —1/4 at each of the twelve poluts like K where two cells meet, and charges —1/2 at each of the six points like L where four of the cells which we have removed meet ; in addition, there are the eight posi- tive charges at the centres of the cubes which have been Ale ble removed. The value of ¢?>~— for these charges is, if d=the side of a cube, r Be a 16" e ne ao ae egy Ga a 734 Sir J. J. Thomson on the Application of the We shall indicate the various cells by the coordinates of their centres referred to O as origin and axes parallel to the sides of the cube. Thus, of the cubes outside the hollow, the nearest to the origin will be those whose centres have for coordinates the 24 permutations formed by (+1, +1, +3) the next by (+1, +3, +3), and so on, the integers which occur being all odd integers. The cells of the type (1, 1, 3) furnish —:0038 x 24 e?/d. (1,3, 3) (e504 =F OOlA 2 eee (1,1,5) ,, —0018x 24 e/d, (3,358 |.) / a OO12 cB: (1,3,5) ,, —00006x 48 e?/d. (3,3,5) ,, +0002 24 e/d. (1,7) 4, 00026 x 24 e/d. (3,5,5) 4, --00012x 24 e/d, (3,1,7) ,, —00020x 24 e/d. (3,3,7) 4, —00004 x 24 e?/d. Adding these to the preceding we find that Xee'/r for the electron 1s oe 3°65 7° This will be the same as the value of & ee’/r for the positive atom, for the arrangement of atoms and electrons may be | represented either as a system of cubes with electrons at the corners and atoms at the centre, or a series with electrons at the centre and atoms at the corner; hence, if there are N atoms and N electrons per unit volume, the potential energy 1% ee'/r due to the forces varying inversely as the square of the distance will be eee Bee a Now consider the part of the energy due to the forces ‘varying inversely as the cube of the distance, and suppose that the supplementary forces between the atoms also varies necording to this law. The potential energy due to forces varying inversely as the cube will for two charges ee’ vary as 1/r?, where r is the distance between the charges; hence when we take the summation for all the charges per unit volume we shall get for the potential energy of unit volume an expression of the form Ne? Electron Theory of Chemistry to Solids. 735 where f is a positive quantity, since these supplementary forces are always in the opposite direction to the corre- sponding inverse square one. Hence the potential energy of a system containing N atoms will be expressed in the form, SOO IS fe — 2 SS Ne ( 7 Z) : But, neglecting temperature effects, for the equilibrium to be stable d must have such a value that the quantity inside the bracket is a maximum, 2. e. DOO 2a ad a =e ° e . e e e 7.) Ces, a d= 5B? 2 so that the potential energy of the system = —1825— N. Now let us find an expression for the work required to compress the solid so that d is changed from d to d—Ad; if the potential energy of the N atoms is denoted by u, the work required to compress them will be udd+4u(Ad)?+...., where w is the first and & the second differential of uw with respect tod. But w=0 and w is equal to LOY Of we a?” dt Ne Hence the work required to ae the cells is if V denotes the volume occupied by these cells Ad _1AV Wate Vik 13°65 Ne ale hence the work = a GOR ( Vv 736 Sir J. J. Thomson on the Application of the but if & is the bulk modulus this work is equal for unit volume to 1}, AV nary. 3°65 Ne? hence Cc arate where N is the number of atoms per unit volume. If M is the mass of an atom and A the density, NMA and.) Ni(@)t= ae ° 4/3 hence k= (a) PAA hn cn The comparison of the values for & given by (8) for tlie monovalent alkali metals with their values as determined by Richards’ very valuable experiments (Kaye and Laby’s Tables) is given in the following table :— Metal. A. M/1'64x10-24. & ealculated. k observed. oithiams) 2) "534 7 14° X10! ee Sodiuny 2.7 oS) 23 “068 x10 > 0Ga>5<0™ Potassium... °*862 mill ‘03° x 10%" *0a2 sme Rubidium ... 1°532 85°5 "022 x10" > 3025 0 Cheeta 0 ea se 016 x10 -016x10" Thus the absolute values of & and not merely the relative ones are in very close agreement. If we take the cell with charges 1/8 at its corners and the atom at its centre as the unit, the potential energy corre- sponding to each cell is ay e?/d. When the solid is changed into a monatomic gas, each of these cells becomes an atom whose energy is —4e?/c. Thus to convert N atoms of a monovalent solid into a monatomic gas requires the expenditure of Soh Raya 2 ae ie (a ad -5,) units of work, or, if 2c/d=1°7, axe 124 If N is the number of atoms in a gramme NM=1, and di=M/A; hence the work required to convert 1 gramme of ee eee te ee : Electron Theory of Chemistry to Solids. tad the solid into a monatomic gas is, neglecting the difference 5 zs fo) p) to) Oo in the specific heats, ce Ais my C7 DA. = Ee M42 If M’ is the atomic weight of the atom whose mass is M, iL SSWily Selaoe << 1G ee and the preceding expression becomes, tet ae : 2: a ieee 3 1:45 x 10 = (ee) ; The energy 1:825¢?/d possessed by an atom in the solid state with its electron is equal to that acquired by a 1/3 charge e falling through 21°25 (a) ~ volts. Latent Heat of Fusion of Monovalent Elements. If the energy per atom in the liquid state is greater than that in the solid the transformation of each atom from the solid to the liquid state will involve the expenditure of a definite amount of energy. This, when there is no difference in the specific heats for the two states, will be a measure of the latent heat of fusion. | The energy per atom when these are at the centres of cubes whose edge is d is represented by Ob. if M’ is the atomic weight and A the density, this is equal to A \'8 ee 7 —12 ot e 34 . 10 7, —1°82. the energy per gramme 1S. ZOet x LO ANE — M’ lar) seus ° 5 ° ° (9) hence if A’ is the density in the liquid state, the difference in energy between one gramme of solid and one of liquid is 20°7 x 10% ¢ /A\I8 / AN\18 ooo) ilnieecllnel ae For potassium M’=39, the values of A, A’, as determined Phil. Mag. 8. 6. Vol. 43. No. 256. April 1922. 3B 738 Sir J. J. Thomson on the Application of the by Vicontini and Omodei (Landolt and Bornstein Taédellen), ce A='851, A’ ="8298. Substituting these values, we find that the difference be- tween the energy of one gramme of potassium in the solid and liquid state is 28 calories, the latent heat of fusion as found by experiment is 16. We have neglected any effect | due to the alteration of the specific heats, ‘and A and A’ are so close together that a small percentage error in either would produce a large error in their difference. We have hitherto confined our attention to cells in the interior of the solid; let us consider yey are the conditions of a cell at the surface. A corner of a cell on the surface is the meeting-place of only four cells ; thus only four corners coalesce, and as each corner only carries 1/8 of a charge, the total charge would be 1/2. Now the charge on an electron is the smallest that can exist, so that we cannot represent the state of the surface by placing electrons at the corners of the cells. We must replace this distribution by one where the number of elec- trons is only one-half that of the corners of the cells, 2. e., one-half of the number of atoms in the cells at the surface. One symmetrical way of distributing the electrons is, if ABCDEFGHIJKL represent the atoms below the surface ta ne. | Sy atari seve beans pals srs layer of electrons, to put electrons over BDF HJL but not over ACHGIK. | We must now consider the forces acting on the surface electrons. Starting from the top of the crystal, we have first a layer of electrons with N/2 electrons per unit area; next below this we have a layer of positively charged atoms with N atoms per unit area; foliowing this we have a layer of electrons with N electrons per unit area; and so on. The excess of positive over negative electricity in all the layers below the surface is the amount of electricity in a layer of atoms when there are N/2 atoms per unit area. It, as a first approximation, we suppose that the effect of the © charges carried by the atoms and electrons in the various layers is the same as if the amount of electricity in the layer Electron T heory of Chemastry to Solids. 739 instead of being concentrated at points were uniformly dis- tributed over the layer, the electric force produced by the electricity below the surface layer of electrons would be an attraction equal to 27o, where o is the surface density of the layer when there are N/2 atoms per unit area. Hence, if d be the distance between two atoms, c=e/2d?; hence the force acting on each surface electron is mc?/d?. This is the attraction due to the forces varying inversely as the square of the distance ; it must be balanced by the repulsive force varying inversely as the cube of the distance which the atoms exert on the electrons. If we suppose that the distance between the surface layer of electrons and the layer of atoms immediately beneath them is the same as that in the body of the crystal between a layer of electrons and the adjacent layer of atoms, we find that the repulsive force is greater than 7re?/d?, the attractive one. The upper layer of electrons will hence move further away from the atoms, so that the distance of the electrons in the oater layer from the nearest atoms is greater than it is for the electrons in the interior. Thus the potential energy of an atom on the surface will be greater than that of an atom in the interior ; ; as the atoms on the surface have an abnormally large amount.of energy, there will be in the expression for the energy of the solid a term proportional to the surface. This term will be proportional to the surface tension, so that the surface tension possessed by the substance can be calculated. I must leave this, however, for another occasion. Another possible way of arranging the atoms and the electrons when they are equal in number is that shown in fig. 4, where the A’s represent atoms and the H’s electrons. Fig. 4, We shall eonsider the conditions for stability and the properties of this system. Let us take the coordinate axes parallel to the axes of the lattices, and the origin at an eae The electrons may be divided into two classes: in the first class the coor dinates a 52 740 Sir J. J. Thomson on the Application of the of the electrons are represented by the equations : Ove pd. Od ne ace. where p,q, 7 are all even and d is the shortest’ distance between an atom and an electron ; in the second class they are represented by the equations | om NO. Y= 0g 2 00, where two of the three integers are odd and _the third even. The atoms are represented by = = (ile S21 where p, g, 7 must either all be odd or two even and one odd. * We shall suppose that the displacements of the atoms may be neglected. We shall denote a displacement parallel to z of an electron of the first type by p, those of the second type by p'. If we wish merely to investigate the stability, it will be sufficient to confine our attention to the case which is most likely to be unstable. A little consideration will show that this is the one where the displacement of any electron is equal and.opposite to that of its nearest neighbours in the plane at right angles to the direction of displacement. This will be the case if all the p displacements are the same and if the p’ displacements are equal and opposite to the p ones in the plane of wy and in all parallel planes whose distance from xy is an even multiple of d, while in those in which the distance is an odd multiple of. d the p and p! are equal both in sign and magnitude. : Tn this case the force tending to increase p due to the repulsion of the other electrons is equal to 3 2 pe” ( 1: 37? a (p? ii g a 72) 32 (p? Sa g a8 =) where » and g must both be odd and reven. I find by a combination of arithmetic and integration that the quantity inside the bracket is equal to 1:04, so that the disturbing 2:08pe? ake Consider, now, the restoring effect due to the attraction of the positive atoms. Owing to the symmetry of their distri- bution round the electron, no term arises from the part of the force which varies inversely as the square of the distance; the whole effect is due to the part varying inversely as the cube of the distance, and if we limit, as before, the effect of force due to the electrons is ii = Se: es Electron Theory of Chemistry to Solids, — 741 this to the atoms which are nearest, next nearest, and next next nearest to the electron the restoring force is De", S64 156, e70 Bored sasha C oa ta tas F) = 820 Hence the equation for p is 9 n a — pap (2°62 os —2-08) : ———_ = dt” so that for the equilibrium to be stable, Ge 20S Le The frequency p of the greatest vibration is given by the equation mp? = 962°. 2 Sea tas To express d in terms of M and A, the molecular weight and density of the solid, we notice that the atoms may be arranged in face-centred cubes with a side 2d, and that each of these cubes will contain four atoms. Hence the number of atoms in unit volume is 4 x (1/8d?), and this equals A/M. Hence aa ; so that the critical frequency is given by A c¢ Md lt as in the case considered on p. 732 the shortest distance between an atom and an electron is much the same in the solid and gaseous state, c/d will be nearly unity, so that mp? = 9°24. Te = 24 2 approximately. This is practically identical with the critical frequency for the other arrangement of electrons, so that the selective photo-electric effect will not distinguish between them. We proceed to consider the value of the bulk modulus given by the new arrangement. We calculate, as before, the potential energy of the system of electrons and atoms. We divide the system up in cells, but now the cells, instead of being cubes with electrons at the corners and atoms at the centre, are face-centred cubes whose side is equal to 2d and which have atoms at the middle points of the sides and also at the centre. I find in this way that the potential energy due to the forces varying inversely as 742 Sir J. J. Thomson on the Application of the the square of the distance is, for N atoms and N electrons, e2 | —N.1°77 a It follows, as on p. 735, that the bulk modulus will be given by Li . e? ad 5) where N is the number of atoms per unit volume. =n Sides’ Nand tee ince =" and | =(51 ‘ 226? (A 8"8 or a Ge This is not much more than half the value for the previous arrangement, which agreed exceedingly well with experiment. Hence we conclude that the arrangement we have just been discussing does not represent that of the alkali metals, while the distribution with one atom at the centre of a cube of eight electrons does so. Compressibility of a trivalent element. We proceed to test the theory by taking another case where the distribution of electrons is of a different character from either of the preceding cases. A trivalent element when in a solid state must have the atoms and electrons arranged so that there are three electrons for each atom. It the solid crystallizes in the regular system this will require each atom to be surrounded by a rhombic dodecahedron of electrons. When these dodecahedra are packed together so as to fill space, the arrangement can be seen to be equivalent to any one of the following :— 1. A system of cubical cells with the atoms at the corners of the cube and also at the centre of its faces. The electrons are arranged (a) at the middle points of the edges of the cube, (0) at the centre of the eight cubes into which ‘the larger cube is divided by planes bisecting its edges at right angles, (¢) an electron at the centre of the large cube. This arr angement gives the equivalent of four atoms per cell: the electrons at the middle points of the edges are equivalent to 3, those at the centre of the little cubes 8; these with the one at the centre of the large cube make 12—three times the number of atoms. J : ) q 4 ; . Electron Theory of Chemistry to Solids; = TAS. 2. Another equivalent arrangement is a system of cubical cells with electrons at the corners and the centres of the faces and at the centres of each of the eight cubes into which the larger cube is divided : the atoms are at the middle points of the edges of the large cube, and one is at the centre of the. large cube. The electrons at the corners of the cube in this arrangement correspond to the octahedral electrons. 3. An arrangement with the electrons at the corners, the centres of the edges and the centre of a cube, and at the centres of half the eight small cubes into which the larger cube is divided: the atoms are placed at the centres of the four little cubes not occupied by the electrons. ‘The centres are chosen so that both the electrons and atoms are at the corners of regular tetrahedra—the electrons on the outside of the cube with this arrangement corresponding to the cubical electrons on the dodecahedron. To calculate the compressibility we proceed, as before, by . calculating 5 for each type of atom and electron. Starting with the positive atom, the arrangement 1 affords the easiest means of calculating BS, where E is the charge on a positive atom and r the behave of the charge e from H. Putting E=3e, I find this sum is equal to 543 2, where d is a side of the large cube. For an octahedral electron, using the arrangement 2 for i 2 calculation, I find the corresponding quantity to be 7 I: For a cubical electron, using ee 3 for caleu- lation, the corresponding quantity is — o Br ee Each atom is associated with one octahedral electron and two cubical ones, so that the value for N atoms with their associated electrons is 2 2 ; —4N 5 (43-24-14 14-4) =—N5 (28°8). Tf N is the number of atoms per unit volume, M the mass of an atom, XM ] where A is the density of the substance. Since there are four atoms in each cube whose side is d, 4A if Deus N=3 or a= (an) . 744. Sir J. J. Thomson on the Application of the Hence the potential energy per unit volume due to: ee forces varying inversely as the square of the distance 1 is ‘ 98: 4/3 ae aun so that (see p. 735) the bulk modulus will be 28°8) @? (ANS? &: 9° 418\M _ Aluminium is a trivalent element, which crystallizes in the regular system; for this metal, A/M is equal to 2°65/27 x 64x 10-™., Substituting this valne in the pre- ceding expression, we find that the value of the bulk modulus is “98x 10; the value found by experiment is *78 x 10”. The limiting Srequency of the vibrations of the electrons. ‘ Following the method used for the monovalent elements, : I find that the limiting period for the vibrations of the octa- hedral electrons 1 is given by the equal oe ING ap? = od: 88 e2 Maes and that of the cubical electrons by | mp? = 36900 he where - ne ; i_aA a2 AMI The stability of the system requires that c/d should be greater than °3. If we compare these expressions with that for sodium, and remember that A/M for aluminium is 2-4 times that for sodium, we see that the wave-length of the critical frequency for aluminium will be less than 1700 A.U., 2. e. far up in the ultra-violet. Thus-it is only in this region that we should expect to find evidence of the selective photo-electric effect with aluminium. Electrons arranged in face-centred cubes. With this arrangement the number of electrons is four times the number of atums. As this is the proportion between the atoms and electrons in all binary compounds Electron Theory of Chemistry to Solids. 745 which satisfy the ordinary conditions of valency, its investi- gation is of exceptional importance. Stability of the arrangement. Take the origin of coordinates at the corners of one of the cubes, the axes parallel to the sides of the cube. Let 2d be a side of the cube. Then the coordinates of the corners of the cubes are given by Lie ae i —- Gs = where p, g, 7 are even integers. The coordinates of the centres of the faces are given by aay Gd ee where two out of the integers p, g, r are odd and the third even. The coordinates of the atoms are given by pa, y= gd, 2s —rd, where p,q, rareallodd. | | Consider a displacement of the electrons parallel to the axis of z, such that all the electrons of one type have the same displacement. Let p,, po, p3 be the displacement of the electrons forming the corners of the cube, the centres of the faces parallel to xy, and the centres of the other faces respectively. | The force tending to increase p;, due to the displace- ment po, is e” E ar (p1— 2) a » ( + 97+ aR oe) for all odd values of p and g and even values of r. The summation I find to be 1:1. The force tending to increase p; may thus be written a(p\— pe), where e2 6; = Teng as Let us now consider the force tending to increase p, due to the displacements p;. If pe is equal to p;, then the symmetrical system formed by the whole of the electrons at the centre of the faces will not produce any force tending to increase p,;; hence the force due to the 746 Sir J. J. Thomson on the Application of the displacement p; tending to increase p, is — a(p1—ps). The total force tending to increase p, is thus a(p3— pz) 3 the force tending to increase pg is a(p2— 1) ; and that tending to increase pz is a (Pop). The force due to the positive charges tending to diminish p; if His the positive charge of the atom is 8,p; when &, = Be(s. CM Nea) ee 8 z) 97 dt 121 2 6s where the effect of the force varying inversely as the cube. of the distance is supposed to be confined to the nearest, next nearest, and nex! next nearest atoms. The effect of the force varying inversely as the square of the distance exerted by the positive atoms is proportional to. 3 iL Ree er (c. + G47 yr (p+¢ ae for all odd values of p, g, 7, and therefore vanishes. The force tending to diminish p2 due to the positive atoms. is 85p2, where He /,¢ B= 3 (65-7 = | 317) a =a(5 960-3 77). The effect of the force varying inversely as the square of the distance is 3r? “a > (ern ge ey cere for all odd values of r and even values of p and g. The sum under these conditions is —3°77. The force due to the positive charge tending to diminish. p3 18 83p3, where £3) Bs =p (1 88— 1-755). Klectron Theory of Chemistry to Solids. 147 The equations of motion are a? m a = a(p3—p2) —B3p1, a? a a = = a(p3—p1) — Bopo, d? + m Ps gle —ps)— Psp, dt \ If p1, Po, p3 are Joel a to e?’; p is given by the equation mp? — By, —? 9? Bs The electrical moment of each cell due to these dis- placements is (py + P2 == 2p3)e. The number of cells per unit volume is 1/8d?.- Hence the electrical moment per unit volume is ny be (p1 + P2 + 2ps) 3? _and if K is the specific inductive capacity this is equal to K-1 Anr Substituting the values for pi, Po, Ps, we find sg por 0) An 8y (474y—252y?— 204) ’ alte where y=c/d and must be between ‘71 and 1:17. : Substituting the permissible values of y, we find that the specific inductive capacity of solids possessing this structure would be greater than four ; so that it is only substances with a high refractive index which can have their electrons arranged in this way. Electron T heory of Chemistry to Solids. 749. Value of the bulk modulus. This can be calculated if we know the value of © e,e,/7. for the system. To make the calculation, consider first the value of the contribution made by one of the electrons at the corners of the cube to this sum. Let A be such a corner. We suppose the medium built up of face- centred cubes: remove the eight that meet at A, and calculate separately the contributions the charges so re-. moved would have made to the sum; then treat each of the remaining cells as a unit, and calculate the contribution, of each of these. This contribution diminishes very rapidly as the distance from the electron increases. In this way we find that the value of the sum of the terms > ¢,e,/7. which contain this corner electron is —4:5e?/D, where. D is a side of the cube. Now consider the contribution to the sum of one of the electrons at the centre of a face. Proceeding by a similar method, I find this equal to. —3°2¢/D; while the contribution of a positively-charged atom is —52°6 e?/D. As the charges carried by the electrons at the centre. of the faces is three times that carried by the electrons,. the value of & e,e,/r for N cells is equal to —2(4°543 x 3°2 452°6)e7/D, t. €. to — 33°35 e?/D. | From p. 735 we see that this will correspond to a bulk. modulus & given by the equation 1 = 38°35 (A 4/3 Sian Eoe* 4. ma So that for similar values of A/M the bulk modulus with?this. arrangement of electrons is about nine times that for the alkali metals. For the diamond for which A=3°52 the value of the bulk modulus deduced from this formula is 8:1x 10. This is much higher than the value 2x 10” obtained by Richards for carbon in the form of diamond. Richards puts a question mark after this value, so that it may be inferred that there is more uncertainty about this value than about those of the other elements. The very high value for the diamond given by the formula is due not merely to the value of the coefficient but to the abnormally high value of A/M for the diamond— - a value far in excess of that for any other element. In this investigation we have supposed that each carbon 750 Sir J. J. Thomson on the Application of the atom acts as a separate and independent unit. There is con- siderable evidence, however, that the atoms in the diamond are arranged in groups. If so, it may be necessary to take the group as the unit rather than the atom ; this would indicate that when the diamond is compressed the com- pression is effected more by bringing the groups together than by diminishing the distance between the atoms in a group. This grouping of the atoms would seriously modify the expression for the bulk modulus, for we have to replace A/M by A/nM, where » is the number of atoms in a group. This will diminish the value of & in the proportion of 1 to n**: thus if there were three carbon atoms grouped together in the diamond, & would be 8-1/3* or 19x 10”; if there were four the value 1°3x10', values which are much more consistent with Richard’s results than that deduced on the assumption that the atoms were not grouped. A point of considerable interest with regard to the constitution of salts is raised by the value of the com- pressibility for electrons arranged in a space-centred cube ; af all the movable electrons go into the walls of the cell the space-centred cube would be the cells enclosing the atoms in a binary valency compound, since in these compounds there are eight electrons for the two atoms. The value we have obtained for k shows that if this were the case these compounds would be characterized by small compressibility and consequently high boiling-points : the oxides answer to this description, the chlorides certainly do not. Though the & has been determined for few simple compounds, its value for NaCl and-for KCl has been deter- mined by Voigt, Rontgen, and Richards. The results are given in the following table; the last column gives the walue calculated by the preceding formula. A. k observed. hk ca'culated. Wal. 22:. Zoli 23 0 1°? x 102 IOI ec eos) “192 x 10% “76 0S It will be noticed that the values for the chlorides indieated by the theory are far in excess of those found iby experiment. "The actual values are comparable with those which would be obtained from the formula (8) on p. 736, which corresponds to the case of simple cubical cells with a single positive charge at the centre of each. This suggests that the -structure of the chloride cells is not a face-centred cube Electron Theory of Chemistry to Solids. 751 with a singly-charged sodium atom in one cell and a chlorine atom with a positive charge of 7 in the adjacent ones, but that the chlorine atom instead of Thain all its seven electrons to form the outer walls of the cell “only uses one, and the other six electrons and the atom with seven charges form a unit having unit positive charge ; so that the structure of the cell for sodium chloride would be represented by a sodium atom at the centre of one cell and a chlorine atom surrounded by six electrons arranged asa regular octahedron considerably smaller than the cube at the centres of adjacent cells. This arrangement is in some respects more symme- trical than the other, for in it we have what is equivalent to a unit positive charge at the centre of each cell ; in the other arrangement there is a unit positive chargein one cell and a positive charge of seven in the adjacent cell. Thus an electron at the centre of a face would be pulled one way by a charge of seven and in the opposite way by a charge of one. The result would be that the electron would be displaced ; the face-centred cubes would become cubes with pyramids on their faces, the pyramids being convex for one set of tubes and concave for the other. If we regard the chlorine cel! as equivalent to a mono- valent one, we can find the difference between the energy of a gramme molecule of sodium chloride and the energy of the sodium and chlorine before combination. By the equation (9) on p. 737 the energy of 23. grammes of sodium is = wfOy ~2-07 x 10"(5; | where A=-971iis the density of metallic sodium. Thus the energy of the sodium before combination is —2°07x 348 x 108 ; after combination the energy is 2 1)3 = 7x10" | as ae where A,;=2-17 is the density of sodium chloride. The energy after combination is thus — 2°07 x 4213 x 10”. The energy of the chlorine before combination is Ao yo kee ye where A, is the density of liquid chlorine, A,=1°558 at —2°07 752 Sir J. J. Thomson on the Application oj the —33°C. Hence the energy of the liquid chlorine before combination is 9-07 x 353 x 108 - ge ae after combination it is the same as that of the sodium. Hence the change in energy is | 2°07 x 108(2 x 4213 —°348 —°353) = 2:07 x 10" x 42 eres or about 70,000 calories. . ‘he heat of combination for sodium and gaseous chlorine is 97°8 x 10%. If we subtract the correction given by Richards of 18,000 calories for the vaporization of the liquid chlorine, the heat produced when liquid chlorine combines with sodium would be about 80,000 calories; so that the theoretical and experimental results are in fair agreement. An interesting point in connexion with the energy of a crystallized solid compound is whether all the atoms in a compound are equally spaced, or whether some may be regarded as collected in a group and more intimately connected together than the others. Chemists have intro- duced the conception of radicles such as CN, CH;, C,Hs, CH, which play the same part in chemical reactions as the atoms of monovalent or divalent atoms. The question is whether in a solid crystalline compound containing these radicles their atoms act as a single group, or whether each of their atoms occupies a separate cell. This question could be answered by a determination of the compressibility of the compound. Thus, take the compound KCN as an example: are there separate cells for the atoms of K, C, N—~. e., are there three cells for each molecule, or are the C and N atoms paired together in a single cell so that each molecule of KCN furnishes two cells instead of three? The atomic volume of KCN is 42°8: if there are three cells the volume of each cell will be 14°3, if there are two it will be 21:4. The compressibility of the substance diminishes with the size of the cell ; the larger cell would give a compressibility not very different from sodium, the smaller one a com- pressibility less than two-thirds of this value. I cannot, however, find any record of the compressibility of KCN, Take ammonium chloride, NH,Cl, as another example. The atomic volume is 35°2: if NH, forms a group at the centre of one cell each molecule of NH,Cl will furnish two cells, each having an atomie volume of 17°6 inter- mediate between the volumes of the cells in solid lithium and sodium and giving a compressibility between the values for those substances; if, however, each of the Electron Theory of Chemistry to Solids. 753 six atoms in NH,Cl had a separate cell the volume of each would be 5:8. ‘This is small compared with that of the cells in any of the chlorides of the alkali metals; so that NH,Cl would have an abnormally small compressibility compared with any of these salts. Though no deter- minations of the compressibility of NH,Cl have been published, there seems nothing to indicate that its com- pressibility is abnormally small ; if it is not, then ammonium, NH,, must behave like a single atom in NH,Cl. Stability of chains of atoms. The possibility of atoms being linked together in long chains is interesting from the point of view of organic chemistry. We shall consider the case of a chain when the electrons are arranged like a line of cubical boxes placed end to end (fig. 5) ; the positively charged atoms are at the centres of the cubes. The chain is supposed to be of infinite length. Fic. 5. Consider first the forces exerted by the electrons on each other. The force in the direction P’’P on an electron P, due to the other electrons, will, if a is the side of one of the cubes, be e” iis cif 1 1 1 aA 1+2(s5at gmat io (At Gy paet ie ie due to the electrons in the line Q'’’R”’S"; and Za i 1: 1 1 =e gat? (srt gat qt -@p jet kins Le ‘ due to those along the line Q'” RS”. The first part is equal to the second ito Phil. Mag. 8. 6. Vol. 43. No. 256. April 1922. aC 754 Sir J. J. Thomson on the Application of the Hence the force on P along P’’ P due to the other electrons is 2 3°04 = We now proceed to calculate the force on P due to the ‘positive charges if the force between an electron and a positive charge Ei at a distance r is represented by Ee fey The force on P along PP” due to the positive charges will be 8HKe (1 1 1 ik “a? Quote ee (5 1 a ) a’ ere ope e sa (189-5 1:95), a a In order that the positive charges on the atoms should be namerically equal to the charges on the electrons, E must equal 4c ; so that the force due to the positive atoms is ery equ pie! 8). Hence for equilibrium 2 : P we 204 = (7-56— i 8) fee a a a so that 6) ieee a Though this value of a would ensure the equilibrium of the system, it can be shown that the equilibrium is unstable. We might do this by obtaining the frequencies of the vibrations of the electrons about their position of equilibrium, but as the equilibrium proves to be unstable it is unnecessary to take this trouble, and we shall content ourselves with showing that the equilibrium is unstable for a particular displacement of the electrons. The displacement we shall consider when P is displaced vertically upwards through a distance p, Q vertically down- wards through the same distance, R vertically upwards. Thus the displacements of the electrons on the line PQR..S are all of equal magnitude, but alternate displacements have opposite signs. The same is true for the electrons on the three Hlectron Theory of Chemistry to Solids. 755 other lines parallel to PQR; the displacements of electrons in the same vertical line are equal, but the displacement of any electron is of opposite sign to those of its three nearest neighbours on a horizontal - plane. The force due to the other electrens tending to increase p, the vertical dispiacement of P is 8 agin eae s PEr rae due to the electrons QRS, plus = i 1 ge Ge ieee )) = Fo (2-408), due to the electrons Q'R’S’, lus Ape? / 1 ik 1 I 1 ya e reo = ews xX ‘Obd, a due to the electrons QRS”, gee fi ofl 1 1 fe linia oy a i pa leant Leman pee tyget-)—-3( gat? (gn it a pers ‘ ty e527 )) \= Bex 1366, due to the electrons Q’"R/”S’"" Taking these together, the force tending to increase p is Li) 2 € ~ Fy 5950. Now consider the forces due to the positive atoms tending to diminish p ; this is Ke 1 1 1 lop—, a is 93/2 b 12 * 9732 a calle je if \ i “ibe J ey a as Gs eee a agg Sa ae snes rns We, pos (08 +5 934), Since H = 4e and c/a=°58, this equals _ x 4:28, 302 756 Application of Electron Theory of Chemistry to Solids. Thus the force tending to increase the displacement is greater than that tending to diminish it, and so the equi- librium is unstable. [t follows from this result that the equilibrium of a long chain of carbon atoms alone must be unstable. With atoms such as those of oxygen, which can have positive charges amounting to six units, a chain such as is shown in plan in fig. 6, where the electrons are at the corners of cubes placed so that each cube has an edge in common with its nearest neighbour, would satisfy the Fig. 6. condition of having six electrons for each atom. An investi- gation similar to that just given shows, however, that the arrangement would be unstable. This is in accordance with experience, as long chains of oxygen are exceedingly unstable. A chain which I find to be just stable as far as the displace- ment of electrons is concerned is that where doubly charged atoms and electrons are arranged in the way shown in fig. 7. The atoms A are arranged at equal intervals along a straight line. The electrons are arranged in pairs, the line joining the constituents of a pair being at right angles to AA. Each pair is atright angles to its nearest neighbour. Each atom is thus at the centre of a regular tetrahedron formed by the electrons which are its next nearest neighbours. This arrangement is stable if the atoms are fixed ; and the stability when the atoms are free to move can be insured by giving a suitable value to the force between two positively charged atoms. If the radicle CH, be regarded as equivalent to an atom of a divalent element with two disposable electrons, then by the preceding result we might expect to get long chains of CH, Fluorescence and Photo-Chemistry. 757 radicie, though, as we have seen, we could not get them of carbon atoms. We know of many compounds which contain chains containing very large numbers of CH. Summary. A theory of the structure of solids is given based on the views as to the nature of the structure of the atom and the mechanism of chemical combination which I have given in previous papers in the Philosophical Magazine. Since the atom of a monovalent element has one disposable electron, that of a divalent element two, and so on, there must in a crystal of a monovalent element be such an arrangement of electrons and atoms that for each atom there is one electron, in a crystal of a divalent atom there must be two electrons for each atom, and so on.. It is shown that this condition determines the crystallographic forms in which the various elements can crystallize, and leads to a con- ception of the structure of crystalline solids which allows us to calculate without further assumptions the values of certain physical constants such as the bulk modulus, the value of the critical frequency, and the dielectric constants. The values of these are calculated for elements of different valencies, and are found to agree closely with those found by experiment. LXXXVIII. Fluorescence and Photo-Chemistry. By R. W. Woon, Johns Hopkins University *. [Plate XVI] HOUGH it has been known for some time that many fluorescent solutions are bleached by the action of light, Perrin appears to have been the first to: definitely © associate fluorescence with chemical change and to regard the light emitted by fluorescent substances as due to the “flashes” of exploding molecules. Perrin showed that thin films of a solution of fluorescein were bleached at the spot on which the light was focussed by the sub-stage parabolic reflector used for the examination of colloidal particles under the microscope. He also examined other fluorescent substances and found that in all cases they were bleached with greater or less rapidity by the action of light. * Communicated by the Author. 758 Prof. R. W. Wood on The experiments which I shall describe in the present paper were carried on in my summer laboratory at Hast Hampton in 1919 and 1920, and, though they were made with somewhat inadequate facilities, they appeared to establish certain points which were at variance with generally accepted views. By operating with a very intense beam of light, obtained by concentrating sunlight with a short-focus lens 6 inches in diameter, sufficient quantities of the transformation products produced by the action of light on fluorescent solutions were obtained to make possible the examination of their optical and chemical properties. The time required to effect the change varies with the nature of the substance : eosine 1s completely changed in two or three minutes, while rhodamine in the same concentration requires several hours. Usually the decomposition of the fluorescent substance by the action of light yields a coloured non-fluorescent sub- stance which is bleached by the further action of the light to a colourless solution. This makes it impossible to obtain the intermediate substance in a pure state; but, by choosing the proper concentration and stopping the illumination at the right moment, it was possible to obtain fairly satisfactory results. The absorption spectra of the various compounds were photographed by mounting a prismatic cell of quartz filled with the soluiion in front of the slit of a quartz spectro- graph, in which the prism was replaced with a plane grating, and illuminating the instrument with parallel rays of light. In this way we obtain the form of the absorption curve, as well as its location in the spectrum. The angle of the hollow prism was about 10°, and the thin edge (in a horizontal position) was brought into coincidence with the bottom of the slit. The refraction of the fluid prism was compensated by an opposed prism of quartz of about the same angle. On Pl. XVI. will be found reproductions of some of the photographs. I have called the intermediate coloured substances photo-compounds, for want of a better name. Fluorescein (uranine), for example, which is lemon-yellow by transmitted light, is changed into a non-fluorescent compound (photo- fluorescein), which is orange-red, with an absorption band of a totally different “form. The fluorescein band is very steep on the red side, sloping down gradually towards the region of shorter wave- lengths. The band shown by the photo-fluorescein, however, is more Fluorescence and Photo-Chemistry. 799 nearly symmetrical, sloping gradually in both directions. It is to this peculiarity that the red colour is due. Continued illumination eventually bleaches. the photo- compound. Hosine behaves in a similar manner—the action of the light being much more rapid, however. Rhodamine behaves in a very curious manner. A solution in water is gradually bleached by the action of the light without the formation of an intermediate coloured body, the absorption band fading away without change of form or position. In solution in methyl or ethyl alcohol, how- ever, a strongly fluorescent intermediate body is formed, which emits a green fluorescence and exhibits an absorption band of totally different form from that shown by the rhodamine (which shows an orange-red fluorescence). ‘Tho rhodamine absorption band has a double maximum, while the band of photo-rhodamine is single and shifted towards the region of shorter wave-lengths (see Pl. XVI). If the alcoholic solution of photo-rhodamine, which shows the green fluorescence, is evaporated, and the residue dis- solved in water, the solution shows the same green fluor- escence—a circumstance which is interesting in connexion with the fact that the photo-rhodamine cannot be formed by the action of light on an aqueous solution. It seems probable that in this case the photo-compound may be formed also as an intermediate product, but its rate ot decomposition is as great as that of the rhodamine. It is clear that to obtain a recognizable amount of an intermediate body it must be more stable under the action of the light than the original substance ; otherwise it will disappear as rapidly as it is formed. It was thought that the rhodamine might be a mixture of two substances—one having a single absorption band at 5500, the other with a band at 5200, the double band actually photographed being the superposition of the two If this were the case the first substance would probably yield a red fluorescence, the second a green fluorescence ; and if the former were decomposed more rapidly by the action of the light we should have observed change in the colour of the fluorescent light without the intervention of an intermediate body: in other words, the green fluorescence would be present initially, but masked by the orange-red fluorescence. This point was tested by ex- amining the fluorescent light of two solutions simuitaneously with a grating—one a solution of rhodamine, the other 760 Prof. R. W. Wood on a solution of photo-rhodamine formed by the action of light on a solution of rhodamine of similar concentration. In this way the fluorescent light was spread out into a spectrum and the possible masking effect was eliminated. It was found that the green fluorescence was enormously stronger in the case of the photo-rhodamine than in the case of the original solution. This makes it seem certain that we have an intermediate coloured body, which is fluorescent. Perrin has recorded similar results. A number of other fluorescent substances were examined, with the following results :— Phloxine: Bleached by the action of light, the absorption band remaining in its original position and gradually fading away. Resorufine : Similar to phloxine. Acridine Orange: Solution yellow by transmitted light, becomes reddish after long illumination, with broadened absorption band. Much like fluorescein. Napthaline Red: Alcoholic solution shows orange fluor- escence. Bleached by illumination, with no trace of other coloured compound. Solution in hot water: Illumination causes the precipitation of a fine suspension which gives a bluish colour by transmitted light, The precipitate is soluble in alcohol and shows fluorescence similar to that of the alcoholic solution of the original substance, but less bright. Influence of Temperature on Fluorescence. Perrin states in his paper (Annales de Physique, 1918) that the intensity of the fluorescent light is independent of temperature, and draws a comparison between fluorescence and radio-activity. I have found, however, that rhodamine is almost non- fluorescent in a water solution at 100°. The observation was first made in the case of a small tube containing about 1 c.c. of the solution, illuminated with sunlight focussed by the large lens. The solution was raised to the boiling- point in about a minute and the fluorescence disappeared, reappearing if the solution was cooled. Fluorescein appeared to be slightly brighter at zero than at room- temperature, and slightly less at 100°. EHosine showed no temperature effect. On account of the peculiar temperature effect in the case of rhodamine, it was necessary to keep the solutions cool by a water jacket during the trans- formation experiments recorded in the previous section. Fluorescence and Photo-Chemistry. 761 Intensity Relation between Fluorescent and Eexeiting Light. It has been pretty well established that the intensity of the fluorescent light is proportional to the intensity of the exciting light. Knoblauch reduced the intensity of his exciting beam to 1/6400 by means of dark glass, and found that the relation held. As his maximum intensity fell far short of the intensity at the focus of my lens (which was roughly 1400 times the intensity of normal sunlight. and as it was very essential for the interpretation of the results of certain experiments about to be described to be sure that the relation held for the very high intensities used, the following very simple and instructive experiment was made. The theory of the experiment will be made clear by fig. 1. The 6-inch lens a was illuminated bv sunlight, and a small Fig, 1. %S Ss = = ss me Si = — = Sass Sa > Ss ae Cf. — = ap tae Sec she. BE — sl = -—_- si oe ¥ “—— 4 ea ent ot Ee ale Eye. Det Mair prism 6 of about 10° was mounted near one edge of it. The area of this prism which was exposed was 4 per cent. of the area of the lens not covered by the prism. This ratio of areas was chosen for reasons which will appear presently. The powerful beam concentrated by the lens illuminated the small bottle c, which was filled with a solution of fluorescein. The beam deflected by the prism was reflected by the silvered mirror @d and concentrated on the bottle e, which also con- tained fluorescein. The ratio of the illuminating intensities was thus 4:100. If the fluorescent ratios were in the same ratio, then the image of the fluorescent spot on bottle ¢ seen reflected in a single surface of glass (4 per cent. reflexion coeff.) should have the same intensity as the fluorescent spot on bottle e viewed directly. The reflexion was taken from the surface of a small prism /, the opposite surface having been painted with black paint to prevent reflexion. As this prism had a razor-edge, it was possible by holding 762 Prof. R. W. Wood on the eye and prism in suitable positions to bring the two images together with a sharp line of separation between them. This line was found to be practically invisible, showing equality of the intensities and proving that the law holds up to the very high intensities used. The lens was then stopped.down to 1/600,000 of its full aperture by means of a sheet of black paper with a small pin-hole in it. Hven with this enormous reduction in the fluorescent spot on the bottle c the fluorescence was distinctly visible, which shows us that we can increase the intensity of a beam of light, which is sufficiently intense to cause the decomposition of the fluorescein molecules, six hundred thousandfold without causing the instantaneous destruction of all of them—in fact an exposure of ten or fifteen minutes to the intense beam is necessary before the transformation is complete. Perrin’s theory is that the intensity of the fluorescent light emitted by a single molecule is independent of the intensity of the exciting light, and that the increase in the intensity of the total emitted light is due merely to the circumstance that more molecules are being destroyed (or .transformed) per unit of time in the case of intense illumi- nation. In support of this, we have the circumstance that sooner or later the fluorescent solutions are bleached by the action of the light. But this is also true of a large number of non-fluorescent solutions of organic colouring- matters. We can make the alternative hypothesis that all of the molecules in the solution fluoresce to an equal degree, and that as the intensity of the exciting light increases the amplitude | of the fluorescent radiation emitted by each molecule in- creases to a corresponding degree. While this is going on the solution is gradually undergoing decomposition by the action of the light, asin the case of a non-fluorescent solu- tion. It seems extremely desirable to devise experiments which will enable us to choose the more probable of the two hypotheses. Two such experiments suggested themselves. Are the Molecules Fluorescent only at the Moment of - Breakdown ? It would appear as if some light might be thrown on tlie two alternative hypotheses mentioned above if we can pre- vent fluorescence and still allow light to act on the medium. There are two ways in which this can be done. As has been already stated, the fluorescence of an aqueous solution of Fluorescence and Photo-Chenustry. 763 rhodamine nearly ceases at 100°. Moreover, fluorescence ceases if the concentration becomes too great, due to what Perrin calls protective action. If, now, it be found that preventing the fluorescence by these means prevents the bleaching of the solution by the action of light, it will be a strong argument for the theory of Perrin. On the other hand, if it be found that bleaching proceeds at the same rate as before, we shall have grounds for suspecting tnat the fluorescence is not directly connected with the breakdown of the molecule. Both of these experiments have been tried. A strong solution of eosine was made which showed no trace otf fluorescence, even in concentrated sunlight. A caretully measured amount of this solution (4 ¢.c.) was introduced into a small cylindrical flat-bottomed bottle and illuminated by a vertical beam of sunlight concentrated by the six-inch lens for half an hour. The bottle was kept cool by immer- sion in a beaker of water diiring the experiment. A similar amount of the strong solution diluted with 5 c.c. of water was subjected to a similar treatment for the same length of time, fluorescing brilliantly throughout the experiment. The contents of the first tube were now diluted to the same volume, and the contents of the two tubes compared. ‘The concen- trated solution had suffered no change from the action of the light, while the diluted solution was largely made up of the photo-eosine, as was shown by its red colour and feebler fluorescence. This indicates that proximity of the molecules protects the solution against decomposition as well as against fluorescence, and argues in favour of the theory that the decomposition results from the fluorescence. The experiment of inhibiting the fluorescence of rhodamine by raising it to a temperature of 100° was next tried. Two small tubes were prepared, of equal size, and equal amounts of a dilute solution of rhodamine introduced into them. They were then sealed in a flame as close as possible to the surface of the solution. One tube was mounted in a beaker of cold water close to the wall, the other ina beaker of water kept boiling by a Bunsen burner. Both were illuminated simultaneously by sunlight concentrated by two six-inch lenses, which formed images of the sun of a diameter about half that of the tubes. In this experiment the conditions were exactly the same, except that one tube was at a higher temperature than the other. The fluorescence of the cold solution was about twenty times as bright as that of the 764 Prof. R. W. Wood on hot solution, yet decomposition proceeded at practically the same rate in the two cases. At the end of half an hour the colour had been considerably reduced, but no difference could be detected, except that the fluorescence of the high- temperature tube, after it had been cooled off, was slightly greenish in colour, while that of the other was orange- yellow as at the beginning of the experiment, only much less bright owing to the partial decomposition. This was a rather disappointing result from the point of view of confirming Perrin’s theory. A control experiment showed that the rhodamine solution was not bleached by an hour’s heating in a sealed tube immersed in a beaker of boiling water. Attention is drawn to the fact that not all samples of rhodamine are sensitive to temperature. I have, in all, about a dozen different rhodamines of which only two show the very marked loss of fluorescent power at 100°. Function of the Intensity of the Haciting Light. A very remarkable relation between the rate of breakdown of eosine and the intensity of the exciting light was dis- covered, which has probably a very direct bearing on Perrin’s theory. Stating it in simple form, we may say that high intensities acting for a short time produce a greater amount of decomposition than low intensities acting for a long time, the time being increased in proportion to the reduction of the intensity, of course. This means that the change is not proportional to the amount of energy extracted from the incident light—a rather surprising result, and one at variance with the Bunsen-Roscoe law. The experiment was made in the following way :—Two circular cells were made, each six inches in diameter and + inch in thickness, by cementing glass disks to brass rings. Hach cell was filled with a solution of eosine of the same concentration. One cell was exposed normally to direct sunlight, the other to sunlight concentrated by a six- inch lens. The solar image was formed near the bottom of the cell, so that the convection would be as complete as possible. In each case the same amount of light is passed through the cell, and through the same thickness of solution (4 inch). The only difference is that in one case it is spread uniformly over the whole area of the cell, and in the other case it is concentrated on an area of very small size. Fluorescence and Photo-Chemistry. 765 Variations in the intensity of the sunlight during the experi- ment make no difference, as both cells are affected by them to the same degree. After an exposure of three hours it was found that the cell receiving the concentrated illumination showed a much greater change than the one exposed to normal sunlight. Its fluorescence was less, and the colour of the transmitted light much redder (due to the formation of a greater quantity of photo-eosine). ‘The gain in the action by increasing the intensity is probably even more marked than is shown by this experi- ment. To perform the experiment in an ideal manner a given amount of the solution should pass enly once through the concentrated light-beam. Asa matter of fact, there Ge strong convection current upward, due to the heating of the liquid at the focus, and the solution which has passed through the beam and been partially decomposed mixes with the fresh solution and dilutes it, so to speak. This will make the in- creased action due to high intensity of the light less marked than it would beif the solution could be kept at rest, and the solar image moved over every portion of the cell in a per- fectly uniform manner, so that each surface element would recelve equal exposure. This relation between the lutensity of the light and the rate of change-is evidently extremely important, and it is being subjected to a careful quantitative investigation at the present time in collaboration with Mr. Subkow, one of ny students. Unstable dyes, which are non-fluorescent, do not behave in this way, the bleaching in the two cells being equal in amount. We now have at our disposal a mirror of 32 inches aper- ture and 18 inches focus, with which platinum-foil can be instantly melted by concentrated sunlight. With this mirror the photo compounds can be made in quantity sufficient for chemical examination. [ 766 | LXXXIX. Note on a possible Relationship between the Focal Length of Microscope Objectives, and the number of Fringes seen in Convergent Polarized Light. By F. lan G. Rawiins*. 1* connexion with a statement by Groth (‘Optical Pro- perties of Crystals,’ Chapman & Hall, 1910), to the effect that the number of interference fringes seen in crystals by means of a microscope 1n convergent polarized light is a function of the focal length of the objectives used, it was considered of interest to investigate this—in as simple a way as possible—with a view to determining whether such rela- tionship was capable of exact mathematical expression: more especially since inquiry into the literature of the subject failed to show that this aspect of the question had received attention. Three crystals were available for trial (quartz, tourmaline, and mica), the approximate thicknesses being 5, 4, and 1mm. respectively. It is clearly necessary to have sections of unusual thickness, otherwise a sufficient number of obser- vations cannot be made. Daylight was used together with ared gelatine filter. The objectives were ordinary achro- matic lenses, and the figures were viewed by means of a supplementary Becké lens above the eyepiece. The following data were obtained :-— QuaRTz. | ‘TOURMALINE. Mica. ae | Focal Length Focal Length Focal Length of Objective No. of of Objective No. of | of Objective No. of in mm. Rings in mm. Rings in mm. Rings =F, == Ni = =—=INi — He =N. 32 ) 32 9 16 1 22 2 22 2 8 4 16 4 16 4 + 7 8 8 8 ‘leh 4 itr 4 16 These results are plotted on the accompanying graph (dotted-line curves), from which it will be seen that a fairly smooth curve can be drawn through the points. Plotting log F against log N was a failure in so far that a straight line was not forthcoming. A trial was then made by taking for variables log, F and N, and the points shown in solid lines were obtained. It is apparent that straight lines can be drawn through these * Communicated by the Author. Interference Fringes and Focal Length. 767 with considerable accuracy. This at once suggests an ex- ‘ponential equation of the form a oD (where A is a constant), or pla aM (2) wots SONY | Lukens from which we obtain = Fye-4% as the equation required. Is in Nn N io} ro —— 2 4oce/ Length 7.7. © rs rN) ° ee *e ro) ——> 409. (Foce/Length) OS ear VIN eras $ n @ 3 = =. = 4 | 2 3 4 5 6 7 & 9 io WW fea 4 BIS Kee hee) el 2 RIGS are) No. of Rings —————> Measurement of the angle made with the axis of N gives the following values for A:— Oirartae this eee SAWAS Pourmaune “2 (202 Pio i) SVC es ae et ee “2493 A is necessarily a mere number, and it is of great interest to speculate as to itsnature. Since monochromatic light was not available, it is evidently useless to attempt to identify A with any function of the index of refraction or bi-refringence of the crystals used. If we invert equation (1), we obtain the result that the rate of decrease of the number of fringes with focal length is proportional to the reciprocal of the focal length itself. These results, though very elementary, seem to show that another case in nature has been found to add to the list of known phenomena following a logarithmic law of decay. ies ORIN XC. On Products of Legendre Functions. By J. W. Nicnotson, F.R.S., Fellow of Balliol College, Oxford *. Ca fact that it is possible to express the product of the two functions P,(w), P..(u) as a series of functions of the type P,(«) was discovered independently by Couch Adams and Ferrers many years ago. Apart from a paper by Sir William Niven, the subject of these products, which can be developed in a manner of some significance for applied mathematics, does not appear to have attracted the attention which it deserves. Niven’s paper made a notable advance, being essentially more general in its scope than the intuitive method of the earlier investigators—afterwards arranged more concisely by Todhunter. But no author has considered the equally interesting and important products in which functions of the type Q,(m#) appear, though these can be included in an investigation by a method which is, after Poincaré, now familiar in the theory of differential equations. An outline of this investigation, with some of the more important properties of the functions, is our immediate object. Let an accent denote a differentiation with respect to y, and let P stand for P,(w) or Q,(), and Q for Pn(w), Qm(p). For convenience, we write also M=m(m+1), N=n(n+1). Then (1—p?) P”=2uP’—NP, (1—27)Q" = 24Q'— MQ. We first form the linear differential equation of the fourth order satisfied by PQ. Its general solution must be A en ae B PQ; = C ena ote D QO. Qe where A, B, C, D are constant:, and from its solution in the form of series we can derive the four typical products. ‘ y=PQ, we have y= GL =Py +r, eee (+ a + QP" SSE ee a NP), whence (1— w*)y" —2py' + (M+ N)y=2P'Q’1—p?). * Communicated by the Author. — . | oo - ie Products of Legendre Functions. 769 The operation d? d d d (1— p? Jt Oye Sey Fie a? = aa eri 5 is sata may be conveniently called D, so that (D+M+N)y=2P’Q'(i—p’). Accordingly d d SDI)! (Ww) 7, (D+M4N)y=(-1*) 7 { 2P'Qa—w) } = —4(1—p?)P’Q! + 20. —p?)?(P"Q' + Q"P') = —4y(1—n")P/Q’ + 24. —p")Q'(2uP’—NP) +2(1 —p?)P'(2uQ’—MQ) Au(L~ 7) P’Q'—2(MP'Q + NPQ’)(1—p?). Substituting for P’Q’, this may be reduced to (=) 7 .(D+M+N)y—2un(D+M4N)y = —2(1—p?)(MP'Q+ NPQ’) Differentiating again with respect to p, D(D+M+Njy-27- (D+ M+4N)y =4(MP'Q +NPQ’)—2(i—p?)(M+ N)P'Q' —2(1—p?)(MP”Q+ NPQ”) =4u(MP'Q+ NPQ’)—2(1—p?)(M+N)P'Q! —2MQ(2uP’—NP) —2NP(2nQ!—MQ) = 4MNy—2(1—p")(M + N)P'Q’ on reduction. Again substituting for P’'Q’, D(D+M+N)y~2— .e(D+M+4N)y—4MNy + (M+N)(D+M+N)y=0, which we may write in the form (D +M4N)y=25 (D+ M+N)y + 4MNy. Phil Mag. 8. 6. Vol. 43. No. 256. April 1922. 3D 770 Dr. J. W. Nicholson on This is the required equation of the fourth order. In this form, itis much more convenient than when expanded in full. Solution in Series of P Functions. A series solution in powers of ~ cannot be obtained in a simple form, but one solution in zonal harmonics of integral order is readily possible, as we know from Adams’ expres- sion for P,(u)Pn»(w), and it must have a simple general term. We may anticipate the existence of other series solutions of this type representing such solutions as A aoe P,() Qin (4) . B o ae a,P,(m), the summation being for integral values of 7, and the limits a and & being at present unknown. Then - : Dy= ie .l—p ae = aS r(r+1)a,P,(p). dpe a @ Write Write and we find that B >, { (M+N—R)?—4MN } a,P,(p) d’ B mr pd, (M+N—Rja,P,(p). Quoting the recurrence formule, (2r+1)P,=(r +1) P,414+7P,-1, _ dF, tel co Gere (2r+1)P, ile de p) we may change this to the form, AP p44 phe: oo} S { Ql+N—Ry 4M t& aa = cant Ay =7-3, aera ° Pri t7rP,_1) t fred. oat +e, m =23 (M4+N-R) oe +1 Products of Legendre Functions. 771 which must be satisfied identically for all values of w. In another form, = >» GP) “22r+1" du Pees aE. > eee e2r+1- dp | (M +N—R)?—4MN—2(0r-+ I(M+N—R)} | QL+N—R)—400N + 2r(M+N—R)}. Clearly the values of r are a, «+2,..., alternate functions being missing. ‘The series on the right contains a term of lower index than any on the left unless its coefficient is zero. The indicial equation, for an ascending series of P functions in which the order increases by two is therefore (M +N—R)?—4MN + 2(M+N—R)=0, where r=a. This determines the possible values of a. The equation is a quartic, and becomes, in full, On? + m+n? +n—a?—a)(m?+ m+n? +nrn—a? +a) =i(n?+m)(n?+n), which can be reduced to at—o?(1 + 2m? + 2n? 4+ 2m+2n)+(m—n)?(m+n+1)?=0, whose roots are a=m—n, u—m, m+trn+tl, —(m+n+1). Whether m or » be the greater, two ascending series beginning with a function P of positive order are available. If we confine ourselves to integral values of m and n, which are the cases of practical value, we select e=m+n+1 and 42=m—n or n—m, whichever be positive. For a descending series, in which the order of the P func- tions decreases by two, we choose § to satisfy, when r=, (M+N—R)?—4MN—2(7+1)(M+N—R)=0 —the extra term being on the other side of the identical relation. This is the quartic (n?+m+n?+n—B—B)(m’+m+n?+n—f?— 3B —2) =4(m?+ m)(r? +n} aD 2 me , Dr. J. W. Nicholson on which is more troublesome, but which factorizes ultimately to (B+ m+n+2)(C—m—n)\(Btm—nst l)(B—m tas) and the possible values of 6 are —(m+n+2), mtn, m—n—1, n—m-l. The second clearly will lead to Adams’ formula. The Ascending Series. We may now write the identical relation in the form B >, (7+ m+ n+ 2) (r—m—n) (r+ m—n4+1)(r—mtn41) Ay Bl cpt 2r+1 dp B =2(rtmt+nt+1)(r—m-—n—-1)(r+m—n)(r+n—m) | a AP, -1 x or its equivalent, B j(7-+m +n+ 2)(r—m—n) (r+m—n+1)(r—m+n+)) x Ap dP .44 2r+1 dp B =D. (r+m+n+3) (r—m—n-+ 1) (r+m—n-+ 2) / Ayro APryy x (r +n—m+ 2) r+ du +(atm+nt+1) (ec —m—n—1)(a+m—n)(a+n—m) hy lees 2a+1l dp xX The last term, not under the sign of summation, vanishes if a is chosen appropriately, as above, for an ascending series. We then deduce, for any value of 7 equal to or greater than «@, Grog 2r+d r+-m4+nt2 r—-m—n a. D2rel rimint3 fom ad rtm—ntl r—mtn4l "ey tm—n+2 > r—m+n+2’ Products 0, Legendre Functions. 113 and the corresponding solution { Oe Aa Ag+2 es ‘ Poe) + = Pata(e) + lean’ dg Pots) + sey 7 where a, is arbitrary. For the first series, write ez=m+n+1, and we find the solution 1 39m+2.2n4+2 2m4+2n4+3 2n4t+2n+7 ¥=Pntntit 5-5 Acie Gas POR ay, ya 2° 2m+3.2n4+3 2m+2n+4 2m+2n4+3 Xx Les “ 1.3 2m+2.3m+4.2n+2.2n4+4 A) Im +3. 2m +5 . Qn +3.2n+5 2m+2n+3.2m4+2n+5 2n4+2n4+7.2mn+2n4+11 "Im +2n+4.2m+2n4+6°2n4+ 2n+3.2m4+2n4+7 X Prins t ae) multiplying by a constant, we may take instead the form yas gum E@ntrt DH Tatrt+) Pomtntr+3) Fhe ae ie Iim+r+3)l(n4+r4+3)0(m4n+r 4-2) Qr! - (rtp (2m +2n+4r+ 3) Bien Pe is which may be called Series A, and to vl we shall return later. For the second ascending series, we suppose m>n. Then a=m—n, and the series becomes y= ae ae 2n 2m—2n+ ae 2nm+3° 2nr—1° 2m—2n+2 ( 2m—2n4+1 x Bes Sylisg eal oan os Dan A On. On 2 i 2.4|2m+3.2m4+5° In—1. Qn—3 2m—2n+1.2m—2n+3) 2m—2n+9, ‘2m—2n+2.2m— un+A4 f Pee 1 Ba Bonn: We notice that it terminates with the term in Py4n, so that it must be a multiple of Adams’ series, and in fact ean be arranged equally asa descending series. We may call it Series B. 174 Dr. J. W. Nicholson on The Descending Series. The identical relations can also be written (B+m+n-+ 2)(8—m—n)(B+m—n+4+1)(B—m+n-+1) i 2G d Past ert dp p-2 +3 (rtm+n+2)(r—m—n)(r+m—n+1)\(r—m+n4+1) ay ad Ppa ay du B => (r+mt+n+1)(r—m—n—1)(r+m—n)(r+n—m) Ap aes 1 * See dp 7 where the first term vanishes if @ is chosen suitably. This is equivalent to , B > (r+m+n)(r—m—n—2)(r+m—n—1)(r—m+n—1) s: Ay—2 is Feo 2r—3 dp ; ! => (r+m+n-+4+1)(r—m—n—1)(r+m—n)(r+n—m) ay ee x — 2r+il du ; ; : dP, The last term, » being an integer like m, n, involves vas or zero, thus defining «. We find a Oyo 2r—3 rtemtntlr—m—n—-1l r+m—n Ge Mee eae Pom ee aot: m—-n—lL Tee = and the series is, a, being arbitrary, ae to las EQ oe Wt ee ot Ped (H) + . Taking B=m-+n, we have (a) ee 2m .2n 2n+2n+1 2m+2n—3 hae 2° 2m—12n—1° 2m+2n *° 2m+2n+1 x Pirtn—2( fe) + eee Products of Legendre Functions. 179 This is essentially Adams’ theorem, and the same as Series B. Since P,,(w), Pa(u) are polynomials of degrees m, n respectively, and the Q functions are infinite series, we may, by equating coefficients of w”*”, derive at once _ an! Im! (m+n!)? Pa(#) Pn(u) = (m!n!)? (2m-+ 2n!) — giving the complete expansion of P,,P,, and thence the - value of the integral 1 i) P,»P,P, dy 1 for all integer values of m, n, and r. The results are known. The Second Descending Series. The second series, if m>n+1, has the indicial equation B=m—n—1 ee and the form defined by Ag ya Pau) t AP, Wt oe with B=m—n—1. Now again, dy 9 2r—3 r+mt+n4+1 r—m—n—-1l r+m—n GQ Ww+tl” rtmtn r—m—n—2'> r+m—n—-l1 r+n—m *rtn—m—l1’ and the series becomes 1 2m %%w+-2 2%nm—QZnr—1 Zm—2n—d BF it 2° I%m—1° Qn4+3° Im—2n—2° 2n—B2n—1 % bee ee Ziti — 2 2Qn+2.2n+4 2.4° 2m—1.2m—3 ° In+8.2n4+5 2m—2n—1.2m—2n—3 2m—2n—-—9 Dat P tnt ink Pen 776 j Dr. J. W. Nicholson on _ We shall denote this as Series C. It stops just before the P’s are of negative order. This series we must proceed to identify. From the order in mw of the functions P which occur, it cannot involve Pin Pr, in which they are of orders m+n, m+n—2, ... or Qn Qn, in which they are clearly of orders m+n 4-2, m+n, 2.3 therefore a linear combination of P,Q, and P,Qm; and since’ lt is a terminating series, it must be of the type A(Pn c= le OD : For this is the only such type which is polynomial. We know from Christoffel’s formula that Ge =r bee = ac 1s where A, is a polynomial in p, of oe n—1. Thus Je m Q;, Sie Pa Qin = iP ae P, Pn, le =o, 108 ima 9 !0 OS} =", a Ng aioe ae Ne = nee which is a polynomial whose degree cannot exceed n+m—1. No other such polynomial can be constructed, all other com- binations being ee Now when n=0, 1, 2,... we-find that the series 7 becomes iL 2m. 2m—5 3° In—2 | %n—7T 1 2m.2m—9 BF Do iy emt A 2m.2m—7 5! Qn 9p Se TSS 6 2 MHD * 2m—)d.2m—11 5.7 ° 2m—1.2m—3 . 2m—4 . 2m—6 Y= Pepin ate Pe tos Pee ae S58 POR 16 2m .2m—9 2.7° %Wm—6.2n— p Pmt... Yo= | ge 3+ = These may be compared with the values of P,,Q,—Q,,P, for the earlier values of n. We have, if P’s of negative —l Products of Legendre Functions. 77 order are properly interpreted as zero, l+yp Soot Pn Qo— QnPo 2m—1 2m—5 en he it eats ee _ 2m—1 2m.2m—) a aS it ee es pPn—3t air “f ee eee Again, Par Qi ae Os, P, 2n—1 2m—5 i =—P,+py Tom be-it gy Pa-st Gas and using the recurrence formula Qr+ it ° pe S=rt+ iL ° Pyait a copes we readily find that P,,, disappears, and the result Im—1L.2%m—3 Pin Qi — Qn P= 7 i : .m—l is obtained,—a series beginning with P,,-2:. Further, by the recurrence formula, ines Q.— QE D emee) ye Qi- Qo) “75 Qn(spP, =) = Foe oe Q: Cha Or P;) ae BE a Qe a Qn Po) _y 2m—1. oat tis 2m—1 3m.m— Reduction of this expression, = use of the recurrence formula in the first term, readily shows that the coefficient of P,,-2 now vanishes, and that the finai result is P,Q Q a 2 2m—-1.2m—3.2m—5 m ede hee iaile iar m.m—1l.m—2 Y2- ee ee 718 Dr. J. W. Nicholson on The next formula obtained in the same manner is tn @;—Q,P.= ee 2m—1 : 2nm—3 5 2m—5 . 2n—7 as Dak We die ee the work becoming laborious at this point. But the nature of the final result is already clear, and a proof by induction can at once be given. If we suppose that, when m>n, 2m—1.2m—3.... 2m—2n—1 m.m—-1.m—2....m—n iP Qn ras Qin P =f (n) Y n9 where yn is the series denoted by Series C, and f(n) is a function only of n, and not m, we can show, by use of the fact that ee (a41)(Pi.Q. O21) | ara (2n its 1)u(Pn oe, ics, OQ; PA) =o ni en Q,=4 cag On Pia) that the formula is true universally, and that nr f(r) = ot = 1) This latter relation is also clearly satisfied by the initial values, for fO=1, fM=3 f@O=—3; (O=5>> which are perhaps in themselves sufficient to suggest the relation. Finally, if m>n+1, : 2m—1.2m—3....2m—2n—1 PnQu— Qn lee A>) m.m—1L.m—Z....m—n nd where ofa It Lape n(n—1) ow TOOT Fea De aie 2 ee nn lon 26n— se VFO) s nm! 2 on+1. Wn Leena ok 3. Bee 22n1yP 7 Qn+11? FQ; — Oren 2”(n 1)? m—n—-1! 2m ! m—n—1! TI On tl tm | | Om = Qe — 2 2a ; ~ \ t ‘ OR eee ee ee ee ee ee ae ee eS Products of Legendre Functions. ato or finally, if n>n+1, Pe Qa— Cs, lig n (24 ie 2m!(m—n—1!)? p 4 —~ 9\m! one : 2m—2n—2! aia ass 2m .2n+2.2m—2Qnr—1.2m—2n—5 2° Im—1.2n+3 .2m—I2n—2 .%n—2n—1 dS 2m .2m—2 2n+2. an+4 2.40 2m—1.2m—3 2n+3.9n+5 2m—2n -1.2m—2n—3 %mr—2n—9Y on om — 2n 4. 2m 2a 1 i Pp —n—3 + a *) Pe ee se eee | If m=n+1, the expression is known to be constant. ii m—n, it is zero. It is constant again if m=n—1l, and if n>m-+1 the factorials clearly only need re-writing. If expressed as Gamma functions, the same theorem is true throughout. Series for Qom Pn. _ We know that (4r+3) Parad QonlH) =~ 2%, Faye Bane SED emlH) Pole). Thus, multiplying by p, and using a recurrence formula, Ar+3 — 4 Qom P; = = 2m—2r—1.2m+2r4+2 2r+2. Porpot2r+1. Po; x | 4r+3 } ee eg | ere ie eee ~ CF a 0 (2m—2r—1.2m+ 2r+2 2 ! * Om —2r +1.2m+2r Po», which, on reduction, becomes —$Qe,P, as” an Vom or 2m +2r+1 ene 1 70 Om =9r—P 2m + Ir +2. %Wn— Ar 4+). Am +2) Bos. the term P, falling naturally into the series. 780 Dr. J. W. Nicholson on Again, — ¥ Qem P = -—-7Q» {3uP; alin BX Ap t+1.2m—2r.2m+2r4+1 Im —2r—-1.2m4+2r4+2.2m—27r4+1.2m+2r fer +1. Po41 +27 Poy as Ay +1 ! 2 Ar +3. Porgy 0 Im—2r—-—1.2m4+2r+4+2° ee After a fair amount of reduction, this becomes -2m+2r4+3. Poets 24r+3.2m—2r.2m—2r—2.2m+2r+1 Lees Ss > em Ps P Xo Im —-2r—3. 2m—2Zr—1.2m—2r4+1 2+ Qr.2m+2r+2 .2m + eae e and indications of a general formula are already apparent. Continuing this procedure, we are able to show that —3 QP .(2m+2r—1.2m+2r+1.2m+2r4+3) ov 4r+3 (2m—2r4+2.2m—2r.2m—2zr—2) 0 (2m —2r—3.2m—Qr—1.2m—IQr4+1.2m— Ir ar (2m+2r.2m+2r4+2.2m+2r+4.2m+2r4+ 6). These forms may be shortened to Ay +3. Po, e+] — 2Qen Po= X 2m .%n+1—Qr+1. W742’ 4rtl. less —2QnPi= 24 oan 2m+1—2r4+1.2r4+2 2m.2m+1—2r.2r4+1 "2m .2n+1— 27 See 47 +3 Po, — Qo, Po= : —S =) 2m.2m+1—2r4+1. Dp pz Qm .2m+1—2r. 2 "2m .2m+1—2r—1. 8r 2m. 2a = 9p 2m .2m +127 ee eee Fi Products of Legendre Functions. 781 ‘The next quotient, appearing first in —4Qom Ps, is 2m .2m4-1—2r—2 .2r—-1 2m.2m+1l—2r—3.2r—2° It is suggested that the general formula is as follows :— f(s)=2m.2m+1—s.st+l=2m—s.2m+s+1, then BP SS arto Pot J 2r) fre 2 of Gr 2) oo 8 Ferrel)” fCr=)) fares) fers) SGr+4) we) if Ar—4) 7 2a) f(Qr+d) f(Qr—5) f2rt+T)’ f2e—2n +2) \ fend 2a “f (Qr—2n4+1) f(2Qr4+2nr+41)’ and the reader will be able to prove by induction, following the above method, that this is correct. The analy sis 1s in no way more complicated than that involved in deducing | the preceding expressions. The coefficient of P24; in this expression is (47+3)S, where g f (Qr—2n+2) f(2r—2n+A4) ... f(2r+2n) ne (2r—2n-+ 1) Bei —2n+3) Lee #941)? and since f(s) =(2m—s) (2m+s+1) this becomes P= Pibs, where g 9m + 2r—2n+3.2m+2r—2Qn+5..... 2m-+ ar+an+1 ee ee pe nee? . Bn or 2a 4). om or Een +2 1 V(mtn4+r4+3) P(m—nt+r4+l) ~ 2° h(m—n+r4+3) Pim+n+r+2) and Q Im —Yrta2n—-2 .Q2m—I2rt+2n—4+4.....2m—2r—22n 2 Im —2r +2n—1.2m—2r4+2n—3. .2m—2r—2n—1 Tan) ne) ~ 9° Mm—n—r) P(mtn—r4+$)’ 782 Dr. J. W. Nicholson on and finally, we derive the remarkable formula We rks! Nimtn—-r) Tom—n+r+)1) Qon(H) Pon(w) = — 9 2 (47 + 3) T(in—n—r)° Dim+n+r4+2) E(m—n—r—-3) Tomtn+r4+3) P(m+n—r+4) Tim—n+r+3) Pe. aes where m and n are integers. There is nothing in our argument which causes it to fail when nis pushed to a value exceeding m, and the formula is therefore general. The only two Gamma functions which can ever become infinite are ['(m—n—r) in the denominator, and '(m+n—r) in the numerator. The former must occur first, and when it does occur, in the first term (when n>m), the earlier part of the series vanishes, and we obtain a series starting with the harmonic Pom+2n+1: If m>n, this series so obtained is accompanied by another of a terminating type,—a gap lying between the two,—and this other is of course found to be a multiple of the series representing ley Qon Ea, Qom leas and already discussed. It is more convenient, however, not to separate the two cases, but to include both in the general formula expressed in terms of Gamma functions. The single series obtained when n>m is found to bea multiple of our Series A, found from the differential equa- tion. This series is therefore recognized in terms of the solution it represents. The preceding discussion has proceeded on the supposition that the order of both P and Q is even. When this is not the case, the analysis is very similar, and we may leave it to the reader to prove the following theorems, where m and x are integers in all cases :— Qom+i() le onti( pe) = 5 = (49 aye ee ee) Ges ae sr iL) T(m—n—r) *T(m+n+r4+3) TVin—n—r—s) POm4+n4+r+3) . Porti(e), "TOom+n—r+3)° Pom—n4+r+4) Products of Legendre Functions. 783 Qo (mn) Pon+i() ees ; Tim+n—r+1) Ton—n+r) =. 32 ale +1) I(m—n—r) * V(m+n4+r+1) T(m—n—r—$) V(m+nt+r+8) Pss(u) *T(m+n—r4+3) ‘T(m—ntr+4) or), (Qom41(f) Pon(m) edie Tim+n—r+1) P(m—nt+r+4+1) a = leer =a ew “Tantn+r+2) Bisnis tT On meE 3\ Teme Gees The Value of an Integral. We can now determine the value of the important integral 1 r={ P, Ps Q, dus a )auk where #, 8, y are integers. ‘The integrand is odd or even in mp according as a+6+y¥ is even or odd. Thus the integral is zero if «+ @+y is even. When a-++ 8+y is odd, various cases arise above. Thus ant a T(m+n—r) D(m—n+r+1) {Gem Pa ee T(m—n—r) T(m+n+r+2) T(m—n—r—$) U(n-+n+7-+3) ; Tim+n—7+4+$) : D(m—n+r+3) ? ey 9 1 i, (P2-+1) : dp = date ( Popay Poa) a= 0. e Again, 1 D(mtn—r+1) TPOn—ntr) ont n Po, d eae ida AYE {" Qo Ponti Par dy Tim—n—r) ~ P(m+n4+r+1) P(m—n-r— 3) Dim+n+r+2) "T(m+n—rt+3)° Cm—n4+r44)' There are four such cases in all. We find that they can bas 784 On Products of Legendre Functions. all be expressed by the single formula Se mae) p(rtetees ( is Lee — ae —— =) T re yer ett) p(vteeett) ; eee) p (yee et 2) which is valid for all integer values of a, 6, y whose sum is odd. This result is symmetrical in « ae 8, as of course it should be. Since y+«+ 8 is odd, the arguments of the [' functions in the numerator, capable of becoming negative, are L{y-a-B, yte-Bt+l, y+R-2+1} b(y+atB)—(4+8), ¥(r+a+8+1)—8, d(ytat B+1)—«. The first cannot be an integer, y+a+ being odd. The others can, but if they are integers of negative sign, the function y—«-B+1 Mga in the denominator is of the same type, so that the limiting value is always finite. There are interesting cases in which the integral is zero when «+8 + ¥ is odd. * This occur s, for example, if or y-a—B+1 2 is a negative | integer (or zero), or if a+B—y is an odd positive integer. This in fact includes all cases, when a+P8+yisodd. Thus the integral | P.P2Q,du = 0 ee when (1) at+B+y is even and‘ (2) a+8+y is odd, and a+ @8--y is positive. It is always zero if «+ 1s greater than y. Foor, Monter, & Meccers. Phil. Mag. Ser. 6, Vol. 43. Pl. XIII. iVOET ARC SPECTRA OF POTASSIUM. 70 ma. i al ox 1a A AS => Der~ it~ aN Yom) Se +R —aA = = Sains == eS © =O o> OD =i SS 7 (ie ; | 1000 ma. Tic. 2.—The are spectrum of potassium obtained with a hot-cathode tube similar to that shown by fig. 1. The pair 1s-3d appears very prominently at 1000 milliamperes and 7 volts. This is an exception to the principle of selection, since the change in azimuthal quantum number is two units. which cannot be attributed to a Stark effect. an eo NARAYAN. Phil. Mag. Ser. 6, Vol. 43. Pl. XIV. reales Spectrophotometer. dice, Wy Spectrophotometer in position. Narayan & SuBRAHMANYAM. Phil. Mag. Ser, 6, Vol. 43, Pl. XV. Woop. Phil. Mag. Ser. 6, Vol. 43, Pl. XVI. (Uranine) Fluorescein. Photo- Fluorescein. Rhodamine, Photo- Rhodamine. Kosine. Photo- Kosine. Y = r “ - a + / . « THE LONDON, EDINBURGH, ann DUBLIN [SIXTH SERIES.] MAY 1922. XCI. Notes on the Kinetic Theory of Gases. Sutherland’s constant S and van der Waals a and their relations to the intermolecular field. By R. H. Fowier, Fellow and Lecturer of Trinity College, Cambridge *. § (1). Lvrropverrov avy Suuuary.—This paper consists of two somewhat loosely connected parts. In the first part, § 2, I point out the conditions under which a strict com- parison may be made between the intermolecular fields deduced from Sutherland’s constant S and from van der Waals’ a. On comparing the numerical results I have then to point out a serious outstanding discrepancy between the values deduced in these two ways; I make a suggestion as to the probable source of this discrepancy, but have not yet resolved it in detail. In the second part, $§ 3-11, I give two simple proofs of the exact formula for van der Waals’ a as a function of the intermolecular field of force which I believe to be illuminating and new, or at least insufficiently known. The calculations are reduced to explicit formule for two particular molecular models :— | (1) Hlastic spheres attracting according to an inverse sth power law. (2) Elastic spheres surrounded by a limited spherical shell in which there is a constant attraction, with no forces at all outside. : * Communicated by ©. G. Darwin, M.A. Phil. Mag. 8. 6. Vol. 43. No. 257, May 1922. 3d i 786 Mr. R. H. Fowler on the § (2). Sutherland’s S and van der Waals’ a.