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Ley: + ° : op: ‘ ™I oe 4 a "4 j ss hy heedte ta i vid hi abet ; ” ¢ “en meet “ J J j ah #9 seedy ‘ i + 4% is ae Ae hea ee Fo WE Ane ‘yt by fat cet 3 een a4 ret oh “: -e € ‘ ) ; ‘ avuLy vg { i yore idl be ‘ Ab be No hy hg 4 i! 6+ np ele ine ‘ gl 4 F u bom 7h hl bee baie hres be i? eat ye oie we nh tenet wad by ag Boe a) Gales eepatrn 4. © bbe Heeb OE pha i ptt ! | ee “LONDON, EDINBURGH, axn DUBLIN PH ILOSOPHICAL MAGAZINE a AND JOURNAL OF SCIENCE. SO == CONDUCTED BY - eee JOSEPH LODGE, D.Sc., LL.D., F.B.S. JOHN JOLY, M.A., DSe., MRS. F.GS. GEORGE CAREY FOSTER, B.A. LL.D., F.RS. AND WILLIAM FRANCIS, F.1.S. foes _ “Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster lior quia ex alienis libamus ut apes.” Jusr. Lies. Polit. lib. i. cap. 1. Not. +1 VOL. XXX.—SIXTH SERIES, JULY—DECEMBER 191503 R A. ‘ | g | eo LONDON: % - TAYLOR AND FRANCIS, RED LION COUR?, FLEHLT STREET, SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND OO., LD. SMITH AND SON, GLASGOW :—IIODGHS, FIGGIS. AND CO., DUBLIN;— AND YEUVE J. ROYVEAU, PARIS, i" “Meditationis 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 ry) AS 99 Tam vario motu, J, B. Pinelli ad Mazonium. - ALERE | FLAMMAM oP ee, OAC) etna POH AMA RAR ° e "Cen Anan e ° e ° CONTENTS OF VOL. XXX. (SIXTH SERIES). NUMBER CLXXV.—JULY¥ 1985. PAL Page Dr. G. C. Simpson on the Electricity of as Pre- TE ene Sa Gh Gey ays cry orale 4 Hin abs sip Miele Me 1 Mr. R. M. Deeley on the Theory of the Winds............ 13 Messrs. ©. C. Paterson and B. P. Dudding on the Estima- _ tion of High Temperatures by the Method of Colour Ne ce Uo tte ah i chai pin BN ee wo Ale d+ Messrs. C. C. Paterson and B. P. Dudding on the Unit of Seeandie-power in White Licht .......-....200-ee-ee005 63 Dr. C. V. Burton on the Scattering and Regular Reflexion of ipant by Gas Molecules.—Part U1....... 0... cceeeeces 87 Mr. G. H. Livens on the Number of Electrons concerned in PA ONOUCHOM Tl). ss. cien dd ves beets nn ctas ween s 105 Mr. G. H. Livens on the Electron Theory of Metallic Con- LE) eis. ee oie a chelsea gin cide Cok as 112 Prof. W. M. Thornton on the Electric Strength of Solid SIME Ce RR) ne ae 124 Prof. P. J. Daniell on the Coefficient of End-Correction.— AEE SO: BA ee > ree 137 Mr. G. von Kaufmann on the General Theory of Corre- sponding States, and the Thermodynamic State-Equation . 146 Dr. L. Silberstein on Radiation from an Electric Source. ELT Dial y Ua ited. ts a WS aie oid sas wa knw ole 163 Lord Rayleigh on the Resistance experienced by Small Plates Berens a iream' Or Pinid) fo. ls seis. eee e ee ce des 179 Mr. T. Harris on the Distribution of Electric Force in the Pieeeeee ab Low Pressureg! oie ee a es 182 Sir J.J. Thomson on Conduction of Electricity through Metais., 192 Mr. H. Pealing on an Anomalous Variation of the Rigidity of Phosphor Bronze when in the form of Strips ........ 203 Notices respecting New Books :— Dr. C. G. Knott’s Physics, an elementary text-book for MHRVEPSioy WIMSGS- EL AA el Mice ney bk cae valenas 208 Dr. H. Stanley Allen’s Photoelectricity: The Liberation ENICCOROMA RY MEM ies eke Ria alee dae aleals etek 208 IV CONTENTS OF VOL. XXX.—-—SIXTH SERIES. NUMBER CLXXVI.—AUGUST. Prof. F. Soddy and Miss A. F. R. Hitchins on the Relation between Uranium and Radium.—Part VI. The Life-period EL OMANI se) 12) CN UNSURE ca ee ie 209 Miss Jadwiga Szmidt (Petrograd) on the Excitation of y Rays TDSVRNB VA UPUViSS ps) sou'e a elas Ge ee Fee ACU aah el ese) Stee 220 Prot. E. Taylor Jones on the most Effective Primary Capacity tor Induction-coils and Tesla Coils os. 0.:.. 2. 20 224 Prof. E. Marsden and Mr. W. C. Lantsberry on the Passage ofa. Particles through Hydrogen.—Il... ......). ae 240 Mr. K. F. Nesturch on the Probability of Ionization and Radiation of Gas Molecules due to Collision with Electrons. 244 Prot. P: J. Daniell on the Coefficient of End-Correction.— Dr. A. C. Crehore on Construction of the Diamond with Theoretical Carbon Atoms (io eyo eee 25 Mr. E. H. Smart on the Third-Order Abeeuions of a Sym- metrical’ Optical instrament) We oo 270 Mr. W. G. Brown on Reflexion from a Moving Mirror .... 282 Prof. W..B. Morton and Mr. J. Vint on the “Pathe of ihe Particles in some cases of Motion of Frictionless Fluid in a Rotating Hinclosure., (Plates J: & IN.) ...... 22228 284 Mr. G. H. Livens on the Electron Theory of Metallic Con- GUNG ley gece ih REMAN RO AH DUAR NKWMN EAN ea o 287 Prof. O. W. Richardson on Metallic Conduction ........ 290) Prof. R. W. Wood: Experimental Determination of the Law of Reflexion of Gas Molecules. (Plate III.) ...... 300 Prof. W. H. Bragg on the Structure of the Spinel Group of Bieiasi tt: er PML Mo ietanah de DRO e) MUL OIN oc 305 Mr. é. EK. Mendenhall and Prof. R. W. Wood on the Effect of Klectrie and Magnetic Fields on the Emission Lines of Dolids.. CElate PV iyi ee ategniaa, a) csi ae ae Ne ee 316 NUMBER CLXXVIL--SEPTEM BER. Sir J. J. Thomson on the Mobility of Negative Ions at Low MPMSSTITOS, lei facidiye, 2 $3) 52 )ul oA Matin Oy 01 Sia lia Gite Oh eR Oe ae en aT 321 Lord Rayleigh on the Stability of the Simple Shearing Motion of a) Viscous Incompressible Plaid. os ae 329 Sir E. Rutherford, Prof. J. Barnes, and Mr. H. Richardson on Maximum Frequency of the X Rays from a Coolidge iobe ton Ditterent Voltages. Suwanee. a Ul ee 339 Sir EB. Rutherford and Prof. J. Barnes on the Efficiency of Production of X Rays from a Coolidge Tube ............ 361 Prof. J. Barnes on the High-Frequency Spectrum of Tungsten. 368 ee “CONTENTS OF VOL. XXX.——SIXTH SERIES. Dr. L. Silberstein on Mutual Electromagnetic Mass........ Prof. EH. P. Harrison and Mr. Sujit K. Chakravarti on the Temperature Coefficient of Young’s Modulus for Electrically EIRENE, oC ke cg wv es Mal lioeeld eo. b W alate ee Prof. Frank Horton on the Zeleny Eiectroscupe .......... : Prof. W. M.'Thornton on the Total. Radiation trom a Gaseous MRE a A SV MA, Land Zia SSIES tae RNS eri we ad Prof. W. G. Duttield: A Comparison cf the Are aud Spark Spectra of Nickel produced under Pressure; with a Note upon the Influence of Temperature and Density Gradients upon the Displacements of Spectrum Lines ............ é Dr. N. Bohr on the Quantum Theory of Radiation and the Pe ALON 00. se ca Selec ae ect ateleees Prof. J. C. McLennan and Mr. C. L. Treleaven on Residual Seene SE CGARER i. eis sp ls ek ele A aw a le es Prof. J. C. McLennan and Mr. H.G. Murray on the Residual Jonization in Air enclosed in a vessel of Ice. (Plate V.).. Mr. G. H. Livens on the Electron Theory of the Optical rere Metals oe oe oa ove nies aieen lee ed as Notices respecting New Books :-- Peete a os a's ba bee wiepetas Uva gem ne as Proceedings of the Geological Society :— Prof. T. McKenny Hughes on the Gravels of East Anglia Messrs. EH. M. Anderson and E. G. Radley on the Pitch- stones of Mull and their Genesis ................ Mr. H.S8. Shelton on the Radioactive Methods of Deter- amine Geolopieal Time... 1c ede ne ee ee weds NUMBER CLXXVIII.—OCTOBER. Mr. C. F. Meyer and Prof. R. W. Wood: A Further Study of the Fluorescence produced by Ultra-Schumann Rays. ER a at sade cana! A aan: dace, wie nial ane eels Ova Mr. E. Talbot Paris on the Polarization of Light scattered by Spherical Metal Particles of Dimensions comparable with RE IOTEDT TN yD IEA. a is. oh veld a ie ON viens ws id's a a lowe Prof. J. C. McLennan and Mr. Evan Edwards on the Ultra- violet Spectrum of Elementary Silicon ................ Prof. J. C. McLennan and Mr. D. A. Keys on the Mobilities pr hous 1 Bar ot Eich Presaures sii. hi ieee dea es : Prof. J. C. MeLennan and Mr. C. G. Found on the Delta Radiation emitted by Zine when bombarded by Alpha Rays. eee, Vile) ered thant. Ais,» aial dibicbic A Weiinhe, ie G eWeek wid y Mr. W. B. Haines on lonic Mobilities in Hydrogen and DY Lic iy, Palin Led litte bie uw ta caw lor ws ela N ta w lis vale v1 CONTENTS OF VOL, XXX.—SIXTH SERIES. Page Prof. E. C. 0. Baly on Light Absorption and Fluorescence.— ai Uh fey hoses bs ak ee ae Sola oo 510 Mr. G. H. Livens on the Electron Theory of the Hall Effect and Ailed Phenomena... ox .'< soporte oes Cs oe 526 Mr. G. H. Livens on the Electron Theory of Metallic Con- Miretvon, = Vieiic gia. a os ae eae ee ee 549 Dr. F. A. Lindemann on the Relation between the Life of Radioactive Substances and the Range of the Rays emitted. 560 Mr. H. G. Savidge and Prof. J. W. Nicholson on the Calcu- lation of Series in Spectra. pike... vs. 563 Mr. C. E. Weatherburn: Problems in Hlectrostatics and the Steady Flow of Hlectricity under the exponential potential Cr oi ORE i La 568 Dr. N. Bohr on the Decrease of Velocity of Swiftly Moving Electrified Particles in passing through Matter.......... 581 Dr. A. C. Crehore on Construction of Compound Molecules with Theoretical Atoms, especially the Systems of Growth of the Organic Compounds of Carbon and Hydrogen. (Plates VITI.—XIL.) . 0 tee pes sche ee ey Sener 613 Prof. O. Steels on a Simple Resonance Experiment ....... 623 Mr. W. F. Rawlinson on the Decrease in Velocity of G-particles in passing through Matter ................ 627 Dr. Allan Ferguson on the Drop-Weight Method for Deter- minine Surtace-Tensions) 2... i) ea eee nee 632 Dr. A. M. Tyndall on the Critical Field at a Discharging | ig 031 chee ig RAR » Gs7 Mr. G. Shearer on the Ionization of Hydrogen by X-Rays... 644 Prof. J. W. Nicholson on Mutual Electromagnetic Mass.... 659 Mr. J. Moran: A Comparison of Radium Standard Solutions. 660 ees NUMBER CLXXIX.—NOVEMBER. Mr. N. P. McCleland on the Electron Theory in Organic Ghemistry 2.5.4. ete ee ee ae sees 665 Prof. J. C. McLennan and Mr. H. V. Mercer on the Ionization Tracks of Alpha Rays in Hydrogen. (Plate XTII.)...... 676 Prof. J. C. McLennan and Mr. R. C. Dearle on the Infra-red Emission Spectrum of the Mercury Arc. (Plates XIV. BIXOVG, o ies la ote ta ese: easel lee o/c elec ee eRe een aoe =F ee 683 Prof. J. C. McLennan and Mr. Evan Edwards on the Absorp- tion Spectra of Mercury, Cadmium, and Zine Vapours. (elabe VE) oe oe oe eile ie RG 20 7 erm 695 Prof. C. V. Raman on Intermittent Vision .............. 701 Mr. A. B. Wood on the Velocities of the « Particles from Thorium Active Deposit. (Plate XVII.) .............. 702 Dr. J. R. Ashworth on the Application of Van der Waals’ Equation of State to Magnetism . ..........-..- 0005 Vil CONTENTS OF VOL. XXX,—-SIXTH SERIES. Prof. W. D. Harkins and Mr. E. D. Wilson on Energy Relations involved in the Formation of Complex Atoms . - Prof. W. M. Hicks on the Calculation of Series in Spectra. . Dr. C. Statescu on the Dispersion of Carbon Dioxide in the imma-seayivecion of the Spectrum ..........0.-00050-5 Dr, A. M, Tyndall on Ionic Mobilities in Hydrogen ga! @! (s7'6) 2 NUMBER CLXXX,—DECEMBER. Prof. C. G. Barkla and Mr. G. Shearer on the Velocity of Electrons expelled by X-rays Prof. P. J. Daniell on Rotation of Elastic Bodies and the emi ee ee PelALIVILY «0... ee ee eee bee e eee Dr. J. R. Wilton on the Solution of Certain Problems of emereeeristonnl Physics .......5..c. cern ee ck use ess Sir J. J. Thomson on a Method of Finding the Coefficients of Absorption of the Different Constituents of a Beam of Heterogeneous Rontgen Rays, or the Periods and Co- efficients of Damping of a Vibrating Dynamical System .. 7 Dr. L. Silberstein on Radiation from an Electric Source, and Tine Spectra.. (Third Paper.) (Plate XVIII.) ........ Mr. H. Smith: A Comparison of the Positive Rays with the Spectrum of the Positive Column in a Mixture of Helium cas 6 iain 4b ain hip gid Raine oo ¥en gs ile Mr. A. B. Wood and Dr. W. Makower on the Recoil of fo) trom Radimm C .... 2... 1. ec eee eve eee ee Prof. R. R. Ramsey on the Variation of the Emanation MEP MCORGHIT SPTINOS ks ee cleat eee Notices respecting New Books :— Dr. T. P. Nunn’s The Teaching of Algebra; and Exer- cises in Algebra, including Trigonometry, Parts I., IT. Index Vli Page wt23 734 737 743 818 1. & LE. Hil. RV: VI. VIL. VITL-XI1. XIU. MLV. & KV. V1. PRAT ES: Illustrative of Prof. W. B. Morton and Mr. J. Vint’s Paper on the Paths of the Particles in some cases of Motion of Frictionless Fluid in a Rotating Enclosure. Hlustrative of Prof. R. W. Wood’s Paper on the Experi- mental Determination of the Law of Reflexion of Gas Molecules. lilustrative of Mr. C. E. Mendenhall and Prof. R. W. Wood’s Paper on the Effect of Electric and Magnetic Fields en the Emission Lines of Solids. . Dlustrative of Prof. J.C. McLennan and Mr. H. G. Murray’s Paper on the Residual Ionization in Air enclosed in a vessel of Ice. | Illustrative of Mr. C. F. Meyer and Prof. R. W. Wood's Paper on the-Fluorescence produced by Ultra-Schumann Kays. Illustrative of Prof. J. C. McLennan and Mr. C. G. Found’s Paper on the Delta Radiation emitted by Zine when bombarded by Alpha Rays. Illustrative of Dr. A. C. Crehore’s Paper on the Construction of Compound Molecules with Theoretical Atoms, especially the Systems of Growth of the Organic Compounds of Carbon and Hydrogen. Tustrative of Prof. J. C. McLennan and Mr. H. V. Mercer’s Paper on the Ionization Tracksof Alpha Rays in Hydrogen. . Illustrative of Prof. J. C. McLennan and Mr. R. C. Dearle’s Paper on the Infra-red Emission Spectrum of the Mercury Are. Hiustrative ef Prof. J.C. McLennan and Mr. i. Edwards’s Paper on the Absvrption Spectra of Mercury, Cadmium, and Zine Yapours. . Illustrative of Mr. A. B. Wood’s Paper on the Velocities of the « Particles from Thorium Active Deposit. . Illustrative of Dr. L. Silberstein’s Paper on Radiation from an Electric Source, and Line Spectra. ERRATA. Page 126, third footnote, for Descriptive read Disruptive. Page 129, fifth line after the table, for equation (4) read equation (8). Page 295: footnote {, for Phil. Mag. vol. xiii. p. 192 (1915) read Phil. Mag, vol. xxx. p. 192 (1915). THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF IGE £915: I. The Electricity of Atmospheric Precipitation. By G. C. Simpson, D.Sc.* ‘il recent years a great deal of work has been done with the object of determining the electrical state of atmospheric precipitation. The results have shown that the whole subject is a complicated one, and different observers do not agree in detail. There are, however, a few outstanding results which may now be taken as estab- lished, and it appears desirable to revise our ideas in the light of these new facts. The following are the results which may be taken as having been substantiated by all recent observers + :— A. Non-thunderstorm Rain. (Ger. Landregen.) (1) Rain is sometimes positively and sometimes negatively charged f. (2) About 90 per cent. of the rain is positively charged. (3) The normal potential gradient is nearly always reversed during the rain. * Communicated by the Author. + Ido not propose to discuss the results of individual observers, but give as an appendix a list of all the workers in this subject with refer- ences to their publications. ¢ In all probability rain has always some charge and uncharged rain practically never occurs; but this cannot be stated with certainty until more delicate methods of observation are devised. Phil. Mag. 8. 6. Vola, No. 175; July 1915. B 2 Dr. G. C. Simpson on the B. Thunderstorm precipitation. (4) The precipitation is sometimes positively and sometimes negatively charged. (5) More positive than negative electricity is brought down by the precipitation. (6) The charges per unit mass of the precipitation and the vertical electrical currents produced by its fall are much larger than with non-thunderstorm rain. (7) The potential gradient undergoes large and rapid changes of sign, and on the whole the potential gradient is more often reversed than not. C. Snow. (8) Snow is sometimes positively and sometimes negatively charged. | (9) In Simla positive electricity was in excess, while in Potsdam an excess of negative electricity was observed. (10) A given weight of snow may be more highly charged than the same amount of rain, even in a thunder- storm. . | (11) High values of the potential gradient, both positive and negative, occur during snowfall. In the above I have only stated results about which there is no difference of opinion ; there are many others which are almost as certainly true, but as there is still some doubt they have been excluded. So far as I know there are at present only two theories seriously put forward to account for the electricity of precipitation. (A) The “influence” theory, first put forward by Elster and Geitel in 1885 *, and revised by them in 1913 f. (B) The “breaking drop” theory, put forward by the writer in 1909 { to explain the origin of electricity in thunderstorms. I propose to consider each of these theories in the light of the established facts detailed above. * Elster & Geitel, Wied. Ann. xxy. pp. 123-124 (1885). + Elster & Geitel, Phys. Zeit. xiv. pp. 1287-1292 (1918). { Simpson, Phil. Trans. Roy. Soc. A. ccix. pp. 379-418 (1909). Electricity of Atmospheric Precipitation. 3 A. ELSTER AND GEITEL’S “‘ INFLUENCE ”’ THEORY. The influence theory is based on what Elster and Geitel ‘consider to be two facts : (a) That small drops, in particular cloud particles, rebound from large drops without uniting with them ; (b) That although the small drops do not unite with the larger ones, electrical contact takes place. Before going further it is necessary to examine these two premises, for it appears to me that they are not sufficiently well established to form the foundation of an important theory. There can be little doubt that (a) is true for pure un- charged water. Every one who has rowed on a river of clear water, or watched the drops falling into the basin of a fountain, will have seen small drops running on the surface for an appreciable time before they are absorbed. On the other hand, the great effect of an electrical charge in causing water-drops to unite has been observed by most physicists since it was first pointed out by Lord Rayleigh in 1879%*. In a thunderstorm the drops are so highly charged rela- tively to one another that it is difficult to believe that there -can be any large amount of impact without the drops uniting more or less completely. Indeed most writers have used this fact to account for the dark colour of the clouds and the heavy rain associated with thunderstorms. The second foundation of the theory, that electrical contact takes place although the drops do not unite, is based on experiments made by Elster and Geitel in 1885. As these experiments have never been repeated, while the results have been accepted without question for thirty years, it seems desirable to examine them in the light of modern knowledge. The experiments were shortly described by Elster and Geitel T, and on account of their importance a literal trans- lation is given here. “The following method was used to prove that small water-drops can come into electrical contact with a large water surface without uniting with it. ‘The metal nozzle of a ‘sprayer’ was directed at an angle towards an insulated water surface which was con- nected to a Thomson quadrant electrometer. Over the * Rayleigh, Proc. Roy. Soc. xxviii. p. 406 (1879). t Elster & Geitel, Wied. Ann. xxv. p. 129 (1885). B2 4 Dr. G. C. Simpson on the nozzle a cylindrical metal tube, 14 cm. wide, was fixed so that the orifice of the sprayer was in the middle of the tube. The sprayer and tube were connected to earth, thus pre= venting any electrical field acting on the nozzle and so charging the drops by influence. ‘“‘ By this arrangement a large proportion of the spray was reflected from the water surface. Above the spray- cloud was suspended an insulated sphere, of 2 cm. radius, which could be connected at will to the positive or negative pole of a Zamboni pile having 2000 pairs of plates. It was. then found that when the sprayer was in action the insulated vessel containing the reflecting water surface took on a charge which was of the same sign as the charge on the metal sphere. The charge was so large that it could not be read by the telescope and scale. At the same time the spray was found to carry a charge of the opposite sign. From the: latter fact we see that there was no transference of charge from sphere to surface through the spray-cloud. The spray particles bouncing from the water surface acted exactly like the drops in a Thompson ‘ water dropper’ and the vessel containing the water surface like the collector itself. This, however, was only possible if the bouncing drops came into electrical contact with the water surface.” Hlster and Geitel’s explanation of this experiment was the only one possible in 1885, but a much more reasonable one: is obvious now. As shown by Eve* and others, when water is sprayed in the manner described the air in the blast is highly ionized. Thus in the experiment the fine water particles were carried along in a stream of ionized air. When this stream came into the strong electrical field between the positively charged sphere and the water surface, the positive ions were driven into the latter, and part of the excess of negative ions became attached to the spray particles. The reverse happened when the sphere was charged nega-. tively. This explanation accounts for the facts without supposing any rebounding of water-drops. In fact the result would have been the same if all the drops could have been removed before the air-blast came into contact with the water surface. When one considers that the cause of a drop bounding from a water surface is the film of air between the two, it is. difficult to see how electrical contact could take place until this layer has been removed and the two surfaces joined up. In this connexion it may be pointed out that C. R. Englund + * Eve, Phil. Mag. [6] xiv. p. 382 (1907). t Englund, Phil. Mag. xxvii. pp. 457-458, March 1914. Electricity of Atmospheric Precipitation. 5 found that there was perfect insulation between two con- ductors only half a sodium wave-length apart. A theory built on such questionable foundations would not be considered strong even if it appeared capable of explaining the phenomena, but it will now be shown that, granting these premises of Hlster and Geitel, the theory tails to explain the chief electrical phenomena during pre- cipitation. The influence theory as revised in 1913 is as follows :— Consider a drop of water falling through the atmosphere. It is then in an electrical field and in consequence becomes electrically polarized. If we assume the earth’s normal field, then the drop will have a positive charge on its under half and a negative charge on its upper half. As this drop falls through a cloud which is raining it will constantly strike, on its under side, cloud particles and smaller drops. These small drops make electrical contact, take on to them- selves part of the induced positive charge from the large drop, and then rebound. In consequence the large drop becomes negatively charged and the cloud particles posi- tively charged. Now the most important point to notice in this theory is that with any given field the rain brings down the same sign of charge as that below it which is causing the field. Thus in the earth’s normal field the charge on the earth is negative, and the rain would bring duwn a negative charge leaving the corresponding positive charge above. This would intensify the field. In fact, according to the theory, the rain would always intensify a field; it could under no circumstances weaken, much less reverse, it. Let us follow the process in some detail. Imagine that the clouds on a given day gradually increase in density. We know from observation that this does not disturb the earth’s normal field. Now let rain commence. The drops have a positive charge induced on their lower halves, which is lost by impact with cloud particles and smaller drops ; hence the rain will arrive at the ground with a negative charge. This intensifies the field and causes subsequent rain to be more highly charged. And so the process goes on with an ever-increasing field. But observation shows that this practically never occurs. With non-thunderstorm rain it almost always happens that as soon as the rain starts the field is reversed and the potential gradient becomes negative. As the rain cannot reverse the field some other factor must have come into play. What is it? We must pause to consider how the field can be reversed. The field 6 Dr. G. C. Simpson on the can only be reversed when the excess of positive electricity which is normally in the air has been replaced by an excess. of negative electricity. We have just seen that the influence effect still further increases the positive electricity in the air; therefore the effect which reverses the field must remove positive electricity from the air faster than the rain leaves it behind. Hlster and Geitel have realized this. difficulty, and say ‘We very much doubt whether the earth’s normal field must always be considered as the exciting field. (a) Selective absorption on the precipitation particles during and after the condensation, and (b) the breaking of drops in the air as investigated by Simpson, act as electro- motive forces. (c) Further, the shifting of electrified cloud masses by wind, and (d) lightning discharges in the later stages, could cause electrical fields which are quite inde- pendent of the earth’s normal field ”’*. We can neglect (d) as we are at present considering the initial stages. At the stage we are considering the cloud masses will be positively charged by the influence effect, hence (¢) may be neglected. If (a) and (6) are sufficiently efficient to produce rain so highly charged with positive electricity as to overbalance the rain charged negatively by influence, then they should be considered as the Bio cause of the electricity of precipitation. It appears to me useless to explain the general oleanieye of rain by one process, and then to have to introduce a more powerful effect to explain the large changes in the electrical field which accompany it. This consideration, which is damaging to the theory with non-thunderstorm rain, becomes more important with thunderstorms. - The most marked feature in a thunder- storm is the rapid reversal of the high electrical field. It does not appear helpful to explain the processes at work by an effect which under no circumstances can reverse a field. The mistake made by Hlster and Geitel is that they have inverted cause and effect. They make the electrical field the cause of the charge on the rain, while in reality it is the charge on the rain which gives rise to the electrical field. One could go much more fully into the failure of Elster and Geitel’s theory, but enough has been said to prove it unsatisfactory. The theory will die a natural death if it can be shown that there is another which fits the facts better. We will therefore now discuss the alternative theory. * Elster & Geitel, Phys. Zeit. xiv. p. 1290 (1918). Electricity of Atmospheric Precipitation. 7 B. THe * Breakinc Drop” THEorRY. This theory is founded on the observed fact that when a drop of water is broken up into smaller drops, the water becomes positively charged and the air negatively charged. If there is any large amount of breaking of drops there is bound to be a large separation of electricity. In thunder- storms, which are known to have associated with them violent ascending currents of air, there is no doubt that the water-drops are much broken and, as shown in the paper in which the theory was formulated*, there is no doubt that all the chief electrical phenomena of thunderstorms can be accounted for by the theory. When the theory was published in 1909 there were not sufficient observations on the electricity of non-thunderstorm rain and snow to test the theory in respect to these. It is proposed now to extend the theory to them. Non-thunderstorm rain—For the theory to explain the electrification of rain there must be a certain amount of breaking of drops. The difficulty, however, is that accord- ing to experiments made by Leonard +, drops of water of less diameter than 4 mm. are not broken by falling through air. As in non-thunderstorm rain the majority of the drops are much smaller than this, it would appear that there would be little or no breaking of drops. That this conclusion is not strictly true can be observed in a simple manner. If one stands during a rain shower in such a position that the falling drops can be seen against a dark background, then after a little practice the following will be noticed. The drops will be seen to be falling in parallel lines inclined from the vertical, the direction being determined by the wind. Suddenly, as a gust of wind comes, the angle of fall will change and the drops will become confused. At this moment a large number of very fine drops will appear. Whether these small drops are formed by the breaking of the drops due to the wind itself, or as a result of collisions, it is im- possible to say, but there can be no doubt of the breaking of the drops by one cause or another. Then again there is the question of one drop overtaking another and colliding with it. Leonard has shown that such collisions are probably rare with real raindrops. At the same time there must be a certain amount of collision through one cause or another. When two drops collide it is very unlikely that they unite and continue as one large drop. On the contrary, they will “splash ” against one another and * Loc. cit. + Leonard, Met. Zett. xxxix. p. 257 (1904). 8 Pr. 65. 40: Simpson on the in all probability break up into smaller drops than either of them was before the collision. It does not appear possible to deny some amount of breaking of drops, the only question is, how much takes place ? Assuming that drops do break, we will now consider what the result of the splashing would be. We will again take the case of the formation of a rain cloud on a day with the normal potential gradient. Until the rain commences to fall there is no separation of electricity in the cloud and the normal potential gradient remains unaffected. ‘The potential gradient indicates a negative charge on the ground, and the corresponding positive charge is in the form of a volume charge in the air above. Now when the rain commences and collisions take place, the rain becomes positively charged and the negative charge remains behind in the cloud or in the air beneath it. Before very long the rain has brought down sufficient positive electricity to the ground and left sufficient negative charge behind to reverse the normal field Thus during steady rain one would expect positively charged rain and a reversed electrical field, and this is exactly what is found by observation. If the air from the rain area is carried over the surrounding country by the wind, it may have such a large volume charge of negative electricity that it will reverse the field by itself. This accounts for the fact that as a rain shower moves across the country, the normal field is often reversed before the rain actually commences at a station. We see from this that the theory accounts for the facts qualitatively, but is it sufficient quantitatively ? The want of accurate measurements makes a definite reply to this question difficult. Schindelhauer * found that with non-thunderstorm rain (Landregen) 92 per cent. of his observations showed a posi- tive charge, of which 81 per cent. gave a vertical current of between 1 and 5x10- amp./cm.? Baldit + found for the same kind of rain 85 per cent. of positively charged rain, of which 86 per cent. gave a vertical current of between ‘1 and 10x 107” amp./cm.?-_ From these figures it will be safe to say that the average current produced by the descending positively charged rain is about 2x 10~- amp./cm.? This is equivalent to 6 x 10~° E.S.U. per em.? per second. Now the normal charge on the ground, assuming a potential gradient of 100 volts per metre, is —3x 1074 E.8.U. per em.? Thus we see that the potential gradient would be reversed in ax 107* 6xi0- * Schindelhauer, Ver. Kon. Preuss. Met. Institut, No. 263, p. 27 (1918). + Baldit, Ze Radium, ix. March, 1912. secs. =50 secs. Hence the reversal of the field is Electricity of Atmospheric Precipitation. 9 easily accounted for by the electricity brought down by the rain. It is more difficult to determine how much breaking of drops would be necessary to give the charge found on the rain. For this we ought to know the charge on the drops ; but so far this is entirely unknown. The next approxima- tion is to know the charge on a c.c. of rain-water. This also is practically unknown for non-thunderstorm rain. From a very few observations Schindelhauer concludes that it is about -) H.S.U. pere.c. This, however, is too high a charge, for it would mean that about 80 per cent. of the “ Landregen”’ he investigated fell at a rate of only -4 mm. per hour, which is unlikely. Probably the mean charge on non-thunder- storm rain is about ‘1 E.S.U. per ec.c. Now the best observations made on the electrification produced by break- ing drops of water are those of J. J. Nolan*, who found that charges up to 1:36 E.8.U. per e.c. could be given to water by breaking it into very fine spray. This indicates that if about 1/10 of the rain was broken up once into fine drops the observed charge would be pro- duced. Future observations must decide whether this amount of breaking does occur, at present we can only say that quantitatively the theory is not impossible. These considerations lead to the conclusion that with non- thunderstorm rain the sign of the charge on the rain and the reversal of the field are explained by the theory, the only outstanding question is whether there is sufficient breaking of drops during light steady rain to give the charges observed. Snow.