— It is well known that both the constant 8 introduced by Sutherland * into the formula for the viscosity of a gas, and the constant a of van der Waals’ or Dieterici’s equations of state | express the effects of intermolecular attractions. On the assumption of ‘spherically symmetrical”? molecules—z. e., that the molecules are spheres and that the attractive force (7) between two molecules depends only on the distance r between their centres and acts along the line joining them— Chapman t has shown how to obtain an exact expression for S in terms of $(7) and known constants. The most complete determination of a in terms of (7) with which I am acquainted has been made by Keesom §, but no attempt appears to have been made to compare together both theoretical and experimental values of aand 8. This is the more to be regretted as the quantities a and 8 may be taken to refer to gases in identical physical states. A strict comparison of theory with the experimental values of a and S is then legitimate, and should lead to results of interest as to the nature of the law of intermolecular attraction under these conditions. | When we come to compare a with experiment, we must remember that our theoretical calculations only apply to gases at reasonably large dilutions, that is to cases in which the departures from the ‘perfect gas laws are moderately small. In order to make a reliable comparison between theory and experiment we must therefore be certain that the observations and calculations do actually apply strictly to one and the same state of the gas. The calculations used really always present the equation of state of a gas in the form po= NET + (A) +0(, a Re * Sutherland, Phil. Mag. ser. 5, vol. xxxvi. p. 507 (1893) and suc- ceeding vols. + Jeans, ‘The Dynamical Theory of Gases,’ ed. 3, chap. 6 (1921). i Chapman, Phil. Trens. A, vol. 211. p. 460 (1912). The actual formula given by Chapman is affected by an_algebraical slip. The correct expressions have been given by C. G. F. James, Proc. Camb. Phil, Soc! vol, xx. p. 447, (192i) and pou in part by D, Enskog, Inaugural-Dissertation, Uppsala 1917, p. 95. § Keesom, Proc. Sect. of Sciences, Amsterdam, vol. xv. (1) pp. 240, | 256, 417, 643 (1912). 1 : : || O (=) is a convenient and comprehensive notation for “terms of order 1/v?, which may therefore be neglected for values of » which are sufficiently large.” This is a satisfactory paraphrase of the strict mathematical definition of O, to which of course I adhere throughout. ae a a ed ST ts oe | ee , f< ra Kinetic Theory of Gases. 787 where p is the pressure, v the volume, & the gas-constant for one molecule, N the number of molecules, and f(T) is a function of the absolute temperature T which can be calcu- lated exactly. Now it is precisely in this form that Kamerlingh Onnes has prepared his extensive data on tbe equations of state of the simpler gases. Actually * he assumes the form oe er Tt See iea &2) vt vw and determines A, ..., F as functions of T to fit the obser- vations. The coefficients A, ..., I he calls the first, second, ete., virial coefficients. It is clear that when the observed second virial coefficient B is obtained in this way it is strictly comparable to the calculated /(T) of equation (1). Moreover, it appears to me that nothing much less than the exhaustive numerical analysis of Kamerlingh Onnes allows a legitimate comparison between theory and experiment to be made. Such comparisons have already been made in certain cases and on certain assumptions by Keesom (loc. cit.), but without consideration of the corresponding viscosity data. I hope to take u» the question in detail in a future paper. Tor the present I must content myself with the following obser- vations. Keesom finds that the observed and calculated second virial coefficients for Argon, between the absolute temperatures 123° and 293° on the centigrade scale, can be reconciled on the assumption that the argon atom behaves like a sphere of diameter, c=3°29 x 107° cm., surrounded by a field of force o(7)=ar~*, of such intensity that the work II(c) done in separating two molecules from contact (r=c) to infinity is 163x107" erg ft. The exponent 3 is a little doubtful, but is probably better than the exponents 4 or 6 and presumably better than any other integral exponent. These values of o and ¢(7) correspond to definite values of b (van der Waals’) and 8, Sutherland’s constant. The values of ) and o are related by the well-known formula b=27No’®, Apne tins ce an cary * Kamerlingh Onnes, Proc. Sect. of Sciences, Amsterdam, vol. iv. p. 125 (1902), or Communications Phys. Lab. of Leiden, No. 71. + Keesom (loc. cit. p. 646) gives 1:46 10-"*, but this is based on the incorrect value 6°85 x 10?* for Losechmidt’s number. I have reworked the calculation with the more recent figure 6°C6 x 10°°. a By 2 788 Mr. R. H. Fowler on the and § is given by Chapman’s corrected formula* ~§$=0°1956II(c)/k. (Inverse 5th-power attractions.) (4) The numerical values deduced from Keesom’s figures are b=0:00201 for 1 ¢.c. at normal pressure and temperature, and S=23°6. But the viscosity data for ordinary temper- atures should be fairly comparable with the preceding ‘‘ com- pressibility ”’ data, and should lead to the same values of b, o, II(c), and S. When analysed, however, by Chapman’s formulee, they lead unmistakably to the values 6=0-00133, S=162. We may exhibit this significant discrepancy thus :— Argon. i II(c) x 10'*. S. 6x 103. o X 108. Compressibility data... 1°63 23°6 2:01 3°29 Viscosity data ......... PZ 162 1:33 2°84 The disagreement, particularly in the values of 8 and II(c), is striking; so far as I know at present similar dis- crepancies occur for other gases besides argon, but the available data are not generally so suitable for a direct comparison. This disagreement is all the more interesting in view of the close agreement between the two values of o (or 6) given by Chapman f himself. But the values of b from the compressibility data there given appear to be obtained from critical data, and must in my opinion give way to values calculated from Kamerlingh Onnes’ second virial coefficient. The discrepancy which thus remains seems to arise from the fact that whereas the theoretical formula for the second virial coefficient B is exact, the formula used for the viscosity is only correct so far as the first power of S/T. The agree- ment of this approximate viscosity theory with experiment is usually regarded as good, at least at ordinary temperatures, so that an exact formula including further powers of S8/T is not likely to improve this fit. It may well, however, bring the facts of compressibility and viscosity into reasonable agreement—in particular at fairly high.temperatures, and so enable a reliable picture to be formed of the range and | intensity (under these conditions) of the intermolecular forces. * Chapman gives S=II(c)/3h in our notation for all power laws. The correction affects only the numerical coefficient which becomes a function of s the power law exponent (James, loc. cit.). ft Phil. Trans. A, vol. 216. Table VIII. p. 347 (1916). Kinetic Theory of. Gases. 789 The existing discrepancies in S and o are qualitatively such as should be properly accountable for by such an extension of the theory; to this point I hope to return in a later paper. Elementary calculations of van der Waals’ a. § (3). My immediate object in the rest of this paper is to present two simple proofs of the exact formula for B, and so for van der Waals’ a. Keesom has calculated B in terms of the intermolecular field by a somewhat elaborate method based on Boltzmann’s expression for the entropy, a method which can be used when the molecules are not spherically symmetrical. But Keesom’s own work shows that molecules with spherical symmetry give the best representation of experimental facts for the simpiest (monatomic) gases. Such molecular models are therefore of the first importance, and for such it is possible to calculate B directly by very simple arguments—(1) By a direct calculation of the boundary field, measured by W, the work to be done in bringing a molecule from the interior to the boundary of the gas; or, (2) By a calculation of the Virial of Clausius. Method (1) has moreover this advantage over all other methods, that it enables adequate account to be taken of surface effects, which is not possible, or at least far less simple, by the entropy and virial methods. In calculating B all the terms in pv of order 1/v? are of course to be omitted. § (4). Zhe meaning of a.—The so-called constant a of ‘van der Waals’ or Dieterici’s equation of state can be specified in various ways. The most satisfactory physical meaning can be attached to it, by first recognizing * that intermolecular attractions must result in the production of a permanent field of force near any boundary of a gas, and then connecting a to the work that must be done against this boundary field in bringing one molecule from the interior to the actual boundary of the gas. At sufficiently . great dilutions this work W must obviously be proportional to the molecular density v, and it is easily proved that Tf oN Wit met. where N is the total number of molecules in the body of gas considered. Thus a will be independent of the volume. * Jeans, loc, cit. p. 159. + Jeans, loc. cit, 790 Mr. R. H. Fowler on the It appears, however, to be generally assumed * that under the same conditions W, and therefore a, is independent also of the temperature, and that a is therefore, theoretically at least, a true constant. When one looks into this assumption more closely it is easy to convince oneself that it has no justification whatever, however great the dilution. The ordinary kinetic theory of gases really demands an a which is a Eunction of the temperature whose variations for most gases are appreciable at ordinary and even at fairly high temperatures. It is the more unfortunate that this theoretical result 1s so often overlooked, since the independence of a of the temperature is not borne out by experiment. In point of fact, both theory and experiment demand variations of a with the temperature which are, qualitatively at any rate, in full agreement with one another. As we shall see directly in calculating a, the error arises from regarding the gas molecules in the interior of the gas as uniformly distributed through its volume. Butif there are attractive forces sufficient to create a boundary field, these attractive forces must also make it correspondingly more probable that any pair of molecules will be close together than far apart. This alters the calculated value of W by an amount which may be called the clustering correction. It appears on calculation that this clustering correction depends on the temperature, and at ordinary temperatures 1s in fact of the same order as the whole boundary field itself. § (5). Lhe dependence of a on the nature of the inter- molecular forces.— We have stated above that ais independent of the volume in the circumstances we contemplate here. It seems necessary even now to insist that this independence holds good whatever the nature of the intermolecular forces, and is in no way dependent on obedience to any special law of variation with distance such as for example the inverse 4th power law. It is still frequently stated Tf that if van der Waals’ a is to be independent of the volume the law of attraction must be the inverse 4th power. This is quite untrue. As we shall see, the law of variation with the distance affects and only affects the form of a as a function of the temperature. In the case of spherically symmetrical attractions following inverse sth power laws generally we can go further, for we find that the integrals that occur do not converge unless s>4, The physical meaning of this failure is easily seen to * Jeans, loc. cit. + #. g., Lewis, ‘A System of Physical Chemistry,’ vol. i. p. 8 (1918), onl many recent papers in the Phil. Mag. Kinetic Theory of Gases. 791 be that if s=4 the attractions of distant molecules in the gas will then be of equal or greater importance than those of the neighbouring molecules. In such a case the recorded pressure at any point would be affected or even dominated by the distant attractions and would depend on the general shape of the whole body of gas. This is of course directly contrary to experience, and no such attractions of sensible magnitude * can be admitted. Finally, it is interesting to observe that we can obtain the case s=4 as a limiting case in which the intensity of the intermolecular attraction between any pair of molecules tends to zero as s->4 from above. In this limiting case and this only, we do in fact obtain a value of a which isa true constant independent of the temperature. This, however, can only be achieved at the price of a zero value for I(o), and therefore for Sutherland’s constant ; this is inadmissible in the case of any actual gas. §(6). The work done in bringing a molecule from the interior to the boundary of a gas.—The molecular model that we consider is that of a bard elastic sphere of diameter o, surrounded by a field of attractive force. Two molecules whose centres are distant 7 apart act on each other with an attraction ¢(7) along their line of centres. It is this molecular model which is so successful in explaining the “transport phenomena” in gases and their variation with temperature at ordinary temperatures—in particular the coefficient of viscosity. When such fieids of force are postulated, the distribution of molecules in the gas is not on the average uniform with reference to any selected molecule. If v is the general molecular density in the gas, the molecular density at a distance r from the selected molecule is, not v, but t a)" Q@(z)dx — ve Meee eel ize (8a) * Gravitational attractions are in this category, and are of course insensibly small. Electrostatic attractions are in the same category, but are so large that they entirely dominate the distribution of charged molecules or ions. + See, e. g., Boltzmann, Vorlesungen tiber Gastheorie, vol. ii. p. 150 or Jeans, loc. cit. p. 182. Jeans’ suggested correction of this formula is here insensible. Owing to the standardization of the use of h for Planck’s constant, I believe that the time has come for the use of some letter other than / for the constant in Maxwell’s Law for the distribution of velocities, and I propose to use 7 for this purpose. I believe, further, that it is desirable to use a notation distinguishing systematically between “‘the gas-constant for one gramme-molecule” and “ the gas- constant for one molecule” (the constant & in 27=1/kT) and to conform to the continental notation of R and & respectively. 792 Mr. R. H. Fowler on the where 2;=1/kT. Consider an infinite plane slab of thickness df, which is small compared with o, and let us calculate the average attraction dF of all the molecules in this slab ona molecule distant z from the slab*. The calculation is a generalization of the classical calculation of Laplace’s theory of surface tension +, generalized in such a way as to recognize the real molecular structure of the fluid. Suppose first that the selected molecule is at P and that z=o. Then the average number of molecules in the slab, per unit area of the slab at a distance » from P, is vdfe aes where we have written, for convenience, My)= |" ee)de, ee ee CD) The number in the annulus at distances between r and »+dr from P is | dr 2711 Qavrsin @.——.vdfen™™ sin 0 i : and their resultant attraction along PO is Javediom@e dr. . a (7) To obtain the attraction of the whole slab, we must integrate the expression for values of r from (ah; ) 07 on this assumption of the equal probability of all positions of the b-molecule, we must exclude a fraction 47a°/Q of the total generalized space Q*. But when there are central forces acting between the molecules we are no longer at liberty to assume that this a priori equal probability for elements of generalized space implies equal probabilities for equal volume elements in ordinary space. The correcting term must be multiplied by the probability of another molecule being at a distance o from the selected molecule— *" Loc, cits p. Lov. Kinetic Theory of Gases. Por that is to say by e”". Making this correction the value of b given by (20) follows at once by the usual arguments. Hquations (19) and (20) used in the ordinary formula N&T6—a must give the exact form for the second virial coefiicient. We may note here the differences between the usual values. of a and 6 and the values given by (19) and (20). The usual values are really obtained on the assumption that 2jII(c) is small so that e’"“ is very nearly unity, and the effect of this factor on a and 0 negligible. The value of a so obtained is of course a=2nN?| w*ll(x) dex. It is most important to recognize that increasing the dilution has no effect whatever on the accuracy or inaccuracy of this approximation, for it does not aftect 2jII(c). The ap- proximation can only be justified if the temperature is sufficiently high. But by equation (4) 2jII(c) is of the order 58/T. For an ordinary monatomic or diatomic gas, whose viscosity is analysed by Sutherland’s formuia, S>100. [The only exceptions are He, Hp, and Ne.| Analysed by a more exact formula, taking account of all powers of 1/T, a considerably smaller value of S (perhaps 5-10 times smaller) might fit the observations, and agree with the compressibility data (see $2). But even so, 2jII(c) is about unity at a temperature of 100° absolute, and can hardly be considered small at temperatures less than 1000° absolute. Jt is not. till we reach these temperatures that the clustering effect can be ignored, and the gas regarded as genuinely of uniform density throughout. § (10). Lhe direct calculation of the Virial of Clausius.— It is possible to caleulate Independently by this method the. exact value of the coefficient of 1I/v in the expression for pv, and so confirm the results cf the last section. The virial method is, however, unsuitable for making allowance, when the need arises, for the effect of the boundary field. We take a molecular model for which the law of repulsion: J(r) between a pair of molecules is fir)=Q, (o-e« and €>0O in a suitable manner. ;The virial argument starts from the formula * 2aN? le fc)de 7, Roe for the number of pairs of molecules in the gas whose centres are at a distance between r and r+dr. Then for the contri- bution $2 27/(7) to the virial we have 1S rf(r = | Caen 21 |, PORE (21) We split this integral up into the two ranges (o—e, c) and (¢, ©). In the former we can replace r* by o® since eventually e>0. On ee /(«)dz we get ‘ssa es o° pumle) his if Qe 8Xe-") Jp pcaanN 3 2m) A? b(re dp, ‘The first integral can be evaluated ; it is 1 — 27Q6€ If therefore we make Q->0 and e>0O in such a way that Qe->oo we may replace the first integral by 1/27, which must be its value appropriate to elastic spheres. We -can write the second integral in the form — (" PI (rye dr Jo with our previous conventions for II(r). This can be integrated by parts in the form ea ie 2711 (r) ve ae EG (¢ se Al -and the integrated part vanishes at both limits. Hence Das DarN2 ("2 s2irf(r) = Rn gue) aN 72 (CO 1) dr, 0 72 (eI _ 1) dr, The complete virial equation is ; pu=NkT +433rf(r), * Jeans, loc. cit. p. 182. Kinetic Theory of Gases. ee OOS so that in this case, with our previous notation, , . po=NkT+ : ' NEATH — a +0 (5) » (22) which confirms equations (19) and (20). This result is in agreement with Keesom’s*, obtained by yet another inde- pendent method—the calculation of the entropy. § (11). Particular molecular models.—The molecular models which are of interest are those in which the attraction (1) varies as the inverse sth power of the distance, d(r)=ar™*, and (2) is constant over a certain range and zero outside fitemmse, o(7)—A; (o 4; the physical meaning of this restriction has already been commented on in § 5. — (i.) The inverse s-power law.—lf we take 6(r)=ar* we have (23) For the second virial coefficient B we combine (19) and (20), expand go. and integrate term by term. We obtain atter reduction an 2 2j11(c))” é B=Nets {§1—3 (2) L L mS, Pin(s—1)—33§$?° ° ) where b=27No’, its classical meaning. (i1.) Young's model.—In this case we have Hi@)=0; ws); ata 7) 5 (GC ememian ys i 2f) (25) =A(d—o), (c>r). _ The integral for y(0) can be evaluated in finite terms by integration by parts. We obtain after reduction, 2arv j (1 a” 2 2 x= 3+ aa + Gat Gay)’ if d? 2d 2 — (—d?+ —— Wao a 5 5 (5?+ gat (Ay * aay) } i) * Loc. cit. p. 264, 27A(d—o) 800 . Mr. C..G. Darwin on the Reflexion of Equation (26) is suitable for use in computations for low temperatures. The expanded form for y(0) analogous to (24) is suitable for use with high temperatures and is more immediately intelligible than (26). This form is (0) = 2a II(o) > z ae Lee ee aa 2(d—a)*d (d— aa a ee 1 Gas n+l n+2 n+ 4 : SY (a7) and the corresponding form of the second virial coefficient is B=NeTb—2nNTT(o) 3 SEA n! (dod? _ 2(d—a)?d o =o (28) me ie n+2 n+3 oe The. Reflexion of X-Rays from ieee Crystals. By ©. G. Darwin, Feliow and Lecturer of Christ’s College, Cambridge *. 1. Introduction. Y@XHE recent work of Bragg, James, and Bosanquet f, on the reflexion of X-rays from rock-salt crystals is of extreme importance in that it promises more directly than any other method to supply information about the actual positions which the electrons occupy in the atom. The method consists in a study of the intensity with which the various faces of the crystal reflect a given wave-length, and is based on the theoretical formule given by the present writer t. These formule showed that such experiments should determine a certain quantity, which is, roughly speaking, the amplitude of the wave scattered by all the electrons in a single atom in the direction of the reflected beam. A study of the various faces of a crystal gives this amplitude as a function of the angle of scattering, and from that the positions of the electrons can be inferred—with this second half of the problem I shall not be here concerned. But the deduction of the amplitude from the experiments encountered certain peculiar difficulties ; for the absorption * Communicated by the Author. + Bragg, James, and Bosanquet, Phil. Mag. vol. xli, p. 309, and vol. xlit) p, 1 (1921): ¢ Darwin, Phil. Mag. vol. xxvii. p. 315 and p. 675 (1914). A-Rays from Imperfect Crystals. 801 coefficient of the crystal is involved in the formula, and there were indications that the actual absorption was a good deal stronger than usual. This was especially the case in the reflexions at small angles, and these small angles are par- ticularly important, for by them is tested the truth of the hypothesis that in rock-salt the sodium atoms have passed one electron over to the chlorine. The difficulty was over- come—atany rate, partially—by finding the actual absorption coefficients for the various directions in the crystal, by a study of the reflexions of the internal planes of a set of plates of rock-salt. But even this method has an unsatis- factory feature; forit was only possible to arrive at a definite result by rejecting the observations from certain of the plates. It is true that the discrepancy was explained by the fact that these had had much rougher treatment than the rest, but still it suggests a certain measure of doubt as to the soundness of the method, or at any rate the necessity for an inquiry into it. Itappeared to me therefore to be worth while to re-examine theoretically the reflexion from crystals in general, in the hope of clearing up the difficulties, and also in the hope that theory would indicate some way of obviating them. The whole point evidently lies in the imperfections of the crystals, and this introduces many complications. [Iam afraid I have not succeeded in welding the parts of the argument rigorously together, but in spite of certain gaps in the theory it seems unlikely that there is serious error in the general views to which it leads. The work has been rather heavy, but the trouble will have been justified, if it helps in determining the positions of the electrons in the atom, one of the supremely important pro- blems in the present condition of physics. There will be frequent occasion to refer to the two papers of Bragg, James, and Bosanquet*. When mentioning them in the text, I shall, for short, use only the name of the first of the authors. The papers themselves will be called B.J.B.i. and B.J.B.ii. Similarly, my own former papers t+ will be denoted D.1i. and D. 11. 2. Previous Theories. Lhe theory of the reflexion of X-rays by crystals was discussed by the present writer in the papers D.i. and D. ii. _In D.i. it was assumed that each atom scattered X-rays just as though it was alone, and the solution of this interference * Loce. citt. + Loce. citt. Phil. Mag. 8. 6. Vol. 43. No. 257. May 1922. 3F ——— Sos exp ik| a (cos y cos —cos €) +y cos y sina—< (sin y+sin €) |. (4:1) To find the intensity multiply by the conjugate imaginary. It == 47 as the intensity of the incident beam, this gives (J /r?) > a SapyJ ‘a'p'y exp tk| (a — x") (cos x cos —cos ¢) apy a +(y— Cerne (z—2')(siny+sin €)]...., (4:3) where 2’ =2a'a,+6'b,+y/c,, etc. jf aud its conjugate f’ will vary from one molecule to ino We must take the mean of (4°3), allowing for the chance variations of /. This will be done by a double averaging of all possible X-Rays from Imperfect Crystals. 807 values of a, 8, y and 2’, 6’, 7’, and as all pairs of molecules oceur in the sum this is the same as taking the mean of f ‘and squaring its modulus. This quantity will be denoted by 7°; it is different from the mean intensity scattered by a single molecule*. Suppose that the crystal is set so that €is near 6, where 6 is given by the equation kasinO=n7, which determines reflexion in the nth order. For angles far from this the reflexion is insignificant, so we may put 6—0 +u,) xy—O- ew, and treat u,v, as small. This approximation excludes all the other reflexions from consideration. The reflected Intensity is then ‘ (J/r?)f? > YJ exprk{(e—z2')(u—v) sind+(y—y')wb cos 8 aBy a'B'y' os 2 ate) casOPe. B24) From this a factor exp—2k(z—z') sin@ has been rejected, as it is equal to unity. Now on account of the smallness of u, v, W the exponential terms only vary slowly with «, B, 9, and so it will be legitimate to replace the summations by integrations. The number of terms contained in a volume dV is NdV and so the intensity becomes apeysepe4 av av’ exp ik{ F(e«@ —2') + G(iy—y') + H(e—2’)f, (4:5) where F=(u—v) sin@, G=wWcos8, H=— (u+v) cos 6, and the volume integrations are each taken over the whole erystal. We shall now suppose that the crystal is put through one of Brago’s experiments. An instrument is placed so that it can catch all the reflected radiation. The element of area at r is cosydydwy or r? cos Odvdy, and so (4:5) must be multiplied by this and integrated over all values of v and . Further, the crystal is made to rotate with angular velocity about the y axis, and the total energy H received by the instrument is measured. This is equivalent to an integra- tion fe du/@ taken over all values of wu. Then HE =| du/w (| r? cos 6 dv dip x (4'5). * See Brage, James, and Bosanquet, Zeitschrift fiir Physik, vol. viii. Dm? p. 77 (1921). —\ > ™ ee SEES cen were, ae a sr re ene ef ee ee 508 Mr. C. G. Darwin on the Reflexion of To perform the integrations change variables from wu, v, wv to F, G, H; the latter will all go from —x to w, and du dv dw =dF dG dH cosec 26 see 0,7 and so Ha/J = N?2f? cosec fff {ar agaH |" dV dV’ exp ik{ F poe i) Eee » (4%) This expression can be evaluated by an inversion of the order of integration ; I shall not attempt to justify the pro- cess rigorously. First, take the F, G, H integrations between the large limits +F,, etc. Then (6) Eo/J=N?2f? cosee 26(2/k)8 Lt | dv dv' sinkF, («#—a') sin kG, (y -y') sn kH, (2—2') ea! y—y! sel Now in the 2’ integration, which is to follow next, the presence of I’, implies es the only important part is near w’=w. Similarly, for y/ and 2. Hence it will be valid to take these three integrations over all space instead of only over the crystal, for the parts outside will contribute nothing. We now have @ °° / sinkE (x#-—w@ { eA ae ete. C— & The final three integrations shied simply yield Ho/J = N?f? cosec 20 (2/k)*. 7°. V. Now 27/k is the wave-length X. Also we shall adopt the notation of B.J.B. 11. and write Q= N? (2A? cosec 20. 7) ieee) Ho/J=QV. UM EER RE MIC entity is. ko (4°8) Q is of the dimensions of the reciprocal of a length. This equation is the same as B.J.B.i. p. 826 (4). A little care is needed in considering it, because its physical dimensions are different from those of other equations which will occur later, though it is similar to them in appearance. The factor () will include the special peculiarities of the crystal, such as the weak (1, 1,1) reflexion of rock-salt. when X-Rays from Imperfect Crystals. 809 The averaging process will introduce the temperature factor e~8sin*@ and, if desired, the meaning of this may be modified so as to include the relative motions of the atoms in the molecule. There will also be the usual polari- zation factor $(1+cos220). If these are put in explicitly Q=N?/? cosec 20 eB sim? 911 + cos? 20). . (4°9) Here f will represent the mean scattering in the equatorial plane of the emergent spherical wave from a molecule ; it is the right quantity for determining the distribution of the electrons. In this paper we shall only be concerned in the deduction of the value of Q and nothing further will be sald about the other half of the problem. The result of this section has been proved without allow- ance for the fact that the incident waves are really spherical and that the Fresnel zones are exceedingly small in X-ray work. It is easy to carry out the whole process, retaining the squares of x, y, z; but the formule are much more cumbrous. As they lead to precisely the same result, it is not necessary to give them. d. Reflexion from a Conglomerate. The next problem to be considered is the reflexion from a small imperfect crystal. It is supposed to be made up of a number of perfect crystals differing slightly in their orien- tations, and the whole is to be so thin that extinction and absorption are negligible. We shall describe it as a con- glomerate and the component perfect crystals as blocks. Suppose that such a conglomerate is put through the same experiment as in § 4. At every point of the observing instrument the intensity will be the sum of the intensities from the separate blocks. Hence the integrated energy will be given by (4°8), where now V is the volume of she whole conglomerate. But this is not enough; it is also necessary to find the actual reflexion when the crystal is fixed at any angle of incidence—a much more difficult matter. However, Bragg’s. experiments showed that there was reflexion for settings of the crystal differing by as much as a degree, which is very much larger than the breadth of the diffraction pattern of a single block, and this fact makes it possible to approximate. We defer the discussion of the size of blocks required for the approximation to be valid. Consider a block of volume W which has normal J, m, —1 (it is convenient to take it in the negative direction), where a — ee i. oe — > i a = a: SS, See ae eo ee ae a a a" Se 44° -° oe 810 Mr. ©. G. Darwin on the Reflexion of — l,m are small and are measured with reference to some standard direction in the conglomerate, not necessarily the mean direction of the blocks. The intensity of the beam reflected by a single block is then given by (4'5) provided that w—/ is put tor u, v+l for v, and yw—2mtan@ for w. To specify the distribution of the blocks let VECW, 1, m)dW dldm > 7) aa be the number of blocks in the volume V of the con- glomerate which are themselves of volumes between W and W+dW and have normals between /, m and l+dl,m+dm. It follows that { dW | {alan weow, im=1i. 2 aoe) The intensity of reflexion in the direction v, w is therefore obtained by taking (4:5) for a volume W (modified as above), multiplying it by (5:1) and integrating over all values of W,/,m that occur. To obtain the reflected power we multiply by r?cos@duvdwW and integrate over all values of v,. The result is ie 9) a,® (3) (6) JN?/? cos @. v | dv ral FCW, 1, m)dW dl am dV dV’ exp ik { (a—a’') (u—v—2l) sin 04+ (y—y’) (cos 9290 sin @) — (z¢—2')(u+v) cos}. 2 2 eae This is a function of the angle of incidence, that is of u, and the fact that we are not to integrate for u alters the proce- dure. We must use the assumption that the diffraction pattern of each block extends over a much narrower angle than the distribution of the blocks. Now, if the shape of the blocks were known, it would be possible to carry out the six last integrations, and, regarding the result as a function of J, the J integrand would then consist of the product of two functions, one of which vanishes except for a narrow peak. It would then be correct to substitute in the rest of the integrand the value of / given by the maximum of the narrow peak—-that is, to substitute }(u—v) for / in F. We may make the same change, even when the order of integrations is altered so as to do that for / first. The same argument also applies for wand m. Ifl,, m, denote quantities which ” A-Rays from Imperfect Crystals. 811 are to be made infinite later, the result of the 1 and m integrations is k? sin? @ 2 £2 if eee (6) ey ( { dv dyp { FU igre oe =) aw i} dV av' sin k(a@ —a') sin@.21, sink(y—y’) sin @.2m eu y-y' | xX exp—ik(z—2')(w+v) cos 6. In this a! has been equated to w and y’ to y in all the terms that do not involve J, m,,. Next integrate for v. With exactly the same argument, we may put —w for v in F, and this makes the integration possible. The result is JN? f? ie) (6) i gag VY dwt |} FCW, u, dy cot —)dW dV dV’ k? sin? 8 ee sin k(a—a') sind. 21, sink(y—y'jsin @ . 2m, ua! aa y Sin k(z—<') eas G0. o—e The argument of §4 now shows that the double volume integration is equal to 77>W. Thus the whole effect is JN? /7X? cosec 20 . v(awf WE (W, u, m')dm’. Let | TK afaw fan wrow, imme \i==G (2) 4...6). 7 (O54) Then, using (4°7), the power reflected is DOG GU eee ley ate Cam) and by (5:2) {eena=a, Bae the aca COD) The relation (5°5) is practically equivalent to saying that the incident beam is reflected by those blocks which are at the proper angle and no others”. In order to test the validity of the assumptions made in this work, we shall simplify the problem by supposing that all the blocks are rectangular of sides &, n, € and orientated * With a little modification the same argument proves the result deduced in general terms in D. ii, p. 686.- a i a a i — a a lie ee ee ee ee sere eae 812 Mr. C. G. Darwin on the Reflexion of according to the error law. Let o be the scatter of the blocks—that is, the departure of mean square of the normal of a block from that of the conglomerate. Then for (5:1) we must write V exp—(?+m?)/20? Ent oes | yee It is now possible, though ‘still tedious, to work out (5°3) down to the last integration, which involves an error function. Approximating for this when wu is small we find eve de one /2ro.ksind.& é as the expression corresponding to (5:5) ; so the validity of (5°)) depends on neglecting the second term in the. bracket. Thus for a first-order reflexion a/£ must be small compared too. In Bragg’s experiments o was of the order of 1°; so to get an accuracy of 1 per cent. E/a must be of the order of 10%. For spectra of higher orders the conditions are less exacting. From the general appearance of the work one may hazard a guess that the approximation will be true over a much wider range of values, and would cover the case of a crystal imperfect by warping. It is, of course, possible always to define a function G(u) so as to satisfy (5°5) and it will JIVQ (5'8) probably be always true that ( G(u)du=Q; but the eg important point is that G(u)/Q should depend only on the structure of the conglomerate, for only so will it be possible to pass from reflexion of one order to one of another and from one set of crystal planes to another. 6. EHetinction. The calculations have so far dealt with crystals which are so thin that absorption and extinction can be neglected. It is now necessary to inquire to what extent this is justified. In D.ii. a study was made of the reflexion from an infinitely deep perfect crystal, and it was shown that the reflexion is practically perfect when the glancing angle differs from @ by less than g/kacos@, where g==N/fdacosec @ is the coefficient of reflexion for a single plane (a quantity of zero dimensions); while on either side of this band it falls X-Rays from. Imperfect Crystals. 813 off rather rapidly to zero. Inside the crystal the trans- mitted beam was found to be extinguished at a rate depending on the exact angle, the greatest factor being e~*v/¢ at the central point. Now if we take the numerical value which g would have in Bragg’s experiments, using rhodium K, rays and the (1, 0,0) planes of rock-salt, we find that g=2x10-* and 2q¢/a=14000. The ordinary ab- sorption, measured by depth, is ~cosec 9=100; so it is quite clear that extinction will be of far greater importance. Moreover, if we suppose the crystal only a thousand layers thick we have 2qz/a=2q x 1000=:0°4, so in even quite a thin layer the extinction may be expected to become con- siderable, and its influence must be examined. We shall see that it cannot be neglected, but that there is a considerable modification in the formule. In D. ii. the phenomenon was studied for an infinitely broad and intinitely deep crystal. The latter condition is to be altered, but to give up the infinite breadth would lead to great difficulty and we shall therefore retain this condition. It requires, however, an alteration in the type of observation, for an infinite plane willalways reflect the rays from some point of its surface and so there will be no definite reflecting position. We therefore take a fixed crystal and find the total power reflected for a point source. ' Take a crystal composed of m planes, and first consider its effect on plane waves. The equations of D.ii. p. 678 are applicable. They deal with the multiple reflexions in the successive planes of the crystal, allowing, of course, for their phase relations. The difference equations connecting T,., the amplitude of the transmitted wave at the rth plane, with §,, that of the reflected, take the form * S,= —igT,+(—)"(1—h—-itka cos 6 .u) S41 | BE + ea) T,41=(— )"d.—h—tha cos 0 .u)T,—i¢ 841 where h=3yacosec @ is the absorption factor for amplitude. The form of the solution will differ from that in D.ii., as the * It has not been possible to retain completely the same notation as D.ii. The following are the chief differences :— Da eee ) co) v Heresy 0 204: Otu 0 kacos@. u. Tam afraid that in D. ii. « was used in two senses; on p. 679 it has the same Meaning as here, but on p. 681 it is the same as w here. nn en ae ee ee ee 2 le i i ae _ Se a a, a > oe — -*; at 2) Se et Se es ae 814 Mr. C. G. Darwin on the Reflexion of end condition now is 8,,=0. The solution is found to be So= —2q Ty] (h+tka cos 6. u+ecoth me) (6-2) T,, =e cosech me. Ty/(h+ikacos @.u+ecoth me) |° ‘ where f=7+(h+tkacos@.u)?. .0. by ae Now, as we saw, fl is very much smaller than g. If we neglect it altogether we have e=r/q?—(kacos@.u)? for |u] < g/kacosé 54) and e=i\/(kacos@.u)?—q*? for |uj > g/ka cos ae It is then easy to verify that for all values of wu [Ti [P+ [So /?= [Tc]? Further, if h is not quite zero, it may be verified that to a first approximation [Tel tl So P= [To (2 (LH 2mh) oe Thus we can always calculate the intensity of the trans- mitted beam by reducing the incident by an amount corresponding to ordinary absorption and subtracting from it the intensity reflected. ‘his will play an important part in the next section, as it gives rise to the secondary extinction. Now, consider the reflected beam coming from a point source. Following the line of argument of D.ii., we may resolve the spherical wave into plane, and the whole re- flexion is given by integrating the plane wave formula over all values of wu. ‘The exact form of the answer involves such matters as the length of the slit in the observing instrument, but here it will suffice to find a quantity that is proportional come oy leis ais { IS) du= ae Tor | du/|h+tkacos 6 .u+ecoth me |?. (es me AN The evaluation of the integral seems to be impossible in general, but our object can be achieved by taking advantage of the smallness of h and using (6°4). But there is a com- plication ; for unless h actually vanishes, e will have a small real part in the outer region, and therefore if m is large X-Rays from Imperfect Crystals. 815 enough cothme will tend to unity. This will bring the denominator to the form it had in D.1ii., viz. : [ika cos 0 .u+ir/ (ka cos @.u)?—¢?|?. But if we put h=0 before allowing m to become large, the corresponding expression is |ika cos 0 .u+/ (ka cos 6 .u)?—q? cot mv/ (ka cos 0. u)?—@?P. The integral still converges, but to a different value. To avoid this difficulty we must suppose that the crystal is so thin that mh is small—it must be Jess than about 10° layers thick. For such a crystal the real part of e will not matter and we may put h=0 and write the relations (6°4) straight into (6°6). In spite of its unpromising appearance the inte- gral can be evaluated and leads to the remarkably simple result * \ | So? du=| To}? 7 tanh mg. a Sawa (Op) Tf in this we allow m to become infinite the result is ls eno whereas the true value from D.ii. should be | Ty hacosQ@> 3° even in this extreme case the error is pe oy 3 ka cos only 18 per cent. for taking h zero before putting m infinite. This shows that the approximation may be expected to hold for quite deep crystals with considerable accuracy. Now take the same problem and work it out on the principles of § 4. It is easily found that l1—exp—2mikacos@.u 0m ae 2ika cos @. u ; which leads to the result (“| sol? du=|T)1? Lao - mg. Page (ius) i090) * This result was first diseovered by obtaining an expansion in terms of ng. The complete proof may be constructed as follows. Write the denominator as the product of two conjugate imaginaries and split it into partial fractions. Next express it as a complex integral with argument é.... necessitating a cut between +q in the e- plane. It may then be proved that the poles of either fraction in the integrand lie entirely on one or other side of the path of integration. Hence the path may be replaced by a circle at infinity which contributes nothing, together with a small circle round e=q which introduces the hyperbolic tangent. 816 Mr. C. G. Darwin on the Reflexion of So we may represent the effect of extinction by introducing a correction factor tanh my ' mg co and this is quite accurate for crystals not so deep that the ordinary absorption would become important, and remains fairly good even for those much deeper. | ‘ Now consider a small block limited in breadth as well as depth and irradiated by plane waves. The multiple internal reflexions will give a complicated system, which will depend on the ecrystal’s shape and will be irregular at its surfaces. But it seems reasonable to represent its effect by calculating the intensity of reflexion as though it were of infinite area, and then selecting from the reflected rays the cross-section which has met the actual crystal. Let d be the mean depth, then V/d is the area. The cross-section of the rays is there- fore (V/d) sin @, and so the power reflected is ISo]? (V/d) sin 0. Then we have J = { [So |? (V/d) sin 0 du/o. liwe pat |i =J; dma, 97 —20a7 cong me we thus get He /J=VQ tanh mq/m¢,. . yee we) which shows that on these assumptions the same correction factor is applicable in (4°8) as in (678). Hxactly the same process may be applied to the argument of §5. For though there the crystal was not rotating, yet the distribution of the blocks was such that there was an integration, equivalent in its effects to the wu integration here. We may thus say that the reflexion of a conglomerate at angle (9+4u) is given by JVG(Y), (s.r ct) provided that G=Q | dW f dm' WE(W, u,m'), . (6°12) where Q'=Q tanh mq/maq.:. » =) ee ee) In.consequence of this (5:6) becomes (Pacna=a. . «6: ae eae momen) X-Rays from Imperfect Crystals. 817 The extinction factor is perhaps more properly expressed in terms of the depth of the block, rather than the number of its planes, because this will be roughly the same in all direc- tions and so will give rise to a formula suitable for comparing reflexions of different faces of the crystal. Then the extinction factor is tanh ,/2Qd? cot O/r s/ 2Qd? cot O/r 1. 2Q@ cot? . 8 Q'd*cot? 6 ero =1l—3 = (5c eae Bee (6°15) Considering the numerical values in Bragg’s experiments, it appears that for a block two thousand layers thick the correction will be about 5 per cent. Thus we see that a conglomerate of crystals of size d will give rise to a Q modified by the extinction factor (6°15). This modification is the primary extinction, and as we shall see it is untouched by Bragg’s method of eliminating extinc- tion. The secondary extinction arises in considering the action of the transmitted beam on the lower blocks. It may be calculated by allowing for the ordinary absorption of the incident beam and in addition subtracting from it the amount of the reflexion. As this last depends on Q’ the secondary extinction will do so too. 7. Reflexion from a Face. Now consider what happens when a beam strikes the face of a thick conglomerate at any angle near the angle of reflexion. Imagine the conglomerate divided into successive layers. In the first layer it will find a few blocks rightly placed, and from each of these a ray will be reflected. In § 6 we saw that extinction would reduce the intensity of the transmitted beam by an amount equal to the intensity of the reflected beam. After traversing the first layer the beam will thus be defective in a few patches, where par- ticular blocks have been able to extinguish it; but in considering the effect of many layers it will be correct to average the intensity after traversing each and so treat it as uniform for the next. To obtain the power transmitted through a single layer, we shall therefore take the power reflected by it, subtract it from the incident and reduce the result by an amount given by the ordinary absorption. The whole reflexion from a deep crystal results from the Phil. Mag. 8. 6. Vol. 43. No. 257. May 1922. 3G — iq ? . Se a a pe ~ —Sh Se aie... Se tee “eee kA i ol 818 Mr. C. G. Darwin on the Refleaion of multiple reflexions in the successive layers. The multi- plicity is of a different type from that of (6°1) because the rays are not now coherent. The problem of these multiple reflexions would be exceedingly difficult if it were treated exactly ; for each layer will, on account of diffraction, spread out incident parallel rays into a certain range of angles and so will continually change the angle at which _they attack the successive layers. But, if (as assumed in § 5) the crystal is so imperfect that diffraction does not — change the direction of the rays to an extent comparable with the scale of variation in the orientations of the blocks, then it will be legitimate to regard the reflected rays as coming plane parallel off the crystal (at an angle exactly 20 to the incident beam). In consequence of this it will be possible to replace a highly complicated system of integral equations by differential equations of a simple type. Suppose that a plane incident beam of total power I strikes a deep crystal at angle 0+u to the face, and let HE, be the total power reflected. Let I,(z) and E,(<) be the powers of the incident and reflected beams at a depth ¢ inside the crystal. Suppose that the area of face they strike is B, and consider the effect of a layer of thickness 6z. The incident beam has cross-section Bsin (@+u) and so its intensity is I,(z)/Bsin(@+u). The power reflected by the volume Béz will, by (6°11), be therefore I,(z)dz G(u) cosec(@+4u). The incident beam will lose the same amount through extinction, . while through absorption it will lose I,(<)dzu cosec (8+u). In the same way the reflected beam will be partly reflected back again. ‘To treat of it we must regard the conglomerate upside down—that is, for F(W, /, m) in (5-1) we must write F(W, —l, —m) and also for u,—u. Thus G(u) will be unaltered in form, and the beam H,,(z+6z), which is coming outwards through the layer éz, throws an amount H(z + dz) dz G(u) cosec (9 — u) back into the incident direction. Corresponding to this there is an amount 3 Hue + 2) 62 {u+ G(u) } cosec (0 —u) absorbed and extinguished. Balancing up the gains and losses we arrive at a pair of equations, ae ee ea 02 sin (+4) Lule) + sin (@—u) E. (2) (71) 3H, (z) w+G(u) G(u) aa Thine an aay! ga 6-Ea eae X-Rays from Imperfect Crystals. 819 These equations will be true even when the scatter of the blocks varies with the depth in the erystal, but to make progress we shall suppose it constant-—that is, G(w) is nota function of z. This makes the equations linear. The end conditions are that I,(z) and E,(z) should vanish for infi- nite z. There is no need to give the solution in detail—— E,, the value at the surface is what is required. It is * B,/1= Taos OCOfl + G(w) sin 0 cos u +4/{[w+ G(u) P— [G(u) 21 —cot? A tan?u)}]. . (7:2) The first factor represents the influence of having a crystal face that is not the true reflexion plane—whether because the surface is covered by a vicinal face or because it has been badly ground. for the layers into which the crystal was divided were drawn parallel to the actual face, and if this is not the true reflexion face G(w) will be unsymmetrical about w=0. Now suppose that the source and point of observation are interchanged—this is the same as observing on the other side of the spectrometer. We must then draw the x axis in the other direction, and so shall obtain a formula involving —w instead of u. But if on this side we take u’ =*-u we shall have sin (0+) sin 6 cos u! [Died bes G(u') /{u+ G(u') + ete.}. Thus if we compare together points where the wu of one side is the same as the w’ of the other, then clearly W',,/H,=sin (8+) /sin (Q—u). In the case of a fairly perfect vicinal face all the settings which give perceptible reflexion will be not far from u=«, the inclination of the face, and so the ratio of all pairs of corresponding powers will-be nearly sin (9+) /sin (@—2), and therefore the same will be true of the integrated reflexion. This factor may also be derived by simple con- sideration of the area of the crystal on which a limited beam would fall in the two cases. The difference of the re- flexions was originally observed by Sir W. H. Bragg +, and * Mutatis mutandis this is substantially the solution obtained by K. W. Lamson, Phys. Rev. vol. xvii. p. 624, by quite a different method. lf (7:1) are treated as difference equations his exact form is obtained. { W. H. Bragg, Phil. Mag. vol. xxvii. p. 888 (1914). At first I thought my explanation was different from his; but through corre- ‘spondence it became evident that we were only regarding the matter from different points of view. I wish to express my thanks to him for his interest in the matter. 3G 2 820 Mr. C. G. Darwin on the Reflexion of explained by considering the absorption of the emergent beams in the two cases. His argument leads to the same factor. The influence of the vicinal face can be completely eliminated by averaging for both sides, and we shall suppose this done. There is then no need to consider the first factor in (7:2) at all. Bragg’s “reflecting power” (which we are here calling the integrated reflexion) was defined by him as Ew/I, where the crystal was turned with angular velocity » through the reflecting angle and E was the total energy obtained, while I was the power of the incident beam. His I is the same as ours, his Fis our | E,du/o@. So we have for the integrated v ry) reflexion =e ae Wer | w+G(w) —e + VW [ wt G(w) |?—[G(u) ]?(1—cot? 6 tan? u) } (73) If G(u) is small compared with w for every value of u, then neglecting the small terms of the denominator and using (6:4) we have pO ae Apart from the difference between Q’ and Q, this is the equation used in B.J.B.1. If G(u) is not always small enough to justify this approxi- mation, it may still be small enough to admit of expansion in powers of G(u)/u. Then we have 1 apa = iA G(u)du— 5 G?(u)du as : { G2(w) (?—cot? 6 tan? x) du. —2 ae [oo26) du=9.0" | Oe ae aie. (7:4) ["crenau= og | = J then gy, and g3 will be constants of the crystal. For most crystals it will be legitimate to neglect the term involving > 4 ~~ fle X-Rays from Imperfect Crystals. 821 tan?u. Then Qi >. Q2) 1Q# Ca), Poe +93 ys be = 39) { uta + (ray =}. (75) If the third term is neglected this is in the form used by Bragg, who calls g.Q! the “ extinction coefficient.” If we had considered that every incident and every reflected beam had only a single reflexion, then we should have had instead of (7:2) H1=3G(@)/ju+ G(u)}, and this would lead to the same first two terms in (7°35). This idea has been used by Sir W. H. Bragg *, and it is clear that there will be a wice region of vaJues in which it will be a very good approximation. It is evident that a knowledge of p by itself is not sufficient to determine the value of Q’; but (7°2) suggests that it may be possible to do so by a study of the shape of the reflexion curve. For if we know E, for all values of wu we may solve (7:2) for G(u). If the first factor is omitted, we have fi 2(H./ I) ee ek cot tan ayy A quadrature will then lead to @’ by (6°14), and so the secondary extinction is eliminated. It thus appears theoretically possible to determine Q! from observation on a single face. There is, however, a serious objection to the method. Jt is not reasonable to suppose that G(u) is really independent of the depth; for grinding or even cleaving must necessarily act differently on the surface-layers and interior, and if G is an unknown function of c, the data are insufficient for a solution. If, in spite of this, the process should be valid, there would still be the difficulty that Q’ may differ from Q. The only possibility of deter- mining Q would appear to lie in finding @! for several crystals, of which the blocks were scattered in various degrees. If then the resulis all came the same, there would be a presumption that primary extinction was not present. Of two discordant values the greater is to be = Ws Brase, Phil. Mae. vol. xxvii. p. 881 (1914), and Proc. Lond. Phys. Soc. vol. xxxiii. p. 304 (1921). 822 — Mr. C. G. Darwin on the Reflexion of preferred, and it would be expected that this greater value would be associated with a greater scattering among the blocks—that is, a broader region of reflexion. 8. Reflexion through a Pilate. To overcome the difficulty of the unknown extinction Bragg sent X-rays through a crystal plate and observed the reflexion from the interior planes. In this case the equations for the multiple reflexions take quite a different form from (7'1). Suppose the crystal cut into layers parallel to its faces, the breadth of a layer being ow. Let L,(z) be the power of the incident beam at depth x from the front face, H,,(@) of the reflected beam. Let B be the area of the face on which the rays fall. The incident beam now makes angle 0+wu with the normal to B, and so the intensity is I,(x)/Bcos (G+ 0). The power reflected in the volume Boz is I,(x) 6x G(u) sec (9+u) and the incident ray is reduced by an amount 1,(a) da{p+G(u)} sec (O+u). Similarly for the reflected rays. The differential equations now are Ol.) w+ G(x) G(w) Ay Oe” cos (6 +4) dake) Ws (O— ye e (8:1) OL, () pt G(w) G(u) ey! Se cos (Ou) one Oe The end condition is that B,, (x) =() for «=0. If H, is the power of the emerging reflected beam, the solution gives G(u) sinh re eo ase {w+G(x)} cos 0 cos u COCs aw) or “EE Gok (O+u}cos(@—u) ’ (3°2) Ae cos (0-+u) cos (@—wu) + (w+ G)? sin? 6 sin? u cos (9+ u) cos (@—u) BH /I= where Apart from the first factor in (8°2), u only occurs as a square. So, as in §7, an averaging with the reversed — beam will eliminate the effect of untrue faces. In most cases this will be far less important than it was in § 7, because the factor occurs as a cosine and in the important cases @ will be fairly small. A-Rays from Imperfect Crystals. 823 The exponential and hyperbolic functions can always be expanded, and if G(u) /m is not large the series will converge — It will usually be right to omit the terms involving sin? u, etc., even though some of these are multiplied by wu and are being compared with others only multiplied by G(u). Then T= (G(u) sec 6 and H,/ l=e-** | 2'G (u) —2”G?(u) + 2a"GF(u)], . (8:3) where wz’ is written for zsec@. As in §7, if the form of H,,/I is found experimentally, it is possible to solve (8°3) and so to obtain G(u), and consequently Q’, from observation of a single crystal. In this case the process will be free from the objection raised in § 7 about non-uniform distribu- tion of the blocks; for, in reflecting, the surface does not now receive preferential treatment over the interior. The primary extinction is again untouched by the process. Bragg adopted a method which assumed that he could get a series of plates of various thicknesses, for all of which the distribution of the blocks was the same. He took the inte- grated reflexion of them, and found for each set of reflecting planes the thickness which gave maximum reflexion and the height of the maximum. Now by (7°4) the integrated re- flexion may be written as p= (E/T) du=e-#* [Q’a! — go Q? ax” + 29, Q v2]. (8°4) a —-eo This has maximum at rae Q’ Q? x mC 19s Beh oe tha I Cenee and the value there is “2 1-92 + (390° + 300) | sg a Sep =e fata +a ten (8°7) Thas Bragg’s work determines g,Q’, and if the distribution of the blocks is the same as in the crystal used for the work of $7, it follows that his correction for secondary extinction is correct to the second order, and from the magnitudes of the quantities involved it is improbable that the third order is sensible. i ‘ h 824 Mr. ©. G. Darwin on the Reflexion of 9. Comparison with Experiment. There is not a great deal of material suitable for testing these formule, and the resalt of the test is not very satis- factory. The first point of comparison is the curve in B.J.B.i1. p. 12, which relates the modified absorption coefficient to the integrated reflexion of a face. The ordi- nate of the curve is given by (8°7) and its abscissa by (7°95). The linear form of the curve means that {0.4 (492-49) but) - yes is practically constant, and this it will be, if g,’Q’, g3Q! are negligible compared with gu. Neglecting these terms we find 2=5'6+ (5:41 x 10-4), whence g,=484. This may be best interpreted by assuming an error law of distribution as in (5°7). Then G(u) = (!e-W/20? | 4/ 2a and OS V7 | 20, which gives c=6'. ‘This is a good deal smaller than would be expected from the general description of the experiments ; for it means that all the reflexion should take place within less than half a degree, whereas the paper implies that the band of reflexion was nearly a degree broad. A part of the discrepancy may be due to the neglect of the further terms in (9’'1), for it is evident that the series is not very rapidly convergent, when, as here, 92Q'/m#=5-6/10°7. A more detailed, but still less satisfactory, comparison may be made with the reflexion curves of B.J.B.ii. p. 13. The experiments dealt with the reflexion through two plates of the same thickness, of which the surfaces had been dif- ferently treated. The information about the curves is not quite complete, but can be supplied indirectly. It is first necessary to find the absolute values cf E,,/I. The curves are drawn in arbitrary units, and a constant multiplier must be obtained for each from the observed value of its integrated reflexion (which is the area of the curve), in terms of that of a standard plate of the same thickness. The reflexion of the standard was calculated (p. 7) from that of a face, on the principle that for a surface the integrated reflexion is Q'/2(w+92Q'), while for a plate of thickness giving maxi- mum reflexion it is //e(u+g.Q’). There is thus the assumption that the plate has the same scatter as the face. X-Rays from Imperfect Crystals. 825 3 However, from these data it is possible to get the numerical values of H,/{ for all values of uw. It is found that the approximation of (8°3) is quite accurate enough, and this equation can be solved for G(u). A quadrature then gives Q’. The results are rather disappointing, for the curve A gives Q’=:0119, while B gives Q’=0146. Moreover, the extinction coefficients go’) come out as 1:01 and 0°47 respec- tively, whereas values in the neighbourhood of 5 would have been expected. The discrepancy i is exactly the same as in the evaluation of g, above. There it was found that the region of reflexion ought to be narrower for the observed extine- tion, here that the observed reflexion curve implies less extinction than is in fact found. It is, of course, pussible that a part of the difference between A and B may be due to a difference of their primary extinctions, and the cause suggested at the end of § 7 may be another source of discrepancy. Finally, our results may be applied to some experiments due to Davis and Stempel*. Here the perfection of the calcite crystal was enormously greater than in Bragg’s rock- salt, and all the approximations are hopelessly wrong. I, nevertheless, we apply our formule to the actual curves we may obtain something of an idea of the perfection of the erystal. The data are directly in terms of H,/I. They were dealing with white X-rays, but there was double reflexion in two crystals with parallel faces and it is easy to see that dispersion will play no part, so that the formule for monochromatic rays are applicable. Taking the most extreme case of all, their fig. 6 (p. 617), we use (7°6) to obtain G(u) and from this we get a scatter c=4/"8. Now this is only a little greater than what should be the region of complete reflexion in a perfect crystal, and the most remark- able thing about it is that not more than half the incident beam is reflected. This suggests that a part of the breadth of the reflexion is really due to imperfection. It does not appear worth while to carry further the comparison with these experiments, both because our methods are not applicable rigorously, and because there must certainly be a great deal of primary extinction in crystals that are so néarly perfect, so that they would be of little use in a determination of Q. * Davis and Stempel, Phys. Rev. vol. xvii. p. 608 (1921). SS ee ee ee 826 Mr. C. G. Darwin on the Reflexion of 10. The Powder Method. From the preceding sections it appears that the pheno- menon of primary extinction is likely to make serious difficulty in determining Q by the method of reflexion, whether from a face or through a plate. The only way to ensure its absence is to use crystals so small that it is bound to be negligible. For example, from the numerical data of § 6, primary extinction would be absent, if the crystals were so small as to be just about invisible under a high- power microscope. The only practicable way of using such - is by the powder method of Debye and Hull, which has recently been used quantitatively by Sir W. H. Bragg * For the sake of completeness we shall apply our processes to this, adopting an arrangement which is probably not the most convenient, but which could easily be modified. We shall suppose a speck of powder is illuminated by rays and shall find the total power thrown off into a cone (of half- angle 20), corresponding to one particular set of planes. Let the volume be V and let it be composed of small blocks, the typical block being of volume W with normal in the direction of colatitude and longitude w,¢. let the distri- bution of the blocks be given by VECW) dW sing dadd.. > 2a iy F is nearly the same as in §5, but is now independent of w, d, as they are pointed equally in all directions. We have then dn | W FCW) dW=1, w is the inclination of the normal to the incident beam, and oO o= = —@—u. Multiplying (4°5) by the appropriate factors we have for the whole reflected power arising from this set of planes (3) *(2) (6) IN? feos. Vf F(W) dW udp | away | dV dV' Ww — expik {(a—«2')(u—v) sin 0 + (y—y)¥ cos 8 —(z—<')(u+v)cosO}. . (10:3) Here J is the incident intensity and of the a cos’ @ one term is due to the sinw, the other to the 7 cos0dudy integration. ‘The integrations follow the same course as * W.H. Bragg, Proc. Lond. Phys. Soc. vol. xxxiii. p. 222 (1921). X-Rays from Imperfect Crystals. 827 those in § 4. The result is * PY COU ey as ee (LOS) This must then be multiplied by a numerical factor given by the symmetry of the crystal according to the following rule. If the crystal has no centre of symmetry, add one on to its symmetry elements. Now construct the “form” corre- sponding to the planes that are being studied. The number of its faces is the required numerical factor for (J0°4) ; by virtue of the centre of symmetry it must always be an even number. For example, in rock-salt the form for (1, 0, 0) is a cube and the factor is 6; for (1,1, 0) it is a rhombic dodecahedron and the factor is 12; for most planes the number is 48. It may sometimes be necessary to apply a correction for absorption. This will depend on the shape of the powder and isa matter of simple geometry. It should not be necessary to make any allowance for secondary ex- tinction, but if it were needed it could be calculated on the principles of §§ 7, 8. 11. Discussion of Results. The tests in §9 were rather unsuccessful, but I do not think sufficiently so to condemn our theory out of hand. Should further tests prove that the discrepancy is real, it appears to me that it would throw doubt, not only on my own work, but also on the validity of the deduction of Qin B.J.B.ii.; for that deduction can only be founded on some theory which must be the same as the present one in principle. It is of course possible that a crystal, imperfect by warping instead of cracking, should obey a different rule, but I should judge this to be very unlikely. For (5°5) may be used to define a function G(w) for such a crystal, though its expres- sion in terms of the imperfection will not be so easy as for a cracked crystal. With this G(u), the work of §§ 7, 8 will all stand good, and will determine the relation between the extinction and the breadth of the reflexion region, without touching the question of the meaning of G(u). As to this last, it is a most natural conjecture that (5°6) will be true for it, in view of the generality which that equation has already been proved to possess, and so will lead to the right value for Q. If, as appears probable, rock-salt is warped rather than cracked, this will have the advantage that primary extinction is unlikely to be important, and so the * The expression does not vanish for 9=7/2 on account of the factor cosec 26 in Q, ee 828 Reflexion of X-fays from Imperfect Crystals. interpretation of experiments will be freed from a source of error, the amount of which must be very uncertain. We therefore conclude that, to establish beyond doubt the validity of the work of Bragg, James, and Bosanquet, it is essential that the work of §§ 7,8 should be verified by tests like those of § 9, and if it should be proved correct, then we may have great confidence in their results. Failing this verification, the theory on which they eliminated the extine- tion is when good foundation and the results must be recarded with some caution. In this case, it seems to me that the most satisfactory way of determining @ is by the powder method of § 10. Summary. The paper is a theoretical inquiry into the possibility of determining the arrangement of electrons in the atom from the intensities of the “X-rays scattered by crystals. This problem falls into two stages: first from crystal to molecule, then from molecule to electrons—only the first stage is here treated. Simple formuls have been given by various writers, and the process has been carried out experimentally by Bragg, James and Bosanquet. They encountered the difficulty of “extinction.” This extra absorption falsifies the formule, but they measured it directly and so obtained a correction. This paper is concerned with seeing whether their correction was valid. The point of the problem was known to lie in the imperfection of crystals. After a general discussion (§§ 1, 2,3), it is shown (§ 4) that if a small perfect crystal of any shape is turned through the reflecting angle for monochromatic rays, the amount of reflexion determines a quantity Q, which is what is required for the second stage of the problem. The reflexion is worked out (§ 5) for a conglomerate of small blocks of perfect crystal all orientated nearly i in the same direction, the conglomerate being so thin that aoe tion and extinction can be neglected. Extinction—that is, the special absorption of rays at the reflecting angle—is shown (§6) to lead to two effects, primary and secondary. The primary diminishes the reflexion from a perfect crystal below the amount given by the simpler theory. It leads to a change in the value of Q depending on the depth of the crystal, and none of the experimental processes eliminate this change. The secondary extinction results from the reduction in the strength of the beam transmitted through the crystal. Scattering and Dispersion of Light. 829 The reflexion from the face of a deep imperfect crystal] is evaluated (§ 7), and it is shown how the secondary extinc- tion may be eliminated. The same process for reflexion from the interior planes of a plate is worked out (§ 8), and the formule are justified whereby Bragg. James, and Bosanquet eliminated the extinction—but only the secondary extinction. The theoretical results are compared with experiment (§ 9). The experimental data are rather inadequate and the agree- ment is not very good. The. corresponding calculations are done for the powder method of observation on crystals (§ 10). The paper concludes with a short discussion (§ 11), suggesting the need of further tests. XOILL. Scattering and Dispersion of Light. By U. Dot, Research Student in the Institute of Physical and Chemical Research, Tokyo*. UTHORS differ in their opinions as to the mechanism of scattering light by a medium through which the light travels. Schuster asserts, however, in his ‘Theory of Light’ (p. 325) that, if a molecule of the medium may be looked upon as a separate source of scattering, the scattering due to it follows undeviatedly the celebrated formula of Lord Rayleigh, whatever be the theory we adopt. It will not be without interest, for instance, to notice that Jakob Kunz+, indeed, derived exactly the same formula from an elementary theory of scattering of light by small dielectric spheres. Ever since the electron theory of matter began its striding progress, and the well-known dispersion formula was deduced by H. A. Lorentz through his electronian analysis of atomic constructions, attempts have been made to interpret the absorption of light from the electronian standpoint of view. Thus, Drude} and Voigt § attribute it to the damping of the oscillations of bound electrons in the atoms of the absorbing medium, the damping pro- cess being caused by a resisting force proportional to the velocity of the electrons. They insert consequently a term of this damping in the equation of motion of * Communicated by the Author. Tt Phil. Mae. xxxix. p. 416 (1920). t P. Drude, Lehrbuch der Optik, p. 358. § W. Voigt, Magneto- und Electrooptik, p. 104. 830 Mr. U. Doi on Scattering an electron. Lorentz* assumes on the contrary that there is no damping, but that the electron can continue to vibrate further and further, restrained only by radiation, until the moleenle collides with another. When this happens, he assumes that all the energy of the vibration will be trans- ferred to energy of translation, or thermal agitation as it is called, of molecules. The effect of this he proves equivalent : 2m dr iy to a damping force — —-—, where m is the mass of an U dt’ electron, and 7 isthe mean time between colluean of mole- cules. Thus, defining the absorption coefficient * by the _ equation dl ae —klI, he finds t ir Ann?gNe?/c 2m» ay m (ny? 65 n?) ae ng? ’ (v= ee The meanings of the notations may need no explanation, as one is, if necessary, rather preferably referred to the original. It come next, to Nagaokat and Planck § that the reaction, = a £, to the radiation of an electron, when it has an acceleration, £, will act frictionally. Along this line of idea Planck derives the coefficient of extinction, which agrees with Rayleigh’s formula for the scattering coefficient. The general process of deriving the coefficient is to insert a term of this frictional force in the equation of motion of the electron, and to calculate the energy dissipated in overcoming this friction. The calculation, however, requires too complicated a process to be summarized here clearly. Be that as it may, the scattering is another thing, in a rigorous sense, than the extinction or absorption ; and the calculation of what part of the energy of the incident radi- ation is dissipated in overcoming any ; frictional force exerted by molecules of the medium thre ough which the light travels, does not offer us any means of deducing the coefficient of scattering, as the energy absorbed by the medium is not all scattered by it,—a part, at least, of the absorbed energy is converted, as we know, to the beating agent of the medium, * H. A. Lorentz, ‘The Theory of Electrons,’ p. 141 (1916). + Phys. Rev. iv. p. 189 (1914). t Proc. Tokyo Math.-Phys, Soe. ii. p. 280 (1904). § Sitzungsber. d. Kgl.-Pr. Akad. d. Wasa p. 748 (1904). and iispersion of Light, Sal though for a gaseous medium it is general to disregard the heating effect accompanying the absorption, and conse- quently to make no marked distinction between the scat- tering and the absorption or extinction. In order to deduce more rationally the coefficient of scattering, therefore, we have to resort to another process of calenlation ; an answer to this request will be obtained through identifying the scattered energy with the energy of radiation itself from the electron which is set in motion forced by the incident light. The amount of radiated 9 (oa oo 2 - energy, then, must be measured by erate * where e is the ~ 67 charge of the electron, and ¢ the light velocity in vacuum, while & is the acceleration of the vibratory motion of the electron. It will be noticed here that no term needs to be added to the equation of motion of the electron corresponding to a reaction to the radiation from it, as it is too small to affect the general features of the problem when we attack it from this side. . Let us take, then, the Lorentz equation of motion of an electron for the case of a plane polarized wave incident on it, in which no term of damping appears, namely, of the form + dé me de — e( H..+ ae.) —fé, where & is the displacement of the electron from its position of equilibrium, a a constant nearly equal to 4 (Heaviside- Lorentz Units of electromagnetism are adopted here), f another certain positive constant that characterizes the elastic force of restitution of the electron, and P,, the polari- zation within the medium caused by the electric vector H, of the incident light. Denoting by N the number of polarized particles per unit volume of the medium, PS Nee Thus m ay =é(H +aNeé)— fe, or m 8 +(f—aNe)F&=ck,. Supposing the light vector to be a periodic function of the frequency n, say H,=X,e’ at a point, and putting the frequency of the free vibration of the electron, as usual, * O. W. Richardson, ‘The Electron Theory of Matter,’ p. 256. + H. A. Lorentz, ‘The Theory of Electrons,’ p. 309. } Ldid., p. 136. SoZ Mr. U. Doi on Scattering we wa =n, we obtain, for the general solution of the above m differential equation. @ De & ~ mno?—n?—aNe?/m in which A and B are arbitrary constants of integration. We have to observe at this point of discussion that there is every reason to suppose that within a medium the radia- tion and absorption due to free vibrations of the electrons compensate each.other, so that in consideration of the present problem of scattering of light incident on them from an external system, we may naturally put the terms of free vibrations out of consideration, limiting our attention solely to the part of forced vibration, ae Xpet Cn a aa NCTE Differentiating & twice as to time, e fie ge” M No? — 1? — aNe?/m not —inot + Acito! + Ben inet, The amount of radiation corresponding to this acceleration is es pe ( i n2X eit ; —, = 670 67re® mne—n?—aNe?/m ent X rerint 5 Ouano 2] \2* 6rrc*m? (n>? —n?—aNe?/m) This energy of radiation must be consumed from the incident energy, which results in scattering afterall. Take, now, a thin lamina of the medium, of a thickness dw and cross-section A, perpendicular to the incident ray of light. The number of scattering electrons contained within this elementary lamina will be NA.dz, so that the amount of energy radiated from this lamina during a time-interval ¢ is ae NA. del A, Eat t e/ 0 =NA.dz ett, ent Xo? rt 61rce?m? (ny? —n? —aNe?/m)? J, But the energy incident on this lamina during the same interval of time is* t MLN AiG a Niet ds : t =A. x. ede 0 - * Of. O. W. Richardson, loc. ext. p. 211. and Dispersion of Light. $33 Thus, the scattered energy 1s 4,4¥ 2 ‘ —dI=NA.dz. erm Rc { ere 62re2m?( m9? — n? —aNe*/m)* ; std Ndwe*n* I ~~ 6mre*m? (np? — n? —aNe?/m)?" ~~ Then, defining the coefficient of scattering « by the equation dl =—kxldz, we have Netné | oc 6rre*m?(n? —n? —aNe?/m)? Again, if we denote by wu the refractive index of the medium, we have * Ne? 1 2 ee asset 2a ei Rae ale se m ny —n?—aNe?/m pom ee 4 1a Lge lg Ie; Hence c= bain 12) ame | > p) .. 6m') 2 ae cee ame Emo) Dee ie A e—-12(u—1), Putting n= If we put further we have finally which is Rayleigh’s formula for the scattering coefficient for a gaseous medium fF. The correctness of this formula is acknowledged generally. and its experimental verifications, too, are not wanting, the latest and most extended treatment of the subject being given by Cabannes in Annales de Physique, vol. xv. Jan.— Feb. 1921. (There will also be found in it a complete list of authors and works in this direction.) As for the theoretical deduction of the formula, the sug- gestion is given in Lorentz’s ‘ Theory of Electrons’ (p. 309, Note 56*), and Natanson{ also derives the form after a long and more or less complicated discussion. But so far as I am aware, the way above indicated seems the simplest one to arrive at the result. * H. A. Lorentz, doe. cit. p. 144. t+ Phil. Mag. [5] xlvi. p. 379 (1899). t Phil. Mag. xxxviil. p. 269 (1919). Phil. Mag. 8. 6. Vol. 43. No. 257. May 1922. dH 834 Mr. G. A. Hemsalech and the Comte de Gramont on It will hardly be necessary to remark that, along this line of argument, when the freqnency of the incident wave is very near to that of proper oscillation of the electrons in atoms, there takes place a resonance phenomenon, andas the result of this there may be expected to occur such pheno- mena as ionization and resonance radiation of the gas, and the incident energy thus being consumed by a considerable amount, the spectrum line corresponding to this frequency will be missing, which is called an absorption line in the ordinary spectroscopy. Physical Institute, College of Science, Imperial University of Tokyo, Sept. 20, 1921. XCIV. Observations and Experiments on the Occurrence of Spark Lines (Enhanced Lines) in the Are.