— Very few observations on the electricity of snow have been made, the chief being those of the writer in Simla f and those of Schindelhauer in Potsdam f. At both Simla and Potsdam the snow was sometimes positively and sometimes negatively charged, and on oeca- sion the snow proved to be very highly charged. In one important respect the results from the two stations were opposite. In Simla the snow brought down a large excess of positive electricity, while at Potsdam the excess, also large, was negative. The chief objection brought against the theory based on the electrification through the breaking of drops has been that it cannot explain the electrification of snow in which there are no drops to break. But there is with solids the exact counterpart of the phenomenon with liquids. Whena drop breaks the liquid takes a positive charge and the air a * Nolan, Proc. Roy. Soc. A. xe. pp. 531-548 (1914). + Simpson, Proc. Roy. Soc. A. ]xxxiil. pp. 402-403 (1910). t Loe. cit. 10 Dr. G. C. Simpson on the negative charge. Now Rudge* has shown that when two solids are rubbed together, the air near them becomes charged and in consequence the sum of the charges left on the solids is the complement of the charge given to the air. In other words, the collision and friction of solids of the same kind produces a charge on the solid and a charge of the opposite sign in the air. When the collisions are sufficient in number the electrical separation can be very large, and by blowing dust into the air a large separation of electricity can be effected. Itis not difficult to imagine that a snow storm is a repetition of Rudge’s experiments on an enormous scale in which the snow-flakes take the place of the dust particles. The constant collision and rubbing together of the snow-flakes causes them to be charged and the air to receive the opposite charge. As. the snow falls it leaves the charged air behind it, so pro- ducing a separation of electricity which accounts for the observed electrical effect. Rudge did not make any experiments with snow or ice, therefore his work does not indicate the sign of electricity we might expect on the snow. The following observation made in the Antarctic, however, confirms his general conclu- sion and gives the sign desired. On December 6th, 1911, there was a heavy fall of snow. Two days later, when the sky was cloudless, a wind sprang up from the north causing a very heavy drift of loose snow along the surface. The driven snow did not extend more than four or five feet abovethe ground. The collector of the potential gradient apparatus was well above the top oz the drift. As soon as the wind arose and the drift commenced, the potential gradient was reversed and high negative values were recorded as long as the surface drift continued. This observation showed that the drift snow hada high positive charge. The drift was blowing over the frozen sea and therefore came into contact with nothing but snow ; hence the charge was produced by the friction of snow on snow. Returning now to the conditions during snowfall, we have shown how the snow can obtain a positive charge. Assuming an ascending air current (this is necessary tor snowfall as for rainfall) the charged air will ascend as the snow descends, so that there will be an accumulation of negative electricity towards the top of the cloud. Snow formed here will start with a negative charge, which, under suitable conditions, it might carry down to the ground ; thus accounting for the negatively charged snow. The high charges found by Rudge on dust would lead one. * Rudge, Proc. Roy. Soc. A. xe. pp. 256-272 (1914). Electricity of Atmospheric Precipitation. 1Y to expect that the electrical effect with snow is very large, probably much larger than can be obtained by splashing water, and this would explain the high charges observed.. But a snow-flake is much less suitable than a raindrop for causing a large separation of electricity, for its downward movement relative to the air is very smali, hence much of the: electricity separated joins up again. This probably accounts. for the absence of thunder and lightning during snowstorms. In Simla we have much thunder and lightning at the beginning of snowstorms, but so far as my observations g0,. c=) during the thunder and lightning the snow is accompanied by much small sof hail. The latter, falling through the snow, becomes highly charged by impact, and its rapid rate: of fall gives the condition for a large separation of electricity. This no doubt accounts for the large excess of positive charge on the snow investigated at Simla. It is questionable whether thunder and lightning ever accompany a pure snowstorm. Certainly they do not occur in polar regions, and are very rare during snow in the winterin Europe. More observations will probably prove that violent ascending currents producing soft hail are necessary for thunder and lightning to accompany a snowstorm. The latter part of this paper may be summarised by stating how each of the results of observation given on pages 1-3 is accounted for by the theory. (1), (4), and (8). The process described gives a positive charge to the precipitation and a negative charge to the air, thus accounting for the positively charged precipitation. The air carrying its negative charge may get removed from the place of origin, and then precipitation formed in it will start with a negative charge. If this falls to the ground without becoming charged positively, it will appear as. negatively charged precipitation. (2) and (5). The chief factor at work gives rise to a positive;charge on the precipitation and a negative charge on the air. As the precipitation falls to the ground, while the air tends to prevent the negative charge reaching the ground, the precipitation as a whole will be positively charged. (3), (7), and (11). The changes in the potential gradient result directly from the large separation of electricity, and from the fact that the charge carried to the ground by the precipitation completely masks the small permanent charge normally on the ground. (2) and (6). During quiet non-thunderstorm rainfall there is not so much breaking of drops as in thunderstorms with their violent ascending currents, hence the charges are: less. At the same time the conditions are more suitable for- 12 Electricity of Atmospheric Precipitation. the complete separation of the electricity, hence the ratio of positive to negative electricity is greater. (10) The separation of electricity produced by the impact and friction of solids is probably greater than that produced by the splashing of drops, hence very Jarge charges on snow may be expected. Simla, March 1915. APPENDIX. Observations on the electricity of precipitation. { | PLAce. | OBSERVERS. PUBLICATIONS. Wolfenbeiitel Elster & Geitel. Wien. Ber. xcix. p. 421 (1890). (Brunswick). | Terr. Magn. iv. p. 15 (1899). | Gottingen. Gerdien. | Miinch. Ber. xxxiii. p. 367 (1908). | Phys. Zeit. iv. p. 837 (19V3), Vienna. Weiss. | Wien. Ber. exv. Abt. iia. p. 1299 (1906). | Porto Ri Rico. Kohlrausch. | Wien. Ber. exviii. Abt. iia. p. 25 | (1909). | I ad Ba oe ee | Simla (india). Simpson. Phil. Trans. A. ecix. p. 379 (1909). | Proc. Roy. Soc. A. Ixxxiii. p. 394 | LOO): |Full details of observations given in | India Met. Memoirs, vol. xx. papi. | | Potsdam. K. Kabler. | Phys. Zeit. ix. p. 258 (1998). | Veroff d. k. Preuss. Met, Inst. | No. 213 (1909). | 9 F. Schindelhauer. | Do. No. 263 (1918). SS | Graz. Benndorf. | Wien. Ber. exix. p- 89 (1910). | Sitz. Ber. K.B. Akad. d. Wiss. | Minchen, p. 402 (1912). Puy-en-Velay. Baldit. | Le Radium, viii. April 1911. | Do. ix. March 1912. Buenos Aires, Berndt. Phys. Zeit. xiii. p. 151 (1912). | Veriff d. Deutsch. Wiss. Vereins in Buenos Aires, No. 3 (1918). Dublin. McClelland & Nolan. | Roy. Irish Acad. Proc. xxix. p. 81 | P@L912). | Do. xxx. p. 61 (1912). age aed II. The Theory of the Winds. By R. M. Deetey, MInst.C.E., F.GS.* ane so far no theory which meets with general acceptance has been propounded to account for the movement of the winds, it is probable that Edmund Halley+ and George Hadley{ have furnished us with a basis upon which such a theory may be founded. Halley attributed the motion of the atmosphere to the condition of unstable equilibrium brought about by temperature differences between one region and another; whilst Hadley showed that, owing to the rotation of the earth on its axis, air currents directed north or south suffer deflexion. Since Hadley’s time it has been shown that all winds are subjected to a deflecting force due to this movement. When Halley and Hadley propounded their views, know- ledge of the actual courses of the surface winds of the earth was very imperfect, and very little indeed was known concerning the upper currents of the atmosphere. However, Hadley was able to show that, although, as Halley had con- tended, the winds move in directions which to a large extent might be expected from the temperature gradients, if the effects produced by the rotation of the earth be taken into consideration, then the agreement between fact and theory is much more satisfactory. With the more detailed knowledge of the actual circulation of the atmosphere which has been gained since Hadley’s time, numerous difficulties have presented themselves. In middle latitudes, for instance, the surface winds generally blow in opposition to the surface temperature gradients. In 1857 James Thomson§ propounded a theory which he considered brought the facts then known into agreement with theory; but to do this he had to assume the existence of currents in the upper atmosphere which have not been found to exist. His view was that in middle latitudes there were three distinct westerly currents of air: a middle current from north and west, and an upper and under current from south and west, the lower current being due to the decrease of velocity resulting from friction with the ground or water. Ferrell at a later date advocated a similar theory. The investigation of the upper atmosphere by pilot-balloons * Communicated by the Author. + Phil. Trans. vol. xvi. No. 185, p. 153 (1686), ¢ Phil. Trans. vol. xxxix. No. 487, p. 58 (1735). § Rep. Brit. Assoc. 1857, Notices and Abstracts, p. 38. Also Bakerian. Lecture, Phil. Trans. 1872, vol. clxxxiii. A. p. 653. 14 Mr. R. M. Deeley on the during recent years has failed to reveal the currents postulated by Thomson, and great doubts are now entertained concerning the correctness of the theory. We are consequently faced with the fact that the directions in which the winds blow -are not such, in many cases, as the temperature gradients on the earth’s surface seem to require. A better knowledge of the temperature conditions of the atmosphere, and of the directions of flow of the upper currents, which have been gained during recent years, instead of bringing theory into accord with fact, has to some extent widened the breach ; for in cyclones the temperature gra- dients are in opposition to the circulation of the winds*. Halley called attention to the fact that the temperature of ‘the air over different parts of the earth’s surface varies greatly owing to the varying amounts of heat received from the sun in different latitudes. These variations of temperature with latitude persist the whole year round, but vary in amount with the seasons. In many cases the temperature along the same line of latitude varies greatly owing to the varying heating of the air over land and water respectively; and such temperature gradients are often reversed with change of season. The atmosphere is thus always in a state of unstable equilibrium, with the result that there are always temperature gradients to cause warm air to flow over the neighbouring colder air, and cold air to flow beneath the warm air from cold to warmer regions. Gradients thus persist throughout the year; but these gradients vary in intensity and often in direction with change of season. The manner in which air currents result from differences -of temperature is shown in fig.1. Here the vertical column Bigt dl: ——{ >>—_—> D B D of air AB, being warmer than the columns of air CD and C'D’ on either side, is the thicker. But the weight of each column isthesame. The top of the column AB, being higher than the columns on each side of it, flows in the directions shown by the upper arrows. The columns CD and C'D’ then * The Free Atmosphere in the Region of the British Isles. Dines and Shaw. Meteorological Office Memoir 2100, 1912. Theory of the Winds. 15 become heavier than the column AB, and the lower portions of CD and C'D’ flow beneath the lighter column of central air, in the directions shown by the lower arrows. If the differences of temperature between the columns persist, we have a central area of low pressure with higher pressures on either side, and a continuous circulation of air goes on from the warm area to the cold area above, and from the cold to the warm below. If the earth’s surface consisted entirely of either land or water, and there were no mountain ranges, the application of Halley’s theory would be much simplified. The earth’s sur- face, however, although mainly occupied by water, has large land areas, and these land areas are peculiarly distributed. In the Northern Hemisphere we have the continents of North America and Eurasia, and portions of South America and Africa occupying a large part of the area. In the Southern Hemisphere, however, the only large land masses are the Antarctic continent, Australia, and the narrow southern portions of Africa and South America. Here, consequently, we have a large water area and comparatively small land areas. Itis mainly to this distribution of land and water that we owe the different meteorological conditions of the two hemispheres. Although a portion of the sun’s heat is intercepted by the atmosphere, a good deal of it reaches the earth’s surface. Over land areas it quickly heats the ground, and the heat is passed in an upward direction by convection currents ; but as the air rises it is cooled by expansion, and we geta vertical gradient of temperature. Dry air rising and expanding in this way cools at the rate of 1° C. in temperature for each rise of 100 metres; but saturated air at the usual temperature of 10° C. only cools at the rate of 0°54 U. for the same dis- placement, the latent heat of the water-vapour being given up to the air on condensation. When, however, the sun’s rays strike the surface of the sea, they penetrate it and raise its temperature but slightly. It thus comes about that during the summer the lower portions of the atmosphere are warmer over the continents than they are over the seas, and during the winter they are cooler. We thus get gradients of temperature between the land and the sea as well as tempe- rature gradients from north to south, and this is one of the reasons why the great masses of land in the Northern Hemisphere produce a climate which differs so much from that met with in the Southern Hemisphere. Comparatively little has been done to determine the vertical temperature gradients of the atmosphere in the 16 Mr. R. M. Deeley on the Southern Hemisphere. From a theoretical point of view this is unfortunate; for if we had as much information of the Southern Hemisphere temperature gradients as we have of those of the Northern Hemisphere, the more simple conditions prevailing there would very probably enable the relationships between temperature, pressure, and wind to be seen more clearly. Over South America, for example, the average monthly pressure gradients are constant in direction the year through. Although the surface winds move from high pressure to low pressure areas, in the vast majority of instances they do not doso in a direct manner. In the case of cyclones in the Northern Hemisphere the winds on the earth’s surface move inwards in rough spirals; the circulation round the centre of lowest pressure being in the opposite direction to the move- ment of the handsof a watch. In the Southern Hemisphere the movement is in the opposite direction to that in the Northern. Similarly, winds moving north from the equator acquire a velocity component directed towards the east, whilst those moving south to the equator acquire a velocity com- ponent directed towards the west. On the other hand, winds moving from the south to the equator have a velocity component directed towards the west, whilst those moving south from the equator have a velocity component directed towards the east. ‘These wind deviations were shown by Hadley to be due to the effects of the earth’s rotation on its axis. The manner in which the rotation of the earth affects the direction in which the winds blow will be seen from the following considerations. If the earth did not rotate upon its axis it would be approxi- mately spherical in form. Owing to its rotation, however, the equatorial regions are caused to bulge outwards somewhat, whilst the polar areas are flattened. Here the gravitational forces tending to make the earth assume a spherical form have superimposed upon them the centrifugal forces resulting from its rotation. As far as the winds are concerned, this flattening of the polar areas has scarcely any appreciable effect ; but the rotation affects the winds very materially in other ways. . Fig. 2 is a section through the earth parallel with its axis. A mass of air at the equator AA is carried from west to east at a velocity which causes it to rotate round the axis of the earth once in 24 hours, or at a velocity of about 1036 miles per hour. At the poles BB the earth and air have no translational velocity, and at intermediate latitudes — Theory of the Winds. 17 the velocities are proportional to their distances from the earth’s axis. Indeed the atmosphere, when there is no wind, is everywhere travelling towards the east at the same angular velocity. If, however, the atmosphere be set in motion at Fig. 2. any point, the air will not continue to move in the direction in which it is urged, for it no longer has the same angular velocity as the surface of the earth below it. It does not matter in which direction the impulse acts. The air will be deflected, and will not, in the absence of other forces, follow a straight course on the earth’s surface. Fig. 3 shows an Fig. 5. area in the Northern Hemisphere. Ifa mass of air at a be forced in a northerly direction by some force, such as might result from a temperature gradient, as it moves north it reaches regions where the earth’s surface is travelling less rapidly than itself, and the air is deflected towards the east, a’. If the air at ais forced in a southerly direction, Phil. Mag. 8. 6. Vol. 30. No. 175. July 1915. C 18 Mr. R. M. Deeley on the then it finds itself in a region which is moving more rapidly than itself, and it is deflected towards the west, a’. When the force acts from the west the movement of the air is more rapid than that required by the latitude, and it is deflected to the south, a,;; but if the force acts from the east, then there is a deficiency of velocity for the latitude and it moves north, aj. These movements have been plotted on fig. 3, and it will be seen that in whatever direction in the Northern Hemisphere the air is urged the deflexion is always to the right. Similar reasoning will show that in the Southern Hemisphere the deflexion is always to the left. If, as is shown in fig. 4, in the Northern Hemisphere, the air 1s Fig. 4. / Ay urged towards any particular centre, the wind deflexions are such as to cause the air to pass on the right side of the centre. In this way a rotation around the centre may be set up, which in the Northern Hemisphere is in the opposite direction to the hands of a watch. These tendencies of the winds to change their directions under the influence of the rotation of the earth are greatest in the polar areas. At the equator a west wind will retain its direction unaltered, but will tend to rise; an easterly wind, however, will tend to move either to the north or to the south. The deflecting force tending to alter the direction of the wind being equally powerful for the same velocity and latitude in whatever direction the wind is moving, as the wind changes its direction, the deflecting forces, the di- rections of which are shown by the arrows in figs. 3 and 4, also change their direction so as to remain at all times at right angles to the wind movement. The deflecting force, therefore, does not accelerate the velocity of the wind, although it changes its direction of flow. Any change in the velocity of the wind must be due to some force acting upon it, other than that due to the rotation of the earth. In a cyclone in the Northern Hemisphere the air circu- lates in the opposite direction to the hands of a watch. Owing to the tendency of the air on all sides of the cyclonic centre to be deflected to the right by the earth’s rotation and | Theory of the Winds. uS the curvature of its path, it moves from the centre until the air pressure at the cyclonic centre falls sufficiently to constrain it to circulate round the centre. In an anticyclone in the Northern Hemisphere the direction of motion is in the di- rection of the hands of a watch. Here the air moving round the anticyclonic centre is deflected also to the right; but as this direction is towards the centre, the air is compressed there and high pressures result. Gold * has carefully compared the actual wind velocities at moderate heights, accompanying actual pressure gradients, and has found that they agree as closely as can be expected with those calculated according to the theories of Ley and Loomis. It must be remembered that every wind must be accom- panied by one moving in the opposite direction. Hven when a wind circulates round the earth’s axis along some line of latitude thisis the case. Sometimes the compensating winds move side by side, in other cases the return current may be above or below the one that is being considered. On this account any force or forces setting the air in motion must deal with the whole circuit, and in doing so may produce variations of air pressure. Perhaps the most simple form of circulation is that of a wind following a line of latitude. In fig. 2aba’b”’ is a section of an air current moving from west to east. The air forming the ring is moving faster than the earth below, and in attempting to turn to the right tends to move bodily to a lower latitude. As a result the air in lower latitudes is compressed, and the air in higher latitudes is expanded. Hyventually the difference of pressure between the two sides prevents further displacement towards the equator. We thus have a high pressure region set up on its equatorial side and a low pressure region on its polar side. This is approximately the condition which exists between latitudes 30° and 60° in the Southern Hemisphere. As long as the westerly wind of this ring is moving at the same angular velocity over its whole sectional area it is quite stable; for although the lower portions of the atmosphere are more dense than the upper portions, and therefore exert a greater pressure towards the equator, the lower air, on account of its greater density, requires a greater force to compress it. When the general circulation of the atmosphere is approximately parallel to the lines of latitude, and blows from the west, the tendency to * Report to the Director of the Meteorological Office on the Calcu- Jation of Wind Velocity from Pressure Distribution. Meteorological ‘Office Report, No. 190, 1908, p. vO i ~ 20 Mr. R. M. Deeley on the create differences of pressure on its two sides is greater im high than in low latitudes. In such a case as that above described the friction of the wind against the earth’s surface tends to reduce the velocity of the lower eg When this occurs the lower portion, a’b' a’'b"’ (fig. 2), tends to flow towards the area of low pressure, whilst the upper portion tends to flow towards the region of higher pressure. This effect of surface friction was pointed out by James Thomson ; and he considered that it accounted for the prevalent surface winds of middle latitudes, especially in the Southern Hemisphere, blowing towards the poles. But as such a lower wind moves in opposition to the surface temperature gradient, he assumed that it was merely a thin current, the main circulation resulting from the tem- perature gradient consisting of two considerable upper currents. The mean annual temperatures along two meridians are shown in fig. 5. The curve A is along 15° W. long.,. Fig. 5. ee aie a i a EOCENE | Mean Annual eared Gradients. and, with the exception of that portion which passes over the extreme westerly side of Africa, is wholly over water ; whilst the curve B passes across Siberia, Central Asia, and India, and then southwards over the Indian Ocean. Both these curves show very considerable temperature gradients from the equatorial towards the polar regions. The mean temperatures on Northern Asia are much lower than those over the North Atlantic. India, however, is. warmer than the corresponding latitudes of the Atlantic. An inspection of a chart showing the mean annual isotherms of the world will show that this feature is a general one. To recapitulate. In low latitudes the mean annual tempe- rature over the land is greater than that over the sea; but poe s Theory of the Winds. 2h in high latitudes the mean annual temperature over the sea is greater than that over the land: there are also steep temperature gradients trom the equator to the poles. Fig. 6 shows the mean temperatures for January along Fig. 6. Latitude the same meridians. Here the low temperatures over the Jand areas, as compared with those on the oceans in similar Jatitudes, are more marked. Fig. 7 shows the July tem- perature gradients. Here the temperature over the land is much greater than it is over the sea in similar latitudes. Fig. 7. Latitude , \ | tal a a a a AW on yp NR, SN ORT 2608 PS Ec GOT EE DN EF en eae Pudsey | a a a a ee ee Pee ee hae a a a N. In figs. 5, 6, and 7 we see that in the Southern Hemi- sphere, which is very largely covered by water, the mean annual temperatures, and the temperatures for each month, do not vary as greatly as they doin the Northern Hemisphere. 22 Mr, R. M. Deeley on the There is thusa very marked difference between the two hemi- spheres. In the more land-covered hemisphere the tempe- rature differences are greater between summer and winter than they are in the more water-covered hemisphere. If the winds are due to temperature differences, then they should likewise show marked differences in the two hemi- spheres. In the Southern Hemisphere the temperature gradients from north to south are fairly regular along all meridians ; whereas in the Northern Hemisphere the tempe- rature gradients along meridians passing over the continents differ widely from those passing over the oceans. In figs. 6 and 7 the arrow-heads on the curves show the north or south components of the prevalent winds. It will be noticed that on the two meridians, both in June and July, at higher latitudes than 30° in the Southern Hemisphere, the winds blow against the surface temperature gradients. A similar state of affairs exists in the Northern Hemisphere during January; but in July along the meridian 75° K. the wind blows in accordance with the surface temperature radient. In the equatorial regions between 30° §. lat. and 30° N. lat. the winds everywhere follow the temperature gradients. We see from this that in middle latitudes the winds do not always obey Halley’s surface temperature gradient theory, but that in the equatorial regions they do. An inspection of a chart showing the mean directions of the winds for each month, will make it clear that figs. 6 and 7 represent with fair accuracy the general relationship between the direction of flow of the winds and the surface temperature gradients. When dealing with the effects of the rotation of the earth on the paths of the winds, it was pointed out that when the air circulates round cyclonic centres it is compressed on one side and expanded on the other, so as to produce pressure variations on the earth’s surface. These differences of pressure are registered by the barometer. By drawing lines of equal mean pressures on a map, and then comparing the map so produced witha map showing the directions followed by the winds for the same period, a very close agreement will be found to exist between the areas of low pressure and the cyclonic winds and between the areas of high pressure and anticyclonic winds. Indeed, as has already been stated, a very close relationship exists between the directions and force of the winds and the trend and steepness of the baro- metric gradients. It must be remembered that we are dealing with the prevalent surface winds and the mean baro- metric gradients, for these show the nature of the genera Theory of the Winds. 23 circulation of the atmosphere. If we take the surface wind directions and the surface barometric gradients for some particular instant the agreement is not so good. Although even on the earth’s surface there is a very fair agreement between the directions of the windsand the baro- metric gradients, it must be borne in mind that the direction of the surface winds is varied greatly by local features, such as hills and surface friction, whereas the barometer measures the weight of an atmospheric column, and gives the sum of the effects resulting from temperature, humidity, &. of the whole column. This subject has been dealt with by Shaw and Dines*. Ata height of about 1500 metres the velocity and direction of the wind and the barometric gradient have been found to be in fair agreement. Fig. 8. Latitude ~~ Longitude| /00°E! 40” 60° 80° N.! Fig. 8 shows the variations of atmospheric pressure along the meridian 100° E. for January and July. The pressure variations are here shown to be much greater between summer and winter in the Northern Hemisphere than they are in the Southern Hemisphere. This meridian passes over Asia and the Indian Ocean. The cooling of the air during the winter over the high mountain ranges and plateaus gives rise to an area of high pressure from which the air flows north and south, the southern current being the Dry Monsoon. In the Indian Ocean there is a high pressure ridge about 30° S. lat., beyond which the pressure falls off rapidly towards the Antarctic Circle. Here the wind moves * The Free Atmosphere in the Region of the British Isles. Meteoro- logical Office Memoir No. 210 8. | be | | | ~- | 2 Me 2 a Inches of Mercury 24 Mr. R. M. Deeley on the from the high-pressure ridge in opposite directions, but on the south side it blows against the surface temperature gradient. In July in Asia there is an area of low pressure, whilst over the Indian Ocean in 30°S. lat. the high pressure area persists. The Wet Monsoon blows from this high pressure area into Asia, in agreement with the temperature gradient, but is deflected to the west before it crosses the equator and to the east on its way to India. The barometric pressures over the Atlantic are shown in fig. 9. Here, except in the extreme north, the pressure Fig. 9. Latitude Longitude 20° Inches of Mercury distribution is much the same in both hemispheres ; but the pressures are higher in July than in January. This arises from the fact that the greater portion of the land area of the earth is in the Northern Hemisphere, and during the summer of this hemisphere the land areas as a whole are warmer than the oceans, and the air is to some extent transferred from the land to the water-covered areas. Here we have a further reason for believing that the main circulation of the atmosphere is due to temperature gradients. The temperature and pressure curves that have been de- scribed show clearly that the great land areas in the Northern Hemisphere have a great disturbing effect upon the atmo- spheric circulation in that hemisphere. In the Southern Hemisphere the curves show a much more simple state of matters, due no doubt to the tapering of the southern con- tinents and the concentric position with respect to the earth’s axis the Antarctic continent occupies. In the Southern Hemisphere the curves show that we have:—1st. A low pressure belt along the equator. 2nd. A high pressure belt along latitude 30° 8. 3rd. A deep low Theory of the Winds. 25 pressure area over the Antarctic Continent. North of the high pressure belt the winds blow in accordance with the surface temperature gradient; but south of it they blow against the surface temperature gradient: and instead of the cold Antarctic Continent being an area of high pressure it is an area of very low pressure. If the winds of the Southern Hemisphere followed ihe surface temperature gradient, we should expect to find an upper current from the north and east, and an under current from the south and east. The lack of velocity of these east winds, as compared with the velocity of the earth, would produce an area of high pressure at the pole. Under these circumstances we should have lower winds moving from the poles to the equator, and upper winds blowing from the equator to the poles. ‘This is the condition Thomson con- templated, but he added a lower thin current, resulting from the friction of the earth, and blowing from the high pressure belt towards the pole with directions from west to east. In the Northern Hemisphere, owing to the presence of great land areas, and the northerly extension of the Pacific and Atlantic Oceans up to the Arctic Circle, the pressure and temperature curves, as we have seen, differ remarkably from those of the Southern Hemisphere. In the Northern Hemisphere there is a high pressure belt at about 30° N. lat.; but it is much broken in the summer by the heating up of Asia and northern Africa. A steady westerly wind round the North Pole, which would produce a very low pressure there, is thus prevented, and we have instead, both during winter and summer, two low pressure areas, one over the North Pacific and the other over the North Atlantic. In the summer these low pressure areas fill up slightly. Around these two low pressure areas the winds circulate, especially in the winter, and mitigate very considerably the winter climate of the extreme north. They also, by carrying the winds over the polar area, keep the pressures somewhat high there. From these two low pressure areas the pressure steadily increases as we approach the high pressure ridge in 30° N. lat., and by producing south-westerly winds in middle latitudes on the east sides of the Pacific and Atlantic Oceans it mitigates the climates of western Europe and the west side of North America. On the western sides, however, we get the cold winds from the arctic area in the winter. Indeed, in the Northern Hemisphere the winds do not depart so markedly from what we might expect from the temperature gradients as they do in the Southern Hemisphere. The action, however, in both hemispheres of some force, other 26 Mr. R. M. Deeley on the than that resulting from the surface temperature gradients, is shown by the low pressure area of which the Antarctic Continent is the centre, and the two low pressure areas in the Northern Hemisphere. In the case of the curves of temperature and pressure we have illustrated, only the north and south components of the winds are indicated by the arrow-heads. The winds, how- ever, seldom blow north and south, and they always show the effects of the rotation of the earth; but the direction of the winds is not, as we have seen, in very many important instances such as the surface temperature gradients would sug gest. When Halley and Hadley propounded their views nothing was known of the changes of temperature which take place with elevation in the upper atmosphere, and it has generally been assumed that the temperature gradients which produce the general circulation of the winds of the world are those which are shown by the temperature gradients at the earth’s surface. [tis to Teisserene de Bort that we owe the discovery that the temperature of the upper atmosphere presents some remarkable features. He found, by means of registering balloons, that at a height of about eleven kilometres the temperature of the air ceases to fall, and sometimes even becomes warmer with increasing height. The lower eleven kilometres, or thereabouts, where, owing to the vertical temperature gradients, vertical currents can take place, has been called the troposphere, while the upper portion, where, owing to the very slight increase or decrease of temperature with varying height, vertical currents of any strength cannot exist, has been called the stratosphere. This peculiar distribution of temperature with vertical height is apparently due, as pointed out by Gold *, to the fact that the sun’s heat is pretty regularly intercepted by the whole thickness of the atmosphere either on its way to the earth’s surface, or