—Part II. Magnesium, Zinc, and Cadmium. By G. A. HEMSALECH and A. DE GRAMONT *. [Plates XVIL-XXI.] CoNTrENTS. §1. Introduction. §2. Influence of the nature of the liquid film upon the emission of various types of lines in the arc. §3. Effect of high voltage currents. §4. Thickness of liquid film at moment of rupture and relative durations of first and second phases. §5. Effects of gaseous media. §6. Are in liquid air. §7. Cause of broadening of the hydrogen lines during the first phase. Stark effect. | §8. Detailed results of observations on the are spectra of magnesium, zinc, and cadmium. §9. Explanation of Plates. §10. Discussion of results. §11. Summary. §12. Concluding remarks. § 1. Introduction. NS a result of an experimental investigation of the A arc spectra of lead and tin described in an earlier communication + we arrived at the conclusion that the emission of spark lines (enhanced lines of the fourth type of Hemsalech’s classification) by the arc is connected with the presence of ‘strong electric fields. Further, we showed that when the temperature of the medium is raised, by * Communicated by the Authors. t G. A. Hemsalech and A. de Gramont, Phil. Mag. ser. 6, vol. xliii. p. 287 (1922). | the Occurrence of Enhanced Lines in the Are. 835 means of a white-hot carbon cathode, the spark lines dis- appear from the spectrum of the metal vapour. Thus the prominence of spark lines in the spectrum of a source appeared to indicate that the temperature of the latter must be relatively low. In fact the spark lines of lead were found to be strongly emitted by an arc burning under liquid air—namely, at a temperature of about —190° C. Encouraged by these results, we first made a general survey of the arc spectra of fifteen different metals by means of the methods previously described, and then devoted our attention more particularly to the spectrum of magnesium. This element was marked out for special examination not only by reason of its cosmical importance but also because its spectrum contains four distinct types of radiations, all of which occupy in turn prominent positions in the various laboratory sources of light. It therefore seemed to us that a detailed study of the spectrum of magnesium might lead to results of a more general interest on account of the possibility of investigating in one and the same spectrum the relative behaviour of four different types of radiations under the special conditions prevailing in our experiments. It may be useful for the better understanding of what follows to briefly recall our present knowledge concerning the emission of these four types of magnesium lines com- prised within that region of the spectrum which is of particular interest to the astrophysicist, namely from r» 3600 to A 7000 :— Flame line 4571. Observed already at the temperature of the air-coal gas flame. ( Series triplets at ) 3838 and 5184. These lines are absent from the mantle of the air-coal gas flame. First type. \ 5184 has been observed in the oxy-hydrogen flame. Both triplets are emitted by the flame pecons IPOS strong under thermo-electrical excitation in the fringe of a single plate resistance furnace. They are likewise brought out under chemical ex- citation in the blue cone of the air-coal gas , flame. Strong in arc and spark. Rydberg’s nebulous series, of which the least Third type. vefrangible line in this region is 15528. With certainty observed only in arc and spark. (Spark lines of which the most prominent and | important is 44481. Not seen in any of the Ramil type é flames nor in the furnace. Traces only in "\ ordinary arc in air between Mg poles. Brought | out strongly in arc under water or in hydrogen. | Very intense in capacity spark. eee: 2 of burning magnesium. Their emission is very . Se hi eee... oo” SR oe eee. ee eee Teen { ; Se eh Se 836 Mr. G. A. Hemsalech and the Comte de Gramont on The second and third type lines constitute the so-called are lines. In addition to a detailed account of our experiments with magnesium the present communication contains also the results of our observations on the enhanced lines of the related elements zine and cadmium. The methods of producing and of observing the various. kinds of ares were the same as those described in our first. communication. § 2. Influence of the nature of the liqud film upon the emission of var 10uUs ty pes of lines in the are. In an are flash passing between magnesium poles through, a liquid film the first phase is generally sharply defined, and the character of the spectrum changes almost abruptly on: passing from the first to the second phase. The relative behaviour of the various types of magnesium lines is pr obably controlled by the conditions which predominate at the various stages of development of an are flash. Thus, if certain chemical reactions took place between the magnesium vapour evolved and the liquid or one of its constituents, such, an event might reveal itself by some special peculiarity of the spectrum and the latter would change on passing from a: liquid of one kind to one of a different chemical constitution. Again, if any particular group of lines were influenced by the electric field set up within the first phase, the effect should be the same for all liquids whatever may be their composition. Of the three liquid films used with maynesiam—. namely, water, glycerin, and paraffin oil, the first two contain oxygen, and we might therefore expect to observe effects of possible oxidation. Indeed we find that whereas in the water and giycerin films the flame line 4571 is well deve-- loped, only faint traces of it are seen in the paraffin-oil film. In like manner the well-known band at 5007, which is. generaily ascribed to the formation of an oxide, is brought out in the water and glycerin films but is absent from the. paraffin-oil film. In the second phase or the arcing stage the flame line as well as the bands are brought out in all three cases. During this phase the arc flash is of course- exposed to the influence of the surrounding air containing oxygen. Thus, it would seem that the emission of the flame line X4571 under these conditions is favour ed, though not: solely controlled, by the presence of oxygen. The lines of the second type, namely the triplet series, on the other hand, are not affected by the nature of the film. the Occurrence of Enhanced Lines in the Are. 837 ‘They are strongly emitted throughout the are flash, though ‘slightly enhanced and symunetrically widened in the first phase ; hence these lines appear to be sensitive to the intense electric actions of which the liquid film is the seat. This ‘observation is therefore quite in keeping with the known behaviour of the triplets in the single plate resistance ‘furnace. The lines of the Rydberg series, which form the third type, are hardly visible in the layer of paraffin oil though very _ intense in the water and glycerin films. 015 ,, 134 [Gans | graphite poles .,, ) es Wisteria bea 0-09 ,, 4°33 ,, >6°6) o%45 an Magnesium chloride | 9200 volts [ solution between 0:08 ,, 2°40 ,, SO9,, ™ graphite poles .. * Continues as a narrow line throughout flash. + After the abrupt drop in intensity at this point the emission continues as a faint narrow line up to 398 mm. Thus, with the exception of the paraffin-oil arc, the thicknesses of the films are all practically of the same the Occurrence of Enhanced Lines in the Arc. 841 order of magnitude. Those obtained with the higher voltage current seem to be the thinnest: this may not merely be an accident, for it is quite conceivable that the application of a higher voltage would tend to accelerate the destruction of the film. Further, the results for the line 7 4481 show clearly that the higher voltage, while reducing the extent of the first phase, raises the critical level of the spark line. The last column gives the length of the are flash. Rather suggestive results were obtained for various types of cadmium lines with an 80 volts water arc, as shown by the following values :— PeMERMESS OL fl)... 0.-see scent ec ees 0:09 mm, Heights from stationary electrode: BENIGN 7.5 coos svs cc du MN 4413 = 5°05. -,, 3rd type enhanced line 4416= 2°41 ,, : Spark lines .........,-. | ee ss, and Pe | ‘Paved Gautly. aid a (first sudden drop in : J meio at a Herphinct about 2°4 mm. intensity) - These results point to the existence of four distinct stages in the spectral development of the cadmium arc flash, namely: (1) the film stage, during which the spark lines are enhanced and symmetrically broadened ; (2) a short region of the second phase marked by a strong emission of the spark lines; (3) a stage during which the spark lines are but feebly emitted and at the end of which both their emission and that of the enhanced line 24416 cease abruptly ; and (4) the last period of the flash, during which -are lines are alone emitted. These facts seem to indicate that the emission centres, which during the first phase emit strongly all vibrations, undergo sudden changes in their constitution at several definite stages in the course of the gradual decline of the forces acting upon them. It is remarkable, although perhaps quite in keeping with the interdependance between the structure of the radiating atom and the character of the spectrum, that the abrupt stoppage of the faint portions of the spark lines 15338 and 2.4379 should coincide almost exactly with the equally sudden cessation of the enhanced line A 4416. The determination of the duration of the various phases connected with the liquid film are requires a knowledge of the velocity with which the upper electrode moves away from the.stationary one. We have made no accurate measurement of this quantity, but a rough estimate of the upward motion 842. Mr. G. A. Eamealoch and the Comte de Gramont on continued over a long range has shown us that the speed of the electrode in our experiments is in the neighbourhood of one metre per second. This result allows us to evaluate the order of magnitude of the duration of the various phases. in the arc flash. Thus the principal events in the course of development of the 80 volts water arc flash between. magnesium poles would be as follows :— Moment of separation of electrodes............ 0°0000 second. Eindvof firstaphase? i... as hevsie iene te eee 00001 __,, Abrupt drop in intensity of A 4481............ 0;00Ia Hind ofeaire Hasler koeicrcsenccee ne een ace 00048 so, In a similar manner the principal epochs during the period of existence of an 80 volts water arc flash between cadmium electrodes are :—. Moment of separation of electrodes............ 0:00000: second . Mndyot first jolnasel ne, eee e tee eee oer 0-C0009 __,, First drop in oe a of spark lines \ 5838 — | BIN DOO age Ee te ey ie eee Ree Ee 0:0013 3 End of emission of spark lines and of _ enhanced line A 4416 20.0... . cc. cee eceee secon 0:0024 =, End of emission of arc line \ 4413, marking extincbion Or Masia yy pucker sachcceen et rere 0:0050 Ms During the first phase of an arc flash the spark lines attain a development which, as regards character and relative intensity, approaches that observed in powerful condenser sparks. It is therefore of interest to compare the duration. of this phase with that of an electric spark. From photo- graphs of condenser discharges through air at atmospheric pressure taken upon a moving film by one of us, in con- junction with Sir Arthur Schuster *, we have calculated that that period of the spark which comprises the initial discharge and the rapid oscillations through the metal vapour lasts approximately 0-000016 second. It is during this short interval of time that spark lines are most strongly emitted. Now we have just seen that the time taken by the upward-moving electrode to pass through the liquid film of a water arc is about 00-0001 second, so that some of the phenomena which happened during the early stages of the progress of the electrode tip through the liquid film must have been as short-lived as the oscillating phase of a condenser spark. * Schuster and Hemsalech, Phil. Trans. Roy. Soc. vol. exciii. p. 189 (1899). the Occurrence of Enhanced Lines in the Arc. 843. § 5. Lffects of gaseous media. The gases which were used in these experiments may be divided into two groups—namely, those in which a steady arc can be established between metal electrodes, such as air and nitrogen, and secondly those in which inden similar electrical ‘conditions only short are flashes can be obtained, such as hydrogen, oxygen, and coal-gas. It seems likely that the kind of are produced in each case is in great measure dependent upon the nature and intensity of the chemical reaction which sets in between the vaporised. electrode material and the surrounding gases. An ordinary seven-ampere arc burning steadily between magnesium electrodes in air emits the “spark line 14481 feebly at the cathode and a little more strongly at the anode. Along the path of the are only the merest traces. of it can be detected. If, however, the observations are begun at the moment of striking the are the spark line is seen to flash up intensely just when the electrodes separate, and it fades away again as the arc becomes established. Photographs taken “of the initial stage of an arc between cadmium poles show clearly that the intensity of the spark lines falls off abruptly after the upward- moving electrode has passed through only a very small distance. In fact both the order of evolution and the general character of the spectral phenomena which mark the early stages of development of the electric are in air are very similar to those observed with liquid films. In order to account for the strong emission of spark lines during the initial phase of an are in air, we venture to suggest the following explanation. As the upper electrode recedes from the stationary one a more or less perfect vacuum ig momentarily left between them. It is through this vacuous space or film of rarefied air that the initial stages of the discharge take place. As no air is present at first to start chemical reactions between it and the metal vapour, both conductivity and temperature of the medium remain relatively low, so that the effect of strong electric forces may develop without impediment. As soon, how- ever, as air has penetrated the vacuous space chemical reactions set in which, by causing increase of both tempe- rature and conductivity of the medium, lead to the establish- ment of a stable arc. If a current of air is blown through the arc from its commencement, ionization of the vapours is rendered more difficult and the brighter spark lines remain visible almost throughout the whole length of the arc flash during the second phase. $44 Mr. G. A. Hemsalech and the Comte de Gramont on A remarkable effect upon the character of the spectrum of magnesium was observed with an are burning in nitrogen. The flame line X 4571 and the triplets at A 3838 and > 5184 show the usual development as given by an are in air. The band at 5007 is absent, requiring probably the presence of oxygen for its emission. The Rydberg series, on the other hand, is far more developed than in air: the individual lines are much better defined and, instead of being winged towards the red, they appear only very slightly widened symmetrically. The spark line 14481, although relatively teeble, passes right across the space between the electrodes and appears quite sharp without any sign of haziness or broadening. Since the are in this case was burning quite steadily, the emission of the spark line under these conditions cannot be attributed to the action of strong electric fields, for these were evidently feeble—in fact, a voltmeter indicated the : 5 ~ It existence of an average gradient of only about 150 _" But it is possible that both the establishment of a stable are between magnesium poles in nitrogen and the peculiarity of its spectrum are caused by some special chemical reaction between this gas and the electrode metal. As is well known, magnesium forms a nitride at high temperatures: it may therefore be that the continuous emission of 14481 as a narrow line and the sharpening of the lines of the Rydberg nebulous series are an optical manifestation of the process involved in the formation of magnesium nitride. In this connexion it is useful to recall the important fact observed by Dr. de Watteville that in the air-coal gas flame the series . triplets of magnesium are emitted only in the explosion region * and, as one of us has shown, the excitation of the characteristic cone lines is most probably associated with the formation of nitrides t. No trace of the spark line 4481 or of any line of the Rydberg series has, however, been observed in the air-coal gas cone, and their emission by the nitrogen arc may possibly be brought about with the additional help of the electric current. When the obser- vations of the are in nitrogen are made at the instant of separation of the electrodes the character of the spectrum is similar to that of the first phase spectrum in air—namely, the lines of the Rydberg series are winged on the red side and the spark line 14481 is symmetrically widened. It is therefore probable that during the initial stages the light _ * C. de Watteville, Phil. Trans. Roy. Soc. A, vol. 204. p. 151 (1904). t G, A. Hemsalech, Phil. Mag. vol. xxxiv. p. 229 (1917). the Occurrence of Enhanced Lines in the Are. 845 radiations both of the are in nitrogen and of that in air are of like origin, and the explanation which we have offered for the air are will equally well bold in the case of the nitrogen are. No stable arcs are formed in oxygen, hy drogen, or coal-gas under electrical conditions similar to aheos used in air Tal nitrozen—namely, with continuous current at 80 volts. The ares consist merely of luminous flashes which show the spark lines strongly during the first phase. The prevention of a stable are in oxygen is possibly caused by the very vigorous e “ rv con) chemical actions which set in as soon as the arc is started. As. a consequence of this the electrodes become rapidly covered with a non-conducting layer of oxide which invariably leads. to the extinction of the arc. With the oxygen are between magnesium poles the flame line 4571 and the bands at X5007 are particularly strong at the beginning of the second phase. ‘he lines of the Rydberg series are winged on the red side, especially at the commencement of the are flash. The spark line 14481 is intense and symmetrically widened in the first phase, but it remains visible as a fairly strong though narrow line throughout the flash. Whether or in how an the emission of the: spark lines in the oxygen are is caused or facilitated by the process of oxidation of the magnesium metal, our present results do not allow us to. suggest. But as for the cause of emission of these lines. during the first phase—namely, whilst there is paucity of oxygen in the space between the electrodes,—it is probably the same as that in the case of the air and nitrogen ares mentioned above. A prominent feature of the hydrogen and coal gas ares between magnesium poles is the relatively o great intensity of the spark line 24481. In the first phase, which also in these two ares is probably caused by a film of rarefied gas, the line is considerably broadened out on both sides. It then suddenly falls off sharply in intensity, although still remaining strong throughout the second phase. The lines. of the Rydberg series are winged on the red side and very nebulous in appearance during the whole of the are flash. The flame line 14571 and the hydride bands are well developed. The series triplets present the usual normal appearance and, as was to be expected, the band at 5007 is absent. The hydrogen lines H, and Hg are very in-. tense during the first “phase but much fainter during the. second. The following data have been obtained for the principal, 846 Mr. G. A. Hemsalech and the Comte de Gramont on » stages in the development of an arc flash between magnesium ipoles in hydrogen :— Thickness of rarefied gas film...................+. 0:18 mm. Heights from stationary electrode: 4481... 1°54 ,, 5184... leo ies H, 6563... 0°89 ,, Thus the hydrogen lines are emitted for a considerably ‘shorter period than the metal lines. The thickness of the rarefied gas film is approximately of the same order of magnitude as that of liquid films. Measurements made on the spectrum of a blown are in air gave 0°17 mm. as the thickness of the rarefied air film. The following table contains a summary of the results of our observations on the appearance of the various types of magnesium lines as given by 80 volt arcs burning in various gases :— Air. Nitrogen, Oxygen. Hydrogen. Coai-gas. “Quality of arc...... Stable. Stable. Unstable. Unstable, Unstable. Flame line \ 4571. Normal. Normal. Normal, Normal. Normal. ‘Series triplets...... Normal. Normal. Normal. Normal. Normal. Rydberg series ... Diffuse. Well-defined Diffuse Diffuse Diffuse and prominent. andwinged. and winged. and winged. — —_~+- —— Spark line (4481. Diffuse, Relatively Strong and Very intense and traces feeble, but symmetrically symmetrically only. narrow and __ broadened in broadened in Ist phase ; sharply Ist phase; fairly narrower in 2nd phase. defined. strong but narrow in 2nd phase. § 6. Arc in liquid air. The electrodes, which consisted of metal rods, were com- pletely immersed in the liquid, so that before the are was struck they were at a temperature of about —190°C. With the current at 80 volts the discharge consisted in short flashes, each one of which was accompanied by the evolution of numerous air-bubbles caused no doubt by the momentary rise in temperature. Since the luminous yapour in this case 1s completely enclosed by liquid air its temperature cannot possibly attain a high value, and this inference is amply borne out by the relative behaviour of the flame line the Occurrence of Enhanced Lines in the Arc. 847 ef magnesium 24571, which is only feebly excited under these conditions. In stable arcs burning between the same metal in gaseous air or nitrogen and in which the tempe- rature reaches a high value, this line is more strongly emitted. The series triplets at 13838 and 5184 show a normal development in the liquid air arc, thus indicating that under are conditions they are little, if at all, affected by temperature changes. The lines of the Rydberg series are well brought out and are markedly winged on the red side throughout the length of the are flash. The spark line r 4481 is strongly emitted, not only at the moment of striking the are, but evenly along the whole path of the flash. Although at no stage of the liquid air are does this line attain the re- markable breadth which characterises it in the first phase of the water or hydrogen arc, it nevertheless shows unmistakable signs of being symmetrically winged. It may also be useful © to place on record the fact that the band at » 5007 is emitted, though only faintly, in the liquid air are. With zine electrodes in liquid air the arc flash obtained brings out the spark lines ) 4912 and 1.4924 strongly during the first phase ; they then fall off in intensity very rapidly and only traces remain visible till the end of the second phase. The red line and the triplet in the blue, which are very intense along the whole length of the are flash, appear strengthened and symmetrically widened in the first phase. Our experiments with a cadmium arc under liquid air were not very successful inasmuch as we could neither observe visually nor obtain photographically any evidence of the emission of the two spark lines 15338 and 25379. It should, however, be mentioned that the whole spectrum of cadmium as obtained by us under these conditions was only feebly developed, possibly owing to the fact that poles of this metal fuse together rather readily when bringing them into contact on starting the are. Although our obser- vations have not brought out any interesting facts bearing on the emission of the principal spark lines of cadmium, they do, however, show up in a striking manner the difference in the behaviour of the arc line 24413 and the third type enhanced line 24416. The are line, which is but very feebly emitted in the first phase, becomes strong as the are flash develops during the second phase. The enhanced line, on the other hand, is strong in the first phase and changes almost abruptly into a feeble line as the second phase sets in. Thus there is here shown up a marked dif- ference in the mode of excitation prevailing at the moment 848 Mr. G. A. Hemsalech and the Comte de Gramont on . of striking the are and during the further course of the: are flash. In order to account for the absence of any marked dif- ferences between the first and second phase spectra of magnesium under these conditions, we venture to suggest that possibly chemical actions intervene in this case. We have already shown in §5 that nitrogen is effective in bringing out the spark line 14481 and in strengthening the lines of the Rydberg series. Remembering then that the boilirg-point of nitrogen (—196°C.) is lower than that of oxygen (—183° C.), we may be justified in presuming: that when the temperature of the liquid air in the immediate- vicinity of the are rises nitrogen will be given off first, so that the first chemical reaction to start along the are flash will be one between magnesium vapour and nitrogen. Thus. the peculiar character of the magnesium spectrum as observed. — in the are under liquid air may be the result of the cooperation of the following factors :— | {. A low temperature keeping down the degree of ionization of the vapours in the are gap and in this way preventing the formation of a stable are. 2. Electric forces acting strongly during the first phase and causing emission of spark lines. Feebler electric: forces acting during second phase but unable by them- selves to keep up strong emission of spark lines. 3. A chemical reaction between nitrogen and magnesium ~ setting in at the end of the first phase and helping (in cooperation with the feebler electric forces) to continue during the second phase the emission of the spark lines initiated by the strong electric forces in the first phase. § 7. Cause of broadening of the hydrogen lines during the first phase. Stark effect. From the results of our preliminary experiments we had concluded that the emission of spark lines by the are depended upon the existence of strong electric fields, and we pointed out that for an electrode distance of 0°05 mm. the intensity of the field set up in the gap with a potential difference of only 110 volts is equal to 22,000 —— An electric field of this. Cc strength should reveal itself by a manifestation of the Stark effect. As will be remembered, the are flash is produced between a fixed lower electrode A’ and an upper movable one B, a diop of liquid (water or oil) being placed between. | * Hemsalech and de Gramont, doc. cit. §12. the Occurrence of Enhanced Lines tn the Arc. 849 them so as to retard arcing. As long as B is in contact with A on striking the are, the intensity of the field between them is of course zero. But immediately after separation as B moves upwards, a very strong electric field is set up, the intensity of which, however, decreases very rapidly as the distance between A and B increases, and more espe- ciully so as soon as arcing has set in after the destruction of the liquid film. Hence we should expect to observe the electric effect whilst the distance between B and A is small, as for example during the first phase. An image of the are flash composed of the two phases was projected upon the slit of the spectrograph in such a manner that the motion of B took place in a direction parallel to the slit. In this way it was possible to distinguish between the spectrum radiations emitted during the first phase and those given out at later stages of the are flash. The height of the spectrum thus obtained obviously depends upon the distance AB, namely the length of the are flash, and that edge of the spectrum which corresponds to the vicinity of A will be emitted whilst the intensity of the electric field is a maximum. Weshould therefore, on passing down a spectrum line from B towards A, observea broadening or decomposition of the line analogous to that obtained with Lo Surdo’s -arrangement*. The liquids which were placed in the are gap in our experi- ments contained hydrogen in combination either with oxygen or with carbon, and the spectra show the red and blue lines of the Balmer series. The red line 6563 was found to broaden considerably near the edge of the spectrum corre- sponding to electrode A. The blue line \ 4862 was, however, distinctly decomposed in the first phase, the distance between the components increasing on approaching A ; in fact, this line presented the well-known Wiffel tower-like appearance, so characteristic of the transverse electric effect, as obtained with Lo Surdo’s method. The effect is especially well developed and defined in a 200 volt are between copper or magnesium poles with water films. Itshows also in hydrogen and coal-gas. The fact that H, was only broadened and not decomposed is undoubtedly due to the low dispersion in the red of the glass-prism spectrograph used by us. With the help of Stark’s table of data for “ Grobzerlegung ” + an approximate determination of the intensity of the electric field was made from the amount of maximum separation * Lo Surdo, Rendiconti dei Lincet, xxii. p. 664 (1913). + Usabura Yoshida, Memoirs of the College of Science (Kyoto Im- perial University) vol. iv. No. 5, p. 188 (1920). Phil. Mag. S. 6. Vol. 43. No. 257. May 1922. 3 I 850 Mr. G. A. Hemsalech and the Comte de Gramont on observed for Hg in the water film arc. Thus in the case of a 200 volt water arc between magnesium poles the separation of the outer components of He at a distance of about 0:05 mm. from the stationary electrede was found to be 20°3 Angstrém units. Hence the shift of each component from the initial line was approximately 10 units. According to Stark an electric field of 10°4~x 10+ — produces a shift of about 19-4 units for the outer parallel component of Hg. Con- sequently, since the displacement is proportional to the strength of the electric field, a shift of 10 Angstrom units would correspond roughly to 50,000 — On calculating the field for a gap of 0:05 mm. with an apples potential of 200 volts we obtain 40,000 ae —a value which is of the same order of magnitude as tlat derived from the amount of separation of the components through the Stark effect. Thus it would seem that the broadening of the hydrogen lines observed during the initial stages of an are flash is caused by the strong electric field set up momentarily when the electrodes separate. § 8. Detailed results of observations. The photographic records of the spectral phenomena exhibited by electric ares under various conditions, which served as basis for our measurements and observations, were obtained in the manner already described in § 9 of our first paper. Also the numbers. and signs used for expressing relative intensities and characters are the same as those adopted there. For reasons already stated in $1 we have made a more exhaustive examination of the are spectra of magnesium. In the case of zinc and cadmium we give _ results only for the liquid air and water arc respectively, i addition to that of the ordinary are. The spectra of Hae two elements do not play an important role in astrophysical problems, and the few results we give will suffice to illustrate the general behaviour of their spark lines under are conditions. Most of the photographs were taken on Wratten’s pan- chromatic plates. A few spectra, however, were obtained on ordinary plates and they are therefore deficient beyond » 5500. In these cases the space in the tables reserved for the relative intensity of the line is left blank. Lines marked S belong to the Rydberg nebulous series, and those marked H to Paschen’s series. the Occurrence of Enhanced Lines in the Are. 851 Taste I.—Magnesium. a. Quartz spectrograph. 22150-23200. Observations were made on the spectra of the direct are, water arc, and capacity spark. The water are brings out a series of hazy lines, all well visible and of which the least refrangible one is even fairly strong. These lines had pre- viously been observed only by Kayser and Runge as very feeble lines in the arc. They are absent from the spark. dA 2605°3 Fairly well visible. _) _ 2680-4 Well visible. | These lines form part of the Ist nebulous series of triplets; no member of the 2nd series has been observed in this part of the spectrum. 2633'0 Very well visible. 2668'2 Rather feeble. 26728 Well visible. r 2732°3 Feeble. | 2733°7 Fairly strong. 2736°7 Fairly strong. ) \ 2777 to X 2784 A group of lines identical in water arc and spark ; in direct ordinary arc, lines are widened and A 2780 is reversed. NA 2795 } 2802 f Reversed in direct are and spark, but not in water are. 2852 Doubly reversed in direct are ; simply reversed in water are. Bie Of equal appearance in the three spectra; but the halos 2937 are less marked in water arc. 3091 3093 3096 Same in the three sources. Same in water arc and spark; reversed in direct arc. b. Uviol spectrograph. 2» 3170-5000. AA 33530 3332 $ Intense lines; same in the three sources. 3337 3722 Strong and very hazy in Ss are; wot given in any tables. ae well seen in direct are and | iwol of Aleit Ghumee 3726 | Fairly strong and very hazy in Hens ‘s ee Se oe | water arc; very feeble in direct | Se taate y 3732 ) are and spark. ) ie 3829 Strong and very hazy ; same in the three sources. 3832 | Intense and very hazy; same in water are and spark; 3858 reversed in direct arc. 3890 | 3892 } Fairly strong and hazy in the three sources. 3896 4058 \ Strong and sharp in water arc; hazy and feeble in direct are 4168 and spark. 4352 Stronger in water arc than in direct are or spark. 4481 Very strong in water arc; very feeble in direct arc; intense and expanded in spark. 4571 Very strong in water arc; only feebly brought out in direct are and spark. 4703 Of identical appearance in the three sources. The green band at 15007 has come out only in the water arc. ol 2 852 Mr. G. A. Hemsalech and the Comte de Gramont on 98 “yuo "que “que “que “yuo “qua “yuo “SyAVuleIy = t JU loa tos 0-69EP 1qUo F 1quo F Tui aqug J ud OT aqua | aque G uo 0 —~ -—- 3 | P-SC6h ——-= acer uud F ud 000 Ud O00 OSGT baat ee, ud () Sa L-€PGP ‘purq uogavd Jo Manaita pout uo } pees eae ee Oe aqua ¢ 9.LO1F atte ud 0 ud OO ia) G Iquoa | uo § €.QCQ0F a 2 00 W) OO G:810F —— aS Sass U0 G uo 0 uuo QO 6-986 900 aa Sises oe: L:GG6§ Ee aes ee SSS ud G aS | 0000 Z-8688 uo ud G wo G 80 of oF L-G68§ TT uu y | : aes pur é { : L-F68E ud 00) uo + 80 o¢ ss S-S688 ud G ud G wits 89 G %G 20 0-G68§ uot : wd T ud % a 2 IL 2 00 F-068E ‘spueq O Aq ud T ear” 6-986 peyseur uuo & ull ZG ag 149 ¢ UDG P-FS8E uud 7 wud T[ uud | a) iy aD) Tl uo & G.6FRE gare ae 20 ae 6-L48S an ees ——— 94 20 5 es 9-9PF8E ae PPO €-GP8S IO CT ID 0G Id CT 20% 906 2 cl G-8E8E 19 OT 10 GI 1d OL ICL 2GL 9 OL G-GE8E 19 Q 10 OL ID Q 9 Of 9 OT | ors F-6686 . : ae *prbry "SnLOoasR.y sai “SVD-1VOO “NUDOUGAFT NQDAXO NVA OIUUN ee ee WoT ; ul er ma liteee sey SYSUeT Bane aEG nT IS es OS ee Se ee eee es -OAG AA ‘ILy pinbry pur sesexyy ut sefod Syy usemgoq 04W 4104-98 ‘ydeadoroods ssvpy “O00L X-O009¢ X | | | | 853 the Are. ines in f Enhanced L . the Occurrence o GH 9H ‘pug Jo ospo pou "pueq Jo adpa pou PS ‘purq jo espo pou |‘pueq jo ape pax |‘pueq Jo espo pa. 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PI 2 00 08 Heese nea —-- of 9.088h ; TI Ut aqu g II Ul aqu F 99g , zquo QT | aqua 9 1quo F ] ui uu & \ ] Ul ua IYyUo G 0.66SP | | P 000 | eis ar cae uo 000 OS3F | ; % eypuene! /) eS igusT ; LS 1quo G | quad g HOG Aq poyseut [ ul uu Et Iquo & 9-L91P I gg uo & 1d ud 0 ee _ seca soo uo ¢ &-890P ud 00 | 2 000 —— eo a are 6-8LOP 6s 19.00 w9 0 a eee ee see wud OO 6 9866 ee —— PO = Sans 0-968 “que PT 2 000 up % up ? pt 9 000 -8688 Mee P9 QE up's up & PS oer L-G68€ pz —- s = G:S68E “yuo PF 20 up £7. up <1 PG 90 | 0-688 ‘quo PG 200 up T up T PI 2 00 | B-068E — — uu & ane te \ aes aaa &-9986 “ue TU aq ‘pF uo | noe oO | ee oT \ aquo ¢ ud Z F-PGGE "que JT Ul 1g ‘PZ ud 9 uuo i pease uud & uo $ ¢-6F8E ei ud 000 9-978 J Ul ard 9 0G ID 0G Id GT ID CT 10 0G eet G:883& J Ut arg “9 GT 19 GT 19 OT 19 OL 10 GT 2 01 G-GE8E JT Ut aaq “ OT 19 OT Id Q T ura ‘0g J Urs 9 OT 2g F-6688 g “uleafT ‘Tlo uve "1948 AA ste RCH are) ; : : Wedd jo doap Wty ore AIVDIPIC ee ack cee : S eae <4 ore ATVUIPAO ur "syle Wry, o doap yt SY}SUOT “i Mee EELS a ee Ue Gene Pages ee —_—__ ———--—/| -0AeM “$3704 006 "89104 08 ‘sajod bry waaayag spunby Huyonpuoo-uouw ybnowy) 80up7 855 in the Are. nes f Enhanced L the Occurrence o "H VS pusq JO aspa pat pueq jo espa pea Gs “qu “eulT OUT "yuo J Ul aq ‘Up QO [I Ul aq ‘p F J Ut aq *9 OT PG uu PP T UPPrT OPP T UPPG Bali Gil 2 ol 2 OL UY Iu q'pt Iquo QT URP 00 é so 0. uu p QO azqP Of uu pp 00 uud 000 des ule epous jv 9 epoyyevo qu F o¢T so ¢ epoue ye uu & ) rin aqua OT ———s a 80 9 dv ur ue9es jou Ppoyyeo 48 YOO epour 4v Og —-— | J ! II 4100 | Tuo J 9 BI 9 OT a}, aS ud 0) oF ax? GT 2 000 —- . TI ur g | I Wf uu 99 | eee J urug 9g a 99 II MT oe ‘spueqQ | hq eee II ™ OT | i" ees aqua I“F | nang. | aaqPp OG uu pp 0 uu pp 0 | TI ut uu II wug | UL UU F 199 0G — ee i> eo) T ur uu TI url uu cata “6 TT J ul ug ITUy LULg OST 26) 9 Ol IT wl [Ure —— 11 #00) qT ul SIE“ OL I 4 0Z uu ppzZ ITM 1] Ture aiqdp OG ‘ = 60 C = epour 4 | apouyed ye | des ur uo 816g 60169 9-81¢ L-GLIG €-L9O1g €-400¢ G:O&LP 0-G0LV IS9P L-$8¢P L-TLSP 8-O1GP 856 Mr. G. A. Hemsalech and the Comte de Gramont on “que S—— ‘ques _—_ “qua, a “yuo oO it) (Scale 4= ay aes oOo S) = 20 ) x "x x va alter a J x =(Scale 1- 10) eS and Poisson’s Ratio for Spruce. 877 the points lie sufficiently near to the straight lines except those corresponding with oz, and oy, but it should be observed that these values were obtained from pieces which were cut with their thicknesses in the direction XOX, that is in the direction of the widths of the rings. The thicknesses of the pieces were thus often comparable with widths of the rings, so that the ratio of Autumn wood to Spring wood usually varied from piece to piece, Fig. 3. Sy Ad Sx Sx LS re Relation between F2y, Fz,oxz, TzX, = yx8 ray and the Density Dry. o o OxXZ y . / / / q Cet N / 59 re) 19 Xe) Density ny { Ibs perco FE ) ; Density dry (Ibs per cu fr) It is significant to note in connexion with these curves that they all obey the symmetrical relations of n, an solo- tropic body having three planes of elastic symmetry *. These relations are : Oyz OFzy Sze Fxz Cry Tyx ee one Be. Hs. By! From an examination of the curves it will be found that approximately these equalities are equal respectively to 700 Sy, 700 Sp and ae - | The degree of accuracy aimed at throughout all the experiments was an error of less than one per cent. It should, however, be noted that in all cases except where the length of the pieces was in the direction of the grain, incipient creeping was evidenced at comparatively small * Love’s Elasticity, Art. 738. x 2x : oe 878 Dr. C. Davison on the Diucer strains. It was therefore decided to load all the specimens at such a rate that the rate of longitudinal straining on surfaces 4 inch from the neutral surface was approximately at the rate of 1 in 4000 per minute. The author wishes to thank the former Principal J. C. M. Garnett, M.A., D.Sc., for providing facilities for conducting the research, and Professor Gerald Stoney, F.R.S., for the interest he has shown throughout. XCVI. The Diurnal Periodicity of Harthquakes. By CHarues Davison, Sc.D.* hie present paper may be regarded as a sequel to two earlier papers dealing with the same subject. In 1896, I applied the method of harmonic analysis to seismographic records obtained in Japan, the Philippine Islands, and Italy J. The results showed a di-tinet diurnal period, with its maximum- epoch about noon, except in two of the nine Italian records when the maximum-epoch occurred about midnight. In the early after-shocks of Japanese earthquakes, the maximum- epoch was also about or shortly after midnight. Returning recently to the subject {, I was able to avail myself of more numerous and extensive records of Japanese earthquakes, and found, by the method of overlapping means §, that, in ordinary Japanese earthquakes, the maximum-epoch of the diurnal period occurred at or shortly before noon, and that of the semi-diurnal period at 8 or 9 a.m. and pM. ; while, in after-shocks, the epoch of the diurnal period was at first about or shortly after midnight, returning later to the neighbourhood of noon, | 7 Since the latter paper was written, I have applied the same method to the registers collected by the Seismological Committee of the British Association ||. As the instruments providing these registers are capable of recording earthquakes from very distant regions, it might be expected that, if the maximum-epoch at every place were to occur at a constant local time, the result would be a marked diminution in the amplitude of the diurnal periodicity. In some registers this may be the case; the amplitude is so small, or the variations in the twelve-hourly means are so irregular, that the existence * Communicated by the Author. + Phil. Mag. vol. xlii. pp. 463-476 (1896), t Loe. cit. vol. xli, pp. 903-916 (1921). § Boll. Soc. Stsm. Ital. vol. iv. pp. 89-100 (1898). See also “A Manual of Seismology ” (Camb. Univ. Press, 1921), pp. 185-188. || As a rule, the registers at my disposal end with the year 1912, Perrodicity of Earthquakes, 879 of the periodicity is uncertain. In others, the period is very distinct, the amplitude of the diurnal period at Trinidad and Christehurch (N.Z.) being about 80 per cent. of the mean hourly number of earthquakes. In the following Tables I have included only those results which seem to me free from donbt. | In Tables I. and II. the time at which the preliminary tremors were first recorded is taken as the time of occurrence of the earthquake. The maximum-epoch is thus retarded, but by an amount that is certainly less than the time required by the tremors to travel from the most distant origin. In measuring the amplitude, the mean hourly number of earth- quakes is taken as unity. In the column headed ,/z/n, n is the number of earthquakes included in the record. Unless the amplitude obtained from isolated records exceeds the value of /7/n, the result, as Prof. A. Schuster has shown *, must be regarded as doubtful. If, however, the epochs obtained for the six winter months (October to March, in the northern hemisphere) agree with those for the six summer months (April to September), the doubt is lessened or removed, even if the amplitudes should fall below the corre- sponding values of /a/n. Diurnal Seismic Period. In Table I. are given the periodic elements deduced trom 17 records divided into three groups. Those obtained from the catalogue of local earthquakes at Tokyo are added for comparison. It will be seen that, with one exception (Baltimore), there is a close agreement in the epochs for the winter and semmer months. ; Thus, in nine records, the maximum-epoch of the diurnal period occurs about noon ; in three, about midnight ; and in five others (three of them in India) early in the afternoon. At Baltimore, the maximum-epoch for the winter months occurs at 1 A.M., and for the summer months at 14 p.m. Other examples of this reversal will be given later. Semi-Diurnal Seismic Period. The number of records giving satisfactory results for the semi-diurnal period is small. In those which are excluded here, the amplitude is usually close to the value of 7/n and the series of six-hourly means vary somewhat irregularly. The results obtained for local earthquakes at Tokyo are added for comparison. (Table IT.) * Roy. Soc. Proc. vol. xi. pp. 455-465 (1897). Dr. G. Davison on the Diurnal 880 Vi “TOUTUUNG AAA 6,81 o, aia oO m= ao oD uoou ——a ‘yood a "LOJUT AA "UpTUL “eT 2) & ETT ‘d 8] ‘yooda —-—— ‘jduay Wee "aeak OfOt AA *pollod oluIsleG [vuIniq—'] @1aVv J, seceebhraseeeh (Cr7onr) Goan Ossrnuy) se eceervernereeusoseoeeeeee ny[njouo HL sVlpBl Ceo cccveeeeeeeerssvcessseoveen Aeq uog Seo reereeeereserso ooo rseoesses BINOTeY) see reese eres seeeeoseoneesevarse *(orrqudery aurquosay) GAOPION Son Btn aos axoung[eq opusuiag urg seeecees eee eeeeeereeresceetee oe rrereeseeeeesee odorT poo jo ody 0}U0AOT, Sa eg are ee Ae OC 6) ROE D meeeseeee(sovenbyydva [BOOT) *" Seetneeettnes seeeeeteecereeerns OLVOT, eeneneeereecanecteeeeeeerers BARTER BY}9]8 A Agistvg MO Spits eee ee eeeerooresreeoseorstoorscoe eeeeeeseeesorsese 'ecevesseo0ae Pears ores seo eFerseeeeosesevraee ee oreo F FF eoeoeer eer FFeencereovsee eeeeere so BSGeoeseeeeeeceoresseones "p1000%y — oe 881 Periodicity of Earthquakes. 60: UG I 10. 66: tI ee] = Ne 6 ol OF lf C= |= 290: t 80 FI: Ol ral OL. 6 we | [day [wap wy | ‘qooday eo—— se ee oo CN a ed —_—_—— —— ‘190 qung 80. , IF: 6 90: LO. 08: tI CO. LO 08 16 80 SI Gs 01 60 CO. SI T FO: 80. Lo | él 90: | Il: Lo: 6 80- | "u/% N ‘dary owe pV “w/a Sp ‘qood nay Sa Nee ee Se ze Se tate eee “TO PUT AA la (AS g eet: tI 1G: II Sites pean Le <0: £8 hes SOT raat 1G: 6 ‘jduy wa 9 KV ‘ood — ——— ‘avad OTOU AA "poldoag OlWISIog [vUANI(]-1Wlog—]] W1dVY, CP SO en ee ere ae RAO plop Aube alee dees | OpuLutay UBg ** exowIy Ug, vip teS matey ements: pepruay, (soyeunbtyjaea ;va07) xf oOYOT, | DOU REDON GOSEUE RuocGhods VyVO[RA ‘plodaryp dL Phil. Mag. 8. 6. Vol. 43. No. 257. May 1922. 882 Dr. ©. Davison on the Diurnal Thus, in four records, the maximum-epoch of the semi- diurnal period occurs at about 9 to 11 A.M. and P.m.; and, in two, at about 14 to 3 a.m. and p.m. In the latter, the maximum-epoch of the diurnal period occurs at about mid- night. Diurnal Periodicity in relation to Intensity. If the origin of seismic periodicity be due to causes which precipitate, rather than prodyce, the occurrence of earth- quakes, we should expect to find the periodicity more marked with weak, than with strong, earthquakes. In many of the records considered in this paper, it is possible to make a rough classification of the earthquakes according to intensity. The earthquakes in the diagrams of which a maximum can be clearly distinguished are not of necessity strong, but among thém are included all strong earthquakes. Those which contain no marked maximum are, as a rule, of much Jess intensity. In Tables III. and LV. the lines marked A give the results for the maximum-epochs of the earthquakes recorded, those marked B give the results for the initial tremors when no maximum can be detected. The following results may be deduced from Table II]. :— (i.) With one exception (Victoria, B.C.) the amplitude of the diurnal period is much greater in the weak, than in the strong, earthquakes. Gi.) In the strong earthquakes, the maximum-epoch occurs about noon at Shide, Kew, Edinburgh, San Fernando, and Cape of Good Hope ; about midnight at Toronto and Vic- toria, B.C. ; and in the afternoon at Cordova. The midnight maximum at Shide in summer may be accidental, as the amplitude is not much in excess of the expectancy. (iii.) In the weak earthquakes, the maximum-epoch occurs about noon at Shide, Kew, Toronto, Victoria, B.C., and Cape of Good Hope; about midnight at San Fernando and Cordova ; at Edinburgh, it occurs about noon in winter and shortly before midnight in summer. (iv.) There is thus an inversion of epoch in strong and weak earthquakes at San Fernando, Toronto, and Victoria, B.C. ; a partial inversion at Edinburgh and Cordova; and a general correspondence in epoch at Shide, Kew, and the Cape of Good Hope. ey ee ee ee eS | | | GI: Gls atl Ol | 20-1 270 80. 06: t £0 Cesc: lh Og. 3 die “ae ii = Me dg 60: ie ig ly ee eee eee eee ee | eee eee Sl. GG: uUuoou ane sod . ad OT. 9T- dy Ole 4 6 dy NO | i eee ee eS IT. Sia hee Ole Bike ‘eT 10: LI- "VOL LD bien nti : iL eC i | ee | gga | en: Iie | eae ey ee Se Il. 98: d Z0."qe Oe LZ. ‘d§ 10: 08: “dT al} AeiaLsoot ocsoda GH = II: 91: ‘d TT qv II: OI: e TO “qe 80: oe ‘2 #0 '4e | y : i) 3 T- Ge |e ee ie Eg. elf 80. bd WL [Le opusuzeg uy S 60: 60: “eTT 60: a VTL qe 90: IT. ‘eit |y GI- 98. ‘dT ST: 6I- TT 60- OI: sO) Cc fal ae caer ee 5) = 90. Ce Se iil 10. iM ff ators tare |p) 80. eV TT a eee 8 OT. 6F- woou ac IF. ‘dg 10. CF. d{ Eaten cccss bes 3 60: LT. woos 60: LT hg 10: 91: ‘dq‘qe | y = | = S OL: IF. dT OL: 66 «| “d&o 10: 6S: LET <5) bs aeiaiee ete Ry oo = gO: vy G0. 80: | woou ‘qu £0: CO. Serie salva as ‘u/ N qdury =| ‘yooda “u/2 | ‘jdury ‘qoody “u/x N ‘jdory ‘woody | *plooayy ——— = | ~-—_______- | a ‘jematung "TOJUL AA "avad OTOU AA "AQISUOJUT 0} WOTyVjor UT poled o1isieg [wuInIq—']]|[ TIAVI, 3) Ly De. ©. Davison on the Diurnal 854 GL- Lc. il OL- SI 1z 79 al Ol. FI: ral OL: II. LI G OL: II- Ol: 6 II- IT- 90): ral OT- IT: GL tP IT OL: QT. Ti OL: II- 2 ae II. 60 80: aac 60: "u/2 NV qduy wa piney) “u/u A ‘yooday em SS Hh Ss WV Se I) SS “CoCUTUNg Gye o8 80: FI. ¢ 60: a el. alles | 6 LO: IG: aI 10: LI: TP 80: LG TG L0- GI: aig 80: va a CT. 90: G qe LO: a Bees 80- CO SS ay 90- ‘jduary | ‘Wd 9 ‘IN’ ‘uj N ‘qoodsy Sa ~-——— -- ————_ — | — "LOJUL AA, ——_— + "1val OTOL AA Tg. _& T+ Tg SI. a 80: ol VI- ah | 80: TV I. 16 OT: TP 61 Ge 30: TG FG. tI 90: TS ‘duty =| (N'd Wp ‘N'Y ‘qooday —_ “am <9 em. The next atom, lithium, has been investigated by Aston * and found to have two isotopes, the one with mass 6 and the other with mass 7. The accuracy claimed for these vaiues is within about one part in one thousand. The isotope of mass 7, Li-No. 2 in fig. 1, is represented as a ring or hexagon, in which the positive individuals are arranged in an equilateral triangle, each having two electrons, one on either pole as in the hydrogen model or in the alpha particle. There are also three connecting electrons to hold the system together located in the same circle with the positive charges, that is on the equator. These electrons approach very close to but do not touch the adjacent electrons occupying planes above and below them. They are each attracted toward the doublet because it lias an effective charge. of 3—2=1. The mass of this isotope is exactly 3 oem 7, if according to Table I. the mass of the positive 3 charge of three is 7/3. The other isotope of Li, namely Li-No. 1 in fig. 1, has but asingle positive charge of 5 units t situated in the centre of the only doublet, and is surrounded by an equatorial ring of three single electrons. The mass of the charge of 5 is 6 according to Table I., so that this form satisfies this requirement. The official weight given for Li is 6°94, indicating that the isotope No. 2 is of the more frequent occurrence. The next atom, glucinum also called beryllium, has not ‘yet been investigated experimentally for isotopes, so far as known to the writer, but its official weight, 9°1, indicates that isotopes are probable. The scheme followed here enables one to construct possible forms of the isotopes of Gl, and it will prove of interest if the masses corresponding to the forms shown are later obtained experimentally. In * Phil. Mag., May and November 1920. 1 See Note, p. 894. 890 Dr. A. C. Crehore on analogy with Li one may suppose that two isotopes of Gl are as in fig. 1, G11 and ,G13, both being based upon a square instead of a triangle form. This, however, makes the masses 9 and 91, the latter being a value with a large fraction. If the positive charges of three units in this form are changed into two twos and two fours, two hydrogen atoms being on one diagonal and two alpha particles on the other, as in G12 fig. 1, the mass is 10°016 instead of 94. These three forms are shown tentatively for Gl. The atom of boron* is known to have two isotopes of masses 10 and 11 respectively. The form, B-No. 2, of mass 1] is obtained by the addition of one alpha particle and two equatorial electrons to the second isotope of Li, giving the mass 7+4=11. | There is an opportunity to add two more such alpha particles and connecting electrons around the Li-No. 2 hexagon. The addition of one makes the mass 15, and corresponds to no known element, but the addition of two makes the mass 19 and is represented by the very sym- metrical assemblage shown at F fig. 1, as fluorine. There are no isotopes of fluorine as yet found. A possible configuration for boron is shown as B—No. 1 in fig. 1, consisting of the Li hexagon to which three hydrogen atoms are attached, thus forming a very sym- metrical figure of mass 10°024. In this connexion it is interesting to note that nitrogen and these less stable isotopes of boron and glucinum are the only atoms shown in fig. 1 that have hydrogen as one con- stituent. Rutherford has succeeded in obtaining hydrogen from nitrogen in the gaseous form ; but, in trying several compounds of nitrogen in the solid form, he noted that boron nitride and para-cyanogen gave between 1:5 and 2 times the number of hydrogen atoms to be expected. The experimental result is in agreement with the supposition that hydrogen may have been obtained from this isotope No. 1 of boron, as has been recognized by Rutherford, who says: “In the case of boron nitride there is also the un- certainty whether boron itself emits H atoms.” According to this form of model the question is answered in the affirmative, that boron does contain hydrogen. The atom of carbon takes a very symmetrical form, con- sisting of a ring of 9 having three positive units of four each in equilaterai triangle formation. The mass is thus exactly 12, there being no isotopes; but an important * Loe. ett. Atoms and Molecules. 891 difference between carbon and the ‘preceding atoms first makes its appearance in that there are two connecting electrons instead of a single one between the doublets con- taining the positive individuals. The connecting electrons do not remain in the equatorial plane exactly, but are staggered the one up and the next down alternating all the way around. This feature is not shown in the model. The oxygen atom is very similar to carbon, but differs in being based upon a square of positive charges instead of an equilateral triangle. There are also double connecting electrons as in carbon. The mass is exactly 16, there being no isotopes. The only remaining atom in the second row of the periodic table is nitrogen. This differs materially from the ring form of carbon and oxygen, and has an alpha particle at its exact centre surrounded by a hexagon of single electrons, which are supposed to be staggered alternately up and down out of the equatorial plane. The circumference outside of this hexagon contains two alpha particles and two hydrogen atoms, and the whole figure has two planes of symmetry. The mass is 3x 4=12 plus 2x 1:008=2-016, making a total of 14-016, thus agreeing with remarkable accuracy with the official weight of nitrogen, 14:01, which has been determined with as much or more care than any other element. The next element, neon, has two isotopes at least, one of mass 20 and the other 22. The form, Ne-No. 1 fig. 1, is thought to be made by the addition of a helium atom in the centre of the oxygen ring, the ee alpha particles cee a mass 20. The isotope of this, 22, not shown in fig. ], thought to consist of the same ee to which is added i hydrogen atoms, one on either pole of the central helium atom, thus making a pile of three coaxial atoms in the centre of the oxygen ring, one helium atom and two hydrogen atoms. The next element. sodium, has no isotopes as yet found and an exact weight of 23. Tt falls under Li in the periodic table, and is thought to consist of the hexagon isotope of Li of mass 7 within the oxygen ring of mass 16, thus making up the total mass 23. This is as far as we shall attempt to go at the present time, although considerable interest attaches to the elements imme- diately following, Mg, Al, Si, P, S, and Cl, to complete the third row in the periodic table. Of these, Mg, Al, Si, and Cl are known to have isotopes. The work of Rutherford in the disintegration of nitrogen and oxygen will immediately occur to the reader, because 892 Dr. A. C. Crehore on he has apparently found that one of the products of dis- integration of both nitrogen and oxygen is a new atom having a mass of three. Since there is no provisicn in the above plan for any atom with mass 3, it seems to be required that the experimental result of Rutherford be reconciled with the views here expressed. And, accordingly, a close study of this question has been made.. When the two hydrogen atoms are knocked off from the nitrogen atom, the whole atom no doubt receives such a shock that some of the electrons are driven away at the same time. Those most easily dislodged are the ones con- cerned in the process of light radiation, namely one of the pair composing the doublets in contact, that is, with the positive Fig. 3. e O Fig. 2.—A possible residue when the nitrogen atom is disintegrated. Fie. 3.—A possible residue when the oxygen atom is disintegrated. This isthe same combination of charges asin fig. 2,and no doubt re- arranges itself into a different form, possibly that of fig. 2. charge, as is shown in a subsequent paper on radiation. It seems probable that what is left of the nitrogen atom after the loss of the hydrogen may be the collection of charges shown in fig. 2, consisting of three positive individuals of charge 4 each together with seven binding electrons. There are no doublets in fig. 2 as in the models, the positive charges being attached at the end of the minor axes of single electrons as in the hydrogen ion, which on this view has one electron and one positive charge of two units. The total charge of Atoms and Molecules. | 893 the residual atom is, therefore, 12—7=5 units, and the total mass is 12. In considering the breaking up of the oxygen-ring it may be supposed that it is cut in two along the line OO’ indicated in fig.3. This will leave a portion which, however, will have lost one electron from each doublet, and thus the number of electrons left is 7 and the positive charge is 12 and mass 12, namely, exactly the same combination as obtained from nitrogen above. The geometrical form cannot remain as a portion of a ring on account of the repulsion of the two ends, and seems likely to adjust itself to exactly the same more stable form of fig. 2. In going over the calculations of Rutherford * undertaken for the purpose of finding the mass of the unknown atom from his observations, it is found that he has begun with an assumption not experimentally well substantiated, that the unknown atom carried a charge of two units. If, for the moment, it be granted that this charge might possibly have been five units, then the ratio of the charge E! of the unknown atom to the known charge H=2 of the alpha particle, which ratio now becomes 5/2 instead of unity, will have to be carried all through the subsequent work. Doing this results in the equation . I MVE = 125 mak, instead of his equation (1). Combining this with his formula (2), which will remain unchanged, results in the values for the unknown atom: igi aa oll ory Ue OFT6 Vi. and z = 9cm.= the observed range. It does not seem a safe procedure to base deductions upon the original energy of the alpha particle and assume that this is the only energy available because, if the energy is sufficient to break up the atom, there becomes available another source of energy which must affect the velocities of the parts of the atom, the residue, and this energy is now unknown. ‘Thus the unknown mass comes out 12 instead of 3 and is in agreement with the conception that this unknown atom is the residue from both nitrogen and oxygen, having a charge of Sunits. The percentage 121/120 is a little closer agreement than that found by Rutherford as 31/30, though this is of little moment. If the above turns out to be a * Bakerian Lecture, 1920, Proc. Roy. Soe. ser. A, vol. xevii. No. 4 686, p. 390. 894 Dr. A. C. Crehore on possible interpretation to place upon the experimental obser- vations of Rutherford, it shows that the unknown atom may have been that of fig. 2 obtained from both nitrogen and oxygen during its disintegration. The exact agreement between Rutherford’s result for nitrogen and the mode} shown for N is truly astonishing when it is borne in mind that he has found experimentally that the distance from the centre of the N atom to the centre of the H atoms in nitrogen is just two diameters of the electron itself. The reason why there has been found to be a critical experimental distance within which the law of scattering suddenly changes is made apparent by this theory. Also the reason why the hydrogen atoms are driven ahead without much scattering within this critical distance is because two electrons, the one in the hydrogen atom and the other in the alpha particle, come inte actual collision and cannot inter- penetrate each other. : Note added February 1922. Since the manuscript of this paper was submitted in August 1921 experimental results of importance have been published. A rule has been adopted for determining the atomic number of the atom from the models which necessitates some changes in fig. 1. The rule may be stated as follows:—the atomic number is equal to the total number of connecting electrons in the atom, provided one is counted for hydro- gen and two for helium, and that one be counted in the more complex atoms when a single doublet, that is either a hydrogen atom or an alpha- particle, occupies the exact centre of the atom, and that two is counted when a helium atom occupies the centre. According to this rule the models shown as Li-], Gl-] and B-1 in fig. 1 do not have the proper atomic numbers for these elements, but all of the others satisfy this rule. The charges of 5e and 6e with masses of 6 and 9 respectively in accordance with Table I. are thus never required. A new model for Li-]1 having perfect symmetry about its centre is shown in fig. 1a, having mass 4+2x1:008=6:016, and Fig. la. Becee atomic number three according to rule because there are two connecting electrons and an alpha-particle at the centre for which one is to ke counted, making three. The isotope Li-2 of fig. 1 may remain as the principal isotope of Li having mass 7, for it satisfies the rule giving atomic number three. G. P. Thomson * reports glucinum as an atom of mass 9, and states * Phil. Mag. Nov. 1921, p. 859. Atoms and Molecules. 895 that if there is an isotope of mass 10 or 11 its existence is not established by his investigation. A new model for Gl is shown in fig. 14, this Fig. 1 6. being simply the octagon ring shown at Gl-2, fig. 1 with one of the hydrogen atoms missing. The atomic number is four according to rule and mass 2X44 1° 008= 9008. The form shown for boron at B-2 fig. 1 gives mass 11 rad atomic number 5 according to rule, but it may be rejected because of its unsymmetrical form, and also ‘because boron probably contains hydrogen as recently obsery ed by Rutherford. The preferred form for boron is the octagon ring Gl-2 of fig. 1 in which a single hydrogen atom is inserted at the exact centre. This raises the mass from 10-016 to 11-024 and the atomic number from 4 to 5 according to rule. It supplies an atom rich in hydrogen, containing three H- -atoms, and is sym- metrical. The atomic weight of boron has recently been revised from 11:0 down to 10-9 in the latest International Table since the work of Aston on isotopes. It seems possible that the peo pot 10 found by Aston may be the octagon form shown as Gl-2 1 having mass 10016, which may enter the boron as an impurity. Ie atomic number is 4 rather than 5 and should be classed as Gl, though it is very similar to the boron atom. Rutherford has recently obtained great range particles from B, N, F, Na, Al, and P. He does not state that all of these particles ‘are hydrogen atoms. In the models of fig. 1 there is no hydrogen in F and Na, but there are particles of mass 22 which must behave much like hydrogen if they exist. If they carry a charge of 2e when liberated from the atom, the value of e/m would be 9/2. as compared with that for hydrogen. 1/1008, which would make it more difficult to distinguish them from hydrogen than it would be if they carry but a single charge. It seems important to ascertain experimentally whether or not all these great range particles are hydrogen. The atoms of magnesium and aluminium may be added to the list above given. There “have been reported isotopes of magnesium having masses 24, 25, and 26, the greater number having mass OA. This ort may consist of an inner ring of two alpha-particles and four connecting 896 Dr. A. C. Crehore on electrons around which is an outer ring like oxygen consisting of four alpha-particles and eight connecting electrons. The total mass is thus 6x 4=24, and atomic number 44+8=12. The isotopes of 25 and 26 are obtainable by inserting a hydrogen atom into the inner ring as in Gl fig. 1 6, and as in Gl-2 fig. 1. These additions do not affect the atomic number, which is 12 for all three isotopes. Aluminium is based upon the boron atom just described to which the oxygen atom is added. The mass is thus 11:024+16=27-024, and the atomic number 54+8=18 as it should be. The experimenta! isotopes of Al have not been reported as yet. One and two hydrogen atoms may be added to these rings without affecting the atomic number. Thus the normal Al atom of mass 27:024 is rich in hydrogen, which is not present in magnesium, and the extreme range of 90 cm. for the particles from Al indicates the presence of hydrogen. As in the boron atom the diameter through these hydrogen atoms is a line of weakness because they are neutral atoms, so here this diameter is proportionally weaker. It seems that the bombarding alpha-particles may easily break this ring into two parts along this line of weakness and thus liberate hydrogen with considerable energy in agreement with the observed facts. Part IT. Computation of the Distances between the Centres o the Atoms forming Molecules. 1, General Considerations. It goes without saying that all of the computations required for the multitudinous combinations known to the chemist of the atoms shown in fig. 1 have not been completed. But the general formule to which they are subject have been developed, and their application to these forms will afford a satisfactory test of these ideas. It seems best to have in mind concrete forms with the understanding that the exact positions of the electrons are subject to investigation. Certain combinations have been calculated, and so far as thev go the results are in agreement with the known facts. Since the discovery by Laue of the diffraction patterns produced when X-rays pass through a crystal, the distances between the centres of the atoms in many crystals have been accurately measured. There are not many dimensions in the whole atomic realm in which one has such great con- fidence as the distances between the centres of the atoms in a crystal. The remarkable thing is that they are all of substantially the same order of magnitude, equal to a small number times 10-* cm. for all known solids and liquids differing widely in atomic weight and number. The greatest known distance between the nearest atoms in any solid is probably within a factor of ten times the least known distance Atoms and Molecules. 897 for any other solid. In the case of such elements as aluminium and lead differing greatly in atomic weight, the edge of the elementary cube lattice is about the same—4 07 for Al as eompared with 4°91 for Pb. These distances might conceivably have been many times less than they are, and possibly greater, since the dimensions of the electron above given are of the order of 107 cm., 100,000 times smaller. The distances between atoms might easily have been 10-*, 107'°, or 107" cm. unless there is some fundamental cause regulating it that applies to all atoms in common. These considerations lead directly to the thought constituting the theme of this paper, namely, that. the general cause contributing to an equilibrium distance in all solids is connected with the properties of the positive and negative electrons themselves common to all atoms. Minor variations in this distance only are brought about by the special configurations of these electrons in the atoms, as in the models. It is shown in this paper that the non-spherical shape of the negative electron is the principal cause of this particular value, 10~§ em., which is known to be confined within such narrow limits. Not much is known with the same degree of certainty about the distances between the centres of the atoms in the chemical molecules consisting of but a few atoms, but the presumption is that the order of magnitude is the same as in the crystal molecule, the whole crystal being regarded as one molecule, and that the fundamental cause operating to hold these atoms apart is no different in the one case from the other. Tt is still an outstanding question what is the nature of the forces holding atoms at a distance from each other in equilibrium at all, and especially at such uniform distances. Modern atomic theories have postulated many things which depart from electromagnetic theory in their fundamentals in order to obtain some cause for holding two atoms together, as in the diatomic molecule, for example. The Bohr form of theory sets electromagnetic theory aside at the outset in postulating electrons moving in orbits at such speeds that they must lose energy by radiation. The Lewis-Langmuir theory, as extended by Langmuir, does the same thing “when it requires an oscillation of an electron back and forth in any form of are or curve. Without some motion of this kind in the latter theory the system loses stability ; their diatomic molecule becomes a system of four point charges in Phil. Mag. 8. 6. Vol. 43. No. 257. May 1922. 3M 898 Dr. A. C. Crehore on effect with two electrons, say, held in common between the two atoms, and the nucleus of each has a single charge. Such a system of point charges without motion of some kind does not form a stable electrostatic system. All such proposals depart in their fundamental assumptions from any form of electromagnetic theory. The investigation in this paper assumes in dealing with the atomic models shown in fig. 1 that. electromagnetic theory is adhered to, there being no exceptions. In view of these circumstances a thorough search has been made for some possible reason that will satisfactorily account for the known distances between the atoms without at the same time abandoning electromagnetic theory at any point, and without altering the fundamental character of the models. It cannot be said to be known whether each atom in a crystal or ina molecule is a neutral atom of smail dimensions, or whether each has some of its electrons at considerable distance from the atomic centre. The latter alternative is assumed in the Lewis-Langmuir theory, and is perhaps the generally accepted idea of most physicists. ‘he conception introduces insurmountable difficulties in conceiving of the phenomenon by means of electromagnetic theory alone without other postulates that depart from it, and it is shown in this paper that the assumption of neutral atoms, which is in accord with electromagnetic theory, does afford a solution of the matter. This is considered to be sufficient justification for making the assumption of neutral atoms, which is made in the following investigation. tAt AC bh Ga Fig. 4.—Representing the centres of the charges in two hydrogen atoms coaxial with each other. C and c are positive charges of 2e each, and A, B, a, and bare the centres of the negative electrons. _ If an atom is pictured as composed of some symmetrical arrangement of charges in close proximity to each other, as in fig. 4 for example, that is close compared with the distance to the next atomic centre, then it has been shown that the total electrostatic force between two such atoms follows the inverse sixth power of the distance law, assuming that each charge is a point charge and obeying the inverse square Se re % ; 7 7 J Atoms and Molecules. . 899 law as regards each other. The inverse second and fourth power coefficients in the development of the force as a series of powers of 7 cancel when the atomsare neutral, leaving the principal term of the series the inverse sixth power, which has a negative coefficient meaning a repulsive force between the two neutral groups. Examples of this are given in earlier papers *, but it is again illustrated here by the use of two of the model hydrogen atoms similar to that described. Let the two atoms be placed at a distance apart, each having a common axis of rotation, as in fig. 4, which shows merely. the locations of the centres of the charges, ABC being one atom and abc the other. A and B, also a and 8, are negative electrons of charge —e, while C and c are positive charges of plus 2e each. The rotations of the charges on their axes are to be neglected in this preliminary discussion. Denote the distance Cc by rv, and the distances AC=CB =ac=cb by 6. The sum of the repulsive forces between the pairs Cc, Ab, and Ba, each at distance r, is according to the inverse square law, 9 F= 7 See. We) ee eae) There are two other repulsive forces, that of A for a, and of B for b, which give respectively 2 F =F (-1)(r+20)-%, per ee and B= (1) (r—26)~ PEF ja P13) There are also four attractive pairs which partly balance the above, that of A for c and of C for a at a distance r+, and that of C for 6 and of B for ¢ at distance r—Jd, which give respectively when added B= (A)(r+b)> Pe ards nicl 9 and ; 1 C5) Clee) a eee a a) The’ expansion of these quantities in series gives (7 + 2b)-2 + (r— 26) -? = Qr-? + 2407-4 + 160b47-8..., (6) (r+b)-2+(7—b)-? = 2-72 + 66?r-* + 100477... . (7) -* Phil. Mag. July 1913, pp. 89, 66, 71; and June 1915, pp. 755, 764. 3 M 2 900 Dr. A. C. Crehore on Hence the sum of (1) to (5) inclusive gives the whole electrostatic force between these two groups of charges, namely 2 Pas (120), oa This is still a very large repulsive force, and attempts were made in former papers to balance this against the force due to the revolutions of the electrons in orbits. With the present model, however, we are deprived even of this pessibility because the electrons cannot revolve in an orbit completely around the centre of the atom according to electromagnetic theory. To estimate the magnitude of this force, let 6 be made as small as possible and take r=10-° cm. The smallest value of b admissible is the minor radius of the electron, say 1071? cm., so that 2 F = © (—0-012) dynés, (a This is equal to the repulsion between two electrons at a distance of about 9em. No forces are available due to the — rotation of the electrons on their axes to balance this great force. For some time this difficulty has been a stumbling block to further progress. , It was not until the idea was used that the negative electrons and the positive charge, as discussed in the pre- ceding paper, do not have a spherical shape that it was perceived the electrostatic force just calculated must be revised, because the assumed form of an oblate spheroid - cannot be treated as a sphere for external points. Let us, therefore, recompute the elecirostatic force for the com- bination of charges represented in fig. 4, the hydrogen molecule. To undertake this brings us at once into unexplored territory, for the solution depends in part upon the electro- static force between two solid oblate spheroids of charge—- a problem that has not been solved in its generality. We shall, therefore, limit ourselves in this preliminary investi- gation to the case of two spheroids having a common axis of revolution, and besides this approximate the spheroid by dividing it into two parts—first the sphere inscribed within it, and second the rest of the spheroid. Jet the charge of the inscribed sphere be denoted by EH, and of the rest by Hy, so that _E,+E, = e for the negative electron. . . (10) ‘The reason for making this division is because the sphere Atoms and Molecules. 901 may be treated as a point charge for all external points, and the rest of the spheroid approximates a ring or circle of a certain unknown radius, say a, and may be treated as a single ring uniformly charged, the solution for whichis known. In view of the uncertainty as to the precise shape of the electron, this substitution of a ring and a point as the equivalent of the negative electron electrostatically may be just as appropriate as to use the solution for the spheroid were it known. 2.° The Electrostatic Force of One Non-spherical Electron upon Another Coaxial with it. Represent the meridian section of the two coaxial electrons as in fig. 5 by ellipses of semi-major axis a and semi-minor Fig, 5.—Representing the meridian section of two’ coaxial electrons, the charge in the inscribed sphere being electrostatically equivalent to a point, and of the rest equivalent to a ring. axis 6, and imagine the inscribed sphere to be replaced by a point of charge H, and the rest by aring of unknown radius, a. The fundamental problem is to find the electrostatic force between two coaxial rings of different radii. Although the radii of the rings for the two electrons are equal, yet to find the force between a ring and a point it is convenient to use the same expression and equate one of the radii to zero for the point. The complete general expression for the average electro- static force resolved along the centre line, 7, between two point charges in circles of different radii but having parallel axes is given by equation (42) ofaformer paper*. In the special case * Phil. Mae. July 1913, eq. (42). g y 7 &G 902 Dr. A. C. Crehore on of coaxial circles the angle of latitude, 1, which occurs in the formula becomes 7/2, and cosX\=A=O0. With this simplifi- cation most of the terms in this equation drop out, leaving in the notation there employed the 7-component of the force of one element of one ring upon one element of the other: ie dK, dK, Aéy 3 ka® a= (Aoi) + As gO A Ay goes + Ag oS°T2AS + Ag .O*T4A®...}, (11) where the numerical coefficients follow the scheme Ayo=l 3 As, o=3Ao,0=3 ; Ag o= Az 9=15/2 3 Ag o= 5A, 9=30/2 3 As, o= Ae, 0= 315/8 3 ete. 5 and the symbols are abbreviations for the following. The distance between the centres of the circles is 7 when measured in centimetres, and is v when measured in a small unit a,, so that r=a,v. The radu of the orbits are oj) amqme, or a.m and a,n, so that ay=a,m and ag=a,n. O=mn, and A = (v?+m?4+n?)—1/2, T=cosy, where y is the phase difference between the two elements of charge. k is the dielectric constant. ; The first process is to integrate for the phase angle, y, around one circle between the limits of 0 and 27, which may be accomplished by replacing all odd powers of the cosine by zero and the even powers by cos?y=1/2, cost y=3/8, &c., as in averaging. These substitutions enable us to derive from (11) the total force between coaxial circles, ni 3 - = 7 HEA Sfp neta mintAe+... } f GE2) 2 kas The expansions of A’v, &c. in series give the following, where M=m?+7n?: : 3 LS 35 Bye tle 20 Suen a dite Sen icine an emene Mey =v 5 Me +—5 Mo iq ee 3 La. ale T {98 M U ood Loney ee OP ie ae 105 2271-8 Gwe a = + minty — = Mnin’e + ue Mem? n2y-?° |... 32 945 945 —— mA = == nee 64 64 so that, when 7, a,, and a, are restored, the whole electro- static force upon one ring due to a second coaxial with it Sas ee i a ee Sa aia Si ts oem Atoms and Molecules. 903 becomes H,E i | a a9 5 (astbas8)r4— G (ay? + 492)? +2atay | r-8 +[ Bae + a”)? + = (a? +432) ay2az" |r* 94 -( (ay? + a9?) * — (ay2+ a )Parae+ Ay*dy ‘| ele i: as)” By making the radii both equal to a and the charges both equal to H, we have the force between two rings of the same radius and charge: F, = BEY 98 4 30% iA =F ar 84 as a Ape -f. on ring. . ; ; (14) To check this result, the force ( (14) may be directly derived from the expression for the force between two elements of the two rings, namely ms —dHjdH, r eet al cdi SE By expanding the denominator in series and integrating each term separately for the phase angle y between the limits 0 and 27, it has been shown that equation (14) results. The general’equation used above, however, will be of much use in other cases. By making a,=0 and a,=a we obtain from (13) the force of a ring of charge, Ky, upon a point of charge Ki, namely HE =e 15 30 a15 . 5 ie — som —_ —Q’r —_ qin ao Oe eens 8 es ag cae te : 16 ring k A ON i ee BS ree | 128 os t ( } On point. To find from these the whole electrostatic force of one non-spherical electron upon the other, as in fig. 5, add to (14) twice (16) and include the force of Hj, in one electron on EK, in ms other as point charges, giving oe = : ' = ey? + 3H.( HB, + E;) a?r—4 on electron. 1 5 — 7 b,(bhn + H,)a* 84.97 B,(10B, + E,)a%r 78 — B,(B5E, +-B,)air 2. , An ioaatyg: 904. Dr. A. C. Crehore on It will be convenient to define the ratio H»/e by the unknown quantity, p, so that e? may be factored out from (17), giving e2 ff = : A Sat spears De ae: o on electron. +5 (0p Iabr-8— =? p(Bdp + Lair. } (18) This equation forms the basis for computing the electro- static force between two coaxial groups of charges, the two atoms of fig. 4. It is to be noted that the force between two electrons does not now follow the inverse square law, but that the forces between elements of their charges do. 3. The Hlectrostatie Force upon one H-atom due to another Coaxial H-atom according to the Atomic Model. Using (18) as the force between two coaxial electrons instead of considering them to be point charges as in the previous example, it may stand without change as repre- senting the force of the electron A on 6, or of B ona in fig. 4, since each distance is equal to vr. To find the force between the most distant pair of elecirons, A on a, the distance must be changed tor+2b. Expanding (r+ 2b)—?, (r+ 26)~* etc. and Bees by the coefficients in (18) gives = © {Ar 24 By-84 Cr-44 Dr-5 4+ Hr $4 Fy? sone +Gr-8+ Hr-94 Te”, ..}, where Ae pe eae D = (320?—24pa2)b ; B= — 8004 + 120pav? © o(2p-+1) at; ee aoe See ee +1)a°d; = — 4481+ 1680pa°b'—315p(2p + L)atd + e(9p+ Ha’; H = 102457—5376pa? + 1680p (2p-+1)a°s?— 70p(9p + La ; I = — 230408 + 16128pa2b$— 7560p (2p + l)a‘l! +630p(9p + 1)atb*—"~" p(34p +a’. . (19) The force between the inside pair of electrons, B on 8, is Atoms and Molecules. 905 the same as (19) in all the even powers of 7, while the signs of the odd powers are reversed. The attractive forces are those between the positive charges and the electrons. Equation (16) gives the force between a ring and a point charge, and the positive charge may still be regarded as a point because of its very small dimensions, 10-16 cm., as compared with that of the electron, 10-%cm. By changing H, in (16) to positive 2e, which changes the sign of the expression, we have the force between the ring of the electron and the positive charge. To this must be added the force of H,, the inscribed sphere, upon the positive charge, each being a point charge, giving at the distance r: ; = electron on 2e, ieee Re, Pera ek 0) 64 P To obtain from this the force of the most distant electron a on the positive charge C, change r in (20) into r+), giving 2 i = {Ar?+ Bro? + Cr-44 Dr? + Er-6 4 Fro? aoncC or © ona + Gr-84+ Hr-94+ | Fag ee ang where A=2; B=—4b; C=60?—3pa?; D=—8&b? +1207 ; E=1004—30pa?b? + > pat k= S120: 60patb? — pat ; G= 141° —105pa%b! fee pats? pa’ ; H = — 1607 + 168pa2b°—210patb? + 35pa° ; 945 315 315 I = 185°—252pab® + >" path'—— pat? + = pa’. . (21) The force of the nearer electron, /, upon the positive charge is the same as (21) except that the signs of the odd powers of r are reversed. Hence the electrostatic force of the whole atom (ABC) upon the distant electron, a, of the other atom is the sum of A on a (19), B on a (18), and C on a (21), giving 2 é — a =@ ak a Fo = —{A’r-*4+ Br? + C'r 64+ D'p-7 4 Er 8 atom ABC k on a, + B’yp- 94 Gong 906 Dr. A. C. Crehore on where A’ = 3(pa?—262); BB! = —125(pa? 20%) : C= 1014 + 90pa%d®§—*? put—15p2a! Dp! =( 18004—420parb? +2 pat + 20%!) Hy! = — 43408 + 1575 pa?bt — oP paul’ + pa! — 630p?a*b? + a as EF’ =10086' — 5208pa7b> + 3360p?a*l? + 1470 patb — 630p?a°d — 35pa%b ; we 75 atta e oath? a pa>— 15120p?ath! + 5670p?a8b? — 280? pat. (22) The electrostatic force of the whole atom ABC upon the near electron b is the same as (22) with the signs of the odd powers of 7 reversed. The electrostatic force of the whole ; atom ABC upon the positive charge of the other is given by (21), for a on C or A one, added to (21) with signs of odd powers reversed for B onc, and added to —4e*/kr? for the force of C on ¢, giving | G’ = — 228688 + 15876pa%* — 6) 52 : F = 28 {Ort 4 B64 Gr 177 es atom ABC where the coefficients have the same values as in (21). Similarly, the force of the atom ABC upon the two electrons of the other atom is found by adding to (22) the same equation (22) with odd powers reversed, giving woe Bo = fAalp44 Or 84 Ble8 + G'r-,..3, (24) atom ABC k on (a+b). where the coefficients are the same as in (22). The whole electrostatic force of the atom ABC upon abe is, therefore, the sum of (23) and (24), namely D p2 Fo= > {(C+A t+ (B+C)r-$+ (G+B)r$ atom ABC v on atom abe. =F (I =P G') Toe Ps Atoms and Molecules. 907 where C+A’=0, K+C'=— 606+ + 60pa?b?— 15p?at= —15(26?