as it is again radiated from the earth and air into space. Indeed, the upper portions of the atmosphere are pierced by all the heat the earth receives from the sun and by the same amount of heat as it is returned to space. On the other hand, the lower portions of the atmosphere receive a good deal of heat owing to convection currents rising from the earth’s surface. This ground, or water-heated air, cools as it expands on rising, and a condition of convective equilibrium is set up in the lower atmosphere. The actual * Proc. Roy. Soc. A. vol. Ixxxil. Theory of the Winds. Ti gradient in the lower three thousand feet of the troposphere. varies throughout the day. Up to the level of the strato- sphere the temperature gradient is more equable, but varies. from place to place and from time to time, owing to the. varying humidity &e. of the convection currents. The stratosphere contains from one third to one fourth of the total weight of the atmosphere. The stability of such an arrangement of temperatures and pressures has been clearly indicated by Helmhoitz* in 1888.. He remarks: “It is well known how very differently the propagation of changes of temperature in the air goes on according as heat is added or withdrawn above or below. “Tf the lower side of a stratum of air is warmed, as occurs at the surface of the earth, by the action of solar rays, then the heated stratum of air seeks to-rise. This is effected very soon ail over the surface in small, tremulous, and flickering streams such as we see on any plane surface strongly heated by the sun; but soon these smaller streams collect into larger ones when the locality affords opportunity, especially on the: side of a hill. The propagation of heat goes on relatively rapidly throughout the whole thickness of the atmospheric layer, and when it has uniform heat throughout its whole depth, and is therefore in adiabatic equilibrium, then also the newly added air seeks de novo to distribute itself through the entire depth. ““The same process occurs with like rapidity when the upper side of a stratum is cooled. “¢On the other hand, when the upper side is warmed and the lower side cooled, such convection movements do not occur. The conduction of heat operates very slowly in large dimensions, as I have already explained above. Radiation can only make itself felt to any considerable extent for those classes of rays that are strongly absorbed. On the other hand, experiments on the radiation of ice and observations of nocturnal frosts show that most rays of even such low temperatures can pass through thick layers of clear atmo- sphere without material absorption. “Therefore a cold stratum of air can lie for a long time on the earth, or equally a warm stratum remain at an altitude without changing its temperature otherwise than very slowly.” According to Gold, the reason why the troposphere exists is that the earth, sea, and the solid matter, vapour and cloud, in the lower atmosphere heats the adjacent air, warms it and causes it to rise. During its ascent it radiates this heat, and, * Smithsonian Miscellaneous Collections, No. 843, 1893. 28 Mr. R. M. Deeley on the therefore, parts with more heat than it receives as it rises. On this assumption Gold found that the thickness of the troposphere is sufficiently great to enable it as it rises to get rid of this heat by radiation by the time the lower level of the stratosphere is reached. This theory favours the assumption that the low stratosphere in cold regions is partly due to the cold ground and sea below. Hitherto no soundings have been made which reveal the temperature of the stratosphere except in its iower portions; nor have many soundings been made which show how its temperature varies with latitude. Such records as we have make it clear that the troposphere is thicker in low latitudes than it is in high ones, and that the stratosphere is cooler in low latitudes than it is in high ones. Thus, although we have temperature gradients in the troposphere which favour winds blowing towards the equator, we have temperature gradients in the stratosphere which also favour winds towards the equator. ‘This is an unexpected circumstance; and if the stratosphere gradient were sufficiently strong to overpower the troposphere gradient, the course taken by the winds of the earth in middle latitudes would be explained. Fig. 10. Absolute Temperature “C. '20+— Pee | DBD REED: 15 i el H 4 : hse 0 5 3 3 180 200 220 NS. 240° S Height in Kilometers In fig. 10 the curve A shows the temperature gradient to a height of 26 kilometres at Batavia, a few degrees south of the equator, where the seasons do not vary much in tem- Theory of the Winds. 295: perature. The curve E shows the summer variation of temperature with height over the British Isles. The differ- ence between the curves A and E is very striking. The temperature difference between Batavia and Britain up to. about 11 kilometres is about 12° C., Batavia being the higher. At greater heights, however, the temperature above Britain is much greater than it is above Batavia. Indeed, if the difference continued into the upper portion of the strato- sphere it would quite neutralize the opposite lower gradient. Balloon observations by Assman on the Victoria Nyanza in Africa also showed the low stratosphere temperatures along the equatorial belt. ‘The curves U and D show the mean annual curves for Milan and for Pavlovsk near Petrograd. Here, although the temperature of the troposphere at Pavlovsk is cooler than it is at Milan, the stratosphere in the more northern locality is the warmer. Table I. shows the temperature of the lower portion of the stratosphere at a number of places in Europe as compared with that at Batavia. It will be seen that the higher the latitude the higher the temperature of the stratosphere is. The figures are by Gold* except in the case of Batavia, where Taste I. Station. Latitude. Temperature A and Height. \ Mean ie eae hin, ep ve) Pee. Fy nimi, | 19 km. | 20 km. | 21 km. | 22 km. || ——— LIAS — ie | 60. 1949 2013 | 2048 | 2670 ) 2020 | 12km. 13km. ! 14 km. | 15 km. | | | LS ise 1] Milan and Pavia ...|| 44.30 2161 2164 | 2177 | 2171 | 2168 oT er eee ! 47 25 | 216-2 9167 | 2159 | 2162 || 216°2 | Munich................| | 47 50 | 2172 | 2182 | 2173 | 2198 || 2181 | WHONIID. Sa vcncrscasxaes | 4810 | 2183 | 419-6 | 219°6 220°1 || 219-4 Strassburg............ 48 30 | 2168 | 2176 | 217°9 218:2 || 217°6 a eee | 48 50 | 2195 | 2193 | 219-1 218°9 | 219°2 ' es | 5045 | 2175 | 2164 | 2153 | 216-4 216-4 | hi ae hoes | 6888. | 293 | 918-7 | 9179 1 S185 | Mambure ...:........ ' 53 30 | 2184 2222 | 219°4 | 220°2 2200 | England............... | 5300 | 2195 | 2195 | 2193 | 2200 2196 | Koutehino............ | 55 40 | 2168 | 2187 | 2216 | 2210) 2195 | MAVIOVER ......0.00:. Proe 40t) SAY} a84 |) 293-5 | 2251 | 2996 | * Meteorological Office Memoir No. 210 e. 30 Mr. R. M. Deeley on the the authority is W. van Bemmelen*. The number of regis- tering balloon ascents made in high latitudes is not sufficiently reat to enable us to ascertain the mean annual temperature -of the stratosphere up to 20 kilometres. One such ascent at Pavlovsk made in May 1905 showed an increase of tempe- rature in passing from 11 to 17 kilometres of 10°°7 C. The point these vertical temperature gradient curves bring out clearly is that there are great temperature differences at similar heights, in}both the stratosphere and troposphere, in different latitudes; but these differences act in opposition to each other. Unfortunately our knowledge of the general temperature distribution in the stratosphere is not sufficiently detailed to enable us to decide whether the increasing general temperature of the stratosphere with increasing latitude is -sufficiently great to cause the lower winds of middle latitudes to move polewards in spite of the adverse temperature gradient -of the troposphere. All that can be said as yet is that the information as to. temperature obtained by balloon ascents shows that there is a strong temperature gradient in the upper atmosphere acting in opposition to the surface tempe- rature gradient. Shawt has maintained that the stratosphere has an im- portant influence upon the winds. He propounded a problem -on this subject in a letter to ‘ Nature’; but considered the case of a horizontal lower stratosphere surface and regarded insolation as resulting in the superior heating of the strato- sphere over low latitudes. The conditions given in fig. 10, however, show that the troposphere is thicker in low than in high latitudes, and that the whole stratosphere has a higher temperature in high than in low latitudes. What is required is further information concerning the temperature of the stratosphere, so that we can compare the temperatures over the equatorial belt with those over the two high pressure belts at 30° N. and 8. lat., and those over the Arctic and Antarctic Circles. That the winds actually blow as though in middle and high latitudes the stratosphere temperature gradient over- powers the low level gradient, is shown by the fact that balloons which reach the stratosphere—although the lower wind may have a northerly moving component—generally fall to the south and east of the starting point. It has been assumed that the cold upper temperatures at the equator are due to the uprising of the air caused by the * Nature, March 5th, 1914, p. 6. + Nature, June 11th, 1914, p. 375. Theory of the Winds. 31 Trade Winds. If, however, the Trade Winds are due to temperature gradients, we must assume that over the high pressure belts in latitudes 30° N.and 8. the air column is stall colder than at the equator, in spite of the fact that along these latter belts the air is descending. Hurope is not a very satisfactory area for studying the effects of the temperatures of the atmosphere at different heights as to their effects upon the winds. Owing to its being a large land area, and the presence of Asia to the east, there are great variations in the pressure gradients throughout the year. Such an area as South America, where the mean pressure gradients are constant throughout the year, would be a more suitable one for the purpose. The fact that in high latitudes the stratosphere is warmer than in low latitudes requires explanation. ‘That it might be warmer in the summer, even at the poles, is not so strange; for the length of the day in summer more than compensates for the low altitude of the sun. In the winter, however, the air receives practically no heat directly from the sun within the Arctic Circle; yet at Pavlovsk, even in winter, it does not fall below the temperature at Batavia. According to Gold’s theory the upper atmosphere, when the sun’s rays pass vertically through it, only intercepts about 12 per cent. of the sun’s heat. It is to the long waves reflected back from the lower atmosphere and earth that tbe stratosphere owes its comparatively high temperature; and in high latitudes the earth’s surface receives less heat from the sun than it does in low latitudes, and should be cooler on this account. It may be, however, that a very appreciable amount of the sun’s heat is intercepted in the stratosphere by solid matter, and that this amount becomes considerable when the sun’s rays pass obliquely through great thicknesses of it. Indeed, the amount thus absorbed may more than compensate for the less quantity received by radiation from the earth and air below. Such solid matter could not be ice deposited by the slow rise which must result in the polar areaasa result of the incoming winds from lower latitudes, for the temperature does not fall. Meteoric dust and ions may, however, intercept some heat. All the sun’s rays falling on the polar regions would pass through the atmosphere obliquely the year round; and it is conceivable that the stratosphere in these regions is actually more heated by the sun’s rays and the rays from the seas and lands below than is the case in equatorial regions, where the sun’s rays pass entirely, or nearly so, through the thin atmospheric covering of the earth. 32 Mr. R. M. Deeley on the Table IT., after Zenke, gives the intensity of the insolation at the earth’s surface for different solar altitudes. Here A is. the altitude of the sun, B the relative length of the path of the rays through the atmosphere, C the intensity of insolation on a surface perpendicular to the rays, D the intensity ‘of insolation on a horizontal surface, and E the insolation inter- cepted by the atmosphere. Tape JI. he B. C. | 10) E. Ea | 0 44-7 0-00 0-00 1:00 5 108 0-15 0-01 0:85 10 57 0-31 0-05 0°69 20 2-9 0-51 0-17 0-49 30 2-0 0-62 0-31 0-38 40 1°56 0-68 0-44 0:32 50 1:31 0-72 0:55 0:28 60 115 0-75 0-65 0:25 70 1:06 0-76 0-72 0:24 80 102 0-77 0-76 0-23 90: 1:00 0-78 0-78 0-22 If these figures be correct, 78 per cent. of the sun’s heat reaches the ground ona fine day when the sun is vertical ; but when the altitude of the sun is only 10°, then only 31 per cent. reaches a surface perpendicular to the sun’s rays, the rest being intereepted by the atmosphere. Column HE, Table II., shows the percentage of insolation arrested by the atmosphere for varying sun altitudes calcu- lated from the figures in column C. If, as these figures. would indicate is the case, the proportion of the sun’s heat arrested by the atmosphere in high latitudes is much greater than is the case in low latitudes, several important conse- quences follow. Ist. In high latitudes, although a con- siderable proportion of the sun’s rays striking the snow or ice-covered surface is reflected, these reflected rays will be short period waves of little heating power; for the atmosphere has absorbed the long ones. 2nd. The greater proportion of the sun’s heat will go directly to Theory of the Winds. 33 heat the atmosphere, and, except over the unfrozen seas, there will be a comparatively small amount of heat rising by convection, and the troposphere will be thin. Balloon ascents in the polar regions have shown that this is the case. It must be remembered that the surface winds in middle latitudes largely blow towards the polar regions, and that the warm ocean-currents go with them. We thus have a considerable amount of heat carried by convection from low to higher latitudes. It is considered that the effect of the greater heating of the upper atmosphere with increasing latitude does not become marked until 30° N. and S. latitude have been passed. The winds of the equatorial belt are, therefore, produced by the surface temperature gradients. North and south of 30° N. and S. latitude it is the superior gradient of temperature in the stratosphere which controls the winds. It must be admitted, as well, that the Thomson friction effect is a valid one, and that the reduction in the velocity of the surface winds by friction will assist to prevent cold surface layers in high latitudes from flowing towards lower latitudes. The action of the upper currents in the stratosphere moving away from the polar areas is similar to that of a basin of water which has a hole in the bottom, and in which the water is rotating. The Antarctic area may be considered as ‘having the air abstracted from the upper portion of the troposphere concentrically with the pole, and the circu- lation is as shown in fig. 2. In the Arctic area, however, the abstraction would appear to be from above the two cycionic areas, one over the N. Pacific and the other over the N. Atlantic, as well as from the Asiatic plateau in the summer. More information is required as to the temperature varia- tions of the stratosphere with latitude. Until our knowledge is more complete any theory based on temperature gradients can only be a tentative one. Ifthe theory here outlined in some degree encourages further investigation into the tempe- rature conditions of the stratosphere, it will not have been given in vain. Phil. Mag. S. 6. Vol. 30. No. 175. July 1915. =D Leet III. The Estimation of High Temperatures by the Method of Colour Identity. By Cutrrorp C. Paterson and B. P. Duppine, A.R.C.Sc. (From the National Physical Laboratory.) * Synopsis. in | Ee experiments are described on the method of “ colour identity” adapted to the estima- tion of the temperature of incandescent substances such as metal or carbon radiating in the open; by this method the “true” temperature of certain bodies as distinct from their “black body” temperatures can be arrived at with a very fair degree of accuracy. 2. By the colvuur-identity method the total luminous radiation (white light) from a black body is made identical in colour with that from the incandescent metal under ex- amination by adjusting the temperature of the black body until there is colour identity in the field of a Lummer- Brodhun photometer. 3. Comparisons are made of the results so obtained with those obtained by other methods, and the colour-identity method is shown to give the correct result for melting platinum. 4, Formule are deduced, based on the fundamental theories of energy radiation and the sensitivity of the eye, connecting the temperature of carbon and tungsten filaments with their lumens per watt, and it is shown that these expressions hold from the lowest to the highest values of lumens per watt. 5. It is shown that the colour-identity method of deter- mining filament temperatures is practically independent of the cooling at the ends of the filaments of ordinary lamps. 6. An explanation is given of the principal factors and limitations of the colour-identity method, in which it is shown that accurate results should be obtained so long as the bodies under consideration act as “ grey”’ bodies throughout the visible spectrum, and that there will be a tendency to error to the extent that they depart from the grey body condition in the visible spectrum. 7. The colour of the radiation from melting platinum is shown to be the same as that from a carbon filament lamp operating at 2°6; lumens per watt, or 4°7; watts per mean spherical candle, or approximately 3°8 watts per mean _hori- zontal candle. * Communicated by the Authors. From the Proc. Phys. Soc. London, April 15, 1915, p. 280. Estimation of High Temperatures by Colour Identity. 35 The work described in this paper is not in the nature of a complete investigation of the subject. It had for its original object the determination of the colour of the light from molten platinum under tie open radiation conditions which prevail in the realization of the Violle standard of light. A good primary standard of light must not only be constant and accurately reproducible, but the colour of its light should approximate to that of the sources which are ordinarily used in practice, so that large colour differences will not be in- volved in the photometric measurements for which such a standard is used. The object for which the investigation was started was completed over two years ago, but the pro- gress of the work indicated some unexpected phenomena which it was intended to investigate further. Pressure of other work has up to now prevented this being done, and the authors desire at this stage to publish this preliminary note onthe subject. The accuracy of the work is the accuracy of preliminary experiments in which all reasonable precautions have been taken. Values given for temperature certainly have not an absolute accuracy of more than 1 or 2 per cent., but the methods described are capable of a higher precision, and this will undoubtedly be attained in the fuller investigation which it is intended to undertake. General Discussion. Optical pyrometry is almost exclusively concerned with the intensity of the light emitted by a luminous body in any given wave-length. The colour of the light thus dealt with is fixed by the wave-length or wave-lengths chosen for the measurements, and colour differences do not occur. In ordinary photometry the sum of the intensities of the light emitted by a source in all wave-lengths over the visible spectrum is compared against the sum of the intensities of the light emitted by another source. The radiation from each source has thus a composite colour whose characteristics will depend on the relative intensity of the light in each wave-length. Although both sources may radiate according to the law of a black body, if there should be a difference of temperature between them the composite colour or hue of the radiations from the two bodies will differ, and it becomes necessary to compare intensities which are not of the same ‘colour. or most solid radiators the colour of the light is a perfectly definite quality, and forms a criterion of the state of incandescence of such bodies. Most bodies are more or less selective in their radiation, but there is a certain group, consisting mainly of metallic substances, which although D2 36 Messrs, Paterson and Dudding on Estimation of apparently selective in favour of the visible spectrum as @ whole, emit light throughout that spectrum without any appreciable deviation from the distribution to be found in the visible spectrum of a black body*. For instance, con- sider a tungsten filament adjusted to a suitable temperature, and compared spectrophotometrically against a carbon fila- ment. The one is mainly selective in favour of the visible spectrum as a whole, and the other acts in this respect as a black body. The spectrophotometer, dealing only with the visible spectrum, cannot detect any relative difference between the two at different wave-lengths throughout the portion of the spectrum with which it deals, and thus a comparison of the total visible radiation is possible with an ordinary photometer, exact identity of colour being obtainable. That is to say, these substances virtually radiate as “ grey” bodies, so far at least as the visible spectrum is concerned, and it is this close approximation to grey body radiation in the visible spectrum which lies at the root cf the method discussed in this paper. Hence, identity of colour can be obtained not only when comparing one tungsten lamp against another, but also when comparing a tungsten lamp against a black body. If the temperature of these bodies is pushed to an extreme value a very slight difference of colour is perceptible at the point where the colour balance is closest, but such differences are too small to prevent an observer obtaining consistent results in judging the colour balance between two radiations. A comparison of colour is made similarly to photometric comparisons of intensity. The current in the comparison lamp is varied so that the colour of the light fluctuates on both sides of the mean, first inclining to be redder and then to be bluer than the light from the test source. The current in the lamp is then readily determined at which the observer judges the colour balance to occur. It must be remembered that in these comparisons it is the hue of so-called white light which is under consideraticn, and not that of spectral or other colours. The colour-identity method depends on the combined effects of the light emitted in all wave-lengths in the visible region. If the intensity is relatively greater at the red end than at the biue end, the hue of the resulting radiation will tend to be red, and vice versa. The radiation from a black body at 1750° C. has a definite hue depending on the relative * Coblentz, “ Radiation Constants of Metals,” Bull. B. S. vol. v. p. 359. Hyde, ‘Selective Emission of Incandescent Lamps,” Trans. Ill. Ene. Soc. 1909. Lhgh Temperatures by the Method of Colour Identity. 37 proportions of the energy in the red, green, and blue regions, and any other radiator emitting light in the same relative proportions will have the same hue of radiation, no matter what the absolute intensity of the radiations. Thus it is that the radiation from a grey body will be identical in hue with that of a black body, and compared on the colour-identity basis the grey body will be given its true temperature. The optical pyrometer, on the other hand, only takes account of the relative intensities of the light from the black and grey bodies, and therefore estimates the temperature of the grey body at a value far below its true temperature. It follows, therefore, that the measure of the accuracy of the colour- identity method is the extent to which bodies radiate as grey (or black) bodies throughout the visible spectrum. Throughout this paper the usual conception of a grey body is adopted—. e., one which, at any temperature, does not radiate as much energy in the various wave-lengths as a black body at the same temperature, but in any wave-length the intensity per unit area of the surface is a constant fraction of that of the black bedy in the same wave-length. By a selective body is meant one in which the amounts of energy radiated in the various wave-lengths throughout the whole spectrum do not bear a constant “proportion to those in the same wave-lengths for a black body at the same temperature. Section 1 of this note deals with the establishment of electric sub-standards of colour, which are intended to serve for defining the colour of the radiation from any incandescent bodies compared against them, and so to fix the temperature of such bodies in terms of the temperature of-a black body whose radiation is identical with theirs in colour. Section 2 gives the determination of ‘colour identity ” temperatures "of carbon and tungsten glow-lamps when burning at different efficiencies, and contains expressions for such efficiencies in terms of temperature based on Wien’s equation for intensity of energy distribution and Nutting’s equation for the sensitivity of the human eye. Section 3 discusses the accuracy of such determinations, and deals with the “ colour identity ’’ temperature of platinum at the melting-point, showing that even for a selective radiator such as platinum this ‘temperature is a measure of the true temperature of the platinum filament, although it is glowing under open radiation conditions. Filament tempe- ratures for carbon and tungsten (vacuum and gas-filled) are also discussed. 38 Messrs. Paterson and Dudding on Estimation of Section 4 deals with the colour of the radiation from molten platinum in relation to the practical usefulness of the Violle standard of light. 1. Electric Sub-standards of Colour for the Determination of Temperature. In spite of the fact that the device of colour comparison by means of a Lummer-Brodhun photometer has been used for many years by various observers for obtaining equality of efficiency of glow-lamps of the same type, it is not generally . realized how easily and with what precision such colonr comparisons can be made. Morris*, Stroud, and Hllis employed this method in 1907, and extensive use has been made of it for investigating selectivity and other properties of radiating substances by H. P. Hydet, with whom Cady and Middlekauff have sometimes collaborated. Hydef, in discussing the question of colour identity and temperature (p. 40, loc. cit.), showed that a colour match with a black body might be regarded as indicating that the temperature of the black body was at least as high or higher than that of the body compared against it, but he expressed the opinion (p. 39) that under the condition of colour identity two different radiators although with continuous spectra would not be at the same temperature. A black body furnace electrically heated and capable of being raised to a temperature of 2200° C., with a clear in- ternal atmosphere, was kindly put at the disposal of the authors by Dr. J. A. Harker, F.R.S. The very excellent arrangements of this furnace need not be explained in detail here. The apparatus is shown diagrammatically in fig. 1. A is a black body kept clear of fumes by a stream of nitrogen admitted at K. Bis a diaphragm with glass window, which permits only light from the centre of the incandescent surface co pass down the photometer bench, and J are screens to cut off extraneous light. C is a carefully seasoned electric lamp, the current through whose filament can be accurately measured by means of ammeter D. Between the two at EH * Morris, Stroud, and Ellis, ‘ Electrician,’ vol. lix. p. 584. + Hyde, Cady, and Middlekauff, ‘Selective Emission of Incandescent Lamps,” lll. Eng. Soc., New York, vol. iv. 1909, p. 884. Hyde, “ Physical Characteristics of Luminous Sources,’ Lectures, Johns Hopkins Univer- sity, 1910. Hyde, ‘“ Radiation Laws for Metals,” Astrophys. Journ. vol. xxxvi. 1912, p. 89. t Hyde, “‘The Physical Production ot Light,” Journ. Franklin Inst. vol, elxx. 1910. Fligh Temperatures by the Method of Colour Identity. 39 is a Lummer-Brodhun photometer head. This photometer is not used to compare the intensities of the two sources of light A and C, but to determine when the hue of their radiations is identical. Two optical pyrometers were used of the Siemens and Féry types, by means of which the temperature Diagram showing arrangement of Apparatus used to obtain the Relation between the Current in Carbon and Tungsten Filament Lamps, and the Temperature of a Black Body at Equality of Hue of their Radiations. of the black body was determined before and after each colour determination by the photometer. Complete deter- minations were made on two separate occasions and the mean result taken—every photometer and pyrometer setting being made on each occasion by two observers. ‘The furnace was first run at a relatively low temperature, and the electric heating adjusted so that it would remain at a constant tempe- rature sufliciently long for both photometric and pyrometric readings to be taken. The hue of the light from the electric lamp C was varied by means of rheostat H until there was exact identity of colour in the photometer. In doing this it is, of course, necessary to place the photometer so that there is also equality of brightness. Two or three settings were made by each observer betore and after which the pyrometers were read. ‘The same process was followed in a series of increasing temperatures up to about 2200° C.—the maximum temperature to which the furnace was carried in these experiments. Two electric lamps were calibrated in this way, one having a carbon and the other a tungsten filament. These two lamps calibrated in the above manner form intermediate standards of colour against which any glow-lamp can be matched, and the tem- perature of the black body hxed to which the colour of its light corresponds. The temperatures so determined are on 40 Messrs. Paterson and Dudding on Estimation of the optical scale, using pyrometers for which the dominant wave-length is X=0°650 pw. Fig. 2 shows the curve connecting current in the lamps and the temperature of the black body for colour identity in the case of each of these lamps. It was found that the sensitivity of the process of colour matching is more than equal to that of temperature measurement by optical pyrometers. Fig. 2.—Carbon (A) and Tungsten (B) Colour Standards. of Black Body ai Colour Meteh. (Cent. absclute scale.) emperature fn 4 Hs TE a ea BS 0° Of «62 403) 04 GS (O6. 07" G8) 09) oye Curreni tn Amperes. Curves connecting the current in the lamps with the temperature of the black body whose radiation is identical in colour with that of the lamps. Duplicates of these two lamps were then made for use in experiments where their continued employment at the higher efficiencies might affect their constancy. 2. “Colour Identity” Temperatures Corresponding with Different Efficiencies. Itis common for the specific consumption of glow-lamps to be stated in terms of the watts per mean horizontal candle. The only rigorous way, however, is to state it in terms of watts per mean spherical candle or in lumens per watt. Throughout this paper the latier designation is used, repre- senting as it does the true measure of the ratio of the total light emitted to the power supplied. In the table of results Lhigh Temperatures by the Method of Colour Identity. 41 the approximate watts per mean horizontal candle of the glow-lamps is also given, since this is the more familiar desig- nation, but it is not rigorous on account of the varying ratios of mean horizontal to mean spherical candle-power to be found in different lamps. It is to be noted that in the deter- mination of the total light emitted from the glow-lamps the light which is obscured by the cap of the lamp is counted as being radiated and not absorbed, but in any case this light is less than 1 per cent. of the total for filaments of ordinary form. The objects of the measurements are:— (a) To find the relation between the various values of Jumens per watt for tungsten and carbon filament lamps and the corresponding temperatures of a black body on the basis of colour identity ; (6) To ascertain to what extent the temperatures so measured represent those of the principal parts of the glowing fila- ments, having regard to the cooling effect of the filament supports ; (c) To find laws connecting lumens per watt and corre- sponding “colour identity’ temperature for tungsten and carbon filaments. Evidence is given later in the paper to show that there seems to be justification for the assumption that the colour- identity method at any rate in certain cases gives within narrow limits a measure of the true filament temperature. This assumption is therefore made in what immediately follows here, and it will be seen that the results which follow from this assumption, whilst not proving its validity, are in agreement with those of Forsythe, who determined tempera- tures by more orthodox methods, (a) A number of carbon and tungsten filament lamps were selected for the measurements. The carbon lamps had both flashed and unflashed filaments. The tungsten lamps had squirted and drawn filaments of different diameters and lengths, so that the effect of the cooling of the ends by the leading-in wires if appreciable might be observed. All the lamps were measured for lumens per watt at different voltages up to the highest they were capable of standing without deterioration. They were then compared for identity of colour against the colour standards, and in this way the temperatures of a black body were determined which corresponded with the various values of lumens per watt. The results are given in Table I. and plotted in fig. 3. 42 Messrs. Paterson and Dudding on Estimation of TasiEe I.—(Reduction factor = Mean spherical candle-power divided by mean horizontal candle-power.) | Temperature Watts Lumens ne | of black body Volts. | Pe sat| ceate. | at identity of | easels | watt. colour (Cent. candle. | absolute). Carbon Filament Lanup ( flashed) No. 1. 100 volts, 16 candles. Reduction Factor=0°85,. 49-2_| 111-, 0:09, 1,515 | 595-1 38°, 0-28, 1,640 po 1S", O77. 1,755 eeot 7-9, 1:35 1,865 | 92:5,| 5:0, 2:14 1,960 1030 | 3:7, 2-85 2,055 1100} 2:5, 4-14 2,120 1300 | 1:5, 6-85 2,250 1350 | 1:3, a 2,300 1400 | 1-2, 8:8. 2,325 100 volts, 16 candles. Carbon Filament Lamp (flashed) No. 3. || Watts | Lumens rei | Volts, | Per mean per ef Dee ‘| horizontal] watt. |@ : entity o candle. colour (Cent. absolute). Carbon Filament Lamp (flashed) No. 2. 100 volts, 16 candles. Reduction factor—0°89, 20°, 0:55, 1,715 On oily A4s 0:80 1,775 Phy ols. 1-09 1,835 80 ae 1-45 1,890 85 61, 1:84 1,935 90 4:8. 