—pa?)’, IG Leen G + EH’ = —42006 + 1470pa?h4— 7 pat — 630p7a*b? + “phat, I+ G! = — 226808 + 15624pa?b— 66 L5patb* + 315pa%b? — 15120p2a4b* + 5670p?a°l? — es pa’. (25) The coefficient of the r~4 term, therefore, vanishes for the non-spherical electrons just as it did in the preceding example for the case of point charges. In the former example the r-® term gave a strong repulsion for the electrostatic force between the two atoms, but now the r~® coefficient, H+(’, depends upon the relative values of b and a, that is, depends entirely upon the shape of the electron. It was pointed out above that the large coefficient of the r~® term in the point- charge example prevented further progress, because there cannot possibly be a force due to the motion of charges great enough to neutralize this repulsion and give an equi- librium distance. In order that the r~® term of (25) shall vanish it must be true that | tea or = (2). (26) p Let us examine this relation in some detail because it reveals the limits between which the eccentricity of the ellipse representing the meridian section of the electron lies. It may be shown that p is equal to the square of the eccentricity of the ellipse, for by definition p=EH,/e, and on account of the assumption of uniform density the two charges H, and e are proportional to their volumes. Hence we have ) Vol. ellipsoid Sma Udy . e e il Vol. inseribed sphere. 47h? 0? KE, e—E, 1-9’ whence ae: 7 ale es pe I ees Can Ag” the square of the eccentricity. Hence the relation to be studied (26) becomes lO = Nice pe Metatata og. 3.2 (29) 908 Dr. A. C. Crehore on This ratio, the unknown radius of the equivalent ring of the electron to the semi-minor axis, is plotted in Curve LI. fig. 6, as ordinates against the eccentricity, e, as abscisse by the use of (29). From the geometrical properties of the | me tev tres De Joe Oe ae E Fig. 6.—Curve I. is the graph of the ratio of the radius of the ring of the equivalent electron to the semi-minor axis obtained from (29). a/b= / 2/e. Curve IT. is the graph of the ratio of the semi-major to the semi-minor axis of the electron, both ratics being plotted against the eccentricity. The curves show that the eccentricity must be greater than «='82 the point of intersection, and they indicate that the eccentricity must exceed about ‘9. The valu used is ‘945, corresponding to a/b =3'058. —) — ellipse the ratio of the semi-major to the semi-minor axis in terms of the eccentricity is ao/b = (Le?) 712, which is plotted as Ourve IJ. fig. 6. The two curves Atoms and Molecules. 909 intersect at an eccentricity about °82. For all eccentricities less than this the unknown radius, a, would have to be greater than the semi-major axis of the ellipse, when the equivalent ring would lie wholly outside of the ellipsoid, a result that is physically absurd. The eccentricity must certainly be greater than *82 in order to satisfy (26) and reduce the 7—® coefficient to zero, and must lie between ‘82 and unity as an upper limit. At an eccentricity °9 it is seen that a/b=1:57, a)/b=2°3, and a/ajy="68. A rough estimation of the position of the equivalent radius would indicate that*9 is certainly a lower limit for the eccentricity. There is no obvious way to compute the radius a from electro- static forces alone. It seems, therefore, remarkable that the eccentricity of the electron as determined in the first paper * by an entirely different method should have yielded a value within the limits above found, namely e="945, corresponding to ratios a/b=1 5, a/ag="5 approximately, and ay/b=3°058, shown in fig. 6 by the Siniats at ‘945. These two independent determinations taken together not only help to establish the non- oe form of the electron, but they approximately determine its value within fairly narrow limits. Let us now proceed with the assumption that (29) is exactly satisfied by the properties of the electron, and examine the values that the various coefficients assume by introducing this relation. From (26) we have pa? = 26°, por = Ap pare Sbo/p*,+i., pta® = 167/27, pare = 1608 /p*, J?) SSC) and these values reduce both (H+C’) and (G+E’) in (25) to zero identically. That is to say, both the r~® and the r-® terms of the series disappear together, while the co- efficient of the r—!° term, (1+ G’), becomes (—31500 +18900/p—2835/p?)8. . . . (31) Using the value of the eccentricity as first determined, namely 945, p is given by (28), and I/o is 1:12 approxi- mately. Hence the whole electrostatic force between the two atoms given by (25) becomes numerically 2 é ~ Nhe De eG bes Naty Suen, (8) atom ABC he on atom abc. * Phil. Mag. Oct. 1921, equation (76). 910 Dr. A. C. Crehore on That this series is so rapidly convergent at distances comparable with 107° cm. that the first term of it represents very approximately the whole force may be shown as follows.. The approximate value of b/r is 10—18/10-8=10~°, so, that, b°/P=107%, d'/rl0— 10)), ete. , amdigcommamar b§y 9 = 10-**, br-2e-10-%, etc., these values decreasine mmtine ratio Of TOMS lin alll succeeding terms the exponent of 6 is always two less than the exponent of » but with opposite sign. The numerical coefficients do not increase at any such rate as 10!°, and the r-! and succeeding terms are of no value. The electrostatic force between these atoms therefore varies as the inverse tenth power of the distance between their centres, when (26) is satisfied. In a similar manner the condition that (26) is satisfied reduces the force of the atom ABC on the positive charge c (28) to F =e t- d(10 —3/p)b4r-§ + (—196 + 315/p — 35/p?) bor-§ atom ABC ee + (= SOOT Neon ; + 1260/p?)b 75). L Senay and reduces the force upon the two electrons (a+6) due to ABC in (24) to F = 2° {5(10—Bjp)b4r-*-+ (196 —315|p +35 /p2) 55-8 atom ABC on (a+6). + (—310 144 17010/p —1575/p? —1260/p°\0er-, These two equations are exactly equal but of opposite sign in both the r~® and xr ® tcrms, and, since the first term represents practically the whole of the force, it may be said that the force upon the nucleus is one of repulsion equal to the first term of (33), while that upon each electron is an attraction equal to one half the first term of (34) electro- statically. ‘These forces are very large, varying as the inverse sixth power in comparison with their sum which varies as the inverse tenth power. It is this smaller force which must be balanced by terms due to the rotation of : the positive nuclei, which will now be considered. _ Terms due to Rotation. Yortunately the solution for the mutual action of two continuous rings of charge in rotation has been solved in its generality, using the Saha equation so far as the inverse square of the distance terms are concerned. At. distances Atoms and Molecules. 911 comparable with 10~* cm. the ratio of the distances, 10~§ cm., to the dimensions of the atoms shown in fig. 1, varies between 30000 and 8000 approximately, being so great that the second atom may still be considered to be at a great distance from the first. It may be shown that the higher order terms than the second are negligible, and that the whole force due to the rotation of the charges is merely that given by the gravitational law, modified where single atoms are considered by terms defining the directions of their axes. This result is given in the general form in equation (57) of the first paper *, and in (59) when the force is resolved along the centre line joining therings. Summing up the Z-component force for the axis case, where Z=1 and X=0, by means of the equation referred to for the whole hydrogen atom acting upon another hydrogen atom, we find e? it Bie i = 7 (2B: ae : : . 3 : (5) Adding this to the electrostatic force found above in (82) gives the complete force upon the first atom due to the second as Q B= 5 (2Bstr?—27776 . Br 0), sy wee (86) whence equating to zero for equilibrium, and solving for 7, we have T = 1AS888 a a Gira! Be" This represents the distance between the two atoms of the diatomic hydrogen molecule. It may be seen that the equi- librium is stable in the direction of the axis, for an increase of the distance 7 makes the gravitational r~? term in excess of the other term, meaning that the atoms are attracted, while a decrease of the distance makes the repulsive force in excess As to stability the other way perpendicular to the axis, this matter must be referred to the X-component of the force which has not been developed above. It may be stated, however, that the general form of equation has been developed, and is discussed in detail in the next paper TY: andy — lot Selim eenns (37) * Loe. eit. j + The numerical values of 6 and 8. used here are the same as pre- viously found in the first paper, 5=1:065 x 10> and B,*=0-6053 x 10-38, 912 Dr. A. C. Crehore on Helium. Hig. 7 may represent two helium atoms similar to that 1, lined up so as to have a common axis of rotation. © A A\BB, bblaa, he iQ Pig. 7.—Representing the centres of the charges in two helium atoms coaxial with each other. The positive charge of 4e is at , the four electrons being Ay, As, B,, and B,. Using the method shown in detail above for hydrogen, the electrostatic force of the atom (A,A,CB,B,) upon the positive charge of the other atom, ¢, is Q Fo = % {(2400?—24pa?)r-+-+ (328004—1200pa%22 He-atom on ec. + 30 pa*)r—€, ee } : 5 (38) The electrostatic force of the same atom upon the four electrons together (a,a,b,h.) is , | F = = {(—2400"+ 12pa?)r-4 + (— 152806! + 3600pa2b? Abid esl | —90p2at—15pat)r-§..$. 2. (39) The sum of these gives the whole electrostatic force of the second He-atom on the first : Q 1 ee 5 {—12pa%r-4+ (— 12000544 24.00 pa2b2—90p?a! He-atom! on He-atom. + l5pa*) imoen ie A (40, Using the relation 2b =pa? as before, (40) becomes yt . Fo => {—248%-4—(7650—60/p)b'r-8,..3, (41) an tledee In the above example with hydrogen all the electrostatic terms vanished up to the r~! term after introducing the relation (26), but here even the r~* term does not vanish and becomes the largest term, showing great repulsion Atoms.and Molecules. 913 between the atoms. No terms in the inverse square due to the rotation of charges can possibly balance this great repulsion. These repulsive terms will persist in other positions of the atoms aside from the particular axis case considered, and it may be concluded that helium must be monatomic if it is as shown in fig. 1 The combination of hydrogen with helium on the axis gives as a final result for the electrostatic force: He on Hq electrostatic. Fo =" {—12b%-#430(241/p)b'r-%,..},. (42) and here again the r~* repulsive force appears, which prevents helium from uniting with hydrogen. In a similar manuer it may, I believe, be shown generally that helium will not unite with any of the other atoms shown in fig. 1. The same statement may be made for neon, which has a helium atom at its centre. The presence of this is sufficient to cause a great repulsive ferce between the atoms of neon and any other of the atoms shown. Summary. In Part I. is presented a concrete picture of the forms of the atoms of the lighter elements including isotopes from hydrogen to sodium inclusive, which are in conformity with the Saha form of electromagnetic theory. Theshape adopted for the negative electron is an oblate spheroid with ratio of axes about three to one, as determined in the first paper. Although Rutherford’s experimental results were unknown to the writer at the time the models were made, the distance between the centre of the nitrogen atom and the hydrogen in nitrogen is just two fremeiet: of the electron, as he has found. General formule not given in this paper have been deve- loped for testing these atomic forms by determining how they satisfy the known results of chemistry in combining with each other to form molecules. In Part II. it is shown that the shape of the negative electron is a most important factor in determining the equi- librium distance between two atoms forming a molecule. The approximately constant distance between the centres of atoms in all solids is directly connected with the shape of the electron, and for this reason it is a universal distance. In order that these atoms may unite with each other at all, it is shown that the eccentricity of the electron must lie between Phil. Mag. 8. 6. Vol. 48. No. 257. May 1922. 3N 914 Research Staff of the G. KE. C., London, on the. the limits of about ‘9 and 1:0. This agrees in a rather remarkable way with the determination of. the eccentricity as ‘945 in the first paper, which was based on an entirely different assumption. It has been shown that the helium and neon atoms of fig. 1 will not form compounds with any of the other atoms shown, and that they must be monatomic. The exact deter- mination of some of the simpler molecules is deferred for a future communication on account of space. XCVIIL. The Disappearance of Gas in the Electric Dis- charge-—IV. By The Research Staff of the General Electric Company Ltd., London*. (Work conducted by N. R. CaMpseiit and H. Warp.) NHE experiments described in this paper are a con- tinuation of those described in the previous papers of the series }. They are concerned primarily with the disappearance of gas in the presence of phosphorus vapour, but some of the conclusions reached are applicable, even when phosphorus is not present. Summary. The main experiments are directed to determine how the quantity of gas absorbed in the discharge in the presence of phosphorus varies with various factors. The experimental methods are described in pars. 3 and 4. The following conclusions are reached concerning the factors :— 1. The nature of the gas—Compound gases and some elements are excluded on the ground of chemical change. Of the rest,.the inactive gases are little absorbed ; hydrogen and nitrogen, on which the experiments have been ee behave almost identically. (Par. 2.) 2. The nature of the discharge has little direct effect upon the amount of gasabsorbed. (Par. 5.) 3. The form of the vessel is eee only in so far as it affects the area of the walls exposed to the discharge. The temperature in the neighbourhood of 20° C. has no effect. (Par. 3.) * Communicated by the Director. + Research Staff of the G. E. C., London, Phil. Mag. xi. p. 585 (1920); xli. p. 685 (1921); xlii. p. 227 (1921). These papers will be quoted respectively as L., IL, III, | Disappearance of Gas in the Electric Discharge. 915 4. The pressure of the gas during absorption has no effect within wide limits on the amount absorbed. (Par. 6.) 5. The relation between the amount of gas absorbed and the amount of phosphorus present is shown in fig. 1. It is Hydrogen absorbed, 0 Ol O02 O03 04 O5 06.07 O08 O39Mgm. Phosphorus introduced. given by a stepped line rather than by a smooth curve. It varies somewhat according to the state in which the phosphorus is introduced. (Par. 7.) 3 N 2 916 Research Staff of the G. E. C., London, on the 6. The state of the walls has no effect within wide limits. All kinds of glass, cleaned or etched by any method or coated with other substances by evaporation of solutions or by chemical action, have the same absorptive power. On the other hand, the absorptive power can be greatly increased by “evaporating ” certain substances from an _ electrically heated filament and so depositing them on the walls. The ionization of the vapour is an important factor in this process. Metallic tungsten is among the substances having this property, but no relation has been found generally between this property and others. These substances act to some extent as substitutes for phosphorus. (Pars. 8-11.) The following subsidiary matters are dealt with :— 7. In the absence of phosphorus or its substitutes, the absorption is indeterminate, representing a balance between true absorption and liberation. (Par. 12.) 8. The absorption of gas mixed with phosphorus vapour takes place in two stages—a very rapid absorption accom- panying the deposition of the phosphorus on the walls, and a slower absorption after it has been deposited. The same two stages are characteristic of the substitutes considered in (6). (Far. 13,) 9. The occurrence of absorption seems determined simply by ionization, and will occur, though very slowly, in the absence of a visible glow. There appears to be no lower limit to the pressure that can be attained by absorption in the presence of phosphorus. (Par. 14.) 10. The gas absorbed is liberated when the phosphorus is evaporated from the walls ; that absorbed under the influence of the substitutes of (6) cannot be liberated because the substitutes cannot be evaporated. There is evidence of the formation of such chemical compounds as PH3;, but none of the formation of compounds which are not formed in other circumstances. The formation of these compounds is a secondary and not a primary feature of the absorption. (Par. 15.) | 11. Arsenic appears to have exactly the same properties as phosphorus, as Whitney has found. Sulphur and iodine were not examined. (Par. 16.) A general discussion of the results is undertaken, in which the view is put forward that the absorption is determined by the formation on the walls of the vessel of electrically polar layers to which ions of the gas adhere in virtue of their charge. It is probable that it is the negative ions that adhere; if this is so, an explanation can be offered of some Disappearance of Gas in the Electric Discharge. 917 apparently unconnected effects produced by the deposition of salts by the method of (6). In all that concerns the forma- tion of the absorhing layer, phosphorus and the substitutes of (6) behave similarly. (Pars. 17-19.) In other respects phosphorus (together with As, and probably 8 and I) has a specific action. The explanation of this action propounded in the previous papers is maintained. (ear 20:) The Quantity of Gas Absorbed. 1. The previous experiments had shown that if a given vessel is filled with gas mixed with phosphorus vapour and a discharge passed through it, the quantity of gas that can be absorbed depends on (1) the nature of the gas, (2) the quantity of phosphorus present, (3) the state of the walls of the vessel. It might also conceivably depend upon (4) the amount of gas originally present, (5) the nature of the discharge. If the form of the vessel is altered, it may also depend (6) on this form, even if its volume is constant. The possible effect (7) of the temperacure has to ve taken into consideration. 2. As to (1), the nature of the gas, the range open to investigation is very limited. Chemical compounds are excluded because they invariably undergo chemical change when the discharge passes, so that the great complications enter which have been fully discussed in the case of carbon monoxide. Of the elements, the inactive gases are known to be very little absorbed. Oxygen and the halozens attack metals, and especially the incandescent filament. The only remaining elements which have sufficiently high vapour- pressures at room temperatures are hydrogen and nitrogen. Hydrogen has been used throughout the greater part of these experiments, and ‘‘ gas”’ will always mean hydrogen unless the contrary is stited. Observations on nitrogen have been made sufficient to show that in the absorption by means of phosphorus its behaviour is almost identical with that of hydrogen. Im all cases a slightly greater volume of nitro- gen than hydrogen can be absorbed ; the difference amounts to about 10 per cent. ; the variations in the amounts of the two gases absorbed as the conditions change are closely parallel, and no distinction between them need be made in respect of any of ihe conclusions which will be put forward. 3. In order to examine factor (2), the quantity of phosphorus present, it is practically necessary to use a new vessel for each experiment on absorption. In the first place, 918 eer Staff of the G. BE. C., London, on the it was found that the absorption of gas might change the state of the walls of a vessel in a manner not easily reversed, while no evidence could ‘be found that all new vessels, suitably treated, were not in the same state ; the separation of factors (2) and (3) could be made with certainty only if no vessel was used for more than one absorption. In ‘the second place, the easiest way to introduce into a vessel a known amount of phosphorus is to coat a wire with a known amount of red phosphorus, mixed with a trace of a “ binder ” to secure adhesion, and to heat the wire after it has been introduced into the vessel. (This process of coating wire with predetermined quantities of material is sometimes used in incandescent lamp manufacture and need not be described here.) Since ground joints are inadmissible in absorption experiments, on account of the inevitable presence of grease vapour, the wire had to be sealed in, and the renewal of the wire implies the renewal of the vessel. Accordingly it was necessary to use vessels that could be readily obtained by the hundred. Ordinary incandescent vacuum lamps proved entirely suitable for the purpose. In such lamps the discharge which causes the absorption passes between the opposite ends of the filament, maintained at a difference of potential by the current which heats the filament ; the negative end provides the requisite thermionic emission. In almost all the experiments 200-230 volt 40 watt lamps of the usual commercial type were used ; they had tungsten filaments about 1 metre long, about ‘(026 mm. in diameter in bulbs of about 210 c.c. volume. t is difficult to coat filaments regularly with weights of phosphorus much less than 0°1 mgm. When smaller quantities of phosphorus were required, they were obtained by filling the lamp with gas, placing it in connexion with a vessel containing white phosphorus at a known temperature, and allowing time for the vapour of this phosphorus to establish its equilibrium pressure in the vessel. ‘The estima- tion of the phosphorus introduced by this method requires a knowledge of the vapour-pressure of phosphorus at tempera- tures below that of the room ; the values assumed have been extrapolated (in a manner which will be described in another communication) from the data of MacRae and Van Voorhis * which deal with temperatures above 44° 0. It has been found subsequently in many of the experiments that this method is not wholly reliable, because if sufficient time 1s * D, MacRae and C, E, Van Voorhis, Am. Chem. Soc. J. xliii. p. 547 (1921). Disappearance of Gas in the Electric Discharge. -919 allowed for true equilibrium to be obtained, there will be a large amount of phosphorus absorbed on the glass walls in addition to that present as free vapour. However, by choosing suitably the time of exposure, so that the equi- librium vapour-pressure is closely. approached without considerable absorption on the walls, the amount of phos- phorus introduced can be regulated with the accuracy required for these experiments. The greatest amount that could be introduced by this method, unless the whole system was artificially maititained at above the room temperature, which never exceeded 25° C., was 0°06 mgm.: bya combina- tion of the two methods the whole range up to about 1 mgm. could be covered sufficiently. 4, Since the contamination of the gas with a continually renewed supply of mercury or grease vapour prevents all absorption, and since access of the gas to cooled traps would have removed all the phosphorus, it was necessary to conduct the absorption ina vessel completely sealed off. Two methods have been used for estimating the decrease of pressure, which alone can be used as a measure of the absorption that occurs. The first involved the use of the lamp filament as the hot wire of a Pirani gauge according to the method recently described by us *. The second depends upon a distinction between pressures which are above and those which are below the value at which the applied potential is equal to the glow potential. (I. 590; III. 228.) The glow potential in a vessel, such as a lamp, having electrodes of which the area is very small compared with that of the walls, is the potential at which the current between those electrodes changes very suddenly from a small value, limited by the space charge and almost independent of the thermionic emission, to the value corresponding to the saturated thermionic emission, and therefore dependent on the temperature of the cathode. Suppose, then, that a lamp in which the pressure can be varied is connected in series with a similar but perfectly evacuated lamp in a Wheatstone bridge, the other arms being constant resistance coils. Whatever the pressure in the lamp, the bridge can be balanced for any constant applied voltage by adjusting the coils ; but if the pressure in the first lamp is above that corresponding to the glow potential, the balance will change if the current through the bridge (and therefore the temper- ature of the filaments) is changed ; for an increase of space * Research Staff of the G.E.C., London. Phys. Soc. Proc. xxxiii. p. 287 (1921). 920 Research Staff of the G. EH. C., London, on the current in the gas-filled lamp, due to increase in thermionic emission from the cathode, will not be accompanied by an increase in current in the vacuous lamp. On the other hand, if both lamps are vacuous, and their filaments approximately similar, the ratio of the filament resistance will hardly change with the current, and the very small space current will not change at all; the bridge will remain balanced while the current through it is varied over a considerable range. By means of this simple device of the two lamps connected in a bridge through which the current can be altered, it is a matter of a few seconds to determine whether the pressure in a lamp is above or below that at which the glow potential is equal to the applied potential. The glow potential varies so rapidly with the pressure in this region, and the pressure corresponding to the potentials used is so small (about "0002 mm.), that for our purpose this pressure may be always taken as zero, independently of the applied potential. It may be remarked in passing that the disappearance of a visible glow is not always a safe test that the lamp is ‘“‘cleaned-up,” and that the pressure is below the critical value indicated by the bridge. In order to measure by this method the quantity of gas that can be absorbed with a given amonnt of phosphorias, a series of lamps, each containing the same amount of phosphorus, is filled to a series of pressures. Hach lamp is then ‘** burnt” on the bridge in series with a perfectly evacuated lamp of the same type, and the division found between those which ‘clean-up ” and those which do not. The pressure P at the point of division can be determined by some ten observations within 10 per cent. of its value ; the mass of gas absorbed is then p V, where p is the density of the gas at pressure P and V the volume of the lamp. 5. It was established that P, corresponding to a given amount of phosphorus, was independent within wide limits of factor (5), the nature of the discharge producing the absorption. P did not depend on the voltage applied to the lamp, at any rate if this voltage was well above the limit at which appreciable thermionic emission from the filament begins. On the other hand, the time occupied by the absorption increased rapidly as the voltage, the temperature of the filament, and the space current were decreased. At voltages near the limit at which any discharge could be made to pass, it appeared that P decreased slightly, probably because, when the discharge lasts too long, the phesphorus accumulates in the parts of the vessel where no discharge Disappearance of Gas in the Electric Discharge. 921 passes. Again, the substitution of a filament of half the length, designed for half the voltage, produced no change in P, nor did the use of a separate anode and cathode similar to the arrangement shown in fig. 1 of (1.). If electrodes of area comparable with that of the glass walls were used, such as those of a thermionic valve, changes in P undoubtedly oecurred ; but they are to be attributed rather to changes in the form of the vessel than to changes in the nature of the discharge. In discussing the other factors, factor (5) may be left out of account. It should be mentioned that experiments made on different batches of lamps, in all respects apparently the same, never agreed so perfectly in the determination of P as those made on a single batch. This variation is unexplained. Factor (7), the temperature of the vessel, does not appear to be important, at any rate if the variation of the tempera- ture is confined between 0° and 30°; but no systematic observations on this matter have been undertaken. It is possible, but not probable, that some part of the unexplained variations are due to a secondary influence of temperature. Factor (6), the form of the vessel, undoubtedly has some effect ; it appears generally that the amount of gas that can be absorbed by a given amount of phosphorus increases somewhat with the area of the walls exposed to the action of the discharge. If the vessel in which the discharge takes place is merely connected by a narrow tube with another vessel, the weight of gas that can be absorbed with a given weight of phosphorus is unchanged ; but if the same electrode system (e. g. the lamp filament) i is placed i in a larger bulb of approximately the same shape, the weight absorbed with a given weight of phosphorus is increased. But again no systematic ‘measurements have been undertaken. 6. There remains the important question whether P is really an adequate measure of the amount of gas that can be absorbed. It will be so only if the amount absorbed is independent of the starting pressures; or, in other words, if, starting from a pressure P+ , the final pressure reached is always p, whether p is 0 (as in the method described) or is comparable with P. The experiments that have been made indicate that the amount absorbed is so independent of the starting pressure within wide limits. They were made by the use of the Pirani gauge. Table I. gives two typical series of results made in vessels of different volumes with different amounts of phosphorus. The first column gives the starting pressure P+ p, the second the final pressure p, the 922 Research Staff of the G. EH. C., London, on the third the difference P. The variations between successive determinations are greater than can be attributed to experi- mental error in measuring the pressure, but there is no consistent variation of P with p. TaBLeE I. : Papp: Ps P. (Original pressure.) (Final pressure.) : eel agis ) 0-491 mm. 0°457 mn. 0-034 mm. 382 “330 052 "293 ‘265 028 "284 "250 034 233 "192 041 ‘174 139 "035 LNG) 081 038 "102 053 . 049 093 "058 "038 ‘061 "032 029 035 000 035 0°150 084 ‘066 ‘109 "040 069 “094 ‘035 059 083 032 051 ‘068 00g 068 It is apparently permissible to speak of a definite amount of gas absorbed with a given amount of phosphorus, independent both of the nature of the discharge and the pressure of the gas threugh which it passes. 7. If such independence is established, factor (2) can be isolated. Vig. 1 shows the relation, determined by obser- vations on some 250 lamps by the method of Par. 4, between the amount of phosphorus introduced into the lamp and the amount of gas that can be absorbed, both amounts being expressed in milligrams. ‘The points (x ) joined by fuli lines were obtained when the experiments are conducted in the manner described. The features to which attention for subsequent discussion is drawn, and which are repeated in all other observations despite variations in detail, are :— (1) The general increase in the hydrogen absorbed with increase of phosphorus. (2) The maximum followed by a decrease. (3) The general slope of the line which, even if it were smoothed into a straight line, would not suggest any chemical combining proportions of the two elements, e. g. 10°3 to 1 (PH) or 3i to 1 (P.H,). (4) The marked “ step ” in the line from 0:09 to 0:27 mgm. ‘ phosphorus ; there are distinct indications of similar but less marked steps at the points for lower values, and even at the higher values considerable variations of the amount of phosphorus about Disappearance of Gas in the Electric Discharge. 923 each point are not accompanied by proportional changes in the hydrogen absorbed ; moreover there are some values of the amount of hydrogen absorbed which seemed never to occur however the amount of phosphorus was varied. (5) An amount of phosphorus below the step never gives a yellow coloration to the walls of the lamp after the gas has been absorbed, whiie an amount above the step always gives such a coloration. The points marked © and joined by the dotted line were obtained by a slightly different procedure. Here, before the lamps were filled with gas, the filament was heated so as to deposit the phosphorus as a coloured film on the walls: the lamp was then filled with gas and the critical value of P determined in the usual manner. The observations were much less consistent than in the first series, the difference being probably due to the much lozger time occupied in the absorption (see Par. 15), but the following features seem clear :—(1) The absorption is less than in the first series, especially at the higher values; (2) the maximum vanishes and is replaced by approximate “saturation”; (3) the marked step does not wholly disappear. 8. We now pass to factor (3), the state of the surface of the vessel. : In all experiments the walls of the vessel were freed from adhering gas (chiefly water and CO,) by baking for five minutes to 400° C. during exhaustion. It this precaution is not taken, the absorption is less but results are irregular ; doubtless the gas which is allowed to remain should be included in the amount absorbed. But if_it is taken, the absorption is independent within very wide limits of the nature of the surface. Thus it is the same whether the vessel is made of soda, lead or borosilicate glass, or fused silica. It is unchanged by washing the surface with strong acids or alkalis ; by etching it with hydrofluoric acid, liquid or gaseous ; by sand-blasting it; or by depositing on it many salts by evaporation of solutions ; itis little, if at all, changed by depositing silver chemically. In the examination of these modes of treatment some anomalies were found that have not been fully explained, but by none of them could the absorption be changed consistently *. On the other hand, there is a method by which the absorp- tion can be greatly increased with perfect consistency. If * In II., p. 702, it was stated that P.O; deposited cn the walls of the vessel prevented absorption. This statement is not true if the material is thoroughly dried by baking and exhaustion. 924 Research Staff of the G. E.C., London, on the the filament is coated with a suitable amount of certain salts, of which sodium fluoride is typical, in the same manner as it is coated with phosphorus for the ex cper iments of Par.7 ; and if, after the lamp has been baked and evacuated, the filament is heated for a moment to bright incandescence ; then it will be found that the amount of gas that can be absorbed with a given amount of phosphorus is very notably increased. Unless two filaments are sealed into the lamp, one coated with the salt and the other with phosphorus, it is necessary in making these experiments to introduce the phosphorus in the form of vapour. That method was adopted in all the measurements described below, and the amount of phosphorus which could be introduced was limited to about ‘06 mgm. But it was found qualitatively that, if salt were mixed with phosphorus on a single filament, so that both were ‘evaporated ” together, the amount of gas that could be absorbed was greater than if the same amount of phos- phorus were present without the salt. But for reasons which will appear in the sequel, this method is not well adapted for quantitative investigation. There is no doubt that the salt produces this increased absorption because it is deposited on the walls. It certainly is so deposited, because, if a large enough quantity is used, their lustre is changed. Moreover, if, after the filament has been heated, the lamp is opened, left exposed to the air for some hours, and then re-exhausted, the salt is still showing its presence by increased absorption. But if deposition of the salt on the walls by this method increases absorption, why does it not ‘have the same effect when the deposition is effected by evaporating a solution? The answer is that the passage of the salt from the filament to the walls is not one of mere evaporation and conden- sation. lonization is also involved; for a visible glow is often seen to accompany the “ evaporation,” and electrical measurements show the passage of a current between cathode and anode, even when the greatest P.D. between them is less than 10 volts. There is also chemical resolution ; for the presence of free sodium, absorbing oxygen in the cold, can be detected after the heating of the salt; but the presence of the sodium cannot be the determining factor, because the exposure to the air after ‘‘ evaporation”’ does not affect the process. 9. Not all salts have been found to produce this increased absorption. Those that show it unmistakably are many Disappearance of Gas in the Electric Discharge. 925 fluorides and chlorides, especially those containing sodium and lithium, silica, glass (the powdered material of the bulb), and sodium silicate. On the other hand, it has not been found with oxides, or with any of the halogen salts of potassium. Sulphates and some other salts cannot be tried because they destroy the filament. We have been unable to discover any relation between this property of salts and other properties, but we have some suspicion that the ionization potential of the ionized salt may be a determining factor, for slight traces of impurity, such as grease vapour or other hydrocarbons, may inhibit the effect even otf those salts which normally display it. Weare inclined to compare their effect with that of hydrocarbons or mereury vapour in inhibiting absorption of gas; the action ceases because the impurity of low ionization potential is ionized in pre- ference to the salt vapour. However, this explanation is speculative. In addition to salts, metallic tungsten from the filament itself may show the effect. If the lamp is burnt while it contains just enough gas to aliow the glow discharge to pass, so that the walls become blackened with sputtered tungsten from the cathode end of the filament, the increased absorp- tion in the presence of tungsten is obtained. In II. T]0e Creeks we attributed the absorption of nitrogen in the absence of phosphorus to the sputtered tungsten; we believe ‘that the presence of this material is the reason why the absorptive power of the walls of the vessel is changed apparently by the process of absorption (cf. I1., p. 703) so thatit is necessary to use a new vessel for every experiment. It should be mentioned that the enhanced absorptive power due to the salt is exhausted by absorption. iia vessel in which salt has been “evaporated” and gas subse- quently absorbed is re-filled with gas, after the usual baking and evacuation, the absorption on the second oceasion will be much less than on the first, and on the third will not be different from that of a normal glass vessel. This factis not inconsistent with the view suggested in the last paragraph of the action of sputtered tungsten ; for in that case fresh sraeeree tungsten is being deposited while the previous layers are being used up. 10. Measurements have been made of the amount of salt necessary to produce the increased absorption. The following figures give the results of a series of trials with 926 Research Staff of the G. E. C., London, on the sodium fluoride, the amount of phosphorus introduced being always ‘(05 mgm. :— NaF coated on wire (mgm.). Hydrogen absorbed (mgm.). 0 ‘0007 0:09 “0007 0-2 "0007 0°28 *OOLI 0°36 0028 2°0 "0023 Almost all the effect is obtained by an increase from 0°2 to 0°28 mgm. Similarly with silica and sodium silicate, it occurred between 0°3 and 0°35, but with lithium fluoride below 0-17 mgm. It is natural to inquire whether the critical amount is that required to form a monomolecular layer on the ee of the glass. The area of this surface was about 200 em.’; the area of a molecule of NaF is about Ax 1057? vem 2; Se to form a monomolecular layer about 0:035 mgm. would be required. This is only about one-eighth of the experimental value ; but it is clearly possible that all the salt introduced is not deposited uniformly on the walls in the active condition. 11. Inquiry was also made how the presence of the active film affected the relation between gas absorbed and phosphorus present, examination being contined to quantities of phos- phorus below the step in fig. 1. The significant result was ebtained that, in the presence of salt, any rot the points below the step (except possibly those corresponding to less than 0-02 mgm. of phospherus) might be raised to the step at ‘0023 mgm. of hydrogen. The maximum effect was not always obtained, but the maximum obtained with any given salt and any given amount of phosphorus appeared quite independent of the nature of the salt (so long as it would produce the effect at all) and almost independent of the quantity of phosphorus ; moreovor, this maximum was the same within the limit of error as the step of the phosphorus line. Some less systematic observations will now be described which provide additional. ey uence on which to base a general discussion. Absorption without Phosphorus. 12. Even when no phosphorus is present, the burning of the lamp and the accompanying discharge produce some absorption of gas. But unless the original pressure is very Disappearance of Gas in the Electric Discharge. 927 low (say less than ‘005 mm.), the pressure never falls below that corresponding to the glow potential, and a complete clean-up is never indicated by the bridge method of Par. 4, even though the amount of gas that can be absorbed, starting from a higher pressure, is much greater than that corre- sponding to a pressure of (005 mm, During the first minute of the burning, the pressure falls by an amount which depends somewhat upon the original pressure, but is about “02 mm.; subsequent changes are much slower. In some cases the pressure begins to increase again very slowly, and will continue to increase until the filament finally fails ; in other cases it will continue to fall very slowly, the fall always being accompanied by a marked blackening of the bulb and being due presumably to sputtered tungsten (c/. Par. 9). The cause of the difference has not been fully ascertained, but it is suspected that the rise occurs when the gas is more highly contaminated with grease vapour, some of which may be admitted during filling. But in both cases there is evidence that gas is being evolved as well as absorbed, for even if no water vapour is initially present, a trace of it can often be found in the later stages, in spite of the presence of the incandescent tungsten. If the wire is originally coated with one of the active substances mentioned in Par. 9, an appreciable absorption of gas occurs during the first instant of burning and while ~ the glow lasts, which is the sign of the “ evaporation ” of the salt. ‘This absorption may amount to ‘03 mmn., and is as much as ‘015 mm. in argon—a gas which shows no absorption comparable with this in the other circum- stances of these experiments. After this initial stage, or if the salt is evaporated from the wire before the lamp is filled with gas, the absorption during the burning of the lamp is definitely greater than it is when no salt is used. Moreover, the absorption is never reversed. It is still true that a perfect clean-up, indicated by the bridge, is never attained, but otherwise the salt acts in much the same way as it does in the presence of phosphorus. In fact, after salt has been deposited, the amount of gas that can be absorbed is approximately independent of the original pressure (as it is when phosphorus is present) ; and the amount of gas that can be absorbed is not very different from that corresponding to the step of fig. 1 which, when salt is present on the walls, gives the absorption for almost all quantities of phosphorus below a certain limit ; it appears, however, that the presence of a little phosphorus is necessary to give the full vaiue of ‘0023 mgm. in fig. 1. 928 Research Staff of the G. E. C., London, on the The Rate of Absorption with Phosphorus, 13. Just as there is a very rapid absorption of gas at the moment when salt is evaporated from the filament in the absence of phosphorus, so there is a very rapid absorption when phosphorus is evaporated from the filament in the absence of salt, or at the moment when phosphorus, present originally as vapour, is deposited on the walls as red phos- phorus by the starting of the discharge. Accordingly, even when phosphorus is used without salt, the absorption takes place in two stages, unless the phosphorus is first deposited on the walls as described in Par. 7 ; the first stage is practi- cally instantaneous and is impossible to control ; the second is very much slower. ‘The ratio between the amounts absorbed in the two stages differs, as might be expected, with the initial pressure; as this pressure is increased, the amount absorbed in the first rapid stage is increased relatively to that absorbed in the second slow stage. But, as noted previously, the sum of the two amounts is approximately independent of that pressure. Attempts have been made to relate the rate of absorp- tion to the ionization in a manner similar to that adopted in (IIT.), but they have met with no success. Indeed, the experiments of (III.) indicate that, even if the experimental difficulties could be overcome, the interpretation of the measurements would be very complex. For it was shown there that, in order to obtain any simple relation between ionization and absorption, it was necessary to suppress, as far as possible, recombination and the arrival of positive ions at the cathode. During the very intense ionization that accompanies the first stage of the absorption, recombination is unavoidable ; in the second, either the ionization is very much greater than that found permissible in (III.), or the absorption is so slow that the disturbing factor mentioned in Par. 5 enters. Moreover, the theory of the action that has been proposed, and is confirmed by the experiments of this paper, indicates that the action which produces absorption takes place, not in the gas as in (III.), nor even at the metallic electrodes, but at the glass walls which are acting as electrodes of both signs. In such conditions the difficulties of interpretation would be enormously increased. All that has been achieved is a demonstration that, in any particular conditions of pressure and state of walls, the rate of absorption is approximately proportional to the ionization. But the only conclusion which it seems permissible to draw from that fact is that absorption is not an action requiring the meeting of two or more charged particles. Disappearance of Gas in the Electrie Discharge. 929 Absorption without the Glow. 14. In (I.) and (1II.) it was stated that (in the absence of phosphorus) no absorption took place unless the potential driving the discharge was greater than the glow potential. Of course this statement, interpreted literally, is not true ; for it is well known that absorption occurs in X-ray tubes and other high-vacuum apparatus at pressures far below that at which the applied potential will maintain a visible glow. But it is true that, at the pressures of a few hundredths of a millimetre with which we were there concerned, the absorption without the glow is so slow that it cannot be detected by the method employed, and that, when the glow appears, the rate of absorption increases enormously. The explanation of the glow potential given in (III.), p. 33, shows that the attainment of the glow potential and the appearance of the glow are accompanied by an enormous increase in ionization, and that the doubt on this point expressed in (I.) was unjustified. Accordingly if, as all our experiments indicate, absorption and ionization are intimately connected, it is to be expected that the absorption which accompanies the glow will also occur, though at a much less rate, with potentials less than the glow potential, so long as they are above the ionization potential. The ratio of absorption to ionization is very much greater in the presence than in the absence of phosphorus vapour *; and it seemed possible that perceptible absorption might occur in its presence, even with potentials too small to cause the glow. The expectation has been fulfilled. The experiments were conducted partly with a vessel like that of fig. 1 in (I.), partly with lamps designed for very low voltages (less than 30). It was found, as was to be anticipated, that the first stage of the absorption discussed in Par. 13 was little affected by the potential between the electrodes, and that a notable absorption always occurred at the moment of lighting the filament, if it had been coated with phosphorus or the gas mixed with phosphorus vapour. But the second stage occurred also, and a perceptible ~ absorption would continue after the first stage was ended, aithough no glow was visible and electrical measurements showed that the current was limited by the space charge * This statement is not certainly established. It is certain that the absorption corresponding to a given electron current from the cathode is much greater if phosphorus is present; but the phosphorus appears to produce some increase of ionization as well as of absorption. Phil, Mag. 8. 6. Vol. 43. No. 257. May 1922. 30 930 Research Staff of the G. HK. C., London, on the rather than by the temperature of the cathode; the process differed from that associated with the glow only in being much slower. The question was therefore raised at what pressure the absorption ceases. If absorption is due to ionization, it should continue indefinitely, even at the very lowest pressures, so Jong as the applied potential is greater than that required for ionization. Experiments, which need not be described in detail, confirmed this view. The absorption of gas in the high-voltage lamps which were used in the main experiments does not cease when the glow disappears and when the bridge of Par. 4 shows a complete “clean-up.” If an ionization gauge, adapted for very low pressure, is used, the pressure will be found to decrease continually, but with - continually decreasing speed, until the limit of the gauge is reached. If the usual precautions against the evolution of gas from the metal parts are adopted, a vessel can be evacuated until it shows the characteristics of Langmuir’s ‘‘high vacuum” by simply connecting it to an ordinary incandescent lamp burnt atits normai voltage ; but the exhaustion, even if the volume of the vessel is small compared with that of the lamp, may take several days. On the other hand, no indication has ever been found of absorption at potentials too small to cause ionization. Experiments on this matter must be made on _ nitrogen, on account of the peculiar property of hydrogen in dis- appearing without ionization (cf. IL., p. 686). It appears that ionization of the gas, and possibly of the phosphorus, is the one condition necessary for the absorption to occur. Some abortive attempts have been made to establish this proposition more certainly by ionizing a gas at atmospheric pressure by ionizing rays; but the proportion of the molecules that can be ionized in any finite time by such a process is so small that success was hardly to be expected. Destination of the Gas. 15. Though all the evidence goes to prove that chemical action in the ordinary sense plays at most a secondary part in the absorption, it was thought desirable to trace as far as possible any chemical changes that take place. The gases in the lamps used in the absorption experiments were analysed by the method recently described by us*. The only gases which have ever been identified “in the lamps, besides * Research Staff of G. E. C., London. Phys. Soc, Proc. xxxiii. p. 287 (1921). Disappearance of Gas in the Electric Discharge. 931 those with which they were filled, are H,O, CO,, PHs, and (when nitrogen was used) NH. Since the “con- densation point” of all these gases is above liquid-air temperature, they can be detected certainly if their partial pressure is above ‘0005 mm. in the presence of 0°1 mm. of non-condensible gas. No other gases with condensation points above that temperature were present. This conclusion is important, for it seems to dispose of the hypothetical compounds of extraordinary constitution that have sometimes been imagined to account for the absorption. If the process of absorption was interrupted after the first stage of Par. 13 (which cannot be subdivided), no gas was ever found except hydrogen or nitrogen. The phosphorus vapour disappears during the first stage, and no compounds are present in the gas in appreciable amount. This observa- tion does not prove that compounds are not formed, but only © that, if they are formed, they are absorbed as fast as they are formed. The restoration of the absorbed gas on heating the vessel was partial'y described in (II.}.. Much more complete observations have now been taken, the results of which will be summarized. The great difficulties in drawing certain conclusions from these experiments are (1) that at the temperatures necessary for the permanent restoration of the gas, there is an almost inexhaustible evolution of water and GO, from the vessel, even if it has been thoroughly baked and exhausted beforehand and no discharge has passed through it *; (2) that the temperature necessary for restora- _ tion may change the nature of the gas restored. (1) cau be avoided hh a great extent by always comparing the gas obtained from the vessel in which gas has been absorbed with that from a precisely similar vessel treated similarly in which no gas has been absorbed. At room temperatures the absorption is quite permanent ; vessels have been kept for a year (without the passage of a discharge which might absorb again the gas restored), and at the end of that period have shown no signs of the restora- tion of gas to the most sensitive tests. Heating the vessel to 100-120°C. causes some restoration of gas, which disappears again when the vessel cools. The first permanent evolution of gas occurs at about 180° C., but the evolution is not rapid until 300° is attained ; at 350° the restoration is effected in a few seconds. Permanent restoration is always accompanied by the appearance of phosphorus * R. G. Sherwood, Phys. Rev. xii. p. 448 (1918). 30 2 932 Research Staff of the G. E. C., London, on the vapour; it seems, therefore, as stated before (II., p. 699), that the evaporation of the deposited phosphorus is necessary for that restoration. In addition to phosphorus, hydrogen, water, and GO, always appear, even if the vessel was originally filled with nitrogen ; if it was so filled, nitrogen appears”. Ihe presence of CO, is an almost certain proof that these gases are derived from the glass. Since they are produced in greater quantity than if the discharge has not passed, it may be concluded that they are produced by the discharge, the hydrogen being reduced from water by the filament. The liberation of gas from the glass, even while absorption is going on, seems inevitable ; it doubtless accounts in part for the fact that, if the absorption is very slow, the amount of gas that can be absorbed is less (cf. Par. 7). PH; appears fitfully in small amounts, never exceeding 15 er cent. by volume of the “restored”? gas. It is known | p yi g that PH, dissociates at the temperatures used in restoration, and it was thought at first that its small and variable amount might be due to varying extents of dissociation ; but when it appears, its quantity is not greatly changed by a further maintenance at the highest temperature reached during restoration ; 1t appears to be in equilibrium with the P and H also present. NH, has sometimes been detected when nitrogen has been absorbed ; but its amount is again small and variable. When hydrogen has been absorbed, it is difficult to tell under what conditions it is all restored, owing to the admixture of hydrogen compounds from the glass. But when nitrogen has been absorbed, this complication does not enter, and it can be established within an error of some 10 per cent. that the gas restored by heating for a few minutes to 300° is equal to that absorbed, so long as the absorption has been effected by phosphorus. If, however, the nitrogen has been absorbed by evaporating salt from the wire, as described in Par. 12, little, if any, of the gas can be restored, * In IL., p. 700, 1t was stated that nitrogen absorbed in the presence of phosphorus could not be restored by baking. But in those experiments the vessel was always filled with more nitrogen than the phosphorus would absorb, and a glow discharge continued through the remaining gas for some time after the absorption was really complete. In these circumstances there is much sputtering of the cathode, and, as sus- pected before, the layer of phosphorus and absorbed yas becomes coated with involatile tungsten. In these experiments, on the other hand, the nitrogen originally present was completely absorbed in a few seconds and there was no appreciable blackening of the bulb; accordingly the gas could be restored by baking. Disappearance of Gas in the Hlectrie Discharge. 983 even by heating the glass to near its softening temperature. The same appears to be true when hydrogen is absorbed ; but, for the reasons explained, the conclusion is less reliable. These observations indicate again that the removal of the solid layer in which the gas is absorbed is necessary to the restoration of the gas. There is similar evidence that, if salt and phosphorus are evaporated together from the wire, some of the phosphorus is so entangled in the salt that it is permanently removed from the action. This is the reason why quantitative experiments in which salt and phosphorus are evaporated together are unreliable (cf. Par. 8). One further fact may be noted. ‘lhe gas restored by heating does not depend on the pressure of the gas in the lamp during absorption. Even when this pressure is as great as 0°5 mm., if the unabsorbed gas is pumped out completely and the lamp then sealed off and heated, the gas restored is indistinguishable from that which would have been obtained if the absorption had taken place from a pressure so low that all the gas originally present could be absorbed. Even ata pressure so high that the free path of ions 1s very small compared with the dimensions of the lamp, the absorbed gas is to be found on the walls of the vessel. i 16. Whitney has shown™* that other substances than phosphorus possess much the same specific action in promoting absorption, namely sulphur, iodine, and arsenic. The first two we have not investigated, but we can con- firm his conclusions that the behaviour of arsenic is indistinguishable from that of phosphorus in most of the matters “eeorecel here. All these substances resemble each other in being elements and so not decomposed by the discharge, and in having more than one allotropic modi- fication. The bearing of the last property will be evident from our suggestions in Par. 20. General Conclusions. 17. We do not profess to offer a complete theory of all the observations that have been described. But it seems to us clear that the suggestions as to the action of phosphorus in promoting absorption, which were offered in III. p. 704, are generally confirmed. The action of this substance is due to * Cf. 8. Dushman, General Electric Review, xxiv. p. 680 (1921). 934 Research Staff ot the G. H. C., London, on the the formation of a layer of red phosphorus * on the walls of the vessel under the action of the discharge, and the gas absorbed is held by this layer. On the other hand, the view that the layer merely acts by protecting the absorbed gas from further bombardment is probably too crude. For the experiments with salts or tungsten evaporated from the filament show that layers of these substances on the walls promote absorption in a manner very closely similar to that characteristic of phosphorus. Since there is no reason to believe that these substances, once deposited, can be removed and deposited again (as suggested in II., p. 698), the gas that is held by these layers must he above rather than below the layer. The view to which we incline now is that solid films deposited on the walls from ionized molecules under the action of the discharge are in an electrically polar condition ; they may perhaps be regarded as bearing a free charge on their surface. When gaseous ions suitably charged approach this polar surface, in virtue of their charges they enter into a combination with it and cannot be liberated again except by a considerable rise of temperature. The action is similar to that which Langmuir has imagined for the explanation of so many surface actions. This supposition will form the basis of our discussion. ; 18. First it should be noted how completely similar, up to a certain point, is the action of phosphorus and of the salts (a term which, for brevity, will be used to include tungsten). The “first dee » of Par. 13 in absorption by phosphorus (which alone will be discussed for the moment) corresponds exactly to the absorption of gases in the absence of phos- phorus by evaporating the salts in their presence (Par. 12). The quantities absorbed in this stage depend so greatly on the pressure of the gas and other factors that quantitative comparison is difficult; but the amounts absorbed in the two cases under comparable conditions are not very different. Again, the amounts of gas that can be absorbed by a layer of phosphorus and by a layer of salt are very similar. It is natural to identify the step of fig. 1 with the amount that can be held by a single complete layer of phosphorus, and this step is very nearly, at least, the amount that can be held * In speaking of “red” phosphorus, we do not mean to commit our- selves to the view that the element is present in the molecular condition characteristic of red phosphorus in bulk. We only mean that it is present in some condition differing, like that of red phosphorus, from the white variety in the fact that the vapour-pressure is inappreciable at room temperature. Disappearance of Gas in the Electrie Discharge. 935 by a layer of salt, whatever its chemical nature.. Further, the phosphorus curve rises to the step with increase in amount of phosphorus nearly as steeply as the amount absorbed by salt ; in fact, we have never been able to satist'y ourselves that there can be absorbed by phosphorus an amount of gas intermediate between that which can be absorbed by plain glass and that corresponding to the step. (The difficulty of deciding the matter is that, if neither salt nor phosphorus is present, the absorption is not determinate, but represents an equilibrium between absorption and liberation by decomposition of the glass under bombardment.) It is not at all clear why the increase in absorption up to the step caused. by salt or phosphorus is discontinuous; for, according to the theory proposed, it might be expected that quantities of material insufficient to cover the walls completely would cover them partially and so give an intermediate absorption. It is possible that, if the walls are not completely covered, absorption isagain masked by liberation of the gas from the glass; but the difficulty, whatever its explanation, is the same for phosphorus as for salt. The increase of absorption with quantities of phos- phorus greater than that required for the step, which is not shown by salt, will concern us presently. The quantity of gas absorbed at the step is not very different from that corresponding to a monomolecular layer. The caleulated amount is 0016 mgm. (or ‘0008 if the layer is monatomic), the observed 0023. The discrepancy is explicable if the molecules present to the surface a diameter less than that estimated by the usual methods, or, in ether words, if the molecules are attached “end-on.’? The view that molecular layers are concerned would explain the close similarity between the volumes of nitrogen and hydrogen absorbed in similar circumstances. 19. If this general view is correct, it is natural to inquire in which direction the polar layer is directed and ions of which sign enter into combination with it. The very small absorption of the inactive gases, which do not form negative ions, suggests that it is negative ions which are absorbed by the layer. We have not been able to devise an experiment to test this point, but there is some indication from another source that layers of salt, deposited as we have described, are able to bind negative charges. For Hamburger and his associates have shown * that layers of salt so deposited have * L. Hamburger, G. Holst, D. Lely, E. Oosterhuis, K. Akad. Amsterdam Proc., Sept. 21, p. 1078 (1919). 936 Research Staff of the G. E. C., London, on the the power of decolorizing tungsten subsequently deposited on them. They attribute the action to the formation of large mole- cular aggregations of the metal which cover but a small part of the surface, but they produce little evidence for their view except a proof that the action cannot be due to the formations of chemical compounds, as had been imagined previously. Now, the decolorizing of tungsten might be effected merely by the binding of the free electron and the transference of absorption to a definite region in the ultra-violet, similar to that of most chemical compounds in which the electrons are “bound”; and itis known that films of tungsten too thin to show appreciable absorption in visible light show strong absorption of ultra-violet-light. We suggest tentatively that this decolorization of tungsten is physically the same process as the absorption of gas; both represent the binding of a negative charge on an electrically polar layer. Whether such binding is to be called ‘‘ chemical combina- tion” is a question of words; we prefer (cf. 1., p. 586) to confine the term ‘‘ chemical” to cases where separable compounds of fixed constitution are found, obeying the law of constant proportions. But the action would, in our view, be “chemical” in the sense employed by Langmuir in his theory of surface actions associated with mono- molecular films. Perhaps we may proceed further. In II., p. 703, we suggested that there is only one reason for believing that the action of phosphorus in promoting absorption is ‘“‘ chemical,” namely that the absorption of gas decolo- rized the yellow film of phosphorus on the walls. If the colour is associated with the positively charged portion of the phosphorus molecule, the binding of negative charges on the layer might well decolorize it. Once more, if this explanation of the matter is correct, absorption by phosphorus is closely related to chemical action; but it is not chemical in the sense that any compound of phosphorus and the gas in definite proportions is ever free from atoms of those elements in the ‘“‘ uncombined ”’ state. 20. We have so far discussed actions in which salt and phosphorus can be mutual substitutes. But the salts do not possess all the properties of phosphorus in the promotion of absorption. With phosphorus, but not without it, pressures well below the glow potential can be reached starting from pressures well above it. On this matter we have nothing to add to the explanation offered in II., p. 697, based on the Disappearance of Gas in the Electric Discharge. 937 wide difference between the rising and falling glow potentials of phosphorus vapour. The other differences are probably due to the fact (1I., p. 698) that phosphorus once deposited on the walls can be liberated and re-deposited under the action of the discharge, while salt cannot. This re-deposition doubtless accounts for the increase in absorption beyond that corresponding to a single layer obtained when the amount of phosphorus is increased beyond that required for a single layer, even if the phosphorus is deposited on the walls before the absorption begins (see the dotted line of fig. 1). ‘The form of the full-line for the larger quantities of phosphorus is more difficult to explain in detail, for here the first stage of Par. 13 as well as the second is concerned ; it is not unreasonable on any theory to imagine that the absorption will be greater when the gas is actually mixed with a large proportion of the vapour being deposited. The decreased absorption with the largest quantities of phosphorus, of which we are sure experimentally, we attribute to the fact that, when such large quantities are used, the number of phosphorus molecules present in the lamp just after evapo- ration is much greater than the number of gas molecules. In these circumstances the phosphorus molecules rather than those of the gas are ionized, and it is the former rather than the latter which are deposited on the walls. 21. Finally, there remains the question of absorption without either phosphorus or salt. Is this also to be attributed te an electrically polar layer on the glass surface? We imagine that it is; for all surface layers are bound to be electrically polar to some extent, since the elementary charges are not coincident. There is also the possibility that the surface layer is rendered polar by the action of the discharge, which also ionizes the molecules and makes them able to adhere to it. Against this latter suggestion is the fact, apparently ascertained, that the absorption takes place on portions of the walls which have not been exposed to the discharge, at least as readily as on those so exposed. But all these matters require further investigation. F938 XCIX. An Attempt to detect Induced Radioactivity resulting from a-Ray Bombardment. By A. G. SHENSTONE, B.A. * ir recent experiments > Sir Ernest Rutherford and Dr. Chadwick have shown that a certain proportion of collisions of a-particles with atoms of some of the lighter elements causes the ejection by the nucleus of charged particles which can be identified as hydrogen nuclei. From the consideration of their energy of emission, it is shown that these particles must have derived a portion of their energy from the internal energy of the original nucleus. Moreover, the work of Dr. Shimizut in the Cavendish Laboratory on the photography of the tracks of e-particles in saturated air suggest that a much larger proportion of a-particles, of the order of 1 in 300, produces some other form of separation of the constituents of the nucleus. In some cases this action is apparently the ejection of a helium nucleus. Such facts have very materially increased our knowledge of the structure of the atomic nucleus. But, because of our almost complete ignorance of the forces existing in and immediately around the nucleus, no definite deductions can be drawn as to the subsequent history of the nucleus from which a portion has been ejected. ° It would seem, however, that such a violent partial dis- memberment of a composite system could not fail to have still more far-reaching results. The stability of the nucleus cannot fail to be upset to some extent ; and lack of stability in the nucleus is, we believe, essentially the cause of radio- activity. The critical point is the degree of that instability. Tt may be so great that the nucleus is completely and instantaneously dismembered ; or it may be so small that the atom almost immediately regains equilibrium in a new form. The existence of isotopes of many elements is strong evidence that the latter condition is quite probable. Between these two extremes, however, is the whole range of degrees of instability exemplified in the radioactive substances. If the effect of the condition of the deranged nucleus is the further emission of high-speed particles after a time of the order of hours or minutes, it would almost certainly have * Communicated by Prof. Sir E. Rutherford, F.R.S. Tt Roy. Soe. Proc. A. xevii. p. 374 (1920); Phil. Mag. Nov. 1921. t Roy. Soc. Proc. A. xeix. p. 432 (1921). Radioactivity resulting from a-Ray Bombardment. 939 been detected before now; but if the time is of the order of a small fraction of a second, it would not have been differentiated from the direct effect by any experiments so far performed. The present experiment was therefore carried out to extend observations down to a time-interval that would correspond to very rapid changes. There were two essentials governing the experimental arrangement used. to attack this problem. The first was a very short time interval between the «-ray bombardment and the observation. The second was the smallest possible space- interval between the bombarded atoms and the observing device in order to detect particles of short range. The scin- tillation method was used, and the problem therefore narrowed to the detection of particles of mass and velocity capable of producing a scintillation comparable with the H-scintillation. The small time-interval was obtained by the use of a nickel steel disk of 8 inch diameter which could be rotated at avery high speed.. The time-interval between the bombard- ment of a point on the wheel and the observation on that point can then be made extremely small by making the dis- tance between bombarding source and observing-screen very ‘small and the speed of the wheel very large. After a few minor difficulties, the final arrangement of the source of a-rays and of the observing-screen was essentially as shown in the attached diagram. A lead bracket having a half-inch hole was screwed firmly to the framework holding the wheel so that its surface was about 1 to 14 mm. from the face of the steel disk. ‘he metal source was of 1°2 cm. diameter, and was arranged to be held in the hole of the lead bracket with its active face about 2 mm. from the surface of the steel disk. It was found necessary. to protect the active material from the draught caused by the disk when at high speed in order to prevent. contamination of the disk itself. A thin sheet of mica of stopping power equal to ‘9 cm. of air was accordingly waxed over the opening in the lead bracket. It was unfortunate that this was necessary, since it cut down the range of the a-particles to 6°06 em. and consequently reduced their effective energy. The distance between the source and screen was limited by two factors ; a certain minimum thickness of lead was required to cut down y-radiation and so fluorescence of the screen ; and it was necessary to separate source and screen sufficiently to prevent the possibility of too many hydrogen particles directly ejected by oblique «-particles from reaching 940 Mr. A. G, Shenstone: Attempi to detect [Induced the screen. This distance in practice was 1°2 cm. Actually with the arrangement of distances such as is shown in the diagram a few hydrogens could reach the screen. With aluminium they were observed about 1 in 6 or 7 minutes whether the disk was running or stationary. To avoid vibration the ZnS screen was mounted on a very stiff attachment on the microscope objective, and remained in perfect focus even when the vibration was at its worst. Bearings of Disc. Mica = cm. tr. ee 1 Source. i \// Nickel Stee! Drse. Zn S Screen. Lead Angle holding Source, hs Microscope Sereived to Objective. Framework (fu of Dise, For Source. APPARATUS FOR INVESTIGATION OF INDUCED RADIO-ACTIVITY. 4eed Brackec, Threaded Hose Yor Source, Difficulty was had at first from the wind from the disk blowing the crystals off the screen, but this was satisfactorily overcome by covering the screen with aluminium foil except’ for a hole over the portion included in the field of the microscope. The microscope used was fitted with Watson’s “ Holos” X 5 eyepiece and Holos objective of *45 n.a. and 16 mm. focal length. A rim of the substance under exami- nation was stuck on to the face of the wheel with shellac, which successfully withstood the huge centrifugal force Radioactivity resulting from a-Ray Bombardment. 941 except in the case of lead foil. After one accident it was not thought advisable to run more than 2 max. speed with lead. With the available driving apparatus it was pcssible to drive the steel disk at a speed of 250 revolutions per second. Taking the circumference for the bombarded portion of the wheel as 60 cm., this gives a velocity of 15,000 cm. per sec. Since the distance between the edge of the active source and the screen was 1:2 cm., ke time for the wheel to move between these points is eT sec. But from the diagram it is evident that the heavily though obliquely bombarded portion of the wheel has to move through only about ‘5 cm. | in order to reach a position where it can un turn bombard the screen. ‘This gives a time-interval of a5 “x 8€C- The observations were first made with the wheel stationary or running very slowly, and the numbers of natural scintil- Jations counted. Except in the case of lead, these never amounted to more than 1 in 2 minutes. The wheel was then gradually speeded up and the screen observed as far as possible continuously. ‘The rcar of the wheel when running at high speed was not, at first, conducive to perfect concen- tration on the observer’s part ; but, after one experiment had been carried out in safety, no further difficulty was experienced. The materials tried included a wide range of atomic weights. The first tried was carbon in the form of paper. No effect at all was found. Aluminium, because of its proven emission of H particles, was considered a very likely material, but fromit also no effect was observed. The wheel itself, of nickel steel, was next tried ; but, like the others, it gave no observable emission. The lower limit of observation in these experiments was probably one scintillation a minute. Two would most certainty have been detected. In the case of lead, however, the number of natural scintillations observed was as high as three a minute, so that the limit of detection of additional scintillations was probably not better than three a minute. With lead no effect at all was observed with the disk running about 160 revolutions a second. A rough calculation shows that the period of decay of the induced radioactivity, if it exists at all, must be extremely short to have remained undetected. Sources of about 25 to 30 mgs. activity were used. Assuming that 5 of the 2-particles could reach the disk and that 1 in 108 ejects a portion of the nucleus, the number of abnormal atomic 942 Radioactivity resulting from a-Ray Bombardment. nuclei produced per minute is 29x37 10x 60 10®x 10 = 95500. Roughly 1 in 250 of the particles which might be emitted by the abnormal atoms would strike the ZnS screen if the - screen were directly over the bombarded portion. Assuming that the number of particles emitted decreases exponentially with the time and that one a minute could be detected, the time between bombardment and observation must have been at least four times the half-period of decay of the substance. This gives a maximum half-life of iam * i=2 x 10-5 sec, The above calculation makes use only of those atoms changed by the ejection of an H-particle. If the variety of disruption observed by Mr. Shimizn is included, the minimum observable half-life is considerably decreased since the number of abnormal nuclei is now 25 x 3:7 x 10" x 60 =)9x 10° per min. 10? x 10 But SOX MLOE ce Ties 7250. Be a Therefore the maximum half-life becomes ie ik Gu Eas ay eee! “Ge 12,500 x 17 =4 1x 10° ° Sec: In order to examine the possibility of a cumulative effect from a long, heavy bombardment by the «-particles, an arrangement was made by means of which a metal strip, atter several minutes’ bombardment, could be brought beneath the ZnS screen after an interval of about ;),5 sec. This likewise gave no observable result. The negative results obtained in these experiments do not, of course, preclude the possibility of radioactive disintegra- tions taking place which involve the emission of @-particles, y-radiation, or of mass particles of range less than 2-0 mm. But the evidence is very strong, in the case of aluminium, carbon, and iron, that no mass particles of range greater than 2-0 mm. are produced after an interval of 8 x 10~° sec., and that none of range greater than 6 mm. are produced after an interval of 3°3x107° sec. The corresponding intervals for lead are 1:2 x 1074 and 5:0x 107°. Onsymmeirical Components in the Stark Effect. 943 It is very unfortunate that time did not permit of further experiments with a wider variety of elements and with devices for the detection of radiation of other kinds. The importance of a complete investigation arises from the fact that the tracing of the subsequent history of the atomic nucleus which has been disrupted by the collision of an a-particle is, at present, one of our few paths to a knowledge of the forces within the nucleus. In conclusion, I wish to thank Sir Ernest Rutherford for giving me this very interesting problem ; and Mr. Bieler for his assistance during observations. Cavendish Laboratory, Cambridge, 1921. C. On the Appearance of Unsymmetrical Components in the Stark Effect. By A. M. Mosnarrara, B.Sc.