9:33 1,980 100 32, 3-43 2,075 105 2-6, 4:17 2,120 107 2-5, 4-44 | 2,185 Carbon Filament Lamp ( flashed) No. 4, 200 volts, 16 candles. ; | Reduction factor=0°86,. Reduction factor=0°85. | 140 18°, 0:60 1,710 65 19°, 0°55 120) de 12, 0°85 1,770 70 13°, OT; PEED 01) GO 9°0;, 1:18 1,835 75 L0:,; 1-06, 1,885 || 180 52, 2°09 1,955 80 (0, 1-40, 1,880 || 200 a2. 3-41 2,070 85 6:0, Li 1,935 210 2:6, 4-05 2,120 90 4:8, 2°20 1,985 220 22, 4°95 2,165 100 373, 3o21 2,070 | 105 nies 3°83 2,115 || Carbon Filament Lamp (unflashed) No.5. 107 2°63 oki 2,130 200 volts, 16 candles. io i 51 a Reduction factor =0°82.. 1130 -| 1-4, 7-50 281g) | | Meer 22r 0-47 1,710 | 135 2: 8-58 2,345 150 | 15, 0°68 1,775 | 3 160 LES 0°95 1,835 Tungsten Filament Lamp (Drawn) No. 6. an ae See aa 115 volts, 30 watts < Akay 2.9/2 ‘Non , ; 210 30, 3°36 2,095 Reduction factor=0:79. 220 2°35, | 4:08 2,140 —) . e/ Pe a dee sae Tungsten Filament Lamp (Squirted) No. 8. ie Ona ae 0-90 1,755 1050 yo #80 were biel 63, 1-57 1,865 ieee 68:7 4-0, 2:47 1,960 Reduction factor=0 78,. 93:2) 19, 517 2,145 Sys al: oa O72 °°:|) Teo 120 %5 40) LiL, 9-00 2,325 40 ils, 085 | 1,746 Oy Uke 1:26 1,800 Tungsten Filament Lamp (Squirted) No.7. || 99 6: 1 6 162 1,860 105 volts, 32 watts. = s 7 3 al oo : = ) “bg 2°03). ’ Reduction factor =0°79. 65 3-0, 3-98 2.010 Joe Lon, 0°63 1,640 70 2°6, 381 2,055 48:2) 93, 1:06 1,755 75 2:2) 4°47 2,095 ay Gr tana: 1:84 1,865 80 ee 52 2,135 6x70)\4), S'Dy 2°78 1,960 90 1-4, 6°65 2,215 90°5 ie 3°66 2,145 LOO ))) vez. 8:19 2,285 LL Oo OF 9°54 2,325 105 1:0 9 06 2,920 eee | Eigh Temperatures by the Method of Colour Identity. TABLE I.—continued. 43 Volts. | Tungsten Filament Lamp (Squirted) No. 9. 135 Tungsten Filament Lamp (Drawn) No. 11. | Watts per mean horizontal | candle, | Lumens per watt. Temperature of black body at identity of colour (Cent. absolute). 105 volts, 60 watts. Reduction factor=0'78.. oo - N a uN OS TN DE TOT Oo Oe oe — O Ret Ss rt et te tO 0 GO HB OUT OD DmoeW pe eR u 0:82. 0-76, | 0-77, 1:01 1°35 1-74 2-18 2-67 3:17 3-73 ‘ 0 7 3 7 12 06 66 951 104 11-2 4 5: 7: 8: 8 | 12-1 13-0, | — — S ore +7 bo ANIoIinqoood ange 200 volts, 20 watts. Reduction factor=0°78.. Sun si ~- aonwnono ~) ac re bok bat est bet BORD CB OT 6) Pe OS 08 SEE ot oe - Oo | 3°87 1 0 Wal 19 oonnncdod b LDR Re ee OO OM ~Is1 Volts. 1] ' 100 110 | 120 | 130 140 150 160 170 180 190 210 | 230 | 240 260 Watts per mean horizontal candle. Lumens per watt. Temperature | of black body at identity of colour (Cent. absolute). Tungsten Filament Lamp (Drawn) No. 10. 100 volts, 15 watts. Reduction factor =0°78.. 8-6, = Oreo bo G2 bo GW OO we CoM aA oF Cm urea: © Oo bom 09 Ro “vu om oF SOS 0 0 bth ret tp by Goi Aanr-aAnmneorw ee Or Cc Or 6 CO OO H= GO Coty DOeAWR WAS ONwucone ote SO Mri 1D MH wwo Ww bv ] | 1,800 1,860 1,915 1,960 2,015 2,055 2,105 2,140 2,225 2,290 2,325 2,360 2,400 2 430) 2,455 2,485 2,505 Tungsten Filament Lamp (Drawn) No. 12. 230 volts, 60 watts. Reduction factor=0°78.. . -- A) a Te Oo oS ~ Set ret — bolo bo Os Ge Ors = DOE OO 09 AT DTS Coe & 1:04 134 1:74 2°17 S = oOo O° 4 4 SDH Ur He C2 CNY DAS ATE OE Ae font | 2,305 2,360 In obtaining the higher temperature values for plotting on fig. 3, it is very useful to make use of a “‘watt-temperature eurve. The carbon lamp, for instance, cannot with safety be run for long periods at temperatures in the region of 2000° ©. obtained by the identity of colour method the resulting curve will be found to be alogarithmice one, and no devia- tion whatever can be detected from such logarithmic over If the watts be plotted against temperatures 44 Messrs. Paterson and Dudding on Estimation of the range between the highest and the lowest observed values. Fig. 3.—Ordinary 100 and 200 volt Carbon (Curve A) and Tungsten (Curve B) Filament Vacuum Lamps. 2600 Z 2500 & ‘ ws = 2400 Es (i a ee pun i Pawn E a ps a wd ~ 2200 ae ~ aS S 2100 S 22000 = oa = i900 cS el = 1800 Ra eS ae cS 8 1700 a © 1600 Bi = 8 1500 via a) * 1400 1300 ea es 2 E20. Ug) ag. ign Ulin), Vg.) Wg) 740.) coal Oat iy en eae Lumens per Watt Curves connecting lumens per watt of the lamps with the temperature of a black body whose radiation is identical in colour with that of the lamps. f This is illustrated in fig. 4, in which the logs of temperature and watts have been plotted for both carbon and tungsten lamps Nos. 3, 5, 6, 8,and 12. It is safe from a knowledge of the watts in any lamp to deduce intermediate temperature values from such a curve, so that the actual number of colour comparisons may be a minimum, and the burning period of the colour standard reduced. The watt-temperature relation is given, for carbon filament lamps, by W ac TH, and for tungsten filament lamps by W ao, T°. Considering again fig. 3, the first point to notice is that no difference can be detected between the various carbon lamps tested or between the different tungsten lamps. Whether the carbon filaments are flashed or unflashed, and the tungsten 4 § P y : } High Temperatures by the Method of Colour Identity. 45 filaments squirted or drawn, appears to leave unaffected the relation between lumens per watt and the corresponding ‘colour identity” temperature of a black body for either of these types. Fic. 4, tf aes | | _—s Ce eT a ad [ink NAS a ° | —“| 22 Log of Watis Consumed. = 06 3°20 3°22 avi ~~ 3/26 3°28 3°30 3°32 3°34 3°36 333 3°40 Log of Temperature of Black Body at Colour Match. Curves connecting watts consumed by carbon and tungsten filament lamps, and the temperature of the black body whose radiation matches in colour that of the lamps. A is tor Lamp No, 5. Carbon filament.) B ” ” No. 3. ( ” ” ) 3a an ae be, (Tungsten ,, ) Py", oon end 8.) (5/5, anes | It will, therefore, be seen that all the results may be taken as lying on two curves, one representing the carbon group and the other the tungsten group, and so closely do the points keep to the curves that very few observations lie more than 1 per cent. in temperature from the mean curve. This implies that in all ordinary lamps of the same cha- racter (vacuum tungsten or carbon) the colour of the radiation from the whole filament, including the cooled ends, is the same for the same value of lumens per watt. Therefore, to considerable accuracy, it may be said that a knowledoe of the lumens per watt of a lamp implies a knowledge of the temperature of a black body whose radiation is the same in, hue as that of the lamp. 46 Messrs. Paterson and Dudding on Estimation of (6) The following considerations show to what extent a temperature, determined as above, may be regarded as the temperature of the main elowing part of the filament. If there were no cooling at the ends of a filament it would be equally bright for the whole of its length. The cooling, however, as shown by Hyde, Cady, and Worthing™ is appre- ciable, but it must be remembered that the colour of the light is governed by the part of the filament giving off most light. The ends of the filament give off actually very little light because the amount emitted falls off according to a very high power of the temperature (T’? to T°). Hence the effect of the dulled ends of the filament on the colour of the total light from it is exceedingly small. In an actual case T the total light emitted below the point where the filament began to become measurably dull was only 5 per cent. of the whole, and a large percentage of this amount differs only very slightly in colour from the light emitted by the remainder of the filament. The following measurements, Table II., were made of the total effect on the measured temperature due to end cooling by determining the ‘colour identity” temperature of the central portion of the filament only and comparing it against that of the whole filament, including the cooled ends. TABLE II, | Temperature of black body for colour identity. | Lamp. Volts Whole filament. Centre of filament. 200 2,245 2,255 INC MEI oe eee ate Oe | 110 1,885 1,898 54 1,556 1,570 105 2,280 2,290 INE is eee Seach aces 60 1,920 1,930 40 1,710 1,722 Motor headlight. ve ane ee 16 volts, 50 watis ey ; 2 ? 2,240 2,285 Lamp No. 11 was an ordinary 200 volt, 20 watt tungsten lamp, and No. 9 was rated for 100 volts, 55 watts; the differences of filament diameter and distance beterean supports * Amer. Ill. Eng. Soc. Trans. 6, pp. 288-257. + See Hyde, Cady, and Worthing, Joc. ect. High Temperatures by the Method of Colour Identity. 47 were, therefore, as large as is usually met with in practice. The motor headlight filament was for 16 volts and 50 watts, and therefore represented an extreme case. It will be seen from this that unless a very thick, short filament be taken with abnormal end cooling, the measured colour identity temperature will be that of the central bright portion of the filament, within about 1 per cent., whilst in the extreme case of the headlight lamp it is of the order of 2 per cent. It is obvious that the cooling effect for carbon filament lamps is considerably less than for tungsten, and is, in fact, quite inappreciable. It is thus clear that the colour-identity method gives results which depend very closely on the temperature of the central portion of the filament. If it may be assumed that the method also gives the true temperature of lamp filaments, the figures in Table I. and fig. 3 indicate the appreciable difference of efficiency existing between the carbon and tungsten lamps for the same temperature, and therefore establish the selectivity of the tungsten filament in favour of the shorter wave-lengths, a subject upon which much has been written, and which has been thoroughly investigated by Dr. E. P. Hyde*. This difference in efficiency would, if anything, be very slightly increased by taking into account the end cooling of the filaments, the tendency of which is to act in favour of the carbon lamp. Also, if the carbon fila- ment is “‘greyer” than the tungsten filament in the visible region the apparent difference of efficiency will be increased. (c) Referring to the curves shown in fig. 3, and bearing in mind what has been said in the foregoing remarks, it becomes of interest to know if a relation connecting lumens per watt and temperature can be deduced from our knowledge of the phenomena involved, and especially to ascertain how nearly the experimental observations conform to such a relation deduced from theoretical considerations. We have to consider, therefore, how the rate of dissipation of energy by a lamp filament, i. e. the watts T, increases with a rise in temperature, and also how the eye estimates the rate at which this energy is radiated. The eye is only sensitive to a small portion of this energy, 2. é. that emitted in wave-lengths lying approximately between 0-3 mand 0°8y. Further, the eye does not appreciate the intensity of the energy radiation in any wave-length over * See Hyde, loc. cit. | The rate at which energy is radiated is power, and is spoken of hereafter as radiant power, = ee a eee ee OOOO es — -—-— ~~~ a a — eee Se — — — SC — ' 48 Messrs. Paterson and Dudding on Estimation of this limited range in direct proportion to the amount radiated, but weights it according to its own peculiar sensitivity to energy of that wave-length. This appreciation of power by the eye is expressed in lumens which may be defined as the measure of the appreciation of the eye for radiant power. An expression must, therefore, be found connecting lumens and the temperature of the radiating body both in terms of the power distribution throughout the visible spectrum and of the sensitivity characteristics of the eye. The theoretical investigation of the problem thus subdivides itself naturally into three distinct parts :—- (a) The rate of energy dissipation of the radiator at any temperature. (6) The quantitative distribution of this radiant power throughout the spectrum at any temperature, with special reference to that range of the spectrum over which the energy stimulates the sense of vision. (c) The relative capacity of equal amounts of radiant power in different wave-lengths for stimulating vision, this being necessarily referred to the average or normal human eye. (a) Relation between Watts and Temperature.—Attention has been already drawn to curves showing the relation between the rate of dissipation of energy by a lamp and its temperature as measured by the colour-identity method. Many lamps of ordinary dimensions have been examined, and in all cases the results can be expressed by an equation of the form watts c I” or (log W= log D+mlogT), . . () m being 4°; to 4°, for carbon lamps and 5:0; to 5°, for tungsten lamps. In no case has any appreciable deviation been observed from this logarithmic relationship for temperatures ranging from 1700 deg. to 2300 deg. abs. This relationship is at once recognized as being identical in form with that ascribed to Stefan and Boltzmann connecting the temperature and radiant watts of the ideal black body, m in the latter case being 4°. (b) Distribution of Radiant Power throughout the Visible Spectrum.—In the case of the ideal black body, the radiant power in any wave-length of the visible spectrum at any temperature below 3000° C. can be expressed according to the well-known law of Wien C2 BE, = Garten) Oe igh Tenperatures by the Method of Colour Identity. 49 Hy being the radiant power of wave-length A at temperature T, C, and ©, being constants, »=5 (for a black body). Seeing that in the experiments described in the following section (3), in which the colour of the radiation of melting platinum was found to be identical with that of a black body operating at the same temperature, it is reasonable to assume that the power distribution, at least over the range of the visible spectrum, can be expressed by a formula of the above form. Lummer and Pringsheim found this condition to be closely fulfilled by radiators having the characteristics of platinum. | When the visible spectrum only is under consideration, the values of C,, C2, and can vary considerably without affecting the shape of the curve by an amount corresponding to a difference of temperature of 10° C. in the region of 2000° C. (c) Sensitivity ofthe Eye to Energy of Different Wave-lengthe. —By examining a large number of persons, Nutting* has obtained data connecting the wave-lengths of radiant energy and the luminous sensation produced per unit of power in that wave-length. He expresses his results in the form pS oe Am Vy=Va(22) AMOR BST where V,, is the photometric value of a unit of power in the wave-length of maximum sensitivity A,,, « is a constant, e is the base of Napierian logarithms. For the luminous intensities ordinarily employed in photo- metry he gives a=181 and An =0°55 pw. Combining the expression for power distribution and the sensitivity curve for the eye (equations 2 and 3), we obtain for the photometric value of radiant power of wave-length X, E,V,, and for the photometric value of the whole of the radiant power ( SELENA BYE TN 2 0 If the power distribution can be represented by Q B,=Pa7"e *7, ‘ e e ° Py - (5) * Bulletin of the Bureau of Standards, vol. vy. p. 261, and vol. vi. p. 337. Phil. Mag. S. 6. Vol. 30. No. 175. July 1915. Ez D0 Messrs. Paterson and Dudding on Estimation of Nutting* shows that the above expression for the photo- metric value of radiant power reduces to L=A(14+ ir)" a where the photometric value of the total radiant power in lumens at a given temperature, T, is represented by L. A=PV,N ean air: . Wp; al Dee p=n+a—l. It follows from what has been said above that the lumens radiated by carbon or tungsten filaments should be capable of being represented by an expression of the form of equation (6). Combining with this expression that for the watt- _ temperature relationship for the filament under consideration (equation 1), we get for the equation connecting lumens per watt and temperature L B as —m W =A,(1+7) bees or expressed for convenience in the logarithmic form L B log Ww =C—m log T—p log( 1+ + ED From the measured values of lumens/watt and temperature, which are plotted in fig. 3, the following values of the constants in the foregoing equation are found. For carbon filaments L 10 Wr and for tungsten filaments log = 21:51 —458 logy T— 185 loguo( 1+ T (8) loging, = 28°31) —5'1 logy T—185 loguo( 1 + Tr) (9) Tne curves drawn in fig. 3 are those derived from these equations, and it will be seen at once how nearly the obser- vations fall on the curves; in fact, it would hardly be possible to find a form of curve which would fit the observations better. The error in temperature rarely exceeds 2 per cent. * Loe. cit, High Temperatures by the Method of Colour Identity. 51 and in most cases is considerably less than 1 per cent.