* 81. Preliminary. ee theory of spectral lines which has hitherto proved most successful in interpreting the results of experi- ment is based upon certain assumptions of a quantum type introduced by Bohr+, Sommerfeld t, and others. Such assumptions are only justifiable in so far as they give satisfactory interpretations of correlated phenomena. ‘The effect of an electric field upon spectral lines emitted by substances subjected to the field was first investigated by J. Stark § in 1913; and an approximate theory was furnished by K. Schwarzschild || and by P. Epstein J inde- pendently in 1916: the two theories are similar and give satisfactory explanations of the phenomenon as investigated by Stark. Now, according to their theory, the components into which any given spectral line is split up are symmetrically distributed about the original position of the line. In the * Communicated by Dr. J. W. Nicholson, F.R.S. Tt See e.g. N. Bohr, “Constitution of Atoms and Molecules,” Phil. Mag. July 1913. a ee Arnold Sommerfeld, ‘ Atombau und Spektrallinien,’ IT. Auf. 1921). " § Bevin Sttzungsber., November 1913; Ann. d. Phys. xliii. p. 983 i i Schwarzschild, “Zur Quantentheorie,” Berliner Sitzungsber., April 1916. q P. 5. Epstem, “Zur Theorie des Starkeffektes,” Ann. d. Phys. 1. p. 489 (1916). /S9AA Mr. A. M. Mosharrafa on the Appearance of vy present paper a closer approximation is worked out, and it is Found [see § 4] that for stronger fields than those used by Stark this symmetry no longer follows from the theory: on the other hand, a pair of components which, for fields com- parable with those that Stark used *, appear symmetricaily situated, would for stronger fields be displaced in the same dur ~ection, so that the symmetry is destroyed. We, naturally, also find that the relation between the strength of the field and the displacements of the lines is no longer represented graphically by straight lines, but by parabolic curves whose curvatures change sign with the displacements (7. e. displace- ments of opposite signs correspond to parabolas of opposite curvatures). It appears to the present writer that an experimental investigation of the Stark effect for fields stronger than those that have already been employed by Stark is highly desirable as a further test of the fundamental hypotheses of the quantum theory of spectra : if such an investigation result in the verification of the predictions already referred to, then this will add to our faith in the foundations of the quantum theory of spectral lines: whereas a negative experimental result would, unless the analysis here presented be at fault, lead us to a reconsideration of our assumptions, and perhaps to certain modifications thereof. § 2. Previous Work. The equations restricting the motion of an electron moving under the influence of an attraction towards a nucleus as well as a fixed foree F can be written in the form n= (vVA® dé=nyh, Soe) (a(n) dn=nsh, ) ving” adv =nszh, f @) where f is Planck’s quantum of action, 7; m2 3 are whole numbers, my is the mass of the electron, and f\(£&) j(m) are given by 20} 5 f= 2(cB+ 8) + 2WE—eF ES, | f - (2) fl) = 2(eH—B) + 2W7? + ent — —, | 0 * For the H lines, e. g., Stark used a field of about 28,500 volt x em.—1 (=95 c.e.s. electrostatic units). We find that a field of about 10 times this strength would give quite measurable effects. Unsymmetrical Components in the Stark Effect. 945 Here (—e) is the charge on the electron, E that on the nucleus, and a, 8, W are constants arising from the-inte- gration of the Jacobian Differential Equation*. W represents the energy of the electron. The coordinates £ and 7 are parabolic coordinates in accordance with the equation Py aera Be Gy where w,y, 2 are Cartesian coordinates at the nucleus, Ox being chosen parallel to the external field F. The limits of integration for the two first integrals in (1) are the maxima and minima of & and 7 respectively. Now these two integrals are both of the same form ; so that we can write: Py se ar 4 eer — 2ahak sep i (ch) thus denoting the two cases for € and 7» by the suffixes 1 and 2 respectively we have ‘ nh \* A,=2m,W, B,=m,(eH+8), G= -(5=) 5 (er) D,=—mel ; nsh\? A,=2m,W, B.e=m,(eH—8), O=—(57) L655) D = + meF. Now Sommerfeld t works out the value of the contour integral on the left-hand side. The value he gives is oer — 5B Do (32 From (4) and (6) we can write, "ee eee elit ie Def.) 3B? B=— vVA( ¥O+=*) + / Bean): Gb) Both Sommerfeld and Epstein have obtained the value of W | which Hpstein denotes by (—A)] by slightly different methods to the first order in F [Epstein’s (—E)]. We shall proceed to a second approximation. * For a fuller treatment of this section, see Epstein’s paper already referred to, also Sommerfeld’s ‘Atombau u.s.w.’ II. Auf. p. 542, and p. 482. Jacobi’s method of integrating the Hamiltonian transformed equations is also given by Appell, ‘ Mécanique rationelle,’ 11. p. 400 (Paris, 1904). + ‘Atombau u.s.w.’ Zusatz vii. p. 482, under f.; we, however, write V6 for his (— VC). Phil. Mag. 8. 6. Vol. 43. No. 257. May 1922. 3P 946 Mr. A. M. Mosharrafa on the Appearance of § 3. Calculations for Comparatively Large Fields. We shall treat the term in D in equation (7) as a corrective 0. 0 PRO term. Let 8, 8B+A8=6', 6'+A'B=B", etc., denote the successive approximations to the value of 8; similarly for A, B, and W. We see that to the first order of small quantities SB Anse See from (5a) Byte Sar, Bo, Re) and similarly 1 0 0 0 | B= B2—2m,BeAB from (50). . 2) 39035) 0 Now the equations for determining A@ could easily be solved, but as we are assuming Epstein’s work we shall merely give here the value obtained on solving his equations (61) *. We have 0 7D N. EF. M(ai t+ 22 73) or 64m,%eK?ar* 0 ails where N= (6nq? + 6ngn3 +73") (2n, +723) + (6217 + 6nynro +737) (2rg+73), (10) so that we have from (8) and (9) 1 0 0 ht(ny tno +nz)N B,? = By? + 2m,B, x arene : (Gn: a) similarly t v 0 h* . (ny +no+ng).N bse — bee SE oa ee . 2 2 m,Bs x 64m 2e Rar! a eee LIL i) 0 also, B? is obtained from (7) on neglecting the term in D, thus : B= A( VO4+ ss ee ie ae d. Phys. |. p. 508 (1916); our AB corresponds to Epstein’s Can Unsymmetrical Components in the Stark Effect. 947 Froin (11) and (12) we have ; ES eee. SS eal See othe a Y ee Ea CLL pm = = VY, iit — l 4 ae The plate is of inner radius a and outer radius b, and is symmetrical about the plane z=0. Let @ be proportional to the thickness of the plate, so that <=te and z=—ta may be taken as the equations of the upper and lower bases respectively. Here ¢ is a small parameter, since the plate is thin. Radial forces are applied to the inner edge, so that points * Communicated by the Author. + I am greatly indebted to Mr. Carl A. Garabedian, of Harvard University, for able assistance in carrying through some of the calcu-. lations and for verifying those [have made. Mr. Garabedian is under- taking the consideration of more general problems by: this method. 954 Prof. G. D. Birkhoff on Circular for which r=a undergo a displacement of amount e. The plate is clamped at the outer edge. Thus,if U and w denote the radial and axial displacements, the boundary conditions are UG O=jSer. UO 0 0 ! UD a. w(b, 0) = Obviously such a nee is unstable, with a tendency to buckle. It is a cardinal fact of the theory of elasticity that the actual displacement of the plate will be such as to yield the minimum potential energy consistent with the constraints imposed (Love, ‘Theory of Hlasticity,’ third edition, p. 169). By Love (pp. 99-100, 141) this potential energy W is given as follows :— Warf ae ao 4U aU Udw OU Ow +uf (2 orale ory rae EN OE or Tae se | be dz dr. It is natural to introduce a new variable z', such that z=t2'. When we replace ¢ by <' and afterward suppress the accents there results :— (A) W= =a | fo+2o[® au u ey 10QU ou) UoU _4U0w ee i tal; Oz * or Te or trode t oroe \rdedr, Our assumption will be that all of the quantities involved can be expanded in ascending powers of t—in particular that WU # Ue a oy } w= Wo + Wit + Wot? Sle akernawa Botea is It is to be noted that if in (A) we replace z by —z and w by —w, ort by —ét andw by —w, the double integral is altered at most in sign. Since these transformations do not disturb our boundary conditions, and since W has a unique minimum, the special relations (B) Unlr,2)—Un(r, —2)=wWnlr, Z) + Wm(r, —2z) =, m= Oe (BY) Usnvi@y 2) = win’, 2)=0," m=O) 1 eee ; must obtain. The case of a plane plate shows that the energy is of Plates of Variable Thickness. Joe order ¢. Hence if W is to be a minimun, it is clear that we must make the leading term of W, namely mu(? (* Tous?» 5. ral) ee S. | r dz dr, vanish if possible. But this quantity vanishes if, and only if, U,=U,(r), and by thus restricting U, the stated boundary Beet ves! wh not violated. When U, is thus restricted, the double integral W will involve no negative powers of ¢, and will have as leading terol =— ni ‘s (A + 2p) | Ue fe 4- Al ee — Uo 0 Gi ote dz dr, where accents denote pans with respect to r. The part of the integrand in braces may be written as the sum of squares :— Ow)? oe Owi\27 rA| U,) + = PSs a 9 2 = [ +s ‘5: | +24 Uo? + 5. We turn next to the choice of ee , which is an arbitr ary Se? function of r and z still at our disposal. If we call this variable 2, the above expression involves x in two terms of the form A(m + x)? + 2yx?. ? | NE Elementary calculation shows that for «=—~—. , the fore- A+ 2p eee : ber 7a ets going expression has the minimum value 5, nv. Hence we must take Tal Ow, —xXr bag i —_ U,+ — Oz A+ 2p ( ce a or == Ne if ae Uo f y= eee e) tse) sir) arbitrary. Bat wir, 0)=O0.by (B); hence s(r)=0. This choice of w, does not interfere with the boundary con- ditions, The terms written above now reduce to give ek? U 5 Us) nif ("4 {o+ p)(U,"+ oS —) vr dz dr, 956 . Prof. G. D. Birkhoff on Circular and an integration with regard to ¢ can be explicitly per- formed. The principal part of Me takes the form Wt, where 8 nw W,= =e) ic + js) (U o7 + opi AU, 7 te dr. (1) This integral is to be pee a minimum subject to the boun- dary conditions ee a) en SUC) = 0: Accordingly our problem is reduced to a simple problem in the Calculus of Variations. The condition 6W,=0 gives at once 426 20, dr fe) Uo fe) UG ‘ } where ® is the integrand in (1). Writing out this equation in full, we obtain Lone : = 2(+ 4) Uy! +2 far— 20+ a) = +2U,! |a=0. Consequently Up) must satisfy the folllowing differential equation :— Mo aU) h' AY A I PO aleO Re tier tes ! 0 Moa P ae h Dos 2 A+ ph) 7 } ya where 2h= 2ta stands for the thickness of the plate. In the case of a plate of constant thickness, / is constant and h'=0, and (3) reduces to a well-known form (Love, p. 141). Thus far we have determined the displacements to be U=U,(7) + U4 + PR A Az 5 — =) Gee eis) ie ° 5 is ee ( Uo + +)i + wt? + : ( where it is to be remembered that the accent on z has been suppressed. We proceed to determine U, from the fact that the body forces F,, Ff, must vanish. We have (Love, p. 141): He OL UN ew VU Ree cas 2) ( Bedr BB =0, (6) Bye S40} oee xe) —p¥ =~ ry et —e seit v) (Re 1P)a0. @ Plates of Variable Thickness. 957 Substituting (4) and (5) in (6) and (7), the terms in ¢~? and t-! vanish. From the constant term in (6) we obtain fet OF dA+4y (U Wy Uy an Rain pe Tp Nar Gage 9 Rye rA+2u r whence, using (B), Bey dA+4u 32 1 Was Ug UE aR Te ea TN a (Uo Se stein +9(r). Also, the constant term in (7) vanishes. Furthermore, the surface tractions must vanish on the free surfaces z=, z= —x; hence (Love, p. 76) the following _ equations must hold on these surfaces :—— X,= X. cos (a, v) + Xz cos (y, v) + Xz cos (z, v) =0, Me cos. (v.07) + Y7, cos (y, v) + Vz cos (2)/v) =O). 2. (8) Z, = Z, cos (#, v) + Zy cos (y, v) + Zz cos (z, v) =0. lf we consider a tangent plane at an arbitrary point of the surface z=a, the direction of the normal is denoted by », and (8) gives the tractions across the tangent plane in terms of the stress-components across planes parallel to the rectan- gular coordinate planes. [For cylindrical coordinates, we have (Love, pp. 100, 141) :— KS AA + Bee =O +249" , E U Y,=NA 2 peg =AA +2, Die =Mbézz =A 2 2 uo, : YC — Ly = pbeyz = 0, toSonmanid BB) ney == Y= Bexy = 0, where To determine the direction cosines, we return to our original variables and observe that tan 0=ta’. 958 Prot. G. D. Birkhoff on Circular Hence we find along the « axis ie! cos (a, v) = —j——.,, cos (y, v) = 0, cos (z, vy) = ——————.. ook v) a/ Lea (y v) k 3 ) a) eee i Hence the conditions that the tractions vanish on the free surfaces become te’ X;,-+ X,=0)) . ee —ta' Let+Z2=0.)-°s 2) ee As we proceed to higher order terms, using (6) and (7), the relations (9) and (10) may be expected to play an im- portant réle in furnishing differential equations to determine the arbitrary functions that enter. In the present case there is no constant term in either left-hand member; and by virtue of (3) the terms in ¢ also vanish. Thus the surplus body forces per unit of volume vanish up to order t, and the surplus surface tractions per unit of area vanish up to order ¢?. Hence the total surplus applied force is of the order ¢?, whereas the total given radial force is of order ¢. It seems clear that when the surplus applied forces are removed, no sensible change occurs in the radial or axial displacement, and that we have a solution of our problem. | It remains to compute the principal part of the radial pressure on the inner edge of the plate. Here we make use of Saint. Venant’s principle and obtain the resultant traction, Ne 2) as , ° on (ta (a) 2 / \ Ndr 0) can go 10) ‘ —ta (a) N+ 2 ; » Teta U)/(a}+ 2 [ 0 (4) Z(r +-) a ! Tf we desire to take t=1 throughout, so that the equations of the bases are: z=a and z=—a, then ¢ disappears as an explicit parameter, but the terms of the solution are still ordered according to the powers of the natural parameter, namely the ratio of the thickness of the plate to its diameter. = Plates of Variable Thickness. 959 Case II1.—The Complete Plate under Anal Pressure. As a second example consider a thin circular plate, nearly plane, clamped at the outer edge and subject to an axial force P. Here a small force P yields a relatively large displacement, and our method is put to a more delicate test. Again adopt cylindrical coordinates (fig. 3). Fig. 3. The plate is not restricted to be symmetrical. Accordingly let 2 and 8 be proportional to the distances of the bases of the plate from the plane z=0, so that z=te and z=¢f are the equations of the lower and upper bases respectively. The boundary conditions are the following :— UO, =0; .w(0, 0), w(a, 0)= oe") =(), Again, we take the formula for potential energy as the point of departure. The term in ¢7! is now a i 1 (V+ 2) Bal +p Ea ry y dz dr. This term can only be made to vanish by setting w=w(r), Us=U,(r), and the boundary conditions are not thereby violated. The constant term in W disappears and we can write the integrand of the term in ¢ as the sum of squares :— Ee ey 9. TT! Ow Ee SE oy 2 + ( ae Ut Oz eens = oe J 2 + py Ee +a! | ; In the present case we are led to attempt to make this inte- grand vanish, since the case of an ordinary plane plate shows 960 Prof. G. D. Birkhoff on Circular that the energy will be of order t?. Thus we must take Uj=0, w=wilr), U,=—2u)' +p), where p(7) is arbitrary. None of these requirements violate the boundary conditions. The physical significance of the conditions thus far obtained is immediate ; the displacement is transverse, and such that filaments perpendicular to z=O remain perpendicular in the displaced position and are not compressed transversely. When these conditions are imposed, it is easy to see that the energy is of the order ¢*. More precisely, the leading term in W now becomes a B - Ps isilh 2 3 f CN) Acs ace emanate he ne { Og H)) Blk, sey) se ‘ 4 v We az pee / : tal (G2 ter J 4 S82) (ant) —4{( i Py ou — 4 — 2109!" +p’) oar) rdz dr. if The part of the integrand in braces may be written as ¢ ee . j Ow: il + 2p] (—2200!" +p)? + 25 (— 200) +P)? + (32) | 2 + | +1;'| Since U, appears only in the last term, and since W is to be a minimum, we choose U,= -—2w,'+ 9(7). We proceed next to the choice of ee and find (by the same method as employed in the first example) :— Ows aa =e at E), ai Sy aU ZWo +p! => Zana q Wo —

08 oo © (oF 0) YO 0) a 0) 70 ° ° fa) GY *G: Oe 0017 Os 10s om Or oO aOM OMe O o) loa ©) ° D (oY (ec) o °o ° ce) ° ° .°) ° (ey) {@) oO, 0) oO OUNOY NOLO ©: OOO (oy 9) C o .°8 OF ORFOR On sOu)-O! (0) OVO" (0. (O00 CON ZOmTO OFO720' OF (On)0) (0) (Os 20:20) ©. 0: 0! O_O 1050 COP Oe ORO OO) 10) Of On On 10) 10) nO OF ©. 10), 2O Oo OF FOF Os Oli OF "O70 7070) -O) 30-6 = Ofc). O-20 CAO Ome ©: A CO OpsOR OM On gOW tO, O55 © - (9) 9 (6) Sea Oe OT OgOL Oy (Oe On On 7 Or (O46) 40. 16),, O41..." 10 OF On eOn OOM Om OL Os Oe 0) 0% On 20> [On Os 'On16 to AB is y, then the average electric displacement over AB in the direction of y increasing may be written as | as K, dV 4m 10" x 8989. dy where V is the average potential, and K, is a constant which depends on the relative values of the radius and the distances between two adjacent layers as well as wires in the same layer, and also on the specific inductive capacity of the medium between the wires. (8°989 is the square of 2°9982.) The constant K, may be called the effective dielectric constant of the composite dielectric formed by the wires and the medium between the wires. In what follows, K, is set equal to 8:°989 x 10" «, 3 . AB, Let Cr — Ae. Then the flux out ae DC is z(3) ae dy / yay Therefore the flux of the electric displacement out a the surface of the prism is as AB (Fe) ; — ( ] baa w. ——_ = ee Ay A dy) ee dy oa dy? ae if Ay is small. But AB. Ay is the volume of the prism. 968 Dr. G. Breit on the Effective Capacity of Hence the charge is distributed as if nh Ke a7V “A i Aq dy’ were its average density. The above treatment applies to the inside of the coil. The contour of the cross section needs special consideration. Let AB (fig. 5) be a small part of this contour. Draw the Fig. 5.— Structure of multilayer coil near edge. (opeiof uroyn ito} (op Ttope Loy | (oy 9} -(o) (e) 000020000 0.0 0 oo Oo °O o- ooo 9s (o} 909 00O0008COWUdUCUCOUCOWUCOOlUCUO Cf fe) (ome * So © © © EL © a o TE © EE © ) (o) ©0o0o0€0C~«wMMC=V. Thus Ne is known over the circumference of a circle. It vanishes at infinity. Therefore, if the value of V, over the circumference could be written as a Fourier series in 0, say as ONG ea ney (bm COS MO + a», Sin m0) terre aa then the value at any point (7, @) (r> d The quantity ao in (2) 1s now simply (S) . Linas > OV, Teliais “(1 sin' nt 16 On =a, dt a9 (2n+1)(2n+3)(Qn—1) This solves the first part of the problem. It now remains mi Vi @V to find the. quantities &. - > The first of these is obtained from (3) as | OV OV. eo 4 ld a Ger sin 0 ee ae J —y’sin@, . er and similarly ie (10) a Nex Jade y dye.) ma? dt Jay Multilayer Coils with Square and Circular Section. 975 _ Remembering now that for the point (a, @) | vo orig ae (11) becomes OV © —2L a on Co me at Now (1) and (12) yield Sin OG comer e chi. 1aQlia) O > Qata?. <7 agate and (2) together with (13) and (10) result in ml di 2 (—)" 1sin Qn41)9 Ta dt n=o (2n+1)(2n+3\(2n—1) Ke Wi dos tra a (20) sa) ee CLD) The latter result as well as (13) can be applied only if g =—4 Bos , because otherwise the radical in (11) must be ae as —acos9. Now the charge in the volume ele” between y and y+dy is made up of two paris. The first is due to p, the second to o. The part due to p is 213,/a?—y? dy, and hence by (14) is ” _ Kel Es dt The part due to o is - | 2lo an . cos 67? and hence o (15) fF Oly. =P on (Ape io Ke 3 di dy a 4 ov Sa CREROB) OE wn 26 Ny eee: Expressing the result in terms of @ and remembering that dy=acos 6 dé, the charge between 6 and 6+d6 is Ke an 2 (—)""! sin (Qn +1)0 ,au -|* sin 20-409 & ta oe | lap eas If this is to be the same as 2(0) dé, then PAG | Kerang Bx SI (—)”-! sin (2n+1)6 i 7 a een pe On Een |! ») 976 Dr. G. Breit on the Effective Capacity of It now remains to find M(@). This is obtained from M(y). M(y) itself is given by MG 2/ a? —y? _ 2cos0 Le oy = Ta? a ae because the area between y and y + dy is 2\/a? —y” dy. Hence M(A) _ Mly) dy _ 2c0s?6 os lp nae oe me Therefore 6=+5 5 @ Co=— 2008 fy | “sin 26 J6=-5 T jo—-. _ OW 7 Biko So ote sin Qe De | ar has (Qn+1)(2n+3)(2n—1) tee o=+5 os. ep Ay 224 26 o= _ C08 | 2 ;(1+ cos 20) rach T +) > 8k, 2 (—)""! cos (2n4+ 10 be 1° n=0 (2n = WR CAP a 3) (2n— 7 3 l o= +3 K as e 2 9 cos 2 Oe: [ sefe(1 +00 ate co 20) 2 ma : (—)"-1{eos See Ie Te eG (2n+1)?(2n +3)(2n—1) 1 73K. , 8k & 2 1 ~ at Ld on 2) Qnt 1)(2n + 3)(2n—1) lees i? 2(2n+ 3) Gay) | Lr3«. . 64K, 2 i | Rl ome 6, Gael (2n+ 3)2(2Qn—1)? forage: | oe u The series a (Qn +1)?2@n+3)(2n—1 convergent. The calculation of a few terms gives for its value 0°1126. yy is very rapidly | Hence Co = 0:07599K1+0:07398 Kl, . (8) or in terms of K,, Kg Cy = 10-2[0-08453 K, +0°08230 Ky |/ farads = (0°08453 K, + 0:08230 K,)/ micromicrofarads. (18’) Multilayer Coils with Square and Circular Section. 977 Coil with a Square Section. The coil with a square section can also be studied by the same method. The side of the square is denoted by 2c. The origin of co-ordinates is placed at the centre of the square. The layers are supposed to be arranged parallel to one of the sides. One of the coordinate axes is drawn parallel to that side and the other perpendicular to it. The first is called OX and the second OY. (See fig. 9.) Fig. 9.—Diagram of coil with square cross section. In this case, if the coil is ungrounded, V at O is zero. Hence V at (a, y) 1s 2a at fale Lee AOE se atl ad ee Ni Ag? LF Wissen! Th (19) because the e.m.f. induced between two layers is proportional to the area of the cross-section subtended by the layers, since the diameter of the coil was assumed to be large in comparison with the dimensions of its cross-section. 2 It is clear that a. Therefore p=0. It remains to compute c. Itis seen from (2) that o is made up of two parts. The first of these contains x) as a factor, and the second similarly involves «,. It is clear that at BC and AD (fig. 9) “== 0. Hence [see equations (1), (2)]| the only contri- n butions to the term involving «, are given by AB and CD, OV dV i.e. by the end layers of the coil. At CD, eee and Phil. Mag. S. 6. Vol. 43. No. 257. May 1922. 3R —— 978 Dr. G. Breit on the Effective Capacity of oy aN at AB, aa ae Hence at CD the part of o involving K, 1S ne ig di 2a dor dt? and at AB this part of o is a 2a Aa dt’ But the potential of DC exceeds the potential of AB by di | —L Te Therefore the effect of this part of o is precisely the same as would be caused by a condenser of capacity Keb Aaa This part of the effective capacity can be interpreted as the capacity between two conducting sheets separated by a medium of specific inductive capacity K, at a distance 2a, the area of each sheet being 2al. ‘he contribution of the above quantity to ( will be called Kel cy == O0796 cb.) ee Now V, must be found. For this purpose it is necessary to solve the potential problem in two dimensions for the case of a square. This can-be done by means of elliptic functions in various ways. The one given here is con- venient on account of the fact that rapidly converging series are obtained for the results. Also it can be understood without any knowledge of elliptic functions. ne ze=atjy (j=/—1), and consider the transformation defined by the differential equation ee (oe ee Pe ey where A is a constant and ¢ is the natural base. It will be shown presently that this transformation has the property of transforming the unit circle in the 7 plane into a square in the z plane. Also it transforms the region outside the unit circle into the region outside the square. , Before proving the above-mentioned properties, a few words must. be said as to the meaning of the square root Multilayer Coils with Square and Circular Section. 979 in (21), because the square root is a two-valued function, and confusion will be caused unless it is known definitely what branch of the function is used. In the following, the only case of interest is that of ba es This excludes the possibility of negative values of the real part of i : ae it ate The square root wil] then be chosen so as to have its real part always positive or zero.. This involves a cut along the negative of the axis of reals in the plane of l—7~*. Since, however, the real part of 1—7~* is never negative, no dis- continuity in (1—7~*)? is introduced by the cut. Fig. 10. --Illustration of proof of conformal transformation of the outside of a square. A a ~ Consider now the value of given by (21) in the case dt of + moving on the unit circle (see fig. 10) in the counter- clockwise direction starting at the point (1). At 1, © =0, UP which shows that 1 is a branch of the transformation (21). The velocity of motion of z is therefore zero as t approaches 1 with a constant velocity. Butif 7 isslightly different from 1 and lies on the unit circle, say if 7 occupies the position B (fig. 10), then the point 7~* has the position Q on the same circle such that 4Z BOL= Z210Q. The stroke Q1 represents then in magnitude and direction _the quantity l—7~*. Now the angie which this stroke makes with the positive direction of the axis of # is readily shown to be 5 Z BO. 3R2 980 - Dr. G. Breit on the Effective Capacity of Hence the argument of (1—7~*)? is ae 01 as long as this quantity is less than 7 and greater than —z (by the definition of the meaning of the square root). Now draw BA tangent to the unit circle at B. The argument of the stroke BA is ee oh Zi Ole But this is the argument of dt in (21). Therefore, by (21), the argument of dz is Tin . A Ti 9 eee Ween This means that as B moves on the unit circle, z moves in a straight line parallel to the axis of realsand in the negative direction. Such is the case as long as the argument of (1—7~*)? is — ZBO1. Consider now the behaviour of arg. (1—7T 4). aD mores teeth IL i vi Q moves from 1 to —1 so that arg. (l—7 ‘) varies from — =e 0. Further, as B moves from ,/) to j, Q follows the wpe part of the unit circle and arg. (l—7~*) varies from 0 to ~ =. Within this region 2 arg. (1—7 *) is ; 5 Zit and, therefore, arg. (1—7~4)? is T mee BOl. But as soon as 7 crosses j, the principal value of the = : GT : é arg. (1—7~*) must again be taken as 5° for otherwise the real part of the square root will be negative. Hence, from 7 on, arg. (1—7~*)2 must be taken as vin T As before, arg. dr is 5+ ZB01, and, therefore, by (21) Multilayer Coils with Square and Circular Section. 981 Thus as t moves from j to —1, z moves in a straight line parallel to the axis of pure imaginaries and in the negative sense. Similarly it may be shown that as t moves from —1 to —j, z moves in a straight line parallel to the axis of reals and in the positive sense. Finally, as r moves from —) to +1, z moves in a straight line parallel to the axis of pure imaginaries in the positive sense. Tt will now be shown that as 7 returns to the value 1, z returns to its original value. In fact by (21), if 7 = e 47, az dz dt! dr\~ Therefore, the length of the straight line corresponding to the values of + between 1 and ) is the same as that of the straight line corresponding to the values of + between 7 and —1. Similarly the remaining two straight lines are equal to the ones mentioned above. Thus the figure is a convex rectangle with all sides equal, and hence is closed and is a square. From the finiteness and continuity of the functions on the right-hand side of (21), it follows that = ‘is finite, single-valued and continuous for t>1. Also if |r| is 4 large, ee approaches e« *. Therefore, changes in 7 call for equal (in absolute value) changes in z. Hence parts of the 7 plane lying at an infinite distance from the origin correspond to parts of the z plane also lying at an infinite distance from the origin. If |r|>1, the right-hand side of (21) may be expanded by the binomial theorem, viz., dz ae x a gen Ae’t [1-3 pe |, ya Oe. (28— 3) een omme. fae Aes Hence by integration a particular solution for z is A al 7 (4s—1) 9 a e 4 a Ps I. ‘ ‘ A (2 ) 982 Dr. G. Breit on the Effective Capacity of This particular solution for z makes the centre of the square coincident with the origin. Other solutions mae be obtained by adding a constant to the above. In the applications to the coil, it is desirable to fccrihaen a; in such a way as to make the real part of the new variable (w=u+ jv) have a constant value at the 7 of the square. This is accomplished by writing = tt, oo al eee so that (23) becomes 2 @-(4s—1(u+je) eo ile aes 2 Se . (25) If u=0, |7|= 1 in virtue of (24) and, therefore, z is on the unit circle. Thus (u,v) will now be curvilinear co-ordinates in the plane (z,y). Only positive values of u are considered because only positive values of [| ee 3 are required. A curve corresponding to w=constant is a closed curve. If uw=0, this curve is the square. If Y=, Too and therefore, as was previously shown, the infinitely distant part of the z plane is attained. Hurshen, there is no difficulty in showing from (25) that larger values of uw correspond to curves enclosing those corre- sponding to smaller values of u. If u is large, (25) becomes 5 *) pee Niet tia) and for varying v represents a circle with a radius Ae’. The value v=( corresponds to a point subtending at the origin an angle ~ with the axis of reals. Similarly on the square w=0, v=0 gives a point subtending an angle 7 ™ with the axis of reals at the origin, are as indicated in fig. 11. dar and the points =) 7, > The value of A in terms of a 1s clearly ics ar/2 Ps 1+ 3 4s—1] le since for v=0, ce F and other expressions may also be given. Multilayer Coils with Square and Circular Section, 983 In the plane of z, the real part of any monogenic function of w+ ju is a solution of Laplace's equation 07 V 07V whe Seas == il) Fig. 11.—Distribution of the parameter v on the square. In the case considered, V must vanish at infinity and Lyd must be 5" oP at the prac of the coil,’ 2..e.at, w=0- But by (25), at u=0 Pt. | ne sin | (4s—1)0—7 | ¥y — sin ot) = p. PW ey gone he a A (26 a) It is clear then that err ee : v=-7,G) |: sin(o+ 7) a [a ee ie ppt $ yu 4 | | os i ae is the solution required because it degenerates into — nee a4 adt for u=O0 in virtue of (26), because it vanishes at infinity, 984 Dr. G. Breit on the fective Capacity of and because it may lee shown to be the real part of 2g Dut i tig ot Daas s; —. At any point on the square the charge density due to Vy is nav Oo = ees On? ° ° . . ° ° (28) ing 8°989 x 10’ > constant of the medium outside the square and. ae being where as before ko 1s Ky being the dielectric the directional derivative along the normal drawn outward at the surface of the square. Since that surface has the equation u=0, ___ % OV dw rey ~ dar Bu | Let as usual J denote the perimeter of one turn and ds an element of length along the perimeter of the square reckoned positive hen in he direction of v increasing. The amount of charge in the element ds is a OV | dw Jk Ar Qu } If now dv should stand for the change in v abe bey ds _ dz ” de [dw formal. Hence the amount of charge between v and v+dv is oo oV Le Bn Ou/ yao Qto) dv = Hence, since the function a(v) is a Qe) , then by (27) a a(v) = ee | [sia(e+a)—. > Asin {(4s—1e—7} iE (29) It is now also necessary to find M(v). This is found Multilayer Coils with Square and Circular Section. 985 from M(y), which is readily seen to be M(y) = 34 because the e.m.f. induced between two layers is pre- portional to the area enclosed by the layers, so that remembering (26 a) ad ; Mr) = My) = = ae [cos(v +" — S peer, f (4s—ljv—F ae (30) s=1 Care must now be exercised to differentiate between the four sides of the square. In formula (30) it was assumed that each layer of the coil is considered once.. For this reason the expression «(v) given in (29) must be modified because a(v) refers separately to all values of v, and, therefore, does not combine the charges on the cle for which Tcv<0 with the equal and similarly situated charges on the side for which oe nase 11s necessary to double «(v) as given by (29), and to discuss only the range of values of v between pOeus and +27. 2 In addition, of course, the charges on the face (see fig. 11) (v=0, v=") as well as those on (v=n, v=) must be computed. These charges are collected on the outside layer, and have simply the effect of a capacity across the whole coil. Thus the effect of the charges on the surface divides itself into two. The first of these is that of the charges on the faces y= +a. The second is that on the faces z=-+a. The contribution of the first to Cy is denoted by Cy" and that of the second by C,'". The computation of C,'” will be effected first, an en 0’ will also be ootained. For C,'" the formula = Moy {-@? -¢(v) Lt fife = Co 7 MST, L : } ‘* or L dv dv ee Saat holds, where, as has been shown in the preceding paragraph, 986 — Dr. G. Breit on the Effective Capacity of M(v) has the value given in (30) and «(v) has double the value of a(v) given in (29), i.¢., a(v) =2a(v). i ge | (Oey =~ | op o5(v+ 9) | ! em) q Ls Ds COS | ( (4s—1 Laisa L | s=1 ails i : a -= fs Sin {(4s—1)»—7} | dv : : 2 Ales ee : sr [cos(e+7 i) na { 4s—1 7} | [cos(v +7 )— Je Bi Pecos (4s — ez : rg a, vin 4 “vy dv, , C08 | (4s— 1l)o— 2 A Ih Ne ae Ko lA? a *(o+ ag pee (v C= |. cos +7) eo +~— 7) ee) Ta Lye ( q > Ps [ cos {(4s— 1)v 7 {cos o+7) 1 T 7 iL T ul cos (vt T)oos { (ts—1)o—T 1, cos(v+ Z) s=1 ao | F508 {(4u—-Lo—7} cos @olpss \ cos {(4)—1)v—4] \ - : 3! s, VS S$}, So=l, os iP : As,—1 bs cos? { (4s —1)v—T } — 008 { (As - Lvs} Moan Ore V2 : | ad, Saal As—1 | i Multilayer Coils with Square and Circular Section. where Dy Si> S.=1, 2, 3 is extended over all values of s,, s. with the exception of the case when s,;=s,. - The integrals in. this expression are all of either of the types: Tw 70 — 27 2 T T : Jon (e+ 7) a a Cos {(s— lju—F } dv, 2 2 27 27 . 7 WT) a cos?(o+ 7 ) dy ii cos? { (4s—L)v — 7} dv, 2 2 227 = TT hs cos (0+ 7 cos {(4s— lju—q } dv, (yr co {(4—1)o—7} cos { (4s:—L)e—7 f dv. For these, direct integration gives 27 is cos (v+7) 7 2 2 i cos { (4s—1)0—7 | do = v2 - am Tv ik Ti: {. cost(vtT)du=T+5, 27 | T 7 1 ‘. cos* { (4s—1)vo—7 dv == ee, 2 27° Tv Tv it for cos (+47) eos {(4r—19-—F} de =— 4, i‘ cos { fan o—7} cos { (48. _ IF} dv anya) 1 4s, +4s,—2° 988 Dr. G. Breit on the Effective Capacity of Substituting these values into the value of C)’” just found, 7 ‘eu Ky lA? [s+ As, zi 2 Co Sn eos as 1) (4s aE + a —1 1% GD Gn t= D! “aoe where now the result is expressed without making use of the notation >’ but using throughout the ordinary &. Now C©,” must be calculated by finding the charge on the side of the square given by values of v included between 0 and — 3" If this charge is Q, C,)/’=— a. 92s ° . ° . ° (32) Ore TER Naoe (1) because -1(5) is the excess of the potential of that plate over the potential of the opposite plate. In virtue of (29), the charge between v and v+dv is Ties. a kp HLA ) : 7 ap he) a0 = — reas (F | sin (0+7) — =p, sin {(4s— Lo—7} | ; so that the total charge on the face considered is molLA di (7 7. nes ces . ee [sin (+ T)— = pesin | (4s—L)o— TF} ao, and therefore by (32) C,"/= a i | sin e+] i) =. 3 Rue sin } }(4s—Do-F A Ae (33) or performing the integrations Kyl AN 2 Ps 4 4 87a 1-3 ie fill es oi Multilayer Coils with Square and Circular Section. 989 Substituting (26) into.(31) and (34), 12 eee je, Kgl S=Al 4s—] Co =F = (35) 1S 8S) 4s—1 Thus formule (20), (31), (35) sive Oyee€y . Cie Hence Cy is obtained as C5 = Co’ + Ca’ + Cae The expression for C,‘ is in a form suitable for numerical computation. It remains to transform (31) and (35) into more convenient expressions. This may be done by formula (26), which gives Denys fake ue nga k= tf A (36) An expression for = may now also be obtained in a different form by making v= in (25). This gives. for z, which is equal to a for the above value of v, so that COM (ip: Memep ame OU) In this series the terms are alternately positive and negative, so that the error committed is always less in absolute value than the last term left out. Thus, taking the first six terms of the series (37), the error committed is less than 0-001. A calculation gives a eee 2. Bla) The value of = thus being known, the double series Ss Ps, Pso (4s;— 1) (4s + 4s. — 2) which occurs in (31) will be expressed in terms of 27 i 2 = ‘990 Dr. G. Breit on the Effective Capacity of and hence by means of (36) through = The trans- formation is effected by means of the identity Ps, Pee 81 a (4s,— 1)(4s, - 4sy—2) Pa Ps. ( _ 1 L aye ee ( As, +4s,.—2 is eat) which ee Ps; Ps2 =—i> Psy Ps ae 2 As; +4s,—2) ” (4s, —1)(4s.—1) af ok: ae (= As—1 ; “or ces (36), Ps Pss L av2 =) a S9 (As, — 1) (4s ar tee Tae j Similarly in (35) the substitution of (36) gives < , Kgl A ay Ole & v2—1) = 00532 ml... GB) Further, the summation 3 48 Ps occurrin : (4s— 1) (4s—2) 8 iin (31) is equal to SPs so Ps = a ee or, using (36), © As Ds Kt ar/2 ps aCe 122 een (4s—2), A calculation of the last, series shows it to be 0°0871. ‘This value together with (37 a) gives ce As Ds 2 aaa) ee Faia 39 he 1) (2) Finally the quantity x has 24 Ae) in (31) must ‘be calculated. The series in ae is rapidly con- vergent, and a calculation gives for it 1-086, so that 7 SON tig: 7 (14% gh) = 0858... . (40) Multilayer Coils with Square and Circular Section. 991 Substituting (40), ae (37), (37 a) into (31), Oo) 44.0°371+0°853 + $(0°198)?] = 0°0412«/. . (41) Now substituting (38), (41), (20) into Os = Cy #9 +00"; Sir a 87 (0°847)? es it follows that Cy = 0:0796 «14+ 0°0944 Kl, OC) = 107"[0°886 K.1+0°1050 Kol] farads = [0:0886 K,1+0°1050 Kol] micromicrofarads. (42) Both (18') and (42) involve the constant K.. As stated previously, a theoretical derivation of K, is, in general, difficult. However, an experimental investigation i is possible. For this purpose, it is only necessary to introduce a slab of the complex medium between two plates of a condenser and measure the increase in capacity caused thereby, treating the slab as if it were a homogeneous dielectric. Of course, care must be taken to have the layers of the complex medium parallel to the condenser plates. One must also remember that the mean vaJue treatment can be applied only in the case of coils having a large number of turns per layer. In some special cases the value of K. may be obtained without difficulty. Such is the case, for example, of a winding such as shown in fig. 6, when adjacent turns almost la In such a case the effective dielectric constant K, is ak. where K is the dielectric constant of the medium between turns and where A is the distance between homo- logous points of two layers, while h’ is the shortest distance between two layers. This principle may be extended also to the case of wires of circular cross section if the distance between layers is large and the winding of each layer is close, provided each layer is replaced by an equivalent layer of uniform thickness, this thickness being made equal to the average double ordinate of the circle. A few words must be said as to the meaning of / in (18’) and (42). Strictly speaking, the derivation given applies only to the case ue the perimeter of one turn is the same as that of any other. However, the considerations as to the distribution of charge will apply approximately even if the turns are not of equal perimeter. In this case 1 becomes 992 On the Effective Capacity of Multilayer Coils. indefinite. If the cross section is symmetrical about a line through its centroid parallel to the axis, | may be taken simply as the perimeter of the turn through the centroid because the mean between the perimeters of two turns symmetrically situated as to the centroid is that value of J, and because the charge density on these turns is the same. Summary. A method of calculating the effective capacity of multi- layer coils has been given. It has been shown that the average density of volume and surface charge may be obtained in terms of a constant Ke which depends on the spacing between the turns of the coil and the insulation used. This constant is analogous to the dielectric constant of a homogeneous medium. The method has been applied to the case of a coil having a circular section, and also to the case of a coil with a square section. The results are: Cy = (0°08453 K, + 0°08230 Ky)! micromicrofarads for the case of a coil with circular section and Cy = (0°0886 K,+0:1050 K,)l micromicrofarads for the case of a coil with square section. Here K, is the dielectric constant of the medium outside the coil, K, is the effective dielectric constant of the medium inside the coil, / is the perimeter of a single turn passing the centre of the cross section. In both cases, the diameter of the coil is taken to be large in comparison with the maximum dimension of the cross section, the winding is assumed to be close, the number of turns large, and the coil itself is taken to be insulated from all objects except the tuning condenser. Bureau of Standards, Washington, D.C., June 25, 1921. CIII. On the Orbits in the Field of a Doublet. By Dorotuy WrincH, Fellow of Girton College, Cambridge, and Member of the Research Staff’, University College, London*, Introductory. Bar, present paper investigates in a somewhat ponaac way the two-dimensional motion of a particle in the field due to a doublet. Several of the results are of course known already as isolated theorems, but the subject does not appear to have been treated by any writer ina comprehensive manner, and it is very difficult to obtain a general view of the motion of a particle around a doublet even in two dimensions by putting together the specific solutions found in dynamical treatises or memoirs. Some new results which may be ot. importance are arrived at in the present paper, for the doublet may be of many types: the analysis is equally _ appropriate to a magnetic pole moving under the influence of an elementary magnet or to an electric charge in the presence of an electric doublet. Physicists are now generally convinced that the structure of a neutral atom of an element follows the lines generally associated with the names of Rutherford and Bohr. Such an atom behaves towards an external electron sufficiently far away effectively like a doublet, and we have little knowledge of its capacity to attract and retain a stray electron and so form a negatively charged atom. In a vacuum tube, where the atoms are con- tinually bombarded -by electrons, we know that negatively charged atoms frequently occur ; but the conditions con- ducive to the presence of a large number in cases in which their existence is chemically possible appear to be quite unknown. The success of the quantum theory has generally established the fact that we must not expect te be able, by the use of classical dynamics, to determine the internal motions of the electrons actually constituting the atom; but if we recall the tacts that (a) the inverse square law of attraction between electric charges appears to remain valid at distances comparable with atomic dimensions, (6) the essential need for the quantum type of dynamical specification only arises when an electron is ‘bound ” or is already a constituent part of the atom, we see that the motion of an external charge must, even * Communicated by the Author. ‘ Phil. Mag. 8. 6. Vol. 43. No. 257. May 1922. 38 994 Dr. Dorothy Wrinch on the on this view, be governed by the formule of classical dynamics. For these reasons the results of this paper should have some bearing on. the conditions under which an electron originally moving in any way in the neighbourhood of an atom may be captured and retained. But in default of a more precise development of atomic theory, we have set out the results as solutions of a formal general dynamical problem in the theory of orbits, for the need for such a scheme of solution is recognized here as well as in electrical theory. : A doublet consisting of two equal and opposite charges— of electrical or other type—subject to the inverse square law of attraction admits a potential V=—pocos 0/7? at all external points. This involves both a radial force R and a transverse force T on any unit particle in the field of the doublet. KR is measured along r increasing, and T in the direction @ increasing. The values are given by ON Tana T=—dV/roé = —ypsin O/9?, The equations of the orbit of the particle are of the well- known form — 2p cos 0/7, ii — 16? oh) 8 1/r d/dt (6) = T. Denoting 726 by h—so that mh is the an gular momentum when the particle has mass m,—we can show that h? = (—R—T du/udé@) | h? (w+ du/d6) = pw (sin 6 du/dé + 2u cos @) | (ut+d?u/dé?); . (1) emdlialse dh? /d0 = 2°) = —2u'sin 6.) These are obtained by the usual procedure of eliminating the time. The field of the doublet is symmetrical about the axis @=0, and we shall find it convenient to consider only those motions which begin on the side of the plane defined b 7202); for it is clear that other possible motions will merely be reflexions of these in the axis; their separate discussion is therefore unnecessary. Orbits in the Field of a Doublet. 995 The equation may be integrated in the form | h? = 2u (cos @—cos 0;) + hy’, where fh takes the value h; when @ takes some specific value 6;. Now motion can only take place when h is real, so that A? must not have negative values. It is therefore evident that no motion can occur when 24cos@,—h,? > 2u, and in the critical case, | 2u cos 6;—h,? = 2u, the motion is restricted to the Jine 0=0 and is given by tulsa eee ae eS) Motion along the axis of the Doublet. We may now proceed to discuss these motions along the axis in some detail. The integrated form of the equation (1) is r= C+ 2Qp/r?. If the velocity U at the point of projection r=r, is + Uj, then ip = — be +f U2 + 2y/7?—2p/r,. EU > 2p/r? ,-the particle describes the line @=0 in the direction r increasing, with a steadily decreasing velocity. | Its velocity at r=a0 is JU? —2u/ry . Further, integrating the equation again, the time is given asa function of r in the form t—ty=( V (U2 —2y/r,?)r? +24 — V/2p)|(U?—2y)r;”), and the time to infinity is infinite. Thus a particle with these conditions of projection will not leave the system in a finite time. The case U?=2y/7? gives a similar result. The particle describes the line @=0 in the direction r increasing with a steadily decreasing velocity, arriving there ge Zero ‘Velocity. The time is given by t—ty = (9? —1)/2 2p ; so that, as before, an infinite time is required for the . description of the line @=0. 352 996. Dr. Dorothy Wrinch on the Now it may easily be seen that ta [7 is the smallest | 4 | velocity of projection which causes a particle to recede from the doublet and ultimately to leave the system. When U, : ; U 1 Py — 4h ae Wher? bf ure + (Ur —2p)r” thus the time to the origin is r/(U, + V2u/7), giving 7,/2U, in the case when U2=2u/r2. The particle can consequently be sent to the origin in any time however small if U; is sufficiently iarge, and attain an infinite velocity in the time; it is further evident that it can approach the doublet with an infinite velocity even if it is projected with zero velocity or a velocity away from the origin, however small its original distance from the origin, provided only that it is not projected away from the doublet at the point 7, with as great a velocity as / 2u/r?. As regards motion on the line 0=7, the equation is et Sal p= O—2Qy/r*. Living Thus r=+ / UP = ple? + ple? if Uj, is the velocity away from the origin at r=7}. Thus 7-Increases as 7 increases and has the value | VU? 4+ 2u/r? at infinity. The time equation can be obtained as before, and it is found that the time of transit of the line is infinite. If, however, the particle is projected towards the origin a VU 2—2p/r? + 2y/ry? and vanishes at 7=7, given by : UP —2y/r.? + 2p/r? = 0; the time to 7, is re Vr —re | V7 2 fb, and the particle recedes from the origin and disappears from the system, though only after an infinite time. Thus no conditions of projection in which U, is finite keep the particle from steadily increasing its distance from the 998 Dr. Dorothy Wrinch on the doublet. If it is projected towards the doublet, it goes to the point r=r, and then recedes to infinity ; otherwise it moves away from it at once. We have therefore two different types of motion, which are typical of the asymmetry of the field of force of the doublet. If the particle is in the line 0=7, so long as che angular velocity is zero, no conditions of projection can prevent the particle from receding from the doublet with increasing speed, either immediately or after an interval | 27s JS rear? | Von If, however, the particle is in the line 0=0, so long as the angular velocity is zero, the particle does not leave the system unless projection is away from the origin and the velocity attains or exceeds the critical value /2p / ™.- If the velocity of projection away from the origin is less than /“2y/7,-—and this includes the case of zero velocity or of projection towards the centre—the particle approaches the centre after an interval 27 Vrs7—7;" | Vv 2u, or directly, according as U,;>0 or U; < 0, 7, being given by bro? = 1/r?—U2/2p, | and the particle arrives at the centre in a finite time with infinite velocity. These conclusions lead to curious effects of electrical doublets in the presence of stray electrons, which have apparently not been completely realized. It is remarkable that these stray electrons, when placed on one half of the axis of the doublet, should arrive from any distance, however large or small, and bombard the doublet with very great momentum ; and that, on the other hand, they must leave the system altogether if they are on the other side of the axis. It follows that a necessary consequence of the field of force created by a doublet is a continuous bombardment on one side by any stray electrons, and a steady ejection of electrons from the system on the other side. General Motions. Thus far, we have considered only the case when particles are projected from some point on the axes of symmetry of the doublet, with no angular velocity. The angular Orbits in the Field of a Doublet. 999 momentum is given, in general, by the equation h? = hy?+2pu(cosO@—cosA,); . - . - (8d) so that if V be the transverse velocity at any point, and in particular V, the transverse velocity at (7, 1), PV? = 7,°V,7+ 2u(cos O—cos 6,). Since A? cannot be negative, motion can only take place in the part of the plane in which cos 8 > cos 6,;—Ay?/2u. It is now evident that there is an important division of the possible orbits in the field of the doublet. For if the initial conditions are such that hy? > 2u(1+ cos 6,), the condition that h? the square of the angular momentum is not negative is satisfied at all points of the plane, and the motion is therefore not restricted to any region of the plane, If, however, —1 1+cos 6,, motion is possible all over the plane; if, however, r?V 7/24 = cos 0; —m and —l r27V2/24—cos 0, > —1 are similar to those on the line =O ; and the characteristics of orbits in which 77°V1?/2u4—cos 6, a () are similar to those on the line @=7. Thus, in the first set of cases, no conditions of projection with respect to the radial velocity can prevent the particle, either immediately or after a finite interval, receding trom the origin with an ever-increasing velocity. In the second set of cases, it is only when the radial velocity is sufficiently big, viz. Uj2z4+/ ‘I um/|r?, that the particle leaves the system. In all other cases the particle arrives at the doublet in a finite time with an infinite velocity. The difference in the characteristics of these two classes of orbits is of course not unexpected. For when | 0|>7/2 the radial force is repulsive, and for |@| <7/2 attractive. On |6|=7/2, it is zero. Consequently, if the particle has sufficient angular momentum to pass the line | @|=7/2, and not sufficient to regain the sector | @| < 7/2 after one or more revolutions, nothing can prevent it from receding from Orbits in the Field of a Doublet. 1001 the doublet, however great its initial radial velocity towards the doublet may be. If, however, the angular momentum is not great enough to enable the particle to get beyond the lines | @|=7/2, or is great enough to enable it to regain the sector |@| < 7/2, it is only in the case of a sufficiently large radial velocity away from the origin that the particle can be prevented from approaching the centre (possibly alter an excursion away from it finite in time and extent) and bombarding the doublet with an infinite momentum after a finite time. Characteristics of the Orbits which lie in a Sector of the Plane. We may now discuss the variation in V, the transverse velocity. Suppose, first, that we limit ourselves to the cases when O< cos 0, — 7 2V 2/2 = cose = m <1, in which the motion is restricted between the lines 0= +a and a < q/2. V is always zero at all points of these bounding rays. Hence, except when U the radial velocity also happens to vanish at a point on the bounding rays, at such a point the velocity of the particle is entirely along the radius vector. Consequently at such a point the orbit touches the bounding rays, except in the special case when the radial velocity is also momentarily evanescent. Motion therefore takes place in general along a curve of wave form touching alternately the bounding rays 0= +a. If the particle is projected at 7,0, with velocities U=U, > 0, V=V,, the particle initially recedes from the doublet, touching the bounding rays alternately. If U, > V(2p 008 @/7,”) it gets perpetually further from the origin. If, however, Vi< V (2p cos 4/7”) the particle recedes from the origin along a curve of wave form, touching the rays alternately until it reaches a point 7, given by 1/r? —I1/r? = U?/2p COS &, At this point its velocity along the radius vector is zero, and it therefore touches the circle r=7.. Subsequently 1002 Dr. Dorothy Wrinch on the it approaches the doublet, converging on it along a wave- like path. In the case of an orbit of this type, the particle approaches the doublet with infinite velocity. The equations of the orbits in these cases can be obtained either by integrating the original equation (1), which with the help of equation (2) may be written 2(cos @—cos «) d?u/d0? —sin 0 du/d0— 2ucosa = 0 or / (cos @—cos ae ( v/(cos § —cos a) oe ) —ucos a = 0, or we may obtain them by using the fact that U/V =dr/rd0é. Proceeding ‘from this, we obtain ss ibe Meats V (u? — uy? + Uy?/2p cos «) —du/udé = dr/rdé = r/rO = ae = ur/ (cos 6/cos a—1) ° Hence du vita on dé is as 7 ee 2 mu? — Zu,” Cos Se J 2per/ COS 0 —cos a Now if the orbit is being described initially with r and 0 increasing, U/V>0. This case needs the lower sign in the above equation. lt sin 6/2 = sin «/2 sn (fp, sin «/2), where sn is the Jacobian elliptic function with modulus sin «/2, and we obtain 2 ee ae ee / 2ur/ cos 0 —cos a J 2 Further, putting u = /U//2p cos a—u,? sinh &, we have du dé — a Wee 2uu?— Du? GOS a “/ 2p cos a Hence omen = Veen Ney where & is some constant; and we obtain the equation of the orbit in terms of a parameter & and constants a, & in the form sin 0/2 = sin (a/2) sn [ (Ey—£)/s/2 cos a, sin «/2 |, u = VW(U2/2wcos «—u,?) sinh &, . Orbits in the Field of a Doublet. 1003 or writing &—F=y, sin 6/2 = sin (a/2) sn (x/1/2 cos a, sin 2/2), u = /U?/2u cos 2—u;? sinh (&)—y). This curve consists of a series of waves touching @= te alternately and going to infinity along the line du > ans Eo, p=(-% BK 0, du PO ioe ear) Wa given by sin 0/2 = sin a/2 sn [ &/ V2 cos a, sin «/2]. _ To obtain & in terms of the velocities of projection U,V, aud the coordinates of the point of the projection 7,0;, we may remark that, if y= Xa» when 6=6,, sin 6,/2 = sin a/2 sn (Wr, / 2 cosa, sin «/2), tanh £&)—y; —¥, = mV 2u cos a/Uj. Bioeelt Thus & = V2cos«sn7—! (sin @,/2 / sin a2) + tanh-!w, /2u cos aU: ; and hence the direction 0=@) is given by sin 6)/2 = sin «/2 sn Es (sin 0,/2 /sin z/2) aor ack uy Vv 2u COS a / Ui]. A typical curve of this class is shown in fig. 1. 1004. Dr. Dorothy Wrinch on the In the case when U,?/2um=v,?, the equation stands 1n the form a tae ie dé UN COS & cos O—cos & and we get I log u/uy = —,/2(abh— WW) COS aie [ts Veil a if u=u, when w=, where sin 0/2 = sin #/2 sn (af, sin 4/2) or sin 0/2 = sin a/2 sn aye log wut yr | = = Sim a2 sn Re -+ — log npr | : This curve oscillates between 0= +a, with r continually | increasing. As r->«o @ is indeterminate, and instead of tending to infinity along a certain line 6=8) after a finite number of contacts with the lines 2 and —¢, as In the previous case when U,?2/2u cose > vy’, this curve oscillates between the rays +a and has no direction at infinity. A typical curve of this type is shown 17 fig. 2. Fig. 2. The case when w?’—U,?/2um=u,?>0 remains. We have du dé / cosa VU? = Us? V cos 0—cos a Putting u=u, cosh &, we get du and consequently, if sin 6/2 = sin «/2 sn (a, sin «/2) = dé; Orbits in the Field of a Doublet. 1005 as before, ££, = — V2cose and the orbit in terms of a parameter W and constants &) and «is sin 0/2 = sin (#/2) sn (vr, sin 2/2), o— Ju —U 2/2 cos 2 cosh (G—W/V 2 COS a). It is evident that in this case, as ~ increases from an initial value Wr, w decreases until p=£&/72cos«: it then increases to infinity; while at the same time 6 is going from the value 6; through the cycle of values (a, 0, —e,0). A typical orbit of this kind is shown in fig. 3. Fig. 3. It is of interest to connect the constants in these equations with the constants given in the initial velocities at the point r,0,. We have, of course, VP = 2pu,2(cos 0;— Cos «), which gives « in terms of x; 0; and V;. Since 6=86, when r=, & is given by the equations sin'0,/2 = sin 2/2 sn(x4/s/ 2 cos 2, sin a/2), tanh E =¥1 =s Vif y/ 2p 5 so that & = 2 cos esn7! (sin 0,/2 / sin 2/2) +tanh-! U,/V2um “cos 2. It is perhaps hardly necessary to get the equation in the case of projection towards the origin, since the analysis is entirely similar to that required in the cases when projection is along the axis, and the geometrical characteristics of the curves obtained have already been indicated in the diagram. For example, the part of the curve (fig. 3) in which the particle is approaching the origin is a typical curve of the case when projection is towards the doublet. 1006 Dr. Dorothy. Wrinch on the Motions which are restricted to half the Plane, including one Periodic Orbit. We have now considered the cases when 0< 24003 0,—h? < 1.’ We have considered in detail the boundary case 2u cos 8,—h,? = 0,. in which motion is restricted to the line @=0. The particular features of the case 2ucosQj—h? = 0, h? = 2ycos 6, also call for attention. Here the motion is restricted to the positive side of the plane, given by 7 ae ee 2 ’ oO Referring to the equation aod = I. we get in this case i=0, b=Us so that the radial velocity is constant. Given then an outward radial velocity initially, the curve touches the lines’ +7/2 alternately, and the distance at any time from the origin of a particle describing it is given by y—r, = Ui(t—-t) if it is projected from 7, with radial velocity U, at &. If +=—U,, the particle approaches the doublet with constant radial velocity along a wave path which alternately grazes 0= +7/2. The equation to the orbit if r=U,>0 results from the equation —du/ud@ = dr[rd@ = Ui/7é = U; Uy/ 2u / cos or U; tw/d = ——>- | vos 8 UU on O in the form sin 2 l= es sn (yr, 1/,/2), uU—U ae 2(4r— 1 rp (yy Wy) Orbits in the Field of a Doublet. 1007 if w=u; when P=. Thus sin 6/2 = 1//2sn | i— 7m) | and sin 0,/2 = 1/72 sn Wr. The equation connecting @ and ¢ can easily be obtained in the form hs We ae) sin 6/2=1 2sn{ sn ECA? isi 04/2) 4 ae ao n@/ | [Vv ( i/ Ish as Ae O LD In the case of 7 being negative originally and equal to — Uj, we have ron = Uilh—0), and progress is towards the origin along a curve waving between 0=+7/2 and cutting the line 0=0 between each two consecutive grazings of these boundary lines. It may possibly be of interest to remark that the reciprocals of the distances of the various points on the rays O=a, 0=0, or d= —z« arein arithmetic progression with common difference A case of special interest occurs, however, when the radial velocity originally is zero. The curve will then be merely the part of the circle r=r, for which —7/2< 0< w/2. The transverse velocity is given by V2 == Qu cos 6/7’. Thus, if a particle is given a transverse velocity ee AV 2p cos 04/7, at the point 0,7, and no radial velocity, it will describe the semicircle r=7,, |@| < 7/2 perpetually. The periodic time 1s easily given by ee ee = =a dis oo — == SSS == ‘: Fae VES WP niNog RC CORE Lak =a \ eee V2 Vis Ja: / Geese 62 /2u (4/2) a yD KS 72" 2p 1008 Dr. Dorothy Wrinch on the We have now. therefore, established the existence of a periodic orbit in the field of force } V=—ypw cos 0, viz. the semicircle r=7r;, in which the periodic time is 4r?7K(1//2)]//p. The path of a particle is therefore this Eide orbit when the conditions of projection at 70 give U, = 0, PIN Ge = 2 COS Gi and only under these conditions. | It is further possible to deduce from our general analysis that when the transverse velocity of projection satisfies the conditions O < 2u cos 0;—7;°V 1? < 2p this semicircular path is the only periodic orbit possible. We next consider the case when 0 0, we see at once that motion must be restricted to the sector —a<@/2. In ne case the (7,¢) equation is | = 2um/r, a ag = Uy? + 2ym/r?—2em/r’. If + is originally positive or zero, 7 increases steadily to the value V(U,?+2ym/r;’) at infinity ; while if it is originally negative, it decreases in absolute magnitude until r=72, where L/ro? = Mfr? + U7 /2um, and then changes sign and increases sieadily. We may integrate the equation in the form | t—t, = r(V ry? — 79 —re@— Vr — 792) // 2m when oe and in the form t—t, = vi) VP? —re[Y 2um ile Ue Whatever, then, the initial radial velocity the particle proceeds either directly or after an interval 2roV r— ro?|V 2um to increase its distance from the origin at an increasing rate, — Orbits in the Field of a Doublet. 1009 At the same time, the variation of @ can be obtained from the equations ne =f" F*n(4, .) B=) r= 1, 8eC ( Vv 2m(b— 1)), or if the relation between 6 and ¢ is more convenient, from the equation — t—t, = (r2/V Quam) tan (“2m (6—¢d))). These equations can be deduced from the relation UW = rdbjdr as before. The path therefore, in general, waves backwards and forwards between @=+e a finite number of times (which may be zero), and has the asymptote @= 6, p= (—duldB ano, given by ; m+1 Bee sin 6,/2 = af Fugit al? V 2m). A typical orbit is appended in fig. 4. Fig. 4. There is, however, one particular case. If hy?—2pcos 0, = 2p, then «=a and the transverse velocity disappears when the line @=7 is reached. Unless therefore the particle goes off Phil. May. Ser. 6. Vol. 48. No. 257. May 1922. 3T 1010 Dr. Dorothy Wrinch on the to infinity before reaching the line @=7, it will approach the line tangentially and pursue the path solely along this line. Characteristics of Orbits not restricted to a Sector of the Plane. The final case to be considered is when hy —2pcos 6;= 2um > 2p. In this case, since h? = 2u(m-+ cos 8) V? = 2u(m-+ cos 0)/7?, the transverse velocity never vanishes in the finite part of the plane. The equation to the orbit in terms of a parameter @ is still given by sin 0/2 = fe an( $, / which is now more conveniently put in the form Bh eA SE m+1 / 2 sin 0/2 = sn (,/ 9 d, “/ =), 7 = 7,s8eC / 2m(b— $y), = ,/2um VJ Ire—1]r+U2/2um. The particle goes to infinity, as before, arriving there with the velocity ,/2um 4/1/r? + U2/2um, and with ./2m(¢—d)) =7/2. Now the equation giving @ in terms of @ will show what angular distance from 6, is described before the particle reaches infinity. It is evident that as m increases this angular distance increases, giving in the limiting case an infinite angular distance. The orbit therefore, in this general case, is a curve described with 7 and @ increasing together ancl circumscribing the origin an increasing number of times as m increases, and tending to infinity along the line 0=6@), p= (-<5) . In the limiting case when m is t u= and and infinite, the path of the curve circumscribes the origin for ever, and only has the circle at infinity as asymptote. If the particle is projected with negative or zero radial Orbits in the Field of a Doublet. HOT velocity, and the trangverse velocity still satisfies the conditions ets r?V1?—2ycos 0; > 2p, the particle approaches the origin to a distance r= r, jes m4 V/ro? = 1/ry? + U1?/2um in times given by tt = (JV rp re —V Pr) | / 2m, and then recedes from the origin according to the law $—t, = 1/2 —r2? |o/ 2m. While the particle is first approaching and then receding from the origin, the variation of @ with ¢ is given by ; fi ae ib 2 ie = Sh (/ See / wea i—t, = Micah v/2um tan V 2m(b—¢y) : so that the orbit circumscribes the _origin until that value of 6 is reached for which (6—d,) V 2m=7/2. A typical orbit is shown in fig. 5. Fig. 5. As in the previous cases, if the initial radial velocity is towards the origin there is a finite excursion towards the origin before the radial velocity becomes positive and increases steadily as time goes on. Summary. The general characteristics of the orbits in the field of a doublet may be summed up as tollows :-— (a) The only periodic orbits are the semicircles r=, —7/2< 8< w/2, for various values of 7, The corresponding periodic time ig re RO /2)/\/m 1012 Dr. Dorothy Wrinch on the These orbits are described when and only when the conditions of projection at the point (7,6,) are such that the radial velocity is zero and the transverse velocity V, is given by To Ne = Ze COs.6 4 Included under these conditions is the case when a particle is placed at rest on either of the lines @6=a/2, 02= —7/2, at any non-zero non-infinite distance 7, from the doublet. The orbit will then be the appropriate semicircle, and the periodic time will be as above. (b) When the transverse velocity of projection is such that 0 < 2ucosO,—7r7ViP = 2pm < 2p, the particle only leaves the system if U,, the radial velocity, is greater than or equal to WZum/r;. The orbit touches the rays O= +2 alternately, where cosa=m, and proceeds to infinity with a certain line G =o,” 1p = (—duide yaa as asymptote, if U,> V2um/r,. If U;= VW 2mm/r, the orbit touches the rays alternately for ever aud does not have a linear asymptote. If, however, U, does not attain this critical velocity, the orbit touches the rays alternately and keeping to the sector |6| 0, and then approaches the doublet converging on it, through a series of waves of decreasing amplitude, which touch the lines which bound it. If U, is. negative, the orbit has similar characteristics, the excursion away from the origin being now deleted. However, whatever the value of ley under the condition 0 < 2 COS 6,—r7Vy — 2p the orbit is confined between the two rays 6= +a, where < 7/2, and paca 2uc0s0,—7rV? = 2p cos a, and is of wave form converging on or diverging from the doublet, the amplitude of the wave varying in such a way that the curve touches the lines which bound it. (c) When the transverse velocity of projection is such that O<77?V,?—2wcos 6, < 2p, the orbit, whatever U, may be, whether it is negative or positive, goes to infinity and the particle describing it recedes from the origin—atter a finite excursion towards the doublet if U, is negative—at a steadily increasing rate, Orbits in the Field of a Doublet. 1013 and it takes an infinite time to leave the system. The path again lies between bounding rays 0= +a, but cos « = cos 0, —7r,?V,?/2p, and therefore « is greater than 7/2; the orbit is therefore confined to an oblique-angled sector of the plane. The path in general touches the bounding rays a finite number of times or not at all and has a certain straight line as asymptote. No velocity of projection towards the origin, however large, enables the particle to reich the doublet, or prevents it continually increasing its «distance from the origin, possibly after it has approached within a certain distance of the doublet. (d) When the transverse velocity is such that 2u<1,2/V? —2p cos 0, the orbit is not restricted to any sector of the plane. The particle again approaches the doublet if the radial velocity of projection is towards the origin, but it never reaches it: in this case it ultimately recedes from the doublet. When the radial velocity of projection is positive it straightway recedes from the doublet. While the distance of the particle of the doublet is changing in this way, the angular distance of the particle from its initial position steadily increases, and it finally proceeds to infinity in the direction of a certain line, after a finite number of circumscribings of the origin. As ; r’V, —2u cos 0, increases, the orbit becomes more and more like the orbit in the limiting case when rPV~—2u COS 0, is infinite; in this case the particle circumscribes the doublet in the same sense, for ever, while its distance from the doublet increases without limit. The main interest of the results les probably in the electrification of neutral atoms in a vacuum tube—the nentral atom being taken, as usual, in its first approxi- mation as a doublet. All the formnle deduced involve, essentially, a critical velocity of an electron, which deter- mines whether it leaves the atom in whose proximity it finds itself, or whether it stays. On the Quantum Theory, the criteria which determine the actual capture of such an electron by an atom have not yet been defined, and the fact that an electron ‘‘stays,” according to this analysis, 1014 =Dr. E. H. Kennard on a Simplified Proof for does not perhaps of itself involve its capture and retention as a constituent part of a negatively charged atom. Never- theless there must be a correspondence and perhaps a rough proportionality between the two classes of phenomena—free electrons with a velocity distribution about the critical value, and the number of negatively charged atoms found in tubes under the conditions of discharge. It would seem that an experimental estimate of the number of such atoms is desirable in relation to the velocity distribution in the free electrons of which we have some definite knowledge. The Quantum Theory of atoms itself, proceeding as it does by the elaboration of successive hypotheses, still requires a hypothesis regarding the capture of electrons and the formation of negatively charged atoms. It would be of interest if it could be shown that a stray electron which by the foregoing analysis could not leave the system really became ‘“‘ bound ” as a constituent part of the atomic: we may, indeed, perhaps anticipate that this phenomenon usually occurs. CIV. On a Simplified Proof for the Retarded Potentials and Huyghens’s Principle. By HE. H. Kennarp, Ph.D.* HE proof usually given in deducing the retarded potentials and Huyghens’s Principle seems to the — author, as it must to many physicists, peculiarly abstract and indirect. ‘This objection is only partly met in Professor Mason’s f modification. The following proof seems to be at least as short and as rigorous as any other, while, at the same time, it seems more natural and easier for a physicist to follow. S 1 WewErooy- The scalar potential @ satisfies the differential equation Eb aV'btsnp, « yi Cou), where p = density of electricity (the units being “ ordinary”’). To find ¢ at time ¢ at any point P, let us surround P by any closed surface S and then, following Abraham, let us * Communicated by the Author. t Max Mason, Phys. Rev. vol. xy. p. 312 (1920). ¥ ne reas! the Retarded Potentials and Huyghens’s Principle. 1015 suppose a sphere with centre at P to contract with the speed of light so as to shrink to a point at P at time ¢. Let us consider the integral meal tt (o+r8 +7 OP \i0, Tee) where 7 = radius a the See C= va of light, w = solid angle about P, and the surface of integration is the portion G of the sphere that is momentarily included within 8S. Clearly at time ¢ the last two terms become negligible (at least, provided the first derivatives of ¢ are all bounded or limited in value) and I reduces to the value of ¢ at P. pa dt We can obtain a connexion between I and p by considering the change produced in I when the sphere contracts from one position to another through a distance — dr =cdt. The change in any term Z of the integrand is Ae bay, amt 87, edt dr or at( r Od val [ -20$¢ =o SS 1a ae | +a Mle+e3e +7 Nie, Beastie) the second integral eet a only over the increment g which is added to G by the displacement of the sphere. Hence 1016 =Dr. E. H. Kennard on a Simplified Proof for Now by (1) [Sten gowns stn. But eat Js V’o)r?dw is, in the limit when dt—0, equal to the volume integral of VV? throughout the shell bounded by the two limiting positions of the “sphere and by the ring s which these cut out of 8, and this volume integral in turn, by the divergence theorem (or “ Green’s theorem is equals the outward normal flux of the vector V@ over the surface of the shell. Hence we can write ed (79) “ie Nl "42 aed (— 5p) g i Fie where n = distance Alone inward normal to S. Substituting from this equation in (4) and then from (4) back into (3), we find that many terms cancel out and Baa l\E)e do + allan Jao -— of) ot as {6) Let us now write instead of dw in the second ieee dS cos 6/7? where 6 = angle between r drawn toward P and the inward normal to 8; and let us write in place of the first term (\\(e/r)ar taken throughout the shell bounded by s and the two limiting positions of the sphere. Wecan then integrate and obtain I. At time t=—o%, I[=—0; at time t, l=, as noted above. Hence oe AVL ae dr + res | ee = $f) cos 0— onl, (7) Here the first integral extends throughout all the space enclosed by § while the second extends over all of S, and the subscript reminds us that values of quantities inside the brackets are to be taken at a time ¢—r/c, the Retarded Potentials and Huyghens’s Principle. 1017 If we now let S recede to infinity, the surface integral may vanish: in that case we have the usual expression, alibi eg ee the integral extending throughout all space. In any case the surface integral expresses the effect at P of all electricity and all field “conditions outside of S and constitutes the usual mathematical formulation of Huyghens’s Principle. The same proof applies at once, of course, to each com- ponent of the vector potential. §2. Heuristic Argument for the Student. This proof is particularly easy to lead up to. Electrostatic analogies suggest seeking a solution of (1) that will repre- sent a train of spherical waves proceeding from a variable point-charge, and (8) is then easily guessed as the general solution (cf. Jeans, ‘ Electricity and Magnetism’). Abraham’s contracting sphere is then introduced as a convenient way of visualizing the solution. It is next suggested, either on the basis of general reasoning or from the analogy of Green’s stratum, that the contracting sphere ought to be able to obtain contributions of potential which shall represent the effect of conditions outside 8 from the elements of 8 itself. Finally, we note that the spherical wave from any element of charge which produces the con- tribution of this element at P at time ¢ just keeps in contact with the sphere as the latter contracts, and we conclude that the contributions from all sources outside 8S must be capable -of representation as an integral expressed in terms of the instantaneous field conditions over the sphere. The study of the integral I then falls naturally into place. HKven the form of I can, if desired, be heuristically obtained - by proceeding from the analogy of the relationship between the mean electrostatic potential over a sphere and its value at the centre. Physical Laboratory, Cornell University. POPnOLe a] CV. On the Forced Vibrations of Bridges. By Professor 8. P. TimosHEnKo * T is now generally agreed that imperfect balance of the locomotive driving-wheels is the principal source of impact effect in bridges of long span. ‘The laws governing this effect have not yet been definitely formulated, and much more information is needed on the experimental side f. Some idea of the forced vibrations which are thus induced. may be obtained by considering the bridge as a beam ok constant cross-section with supported ends (fig. 1), Tne deflexion of the vibrating beam may be Ba as follows :— Uv 2Qarx - OT y= sin7- + dy sin = + 3 ‘in $35 Sey where 1, #2, ..., etc. are functions of ¢ only. Then if HI denotes the flexural rigidity of the beam, and w its weight per unit length, the expressions for the potential and kinetic energies will be Va gBtl ($2) dea jog S [e'pe'], | 0 We suppose that a single variable force P cos 2art/t moves. along the beam with a Neonat velocity v (fig. 1). The corresponding differential equations may be written fin ithe. form : ont ae 7 thy =P cos ae 0 in Bae Ce Then taking @, =,=0 at the instant t=0, and writing hee ght W ‘ Communicated by Mr. R. V. Southwell. + Cf. ‘Engineering,’ vol. exii. p. 80 (1921). On the Forced Vibrations of Bridges. 1019 we obtain ERIE 6 Ws Qat, . narvty .. n?ar°a(t—t,) =}. P cos "sin Shp ee ee 4 Pn nna wl eter l Us m4) 20 and the expression (1) may be written as follows :— ee nara.» Dae nine. 2a 5 - NTL SID sea ep aa ee —— ]t = “aT es Sl) < l TF l a mil UD oe aaa ene ean aa ae P n=] nt—(B+ne) | nt—(B— ne) & ee n? qr at 5 n= arat sin —5— in he + — P P a tt Phy (Oe) WORD) a ros Sohn Ge S 5 Wa? — (rn? —B) | We (e+ 8) vl Zhe where a=: B= =, amr TAT If the period 7 of the force is the same as the period Ty i =) of the principal mode of vibration of the beam, resonance will occur, and the amplitude of the forced vibra- tion will increase with ¢. Under these conditions, we have P=1, and at the instant when the periodic force ceases to act upon the bridge we have (= l/v, ik TE eyo ae so that Beye (in general, a small quantity). 2 Then the first term (for n=1) in the series on the right of (5), which is the most important part of y, may be reduced to the form 2 ral ede sin te L . 20 7m HI l and the maximum value of the deflexion is given by the formula . T ree ARI pes. = aE AAR ye - < - . ° (6) Since, as we have seen, « is usually a smail fraction, we may conclude from (6) that the forced vibration produced by want of balance in locomotives mav be of practical importance. The same method can be applied in other cases, where more complicated expressions are required for the forces which produce the vibration, and also in cases where variable horizontal forces act on the beam. Zagreb, Yougoslavia. Dee. 11th, 1921, f LOZOE Ul OVI. The Occurrence of Ionization by Cumulative Effects. By FRANK Horton and ANN CATHERINE DAYIES*. N the Philosophical Magazine for March 1922, Professor K.T. Compton replies to criticisms we made f of the experiments by which he claims to have demonstrated the ionization of helinm below its normal ionizing voltage, as the result of electron impacts against atoms which have absorbed resonance radiation from neighbouring atoms ¢. In this reply Prof. Compton produces evidence which he states ‘‘ proves these criticisms to be unfounded.” The experiments which we criticized as giving inconclusive results were made with an apparatus in which the collecting electrode was in the form of a hollow cylindrical box having one plane end of platinum gauze and the other plane end of platinum foil, arranged so that each end in turn could be faced towards the glowing filament. The ratio R of the : be are Age currents measured in the two cases 1s ae where 2 is the i+er part of the measured current which is due to ionization (which is the same whichever end of the box faces the fila- ment), and 7 and cr are the parts of the measured current which are due to the photoelectric effects of radiation when the foil-end and the gauze end, respectively, face the fila- ment. Hence,if the constant ¢ is known, the ratio of i/r can be calculated from the ratio R of the two measured currents. For pure radiation the value of R is 1/e, and for pure ioniza- tion it is unity. Prof. Compton assumed that the value of ¢ was given by the ratio of the area covered by the wires of the gauze end to the total area of the plane end, and checked his calculation of this by comparing the photoelectric currents measured when using radiation from a quartz-mercury are outside the apparatus. The value used by him in estimating w/r was 0°59. The following table, which is taken from Prof. Compton’s paper, gives the values of R and 2/r obtained for electron energies between 20 and 25 volts, at different pressures of helium :— 3 * Communicated by the Authors. + F. Horton and A. C. Davies, Phil. Mag. vol. xlii. p. 746 (1921). } K. T. Compton, Phil. Mag. vol. xl. p. 558 (1920). Occurrence of Ionization by Cumulative Efjects. 1021 p (mm.). R. a/?. 0:0005 1:90 0:055 0-001 1°81 0116 0-005 ie 0°176 0-012 160 0333 0-015 1:57 0°378 0:044 155 0410 OG 1°40 0-75 1-00 1-19 2-22 8:00 1:07 6:3 25°00 1-04 11-4 In considering the origin of any ionization obtained below the normal ionizing voltage of helium under the conditions of his experiments, Prof. Compton dismisses the possibility of its being due to electron impacts against helium atoms in such rapid succession that the energies of the impacts are additive in their effect, for reasons given by him in another paper *. He explains the ionization by supposing that part of the necessary energy 1s acquired from the radiation produced at the minimum radiation voltage, which is assumed to be absorbed and re-emitted by other atoms and thus passed on fromatom toatom. That this assumption is justified has been shown by recent experiments of the writers t. Itis, however, upon this very property of the radiation (which is so essential to Prof. Compton’s interpretation of bis results) that the writers based their main criticism of his deduction that ionization of helium was occurring at all the pressures given in the table, and that the ratio of the ionization current to the radiation current at each pressure in his experiments is that stated in the third column. Prof. Compton’s criterion for the presence of ionization was the deviation of the experimentally determined value of the ratio R from the value which he took to be the standard one for a pure radiation effect, viz. R=1/e=2. Any error in the value taken as the standard seriously affects the in- terpretation of the experimental results, and it was pointed out in our criticism of these experiments that the occurrence of the process of the handing on of the helium radiation from atom to atom of necessity means that the area of the collecting electrode acted apon by the radiation in any given case is not simply the area of the covered part of the particular plane end of this electrode which is facing the filament, but may include also the area of the curved surface. of the cylinder and possibly even the area of the plane end * K.T. Compton, Phys. Rev. vol. xv. p. 476 (1920). + F. Horton and A. C. Davies, Phil. Mag. vol. xlii. p. 746 (1921). 1022 Occurrence of Ionization by Cumulative Effects. remote from the filament. In his reply to this point Prof. Compton admits that some uncertainty exists in the value of R which can be taken as denoting a pure radiation effect, but asserts that the observed variation in R with variation of the pressure of helium is entirely too large to be accounted for in this way, quoting as an instance the result he obtained at 25 mm. pressure. It follows, from the additional data now given by Prof. Compton, that in the case of eight out of the ten results in his table (quoted above) the observed variation in R is not such as to justify the statement that it cannot be adequately accounted for without assuming the presence of ionization. The area of the cylindrical surface of the collecting electrode is started to have been equal to four times the area of the plane end of this electrode. Hence the limiting value of R for pure radiation, a value to which R will approximate when the radiation is so scattered by repeated absorption and re-emission that it is as likely to fall on the curved Ar+r 17 ae Lt where 7 is, as before, the photoelectric current. from the foil- covered plane end of the collecting electrode. In deducing this limiting value of R for pure radiation we have neglected the photcelectric effect from the plane end of the coilecting electrode remote from the filament, as this is unfavourably situated in regard to the electric field, whereas the cylindrical surface is, according to the diagram of the apparatus, in a situation comparable with that of the front plane end so far as this field is concerned. At very low pressures the radiation cannot suffer much deviation from its original direction through absorption and re-emission by helium atoms, and hence the front plane end of the collecting electrode will, in these cases, receive nearly all of the radiation which falls on the outside of this electrode, and the value of R will approximate to the value 2 assumed by Prof. Compton. At higher pressures, however, the curved surface receives a larger proportion of the total radiation falling on the collecting electrode, and hence, as the pressure is increased, R decreases from the value 2 and approaches more and more nearly to the limiting value 1-11. From the table already quoted, it will be seen that the decrease in the experimental value of R with increase of the pressure of helium is such as might be anticipated for a pure radiation effect, and that only in the two cases in which the pressure was very considerable do the experimental values of R fall below the limiting value for pure radiation. Thus in eight out of the ten instances the experiments do surface as upon the front plane end, would be On the Buckling of Deep Beams. 1023 not give conclusive evidence of the presence of ionization below the normal ionizing voltage of helium. There is, of course, ample evidence from spectroscopic observations * that when the pressure of the gas is consider- abie, ionization, which is presumably produced by cumulitive action, occurs below the normal ionizing voltage, but this does not show, as Prof. Compton implies in his recent paper, that the results of the experiments we have criticized are reliable. In regard to our criticism of Prof. Compton’s original contention that the neon which was present in the helium used in his earlier experiments would be incapable of causing the effects he obtained at different pressures, it must be pointed out that the interesting new evidence of the purity of the helium used in the later experiments does not affect our objection to his original contention. — CVIL. On the Buckling of Deep Beams. To the Editors of the Philosophical Magazine. GENTLEMEN,--— N. connexion with the two papers on the Buckling of Deep Beams published in your periodical (see Dr. J. Prescott, xxxvi. p. 297 and xxxix. p. 194), [beg to communi- cate the following :— The question of the buckling of beams can be regarded as solved for a long time already. The first paper on this - subject was published in your periodical by A. G. M. Michell, 1899, vol. xilviii. The same problem was investi- gated with more details by Prof. L. Prandtl, 1899 (Alunich Dissert.). In both these papers, as also in the paper of Dr. J. Prescott, the bending of the flanges of girders by sideways buckling is neglected, and in consequence of that the results cannot be apphed to the calculation of I girders. The influence of the flexion of flanges of the girder was studied by me, and the results were published in Russia, 1905 (Bulletins of the Polytechnical Institute, Petersburg). The translation of this paper in German can be found in Zeitschr. f. Math. u. Phys., Bd. lviii.(1910). Other manner of * O. W. Richardson and C. B. Bazzoni, Phil. Mag. vol. xxxiv. p. 285 (1917); K. 'T. Compton, E. G. Lilly, and P. 8. Olmstead, Phys. Rev. vol. xvi. p. 282 (1920); A. C. Davies, Proc. Roy. Soc. A, vol. c. p. 599 (1922). 1024 On the Measurement of Absolute Viscosity. solution is given in my memoir published in French (Annales des ponts et chaussées, 1913, Fasc. iiiv.). There are given the numerical tables, ecu l enable us to calculate very easily I girders under different conditions of loading and fastening of the ends. 70 Sinclair Rd., W. 17. Nonnsiteaeg August 30, 1921. _ §. TIMOSHENKO. CVIII. Notes on the Measurement of Absolute Viscosity. To the Editors of the Philosophical Magazine. GENTLEMEN, INCE the publication of my paper ‘“ Notes on the Mea- surement of Absolute Viscosity’ in the February number of this Magazine, my attention has been drawn to the fact that the variable head correction had already been noticed by Meissner (Chem. Rev. tiber die Fett & Hartz Industrie, vol. xvii. p. 202, 1910), by Simeon (Phil. Mag. GMS Oy axel JESNUAE) atavel by Bingham, Schlessinger, and Coleman (Journ. Amer. Chem. Soc. 1916, p. 27); also the value of unity for the kinetic energy coefficient was found by Couette and Finkener, and again by Kohlrausch (Lehrbuch der Praktischen Physik, p. 258, 11th edition, 1910). Ican only say that for my own part the work in my paper was entirely original, and I had no idea the ground had been covered before, all of which, of course, only serves to emphasize the necessity oF keeping in touch with current literature. Yours faithfully, ee March 81st, 1922. Frank M. Lipstone. CX "tatligenes and Wscdonecue Articles. ON THE MEASUREMENT OF ABSOLUTE VISCOSITY. To the Editors of the Philosophical Magazine. DEAR SiRs,— M4* I point out with reference to the paper by Mr. Lidstone in the February number of the Philosophical Magazine, p. 804, that the equation deduced there for the viscosity of a liquid can only be used if the flow is from a uniform tube on the “ high” side of the capillary tube to a uniform tube (not necessarily of the same bore as the other uniform tube) on the *‘ low” side of the capillary tube. I am, University of Toronto, Yours truly, Physics Dept. JOHN SATTERLY. April 7, 1922. ) —<— Vol. 43, Pl. XVII. J . Mag, Ser. 6 Phil AMONT, HemsaLecH & DE Gr ‘seposzoaja XN Sy S SS hi S n? S ate "QOOWe 0 YOIZQOW LO UVOWZIDMIGZ S, i i | by +e AH SfOPLINZESY ONS pue ubro RW BAF OHA gic. S S N "(S}[OA QQ) UINISOUseIT JO eI}DAdS sIV UBODOLONY L// ISO V/GLZSUL) ‘O UIBOSZIY UL IIE AVYQVCZO 7) we pinby, UW ISP AGP2SU/) 7 HOSPOASCL U/ IE AGJEIS ee. , Vol. 43, Pl. XVITI 6 . Mag. Ser. il Ph E GRAMONT?T, HEMSALECH aND D Wek eS USALBINO ‘oD VPIPSIG. OQ0UE YO VO/ZOW YO Vo v S ly+ ‘SQPOSZII/I £2 PINZEY pue ubig "(S}JJOA 08) UINISoUSsey JO EIZDOdS o1y ‘ILE 10 LS SEES 9 ‘IIMNOS YO IINZE/Y Phil. Mag. Ser. 6, Vol. 43, Pl. XIX: Hemsatrecu & pb Gramonr. ‘IIe AseUulp JO "yseods AQI2ede2 oh ‘asey” | =z” Wy ge S 3 DS g 8 DIE SIQOM bre aseyd S2/9A 007 o * pue 9 $8 ay * b+ bya ‘IIe Aseulpso S2/9A 002 e@ tw+e | SE Se XY nN A DIINOS f0 pue ubig N N ek ‘unIsousSey Jo e1izoodg yIedg pue osIVy Phil. Mag. Ser. 6, Vol. 43. Pl. XX. Hemsatece & DE GRAMONT. Arc and Spark Spectra of Zinc. Spark /1res eo) Nature 2 OF Source. NX (Capa city spark. Water a\ arc. f Ordinary, ere. i | &. Oromary arc. G Vfeter arc. LDurectior OF MOTION? OF 27002 NY Are lines. D x et Spark lines. HEMSALECH & DE GRAMONT. Phil. Mac. Ser. 6, Vol. 43, Pl. XXII. Arc and Spark Spectra of Cadmium. spark lire. Nature = of source. N Capacity spark. Weter =. alc. Ordinary. are. b. Water ere. Direction of 8 9 iS % aN S Ss aS oy) 9 \ spark lines. ss CARRINGTON. Phil. Mag. Ser. 6. Vol. 43, Pl. XXII. scale (ins) YY THE LONDON, KDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH BERIEB.] “ +! 5 a> ) c JUN IG P JUNE 1922: CX. The Hydrogen Molecule. III. By Apert C. CREHORE™*. ie the preceding paper f (II.) the case of two hydrogen atoms of the kind described { in the first paper (1.) was treated in the special position where the two atoms are so aligned as to have a common axis, and the forces acting along this common axis alone were computed. A stable equilibrium in this direction was found when the distance between the atoms is 1:18x 1078 cm. Since this problem is typical of that with other forms of more complicated atoms, it seems important to investigate the character of the whole field around the central atom as affecting a second atom. The method followed in the previous paper has been applied here to cover the general position in space, always keeping the axes of revolution of the two atoms of hydrogen parallel, but no longer coincident or coaxial. The only two forces that have any existence with this form of model are the electrostatic force together with the force due to the revolution of the positive and negative charges (electrons) upon their axes. The latter was completely developed in the first paper cited, equation (57), and may be called the gravitational attraction. It has been shown that no terms of higher order than the inverse square can be * Communicated by the Author. + Phil. Mag. May 1922, p. 886. t Phil. Mas. October 1921, p. 569. Phil. Mag, Ser. 6. Vol. 43. No. 258. June 1922. 3U 1026 Dr. A. C. Crehore on effective in this latter force, and hence we possess the com- plete solution. The rigid solution for the electrostatic force between two oblate spheroids ef charge in general positions _ js unknown, hence an approximation has been made to it, as explained in the second paper, by substituting for the negative electron a ring and a point, the solutions for which are known. ‘The charge on the point situated at the centre is By, and that on the ring E,, their sum being equal to the electron’s charge, ¢. The ratio of the charge BH, to e is defined as p, which evidently must be less than unity. This ratio is obviously determined by the eccentricity of the oblate spheroid, and it was shown to be equal to the square of this eccentricity in the previous paper. Taking the eccentricity ¢='945 as determined in the first paper, the value of 1/p is, therefore, about 1:12. — The process followed has been indicated in detail in the second paper, and it is not practicable to give here more ihan a bare statement of the results obtained. These are all, however, derived in the manner described by the use of the inverse square law between infinitesimal elements of charge according to the principles of electrodynamics. A starting point may now be said to be the electrostatic force upon one electron due to a second, and, in the nomen- clature employed before, this is given below as resolved into w- and g-components. The y-component has the same form as that of x, provided the direction cosine X outside the brace is exchanged for Y.- The symbol U?= X?+4 Y? has been employed for brevity, and this may be converted into a function of Z only, since X?+ Y?=1—Z?. 2 F, = 7 X{—r-?+paf,«(U)r*+ patf,o(U)r* electron seiton + pa*fia(U)e-*-+ pa®f o(U)r- + p2atfi,re(U)r-* Teron pear (UD as. + p’a*fz.ie(U)r—™ oa qe . (1) The z-force has the same form, but in place of the x-func- tions of U must be substituted z-functions, namely, instead of fi,4(U) write f..4(U) and so on: also the direction cosine Z takes the place of X outside the brace. ‘These functions have the following values :— fas (U)=6— PUY UR ee AD OL) 945 Vase (U)= a as + eu v?— 39 Us Pe Sap earn Minean yee S eh (3) the Hydrogen Molecule. - 1027 85 945, 3465... 15015 a ee oy a, oe, wee todd. 81320—_, , 220,2c0—e, eke gat ga ere Wee: (4) 4 M212 y,_ 382882, ee en et) pa i ee. fino) =— 28775, USL yy 10495485 py, 191125, _ 65,000.02 45 ee ee OO a me. sO) fis (U)= Feu eu oe, | a) ee. Oe fe te G8) 3828.825 5, 65,0002, gy The next fundamental expression is the electrostatic force of the negative electron upon a point positive charge, ne, 3U 2 1028 Dr. A. C. Crehore on where nis a whole number. This is 1h electron = nex {rn pre + pa* fr, is( U) r4+ pat Z, 20( U )r~ g on-+ne. + pa f,n(U)r“*-+ patfina(U)r™. 4}, - (16) with a similar expression for the z-component, having Z instead of X outside the brace, and z-functions within. These functions are Jz,1s( U ) = —3 fo,s (U) = —4 of (2) above. Se, 2o( U =: —3 f2,6 (U)= a (3) ) Fz,2X(U) = —$fz,8 (U)=—3,, (4), jan l= Se =$.,6) 29 Se, is( U) = — a Jz4 (U)=— De (9) ” Fz,20( U) = — 3 + 2,6 (U)= ae oP) (10) oe) f2(U)=—% 2.8 U)=— Z 99 (11) oP) Fz,24(U) aren Jz, 10( U) = Fey =F 2 (12) 29 Fig. 1. a-———— Sectional view of two hydrogen atoms with parallel axes, ABC and abe. The electrons are represented as straight lines, since the ring and point approximation projects into a straight line. The diagram, fig. 1, may represent the sectional view of ~ the two hydrogen atoms now under consideration. ABC is the first atom, and abc the second. The distance, 7, may be chosen as the distance between the centres of the positive the Hydrogen Moiecule. 1029 charges C and c. The displacement of the centre of the electron A from the centre of C is the sum of the two minor axes of these spheroids, but since the minor axis of the positive charge C is very small in comparison the whole displacement is practically equal to the minor axis of the negative electron, or numerically, 1:065 x 10-¥ cm. The forces according to the above equation must then be written down for each pair of charges, one from each atom, with due regard to their actual distances, since the small dis- placement, b, is not negligible. This fact makes the process quite laborious, and we must content ourselves with giving the final result only, namely the whole force of the second atom upon the first. So far as the electrostatic part of the force is concerned the coefficients of the r~? and r~* terms of the series are zero, and only even terms appear tliereafter. The gravitational equation (57) of the first paper above mentioned contributes, however, something to the r~? term, and in the following result the term thus obtained is included. The force of the second atom upon the first may be written F = 2x ] Z) Baty 72 ae J 3 faze Bor ee er + = G a 50°) ‘Fare(Z)r-® 3 ce “P(e pa?) | (—8pa") b? + 5 pat | fase (LN 9 5 5 a= = 5 ae oF path — =? patbt+ 7g pure? — L5p?ath* 4d Sons, 43 ys p7ash? — seg Pt’ |fnm AN gaa oe (Li) A similar expression answers for the z-component when X is changed to Z before the brace, and z-functions are substituted for x-functions, as for example, j2,2s(Z) instead of f,,03(Z). These functions follow. (Reh SURE EE i Ree ee eae a (18) fnos(Z) =—3(1—- 142° 421274), . 1 ew ee es (19) Fu,s0(Z)=5—1352?-+ 4952442978, . . 1 2 ee (20) fe 2(Z) = 5(7 — 3082? + 2002Z4 — 400428 + 24312"), . (21) Fz,26(Z) = —14 82, a pee Se rma aise ee 2), fe i = O0E GOT aa Res 28) fe30(Z) = 35—31527 + 693Z!—429Z78, . . . « « « CA) fr, 29(Z) = 315 — 46202? + 18018Z4— 257402 + 121552", (25) 1030 Dr. A. C. Crehore on In the preceding paper the coefficient of the r~® term containing the factor, say P, namely PEP—ipa,) was taken as zero, and this supposition was shown to be quite in harmony with the previously determined shape of _ the electron. This expression equated to zero requires that the eccentricity of the oblate spheroid of the electron shall exceed about °82, for, if it is less than this, the equivalent ring of radius a, that is employed in part to approximate to the electron, is shown to be greater than the equatorial radius of the electron, whichis absurd. An inspection shows that the eccentricity must probably be greater than about °9, and must fall between this figure and unity in order that the above factor shall vanish. And, because the eccentricity was determined to be °945 by another method and at an earlier date, it seemed justifiable to assume it to be exactly zero and examine the result. Let us now again assume, first, that (26) is zero because of the shape of the electron and examine the result. Since this quantity is a factor of both the coefficients of the r—® and 7-8 terms in (17), but not of the r—'° term, (17) becomes simplified as follows eee. {4 +32°)B4r? H-atom b on 63 Dato 45 ret H-atom. Ebi, | NIN Bg Rie Rg mp ee eS ae 7, b8 r—10 eee - + 3 ( rl Ag jeg) 3o(Z) b8 } (27) The ¢-force is similar, having Z for X before the brace and f:,32 (Z) for f:,32(Z) substituted within it. Using the « numerical values given above, namely 1 Ye 8,*='6053 x 107°*° (see (82) first paper), b=1:065 x 107 em. (see (79a) first paper), (28) (27) becomes Lie = a X{4(1+ 8Z?)°6053 x 107*%r-? Hectom, —942°5(7 —308Z? + 2002Z4 — 400428 + 243128) b'r- 10}, | (29) F,= a Z{4$(—14+3Z?)-6053 x 10-8r-2—108°5 (319 462022 + 18018Z!—25740Z' + 12155Z8)br-}, (30) the Hydrogen Molecule. 1031 The chart, fig. 2, is obtained by equating to zero the functions of Z and r within the braces of these equations. The caption under the figure is sufficient to render further description unnecessary. There is a point on the rotation axis (z) at a distance 1:18 x 107% cm. corresponding to the result at the conclusion of the second paper showing stable Fig. 2. .o) Z=10 927 B57/55 532 ¢ DBE? aaa — é a, ray, Su = = KI a ay, fi ) The scale is indicated by the central circle with a diameter of 10—8 cm. The first H-atom is supposed to have its centre at the centre of this circle, and the second H-atom may be located anywhere with its axis parallel to the first, the z-axis. There are two sets of curves, the one where the 2-component of the total force between the atoms is zero, and the other where the z-component is zero. These are indicated respectively by the horizontal and vertical hatching, which terminates on these two sets of curves. In the region of the horizontal hatching the 2-force on the second atom due to the first is in a direction awayfrom the axis ofz, and in all other regions towards this axis. Similarly, in the region of the vertical hatching the z-force is directed away from the axis of 2. The chart is plotted from equations (29), (30), the gravitational term being balanced by the electrostatic »—19 term, the coefficients of the 7-6 and 7-8 terms being zero. There is no position of stable equilibrium here, since the z- and z-curves nowhere inter- sect. The angles where the tangents to the 2-curves fall (the dotted lines) are given by the values of Z? in the margin. The tangents to the z-curves are shown by the dash-lines. 1032 Dr. A. C. Crehore on equilibrium in the direction of the axis, but there is no stability in a direction perpendicular to the z-axis at this distance. Before there is stability on this axis the distance must be greater than about 1:°32x107°8 cm., where the w-curve crosses it. Indeed there is no point on the chart where the w and z curves intersect, and it is requisite that they should intersect for complete equilibrium, since the whole force must be zero; and there must also exist a restoring force for small displacements. The hypothesis made above that P (26) is exactly zero is not, however, necessary, and it is quite unlikely on the theory of probability. All that has been shown is that it is probably small. This small value is entirely unknown unfortunately, so that the best that can be done is to assume different values for it large enough to have some effect upon the curves and see how the chart is thus modified. ‘The chart of fig. 3 is the result obtained by taking P=378x 10-3: ern The value is given to three figures simply because this is the particular value that happened to be used in making the chart. It hes between J? and 6%, not far from 6*®, and is: such as to make the r~° and r~8 terms of the series compare in magnitude with the other two terms at these distances, the r-? and r~° terms. To obtain a single point on one of the curves shown now requires the solution by approximation of a fifth degree equation, a much more tedious process than that required to chart fig. 2. The result shows that there is now a definite position of stable equilibrium for the two atoms at tke point where the «- and z-curves intersect, at an angle of latitude whose sine square is ‘41 and latitude X=39° 49’. The distance between two atoms in the stable position is ‘81 x 107* cm. A very great change has thus taken place in the result, changing fig. 2 into fig. 3 merely by using a value of P so small that it does not appreciably affect the approximation that ee Although no other complete chart has been computed corresponding to another value of P than that used for fig. 3, it has been proved that when this value of P is doubled, nevertheless the angle of latitude at which the stable equi- librium occurs is not perceptibly changed, but the distance 7 1s reduced from, (Sillsalin® to mode Ome acim The conclusion of this investigation seems to be doe in the Hydrogen Molecule. 1033 order to obtain a correct value of P with a high degree of accuracy, the distance between the two hydrogen atoms in the molecule must be sought from some other source. When the problem can be managed using some crystal, like the diamond say, where these distances are well known, this will afford a good value for the small residuum, P, and fix the property of the electron with considerable approximation. suUSEEESGREGEEE my ~TSs Lin ar x CoH |e Givici ee oeebaebamiae7 2. ? N fala afmtertol to leita ataCle OCP, PCCD he Bez NEN | fo se {_ | Se li | : No PEE EEE NEE GEBaSSanZ [— —Z2 Fig. 2 changes into this figure when the value of P (26) is that given in (31). The coefficients of the 7-6 and 7-8 terms are now small but do not vanish. The curves are plotted from equation (17), the scale being the same as that of fig. 2. There is now a stable equi- librium position where the «- and z-curves of zero force intersect at an angle of 39°49! (Z?="41), and at distance »=-‘81 x 10-8 cm. The equilibrium distance for the z-force on the axis has increased from 1:18 10-8 em. in fig. 2 to 1:83 10-8 cm. here. In the case of gases it is believed that not much more is 7 x b known about the distances between atoms in molecules than a rough approximation to their order of magnitude. 1034 | Dr. A. C. Crehore on Perhaps as good an estimate as any of the distance between the two atoms in the hydrogen molecule has been made by K. C. Kemble *, who, however, assumes a dumbbell form of molecule, which by hypothesis has a different amount of freedom from the molecule above described. The above molecule is free to rotate around a circle of latitude at 39° 49’, but is not free to change this latitude without en- countering a restoring force. Kemble makes use of the kinetic theory of gases combined with the quantum theory and arrives at a moment of inertia for the dumbbell, 2:0 x 10-* om.cm.’, from which value it appears that the distance between the two atoms is about:5x1078cm. A slightly larger value of P than double that used for fig. 3 will give the equilibrium distance that is estimated from Kemble’s result, but it seems desirable to revise his estimate in order to adapt it to this form of atomic model. In concluding this part of the subject attention is directed to the fact that, except for numerical coefficients which are not very large, the values of the successive terms of the series for the electrostatic force are as b47~®, b®~8, 68-70, etc., thus making the ratio of any term to the next following 1 constant, b-?r? at a given distance, 7. If r is of the order of 10-8 em., since 6 is of the order 107!* cm., the ratio is of the order 1076-1" = 10", and the terms, therefore, decrease at a very rapid rate. Were it not for the fact that there exists a peculiar reason connected with the shape of the electron that makes the numerical coefficients of the »~® and 77° terms simultaneously very small, it would be quite unnecessary to consider the use of the r~! term at all. This same pecu- liarity does not apply to the coefficients of the r~!°, r-¥, and subsequent terms, and as a consequence they follow the general rule above cited. Hence the r~' and all following terms are quite negligible. The following Table I. gives the approximate values of the terms within the brace of (17) near the position of stable equilibrium for the two cases where P has the value used in making fig. 3 and for double this value. For the larger of the two values cf P it is seen from this Table that the gravitational or 7~*? term is becom- ing negligibly small in comparison with the other three terms. That is to say, the use of the electrostatic force alone is quite sufficient to cause a stable position to exist in the region of °5 x 107° cm. so far as the translational forces between the two atoms are concerned. The internal moment of the force which tends to keep the two atom’s axes parallel * Phys. Rey. Feb. 1918, p. 156, the Hydrogen Molecule. 1035 does, however, depend upon the rotation of charges upon. their own axes, the @-terms. Aes pope ie Bo 16 xO, io erie r. Fae LO pr OL OR rae LORS ae 1? Sc 1070: term. term. term. term. aw-force... *79X10-8 1-067 1G ies = 07 6in =) 9595 z-force... °81x10-8 0-922 11:075 17°39 29°26 PZT Ox Om Die ae x-force... *56X10-8 2°124 501 —253'1 — 289°9 patereece:.)°O1, < 10-8 0-186 365 578°6 — 983 This is rather a startling proposition, for it is well-known that no system of point charges alone without motion postu- lated can be so arranged as to produce a stable configuration for small displacements. The Energy of Dissociation. In the example of the two hydrogen atoms the forces have been given for the action of the second atom upon the first only, the individual parts not being expressed. The force, however, of the second atom upon the positive charge of the first is equal and opposite to the sum of the forces upon the two electrons at the position of equilibrium. These forces are each very great compared with their difference, which represents the whole force upon the atom. This may be proved by an examination of the equations above given, from which it will appear that the coefficient of the r~® term is not small. The effect of these forces is to increase the pressure between the positive charge, or nucleus, and one electron, and at the same time decrease the pressure between the nucleus and the other electron. ‘The result must be to deform the shape of the four electrons very slightly, flattening one pair and increasing the minor axis of the other pair in the molecule. The area of an oblate spheroid in terms of its eccentricity and semi-axes a and b may be written mb*, I+te log. rdehiore ieee € a 2rra,* = 1036 Dr. A. C. Crehore on The area of a sphere of radius + having the same volume may be written S =A, where a°b=1.° aaa Dividing (32) by (33) the ratio of these areas is obtained as a function of the eccentricity alone, Sous 1 ees ihe 2 l+te gq = a(1—e’) + g.\ ne 310 ae Again, it may be shown that if the spheroid always main- tains a volume equal to this sphere of radius 7 while its axes ad) and b are changing, then : (34) de een i 10 ae" Now differentiating (34) with respect to e¢ and multi- plying the result by (35), we find the rate of change of the area of the spheroid with Ae to the minor axis AS a ie b 3bt eo db de =s{—5 D2 ie +; 1 i (36) Restoring the value of 8 (33) and using AS fur dSp and Ab for db, and multiplying the change in area of the spheroid by the surface tension T, (see (73) first paper), namely (35) og 16 rrek? we thus find the energy required to change the shape of the spheroid by the small amount Ad as follows : = (37) e2 9b AE=(A8). =F Ad 4 — ote 3b? 9b Ll+e} + (ga, + apap lB = } ee Using numerical values _ r=2°244x 107% cm. 6=1:065 x 10-8 em. eee e=0°945, | we find A © Ab(—0-236 x 102%)... 3 een the Hydrogen Molecule. 1037 Two of the electrons of the molecule are contracted (or flattened) say by the amount Ad, while the other two are expanded by a slightly different amount, say Ab,. Hence the change in the internal energy of the whole molecule from the state of free atoms may be written 2 AE=2(—0-236 x 10%) 5 (Ad; — Aly). ie Cay The expansion of the minor axis Ab, may be assumed to be greater than the contraction Ad,, so that one must supply energy to the molecule to restore it again into two separate atoms. This is conceived to be the place where energy must be supplied to separate the two hydrogen atoms of the molecule, and the energy on this view is not only interatomic but also interelectronic. Using the experimental value of the energy of dissociation per H molecule as 5°8x10- erg as the value of AE in (41), it appears that MAb em. 1.05 is (4D) That is to say, the difference in the change of the minor axes of the two electrons in the one atom is very small indeed in order to correspond with what may be regarded as a rather large energy change. This is because the total energy content of the electron, mc’, is 8-08 x LO~! erg, and far exceeds the order of magnitude of the energies concerned in chemical transformations, of order 10- erg. Since the actual radius, 6, of the electron is of an order a million times larger than (42) it seems most probable that the change in the shape of one electron(A 5) is of an order jess than say the ten thousand part of the original radius. Any change in the shape of the electron involves a corre- sponding change in its mass, but this change is so small that the change in mass must escape experimental detection, and this is true for the additional reason that the mass of the electron is already an almost negligible part of the atom itself. But this minute change is, however, entirely sufti- cient to manifest itself through the energy liberated or absorbed. The small change in shape of the electron caused by the forces above mentioned when the atoms unite into one molecule, and which act in the proper sense to effect such a change, may account entirely for the observed energy of dissociation in all chemical combinations. This example using hydrogen as an illustration points to the seat of all 1038 Mr. Shenstone and Prof. Schlundt on the Number such energies of combination well known to the chemist. Considering the character of the atoms described in the first part of the preceding paper, it is not difficult to understand that this energy may have a positive sign in some cases and a negative sign in other cases. Assuming that these views — represent the truth of the matter, it is not difficult to see why such results as fill Thomsen’s extensive work on Thermo- chemistry have never been successfully connected as yet with any sound theory. At the same time it revives a hope that this vast store of experimental evidence may soon be -correlated with theory. The important conclusion has been reached in this paper that the chief result in obtaining stable equilibrium for the two atoms of the molecule is independent of the special form -of any electrodynamical theory, as for example the Saha or the Lorentz theory, since it depends chiefly upon the electro- ‘static force. All theories agree in making the electrostatic force between infinitesimal elements of charge follow the inverse square law. Cleveland, Ohio. Nov. 1, 192i. *CXI. A Determination of the Number of a-Particles per Second emitted by Thorium C of known y-Ray Activity. By Auven G. SHenstone, M.A., and Professor HERMAN SCHLUNDT * ECENT M cniere in the Cavendish Laboratery by Sir Ernest Rutherford and others have involved the ‘use of the long-range «particles from thorium C. The -activity of the large thorium C sources used in those experi- ments was measured, as is customary, by means of a y-ray electroscope. It has therefore become necessary to make -an accurate determination of the number of a-particles emitted per second by thorium C of known y-ray activity. Since that number is known for radium C,a direct com- parison by count and by activity of radium G and thorium C will fix the number of a-particles per second emitted per mg. of thorium C. We have jointly carried out the experimental -determinations. Thorium C and its products produce, in the course of dis- integration, a-rays of ranges 8:6 cm. and 95:0 cm. in air, in the proportion of 65 to 35. At the same time they produce shard y-radiation. Radium C similarly emits hard y-radiation * Communicated by Prof. Sir E. Rutherford, F.R.S. of a-Parlicles per Second emitted by Thorium C. 1039 and a-particles of range 6°96 cm. and 3°8 em., the latter, however, appearing in negligible numbers. In this experi- ment the «particles counted were those of 8°6 cm. and 6°96 cm. range. The deposit of thorium C was obtained in the usual way by rotating a nickel disk for about half-an-hour in a solution of radio-thorium. To obtain radium (© a quantity of emana- tion was allowed to decay overnight, compressed in a small glass tube. The active deposit on the walls of the tube was Fig. 1. _ Class Tube. Centring Piece. ‘Source, Mica = —> 8-OMmMs. Fo casAir. Zn S Screen. Microscope Objective. Glass Wheel wIth Brass SW. Q-PARTICLE COUNTING APPARATUS. then dissolved in 10 per cent. hydrochloric acid, andthe radium C obtained by stirring the solution for about three minutes with the nickel disk. The disk used was of 8 mm. diameter, and was carefully protected by sealing-wax on the back and edges while depositing the active material. Check experi- ments on the back of the source showed that an inappreciable number of e-particles was emitted from it. The experimental arrangement for counting is shown in the accompanying diagram (fig. 1). The nickei source was 1040 Mr. Shenstone and Prof. Scklundt on the Number held centrally in a glass tube which could be exhausted by means of an aspirator. ‘he end of the tube facing the’ source was closed by a brass plate having a central bevelled hole of 3 mm. diameter. This hole was closed by a thin sheet of mica of stopping-power equivalent to 4°3 em. of air. The zinc-sulphide counting-screen was mounted on a bracket. 8 mm. from the end of the tube. The total stepping-power in the path of the e-particles was therefore 4:°3+°8=5'1 em. of air *, which is just sufficient to prevent the 5 cm. a-particles from thorium C from reaching the screen. The efficiency of a zine-sulphide screen varies considerably over its surface. To ensure permanence of experimental conditions, it was therefore necessary to be certain that the same portion of the screen was always used. This was accomplished by mounting the screen with its crystal face against a piece of thin brass having a hole of diameter slightly less than the field of the microscope. The micro- scope could then always be focussed on the identical portion of the screen. The microscope used gave very bright images of the scintillations. It was fitted with Messrs. Watson’s “‘ Holos” x 5 eyepiece and ‘‘ Holos” objective of "45 n.a. and 16 mm. focal length. The usual “pea” lamp was used to slightly illuminate the screen. The“ Wheel” Method of Counting. To enable large counts to be made with ease, a method of counting devised by Sir Ernest Rutherford was adopted. The essential feature of this method is the use of a disk of about 20 cm. diameter mounted with its edge intercepting the beam of a-particles. The disk is provided near its edge with a slit of suitable width which was 1°6 mm. in our ease. The disk must be mounted on good bearings and must be fitted with a driving device capable of rotating it at a very steady rate of from 50 to 200 revolutions per minute. When the disk is rotating, the particles can reach the screen only when the slit is passing through the beam of particles. Moreover, the number of a@-particles which can pass per minute is not a function of the speed of the wheel, but. merely of the ratio of the width of slit to the circumference of the disk. The method of counting and recording also differs from the ordinary method. The observer does not mentally add the number of scintillations over a minute, but calls out the number which appears at each revolution of the * To this should be added about 0:2 cm. for the air left in the tube between the source and the mica disk. of «-Particles per Second emitted by Thorium C. 1041 wheel. The recorder writes down this series of small numbers, and they are added by minutes later. The power of the method lies in the fact that calling out a seen number is merely a reflex action, whereas addition requires mental effort. The best observer cannot count with certainty above 65 or 70 scintillations a minute by the ordinary method; but with this wheel method up to 160 a minute can be recorded. Practice is, of course, required before the actions involved become quite mechanical. For a beginner, the wheel should be run as iow as 50 a minute for counts averaging about 75 a minute, or about 1°5 per revolution. Later, the observer should have no difficulty calling out the numbers at a rate of 80 a minute for counts averaging two a revolution. Ability to judge rather than to count the number of scintillations in a group is essential. An experienced observer can judge accurately up to groups of 7, mainly, in our experience, by the geometrical figures formed by the scintillations. Beyond 7 it is not usually possible to be absolutely certain of the number, but one must hazard a guess. Such large numbers should appear, however, extremely infrequently if the speed of the wheel is properly adjusted. We found that it was quite unsafe to count when the scintillations were appearing at a rate of over 2°5 per group on the average. When counting the particles from a decaying source and the number drops below 60 per minute, we found it easier to run the wheel about 200 revolutions per minute and to add the numbers mentally. It is of interest to see the operation of the law of pro- bability in the following examples of sequences of numbers called out during this work. A certain number of zeros should appear, but they were not noted. i minute:-—_123114221131443822233133312141 ays tal Oa id Mmmute:—2 4042 2°93 sa2-2 21 lb 2 1 3322 3-53 4432 eee et on lenlvae 2. Qals Pel dS 4 22205. 4 les. Somme: —_ 22239.) 252 2122 2221 0 245 81 38 2b 222. y-Ray Measurements. Since the comparison of the a-ray activities of the radium CG and thorium C was to be based on equal y-ray activities, it was necessary to have accurate y-ray measurements at the same time asthe counts. A y-ray electroscope was mounted, therefore, about 30 cm. from the active source when in its counting position. The activity and decay of the source could then be measured by taking groups of readings while Phil. Mag. 8. 6. Vol. 43. No. 258. June 1922. 3X 1042 Mr. Shenstone and Prof. Schlundt on the Number the count was in progress. Thorium © should fall to half value in 60°5 minutes and radium CG in 19°5 minutes. Actually we obtained decay curves giving periods of 60 to 61 minutes for thorium C and 19°5 to 20:2 for radium C. To calculate the result of a day’s count, we first plotted the logarithms of the y-activities against the time. Tie best straight line was then drawn through these points, and the yeactivity calculated from it for each haif minute during which counting was carried on. The number. of particles counted per half minute per arbitrary unit of y-activity is then obtained by dividing the total number of «’s counted by the sum of all the y-activities calculated. For counts on both thorium C and radium C the ratio of the numbers of «-particles emitted per 4 minute for equal y-ray activities is then obviously Pe for thorium C + So particles ‘ ements aaa f 5 ° : > y-activities S y-activities radium © Such a ratio is independent of the dimensions of the apparatus as long as those dimensions remain unchanved during all the counts. The work was divided, therefore, into groups, and necessary adjustments to the apparatus were made only between groups. It will be seen in the following table that the ratios 3 vary very considerably a radium © a thorium C between groups, but that the ratios do not differ by more than the probable error. rm THoRtIvum C. Rapium C. Lhe, | Ra, | | | Group. | Count. | Activities. | Ratio. Count. | Activities. | Ratio. | Ratio. iL 2700 2506 1:08 | 2953 1993 Letra) aa JEL 1019 780 1:306 913 545 1685) 7S Ill. 3191 2722 117 3515 2285 1 34 LVR 3105 2231 59 3803 2100 181 Un 7 Ve 3250. 2411 1:35 | 2316 1246 1°86 | 13265 13500 Weighted mean = a of a-Particles per Second emitted by Thorium C. 1043 eattnal takocot «thorium OC a radium C computed as the mean of the group ratios, each weighted by the root of the number of a-particles included in the group. for equal y-activities w s Probable Errorin Mean Ratio. The probable error in the ratio *75 arises from two factors. First, the error ie to the finite number of particles counted. This is 100 (545 ae : "= 1:22 per cent. The other 13265 15500 source of error is the electroscope and the plotting of the curves from the electroscope readings. The number of readings taken during a count was, on “the average, 12. The probable error from a combination of 12 readings was computed by the formula 2/3 WROD from a large number of y-ray measurements made on a radium standard. The highest value of the probable error obtained in this way was ‘6 percent. The error in drawing the logarithmic decay line through the plotted points is unfortunately impossible to calculate. It was certainly not greater than 0°95 per cent. Both these last errors enter twice, so that the probable error is given by {(1:22)?4+2 x °6?+2 x ‘5?}2=1'65 per cent. aie probable error calculated by the formula 2/3 ye N—1)’ using the group pales is *9 per cent., which is well within the error calculated above from the various factors separately. It is of interest to note in this connexion the agreement between counts by the two observers. We differed by as much as 9 per cent. within groups, but our final ratios are only *3 per cent. apart. ‘The agreement is excellent evidence of the accuracy 2f the counting of both observers. y-ltay Absorption Measurements. The y-rays, in order to reach the interior of the electro- scope used in this experiment, had to pass through a thick- ness of glass, lead, and aluminium having an absorbing power for the rays equivalent to 3°3 mm. of lead. In order to extend the application of the ratio of e emissions to com- parisons through different thicknesses of lead, it was necessary to compare the absorptions in lead of the y rays from thorium C and radium.C, This was necessary because or we 1044 Mr. Shenstone and Prof. Schlundt on the Number the coefficients of absorption given in the lilerature on the subject apply only to thicknesses of lead greater than 2 cm., and are far from the truth in the region below a centimetre. Curves I. and II. of fig. 2 are the result of y-ray measure- ments on thorium C and radium C sources when increasing thicknesses of lead were placed between the source and the electroscope. It has been seen that the ratio of a-activities Da for equal y-activities was computed as ze thorium C + Sy zy radium (©, t TAC. scfe. Il 4ea.c0. tr Wl Ae. S€Zandard’. IV Ratio R2.C37h.C. Vv ’ fra. St: Pra. C. VI fra. St: Th.C. 2 : : = SS) 473 63 8°3 10-3 12:3 14-3 y-Ray Absorption Curves. Sy radium C Sy thorium C fore if, from curves I.and II., we plot curve IV., the ratio of y radium C y thorium C° ratio of a-emissions for equal y-activities will be changed when comparisons are made of y-activities between radium OC and thorium © through greater thicknesses of lead. The curve obtained in this way, by multiplying the ordinates of curve LV. by °75, is given in fig. 3. The activity of a radioactive source is usually measured by comparing its y-activity with that of a radium standard. The radiation of a radium standard consists in the maiu of two y-radiations—a soft radiation from radium B and the hard radiation from radium C. The soft radiation is reduced In this expression the ratio appears. There- that curve wiil give us the factors by which the of a-Particles per Second emitted by Thorium C. 1045 practically to zero by passage through 1°5 em. of lead. The absorption curve of the radium standard, therefore, should drop rapidly at first, and should finally attain a rate of fall identical with radium C. Consequently, if the ratio of the activities of a radium standard and radium C for different thicknesses of lead is plotted, it should show an initial fall, Bie’ 3: “ea ia ee 6-3 8-3 10-3 12:3 14-3 Ratio of number of e-Particles ThC: RaC for equal y-Activities. FAS) 725 -70 Mins. Lead. 3-3. 43 but a final constant value when all the radium B radiation has been absorbed. Curve III. fig. 2 is the absorption curve of the radium standard and curve V. is the curve of the ratio, radium standard/radiaum C, plotted from II. and III. The truth of the statements above is evident from this curve. ‘The curve becomes flat at a value 91. Consequently the ionization which was produced at 3°3 mm. of lead by the radium B in the standard, and which has been eliminated by 1-4 cm. of lead, must have represented 9 per cent. of the total ionization produced in the electroscope by the radiation from the standard. The following table gives the y-ray measurements from which the curves were plotted :— Mm. of Lead. | Radium Standard. | Radium C. | Thorium C. wie cee ee aes Penny setts |L SE oy See 33 1-000 1-000 1:000 5°52 "836 "884 ‘890 7:62 “21 ‘T77 ‘794 9°87 ‘631 679 ‘705 11°95 D902 "597 640 14:15 "4E4 539 565 The determination by Hess and Lawson* of the number of particles emitted by the amount of radium C which is in * Publication 105 of the Radium Institute at Vienna. 1046 Number of a-Particles emitted by Thorium. C. equilibrium with 1 mg. of radium is 3°72 x 107 per second, with a probable error of ‘54 per cent. But we have just seen that this amount of radium © gives only 91 per cent. of the total ionization produced by radium and its products. Therefore the rate of emission of e-particles by the amonnt of radium © which will give the same ionization as 1 mg. of radium when measured through 3:3 mm. of lead is 3°72x LOSS = — 4:09 x 107 per sec. Consequently, thorium © of | equal y-activity emits at the rate of 4°09 x 10° 750i 10° per second. This number involves the error of 1°7 per cent. in the ratio’75 ; the error of ‘54 per cent. in the deter- mination of 3°72 x 10"; the electroscope error of *8 per cent. again entering twice in the absorption measurements ; and Fig. 4. Number of 8°6 cm. #-Particles emitted by ThC of y-Activity = 1 mg. Radium. the error due to the weighing of the lead sheets and varia- tions in thickness of those sheets. This last error can only be estimated, but it should not be greater than 1°5 per cent. Combining all these factors gives a probable error of 2°5 per cent. The amount of thorium C which will give the same amount of ionization in an electroscope as 1 mg. of radium and its products will vary with the thickness of lead through which the y-ray measurements are made. In fig. 2, curves I. and III. are the absorption curves for thorium C and for a radium standard, and curve VI. is plotted from the ratios of the ordinates of curve III. to curve I. Starting with equal y-activities of thorium C and radium at 3°3 mm. of lead, this curve VI. evidently gives the amounts of thorium C which will give at greater thicknesses of lead y-activities equal to Movements of Molecules at very low pressures. 1047 the y-activities of the radium with which it is being com- pared. But, at 3°3 mm. of lead, thorium C of y-activity = 1 mg. of radium and its products emits 3°07 x 10‘ a-particles per second. Consequently, when comparison is made through a greater thickness of lead, the rate of emission of a-particles by thorium C is obtained by multiplying the corresponding ordinate of curve VI. by 3:°07x10". The curve obtained in this way is given in fig. 4, from which may be read the number of particles of range 8°6 cm. emitted by thorium C of y-activity=1 mg. of radium and its pro- ducts for comparison through any thickness of lead between 3°3 mm. and 14°3 mm. Our thanks are due to Mr. G. A. R. Crowe for the pre- paration of the sources used; and to Sir Ernest Rutherford and Dr. J. Chadwick, whose valuable advice rapidly eliminated our initial inexperience. Cavendish Laboratory, Cambridge, 1921. CX. lestricted Movements of Molecules at very low pressures : A Limit of Applicability of the Second Law of Thermo- dynanucs. By ARTHUR FAIRBOURNE, Lecturer in Chemistry, King’s College, London *. T has long been realised and has often been stated (compare Maxwell, 1871, ‘Theory of Heat,’ p. 308) that if sufficiently minute apparatus could be constructed to restrict, selectively, the movements of certain individual gaseous molecules so as to direct, for example, those with high velocities into one particular portion of a system and those with low velocities into another portion, then there would result in consequence of this selection a continuous creation of potential of temperature between these two portions of the system ; a continuous creation of potential which would render the kinetic energy of the molecules continuously available for the performance of external work. Purely hypothetical apparatus for the satisfactory theoretical consideration of this possibility has been described, and much discussed (compare Maxwell, loc. cit.; Johnstone, 1921, ‘The Mechanism of Life,’ p. 215; Jeans, 1921, ‘The Dynamical Theory of Gases,’ p. 183), since the * Communicated by the Author. 1048 £=Mr. A. Fairbourne on Restricted Movements possibility of continuous creation of potential is denied by the second law of thermodynamics. It must be agreed, however, that any such continuous experiments as this, whether actual or hypothetical, fall definitely beyond the jurisdiction of the second law of thermodynamics, since they depend entirely upon the deliberate selection of particular molecules with specially desired velocities, while the second law considers and deals only with the average effects of large numbers of molecules of every variety of haphazard movement, acting together merely under the play of chance ; and the conclusion stated by this law is consequently entirely subject to the condition that no such deliberate selections of molecules with particular movements shall be made. No record of any experimental verification of the effects of such selective restriction of certain molecular movements as are involved in these considerations has ever been put forward, nor has any means for performing such ever been suggested, and experimental verification has come in con- sequence to be regarded as unlikely, or, perhaps, impossible. Although it may be impossible to construct an apparatus capable of selecting molecules travelling with certain desired speeds, yet it can be shown that it is not impossible to construct an apparatus capable of selectively affecting molecules travelling in certain desired directions, provided that portions of the apparatus to be used for this purpose are made smaller than the free paths of many of the molecules of the gas employed ; and, since the mean free path of certain’ common gaseous molecules at the easily realizable pressure of 1x 10-? mm. is calculated to be in the order of a centi- metre in length, this requirement can be satisfactorily fulfilled without such difficulty as might have been anticipated. Just as the deliberate segregation of molecules of parti- cular speeds has been considered by Maxwell to cause the creation of potentials of temperature between the different portions of a system, so the continuous affecting or re- directing of molecules moving in certain desired directions only will cause corresponding differences (or potentials) of pressure on certain surfaces in the system. The principle by which such selective directional inter- ferences may be accomplished can be readily indicated by considering a long and narrow tube, ideally smooth and open at both ends, suspended in a gas whose inean free path is much greater than the length of the tube. Molecules enter freely at both ends, and most of those which happen to be moving in directions parallel or substantially parallel with of Molecules at very low pressures. 1049 the axis of the tube will pass through and out without any interference whatever to their motion, since the length is much less than the mean free path of the gas. Those, however, which enter at angles sharply inclined to the axis of the tube cross from side to side many times during their passage through the tube, and receive in this way many impacts from the walls. In fact, not only do they receive more impacts in unit time than do those which pass directly through the tube, but they also receive more impacts than do those outside the tube, since an extra restriction has been imposed * upon these molecules selectively. With a suitable shape of vessel in place of the tube, it may be proved as follows that this extra restriction on the move- ments of the molecules can be applied selectively in such a manner that molecules which enter the vessel moving in haphazard directions will be diverted to leave it moving in one preponderating direction. For the sake of simplicity, although the case is general, it will be assumed for the moment that the molecules are moving in two dimensions only, and it will also be assumed that the mean free path of the gas is so great in proportion to the size of the vessel that intermolecular collisions during the short journeys in the vessel are occurring with only a negligibly small fraction of the total number of molecules which pass through in any representative period of time. Z B C Let ABCD be a two dimensional vessel, the shape of which corresponds to the cross sectional elevation of a truncated right angled cone. Let it be open at the top, AD, and bottom, BO, and, moreover, let BC be twice the length of AD. * Any gas in a hollow cube, the length of whose face is many times less than the mean free path of the gas, receives more impacts per molecule, in unit time, than does the same gas, at the same temperature and pressure, outside the cube, since, inside, no molecule can travel the calculated free path without receiving extra impacts from the walls. 1050 =Mr. A. Fairbourne on Restricted Movements Molecules enter through all points along both AD and BC, and at each of these points they enter in all possible directions. If the walls are assumed to be ideally smooth, or to behave as such with respect to impacts upon them ~, then every molecule entering through AD, through any point, and in any direction, will necessarily pass out at BC, except in the very rare case where intermolecular collision occurs during its journey through the vessel. Of the molecules entering at BC, some pass out at AD, and others are returned through BC, two possible cases presenting themselves for consideration : (1) molecules entering through a point in BC under the opening AD; and (2) molecules entering through a point under either of the inclined sides, BA or CD. Case 1: Any point under the line AD. If the point be in the centre of BC (fig. 1), then all Fig. 1. | A D ’ ’ ’ / s) \ molecules entering in directions included by the angle AOD will pass out through AD, while all those entering in directions included by angles AOB and DOC will be returned through BC, owing to impacts on walls inclined at 45°. Since AOD is a right angle, and since molecules will enter equally in all directions throughout 180°, therefore half the total number of those entering through this point in any representative period of time will leave the vessel through the top, AD, the other half being returned through BC. If the point be not in the centre (fig. 2), but still under AD, those molecules entering in directions included by angle AOD, which is less than a right angle, wil] escape through AD. * The effects of irregular surfaces, and of adsorption on the surfaces, are considered later. of Molecules at very low pressures. 1051 All those whose directions are included in angles DOU and AOB will either be returned through BC directly, or will be directed on to the opposite wall, and thence returned through Fig. 2. A D BC. Thus more than half the total number of molecules entering through such a point as this will be returned through BO. Case 2: Any point under AB or DC. A perpendicular through O will cut AB (or DC similarly, if it be under DC).. Let this point of intersection be called P. Since POB is itself a right angle, and angle DOC not small, therefore angle POD is considerably less than a right angle. Molecules entering in directions included in angle DUU are returned directly through BC. Those entering in directions Fig. 3. A E > 4 ra B 2) ! 1C \ ' U i} included in angle POB are also returned through BC, either directly or after impact with DC. Only molecules entering in directions included in angle POD * succeed in reaching the top. Thus more than half of the total number of molecules entering through such a point as this will be returned through BC. Thus of all possible points of entry through the line, BC, * And only some of these, namely those which enter in directions included in angle DOX, where X is the point on BA such that angle OXB is equal to angle DXA. 1052 Mr. A. Fairbourne on Restricted Movements both those considered under case 1, and those considered under case 2, only that one point, which is midway between B and C will allow molecules entering equally in all possible directions to escape through AD to the extent of fifty per cent. All other points of entry on BO are placed so that the walls return through B© more than (and in most cases very considerably more than) half the total number of molecules which enter. | Let N molecules enter the vessel through AD in any representative or sufficiently long period of time, when, as has been shown, these all pass out through BC. Since BO= 2AD and two dimensions only are being considered, therefore 2N molecules enter the vessel through BOC in this same period of time, but of these, whatever points they enter through, and whatever be their directions, more than half are returned through BC and less than half pass out through AD. Thus, while N molecules pass from AD to BO, less than half 2N, or less than N, pass from BC to AD: that is a downward flow has been created. This result (which is dependent upon the mean free path being so great in proportion to the size of the vessel that intermolecular collisions inside and in the immediate neighbourhood of the vessel are occurring with only a relatively small number of molecules) is due, as has already been explained, to a deliberate restriction or re-direction of movement applied selectively to molecules having certain velocity directions only (directions other than those included in angle BZC), and the effects of such restrictions have been shown to be theoreticaily independent of the second law of thermo- dynamics. This flow will produce a pressure potential on surfaces exposed to it, and external work can consequently be obtained at the expense of the kinetic energy of the gas enclosed in the system, precisely as in the case of the purely hypothetical experiment described by Maxwell. The same effect may also be proved with equal simplicity for vessels with-walls inclined to each other at smaller angles. For example, if BZC is an angle of 30°, from any point O on the base BC draw OX and OY, each at 45° to BC, cutting AB at X and DC at Y. OX and OY then meet AB and DC each at 60°, and are consequently reflected to strike the opposite side at 90°, whence it will be seen that all molecules entering at O,and travelling in directions included in angles XOB and YOC are returned through BC. This is true even in the extreme case where O almost coincides with B or C, since Y’'BC and X'CB are themselves then each 45°. Angles YOC and XOB are together always equal to a right angle, of Molecules at very low pressures. 1053 wherever O may be situated, so that at least half of the total number of molecules entering through BC at all points, and in all directions equally, are returned through BC. More- over, many molecules travelling at greater angles than 45” to BO are returned also (for example along OM, where angle Fig. 4. MOB=60? and angle YOC=45°), and therefore the resultant effect is as has already been proved for a vessel with an angle of 90°. It can be shown further by similar reasoning that these results are true for vessels in which the height is greater than in those which have been dealt with, that is in which BC is greater than twice the length of AD. In a consideration of three dimensional molecular move- ment, it will readily be seen that truncated pyramids and truncated cones must necessarily behave in the same manner, the additional velocity component producing its corre- sponding downward movement. Only ideally smooth walls have so far been considered. Since polished surfaces are known to reflect even light, in the main, according to the usual-laws of reflexion, although they are always very imperfectly smooth as seen under the microscope, it would consequently appear not unreasonable to suppose that this may be the general effect also where gaseous molecules are concerned, especially if the so-called molecular impacts be merely the bottoms of parabolic curves caused by the repulsion of the wail, but, in view of adsorption phenomena and other differences, such similarity of regular reflexion, even as an average effect, perhaps ought not to be premised, and it becomes desirable therefore to consider the eftects which would be caused if the angles of reflexion were different from those of approach. 1054 Mr. A. Fairbourne on Restricted Movements Actually the result must le intermediate between the two extremes, (1) perfect reflexion, and (2) absolute inde- pendence of the angles of approach and departure. The first of these extremes has been considered. In the second it is desirable to consider a vessel whose height is greater than in those previously drawn, and whose angle, BZC, is small (fig. 5). Fig. 5. B Cc Let the vessel be imagined filled with a gas whose mean free path is relatively so large that intermolecular collisions during the short journeys in the vessel are occurring with only a negligibly small fraction of the total number of molecules which pass through in any representative period of time; and, at the first instant of consideration, let the molecules in the vessel be moving in all directions equally. If the walls were ideally smooth and the usual laws of reflexion were in operation, then any impact of a molecule, whatever its direction, with either wall (on the inside of the vessel) would necessarily increase the downward motion of that molecule, since all impulses caused by the smooth walls must be normal to their surfaces. Further, in this case, since the distance from wall to wall is less in the neighbour- hood of AD than in that of BC, and since molecules pass from side 1o side without intermolecular interruption, there- fore laterally moving molecules near the top of the vessel would receive more impacts from the walls in unit time than of Molecules at very low pressures. 1055 would those near the bottom *, thus suffering a greater total downward impulse than those in the lower layers; and as there is no balancing effect in their particular neighbour- hood a downward flow must be produced, this conclusion being identical with that already obtained. The cause for such an effect has been explained in terms of an extra restriction selectively applied to molecules with particular movements, and would appear to be in strict accordance with the statements of Maxwell to which reference has been made. If, however, perfect reflexion were not in operation, and absolute independence of angles of approach and departure existed, then any point on the wall would discharge molecules haphazardly on either side of the normal to an equal extent, this being true, whatever theory of ad- sorption be adopted (compare Langmuir, J. Amer. Chem. Soe. 1916, xxxviil. p. 2221). Since the normal to either wall (inside the vessel) points downwards, and since this must be the mean direction of all discharge, and also of all approach so long as the molecules are moving equally in all directions, therefore hits upon the walls must still produce, in the main, downward impulses on the molecules concerned ; and, for the reason given, these impulses are inflicted to a greater extent in the higher and narrower portions of the vessel than in the lower layers, whence the ultimate effect must be the same, qualitatively, as with perfect reflexion, although quantitatively it may not be so great. On the other hand it may be greater, for, if the theory of Langmuir (loc. cit.) on the repulsion of adsorbed particles be accepted, then these will probably be discharged from the sides mainly in directions substantially normal to the surface and will consequently leave the vessel through the base, thus giving an exalted effect above that to be expected in the case of ideally smooth walls and perfect reflexion. A vane mounted with sufficiently minute friction inside or below one of the vessels described, while the vessel itself be suspended in a gas at a suitably low pressure, should there- fore be capable in all cases of realizing experimentally the * Compare note p. 1049, which shows that although the pressure may be the same inside and outside the walls ofa vessel, yet more impacts per moiecule can occur inside than outside the vessel in the same period of time. So, in a vessel with converging walls, more impacts per molecule may occur in‘one portion of the vessel than in another. Assuming the concentration to be the same throughout the vessel, and the molecules to be moving equally in all directions, then the more constricted any portion of the vessel may be, the more frequently will molecules in that portion collide with the walls. 1056 Mr. E. J. Hartung on the Construction and theoretical conceptions put forward by Maxwell, and should consequently yield mechanical work at the expense of the kinetic energy of the molecules ; that is of the temperature of the gas as a whole, a result which could not be obtained in any experiment coming within the application of the second law of thermodynamics. General experience would indicate, how ever, that caution is desirable, in accepting this conclusion. Kine’s College, University of London, Strand, W.C. 2. CXILI. Observations on the Construction and Use of the Steele-Grant Microbalance. By KH. J. Harrune*. VY @NUE Steele-Grant microbalance has recently been applied bythe writer to the study of the action of light onthe halides of silver. During the course of the work some observations have been made on the construction and manipulation of the balance and on possibly unsuspected sources of error. It is hoped that these are of sufficient interest to be given here, as they have not always been empha- sised, even in such an excellent article on microbalances as that of F. Emich (Handb. d. biochem. Arbeitsmethoden, 1x. Pao» LOU: The microbalance was described by its inventors in 1909 (Proe. Roy. Sec. A. Ixxxii. p. 580, 1909). It) ies ame entirely from vitreous silica ‘and consists of a rigid beam oscillating on a single central knife-edge in the more sensitive forms, or on a pair of edges for greater stability in the less sensitive forms. To one end of the beam a counterpoise of constant weight is fused, to the other a suspended system is attached by a fine silica thread. This suspended system consists of a light silica grid carrying the object to be weighed, as well as certain ‘silica weights and a bulb containing a known quantity of air sealed Mer ae Equilibrium is ‘attained, at first roughly by selection of the proper weights, and then accurately by adjusting the pressure in the balance case. The constant load is then made up of the object to be weighed, the weights and the apparent _ weight of the air-bulb, which varies with the density of the medium in which it is immersed. For the investigation mentioned above, a balance of high relative sensitiveness was required. Pettersson (Proc. Phys. Soc. Lond. xxxii. p. 209, 1920) has modified the Steele- Goat * Communicated by the Author. | Use of the Steele-Grant Microbalance. 1057 instrument by introducing a bifilar suspension in place of the knife-edge, and in this way he has succeeded in detecting differences in weight of only 0°25 1 (1u#=107° mg.) in a load of 250 mg. This gives the almost incredible sensitiveness of 1 part in 10°, though presumably the actual working sensitiveness would not be so great. The Pettersson type of balance was therefore eminently suitable, but it was not used because the balance cases available were designed for the original knife-edge type of Steele and Grant. The most sensitive balance of the inventors was able to detect changes in weight of about 4u, but the load is not stated and apparently it was not great. Their type-B balance with double knife-edges would measure differences of 100m in a load of 100 mg., giving thus a relative sensitiveness of 1 part in 10°. This appeared to be sufficient for the proposed work, and an instrument of this type was therefore constructed. Its constants were as follows :— Period 33 seconds. Maximum load 43 mg. Minimum indication 204. Accurate weighings could be made to within 40m, giving a relative working sensitiveness of 1 part mi 10°. Experiments were also performed with some other types of balance, differing in structural details, and the general conclusions reached through a comparative study are given in the following paragraphs. The Beam. The correct design .of the beam is of the first importance, as it must combine rigidity with lightness and therefore with economy of material. The best form is undoubtedly the rhombus with vertical diagonal (fig. 1, A), as adopted finally by Pettersson, but it is not suited for knife-edges. The original design of Steele and Grant is exceedingly good, and it is not easy to see how to improve it. Gray and Ramsay (Proc. Roy. Soc. A. Ixxxvi. p. 270, 1912) evidently considered that it was not sufficiently strong, and in their classical work on the atomic weight of radium they employed a new type of beam (fig. 1, B). The radium balance which they con- structed was a very trustworthy instrument, and the results they obtained Jeave nothing to be desired; the maximum load was 24 mg., which could be weighed to the nearest 14 p, thus giving a relative sensitiveness of about 1 part in 2x 10°. However, from a theoretical point of view, it is not by any means clear what advantages the radium balance beam possesses over the simple double-triangle type of Steele and Phil. Mag. 8. 6. Vol. 43. No. 258. June 1922 3Y 1058 Mr. E. J. Hartung on the Construction and Grant (fig. 1, C). It is inherently heavier for the same strength because the material is less economically distributed, and the absence of the vertical rod connecting the upper and lower portions probably increases the tendency to fail laterally under load. Further, the placing of the knife-edges at the end of a comparatively long pillar attached by only one Big, ab. complex joint to the rest of the structure is a device of problematical value. In order to test the limitations of the double-triangle beam, a balance of this type was constructed with a single knife-edge about 0°4 mm. long. The members from which the beam was made were carefully selected, and special pains were taken to fuse them together strictly in one plane. The instrument was adjusted so that its period of oscillation was about 90 seconds, and it was then capable of detecting a difference of ly in a load of 40 mg. This Use of the Steele-Grant Microbalance. 1059 corresponds to a relative sensitiveness of 1 part in 4x10° ; in use, however, the resting-point was found to vary slightly from day to day, so that trustworthy weighings eould not be obtained beyond the nearest 4. This gave a relative working sensitiveness of 1 part in 10’. One may therefore venture to state that the simple Steele-Grant type of beam leaves little to be desired as to trustworthiness and constancy of behaviour, and there is no reason to discard it in favour of other types. Hor comparatively heavy loads, the thick- ness of the members may be proportionately increased and the requisite sensitiveness attained by suitably raising the centre of gravity. the Knife-edges. Steele and Grant, in their original memoir, described the process by which the knife- edges had been arias by them. They were ground in sets of face by means of a small rect- angular metal holder into which silica rods had been cemented. By resting the ends of the rods and the long edge of the holder on a ground glass plate supplied with a little fine abrasive, alternate grinding on both sides resulted in chisel edges. The sie eneee on the holder was then increased by securing steel slips on either side, and the operation was finished on a very finely-ground oluss plate with a little glycerine and water but withoutabrasive. This method requires only comparatively simple apparatus and is capable of giving very satisfactory results ; it has been employed in the present work. ‘The oreatest difficulty is found in the finishing touches. The glass plate employed must be heavy and very steady, truly flat, tad with so finely eround a surface that it is almost transparent when dry. It fee been found advantageous to use a very light forward stroke with abundance aR lubricant for the ell polishing and the edges must slide over the glass without the slightest harshness or hesitation. The progress of the work is watched with a microscope, the edges being carefully dried before examination ; they may be taken to be good enough when the irreg ularities are barely visible with a “good d-mm. objective. Pettersson criticizes rather adversely the knife-edge type of balance on the ground that dust collecting on the quartz bed-plate will enone with the free movement of the edge and that it will gradually become blunted through use. These disadvantages do not attend the bifilar suspension which he adopts, and he makes out a strong case for the oe 2 =? 1060 Mr. E. J. Hartung on the Construction and superiority of his method, in spite of the greater difficulties in construction and adjustment. However, the thecretical and practical simplicity of the knife-edge as a point of sup- port is a strong recommendation in its favour, and, moreover, the results of the present work show that Pettersson’s criticisms are largely unfounded for balances of minimum indication not less thanlw. No trouble has been experienced from dust and, far from deteriorating, the edges seem actually to improve with careful use ; this is no doubt due to the gradual removal of the infinitesimal raw edge left after grinding. It is necessary to arrange the arrestments of the balance so that the beam is merely arrested in its movement when not in use and the knife-edges are not lifted away from the plate. If the beam is lifted when the instrument is not in use, the resting-point will not remain steady, particularly in the more sensitive types. Steele and Grant mention this in their paper, but donot explain the reason ; probably it is due to irregular distortion of the knife-edge. Calculation shows that in the sensitive balance described above, the centre of gravity of the system is not more than ‘001 mm. below the edge, while, in spite of the lightness of the beam and its accessories, the pressure on the edge is intense owing to its very small beari ing surface. A slight distortion may there- fore have serious influence on the equilibrium position of the beam, and it is probable that the wandering of zero after lifting the beam is due to slow alteration in this distortion, which only becomes permanent and constant when the knife- edge rests continuously on the plate. The Fibre Suspension. Steele and Grant found that a fine silica fibre suspension from one end of the beam was more satisfactory and much more easily managed than the knife-edge and plane suspension used on ordinary analytical balances. They drew the fibre directly from the beam by hand. An improved process is due to Mr. G. A. Ampt, of Melbourne University. He fuses a weighted hook to the beam by a small, very thin rod, and then passes a minute oxy-gas flame rep eatedly and very rapidly across this rod. - The result is a fine fibre which gradually extends and can be drawn to any desired thinness at any point. Gray and Ramsay (Proc. Roy. Soc. A. lxxxiv. p- 536, 1911) fused a long fibre direct to the beam, and obtained in this way a very free suspension for their niton balance. The desideratum of a good fibre is that it bends Use of the Steele-Grant Microbalance. 1061 very sharply at one point so that it approximates to the ideal point and plane suspension. The bending, however sharp, must necessarily take place through a small are, with the result that as the end of the beam rises it lengthens slightly, and as it falls it shortens slightly. This departure from the ideal suspension may show itself in three ways:— (i.) The sensitiveness of the balance will depend to some extent on the positions at which weights, ete., are hung on the rack carried by the fibre. (i.) The actual resting-point of the balance will depend to some extent on these positions. (ii.) The values obtained by calibrating the weights against the «ir-bulb will depend to some extent on the relative positions of weights and bulb on the rack, All these points could be observed with the double knife- edge balance before described, which was tested for the purpose. The suspended system here consisted of a short fibre about 2 cm. in length which carried the air-bulb of volume 332°8 cubic mm. Hanging from the lower end of this was a fine rod about 10 cm. long, fashioned below into a rack and terminated by a small hook. The whole system was about 14 cm. long and is shown in fig. 2. Table I. shows the alteration in resting-point and in sensitiveness consequent on hanging a standard weight of 48 mg. in positions A, B, or C Tasue I. Position of Sensitiveness in Equilibrium case Change in weight. divisions per 100 p. pressure. weight. Bt ahs 50 542°1 mm. 7 100 p jo aoe 3°8 3) ie s 50 p Cts 3°5 541°8 99 It will be observed that the sensitiveness increases with the height of the load. The resting-point has also altered, for a vertical change in position of the load of 2 em., by 150 in a load of 43 mg., i.e. by 1 part in 3x10°. Though small, the alteration is appreciable, but it becomes negligible if the load is always hung at the same level. Table II. shows the alteration in apparent value of three selected weights as calibrated against the air-bulb when two different fibres were employed. As these fibres differed 1062 Mr. EH. J. Hartung on the Construction and somewhat from one another in length, the relative positions of air-bulb and weights in the suspended system were not the same in the two cases. Avrora OM Bh Value with Value with Percentage Weight. fibre 1. fibre 2. Bo eet Q:2590 mg. 0°2596 meg. 0-23 Se 0:5910 A OO 25) tae 0:22 eet ae COS Os: eG BSD 0:24 Fig. 2. Koes B Cc The difference between the two sets of values is consider- able, and neither would agree with the true values obtained when bulb and weights are hung at the same level. This last procedure is therefore to be recommended, although it is not really necessary unless absolute weighings are required. Use of the Steele-Grant Microbalance. 1063 The slight variation in the percentage alterations as given in the fourth column is doubtless due to an unfortunate accident during the two sets of weiglings. The stand carrying the silica weights was overthrown, and the weights had to be cleaned from dust and fluff which adhered to them. Such cleaning operations, even if very carefully carried out, are always liable to leave slight residual changes in weight. It is very likely that the variations shown in the tables could be materially diminished by the employment of thinner and longer fibres. The long and free fibre suspension used by Gray and Ramsay is probably much superior in these respects, but no tests have been made with it. Attention is merely directed to possible sources of error in the use of the balance. It should also be borne in mind that the risk of breaking the fibre during the ordinary manipulations con- nected with weighing is considerably increased with very thin fibres, and the delay consequent on the insertion of a new fibre and the readjustment of the whole baiance may be very annoying. In this connexion it may be worth mentioning that a very neat device, due again to Mr. G. A. Ampt, renders such manipulations comparatively safe. All weights, counterpoises, and objects to be weighed are pro- vided with two hooks, one above the other, as shown in fig. 2 They then hang on the balance or on their stands from the upper hook, and are carried from place to place on a slender silica rod fon) thelower hook. When it is desired to hang an object on the balance, the upper hook is allowed to engage with one of the arms of the rack, and the siJica rod is slipped out of the lower hook without straining the fibre in the least. The operations are reversed when an object is removed from the balance. The adoption of this simple device will save much worry and trouble. General Arrangement of Apparatus. It is almost needless to remark that the precautions against dust, mechanical shocks, and temperature changes, empha- sised by various workers, must be carefully followed if accurate results are required. In the present work the balances were mounted on rubber pads in a cellar with con- erete floor which was almost free from vibration and maintained a fairly steady temperature. The cases were surrounded by metal boxes as advised by Gray and Ramsay, and the indicating light was only allowed to impinge on the mirror momentarily and after passage through at least 25 cm. of distilled water. The manometer was connected to the 1064. Mr) Co By Bieleerdulelon tan cases by means of long, narrow glass capillary tubes bent in the form of long loops ; this served the double purpose of protecting the instruments from shocks and from air-currents which would be engendered by rapid alterations of pressure in the case. The air which is allowed to enter must be dried by passage over suitable reagents, and subsequently filtered through tightly-packed cotton-wool. Itis not advisable to introduce drying agents into the case itself. Gray and Ramsay noted that when air was allowed to enter rapidly, slight shifts in resting-point were observed, and they attri- bute this to the lodgment of dust particles on the balance ; the source of this dust was doubtless the solid barium oxide which they put into the case for drying purposes. Phosphoric anhydride is particularly liable to canse such trouble. My thanks are due to Mr. J. A. Smith for much useful information on the design of balance beams. Chemical Laboratories, University of Melbourne , October 1921. CXIV. The Interaction between Radiation and Electrons. By C. ¥. BicKERDIKE *. re a “Report on Radiation and the Quantum Theory” _ to the Physical Society (1914) Jeans examines the question of the equilibrium distribution of energy between electrons and the ether. Ina paper in the ‘ Philosophical Magazine’ June 1914 he points out that the substantial difficulty of reconciling the facts with dynamical theory arises not in connexion with the mere propagation of radiation in the ether conceived asa continuous medium, but solely in the interchange of energy between radiation and matter. He applies the equations appropriate to the motion - of a charged particle acted upon by radiation, and deduces that the only final state of equilibrum would be one which is not to be reconciled with the facts observed experimentally. His inference is that “the departure from the classical mechanics is to be looked for in the fundamental equations of ether and electricity.” The writer ventures to suggest that possibly the classical conception of Faraday lines may afford a clue to the direction in which to seek for the solution of the difficulty. * Communicated by the Author. Interaction between Radiation and Electrons. 106) The equations used by Jeans were originally derived with reference to ihe motion of a charged } particle, tle mass being that of the particle, the electrical inertia of the charge being negligible. In applying the equations to the electron, the material mass of the particle is dispensed with and the electrical inertia of the charge substituted. If, instead of dealing only with mathematical equations, we picture the charge as consisting of radiating Faraday lines which extend indefinitely through the ier and which presumably are states of the cether itself,—this picture at once suggests the question whether the eee acts directly on the lines, the centre or body moving as a result of motion first imparted to the lines, or whether, on the other hand, the motive force is applied directly on the centre or body of the electron and the lines are dragged after it. In the theory of the effect of an acceleration applied to a charged body as causing emission of radiation, it is always conceived that the force is applied to the body and the lines of the charge dragged after it. When there is no material “ body,” it may still be the case that that is the mode of application of an acceleration to a negative electron when the force is the attraction of a positive, which is conceived as a tension along the length of some of the lines. It is rather to be expected. however, that when one negative electron acts on another, by radiation, the case will be substantially different, and the action may he directly on the lines of the receiving electron. The centre may then move in conformity with the motion already imparted to the lines, and it may be quite inadmissible to infer that acceleration of the centre so produced would result in the emission of radiation in the manner in which emission would result if the aceeleration of the centre has been produced by a force acting directly on that centre. Ti may be objected that the radiation, on first approaching the electron, must reach the centre at least before it has reached the lines which extend in the direction away from that in which the emitting electron lies. In reply it may be pcinted out that the first arrival of the front of a wave of radiation is a very brief moment. Actually, in all experimental work, we have to deal with a long train of radiation, and ie question what exactly happens just at the eanierd of the arrival of the first wave is relatively unimportant. The receiving electron is very soon surrounded on all sides equally by ether affected by the radiation, and it is suggested that, for a free electron, the receipt a radiation is the converse of the emission. kd the 1066 Mr. C. F. Bickerdike on the case of emission we conceive the centre as moving first, dragging on the ends of the lines and sending flexures out- wards. Conversely, when radiation is absorbed, flexures travel inwards from the outer parts of the Faraday lines, and the centre moves in accordance ; but that motion does not itself produce an outward flexure so long as it is unopposed. At all events, the kind and degree of such action must be quite different from what it would be if the centre were itself first acted upon independently of any pre-existing motion of the lines. When, however, the electron comes up against the inertia of a positive the situation is necessarily different. There may be either reflexion or transmission. Prima facie one would associate transmission with a state of things in which there is no scope for free motion at all, and complete reflexion with a state of things in which there are free electrons which have already absorbed radiation up to a critical point. The general nature of the argument may be illustrated by a comparison with a vessel floating in water—notwithstanding the fact that such analogies are liable to be very misleading. A wave generated in the water moves the vessel up and down—1. ¢. the vessel receives accelerations. We should be very much in error, however, if we were to infer that these accelerations in turn produce waves in the water in the same degree and manner as if they were produced by some force acting directly on the vessel and plunging it up and down in water which was otherwise at rest. It is a different situation again, however, if the vessel is tied by a string to a weight at the bottom of the water so that it is not free to move in conformity with the wave ; but whether that offers any useful analogy with the case of the bound electron is difficult to say. We have to recollect that the vessel is something existing independently of the water, and substantially not affecting the state of the water, whereas the electron really consists of lines which are presumably states of the sether and not something existing independently. We are not entitled to liken the lines to rods stretching out from the vessel into the water, but not forming part of the water. The analogy of a vessel in water has, however, the limited use of calling attention to the importance of distin- gaishing between the conditions under which an acceleration is produced as bearing on the effects to be inferred. If the suggestion is tenable that the perfectly free electron will receive and absorb radiation up to some critical point without emission at all so long as its motion is unhampered, Interaction between Radiation and Electrons. 1067 it seems clear that in general the shorter the wave-length of the radiation, the oreater the possibility of its being so absorbed, because the length of free path necessary to fulfil the ep endition would be correspondingly small and would the more frequently occur. ‘There would therefore be selective absorption of small wave-length vibrations, and the tendency for the energy all to be absorbed into Sole and smaller vibrations vould be prevented. The electron which constitutes part of the system of an atom also may be able to absorb similarly, since, w vithin limits, it may be free to move in the direction per pendicular to the plane of its orbit. In that way it alxe may be a selective absorber of the smaller wave-lengths of radiation. The writer is disposed to go further, however, in criticism of the argument that energy must, on ‘Newtonian principles, fritter away into the minutest chaotic motions. It is submitted that the possibilities of thermo-dynamic analogy have not been exhausted, and that even if we forget the Faraday lines and treat the electron in the ether as analogous to a heavy particle in a kind of gas composed of far more minute particles than any ordinary material gas, the argument that the heavy particle must lose all its energy does not necessarily hold good when we proceed from the finite case to the infinite, or when we at least entirely alter the order of magnitude of our gas molecules. Jeans* takes the formula for distribution of ener gy between N moiecules of air at temperature T and n heavy spheres. “The total energy in the steady state will be ({3N+4n)R', and T will be determined from the peat eee that this quantity must be equal to the total energy of the original oscillations of the spheres. Since N, the number of mole- cules, will be enormously large compared with n, the number of spheres, it is clear that pr actically all the energy will be contributed by the term 3NRT. By the time the steady state is reached the energy is almost entirely transferred from the spheres to the gas.” The principle is the same if the energy is dealt with in terms of sound-waves. The question is, however, whether, if we are going to make N enormously greater then it would be in the case of any actual gas, there are not some considerations to be taken into account which can be neglected as unimportant so long as we are dealing with familiar gases. Starting from the conception of a familiar gas, such as air or hydrogen, having N molecules in a given space, we must * Report, p. o. 1068 Mr. C. &. Bickerdike on the first have a conception of how the change is being made when we imagine N to be increased ad infinitum. If we simply add to the number of air-molecules, with the average kinetic energy per molecule unaltered (except so far as may be the result of abstracting energy from the heavy particle) we are packing our finite space with an infinity of mass and of energy. No one can conceive what this would mean—whether the gaseous molecules have any room to move at all, in fact. Let us, first of all, therefore, conceive the air-molecules as diminished in mass just in proportion to the increase in their numbers. We can imagine each molecule—supposed originally as uniformly of mass m—to be divided into z parts, and wcan be made aslargeas welike. The original mass and density of the gas are then unaltered, and its total energy is unaltered excepting for transference from the heavy particles *. The accepted laws of distribution of energy are arrived at on the supposition that the actual time of any collision is so small in comparison with the time of free motion of each molecule that it can be neglected, and the heavy particle is conceived as colliding successively with individual molecules of the gas. | If the mean length of free path is / originally and the radius of the molecules is 7, / is assumed to be so much larger than 7 that r is negligiblet. When we suppose each molecule to divide into w parts, { and r are diminished indefinitely, but remain of the same order of magnitude relatively to one another. | On this supposition it is possible that the time required * It is clear that on this hypothesis the formula of Jeans (Report, p. 6) for total energy in terms of energy of waves would not have the meaning which he attributes to it. Am is the limiting wave-length, which may be made as small as we like provided it is not comparable with the average distance between molecules. The total energy 1s 4R1 BU ee Ma 2 clea 3 me Jeans says this gives infinite energy when )z is infinitely small. This can only be on the assumption that RT is not similarly reduced in magnitude and must imply that M is increased ad infinitum by mere addition of molecules with the same mass and energy as the original ones, 2. e. packing the finite space with infinite mass and energy. Of course, if that could be conceived, the energy of the heavy particles would be small in comparison. If we multiply N, however, by the method proposed above, the total energy is unchanged and RT must be reduced as much as A This consideration alone, however, does not suffice to dispose of the argument that the energy of the heavy particles would all go into the gas. + Vide Edgeworth, “On the Application of Probabilities to the Movement of Gas Molecules,” Phil. Mag. Sept. 1920, pp. 249, 260 and footnote. Interaction between Radiation and Electrons. 1069 for the interchange of energy at collisions would diminish in the saine ratio, and, if so, the principles applicable to the partition of energy would possibly be unaltered, provided that the velocities attributed to the heavy particles were not much smaller than the average velocity of the gas particles. There is, however, an impor tant new condition introduced. The pr obability laws governing the distribution of energy in gases are arrived at by conceiving encounters to take place between heavy and light particles successively. When we have multiplied the light particles ad infinitum, however, by the process of division above suggested, we have now to conceive that instead of the mass M having successive encounters, altering its velocity as stated, for ‘instance, by Edgeworth, there would be semultaneous encounters on all sides. In a direction at right angles to the motion of M there would be no momentum given to M. When the velocity of M is comparable with the velocities of the ligit particles, however, there would be more particles in co:tact with M in front than in the rear, and M would lose momentum. 3 In working towards something supposed to be analogous to the zther, however, we may postulate very high velocities for the ultimate particles—velocities, say, of the order of magnitude of the velocity of light. In that case, a heavy particle moving in such a medium, with velocity of a lower order of magnitude, would have practically as many light particles, at any moment, in contact in front as in contact behind. The impacts saul ne transmitted through M, which would have its velocity unaltered. M, then, once in uniform motion, would move through such a gas as through a perfect fluid, without loss of energy—or only extremely slow loss. -It may be objected that electrons also have very high velocities, and ee they are the “ heavy particles ”’ dealt with by Jeans. The introduction of high velocities involves, however, a further consideration which can be neglected in dealing with ordinary gases. The length of the free path is made indefinitely small, although, if we keep to the idea of a limitation of the amount of mass (7. e. conceiving the masses to be diminished in proportion to the increase of numbers), there is still the possibility of motion of the individual particles. The time occupied | by free motion may be so reduced, however, that it is no longer small compared with the time occupied by an encounter, and may be less than such time. In that case, the heavy particle, being impinged upon by a 1070 Mr. D. Coster on the Spectra of X-rays light particle, the two would have parted company before other light particles had impinged upon the rear of the first one. The heavy particle would, in fact, be charged not successively by single light particles, but by battalions of them in column. Equilibrium would be reached when, on an average, the columns were of mass equal to that of the heavy particle. The equilibrium could occur even though the velocities of the heavy particles were of the same order of magnitude as that of the lighter particles. Under those conditions the law of final partition of energy would be entirely different from that which would result merely by making N large in the formula for gases. Whether this would give a credible picture of the eether which would conform with other facts, it is not the intention here to discuss. It would seem to imply a medium which would not be quite like ether, a gas, or a liquid, ora rigid solid, but having some of the properties of each. The only point insisted on here is that such a thing is conceivable, and, although the Newtonian laws are assumed, there would not be that frittering down of all energy inte the smallest movements. One exception alone is all that is necessary to disprove the generality of the argument that the Newtonian laws must always lead to this conclusion about dissipation of energy. Conceptions of the esther on these lines may have been thought of before. Novelty is not claimed, but the writer is not aware that the bearing of such aconception on the theory of partition of energy has been pointed out. CXV. On the Spectra of X-rays and the Theory of Atomic Structure. By D. Coster: [Plate XXIII] Part I. Introduction. §1. CCORDING to Bohr’s theory of spectra the fre- quencies of the lines in the X-ray spectrum of an element may be represented by the difference of two terms which correspond to the energies of the atom before and after the emission. Following Barkla’s original notation for the different kinds of characteristic X-radiation, the various * Communicated by Prof. Sir E. Rutherford, F.R.S. and the Theory of Atomic Structure. 1071 groups of energy-levels whose existence may be inferred from the X-ray spectrum are usually denoted as K, L, M,N.. levels. As pointed out by Kossel, these levels may be assumed to be connected with the different groups of electrons in the atom, in such a way that the various energy terms correspond to the work required for the removal of an electron from one of these groups. ‘The various groups of electrons of the atom are in consequence often termed the K-, L-, M-, N- shells. This interpretation of the levels affords an explana- tion of the laws governing the absorption in the X-ray region: it is well known that this absorption is not connected with the single lines in the spectrum but extends over. spectral regions which are sharply limited by the so-called absorption edges, the frequencies of which correspond to the energies of the levels concerned. Corresponding to the ditferent ways in which the removal of an electron from a shell may take place we obtain several levels for each shell. As has been pointed out by Sommerfeld, part at any rate of this complexity in the groups of levels may be connected with the complexity of the ensemble of the stationary states of the hydrogen atom. According to Sommerfeld’s theory of the fine structure of the hydrogen lines, the stationary states of an atom containing one electron are characterized by two quantum numbers. One of these numbers, which we shall term the total quantum number n, is the same as that oceurring in Bohr’s interpretation of the simple formula for the hydrogen spectrum. The other number is the so- called “‘azimuthal”” quantum number, which determines the value of the angular momentum of the electron round the se and which we shall denote by &. In his recent publications* Bohr has developed a ae of atomic structure which contains certain essentially new features, and which seeins to give a natural interpreta- tion of the periodic system and at the same time to offer an explanation of the results of Kossel and Sommerfeld on the X-ray spectrum. According to Bohr’s theory, the orbits of the electrons in the different groups of the atom are charac- terized by different total quantum numbers, this number being equal to 1 for the innermost group (K-shell), 2 for the next group (L-shell), and so on, every time increasing by one unit until the surface of the atom is reached. Within each group the electrons are again divided into sub-groups, corre- sponding to different types of orbits and characterized by different values of k. A survey of the gradual development * ‘Nature,’ March 1921 and October 1921. See also for a fuller account, Lettschrift f. Physik, 1x. p. 1 (1922). 1072 Mr. D. Coster on the Spectra of X-rays of these groups and sub-groups with increasing atomic number is illustrated by the following table, which indicates the proposed constitution of the atoms of the inert gases. Number of electrons in 2; orbits. Element. l 1,|)2,| 20})8,|3 |35||4,]45}4,| 44] 5,| 5a| 5s! 54} 5s||6,| 62164] 64, 6516, abade —||—|— |||} |--|- |---|] ||| |— Bea id HS Helium 2 ...[2 | | Neon 10 .../2|/ 4/4] Argon 18 ...|2|/4/4|/4/4/— | Krypton 36 ...)2|/4/4|6]6/6]/4)/4/—|-] . | Xenon 54 .../2||4/4/6/6/6|/6/6|6/—|4/4/—i-|—l] |. Niton 86 ...|2|/4/4| 6}6/6/ 8/8]8/8] 6 6|6|—|-||4/4|—|— —|- pees tals) 3. In previous papers* I have been able to show that nearly all the lines of the X-ray spectra of the heavier elements can be arranged in a simple scheme, involving the existence of one K-level, three L-levels, five M-levels, and seven N-levels+, and that in continuation of the work of Sommerfeld it is possible to characterize every level in a definite way by means of two quantum numbers n and k as defined above.. Further, the appearance of the observed X-ray lines was found to be governed by two simple “rules of selection.”” According to the first rule only those com- binations between two levels. will appear, for which the quantum number & remains unaltered or changes by one unit. The second rule tf states that the levels may be divided * Zeitschrift f. Physik, v. p. 139 (1921), denoted in the followmeg by L, and vi. p. 185 (1921), (denoted by Il). Compare also: A. Smekal, Zeitschr. f. Phys. v. p. 91 (1921), and v. p. 121 (1921); A. Dauvillier, C. R. elxxii. p. 1850 (1921), C. &. clxxin. p. 35 (1921), C. R. elxxiii. p. 647 (1921); G. Wentzel, Zertschr. f. Phys. vi. p. 84 (1921); A. Som- merfeld and G. Wentzel, Zectschr. f. Phys. vii. p. 86 (1921). | This also suggests that we must expect the existence of one K-, three L-, five M-, and seven N-absorption edges. These absorption edges have actually been found in the K- and L-series, by several authors for different elements, and recently I have been able to establish the existence of five absorption edges in the M-series for U and Th. (See Phys. Rey. II. xix. p. 20, 1922.) { This rule of selection has been stated independently by Sommerfeld and Wentzel in another way. They introduce a third quantum-number, “ Grundquantenzahl.” This number is equal to or one unit larger than the azimuthal quantum number. ‘The transitions are subject to the condition that this ‘“‘Grundquantenzahl” must change by one unit. As we do not yet know the physical meaning of this ‘“ Grundquantenzahl,” it seemed to me better to state this rule of selection in the same way as it was first suggested to me by Mr. H. A. Kramers. and the Theory of Atomic Structure. 1073 into two types, a-levels and 0b-levels, in such a way that every observed line appears as a combination of one a- and one b-level, while combinations of two a- or two b-levels do not appear *. It will be seen, that in consequence of the latter rule we never observe lines, the frequencies of which are the exact sum or difference of the frequencies of two Diagram I,—Niron. Se er aera eT) a ee yg“ b 5 4, Raia eae oe | Soe ee Pe eT ae | Li iS ae ae eee |_| Tes X eae PEROT B Ma a ytd RT EE Sn SE a Os WO be 6 Oe SPOOEE 7 FE Ss aa a pis M FS | faut | é 41 | cine Peer | | I | M co c/} yee a oy ltt ee AM a BT men eee SG Riera a | | L4A kb fp €2 % ahh fap bop Fe hop other lines. An illustration of this classification of the X-ray spectra is given in diagram I. for niton, which is based * It should be mentioned that some very weak lines have been ob- served which represent a transition d~b. They are the transition L,—-K observed only for W by Duane and Stenstrém and the lines Lf, (transition M,-L,) and L6,, (transition M,-L,) observed by several authors for various elements. (See Zevtschr. f. Phys. ii. p. 200, table 9.) Phil. Mag. 8.5. Vol. 43. No. 258. June 1922. 32 1074 Mr. D. Coster on the Spectra of X-rays on measurements for elements of atomic number preceding or following that of niton (86) *. Comparing this diagram with the above table representing the results of Bohr’s theory, we see that it has been possible to characterize the levels by the same quantum-numbers as those which in the niton atom characterize the different groups and subgroups. We may now assume that the existence of the levels indicated in diagram I. is connected with the appearance of the various subgroups of electrons in the atom. Further, we may assume that in those cases where more than one level is characterized by the same quantum numbers, we witness different processes of removal of an electron from the same subgroup, the remaining electrons in the group arranging themselves afterwards in different ways. §4. These conclusions obtain strong support from an inspection of the way in which the energy differences of the levels corresponding to the various values of n and k vary with the atomic number N of the element. The considera- tions in question rest upon a comparison with the theory of the stationary states of an atom consisting of a single electron revolving round a positive nucleus. In the first approximation, where the motion of the electron is calculated according to Newtonian mechanics, the energy necessary to remove the electron from one of these states to an infinite distance from the nucleus depends only on the total quantum number n. This energy is given by Bohr’s formula, oe hein W=N-], Mares CI) where N is the number of unit charges on the nucleus, / Planck’s constant, ¢ the velocity of light, and R a universal constant appearing in the theery of spectra and called the Rydberg constant. The theoretical value of the latter constant is given by the relation 0724 = aoe = 109737, (number of wave-lengths per cm.), where e is the charge of the electron and m its mass for velocities small compared with the velocity of light. * As to the nomenclature of the lines, which often differs very much with different authors, I have in this paper in general used that proposed by Prof. Siegbahn. Still, for a theoretical discussion it may be advantageous to use sometimes another nomenclature analogous to that used for the visible region. According to this latter one, e. g., the lines Ke, and Ly. may be called KL, and L,N, respectively. and the Theory of Atomic Structure. 1075 Taking into account the variation of mass with velocity required by the theory of relativity, the energy necessary for the removal of the electron from one of the stationary states is, to a first approximation, given by Sommerfeld’s formula, eke hes > N helwc 3 Wao (| i) (2) m4 x where & is the azimuthal quantum number and «a numerical constant small compared with unity, the theoretical value of which is given by Dear p22 = a=) = 530.107. he Formula (1) gives an interpretation of the general laws which had been revealed by Moseley’s Pandan enol dis- coveries. According to these laws the energies of the observed levels are closely proportional to the square of the atomic number, the energies of the K-, L-, and M-levels being approximately represented by the formule Wxr=Rhe(N —ax)’, eee R= (3) where ax, ap, and ay are constants which are different for the different levels. The appearance of these constants in formula (3) is simply explained by taking into consideration the interaction of the electrons in the atom; the main effect of these is virtually to reduce the attractive influence of the nucleus on the electron whose removal corresponds to the level under consideration. The constant a is therefore often termed the “screening-constant’’ belonging to the level ; und the quantity (N—a) may be called the effective nucleus charge. Fron formula (2) we obtain an explanation of the circum- stance that certain pairs of energy-levels corresponding to the same value of 2 vary to a close approximation as the fourth power of the atomic namber. Following the notatioa of Sommerfeld, such pairs of levels may be termed “ relativity doublets,” since the energy-difference between the two orbits is due to the differential effect of the relativity modification on orbits having the same value of n but different values of k. In the diagrams these pairs of levels are denoted by {. As stated above, for levels having the same values of nand k we should expect that differences in the energy necessary 3242 1076 Mr. D. Coster on the Spectra of X-rays to remove an electron from the atom would arise from dif- ferent orientations of the orbits of the remaining electrons within the group concerned. In such levels it may be said that the remaining electrons can have different. screening effects. ‘This offers a simple explanation of a fact revealed by an inspection of the measurements, that the energy- differences between two such levels to a close approximation vary linearly with the atomic number. Such a pair of levels may therefore be termed “screening doublets ” *. In this connexion, however, it must be pointed out, that the numerical values of the screening constants for the different levels calculated from the relativity doublets do not agree with those calculated from the screening doublets f. This is just what we might expect from Bohr’s theory of atomic structure. According to this theory the electrons of the outer shells come during their revolution round the nucleus wholly inside the orbits of the inner shells, so that they are moving in a varying field of force. Now it is easily seen, that the effect of the relativity change of mass on the orbit of the electron is mainly due to that part of the orbit which lies close to the nucleus, and where the velocity is very great. We thus understand that the screening constant for these electrons appearing in the relativity term of formula (2) has another value than that appearing in the main term which in first approximation gives the whole energy of the orbit. § 5. In general, corresponding to a given pair of values for n and k there exist two levels, of which one is of the type denoted in the above diagram as an a-level, the other a b-level. For the largest value of & corresponding to any given value of n there appears, however, only one level. I am indebted to Prof. Bohr for the remark that this circumstance may be brought in suggestive connexion with his theory of atomic structure, which rests upon a considera- tion of the way in which an atom may be formed by the successive binding of the electrons by the nucleus. In fact, in such a process the subgroups corresponding to the highest value of & will correspond to the electrons bound during the last stage of formation of the group, and, in contrast to the removal of an electron from subgroups corresponding to * Sommerfeld divides the doublets into “regular” and “ irregular” doublets. As the screening doublets show no irregularity at all, these names seem not to be well chosen. t+ Compare Sommerfeld and Wentzel, Zeitschrift fiir Physik, vil. p. 86 (1921). and the Theory of Atomic Structure. 1077 smaller values of 4, the removal of an electron from this sub- group may therefore be expected to represent a simple and well-defined reversal of a step in the process of formation of the group. In this connexion it must be pointed out that the proposed explanation of the origin of these levels requires that the screening constants for a pair of levels corresponding to a relativity doublet should have approximately the same value. No simple explanation of this, however, is offered in the present state of the theory. §6. Though there still remain some difficulties, we may say that the X-ray spectra are built up in a simple manner and that there exist many analogies between these spectra and the series spectra in the visible region. ‘There are also, however, some striking differences. In the visible spectrum, transitions in which the azimuthal quantum number remains the same do not occur under ordinary conditions. In the X-ray spectrum, however, there are several lines for which ‘k remains constant. A few of them are fairly intense Imes (e. g., in the case of the heavier elements L@, and L®; and also Ly, and Ly; are of about the same intensity). Recently I have found that there is another difference between the X-ray spectrum and the visible spectrum. In the latter there exist also transitions for which the total quantum number n does not change at all. To these belong for instance the first line of the principal series of the alkali metals. From this we might expect that the transition L,-L; would give rise to a line in the X-ray spectrum which might easily be detected. An investigation with a tungsten anticathode showed, however, that this line does not exist at all, or at any rate must be very weak. Hxperimental particulars are given in Part II. . These various differences between X-ray spectra and optical spectra need not be surprising in view of the funda- mental differences which exist (in spite of analogies) between the origins of the two types of spectra. This difference is due to the fact that in the emission of the optical spectrum we have to do with the change of the motion of an electron whose orbit is characterized by higher quantum numbers than the orbits of the other electrons in the atom. In the emission of the X-ray spectrum, however, we meet with a change in the motion of an electron which must be expected to be in intimate interaction with the electrons of the same shell moving in orbits with the same quantum numnbers *. * Compare Bohr, Zeitschr. f. Physik, ix. p. 1 (1922). 1078 Mr. D. Coster on the Spectra of X-rays Part II. The New Measurements of the L-series in the X-ray Spectra of the Hlements from Rb to Ba”. § 1. As mentioned above, the results indicated by diagram I. have a direct bearing only on elements of atomic number comparable with that of niton. We should expect a change in this diagram for elements with lower atomic number, since the formation of new shells in the outer region of the atom with increasing atomic number must be accompanied by the appearance of new levels in the energy diagram which find their expression in the appearance of new lines. My previous work was based on an investigation of the L-series of most of the elements from W to Uf, which I carried out in the laboratory of Prof. Siegbahn. In view of a comparison with the theory of atomic structure, it was desirable to extend this investigation to elements of lower atomic number. In continuation of the former work I have therefore undertaken in the same laboratory an. examination of the L-series of such elements. Though this work is not yet finished, the results already obtained seem to be sufficiently interesting to justify publication. In the present paper only the results obtained for the elements Rb-Ba will be discussed. §2. The apparatus used for the experiments consisted of an X-ray vacuum-spectrograph, and the metal X-ray tube of the Coolidge type described by Prof. Siegbahnt. The tube was driven by two similar induction-coils of medium size. ‘The primaries of the coils were connected in series to a source of alternating current of 50 cycles. As the tube itself acts as a current rectifier, no other rectifier was used in the secondary circuit. For most of the work the secondaries of the coils were connected in series. In this way a current of about 30 m.a. with a maximum tension of about 30 k.v. could easily be obtained. This tension is at the same time about the highest which can be sustained by this tube. The tension was estimated by an adjustable parallel spark- gap. If the vacuum is not very good, the discharges through the tube give rise to great fluctuations in the tension. Under these circumstances no simple relation exists between the maximum tension measured by the spark-gap and the mean value. For a very high vacuum, which could be obtained easily with the molecular pump, the maximum tension was * Part of these experimental results have recently been published in Srchives Néerlandaises (Serie III. A, tome vi, 1* livraison, p. 76.) | Zevtschr. f. Phystkh, iv. p. 178 (1921), and Zettschr. f. Phys. I. and II. ft Phil. Mag. xxxvii. p. 601 (1919). and the Theory of Atomie Structure. 1079 about 1-4 times the mean tension. Up to 10 k.v. a Braun electrometer was used. At times it was necessary to work with a rather low tension. In this case the secondaries were connected in parallel, so that a larger current could be obtained. In photographing the absorption discontinuities of silver a maximum tension of not more than 5400 v. was used, and with this tension the influence of the ‘‘space charge” in limiting the magnitude of the maximum current could readily be observed. As is known, this effect has been experimentaliy and theoretically studied by Langmuir *, who clearly showed that in an extremely high vacuum the charge due to the electrons moving in the field between the hot wire cathode and the anode diminishes the rate at which electrons may escape from the hot wire and enter the field. For every tension there exists a maximum “ saturation current” which depends on the dimensions of the tube and is independent of the temperature of the hot wire, if once a certain temp2rature has been surpassed. In this experiment the saturation current was less than 10 m.a. fora tension of about 5400 volt. As with this small current a very long exposure is required, it was desirable to get rid of the space-charge effect. This may be done by working with a somewhat lower vacuum, as in this case the positive ions of the gas neutralize the influence of the electrons in the field. Fortunately I was able to reduce the vacuum just sufficiently by an imperfectly sealed join in the tube. Otherwise it would have been necessary to lower the vacuum by regulating the speed of the molecular pump. Under these circumstances it was possible to get a current of about 50 m.a. at a tension of 5400 v. As fairly long wave-lengths were measured, it was necessary to have also a vacuum in the spectrograph. The spectrograph was exhausted by a Gaede pump which gave a pressure of about 0-1 to 0°2 mm. The slit of the X-ray tube, which was immediately connected with the spectrograph, was covered with goldbeater’s skin in order to separate the high vacuum from the low vacuum. This goldbeater’s skin was coloured with erythrosine to prevent the visible light as much as possible from entering the spectrograph. The tube itself was exhausted by a molecular pump in series with the low vacuum pump. The pumps were able to give a very good vacuum in about 10 minutes. The time of exposure for one plate varied from 4 to 13 hour. §3. To get also the fainter lines of the spectrum, it appeared to be advantageous to use tensions several times higher than the critical exciting tension. Thus in several cases the maximum tension obtainable with the apparatus * Phys. Rev. II. ii. p. 450 (1918). 1080 Mr. D. Coster on the Spectra of X-rays was used. On the other hand, the high tension gave some trouble in identifying the lines, as several lines due to other elements appeared in higher orders on the plates (e.g., on’ many plates taken with a gypsum crystal, Cu K-lines and W L-lines were obtained in as high as the 5th order). In general, the best plates were obtained when the element could be placed in the form of a metallic sheet on the anti- cathode. For this reason Sb, Sn, In, and Cd were melted in a fairly pure state on the copper anticathode, while the elements Ag, Pd,and Rh were attached with solder ; Mo was pressed in a copper ring which was soldered on to the anti- cathode. There was no particular difficulty in obtaining good plates with these elements. The other elements examined on this occasion (i. e., Ba, Cs, Te, Ru, Nb, Zr, Y, Sr, and Rb) were used in the form of salts or oxides, which were pressed into the roughened surface of a copper plate soldered on to the anticathode. For every element this copper plate was renewed. As the salts are sputtered from the anticathode somewhat quickly, it appeared to be better not to use the highest energy which could be obtained from the apparatus. In general, for these elements the best plates were obtained with a maximum tension of about 20-25 k.v. and a current of not more than 15 m.a. For each element from 6 to 10 different plates were taken through the whole region of the spectrum. Between two exposures the tube was opened and a fresh quantity of salt was brought on the aiticathode. Very often it was necessary to take several plates of the same part of the spectrum before a good one was obtained. §4. The distances of the lines on the plates were meas- ured under the microscope, and from this the wave-lengths of the new lines could be measured relatively to the lines a, (1, and yy, which have previously been determined with great pre- cision by Hjalmar*. The other lines mentioned by Hjalmar (a, @3, 84, 83, Bo, and in a few cases yo, y3, ys) Were usually determined by this author either relatively to a, 81, 71, or from the old measurements of Friman, a correction having been applied. In the following tables the lines determined by Hjalmar by the precision method as well as those measured relatively to them on his own plates have been taken from this author without change. These comprise nearly all the lines whose wave-lengths appear in his tables to one or two decimal places. The other lines have been determined in the present investigation. * Zewtschr. f. Physth, iii. p. 262 (1920), and vii. p. 841 (1921). G-1L06 1081 L-GOSG 9-89% S-TLLZ 1-616 Structure. tk G-661Z 1-6916 | 0-266 O-FEIG | 6-589G 6-LE86 9-€166 9-TE1E L-6666 6-08F§ 0-L195 6-L88¢ €-19&P G-GEGG | 6-P9GG 669P GL h6P P-669¢ 6:8609 and the Theory of Atom eh oh 09-9866 G2-GPEG LV-90LG L0-GP8E 66-662 66:99TE 00-8665 ¢8-P1S8 96-9TLE L-GE6E G8:GLIP LIT Lh L-F60¢ 0-ELE¢ ee th 66666 ¥-90G6 | T-9L86 99108 6-LOTS G- 1886 F-90968 68-669 10068 L-261P 6: 1987 G-606P §-9669 P-ELGG G-LLVS G- L896 P-F966 8- LOTS 'G:G9CE 0-867E 6-L096 b-66L6 0-L00F L-0&8GP P-OLbP L-9FES ¢-§699 0:€099 ¢-1969 6) 0-119 | 86-696 &-1008 FI-GhIS 6:862E 6-LOFE GP-9G9E Ch-FZ8E L-GGOF &: LPOP F-OLPP Z-000¢ 6-46GE S-819G 8+ 1964 6-6FS9 6-69L9 tg) 8-6796 ¢-0996 0-0P08 6-F8I& 6-9666 0-66F& GG-PLOE 1: 198& 6-G90P 8:-LLGP 9-G1SP 8-809 P-1E&G L-Gg9¢ 6-1009 ¢-G8§9 38-6089 ro) VG:G9GG b8-L196 16-6906 96-816E C6-LLEE 68: Lhe 80-0626 ¥9:966§ 08-LELF 00-F9EF 00: LL9F 8-G9TG 9-6Lb9 8 6689 P8619 G-6099 F-0904 9] 9.6922 t O1-988 66-1828 LL-IShE SI-264E 19-916 G8-LF68 9-CFIF 0G-898F 8L-L89P 19-GE8F -F68E G.I TLS 6-909 6-FSF9 8.LP89 L-GOGL —<— | ——— ———— ” “(HO OT) “AX UF SyzSUET-oAe MA ‘T @Iavy, GO-6LL16 09-4686 00-1668 GL-OFPE 80-1096 GP-GLLE 96-9968 c8-ES1PF 09:996P 9¢-C6eP LO-Eh8h OOFG LIL eeeecee eeeeee ereeee T-L686 | L8c1€ e.96e | soe | Lose 9-6696 | &-O88€ 8-18LE | &-S90F 1-916§ | &-69¢F G-L8IP | &-LLbP L-OLPF | 9-L69P G-OG9PF | 9-686F GII6P |0-L069 cego | C619 | 609 T-€6¢9 | 8689 GOGL | 1268 0:6608 | ~"" u 7 “emer 9G He BQ Og 97 2G ag Te ag 9g UT BF DO OF Dede Nf LP Da OF UT OF eeenaT Ey, OW oP GaN LP 9 a7 OF eos or ge HH Ag Qe oe eae Mr. D. 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Coster on the Spectra of X-rays Hjalmar and Friman found fy, 83, and ®, as far down as Pd, y, appears for the last time for Mo in their tables, and the 7-1 and y;-8, doublets were not observed at all in this region. But I could still detect 6, and ; for Sr and Rb, @, and y; were found as far down as Zr, although very faint (Hjalmar’s value of X=5295:1 for Nb, corre- sponds to my value of 7X=5295-9 X.U. for 3 for the same element). The lines ys, v3, and y, in this region had not been identified correctly by the former authors. The line Y2, 3 (2 and y,; could no longer be separated in this region) has still been observed for Rb and yy, still for In. Both doublets 7-1 and y;-8, could also be measured down to Rb and Sr. . The spectra of the elements Ba and Cs were taken with a rocksalt crystal, Te—Pd with calcite, Rh partly with calcite, partly with gypsum, and Ru-Rb with a gypsum crystal. For these elements the breadth of the lines was considerably greater than the width of the slit, implying that the lines represent a finite frequency interval. Strongly exposed lines obtained with a rocksalt or calcite crystal, especially, were diffuse at the edges, which obviously reduced the accuracy of the measurements. Extremely broad and diffuse were the lines, 7, J, and yo,3. For 8, for some elements a broadening could be observed at the short wave-length side. The lines a, and @; were accompanied by satellites (see _ Part II. $7), while for the lightest elements a, could not be separated from a, All this reduced the accuracy of the measurements. Usually only reference measurements were made. The distance on the plate to the reference line was often fairly large, sometimes over 10 mm., and in a few cases even between 15 and 20 mm. Thus errors in the measurements of the line-distances and irregularities in the structure of the crystal may be quite appreciable here. In the ease of the lines taken with a rocksalt or calcite crystal, however, the error in the wave-lengths must be in nearly all cases considerably smaller than one X.U. This is especially true of the lines lying in the neighbour- hood of a reference line. In a few cases (sometimes for y, 1, andrye,3) this error may be a little larger. Some lines could not be measured under a microscope and their distance was estimated with a millimetre scale. They stand in the tables without decimals in the wave-lengths. Their errors may be more than 2 X.U. Using gypsum as analysing crystal, we have on the one hand the disadvantage of the greater space-lattice constant, on the other hand the ad- vantage of getting sharper lines, which usually le nearer and the Theory of Atomic Structure. 1085 together on the photographic plate. For the lines measured with a gypsum crystal we have to multiply the above given limits of error by 13-2. Thus far I have neglected the errors in the wave-length which are due to the remarkable fact detected by Stenstrom7 that Brageg’s relation nX= 2dsing is only approximately true. specially with a gypsum crystal the deviations are fairly large and may give rise to an error of several X-units, but as this error is nearly the same for wave-lengths of about the same value it does not affect the conclusions drawn in this paper. Tables I., I1., III. contain only the lines which can be arranged in diagram I. §5. The right classification of the lines cften gives trouble. Some indications were supplied by the relativity L-doublets, which are given in Table LV. Down to Ag the y,-@, doublet TasueE IV. Relativity L-doublets. | | nl. | B-a. | y5-Be- 11-B.: | | | L,-L,. | Pains... 4:28" | 4-48 | | Boise cts... 492 | 480* | 5-0U | Bey es: 540% |... | BG) 615 | 603* 621 | 613 | AND ns .|. 710° | 6:89 ae 6:99 | 49 Mose... Ete ae, 7-70 ish 7-80 Ba | sa 949 | 953 | 9-47 : a5RH 10:54 | 10:48 | 10-41 |-10-46 Pass... 11-48 | 11:56 | 11-66 | 11°57 ce ae 12-64 | 1269 | 1277 | 1246 | 12°68 Ae Gd. o... 1381 | 1397 | 1398 | 13-92 ci Ton oe 15-23 | 15-29 | 15-26 | 15-24 Giga ee. 16-69 | 1671 | 1677 | 1660 Bib e.. 1831 | 1829 | 1826 | 18-21 52 Te <....... ... - | 19:94 | 2008 | 19°85 Bots oe. | 2559 | 25°76 | 25-42 | (25-8) BeBe! a 9769 | 27-74 | 27-95 | 27-61 | 28-00 | * As a, could not be measured for this element, the value B,-x, has been used. ‘This value is smaller than 6,-,. difference is apparently smaller than the other ones f, lower down this ‘doublet defect” disappears in the accidental errors. The cause of this defect will be seen from diagram J. The pairs of lines 7-1, Qy-a, and y;-@8, come each from the same initial level. whereas y; and 8, come from different initial levels. I have never observed the transition N,-L,, + Stenstrém, Dissertation, Lund, 1919; Hjalmar, Zeztschi. f. Phys. i. p. 489 (192u) ; Siegbahn, Comptes Rendus, clxxiii. p. 13850 (1922). { Compare also Zeitschr. f. Phys. I. table 3, p. 191. 1086 Mr. D. Coster on the Spectra of X-rays which might be supposed to exist by analogy with the transition M,-L,, which gives rise to the line ap. Pairs of lines the frequencies of which show a relativity doublet difference occur very often in the X-ray spectrum, but those which would form a screening doublet do not occur at all in virtue of the a-b rule (see diagram [.). But we may find lines the frequency difference of which is either the sum or the difference of two screening doublets. To the former class belong B.-n and y~¥;, to the latter 8;-8, and y;-y,. As the screening doublets are approximately proportional to the effective nucleus charge of the atom, the differences of the TABLE V. e e Vv Differences in Re | Bs-n ya-7s | Bs-By | aan | 37 Rb ails 0-921 | 0666 | 0-242 | 88 Sr sees: | 927 676 237 BON Fora): i Mis 232 RADA ee eee) 941 686 226 | 0:550 | AGN oe ieee | 946 a 223 | 547 | 42 Mo......... | 955 704 218 547 44 Ru... kos 712 210 581 45 Rh 973 715 209. |. 527s AG Pde 979 718 204 522 RAT Ae ae 989 724 203 520 AB Od os ccces. 996 731 200 512 Gsm a | 999° | “740, | “ies 512 On 1004 745 | 196 510 (ol Sb ae) 1-) sLOOS 760 194 511 52 "Meee eee 756 | 196 510 RBCS eee 1027 Tal) |. 402 505 BOMBA oe... | 1045 770 | 192 502 VMN io. | 1136 e22mnl | 209 517 Sombie... 1 1200 O10 0) 924 495 | | | square roots of the frequencies of these lines are easily seen to be nearly constant for the different elements. These differences are shown in Table V. This table in connexion with Table 1V. most strongly supports the identification of the lines which has been proposed by the author. The dif- ferences 8,-7 and y.-y; increase slowly with increasing atomic number. This is due to the fact, that in every line-frequency the screening constant of the initial as well as that of the final level is involved, and these screening constants in general will be different. For the heaviest elements the relativity correction here also plays a part. | The values which I have given for y; may be verified and the Theory of Atomic Structure. 1087 by some measurements in the K-series. As is seen from diagram I. the following relation must hold for the fre- quencies : K6,—KB,= Lys — LB. Table VI. gives some numerical values. TABLE VI. Ly,-L6,. | KB.-KB,. decane 4041 | 39:0 i 20 EVER 30°44 32°4 BSH. J susee 32°97 34:7 Me beer saan 26°70 26°5 { The values for the K-lines of Rh and Mo are calculated from the wave-lengths given by Duane * (K@, is called Ky by Duane). The values for Cd and Pd have been kindly furnished me by Mr. A. B. Leide, who has measured some K-lines in this laboratory. Asan error of 0°5 X.U. in one of the K-lines involves an error of more than 4 per cent. in the differences under consideration, we may say that the agreement is very good. ; The line 8, crosses 8, and 83 twice, a circumstance which in the beginning gave rise to some difficulties in the identi- fication of these lines. , crosses 6, once for Pt and ; for Dy and crosses both lines again between Rh and Mo. In this region the @;-curve is nearly a straight line in the Moseley-diagram, whereas the §3- and §,-lines are noticeably curved. § 6. As regards the intensities of the different lines, it has been found that for the elements U-W the lines @, and 8, are not much different in intensity, but in the region Ba-Rb 83 is much more intense than 8,._ An illustration of this fact is given by figs. land 2 (Pl. XXIII.) +. Fig. 1 gives * Phys. Review, LI. xiv. p. 373 (1919). + The reproductions are about twice the natural size. The black lines in the original photographs are represented as white lines in the reproductions. — 1088 Mr. D. Coster on the Spectra of X-rays the lines of the @-group for Pb taken with a rocksalt crystal ; fig. 2 gives nearly the whole L-spectrum of Rh* taken with a gypsum crystal. For the lines y, and y, the case is much similar: for the heaviest elements they are approxi- mately of equal intensity ; for Ta y3; seems to be prominent ; for Ba and Cs ysis, still fairly strong, whereas y, can scarcely be perceived. ‘This fact may imply that the disturbances of the inner atomic field which give rise to 8, and y, (the azimuthal quantum number does not change for these lines) are less important for the lighter elements. But the change of relative intensity of 83 and 8, must partly be accounted for as a change in the intensity of @,. This line seems to correspond to a transition less probable for the heavier elements than for the lighter ones. fea In the case of y-l and y;-8, doublets a corresponding change of the intensities could not be observed. For all elements / is about 2-3 times as strong as 7 and £, 2-3 times as strong as y,- The lines » and y; like §, and y, correspond to transitions for which the azimuthal quantum number does not change; the lines / and @, correspond to transitions for which the azimuthal quantum number decreases, whilst the lines 83; and 3 correspond to transitions for which it increases. | § 7. Another interesting result of this investigation of the L-spectra of the lighter elements was the discovery that the two most intense lines of this spectrum, a, and 8), show a complicated structure, which has not been observed for the heavier elements, or is at any rate much less pronounced. Both lines in question show a broadening on the short wave- length side. This cannot be ascribed to any peculiarity in — the experimental conditions, asit has a very regular structure which is independent of the time of exposure. On the other hand, the line @; which on several long exposed plates was very intense showed no such structure, appearing merely somewhat diffuse on both sides. For the «,-line this broaden- ing ends fairly abruptly, so that the edge could be measured with sufficient accuracy. From this the wave-length corre- sponding to this edge could be calculated. (A correction must be made for the half width of the slit.) In the following tables this wave-length has been called «;'. This must be * The satellites of «,and B, (see Part III. § 7) have not been well reproduced on this plate. They were more easily visible on the original photograph. On the reproduction Sn La stands erroneously for Fe Ka 2nd order. and the Theory of Atomic Structure. 1089 understood to mean that there is an emission-band extending from a, to «,'.. When one passes from Cd to Ag this band suddenly changes. [or Ag two new lines could be detected : one, which is within this emission-band, is called a3, the other, a short distance outside, is called a,*. The intensity of the line «; changes very little from Ag, where it is found for the first time, down to Rb. The line a, seems to become somewhat stronger for the lighter elements Tf. The satellites of 6, takea different course. The proximity of 8, makes it impossible to study these satellites for the elements Ba and Cs, but from Sb (where they were observed for the first time) down to Rb they do not essentially change. Firstly there is an emission band, the short wave-length-edge of which has been denoted by @,'. This band is fainter than the «,-band and not so sharply limited. Furthermore, out- side the band a line has been found, denoted by Aj, which is faint for the elements Sb—Rh, but becomes more intense for the elements Mo—Rb. It is a well known fact that the human eye does not always form a correct judgment of the blackening of a photographic plate. In particular a sudden change in the gradient of the blackening gives the impression of a white or a black line. It therefore seemed to be advisable to study the ~,- and §,-sateilites with the photometer also t. It must be borne in mind, however, that even the photometer does not give a wholly correct impression of the blackening of the plate. With this photometer part of the photographic plate is projected by a microscope-objective on the slit of the thermopile. In these measurements 0°03 mm. on the photographic plate corresponds to the breadth of the slit. Further, it is impossible to have a sharp image for the * (Note added during the proof.) —Recently, however, by taking some other powerfully exposed plates, I was able to establish the existence of the lines a, and a, for Cd also, but for this element they are very faint. For In the existence of these lines remains very uncertain, for Sn no trace of them could be observed. In this connexion it might be of interest to remark that the appearance of these satellites is within wide limits independent of the tension used on the tube. t I have previously observed the broadening of «, for the elements Ta-U, but have interpreted it erroneously. See Zs. ft lys: Ah, p. Woke t This photometer has been described by M. Siegbahn, Ann.der Physik, (4) xlii. p. 689 (1913), and A. E. Lindh, Zs. f. Phys. vi. p. 303 (1921), See also W. J. H. Moll, Proceedings Phys. Soc. London, vol. xxxiil. part 4, p. 207 (1921). Phil. Mag. 8. 6. Vol. 43. No. 258. June 1922, 4A 1090 Mr. D. Coster on the Spectra of X-rays visible and the infra-red light at the same time. Hencea blackening, which may be represented thus : | is regis- tered by the photometer thus : / \ et Small irregularities of the plate which are not readily noticed by the eye are registered by the photometer. Most of the small abrupt changes of the curves must be explained in this way. Asa rule the more continuous changes corre. spond to changes in the blackening of the plate. This may be verified by comparing photometer-curves taken across the plate at different heights. Fig. 1 gives a typical curve for Rh (taken in the opposite Fig. 1. ay Rhodium La, with satellites (calcite crystal). direction as compared with the other curves). Fig. 2 shows the great difference between the Rh jf «, and the In a, line. Fig. 3 clearly demonstrates that this difference suddenly appears between Cd and Ag. (The numerous accidental * Another question is, whether the blackening of the plate really gives a correct impression of the emission-spectrum. It might be supposed, that for these strongly exposed lines the secondary radiation in the photographic plate has some effect in broadening the lines, but as the breadth of the lines largely depends on the space-lattice constant of the analysing crystal (with gypsum rather sharp lines were obtained) we may conclude that this influence cannot be very great. + This curve is taken at a height different from that in fig. 1. In figs. 2, 3, and 4 Tn stands for Jn. and the Theory of Atomic Structure. 1091 irregularities in the latter curves must be ascribed to some mechanical disturbance.) Fig. 4 gives the 8, curvesfor Ag Fig. 2. Rh La, and In La, with their satellites (calcite crystal). Fig. 8. wa an ght Pes rhe par g et ; frtmey 7 : ft Ard aa dies a, yr caf oe eed ies a pany a, Aen f The Lay, lines of Pd, Ag, Cd, and In, with their satellites (calcite crystal), and In; 8,/ and 8,; are much less pronounced than the « satellites. No sudden change is here observed for Ag. AA? 1092 Mr. D. Coster on the Spectra of X-rays Fig. 5 shows a, and 8, with their satellites for Zr, taken at two different heights of the plate. On the plates of the elements Mo—Rb taken with a gypsum crystal, o; could not Fig. 4. 3 Suh i | wit eve, i Ag L8, and In Lf, with their satellites (calcite crystal). Fig. 5. a ane — RE NT NA cn Zr La, and L@, with their satellites (gypsum crystal). be separated from a,'. This gives a somewhat asymmetrical form to this line. @, and 8,', which could be detected easily with the eye, could not be separated from a, and ®, with the photometer. Fig. 3 (Pl. XXIII.) gives a reproduction of the a,- and §,-lines of Rb. The wave-lengths and frequencies of the «- and (,- satellites are given in Tables VII., VIII.,and IX. The wave- lengths of a, for Mo—Rb are probably too great, since this line Taste VII. B43 | By! ee Se — =- — Vv Vv Vv Voi ie R RE Bey Wy abe | R R R orb... 7022-4 | 129°77 0-032 | 7041 | 129-42 0-016 Be Sr ool. 6573°4| 138°63 32 || 6590 | 138-28 17 40 Zr ...... 5793°3 | 157-30 32 | 5807 | 156-98 17 41 Nb......| 5449°7) 167-21 36 5465 | 166-76 18 42 Mo......| 5138°4| 177-34 35 5150 | 176:95 20 a5 Bh:.:.. | 4342-5 | 209°85 ay 4351 | 209-45 23 46 Pd......| 4117-1} 221-33 36 | 4125 | 220-91 22 47 Ag ...... | 3906-9 | 233-24 39 3914 | 232-84 26 45Ca -2.... 3711°6 | 245-52 39 3719 | 245-04 24 26% 2k 3580-4 | 258-12 40 3539 | 257-48 20 50 Sn ...... 3360°7 | 271-15 | 43 | 3367 | 270-64 27 Se end 3202°6 | 284:54 | 4i 3210 | 283-91 23 TApiren VAULT: Aye as. ‘e | en Yar DN ch Yay — a : R ve R Bis aN \/ R 35 Bb...:.: 72488 12571 0-042 7271-0 | 125-33 0025 SSSr ....-- 6797°3 134-06 43 68183 | 133-64 | 24 patwen.. 5. 6013-2 151°54 43 6029:1 | 151714 27 | 41 Nb...... 56711 160-68 45 5684-9 | 160-30 30 42 Mo...... 5356°4 170-13 | 45 5370°3 | 169°68 | 28 | 45 Bh...... 45588 199-89 A5 4571°3| 199°34 26 '46Pd...... 4330°9| 210°41 46 4349°7 | 209-84 | 27 agp Ae... | 4119°4| 221-21 AT | 4131-0] 220-59 | 26 TABLE JX, a, | »\ a Bar one | | R: The | 45 RBh.........| 4564-9 19962 0-03 | 146Pd.........| 48362 | 210-15 38 Bye Ua. sper 41254 | 220-90 36 | [Ag Cay 2350 39283 | 231-95 37 | AG rac. 3744-7 | 243-35 40 | 50 Sn .........| 35740 | 254-97 4) es) aes 34138 | 266-94 43 Pe Os tay. .se| 2870°8 317-43 47 | Bey Bate los ero k 330°76 48 and the Theorm of Atonie Structure. 1093 1094 Mr. D. Coster on the Spectra of X-rays could not be separated from «,'. The last columns give the difference between the square root of the frequency of a or 8, and that of a satellite. These differences are nearly constant but decrease slowly with decreasing atomic number. § 8. Two very remarkable lines are the new lines @,, and Biz. They may be seen on fig. 2 (Pl. XXIM.). For Ba, Cs, Te, these lines probably do not occur ; they could be detected for the first time for Sb. From In down to Mo they are fairly strong lines which could be easily measured under the microscope. They could be detected also for Nb and Zr, although for these elements they nearly coincide with y;, which is a much sharper linein this region. For Sr and Rb they are very faint or perhaps they do not occur at all. Apparently these lines have nothing to do with the lines §;, Bs, 89, and Bi which I have previously measured for the heavier elements *. On the short wave-length side of y; a new line (v7) could be detected. This line is weaker than 6,, and 8, and could be observed as early as Ba. It might be supposed that this line forms a relativity-L-doublet with 8), or Bj,. The differ- ences, however, between the frequency-differences y;-6i TABLE X. | P y) Wave-lengths. | Frequencies. Differences ve ; a | ; ip a By, cre ia | Bu Bi. a a Ba-Bo | Bi onBo lia Ad -Nib ies 5161 ee 7G HEM, 6 bee | 0-081 4) Mio ae: 4859°7) 4841-71 2.2... VS DDI ASS Qe 0-071 | 0:095 Ay la 4084'8| 4072°5| 5896°8 || 223 09| 223°76| 233°85 68 91 | 0-076 AG Peder. 3867°6 | 8856°7 | 3676 | 235°61| 236°28| 247-87 65 86 A Bee. 3663'3 | 3653°7 | 3479°5 || 248-76 | 249:40| 261-89 66 87 48 Cd..... 3477°5 | 3468°4| 3302 262:05 | 262°73 | 275-98 67 88 AON iui: 3304-0 | 3295°9| 3125 || 275°81 | 27648} 291-57 68 89 5O Sa, 31426 | 313847 | 2968°5 |; 289-97 | 290°69 | 306:98 69 99 INS ON ASesHE 20934) 2985:8) i eons. | 804:43 | 805:20| ...... 67 89 1, Gyr Bets HADRON st ae Sa ie DOME dl Ae a asta ALO S22. cu i enenees oY Y7-Big and the B,-a, doublet are in most cases greater than the limits of experimental error. As is seen from the last columns of Table X.. 8;, and @, forma screening-doublet, also B,,.and 8. The same is true for y;andy;. Some other * (Note added during the proof.)—From a paper recently published by Mr. Wentzel (Annalen der Physik, \xvi. p. 487 (1921)), however, we are inclined to suppose that the lines 6,, and @,, might be connected with the line 8, of the heavier elements. and the Theory of Atomic Structure. 1095 new lines, which possibly may exist for some elements (one between 8, and 8, and one between a, and §,) will be studied in connexion with an investigation of the elements Ta—Ba and Rb—Cu. § 9. Tables XI. and XII. give the characteristic absorption discontinuities in the L-region for Ba, Cs and Ag, and the TABLE XI. Wave-lengths. TL, | B. L, N L, Ys Y4 2356-7 2198 56 Ba ...... 23993 2236°60 | (2063) | 2129°5| 2071-5 (2348) (2194) 2466 a Rae fk 55 Ox 1... 25064 | (2299) | 2342°52 | (2157) | 2227-1| 2169-4 (2459) ay Ae... 3684°4 sous 3504-7 |3514:85 | $260°5 | 3299-7 { TaBLeE XII. 5 V Frequencies —. 4 R L, Be L, | Di L, | i eae | 38667 | 414-67 | 56 Ba ......| 379:80 407-42 | (441°7) | 427-92) 439-91 | (888-1) (415°3) | 369-50 | 55 Cs 1.0... 363-58 | (396'4) | 389-00 | (422°5) | 409°18/ 420-11 | (8706) | | a7 As ...... 247-33 | 24669 | 260-01 | 259:15 | 279-48 276-17) shortest wave-lengths of the emission spectrum which belong to each of them. The absorption wave-lengths of Ba and Cs were accidentally found on the same plates on which the emission spectra were taken. ‘They were obviously due to the selective absorption of tne heterogeneous radiation of the copper anticathode, in the Ba and Cs salt used on the anti- cathode. Hertz’s values * are added in parenthesis. They differ from ours by about 7 X.U.in the mean. The absorp- tion spectrum of Ag is determined with a gypsum crystal, * Zetschr. fiir Physik, iii. p. 19 (1920). 1096 Mr. D. Coster on the Spectra of X-rays making use of the increased absorption in the silver of the photographic plate. For this region of wave-lengths, especially if the rather faint discontinuity L; is to be photo- graphed, it is desirable to exclude totally the spectra of higher order. For this reason the maximum tension on the tube should be not more than twice the critical exciting tension. Therefore in taking the L, absorption the mean tension on the tuhe, as read with the Braun electrometer, was fixed at about 5400 volts. Asa fairly large current was used (50 m.a.) a very good plate was obtained in four hours. The discon- tinuities L, and I. were both found on one plate after an exposure of about three hours. § 10. As has been stated in Part I. § 6, I have tried to obtain some experimental information about the existence of the line L;Ly, 7. e., the transition L,-L;. For tungsten this line should lie in the M-region, between the lines M8 and My which have been measured by Stenstr6m. We may calculate the wave-length of this line for W from the following data. The frequency of the line L,h, is equal to the frequency difference of the absorption discontinuities L,* and L;*. These have been measured by Duane and Patterson. They found 1213°6 and 1024 X.U. respectively. From this we find for the frequency difference L;—L, 139:07 in multiples of the Rydberg number, from which we may calculate the required wave-length as 6553 X.U. Taking into account the limits of experimental error given by Duane and Patterson, we find ‘that the error -in the wave- length in question must be less than 3 per cent. We may calculate the same wave-length with somewhat greater accuracy in the following way. From diagram I. we see that the following relation must hold between the frequencies : L8, + (lL; — L;) = LB; + My. Putting into this formula the frequencies measured by Siegbahn for L@, and L@; and the value for My measured by Stenstrém, we find for the same frequency difference Lz;—L, 139:05, giving the same wave-length 6553 X.U. Taking into account the limits of error of the My line and of the L-lines, which can be measured with greater accuracy than the discontinuities, we find that the error in this value is less than 0°3 per cent. We should expect that this line would only arise if an electron is removed from the L-shell in sucha way that the * By these symbols the levels as well as the absorption-discontinuities corresponding to these levels are denoted. and the Theory of Atomic Structure. 1097 remaining electrons form a configuration corresponding to a L; -level. Using a tungsten anticathode, the tension on the tobe must according to Hinstein’s relation be more than 12000 volt. As the expected line might not be very strong, it is desirable to work with a tension which is at least twice as great. The tube was therefore driven with the maximum tension which could be obtained. With this tension a very strong L-spectrum in the first order was obtained in ten minutes. As analysing crystal a gypsum crystal was used ; the time of exposure was one hour. Besides the lines M8 and My several other lines which could not be identified at first sight were found on the plate. Now the linesin the M-series have a very typical structure, being rather diffuse and broadened on the short wave-length side. Therefore no doubt could arise as to the identification of these lines. As to the new line L,L,, however, we should expect that: i would be distinctly. different, oad especially that it would be fairly sharp for W. Fora right interpretation of the plate it was therefore desirable to exclude all the lines which appeared on the plate in higher order. This could be done in the follow ing simple way. ~ A new plate was exposed under the same conditions. Half of this plate was covered with an aluminium sheet of 7 uw thickness. By this sheet wave- lengths of more than 6000 X.U. are totally absorbed. Fig. 4 (Pl. XXII.) gives a reproduction of the spectrum obt ained in this way. Most of the lines are not rey absorbed by the aluminium, but the lines MB and My have been totally absorbed. Copper and tungsten lines were found in 4th and Sth order. Between the LB and Lf, line of W the line L3 was seen on the plate alsoin 5th order. This line comes from the L, level. As L@3 is not a very strong line for W, it was very faint on the plate, and it cannot be seen on the reproduction. Further the Ca K absorption line in the second order may be seen as a dark line quite near the My line of W. This line is due to the selective absorption in the. Ca of the crystal*. From the appearance of the tungsten lines in such high ‘order we may conclude that the experi- mental conditions were such as to excite the Lz level quite sufficiently to give rise to the line L;L,, unless this transition is very improbable. The place w here we should expect the line is marked with a + on the reproduction. In the neigh- bourhood of this place ‘only one very faint line was seen on the plate. This line, however, was not appreciably absorbed * In general, absorption discontinuities are obtained in a very easy manner if the absorbing element forms part of the analysing crystal. 10!'8 Mr. D. Coster on the Spectra 0) X-rays by the aluminium sheet (the reproduction does not give a very good impression of this fact). It is quite possible that it is the tungsten Ly, in the 6th order. In the same way I was able to show that the transition Li, fy does not exist ; this, however, is what we should already expect from the a—é rule. ; é Part ITI. Discussion of Results. § 1. As has been stated in Part IL. § 1, the object of this paper is to compare the changes in the characteristic X-ra spectrum as the atomic number decreases with the changes in the structure of the atom as given by Bohr’s theory, of which a brief account has been given in Part I. In order to do this, we will deal with the diagrams of the energy levels of the inert gases, which are based upon measurements of the elements preceding or following the inert gases in the periodic table. For niton the diagram has already been given in Part I. of this paper. As regards this diagram, the following remarks may here be made. Only the lines belonging to the L-series have been systematically investigated for the elements in the neighbourhood of niton*. The complexity of the line Ly which is suggested by analogy with the appearance of the pairs of lines L@;-L8, and ly3-Ly, could not be proved experimentally, because of the small difference of energy between the levels O; and O,. The energy-difference between the levels O, and O, could not be obtained experimentally for the same reason. As yet there is no experimental evidence for the existence of the three P-levels inserted in the diagram. Perhaps the line 568-9 found by Dauvillier 7 in the spectram of U represents a transition Py>L;. But since in general it is very dangerous to draw conclusions from the measurements of only one element, it seems better to postpone a discussion of the P-levels until a thorough investigation of the X-ray spectrum of the radioactive elements has been made. The K-spectrum of the elements -in the region of high atomic number has been measured with sufficient accuracy for W only. But also different measure- ments of the K-series, made by several.authors for elements of lower atomic number, support the arrangement of lines * See D. Coster, Zeitschrift f. Physik, iv. p. 178 (1921) and I. and IT. In I. and Il. the arguments are given in favour of the arrangement of the lines in the diagram 1. + Comptes Rendus, clxxii. p. 1350 (1921). and the Theory of Atonuc Structure. 1099 given in the diagram. ‘The line K@; has been detected by de Broglie * only for the elements tungsten and rhodium. The frequency-difference found by de Broglie for the lines K£@, and K£, agrees very well with that for the lines LA; and L®,. The complexity of the line K®, could not be observed but is suggested by analogy with the pairs of lines KB,-K83, L8;-LA,, and Ly;-Lry,. In any case it is probable that most of the energy of this line is due to the transition N;>K. The lines in the M-region have been measured by StenstrO6m and Karcher for different elements. It is highly robable that there exist further transitions in this region, other than those detected by Stenstrém. Diagram I1.—XENon. — ——\— | | lela | Lh is h 7 wah hp hep 4 Bie uri 1 Qe SiG. Qk, Ls——- 62, MT ieee Kaa hh /h Capes i tae Pa ate K O4 § 2. The diagrams IJ. and III. for the inert gases, xenon and krypton, are derived from the experimental results given in Part IJ. of this paper. As for the L-series we see from the tables in Part II. that all the lines inserted in the * Comptes Rendus, clxx. p. 1053 (1920) and elxx. p. 1245 (1920). 1100 Mr. D. Coster on the Spectra of X-rays diagrams were actually measured for the elements with higher atomic numbers than xenon or krypton respectively. The single exception is the line L8;. Though there are some indications that this line has been found for Ba and Cs, its identification is not quite certain. An investigation of the rare-earth metals, as yet unfinished, will possibly settle the question. The lines «, a, 8, and B, of the K-series have been measured by several authors for different elements Diagram III.—Krypron. | | Baie | in this region ; the line 83, which lies at a very smail wave- length distance from §,, has been measured only for Rh, by de Broglie, as stated above. The lines My and M6 have only been inserted in the xenon diagram for the sake of complete- ness. They have only been found for the elements U, Th, Bi, Pb, Au, and Pt* and they have not yet been studied systematically. It will be very difficult to measure them in this region, as even for Ba they should have wave-lengths of about 13000 and 12000 X.U. respectively. The largest wave-length hitherto measured with the aid of crystal re- flexion is the line Le for copper (13309 X.U.). * See Stenstrém, Dissertation, Lund, 1919; Karcher, Phys. Rev. IL. xv. p- 285 (1920). These lines have been called My: and My2 by Karcher. and the Theory of Atomic Structure. 1101 On the other hand, we may see from a comparison of the diagrams I. and IJ. and the tables of Part LI. that the lines, which according to the diagrams should fall out between niton and xenon (7. e. in the L-series @; and yg) were actually not found for the elements in the neighbourhood of xenon. In the same way we may see from the diagrams IT. and III. compared with the tables of Part II. that the lines Ly,, LA, and Lry,, which according to the diagrams should fall out between xenon and krypton, were not observed for Rb and Sr. _ § 3. We will now proceed to a closer comparison of Bohr’s theory of the development of groups of electrons in the atom with the experimental results as regards the first appearance of the lines in the X-ray spectra. Starting from niton the lines which first disappear are 8; and ye. For the elements in the neighbourhood of niton both the lines 8; and yz are fairly strong : ; In the neigh- bourhood of the Pt metals their intensity decreases very appreciably ; for W,and especially for Ta, which elements both gave very good plates, they belong to the faintest known lines in the X-ra ay spectra. In the rare-earth metals and for Ba and Cs not the slightest trace of these lines has as yet been found. On several plates, however, I found white lines on a dark background in the place where we should expect to find 8; or ye. As mentioned in II. § 9, these white lines are the characteristic absorption lines * L,; and L, due to the absorption of the Cu radiation in the salts which were used upon the anticathode. Though even this is not an absolute proof + that the lines @; and vy, do not exist, it seems on the whole to be very probable that these lines disappear for the rare-earth metals. This isin agreement with the theory. According to this 53 orbits prob: ibly appear for the first time for La (57) 3 {, but in the rare earths the electrons moving in these orbits are very loosely bound in the form of valency electrons, which are har dly present in the salts used in the experiments. It is only in Ta (73) and the following ele- ments, that a 5; electron can first be expected to be present under the conditions of the experiments and to correspond to a binding sufficiently strong for the 5; level to be detected. * On carefully exposed photographs of an absorption edge usually a w hite line may be seen. This white line implies that the electron ‘< prefers ” the absorption of a frequency which is just able to bring it outside the atom. + In studying the L, and L, absorption edges of tungsten, Duane and esos have also made use of the absorption in the tungsten of the anticathode. (See Proc. Nat. Ac. Sci. Washington, Sept. 1920.) +t Compare N. Bohr, Zeitschr. f. Physik, ix. p. 1 (1922). 1102 Mr. D. Coster on the Spectra of X-rays We should expect the line Me to disappear at the same time as the lines L@,; and Ly, as the existence of this line is also bound to a 53 subgroup of electrons. This line has as yet not been studied systematically. It has been measured by Stenstrém * for U and Th and by Karcher f for the elements Bi, Au,and Pt. R 4. The appearance of the lines Mon, May, and MBP is connected with the existence of 4, electrons in the N-shell. According to Bohr, these electrons: occur for the first Hime in the rare earths. For the elements Dy (66) there must already be electrons present in 4, orbits, as the appearance of the lines Ma (a, and «a, could not be separated for the elem:nts with lower atomic number than Tl) and M has been established by Stenstrém for this element. The wave- lengths of these lines are respectively 9509 and 9313 X.U. It would be of interest to try to find these lines also for elements with lower atomic number than Dy. § 5 The existence of the line Ly, depends on the presence of 5, electrons in the O-shell. As pointed out to me by Professor Bohr, it appears from comparison with the optical spectra of Cd and In, that 5, orbits appear for the first time in the neutral atom of the latter element. The experimental results seem to indicate that for the line y, there are certain complications. The very striking changes in the intensities of the pairs of lines 8;—yg (see Part III. § 3) and 8.-y; (see Part IIT. § 6) cannot be questioned. But usually it is very dificult to make any definite assertions concerning changes ‘of the relative intensity of the lines with different elements, as we are not certain that they have been exposed under the same conditions. However, there is evidently a considerable change in the intensity of the line Ly, between the elements Sb and Sn. Down to In (49), this line could be measured but here it had become very faint. For Cd, where we according to Bohr should expect y, to disappear, no certain information about this tne could be obtained from the experi- ments. Extrapolating according to Moseley’s law we find for the wave-length of y, for Cd about 3081 X.U.. which is at the same time about twice the wave-length of the copper Ka, line. Using an X-ray tube of brass and working with > a fairly high tension which appeared to be necessary in pe aphing the very weak lines, it 1s impossible to avoid wholly the characteristic Cu- radiation. Actually the copper * Stenstrém, Dissertation, Lund, 1919. + Karcher, Phys. Rev. II. xv. p. 285 (1920). The line Me has been called My3 by Karcher. and the Theory of Atonue Structure. 1103 Ka, and Ka, lines though very faint were in second order observed on the plate. From a thorough examination of the plates, however, we are inclined to conclude that the line y, does not exist for Cd or at any rate must be still weaker than in the case of In. For the next element (Ag) we are troubled by the increased aksorption in the photographic plate (L; discontinuity), so that for this element nothing can be stated about the existence of y,. A complication showed in the photographs of the Pd (46) and Rh (45) spectra. Besides a very faint line at 3450 X.U. for Pd and at 3651 X.U. for Rh, which might be considered to be y,, another line of the same appearance was found for both elements. (For Pd X=3433 and for Rh 7A=3631.) These lines, however, have shorter wave-lengths than the extrapolated values of the respective L, discontinuities. § 6. The lines 8, and y, depend on the existence of the 4, electrons in the N-shell, which electrons according to Bohr are found for the first time in the neutral atom of Y (39). Relatively to the other lines of the L-spectrum §) and y, have their ordinary intensity as far down-as Mo. For Nb however they are considerably weaker compared with the other lines, for Zr they are both very faint. For Y, 8, and y, were not visible, but here the plates were not especially good. For Sr and Rb, for which the plates were quite good, y, had wholly disappeared. If we extrapolate @, according to Moseley’s law, this line almost coincides with @, in the case of Sr, while for Rb it should lie on the long-wave-length side of §, branch of the network, the condition for no deflexion in the galvanometer can be shown in a similar way to be —nU= A ~ Loli FE Ang Tie Ani ae Ang) | (A1a— Ary — Aza + Ags) 2 [Bmn Knn(Ama— Amb — Ana + And) |” It is noticeable that the junction X does not appear either implicitly or explicitly in these last two equations, and hence we may draw the following interesting conclusion :— In a general network like that shown in fig. 3 (but also having several sources of em.f) for which the variables have been adjusted so as to give no galvanometer deflexion, the per- manently-connected plates of the condenser may then be connected to ANY point in the network without disturbing the galeanometer’s state of no deflexion. of Maxwell’s Capacity Bridge. Litt This theorem evidently applies to the cases illustrated in figs. 1 and 2. We can also show that this theorem may be extended to include inductances in the network. For, by Ohm’s law, Cine dt The initial and final values of ing and ts are the same. Hence, by integration : Ry Ipq =i (Ey, + Vp—Va)dt, and similar equations for all the other pairs of junctions. But this set of equations is identical with the set obtainable in the case of no inductance being present. Hence, in- ductances do not affect the distribution of the total charges, and neither, of course, do they affect the steady current distribution. : ize wl Experimental Verification. The arrangement shown in figs. 1 and 2 was set up (n=50 per second, C=1 microfarad) and “ balanced.” “Balance” was unaffected by connecting the condenser- terminal to a large number of points in the network taken in succession and at random. A difference of 1 in 5000 would have been noticeable if it had existed. Another network examined was that shown in fig. 4. 150 aw 3 A condenser of 4 microfarads made contact alternately on junctions 1 and 4 (n=50). The “ balance” was found to be unaffected to less than 1 in 5000, as before. My thanks are due to Prof. J. 8. Townsend for his kind permission to carry out the experiments in the Electrical Laboratory, and also to Mr. R. I. Mincovitch for carrying out the arithmetical verification of formula (5) as applied to the last experiment. Phe Prof. N. Bohr on the Selection Nore.—Mr. H. F. Biggs has pointed out an extremely simple way of viewing the action of the condenser, which immediately leads to the conclusions already obtained. Since there can be no conduction-current in a perfect insulator, the total charge that passes into the network at the junction X during a complete period of the tuning-fork must be zero. The condenser-plates attached to the vibrating reed may be then regarded as conveying a charge C(Va—Vz) from A to B n times per second. On this view, it immediately follows that the permanent connexion may be made with any part of the network with- out affecting the galvanometer “ balance ” when once made. The Electrical Laboratory, Oxford. June, 1921. CXVII. On the Selection Principle of the Quantum Theory. To the Editors of the Philosophical Magazine. GENTLEMEN,— N a note titled “A Significant Exception to the Principle of Selection,” and published in the April number of this magazine, P. D. Foote, I. L. Mohler, and W. I’. Meggers describe some interesting experiments on the excitation of the lines of the are spectrum of potassium, and conclude that the results of these experiments throw doubt on the general principles, on which the laws of the series spectra are interpreted on the basis of the quantum theory. As I shall try briefly to explain, the results in question do not seem, however, to offer sufficient basis for such a conclusion. According to the ideas of the quantum theory each of the lines of a spectrum like the arc spectrum of potassium is emitted by the atom during a process of transition between two among a multitude of stationary states, in which one electron moves in an orbit, the dimensions of which are large compared with the orbits of the other electrons in the atom, which together with the nucleus may be said to form the inner system. In first approximation the orbit of the outer electron will be a plane central orbit, which may be described as a plane periodic orbit, on which is superposed 2 uniform rotation in its plane. Jn the stationary states of the outer electron the motion is therefore in first approxi- mation fixed in the well-known way by means of two quantum numbers, which may be denoted by nm, and n. While 7, Principle of the Quantum Theory. pL refers to a certain condition concerning the radial motion of the electron, n, fixes the angular momentum of the electron round the centre of the orbit through the condition that this angular momentum is equal to moh/27. These numbers are assumed to be related to the spectral terms in such a way that n, increases by one unit when, within each series of terms, we proceed from one member to the next, while mn, remains constant within each series of terms and increases by one unit when we pass from the S-terms to the P-terms, and from the P-terms to the D-terms, and so on. This classification of the terms refers, however, only to the structure of the are spectra in large features. I» order to account for the complex structure of the lines (doublets, triplets) a higher degree of complexity of the multitude of stationary states is claimed. ‘This is assumed to arise from a complication of the motion of the outer electron due to a small departure from central symmetry of the inner system, which causes the plane of the orbit of the outer electron to undergo a slow precession round an axis coinciding with the axis ofangular momentum of the atom. Due to this compli- cation of the motion there will in the fixation of the stationary states appear a third quantum number 3, which fixes the orientation of the plane of the outer electron relative to the axis of the inner system through the condition that the resultant angular momentum of the atom is equal to n3h/27-. This third quantum number is related to the complexity of the multitude of spectral terms in such a way that the components of a set of complex terms corresponding to the same values of n, and mn, are distinguished by different values of nz. Now the so-called principle of selection originates from considerations dealing with the limitation of the possibility of transition between stationary states. Such considerations are based on two entirely different types of arguments. One argument rests upon the so-called principle of corre- spondence according to which the possibility of a transition between two stationary states, giving rise to the emission or a train of harmonic waves, is sought in the presence in the motion of the atom of a certain “corresponding” con- stituent harmonic vibration. For stationary states of the atom of the type described above, this argument leads to the conclusion that, at the same time as no limitation is imposed on the variation of the quantum number n,, the number v, must by a transition always change by one unit, while the number n,; may either change by one unit or remain unchanged. Another argument is obtained from 1114 Prof. N. Bohr on the Selection the fact that it is possible in certain cases to exclude tran- sitions between stationary states from the condition of con- servation of angular momentum during the radiation process. As regards the problem under consideration this leads to the conclusion, that the quantum number ns, which, as mentioned, is directly related to the resultant angular momentum of the atom, by a transition cannot vary by more than one unit, while no direct information as regards a limitation of the quantum numbers »,; and mn, can be obtained from this argument. As regards the comparison of these conclusions with experiments, our insight into the origin of the complex structure of spectral “lines is at present hardly suff- ciently developed to provide a definite test as regards the detailed interpretation of the quantum number n; and there- fore of the rules of selection as far as they can be based entirely on considerations of conservation of angular momentum. At present the main problem, with which also the experiments in the above-mentioned note deal, is the test of the conclusions as regards the variation of the quantum numbers n,; and v , on which in first approxima- tion the spectral terms depend. Now previous experimental evidence as regards excitation of series spectra has seemed convincingly to support the conclusion drawn from the principle of correspondence, that under circumstances where the emitting atoms are not influenced by external agencies, only such spectral lines can appear as correspond to a combination of terms for which n, differs by one unit. This may be considered as a very important result, because we may say that the quantum theory, which for the first time has offered a simple interpretation of the fundamental principle of combination of spectral lines, at the same time bas so to say removed the mystery which has hitherto adhered to the application of this principle on account of the apparent capriciousness of the appearance of predicted combination lines. Especially attention may be drawn to the simple interpretation which the quantum theory offers of the appearance observed by Stark and his collaborators of certain new series of lines, which du not appear under ordinary circumstances, but which are excited when the emitting atoms are subject to intense external electric fields. Tn fact, on the correspondence principle this is immediately expl lained from an examination of the verturbations in the motion of the outer electron, which give rise to the appear- ance in this motion—besides ‘the vibration already present in a simple central orbit—of a number of constituent Principle of the Quantum Theory. 1115 harmonic vibrations of new type and of amplitudes propor- tional to the intensity of the external forces. To call such an effect, as is often done, a “ breaking through” of the principle of selection is a terminology which would not seem to be quite adequate in view of the character of the theoretical interpretation which can be given of the pheno- menon under consideration. In the new experiments published in the note, mentioned in the beginning of this letter, precautions were taken to screen the radiating atoms from the effect of external electric forces due to the potential applied to the discharge tube. In spite of these precautions it was found that under certain conditions the spectra observed included, besides the usual arc lines of potassium, certain lines corresponding to combi- nations between two terms, for which n, differs by two units, and which correspond to the new series mentioned above, which appear when the radiating atoms are exposed to intense electric fields. This observation is by the authors described as a significant exception to the principle of selec- tion, since the appearance of the lines could evidently not be caused by external fields of the same type as claimed for the production of the ordinary Stark effect. From a closer consideration of the experimental conditions it would -appear, however, that the observations published rather support than disprove the theory. First of all it was found that the lines in question appeared only if very dense currents were sent through the apparatus, while these lines did not appear when a discharge of less current density was passed through the tube. Thus under the circumstances, where the condition ef non-disturbance of the atoms by external agencies, claimed by the theory, was satisfied to the highest degree, no exception from the simple selection rule was observed, indicating that the presence of the new lines for higher current densities was caused by some agency influencing the usual motion of the electrons in the atoms. Looking for some such effect, it would appear that, just due to the screening from external forces, the experi- mental arrangement described would be especially favour- able for the accumulation of ions in the region of the discharge tube used for the observation of the spectrum; an effect to which already the authors themselves incidentally refer as a possible cause of the origin of the new lines. Without closer information about the dimensions of the apparatus and the details of construction than that given in the note, it is not possible to form an accurate estimate of the density of ions which may have been present 1116 Prof. R. Whiddington on — under the conditions of the experiments; but already a rough calculation makes it very probable that the fields, due to neighbouring ions and free electrons, to which the emitting atoms have been sabject, have been of the order of magnitude claimed by the quantum theory for the appear- ance of the new lines with the intensity observed. Hspe- cially it must be noted, that the intensity of these fields . may easily have been many thousand times larger than the intensity of the external electric forces present in the unscreened part of the apparatus, as a direct consequence of the applied potential. On the whole it seems that the theory of series spectra based on the quantum theory allows to account in a general way for the experimental evidence, and that especially the relative intensity, with which under various experimental conditions combination lines appear, may not be considered as presenting the theory with serious difficulties, but rather as providing a means of investigating the conditions under which spectra are emitted. For instance, investigations of the presence of combination lines may perhaps offer the most direct means of estimating the density of ions in the different parts ofa discharge tube. As regards a more detailed account of the theory of series spectra, and for references to the literature on this subject, the reader of this Magazine may be referred to a paper on the effect of electric and magnetic fields on spectral lines, which was read by the writer as a Guthrie lecture at the Physical Society of London on March 24th, and which will soon appear in the Proceedings of this Society. Yours faithfully, University, Copenhagen, N. Bonr. April 11th, 1922. CXVIII. X-ray Electrons. By RK. Wurppineron, J.4., D.Sc., Cavendish Professor of Physics, University of Leeds*. [Plate XXIV.] Introduction. if has been known for many years that X-rays have the power of ejecting high-speed electrons from the surface of materials on which they are incident. The earliest successful attempts to measure the speed of these X-ray electrons were made by Dorn in 1900 and * Communicated by the Author. X-ray Electrons. LTE Innes in 1907, and it appeared from their results that the speed lay between 6 and 8x 10° cm./sec. and was inde- pendent of the intensity of the X-rays. In 1912 the writer attacked the problem from a different angle by interpreting the absorption experiments of Beatty * in the light of the fourth-power law of velocity diminution previously established +. Beatty having measured the range in air of X-ray electrons, it was possible to translate the range into terms of initial speed, and it was shown that this ejection speed was nearly equal to 10°x A, where A is the atomic weight of the element supplying the X-rays f. It was pointed out recently that this result—frankly approximate in view of the nature of the experiments—was expressible in terms of atomic number, and that for these fastest X-ray electrons a formula similar to that of Moseley for X-rays was approximately true f. It is to be noted, however, that the method just mentioned is only capable of determining, approximately, the speed of the fastest electrons, and is incapable of proving with cer- tainty the existence of independent groups of smaller speed—- unless, of course, such groups are very widely separated §. The present work was undertaken with the object of extending these results, and during its progress de Broglie || has published most important results bearing on the same problem. His results, which will be referred to in more detail later, are confirmed in the main by my own ob- servations. The Apparatus. The apparatus used to determine the speed of the X-ray electrons is indicated in fig. 1 ¥. A quartz bulb B is fitted with cathode © and rhodium- faced target T, water-cooled through the tubes D,, D,. The cathode also was air-cooled by the copper radiator fins R. Just under T and about a centimetre above it is mounted the aluminium slit chamber J (shown enlarged to the right), * Beatty, Proc. Roy. Soe. (1910). + Whiddington, Proc. Roy. Soc. (1912). t Whiddington, Phil. Mag. vol. xxxix. p. 694 (1920). § Simons, Phil. Mag. (1921), has repeated Beatty’s experiments very carefully, and has shown the existence of such groups. || de Broglie, Comptes Rendus (1921). q A preliminary account was given in Proc. Camb. Phil. Soc. (1921). 1118 Prof. R. Whiddington on on the three inside faces of which the element providing the X-ray electrons was spread in the form of thin metal sheet or powder. The upper side of J was a sheet of aluminium thin enough to transmit the X-rays from T with small absorption, but thick enough to absorb completely any direct electrons produced within B *. ALUMINIUM Fol E ALUMINILI4 BLOCK ENLARGED SECTION OF J. me ee eee ae A is the camera chamber, made on the plan adopted by Rutherford and Robinson + in their B-ray speed measure- ments, so that, with a uniform magnetic field applied at right angles to the plane of the figure, the electrons from J came to a focus on the photographic film I. . The slit was -05 em. in width. K is a thick brass partition stretching right across the camera so as to prevent any secondary radiation from the inside of the camera from reaching F. ? Lis a light trap to minimize the effect on F of any stray light from outside or from accidental discharge passing down the glass evacuating tubes. G is a collimator for projecting a registration spotof light on I whenever necessary. The usual pump and liquid-air charcoal tube method of evacuation was adopted. The discharge was produced with an ordinary induction * This necessary thickness d is calculable from d=v‘/a given in Proc. Roy. Soc. (1912). . t Phil. Mag. vol. xxvi. p. 717 (1918). | 1 X-ray Electrons. EELS coil and mercury break, with an auxiliary mechanical rectifier. A maximum input of 1000 watts was used. The pee field was provided by two large air-core coils (H,) placed such a distance apart as to produce a sensibly uniform field over the area of the camera. Auxiliary small coils H, in series with H, were so arranged as to compensate the ext-rnal field of H, over the path traversed by the cathode rays between C and T. The uniformity and value of the field was investigated by search coil and ballistic galvanometer in the usual way. A current of 7 amperes in the coils gave a uniform magnetic field of 101-4 Gauss *. It is clearly an advantage in this kind of work to use air- rather than iron-cored wires, since (1) The value of the field is always queasy proportional to the current traversing the coils, and it is therefore easy to adjust the fieid to any required value with- out troublesome remeasurement. (2) A small accidental variation of current produces only a proportionate change in the field. (3) The compensating coil having been set for any one value of the field is set also for any other value. The disadvantage is that rather large electrical energy is consumed by the field coils—in the present case, about 500 watts. The radii of curvature of the electrons varied from about 3 to 6 cms. Taken as a whole, the apparatus just described is very troublesome to manage owing to the difficulty of keeping the numerous metal, quartz-g -glass joints air-tight. In spite of water-cooling, the Joints ‘too frequently give way. Its main recommendation is on the ground of economy, since the expensive Coolidge tube and “alternator transformer is not required on account of the high efficiency of the arrangement. This efficisney is obtained (1) Because the X-ray electrons are liberated from three surfaces within J. (2) Because the target T being very near J, the intensity - of the X-rays is very great. The second reason just given is particularly important. An ordinary X-ray bulb could not easily be arranged nearer than siy 10 ems. to J, with a consequent 100-fold loss of intensity. * Since H, is small compared with I],, one would expect its external field to be negligible in H,. This was proved by showing that the value of RH was constant for some definite speed group of electrons and thus independent of the actual path traversed in H,. i i AP Rt Nt A TI 1120 Prof. R. Whiddington on Method of taking Photographs. The film used was the Duplitized X-ray film of Messrs. Kodak. A faint impression could be obtained with 5 minutes’ exposure from a copper foil, and a good photograph in 30 minutes. , It was observed that the intensity of the photographs obtained was not proportionately increased by longer exposure, while at the same time the contrast became less pronounced. Without further discussion, it may be of interest to mention that excellent results were sometimes obtained from these duplitized films by dividing the ex- posure into two parts, one on each side of the film. It was necessary, of course, to ensure that the film was placed in exactly the same position relative to the camera during each exposure, and this could be tested by the collimator G of fig.1. The lower photograph in Pl. XXIV. was obtained by this means. In order to test the resolving power, so to speak, of the apparatus the chamber J was removed, the apparatus exhausted with film in place, and a discharge passed from an electrostatic influence machine working with no con- denser. Under these conditions the cathode rays were reflected from 'T through the slit into A and then bent round to foci on F. The discharge was intermittent, so that a velocity spectrum was produced giving the result shown in the top reproduction in the Plate. It is clear that the focussing is excellent, and that the individual lines are sharply defined and practically as narrow as the original slit *. The fundamental product pH was measured from the photographs obtained with an accuracy, it is believed, of 4 per cent. It is to this accuracy, then, that the velocity of the electrons corresponding to an intense well-marked line can be guaranteed. Since, however, it is convenient to express the results in terms of energy rather than velocity, it is proposed to define an electron in ‘‘equivalent frequency ” v, defined by the usual relation hy=}mv’, the relativity correction having been introduced in calculating v from pH=v.m/e. It is clear, therefore, that v will be subject to a possible é ni Searle ‘ * Neglecting the relativity correction, = =A" me, where p is the radius of curvature of an electron velocity vin field H. It can easily be shown from this that the “ dispersion” and ‘resolving power” for the apparatus is nearly independent of v--or the spectrum is normal for a given value of H. X-ray Electrons. 1121 error of 1 per cent.—an error which is likely to be even greater in the case of faint lines preeine difficulty in measurement. Heperimental Results. Hight elements have so far been examined for X-ray electrons under the action of the radiation from Rhodium [45], viz.: Aluminium [13], Copper [29], Zinc [30], Arsenic [33], Strontium {38], Molybdenum [42], Tung- sten [74], Bismuth [83]. One typical photograph is shown (lower of Pl. XXIV.) of copper. As noted by de Broglie, there are a number of lines or rather bands coming to a more or less sharply-defined head on the high-velocity side. To each band de Broglie attaches a maximum v and therefore v corresponding to the pH value of the head *. This maximum value, in fact, is associated with electrons escaping from the surface atoms of the material with a speed unaffected by subsequent passage through or near other atoms: it may, in fact, be taken to be the value corresponding to emission from a free atom. It is these maximum values of v which have been tabulated below. TABLE. v expected from Agency ef origin and agency. Origin of electron. ejection. CopPER (metal leaf). rs; 4 12 L Ring CuKa CuK,—CuL=173 L Ring CuKg CuKg—CuL=194 4 Bie abo or or | outer rings or free CuKae CuKa =195 ' i. alT outer rings or free CuKg Cukg =216 6 3 210 K Ring RhKg RhK,—OuK=273 eth cea 2B7 2 ? 4s. B13 PK Ring RhKg _ BhKg—CuK=334 as 461 L Ring RhKa RhK,—CuL=468 f 502 L Ring RhKeg RhKg—CuL=529 ( z outer or free RhKa RhK, =490 \ vd. 2652 outer or free RhKg Rhg = 551 * T have applied = obviously necessary correction in measurement instead of pH. I use [p—d/4]H where d=slit width. Phil. Mag. 8. 6. Vol..43. No. 258. June 1922. 40€ 22 Prof. R. Whiddington on + oat vy expected from v. Origin of electron. origin and agency. ejection. Zinc (oxide). s. 4189 L Ring - ZnKa ZnKa —ZnL=185 i f. 234 outer or free ZnKg 2nKg ==. 232 s. 260 K Ring RhKa RhKa—ZnKk=267 | mo rs) % s. 318 ?K Ring RhKg RhKg—ZnK =328 s. 456 L Ring RhK, RhKa— ZnL =466 Other bands too nebulous to measure of higher frequency. ALUMINIUM (metal). f, 448 K Ring RhKa RhKa— AlIK=452 v.f. 502 K Ring RhkKg RbKg— AIK=513 ARSENIC (oxide). 222 L Ring AsKa AsKa—AsL=221 951 f L Ring AsKg AsKg — AsL=250; °" | outer or free . AsKa AsKa=252 see = 2478) outer or free AsKg AsKg = 285 s. 442 L Ring » RhKg Rhi.—AsL=455 f 507 _ L Ring RhKg RhKg— AsL=516 : ! outer or free Rhk, RhK, = 490 v.f. 563 outer or free RhKg RhKg = 551 STRONTIUM (oxide). s. 292 L Ring SrKa SrK. —SrL=296 s. 334 L Ring SrKeg SrKg -SrL=3387 j outer or free SE SEG =335 v.f. 370 outer or free SrKg Srg = 385 s. 426 L Ring RbKe RhKa — SrL=442 f, 449 L Ring RhKg RhKg—L=503 ‘ outer or free RhKg RhK, =490 f. 544 outer or free RhKg RhKg = 551. X-ray Electrons. 1123 Seey v expected from of : rigin of electron. eee cs Origin of e Sor origin and agency. ejection. | Mo.yspENUM (oxide). Ra oul. L Ring Mok, MoK,—SrL=364 49] { L Ring _ MoKg MoKg-—SrlL=416 = | outer or free Mock, MoKa = 424 ei AD outer or free MoKg MoKg = 476 TUNGSTEN (oxide). s. 164 M Ring WlLa WL.—- WM=162 s. 208 M Ring WLeg WLe—WM=202 outer or free Wa WLa = 202 te onet ste Wik’) Wie Be eae ad outer a free WLy WL, = 282 f. 380 ? s. 445 M Ring RhKy RhKa—M=450 f. 494 outer or free RhKg RhKa = 490 v.f. 544 outer or free RhKg RhKg = 551 (Much general fog.) BisMuTH (oxide). 207 M Ring WLa Bil, — BiM = 204 260 M Ring WLg BiLg—BiM=260 i outer or free WlLe Bile =260) 319 i M Ring WLy BiL,—BiM=316 { outer or free WLg BiLg =316 ti 363 outer or free WIL, Bily = 372 3 N Ring { RhKa—BiN=484 f, 479 outer or free } Rbk, | BhKa =490 Y N Ring RhKg— BiN =545 ys { outer or free } ae | RhKeg =5o! (Much general fog.) The explanation of most of the bands observed is possible by applying the energy-difference relations suggested by Rutherford in 1914 for the @-rays and applied by de Broglie to his experiments. There are some lines, how- ever, which are somewhat ambiguous. It will be assumed 4 C2 1124 Prof. R. Whiddington on that the rhodium target emits—in addition to ‘ white” radiation—a very strong monochromatic radiation RhKe [v=490x 10%], together with a much feebler frequency RhKg [v=551 x 10"*]. If the radiations characteristic of the element under consideration are excited, it will be necessary to take into account also the radiation frequencies K,, Kg... in the K series and Le, Le... in the I series, Ma, Mg... in the M series, etc., and also the corresponding absorption-. limit frequencies K, L, M ... in the following manner :— The X-ray electrons—. e. the electrons liberated by the X radiation—will have energies corresponding to their origin. Hlectrons liberated from the more deep-seated rings nearer the nucleus will have less energy than those liberated from the outer rings. Thus the Rhe radiation may liberate electrons with energies proportional to the frequency differences : RhKa—K (electron from K ring), RhKea—L (electron from L ring), RhiKa—M (electron from M ring), etc., ete. ; and if it be possible for the free electrons to absorb the X-ray energy : RhKa« (free electron). | There may also be a much-smaller liberation of electrons corresponding to : RhK,—K, RhK,—L, RhK,—M, RhK producing much fainter impressions in the photographs. In addition to these sets of electrons, there may be other sets liberated by the K, L, ete. X radiations excited in the element under examination. The K. line cannot liberate an electron from the K ring, since its energy is not sufficient; but there may be K,—L (from Lring) Kg—L, K,—M (from Mring) K,—M, K, (free electron?) Keg; and similarly for the L radiation, although in this case the spectrum is somewhat more complicated by the presence of a comparatively strong Ly radiation. A-ray Electrons. 1125 In the table are shown the equivalent frequencies of the X-ray electrons for the eight elements examined. In the second column is indicated the position within the atom from which the electron may be supposed to come; while the third column shows by what agency the ejection is effected. The fourth column gives the value of v to .be expected from the suggestions of columns 2 and 3. Discussion of Results. I. Broadly speaking, there isa general agreement between the observed and calculated values of the effective fre- quencies (v) for these X-ray electrons. | Taking copper for example: The three lines of smallest frequency are to be ascribed to the action of K, and Kg of copper on the rings outside the K ring, and it is to be noted that the central line of the triplet is to be regarded as due to two sets of electrons—one from the L ring, the other from the outer rings or even from free electrons. Further, the line 217 ascribed to the ejection from the outer rings (or possibly the lifting out of free electrons) * is very taint— an observation in agreement with Rutherford and Ellis’s 8-ray results. This characteristic K triplet is repeated in the case of zinc, arsenic, strontium, and molybdenum. In the case of tungsten and bismuth, however, the K radiation is not excited, and in place of the K triplet we get the L quadruplet, the additional line being due to the presence of three strong lines L,, Ls, and L, in the L X-ray spectrum. At the other end of the copper electron spectrum are two lines, 502 and 552, very faint and probably due to RhK, and RhKg, ejecting electrons from the outer rings or lifting free electrons from the surface of the material as a whole. The apparatus, of course, is not capable of differentiating between these two possibilities. Of the four remaining lines, 270 is clearly due to a K-ring electron ejected by the RhK, radiation, while 461 is an L-ring electron ejected by the RhKg radiation. The remaining two lines are rather puzzling. 287 is a faint line, and may therefore be inaccurately measured, but 313 is a strong well-defined line. The nearest straight- forward interpretation is to place it as a K-ring electron ejected by RhKg, in which case the expectation for v would be 334. This is a far bigger difference than is to be expected * See Millikan, Phys. Rev, 1921, p. 248, 1126 On X-ray Hlectrons. for so bold a line; moreover, its intensity is greater than the RhKg radiation should give *. Ii. A point which is not brought out in the table and which should be mentioned is that the electron emission increases as the atomic number increases. In fact, the time of exposure necessary to produce a photograph of a certain intensity is very roughly inversely proportional to the atomic number. With the elements of higher atomic number, also, there is a very considerable amount of continuous spectrum, due possibly to the preponderating importance of X-ray electrons ejected by the “ white” radiation from the target. Ill. It is clear from reference to the copper X-ray electron spectrum that, although each line shows a more or less sharply-defined head or boundary on the high- velocity side, yet there is in addition more than a suggestion of breadth. Measured in terms of equivalent frequency, this breadth is explicable on the supposition that these lines are really doublets, unresolved by the apparatus and separated approximately by frequency M. IV. It is to be noticed finally that whenever a frequency difference less than that corresponding to the K (or L as the case may be) might have been expected, no corre- sponding line is observed. Thus, in the arsenic spectrum RhK,—AsK = 202, which is less than AsK = 288, and no line appears at 202; or, again, in the bismuth spectrum RhK,—BilL=91, which is less than BiL=399, and no line appears at 91. Further, although the L electron lines are strong in the cases of tungsten and bismuth when the K electrons are not disturbed, yet in other cases—copper, for example—no trace of L electrons could be found, although a careful search for them was made. Experiments are being continued. My thanks are due to the Government Grant Committee of the Royal Society for assistance in defraying the cost of the quartz apparatus. The University of Leeds, April 3rd, 1922. * Zine impurity in the copper-leaf would reduce the discrepancy somewhat, but hardly sufficiently. Whatever the explanation—and further experiment is in progress—it may be fointed out that the triplet (270, 287, 313) is complementary member for member. with (217, 193, 172), in the sense that when corresponding members are added together the total approximates closely to 490 (RhK,). ere CXIX. The abnormally long Free Paths of Electrons in Argon. By J. 8. Townsenn, M.A., F.RS., Wykeham Professor of Physics, Oxford, and V. A. Battzy, I.A., ~— Queen’s College, Oxford *. N the March number of the Philosophical Magazine we gave the results of experiments on the motion of electrons in pure argon which showed that the mean free path of the electrons is much greater in argon than in nitrogen or hydrogen, and that the loss of energy of an electron due to a collision is much less in argon than in these two gases, for velocities of agitation of the electrons between 8°5 x 10‘ and 14x 10‘ cm. per second. Within this range of velocities the free path in argon increases as the velocity of agitation diminishes, and it is of interest to find whether the free path continues to increase when further reductions are made in the velocity. With the same electric force and gas pressure the velocity of agitation of the electrons is much higher in argon than in the other gases, and it would be necessary to prepare a very large quantity of pure argon or to use extremely small electric forces in order to obtain small velocities of agitation in the pure gas. It is undesirable, for experimental reasons, to adopt either of these methods. . The velocity of agitation may, however, be reduced to very low values by adding hydroven to the argon, and the mean free paths of the electrons may be found for the mixture by determining the velocity in the direction of the electric force and the velocity of agitation, by the method we have already described. The mean free path in argon may thus be found in terms of the mean free path in hydrogen. Hixperiments on this principle were made with an improved apparatus which was first carefully tested with pure hydrogen admitted through a palladium tube. The velocities in hy- drogen which were obtained, agreed with those already given within experimental errors which may be taken as not exceeding 3 per cent. The results of some experiments with the mixture are given in the following table, Z being the electric force in volts per centimetre, W,, the velocity in cms. per second in the direction of the electric force, u, the velocity of agitation of electrons in pure hydrogen at 5:1 millimetres pressure, W,, and wm the corresponding velocities with the same electric force in the mixture consisting of hydrogen * Communicated by the Aut ors. 1128 Abnormally long Free Paths of Electrons in Argon. at 5°l millimetres pressure and argon at 20 millimetres pressure. Z. W,X10-5. | W,,x10-%. | «10-7, | «,,x1077, 17 21 TOA 5°15 53 8°5 151 14 au 4:2 41 10-9 10:2 3°2 31 The addition of the argon to the hydrogen does not reduce the velocity of agitation, under these conditions, by an — appreciable amount, as the figures in the last twe columns are the same within the limits of experimental errors. The proportion in which the velocity in the direction of the electric force is reduced, by adding the argon to the hydrogen, is small, and diminishes with the force. Since the velocities w are not appreciably different in the pure hydrogen and in the mixture, the ratio of the mean free path J, in the hydrogen to the mean free path l,, in the mixture is [;//m= Wz2/Wm. The mean free path (, in argon at 20 millimetres pressure in terms of J, is obtained from the relation 1/l,,=1/l,+ 1/la. Taking the results of the experiments with a force of 8°5 volts per centimetre it is seen that, with a velocity of agitation of 4°2 x 10’ em. per second, the mean free path in argon is about 60 times the mean free path in hydrogen at the same pressure. For this velocity the mean free path, previously obtained, in hydrogen at one millimetre pressure is 0°26 millimetre, so that the mean free path in argon at one millimetre pressure is about 15 millimetres, or fifty times the mean free path deduced from the atomic radius which is obtained from the viscosity. The experiments show that the reduction of the velocity W in hydrogen by adding argon having four times the pressure of the hydrogen, is too small to permit of an accurate deter- mination of the mean free path in argon. We intend to make further experiments on this principle using mixtures in which the pressure of the hydrogen is about 5 per cent. of that of the argon, and to determine the free paths for velocities of agitation much below 4:2 x 10’ cm. per second. 1129.4 CXX. Viscosity of Air ina Transverse Electric Field. By SATYENDRA Ray, W.Sc. Allahabad University, Research Student, University College, London *. UINCKE? found a change in the viscosity of liquids in an electric field. Spheres of crown and flint glass &c. lem. in diameter were suspended by fine silk fibres attached to the arms of a balance. The logarithmic decre- ment was observed with and without a field due to 2000 volts, the distance between the plates being 15 cm. ‘The difference, which he calls electric viscosity, was 0°0398 in the case of a crown glass sphere in liquid ether when the force was transverse to the motion. Parallel to the lines of force the viscosity also increases but to a smaller extent. Again in the case of gases the Faraday Mosotti theory of dielectrics is very well satisfied as has been tested by Boltz- mann ; and if polarization physically means an orientation of molecules or a kind of tidal distortion of the molecule, a change in the effective area of the cross section and a consequent change in the viscosity with the electric field might be expected. To test whether there is any change in the case of gases the transpiration method was employed. The usual capillary tube was replaced by two steel plates 2 feet long, 2 inches broad, and 1 inch thick. One face of each was planed and these plane faces were piaced parallel to and facing each » other. The plates were separated from each other by three bits of drawn glass thread nearly 1 cm. long on which the top plate was laid as on rollers. The sides were painted with molten resin and wax, this method being found the best to make the tube simultaneously electrically as well as pneum- atically tight. The tube thus formed by parallel plates with lateral walls was connected with the usual Mariotte’s bottle arrangement by means of a tin box fitted on one end of the tube with wax and resin. Paraffin oil was used in the Mariotte’s bottle in the first instance, and the time of trans- piration of a definite volume under constant pressure was noted with and without the electric field. Latterly, however, to eliminate complications arising from the use of paraffin oil, dry air was sucked through instead. For this purpose a second box was attached to the other end of the tube and connected up with a sulphuric acid wash-bottle through which the air was drawn in. The time for running out * Communicated by Prof. A. W. Porter, I. R.S. ¥ Quincke, Ann. d. Phys. u. Chem. \xii. (1897). 1130 Notices respecting New Books. water between two marks on the aspirator gauge (and after- wards for running out a definite volume of water, equal to 800 c.c. nearly), starting from the same initial conditions, was taken with and without the field. Assuming the motion to be non-turbulent, which we reasonably can do, the viscosity must vary directly as the time of outflow. In the experiment a potential difference of 580 volts between plates -31 mm. apart did not produce a change of even 1 sec. in 9 min. 39 sec. The potential gradient was 18000 volts per cm. nearly, the sparking potential gradient being 31000 volts per cm. In setting an upper limit to a possible change in the viscosity due to the electric field, attention must be paid to the ‘ resistance ” of the other parts of the channel besides the tube. The time of flow without the field was to the time of flow with it as 4°5: 10 nearly. The resistance of the tube may therefore be taken to be (10—4:°5)/10 of the total resistance. The present experiment may therefore be inter- preted to prove the absence of any change in the viscosity of air greater than or equal in amount to °3 per cent. I am grateful to Prof. A. W. Porter for his kind interest in the experiment and for his valuable suggestions. Ist May, 1922. CXXI. Notices respecting New Books. Multilinear Functions of Direction and their uses in differential geometry. By H. H. Neviune, late Fellow of Trinity College, Cambridge, Professor of Mathematics in University College, Reading. (Cambridge: At the University Press. 8s. 6d. net. 1921.) N this small book Prof. Neville has given a systematic discus- sion of functions of several independent directions. In the course of the development of ideas which are to some extent generalizations of the ideas of differential geometry, a number of propositions which actually occur in modern differential geometry are included. The way in which these theorems are involved in the wider theory which is here introduced, gives an insight into the co-ordinating power of the more comprehensive lines of development which the book describes. The exposition is throughout clear and interesting: The sub- ject matter provides one more example of the way in which mathematics covers the domains of abstract thought with ideas of ever increasing generality. Notices respecting New Books. 1131 Introduction to the Mathematical Theory of the Conduction of Heat wm Solids. By H.S. Carstaw, Professor of Mathematics in the University of Sydney, formerly Fellow of Emmanuel College, Cambridge. (Macmillan & Co. 2nd Edition, 30s. net. 1921.) THs first edition of the work, entitled ‘ Fourier’s Series and Integrals and the Mathematical Theory of the Conduction of Heat,’ was first published in 1906. A new edition of the part dealing with Fourier’s Series and Integrals was issued early in 1921, and the present volume forms a new and revised account ot the part dealing with the Conduction of Heat. There are various additions to the subject matter, including a discussion of all the important boundary conditions associated with the Hqua- tion of Conduction, which is of interest also to those who are concerned with the application of modern analysis to the solution of the differential equations of mathematical physics. We also remark the inclusion of new matter in the chapters which deal with the sphere and cone, the circular cylinder, and the chapter about Green’s theorem, amongst others. Beyond the many minor improvements which make this new issue of the work of 1906 of added value, the considerable addi- tions and the inclusion of a large amount of new material make it of real importance as a new work. The Aggregation and Flow of Solids. By Sir Gnorcr Brinpy. Pp. xv + 256. Macmillan. Price 20s. net. In this book Sir George Beilby has collected together the very interesting work which he has carried out during the past twenty years on the coexistence of the amorphous and the crys- talline states in solids, and the relative influence of the two modifications on the mechanical and other physical properties of metals. He has worked away from the beaten track: the pro- blems which he attacks receive little attention in the ordinary text-books on the properties of matter, although they are of great interest to the physicist as well as to the metallurgist. The study of the optical properties of thin metal leaf, to which he contributes some instructive experiments, has received little attention since the time of Faraday. His careful work on the hardening by wire-drawing, and the effect of annealing hard-drawn metals at different temperatures—work which incidentally brings out a close parallelism between the thermoelectric force against a standard metal and the mechanical strength of the metal in its different modifications, and so indicates the importance of the thermoelectric force as an index of the change of state—simplifies a complex problem of considerable practical importance which was long neglected in this country. The extreme hardness of the amor- phous phase and the effect of dissolved gases on the crystalline 1132 Notices respecting New Books. state are noteworthy facts which are well brought out by ex- periment. Sir George Beilby’s best known work is, perhaps, that on the nature of polish, in which he showed that the effect of polishing is to flow a thin layer of hard amorphous material over the otherwise crystalline surface of the solid. His beautiful coloured photo- graphs showing how a film of the amorphous modification, thin enough to be transparent, can be actually made to flow over empty pits in the surface of copper by the precess of polishing are reproduced. The depth of the disturbance produced by polishing, the increased hardness of the surface, and the transmission of the directive action of the crystals through a layer of non-crystalline material were all investigated, and the whole question of polish was put in order. Two other sections of the book deal with the flow of rocks and ice, and the phosphorescence of crystals respec- tively. The latter shows. a certain disregard of work which had been previously carried out by Lenard and others. The book is beautifully printed and produced, and adorned with over a liundred reproductions of microphotographs printed as thirty-four plates at the end of the book. It is very moderately priced as books go nowadays, and we congratulate Sir George ri, on having brought together his work in so attractive a orm. The Physical Properties of Colloidal Solutions. By EH. F. Burron. Pp. viit+ 221. Second Edition. Longmans. Price 12s. 6d. THE second edition of Professor Burton’s book contains some twenty pages more than the first, which appeared in 1916. The chapter on Coagulation of Colloids has been rewritten and greatly extended : it now contains a table showing the relative coagulating power of different ions. Changes and additions have been made in other parts of the book, one of the most interesting being the discussion of limitations of Perrin’s distribution law, which is known not to hold for large depths; in such cases there is a limiting concentration. By taking into account the charge on the colloidal particles a correction term is introduced. Perrin’s original formula was applicable, however, to such conditions as he employed, and his numerical results are not questioned. The book is warmly recommended. Series Spectra. By Norman R. CaMppett. Pp. vii + 110. Cambridge University Press. Price 10s. 6d. net. THE second edition of Dr. Campbell’s well-known ‘ Modern Electrical Theory,’ which appeared early in 1913, is now necessarily largely out of date—the theories described are now no longer, in many cases, modern. The author has decided not to issue a fresh Notices respecting New Books. 1133 edition, but to supplement the book by a series of monographs dealing with the branches of the subject which have been most affected by the work of the past nine years. The modern electrical theory of spectra has been entirely developed during this period, so that Dr. Campbell has naturally decided to devote his first monograph to that subject. There were fourteen chapters in his book ; consequently the book before us is entitled Modern Electrical Theory, Chapter XV. Dr. Campbell’s book is really needed, and he has acquitted himself well of the very difficult task before him—to describe in some hundred pages the experimentally established regularities of series spectra, and the explanation of them on the lines first in- dicated by Bohr, when he showed how the series formule for hydrogen and helium could be derived by an application of the quantum thecry to Rutherford’s nucleus atom. Bohr’s fundamental theory of stable non-radiating orbits is well-known to readers of this magazine, where the papers de- scribing it were first published; it has been considerably extended by Sommerfeld, Kossel, and especially by Bohr himself. Dr, Camp- bell begins by deseribing briefly the properties of series, laying particular stress on the Principle of Combination, which, of course, finds so simple an explanation on Bohr’s theory, and indicating the bearing of the ionization potential. After having outlined, then, the fundamental facts which call for explanation he gives an account of Bohr’s early theory, with circular orbits, in all its applications. He next deals with conditionally periodic orbits, the Zeeman and the Stark effect, and Sommerfeld’s explanation of the fine structure of the hydrogen and helium lines by applying the relativity correction to the motion in the orbits. After this he passes to the question of intensity of the lines, and treats of the correspondence principle, which has assumed such great importance in Bohr’s latest work, which bad been only hinted at when the book was published. The book closes with a brief discussion of band spectra, the study of which may be anticipated to give im- portant results in the near future. In general the treatment is excellent. The discussion of the experimental facts might have been a little fuller, but the reader seeking more information can now turn to Professor Fowler’s ‘Report on Series in Line Spectra,’ which has just appeared, and should be read in conjunction with Dr. Campbell’s book. The correspondence principle is not as clearly expounded as could be wished, which is regrettable in view of its increasing application. We note with sorrow a large number of misprints, of which, perhaps, the most serious and inexplicable is 6:17 x 10! printed instead of 8:42 x 10° as the frequency for mercury vapour on p. 24. A statement that there is only one electron in the inner ring is an obvious slip. ‘The positive merits of the book undoubtedly out- weigh very heavily any minor defects, and if it does not achieve 1134 Notices respecting New Books. the extensive circulation which it certainly merits, the rather high price set on it by the publishers rather than any fault of the author’s will be the cause. Moderne Magnetik. By Frrix AvERBacH. Pp. viii+ 304. Leipsig: J. A. Barth, 1921. Price 55 M. THis book gives a remarkably comprehensive account of magnetism in all its bearings. The properties of magnets and ferromagnetic bodies, and of the various methods employed for measuring the strength of magnetic fluids and the permeability of specimens, are expounded in a clear and logical manner, all practical experimental devices, including Lenard’s bismuth spiral, being described, if only in outline. ‘This is the ground covered in more detail in Ewing’s well-known book, but much matter which has appeared since the date of that work finds reference here. In addition we have chapters devoted to the effect of mechanical deformation (tension, torsion, and so on) on magnetic properties, to magneto-optics, to recent theories, and to phenomena connected with terrestrial magnetism. The Jast mentioned contains an all too brief account of Birkeland’s and Stormer’s beautiful work on the aurora, with a few of their pictures. The modern electron theories of magnetism of Gans, Langevin, and Weiss are ably expounded, but no reference is made to the bearing of Bohr’s work on the Zeeman effect, Voigt’s theory of the effect, which has had no outstanding success, being given instead. It will be seen that a great deal of modern work, which has not yet found its way into the ordinary English text-books, has been compressed inte this book. On account of the very wide scope no detailed treatment is possible, and intriguing phenomena, such as those exhibited by the Heusler alloys, are dismissed in a couple of pages. This is inevitable in a book of this size, and is not mentioned asa hostile criticism. Considering the limits which the author has set himself, everything is very clearly treated. The book is written in a pleasant, human, and picturesque style, with considerably more liveliness than a German scientist usually allows himself. There is an amusing digression on text-books at the beginning of Chapter IX. We commend this book, and think that many readers of the Philosophical Magazine will find it useful. iy, Lets oye| XXII. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 640. ] December 21st, 1921.—Mr. R. D. Oldham, F.R.S., President, in the Chair. HE following communications were read :— 1. ‘The Nature and Origin of the Phocene Deposits of the County of Cornwall and their bearing on the Pliocene Geography of the South-West of England.’ By Henry Brewer Milner, M.A., DAE. 2-G-S. The author discusses the petrography of those Tertiary deposits of Cornwall which, by reason of similar mode of occurrence and lithological resemblance to the well-known fossiliferous beds of St. Erth, have been provisionally assigned to the Pliocene Period. Such deposits occur at St. Agnes, St. Erth, Lelant Downs, Polerebo, and St. Keverne; except those of St. Erth, all are unfossiliferous. By petrographic methods of correlation, using St. Erth as a type-locality, the relative age of the St. Agnes and St. Keverne beds is established. At Lelant Downs no deposit occurs zn situ, but scattered blocks of an indurated ferruginous grit are found at an elevation corresponding to that of the St. Erth beds on the opposite side of the valley, and a potential relationship is accord- ingly admitted. The Polerebo gravels differ from the other deposits, and are referred to a later period. The average composition of the St. Agnes, St. Erth, and St. Keverne deposits is shown to be substantially the same, the important ‘heavy’ minerals common to all being magnetite, ilmenite, tourmaline, staurolite, epidote, zircon, anatase, rutile, topaz, andalusite, and cassiterite; garnet was found only at St. Erth, kyanite at St. Agnes and St. Erth. On this basis, correlation of the deposits is effected by considering (1) the frequency of occurrence of individual species ; (2) their persistence or distribution; and (3) the constancy of crystallographical, physical, and optical properties of grains of the same mineral, wherever met with. The source of the material is essentially local, although kyanite and staurolite are ‘foreign’ species; the former, from its striking Lower Greensand habit and absence from St. Keverne, is thought to be derived from the north-east, the latter from a land- mass lying on the south, probably united to Cornwall and 1136 : Geological Society. Devon in late Miocene times, before early Pliocene submergence set in. The gradual ‘swamping’ of sediment-bearing rivers by the advancing Phocene sea from the south-west is correlated with certain physical features apparent, especially the ‘ 400-foot plateau.’ 2. ‘The Phosphate Deposit of Ocean Island.’ By Launcelot Owen, A.R.S.M., A.R.C.S., F.G.S. | Ocean Island (lat. 0° 52’ S.; long. 169° 32’ E.), in the Western Pacific Ocean, consists of a mass of terraced and dolomitized coral-limestone which rises to a height of 300 feet above low water, spring tide. | Its surface is almost completely covered by a capping of calcium phosphate of exceptional purity. This phosphate can be divided into three varieties:—(a) amorphous calcium phosphate, formed of the insoluble residue of the original guano; (6) detrital coral- limestone, converted into calcium phosphate by solutions leached from the guano; and (c) phosphatized coral 7m situ. The dolomitized limestone base shows evidence of having suffered considerable subaérial and marine erosion prior to the deposition of the guano, this deposition having occurred during the final emergence of the island. A detailed study of the composition of the deposit shows that the percentage of tricalcium phosphate (Ca,P,O,) at any point varies in a remarkably regular manner, according to the position of the point on the island. So regular is this variation, in fact, that, if the position and height above sea-level of any point are known, the percentage of tricalctum phosphate in the deposit at that point can be foretold with considerable accuracy. The trend of this variation suggests that:—(q@) the originai guano was deposited on the coral base while a slow negative move- ment of the strand-line was in progress, and no sensible break occurred either in the deposition of the guano, or in the movement of emergence ; (4) subsequent to the deposition of the guano and its conversion into phosphate, the island was tilted at about a third of a degree south-south-eatwards. This supposition is strengthened by the occurrence of raised beaches in the north of the island, and of phosphate zn sztw below sea-level on the south. A study of similar deposits may help to throw some light on the post-Tertiary movements of the floor of the Pacific Ocean. Cosrmr. Phil. Mag. Ser. 6, Vol. 43. Pl. XXIII. EGS 3: Rh L-spectrum (gypsum crystal). Rb La, and L6G; with their satellites (gypsum crystal). Fie. 4. Aluminium No 7M Aluminium V Wave-/ength 4X 1973,48XU 4X 1484, 52 / xX 6090 2X 35063, 3 4X/537,30 4X/54/, 16 5X/24/, SIS IONOD IS ISIS Wi 2/917. W (2) 6X 1095, 53 W /xX 6750, W Land M spectrum (gypsum crystal). WHIDDINGTON. Low Velocity Phil. Mag. Ser. 6, Vol. 43, Pl. XXIV. High Velocity. E listen INDEX to VOL. XLITII.. ——<-——- ADDENBROOKE (G. L.) on the leyden-jar with movable coatings, 489. Aether, on simultaneity and the, 528. 3 Air, on the convection of light in, 447; on the viscosity of, in a trans- verse electric field, 1129. Alpha-particles, on the number of, emitted per second by thorium CO, 1038. Alpha-ray bombardment, on induced radioactivity from, 9388. Alpha rays, on active modifications of hydrogen and nitrogen produced by, 455. Ammonia, on the specific heat of, 369. Anderson (Prof. A.) on scalar and vector potentials due to moving electric charges, 131. Anemometer, on the thermometric, 688. - Appleton (E. V.) on a type of os- cillation- hysteresis in a simple triode generator, 177; ona graphi- cal solution of some differential equations, 214. Are, on the eccurrence of spark lines, in the, 287, 834. Argon, on the motion of electrons in, 593, 1127. Ashworth (Dr. J. R.) on the theory of the intrinsic field of a magnet, 401. Aston (Dr. F. W.) on the mass- spectrograph, 514. Atomic number, on beta rays and, 393 ; on stopping power and, 477. structure, on the theory of, 1070. Atoms and molecules, on, 886, 1025. Bailey (V. A.) on the motion of elec- trons in argon, 593, 1127; on a development of Maxwell’s capacity bridge, 1107. Bars, on the transverse vibrations of, , 258, June 1922. Beating tones of overblown organ pipes, on the, 72. Benzophenone, on the latent heat of fusion of, 436, Beta rays and atomic number, on, 393. Biaxial crystals, on a new optical property of, 510. Bickerdike (C. F.) on the interaction between radiation and electrons, 1064. Birkhoff (Prof. G. D.) on circular plates of variable thickness, 953. Bohr (Prof. N.) on the selection principle of the quantum theory, 1112. Books, new :—Kaye & Laby’s Tables of Physical and Chemical Con- stants, 237; Carslaw’s Introduc- tion to the Theory of Fourier’s Series and Integrals, 238; Bro- detsky’s The Mechanical Prin- ciples of the Aeroplane, 240; Mallik’s Optical Theories, 240; Geddes’ Meteorology, 398; Cun- ningham’s Relativity, the Electron Theory, and Gravitation, 399; Lémeray’s Legons Elémentaires sur la Gravitation d’aprés la Théorie d’Einstein, 399; John- stone’s The Mechanism of Life, 399; Richardson’s The Emission of Electricity from Hot Bodies, 400; Bibliotheca Chemico-Mathe- matica, 400; Edwards’ Treatise on the Integral Calculus, 636; Neville’s Multilinear Functions of Direction, 1130; Carslaw’s Intro- duction to the Mathematical Theory of the Conduction of Heat in Solids, 1131; Beilby’s The Ageregation and Flow of Solids, 1131; Burton’s The Physical Properties of Colloidal Solutions, 1132; Campbell’s Series Spectra, 1132; Auerbach’s Mederne Mag- netik, 1134. Boron, on X-rays from, 145. Bowen (E. J.) on an attempt to separate the isotopes of chlorine 430, 4D) 1138 3 INDEX. Breit (Dr. G.) on the effective capacity of multilayer coils, 963. Brewer (H. B.) on the pliocene deposits of Cornwall, 1185. Bridge, on a development of Max- well’s capacity, 1107. Bridges, on the forced vibrations of, 1018. Bromwich (Dr. T. J. Va.) on kinetic stability, 70. Bronsted (Prof. J. N.) on the separa- tion of the isotopes of mercury, ol. Broughall (L. St.C.) on the frequency of the electrons in the neon atom, 339. Buckman (S. 8.) on Jurassic strata near Hype’s Mouth, 639. Burbidge (P. W.) on the absorption of the K X-rays of silver in gases, 381; on the absorption of narrow X-ray beams, 389. c, e, and h, on the relationship hetween, 698. Cadmium, on the spectra of, 858. Campbell (Dr. N.) on the funda- mental principles of scientfic in- quiry, 397. Cant (H. J.) on the specific heats of ammonia, sulphur dioxide, and carbon dioxide, 369. Capacity, on the effective, of multi- layer coils, 963. bridge, on a development of Maxwell’s, 110. Carbon, on X-rays from, 145. dioxide, on the specific heat of, 569. Carrington (H.) on Young’s modulus and Poisson’s ratio for spruce, 871. Caustic ‘curves, on a method of tracing, 258. Cavanagh (B.A. M.) on molecular thermodynamics, 606. Chemistry, on the application of the electron theory of, to solids, 721. Chlorine, on an attempt to separate the isotopes of, 450. Chuckerbutti (B. N.) on the rings and brushes observed through a - spath hemitrope, 560. Circuits, on a mechanical illustration of three maguetically coupled os- cillating, 575. Circular plates of variable thickness, on, 953, Close (L. J.) on Hertz’s theory of _ the contact of elastic bodies, 320. Cochlea, on the analysis of sound waves by the, 349. Coils, on the effective capacity of, 963. Compton (Prof. K. T.) on ionization by cumulative action, 531. Conduction, on the form of the poe wave spreading by, 309, Contact potential difference and thermionic emission, on, 557. Convection, on the forced, from a pair of heated wires, 277. Convective cooling of wires, on the natural, 329. Coster (D.) on the spectra of X-rays and the theory of atomic structure, 1070. Coupled circuits, on, 575. vibrations by means of a double pendulum, on,.567. Crehore (Dr. A. C.) on atoms and molecules, 886; on the hydrogen molecule, 1025. Crystallization, on the probability of spontaneous, of supercooled liquids, 78. Crystals, on rdéntgenograms of strained, 204; on a new optical property of biaxial, 510; on the yings and brushes in, 560; on the application of the electron theory of chemistry to, 721; on the re- flexion of X-rays from imperfect, 800. Currents, on the mutual induction between two circular, 604. Curvature of the world, on the sig- nificance of Kinstein’s gravitational! equations in terms of the, 174. Dale (J. B.) on the analysis of micro- seismograms, 463. Darwin (C. G.) on the theory of radiation, 641 ; on the reflexion of - X-rays from imperfect crystals, 800. Das (Prof. A. B.) on certain types of electric discharge, 216. Davies (Miss A. C.) on ionization by cumulative effects, 1020. Davis (A. H.) on the convective cooling of wires, 329. Davison (Dr. C.) on the diurnal periodicity of earthquakes, 878. Dielectric constants of esters at low temperatures, on the, 481. INDEX. Dielectrics, notes on, 489. Differential equations in wireless telegraphy, on a graphical solution of, 206. Disintegration, on mechanical, caused by positive ions, 226. Dispersion of light, on the scattering and, 829. Doi (U.} on the scattering and dis- persion of light, 829. Doublet, on the orbits in the field of a, 993. Dowling (J. J.) on the resistance of electrolytes at high frequencies, 537. Dyes, on the dependence of the fluorescence of, upon wave-length, 307. e, h, and ec, on the relationship be- tween, 698. Kar, on the analysis of sound waves in the human, 349. HKarthquakes, on the diurnal period- icity of, 878. Eddington (Prof. A. 8S.) on the sig- nificance of Einstein’s gravitational equations in terms of the curvature of the world, 174. Edgeworth (Prof. F. Y.) on the application of probabilities to the movement of gas molecules, 241. Hinstein’s gravitational equations, on the significance of, in terms of the curvature of the world, 174; spectral shift, on, 396; theory, on the -identical relations in. 600. Elastic bodies, on Hertz’s theory of the contact of, 320. —— plate, on the equations of equi- librium of an, under normal pressure, 97. Electric charges, on scalar and vector potentials due to moving, 131. discharge, on certain types of, 216; on the disappearance of gas in the, 914. field, on the viscosity of air in a transverse, 1129. properties, on the field of a mag- net and its, 401. Electrolytes, on the resistance of, at high frequencies, 537; on the theory of, 625. Electron theory of chemistry, on the application of the, to solids, 721. 1139 Electrons, on the relative affinity of some gas molecules for, 229; on the frequency of the, in the neon atom, 339; on the motion of, in - argon, 0938, 1127; on the inter- action between radiation and, 1064; on X-ray, 1116. Iinhanced lines, on the occurrence of, in the are, 287, 834. Listers, on the dielectric constants of some, at low temperatures, 481. Ewing (Sir J. A.) on a new model of ferromagnetic induction, 493. Eye, on the minimum time necessary « to affect the human, 345. Fairbourne (A.) on restricted move- ments of molecules at low pres- sures, 1047. Ferromagnetic induction, on a new model of, 493. Field, on the intrinsic, of a magnet, 401. Fizeau effect, on the, 447. Fluorescence, on the dependence of the intensity of, upon wave-length, 307; on, and photo-chemistry, 757. Focal length, on a relationship be- tween, and the number of fringes in convergent polarized light, 766. Foote (Dr. P. D.) on an exception to the principle of selection, 659. Fowler (f. H.) on some problems of the mass-spectrograph, 514; on the kinetic theory of gases, 785. Franklin’s experiment on the leyden- jar with movable coatings, on,489. Fusion, on latent heats of, 436. Gases, on the effect of, on contact difference of potential, 162; on the effect of variable specific heat on the discharge of, through noz- zles, 589; on the kinetic theory of, 785; on the disappearance of, in the electric discharge, 914. Gas-molecules, on the relative af- finity cf some, for electrons, 229 ; on the application of probabilities to the movement of, 241. General Electric Company’s Re- search Staff on the disappearance of gas in the electric discharge, 914. Geological Society, proceedings of the, 639, 1135. Glasson (Dr. J. L.) on beta rays and atomic number, 393; on stopping power and atomic number, 477. 1140 INDEX. Gramont (Comte A. de) on the occurrence of spark lines in the arc, 287, &34. Graphical solution of some differen- tial equations, on the, 214. synthesis of the linear vector function, on a, 545. Gravitation, note on, 1388. Gravitational equations, on the significance of Kinstein’s, in terms of the curvature of the world, 174. field, on the deflexion of a ray of light in the solar, 580; on the equations of the, 600; on the de- rivation of the symmetrical, 19. h, c, and e, on the relationship between, 698. Harmonics, on zonal, of the second type, 1. Hartley (H.) on the spontaneous crystallization of supercooled liquids, 78; on an attempt to separate the isotopes of chlorine, 430. Hartung (i. J.) on the Steele-Grant microbalance, 1056. Heat, on the forced convection of, from a pair of heated wires, 277. Hemsalech (G. A.) on the occur- rence of spark lines in the are, 287, 834. Hertz’s theory of the contact of elastic bodies, on, 320. Hevesy (Prof. G.) on the separation of the isotopes of mercury, 31. Hinshelwood: (C. N.) on the spon- taneous crystallization of super- cooled liquids, 78. Horton (Pref. F.) on ionization by cumulative effects, 1020. Hughes (Prof. A. LI.) on X-rays from boron and carbon, 145. Huyghens’s principle, on retarded potentials and, 1014. Hydrogen, on the effect of, on con-, tact difference of potential, 162; on active modifications of, pro- duced by alpha rays, 455. molecule, on the, 1025. Hysteresis, on oscillation-, in triode ~generators, 177,700. Identical relations in instein’s theory, on the, 600. Induction, on a new model of ferro- magnetic, 493; on the mutual, between two circular currents, 604. Insulation of wires in platiniim resistance thermometers, on the, 95. Interference fringes, on a relation- ship between focal length and, 766. Interferometer method of deter- mining the phase difference from metallic reflexion, on an, 471. Ionization by cumulative action, on, 531, 1020. Ions, on mechanical disintegration caused by positive, 226. Tron, on the magnetic field of, 401. Tselober on methods of separating, 3 Jackson (L. C.) on the dielectric constants ef esters, 481. Jeffery (Dr. G. B.) on the identical relations in Einstein’s theory, 600, Jeifreys (Dr. H.) on fundamental principles of scientific inquiry, 398. Joffe (A. F.) on réntgenograms of strained crystals, 204. Jones (Prof. E. T.) on the tempera- ture wave spreading by conduction from point and spherical sources, 309. K X-rays of silver, on the absorp- tion of, in gases, 381. Kennard (Dr. E. H.) on retarded potentials and Huyghens’s prin- ciple, 1014. Kinetic stability, on, 70. . theory of solids, on the, 672, 683; of gases, on the, 785. Kirpitcheva (M. V.) on roéntgeno- grams of strained crystals, 204. Krishnaiyar (N. C.) on the ampli- tude of vibrations maintained by forces of double frequency, 503. Latent heats of fusion, on, 436. Lead, on the are spectra of, 287. Legendre functions, on products of, 768. Leyden-jar with movable coatings, on the, 489. Lidstone (F. M.) on the measure- ment of absolute viscosity, 354, 1024. Light, on the time necessary for, to excite the retina, 345; on the convection of, in moving gases, 447; on the scattering and dis- persion of, 829; on the deflexion of a ray of, in the solar grayita- tional field, 580. INDEX. Linear vector function, on a graphi- cal synthesis of the, 545. Liquids, on the probability of spon- taneous crystallization of super- cooled, 78. Loeb (Dr. L. B.) on the relative affinity of gas molecules for elec- trons, 229. Lough (V.) on the beating tones of overblown organ pipes, 72. McLeod. (A. BR.) on the lags of thermometers, 49. Magnesium, on the spectra of, 851. Magnet, on the intrinsic field of a, Oe Magnetically coupled oscillating circuits, on a mechanical illustra- tion of, 575. Mallik (Prof. D. N.) on certain types of electric discharge, 216; on the mutual induction between two circular currents, 604. Manley (J. J.) on the insulation of wires in platinum resistance thermometers, 95. Mass-spectrograph, on the, 514. Maxwell’s capacity bridge, on a development of, 1107. Meggers (Dr. W. F.) on an excep- tion to the principle of selection, 659. Mercury, on the separation of the isotopes of, 31. vapour, on the effect of, on contact difference of potential, 162. Merton (Dr. T. R.) on an attempt to separate the isotopes of chlo- rine, 4380. Metallic reflexion, on the phase difference from, 471. Metals, on the effect of gases on the contact difference of potential be- tween, 162; on the kinetie theory of, 672, 685. Microbalance, on the Steele-Grant, 1056. Micrometer, on the application of the ultra-, to the measurement of small increments of temperature, 223. Microscope objectives, on a relation- ship between the focal length of, and number of fringes, 766. Microseismograms, on the analysis of, 463. Mohler (Dr. F. L.) on an exception to the principle of selection, 659. 1141 Molecular thermodynamics, on, 606. Molecules, on atoms and, 886, 1025; on restricted movements of, at low pressures, 1047. Morgan (J. D.) on the temperature wave spreading by conduction from point and spherical sources, 359. Morton (Prof. W. B.) on Hertz’s theory of the contact of elastic bodies, 320. Mosharrafa (A. M.) on unsymmetri- cal components in the Stark effect, 943. Multilayer coils, on the effective capacity of, 968. Murnaghan (Prof. F. D.) on sym- metrical gravitational fields, 19; on the deflexion of a ray of light in the solar gravitational field, 580. Narayan (Prof. A. L.) on coupled vibrations by a double pendulum, 567 ; on a mechanical illustration of three magnetically coupled os- cillating circuits, 575; on a double slit spectrophotometer, 662; on the surface tension of soap solutions, 663. Neon atom, on the frequency of the electrons in the, 339. Newegass (G. A.) on a physical inter- pretation of Lewis and Adams’ relationship between h, c, and e, 698. Newman (Prof. IF. H.) on active modifications of hydrogen and nitrogen produced by alpha rays, 455. Nicholson (Dr. J. W.) on zonal har- monics of the second type, 1; on products of Legendre functions, 768. Nickel, on the magnetic field of, 401. Nitrogen, on the relative affinity of molecules of, for electrons, 229; on active modifications of, produced by alpha rays, 455, Optical property of biaxial crystals, on a new, O10. and thermodynamic fields of radiation, on the connexion be- tween the, 641. Orbits in the field of a doublet, on the, 993. Organ pipes, on the beating tones of, 72. 1142 INDEX. Oscillation-hysteresis, on, in triode generators, 177, 700. Owen (L.) on the phosphate deposiis of Ocean Island, 1156. Parallels, on the theory of, 401. Partington (Prof. J. R.) on the specific heats of ammonia, sulphur dioxide, and carbon dioxide, 869; on latent heats of fusion of benzo- phenone, phenol, and sulphur, 436. Pendulum, on coupled vibrations by means of a double, 567. Percival (A. 8.) on a method of tracing caustic curves, 258. Phase difference from metallic re- flexion, on the, 471. Phenol, on the latent heat of fusion of, 436. Phosphorus vapour, on the disap- pearance of gas in the electric dis- charge in the presence of, 914. Photo-chemistry, on fluorescence and, 757. Photo-electric effect of the radiation from boron and carbon, on the, 145. Plates, on the equations of equi- librium of elastic, under normal pressure, 97; on circular, of variable thickness, 953. Platinum, on the effect of gases on the contact difference of potential of, 162. - resistance thermometers, on the insulation of wires in, 95. Poisson’s ratio for spruce, on, 871. Pol (Dr. B. van der, jr.) on oscillation hysteresis in triode generators, 177, 700. Polarization phenomena in X-ray bulbs, on, 193. Ponder (A. O.) on an attempt to separate the isotopes of chlorine, 430. Poole (Dr. J. H. J.) on the minimum time necessary to excite the human retina, 345. Positive ions, on mechanical dis- integration caused by, 226. Potential difference, on contact, and thermionic emission, 162, 557. Potentials, on scalar and vector, due to moving electric charges, 131; on retarded, and IHuyghens’s principle, 1014. Prescott (Dr. J.) on the equations of equilibrium of an elastic plate, no Preston (Miss K.M.) on the resistance ¥ glee at high frequencies, Priestley (Prof. H. J.) on the Einstein spectral shift, 396. Probabilities, on the application of, s the movement of gas-molecules, 241. Projective and metrical scales, on the relation between the, 420. Quantum theory, on the selection principle of the, 1112. Radiant spectrum, on Brewster's, 357. Radiation, on the theory of, 641, 698 ; on the interaction between, and electrons, 1064, Radioactivity, on induced, from alpha-ray bombardment, 938. Raman (Prof. C. V.) on the pheno- menon of the radiant spectrum, 357; on the convection of light in moving gases, 447; on a new optical property of biaxial crystals, 510. tatner (S.) on polarization phe- nomena in X-ray bulbs, 193. Rawlins (F. I. G.) on a relationship between the focal length of objec- tives and the number of rings seen in convergent polarized light, 766. Ray (S.) on the viscosity of air in a transverse electric field, 1129. Reflexion, on an _ interferometer methed of imeasuring the phase difference from, 471; on the, of X-rays from imperfect crystals,800. Relativity, on the identical relations in the theory of, 600. Resistance of electrolytes at high frequencies, on the, 537. Retarded potentials and Huyghens’s principle, on, 1014. Retina, on the minimum time neces- sary to excite the human, 345. Richardson (Prof. O. W.) on gravyi- tation, 188; on the effect of gases on contact difference of potential, 162; on contact difference of potential and thermionic emission, 557. Rings and brushes observed in a spath hemitrope, on the, 560. Roaf (EH. E.) on the analysis of sound waves by the cochlea, 349. Robb (Dr. A. A.) on a graphical solution of some differential equa- tions, 206. = en a ee eS ae NE habe a eevee, ee ae SS a - ” EN-DE Robertson (F. 8.) on the effect of gases on contact difference of potential, 162; on contact differ- ence ot potential and thermionic emission, 557. Rontgenograms of strained crystals, on, 204. Sano (S.) on a_ thermodynamical theory of surface tension, 649. Satterly (J.) on the measurement of absolute viscosity, 1024. Seales, on the relation between the projective and metrical, 420. Schlundt (Prof. H.) on the number of alpha-particles emitted per second by thorium C, 10388. Scientific inquiry, on the funda- mental principles of, 397, 398. Selection principle, on the, 59, 1112. Sen (Prof. B. M.) on the Ikinetic theory of solids and the partition of thermal energy, 672, 683. Sethi (Dr. N. K.) on the convection of light in moving gases, 447, Shenstone (A. G.) on induced radio- activity resulting from alpha-ray bombardment, 938 ; on the number of alpha-particles emitted per second by thorium C, 1088. Silberstein (Dr. L.) on the relation between the projective and metrical scales, 420. Silver, on the absorption of the K X- rays of, in gases, 581. Simons (Dr. UL.) on an apparatus for observation of the gravitational electrostatic field, 143. Simultaneity and the ether, on, 528. Slate (Prof. F.) on a graphical syn- thesis of the linear vector function, 545. Soap solutions, on the surface tension of, 663. ; Solar gravitational field, on the de- flexion of a ray of light in the, 580. Solids, on the application of the electron theoryof chemistry to, 721. Solutions, on the thermodymics of, 606. Sound waves, on the analysis of, by the cochlea, 349. Spark lines, on the occurrence of, in the arc, 287, 834. Spath hemitrope, on the rings and brushes observed in a, 560. Specific heat, on the effect of variable, on the discharge of gases through nozzles, 589; on the, of ammonia, 1143 sulphur dioxide, and carbon dioxide, 369. Spectra, on the, of X-rays, 1070. Spectral lines, on the effect of an electric field upon, 943. shift, on the Einstein, 396. Spectrograph, on the mass-, 514. Spectrophotometer, on a double slit, 662. Spectrum, on Brewster’s radiant, 357. Spruce, on Young’s modulus and Poisson’s ratio for, 871. Stability, on kinetic, 70. Stansfield (J.) on banded precipitates of vivianite, 640. Stark effect, on unsymmetrical com- ponents in the, 945. Steele-Grant microbalance, on the, 1056. Stopping power and atomic number, on, 477. Stratton (Mis. K.) on latent heats of fusion of benzophenone, phenol, and sulphur, 4386. Subrahmanyam (G.) on the surface tension of soap solutions, 663. Sucksmith (W.) on the application of the ultra-micrometer to the measurement of small increments of temperature, 223. Sulphur, on the latent heat of fusion of, 436. dioxide, on the specific heat of, a69. Surface tension, on the thermo- dynamical theory of, 649; on the, of soap solutions, 663. Sutherland’s constant 8, on, 785. Synge (1. EH.) on simultaneity and the ether, 528. Tamma (V. 8.) on a new optical property of biaxial eryst:ls, 510. Temperature, on the application of the ultra-micrometer to the meas- urement of, 223. wave, on the form of the, spreading by conduction, 359. Thermal energy, on the partition of, 672, 683. properties, on the field of a magnet and its, 401. Thermionic emission, on contact potential difference and, 557. Thermodynamic and optical fields of radiation, on the connexion be- tween the, 641, —— theory of surface tension, on the, 649, nN RR RAE = 1144 UN D:EX. Thermodynamics, on molecular, 606; on the second law of, 1047. Thermometers, on the lags of, 49; on the insulation of wires in platinu resistance, 95. Thermometric anemometer, on the, ~ 688. . Thomas (Dr. J. S. G.) on the forced convection of heat from wires, 277; — on the thermometric anemometer, Usiee. Thomson (Sir J. J.) on the applica- tion of the electron theory of chemistry to solids, 721. © Thorium C, on the number of alpha- particles per second emitted by, 1038. , | Timoshenko (Prof. 8S. P.) on the transverse vibrations of bars, 125; on the forced vibrations of bridges, 1018; on the buckling of beams, 1023. Tin, on the are spectra of, 287. Townsend (Prof. J.S.) on the motion of electrons in argon, 593, 1127. Triode generators, on oscillation- hysteresis in, 177, 700. Tungsten, on the effect of gases on the contact difference-of potential of, 162. Ultra-micrometer, on the application of the, to the measurement of small increments of temperature, 225. Vavilov (S. I.) on the dependence of the intensity of the fluorescence of dyes upon the wave-length of the exciting light, 307. Vector function, on a_ graphical synthesis of the linear, 545. Vibrations, on the transverse, of bars, 125; on the amplitude of, main- tained by forces of double fre- quency, 503; on coupled, by means of a double pendulum, 567 ; on the forced, of bridges, 1018. Viscosity, on the measurement. of absolute, 354, 1024. Walker (Dr. W. J.) on the effect of ‘variable specific heat on the dis- charge of gases through orifices, = 5890 Waran (H. P.) on mechanical dis- integration caused by positive ions, _. 226; on an interferometer method “of determining phase difference from metallic reflexion, 471. Wave, on the form of the tempera- ~ ture, spreading by conduction, 309. Wheeler (Prof. R. V.) on the tem- perature wave spreading by con- duction from point and spherical sources, 309. Whiddington (Prof. R.) on polariza- tion phenomena in X-ray bulbs, 720; on X-ray electrons, 1116. Wireless telegraphy, on a graphical solution of differential equations in, 206. Wires, on the forced convection from heated, 277; on the natural con- vective cooling of, 329. Wood (Prof. R. W.) on fluorescence and photo-chemistry, 757. Wrinch (Dr. D.) on fundamental ' principles of scientific inquiry, 398; on the orbits in the field of a doublet, 993. . X-ray beams, on the absorption of narrow, 389. bulbs, on polarization pheno- mena in, 193. electrons, on, 1116, X-rays, on characteristic, from boron aud carbon, 145; on the absorption of the K, of silver in gases, 381; on the reflexion of, from imperfect crystals, 800; on the spectra of, and the theory.of atomic structure, 1070. Young’s modulus for spruce, on, $71. Zinc, on the spectra of, 858. Zonal harmonics of the second type, on, 1. END OF THE FORTY-THIRD VOLUME. Printed by Taytor and Francis, Red Lion Oourt, Fleet Street. re gene cere ee LA Ned aes i. a ai DAUM eee ii 3 9088 0120