—1. e. within the possible error of the experiments. Further, it is shown later that a very large extrapolation of the tungsten eurve by this formula indicates a value for the melting-point of tungsten which is not inconsistent with that found by other observers. The formule indicate that the maximum attainable efficiency would occur in the region of 6000° C., which is quite in accord with accepted theories. The origin of the constants in equations 7, 8, and 9 should be particularly noted. The watts-temperature relationship for a lamp has been found to be of the form W « T” (equation 1), m being a constant which appears in equation (7). The constant p of equation (7) is equal to n+-«—1, where “—n”’ is the index of X in equation (2) of the ‘* Wien ” form, which is assumed to give the power distribution curve for the filament throughout the visible part of the spectrum, and where “a”’ sitivity of the eye, and has the value 181. B= Q aXm equation 6), where Q is the other constant in the assumed Wien equation for power distribution, “ta” has the value as before of 181, and A,, is the wave-length of the energy to which the eye is most sensitive, 7. e., 0°55 wu. Hence (see org a OOD p=185, Q=14,500, and B=145-0 approximately. Before leaving the consideration of these equations con- necting lumens per watt and temperature, it is desirable to discuss one or two points which at first sight may appear to have an important bearing on the deductions that can be made from the foregoing results. Firstly, as regards “‘n” in equation (5) and “m” in equation (7). Fora black body the value of “n” in equa- tion (5) is 5, and it will also be 5 for a true grey body whose radiation in all wave-lengths bears a definite proportion to that of a black body. It will not necessarily be 5, however, for selective bodies, although, as in the selective bodies under consideration, they appear to radiate very much like grey bodies over the visible spectrum. In equation (7) the constant “ m,” which is derived directly from equation (1), can only be regarded as connected with “n” (n=m-+1) in equation (5) if the latter represents the distribution of power throughout the whole spectrum, and not merely in the visible spectrum. This latter is the assumption made in using equation sige this investigation, and the For a true black or grey body m=4, n=5, is derived from Nutting’s equation for the sen- 52 Messrs. Paterson and Dudding on Estimation of extent of the work described here does not justify the wider application of equation (5) to the whole spectrum for substances which do not behave as true black or grey bodies. Lummer and Pringsheim*, investigating platinum, state that the distribution of power throughout the whole spectrum for platinum is given by the following equation :— 15,600 he Ga corresponding to the form for an ideal black body of 14,500 E,=—CvA 2 CF. If this assumption were justifiable, the authors’ values for tungsten work out very nearly the same as those given by Lummer and Pringsheim for platinum; but, for the reasuns just stated, too much significance must not be attached to this agreement. Coblentz t expresses the opinion, based on an investigation of several metals, that ‘“‘n”’ in equation (5) is not a constant, but is a function of wave-length and temperature. There is nothing in the results discussed above which is in- consistent with either Lummer and Pringsheim’s or Coblentz’s suggestions. Throughout this work the practical case has been dealt with of filaments mounted in exhausted globes in which there is undoubtedly some loss of watts and efficiency due to the cooling effect of the leading-in wires. If conclusions of a fundamental nature are to be drawn from the results obtained, it is necessary to know to what extent this cooling is likely to affect the constants given in equations (8) and (9). In the work already referred to by Hyde and Cady, figures are given for the loss of watts by conduction at the ends of filaments, and columns I. to IV. of the following table are from this paper. In columns V. and VI. are tabulated the values of lumens per watt calculated from column II., using average values for the reduction factors of the types of lamp under consideration, and the corresponding temperatures taken from the curves in fig. 3. * Lummer and Pringsheim, Verhandlungen der Deutschen Phys. Gesell. pp- 23-25 (1899). + Bulletin B. S. vol. v. pp. 388-879 (1908-1909). Fligh Temperatures by the Method of Colour Identity. 53 TaB_e III. i. II. TEL, IV. v. VI. | ? Ca as a Watt Effi- Lumens Temp. ; Lamp. * Imean horizontal face ciency | Waits Sai candles. ; loss. ; ; ee es Ae | Carbon 115 volts... id | 2pec. | 4p.c. 3-4, 2,085 | 5 is 180 3 p.c. 5 p.c. 0°5, 1,730 | Tungsten 115 volts,’ 125 4 p.c. @ pe. 8:0 2,280 60 watts . 3 ies Spe. f 1G pic. 0°9, 1,745 | These losses are calculated for different efficiencies as a percentage of the | watts which would be required to maintain the filament throughout its whole | length at the temperature of its midpoint, assuming no loss by conduction. " ! The ditference between the temperature corresponding to the colour of the light radiated from the centre of the filament of any lamp of the above types, and that radiated from the whole filament are given in Table II. The following results are obtained by using the values in Tables II. and III. for ascertaining what would be the behaviour of the filaments used in this investigation had there been no cooling. Carbon Lamps. In ordinary Jamps the watts and temperature are connected cy p by the relation logioW =C,+ 4°58 logo yl ° . . . (12) Allowing for the watt loss as per column II., Table III., due to conduction, the watt-temperature relation for a tilament kept at uniform temperature throughout its length and having no conduction losses is : logy)W,=C,—0°191+ 4°63, logyo Ls = (13) Likewise for ordinary lamps the relation between lumens per watt and temperature is expressed by 5 155 logos = C, —4°58 logy9 T—185 logy, (1 + 7 ). (14) Allowing in a similar manner for the efficiency losses (column IV., Table ILI.), the lumens per watt and tempe- rature relation for the ideal filament is given by 155 log sy’ =C, +0°199 4-63; log T—185 log(1+ T): (15) 54 Messrs. Paterson and Dudding on Estimation of. Tungsten Lamps. Similarly the equations for tungsten filaments are changed when cooling is allowed for, from los WHC, 45-11dgT, 2)... 2 Bair 3 155° and log ww =C,—5'Llog T—185 log (1+ a . (24) to log W =C;—0°700+5°3 log T, . . . 2. [ieee and log = C,+0°703 —5°3 log T—185 log (1 a a , (26) and the same phenomenon is observable as for the carbon filament lamps. By comparing equations (13) and (15) and (25) and (26) respectively, and remembering that (13) and (25) deal only with the watt relationship, omitting lumens altogether, it will be seen that practically all the change produced by cooling is in the watts. That means that the loss of lumens at the ends of a filament is virtually equal to the gain in lumens due to the centre of the filament (as shown in Table II.) being at a slightly higher temperature than that ascribed by the colour-identity method to the filament as a whole. The fact that temperatures based on the colour-identity method and covering so wide a range are thus found to fit in so well with fundamental theory seems to afford presumptive evidence that the colour-identity method gives values of temperature which are accurate at least from the relative point of view, and affords further experimental support for the conclusions which are arrived at on pages 35-37 asa result of a general discussion of the phenomena underlying colour tdentity. 3. Temperature of Lamp Filaments and of Melting Platinum. The foregoing experiments assume that a temperature ascribed to a filament by comparing the hue of its total radiation with that of a black body of known temperature approximates to the true temperature of the filament although the latter is glowing under open radiation conditions. If this approximation can be shown to be a very close one, the method might be of considerable use in certain branches of practical pyrometry. In what follows enough evidence is given of the correctness of the temperatures determined by the colour-identity method to justify the assumption for certain substances, and to warrant a more complete investi- gation of the subject. High Temperatures by the Method of Colour Identity 55 As regards the determination of filament temperature by previous observers, only three sets of determinations, those of Forsythe*, Von Pirani and Meyer f, and Langmuir, are given in comparable form. Tn some other determinations no mention is made of the watts per candle or lumens per watt of the lamps tested. In others “ black body” temperatures and not true temperatures are given of the filaments radiating in the open§. In Table IV. the results obtained by the authors are com- pared with those of the above-mentioned observers. Both Forsythe and Von Pirani only measured the mean horizontal candle-power of their lamps, and a reduction factor of 0°85 for carbon and 0°79 for tungsten has been assumed in both eases for the ratio MECP: It will be seen that the TABLE LV. True Temperature Filameut °C, Type of Lumens * * watt. R ep oMeas Patccthe Pirani and The Ai | Meyer. Authors. Tungsten ......... itr) Sa 1,980 | 2,069 2.010 ee 8:1, 1,982 / 2,072 2,014 > See 8-4, 2,008 2,084 2,027 EY Paveucuse 88, 2,020 2,100 2,041 IN pics ia 90, 2,025 2,109 2,051 ae 88, 2,035 2,101 2 044 eas tne 93, 2.040 2121 2,063 ePHOM f5.6...5.... 3°5, 1,820 1,935 1,818 la: a a 3°9. 1,847 1,966 1,846 SE Hie onl ch's sin 2s 3°9, 1,843 1,965 1,845 The lumens per watt are obtained from the values of watts per candle given by the authors, by assuming ratios of 0°79 and 0°85 respectively for the reduction factors Sa for tungsten and carbon lamps. The value of 0-9 is taken for the ratio of the Hefner to the British unit of candle-power. * Phys. Rev. vol. xxxiv. May 1912. + £. T. Z. 1912, May 2, p. 457, and July 11, p. 725. t Proc. of Amer. Inst. of Elect. Engineers, vol. xxxii. p. 1895. § Dr. H. Lux, #. 7. Z. May 28, 1914, gives tables connecting tempe- rature and watts per mean spherical candle of tungsten lamps. The values given in the table would appear to approximate to true tempera- tures, but the method described for determining the temperatures is that ordinarily used for obtaining black body temperature. Without further information of the methods used by Dr. Lux for the determination of true temperature, a useful comparison with readings of other observers is difficult to make. 56 Messrs. Paterson and Dudding on Estimation of authors’ results agree very closely with those of Forsythe, and it should further be noted that whilst agreeing ‘with Forsythe, who used the usual optical methods, in the case of comparatively non-selective carbon filaments, they also virtually agree with his results for tungsten, although the latter is admittedly selective. Von Pirani and Meyer found values of true temperature which are appreciably higher than Forsythe’s, and therefore also higher than by the colour-identity method given here. They are given in column IV. of Table IV., the values being taken off a curve through Von Pirani and Meyer’s values and reduced to the same basis of lumens per watt. stl Fel PCCP EEE Bee eee te Pb ob bp eh Pere A hel bd oe Pee td] Oy Os Sa a FR a TR a DW Pee Pt Humidity. (Litres of Water Vapour per Cubic Metre of Air). 0 4 4s 6 & 10 12 14 16 18 2 26 Temperature (C*). Diagram showing Values of the Temperature and Humidity of the Air over the Period during which observations were made on the Candle-power of the Pentane Lamp. obtained whilst observing the candle-power of the pentane lamp both in 1904-06 and 1912-14 are plotted. It is seen that a loose general relation exists between these variables, the lines A and B representing the best linear relation for the two sets of experimental data, and so to some extent condition (b) is fulfilled. If, now, it is assumed that the linear relationship shown by lines A and B in fig. 4 actually exists between these values of humidity and temperature, the following are the values which result for the coefficients A, B, and “C in the above equation (3) : C.P.=10 + 0°187(8—e) —0°008,(760—6) —0°076,(12—2) (4) G 2 84 Messrs, Paterson and Dudding on the Two facts should now be observed. Firstly, the tempera- ture and humidity effects act against one another, and in practice it is the difference between the two which is operative. Secondly, as at Teddington an increase of one unit of water-vapour is accompanied on the average by a rise of 1°-6, C., the combined humidity-temperature coefficient becomes 0°187 — (0°0076; x 1°6,) = 0-063, viz., the coefficient in equation (2). Now the coefficient given by Rosa and Crittenden is 0°056;, and it is readily seen that this would result from a prevailing climatic condi- tion in which an increase of one litre of water-vapour per cub. metre corresponds with an increase of 1°-7 C., instead of the 1°°6, C. observed at Teddington. It is obvious that too great a significance must not be attached to the actual values of the coetiicients in equation (4), since they depend on the assumption that a linear relation connects humidity and temperature. The difference between the “ humidity ” coefficients deter- mined (by neglecting the temperature coefficient) in England and America being much larger than that which could be attributed to the error of the experiments, tends to support the suggestion of the authors that the pentane lamp has a temperature coefficient, but that the usual method of making the observations and deducing the results does not allow of its determination. Thus, it would appear that the “ humidity ” coefficients determined for flame-standards are really com- bined humidity-temperature coefficients. Whenever, there- fore, a lamp is used under conditions of humidity and temperature which approximate to those existing at the locality where the original determination was made, the constant so determined will apply rigidly. If, however, the determination of the combined humidity-temperature -coefficient be made under different climatic conditions, a slightly different constant may be expected. If, for instance, the humidity at Washington tends on the whole to increase at a different rate with temperature than it does at Tedding- ton, a different factor for the combined effects would be expected to result. The authors are not in a position to know if this is actually the case, but it is not unreasonable to suppose that differences of the order indicated might be found to exist, and if this should be so it would afford an explanation of the difference which has been found between the “humidity ” coefficients determined in the two localities. Unit of Candle-power in White Light. 85 The conclusion is that, if work of the very highest accuracy is to be carried out with flame-standards under abnormal humidity conditions, the combined humidity-temperature coefficient should be determined for the locality in which the work is to be conducted. It should be pointed out that the difference between the American and English determi- nations for the pentane lamp amounts to less than 1 per cent. in candle-power for a rise of humidity of 10 litres per cubic metre above the normal. Values have seldom been observed greater than this in Teddington. The table at the conclusion of the paper by Crittenden and Taylor shows the average humidities in Boston and New Orleans to be 9°9 and 19-1 litres per cubic metre respectively. Where such large differences exist in climatic conditions the question of the variation of the combined temperature-humidity coefficient might with advantage be further investigated. (6) The Constancy of the Unit of Candle-power held in Llectric Sub-standards. The continual use which has been made of electric sub- standards since the first determination of the unit of candle- power in 1904, has afforded opportunities of watching the behaviour of such lamps and particularly of observing their constancy. Several observers have written on this subject, amongst whom are J. A. Fleming, C. H. Sharp, P. S. Millar, E. B. Rosa, G. W. Middlekauff *, and others, and all witness to the constancy of properly prepared and seasoned electric sub- standards. Their observations have been mainly concerned with carbon filament glow-lamps, and the present authors are able to endorse the views they express. In order to keep records of their behaviour, an annual analysis is made of all the photometric records where sub-standards have been used during the year. When an ordinary routine standardization has to be made, three to six sub-standards are put on the bench in turn, and thus for each day’s work a value is obtained for any one standard lamp in terms of others of the same set. The analysis of such results over a year’s working indicates if any individual lamp shows signs of differing from the mean of the others. It has been * Fleming, Proc. Brit. Assoc. 1904; Sharp and Millar, Trans. Ill. Eng. Soc., N.Y., June 1910; Sharp, American Gas Inst., Oct. 1913; Rosa and Middlekauff, Proc. Amer. Inst. Elect. Eng., July 1910, p. 1911. 86 Unit of Candle-power in White Light. unusual to find that a lamp has appeared to change during 12 months by more than 0:1 per cent., and the majority of them show no change which can be detected. If any lamp in a set shows a difference from the mean of 0:1 per cent. no change in the value assigned to it is made until such difference is repeated in the following year’s analysis. Of the one or two lamps whose values have had to be adjusted in this way, some appear to have risen and some to have fallen in candle- power, and there is no sign that any fundamental set of lamps is undergoing a progressive change. This can be said equally of the tungsten filament lamps as of those with carbon filaments. A set of 1:5 watts per candle tungsten standards (set 6) has to be used nearly every day of the week, and a fundamental as well as a working set of these lamps is therefore kept. The working set fell in candle- power about 0°5 per cent. during the continuous use of the past two years. As the fundamental set will only be used perhaps two or three times a year, its constancy for very many years is assured. It could at any time be compared against set 5 (2 watis per candle), which has never to he used in ordinary routine work, and which, as a matter of fact, has not been used since the values of its individual lamps were first fixed by the cascade method. Similarly, the use of set 1 (matching the pentane lamp in colour) will never be necessary in ordinary work because of the redness of the light. When it is remembered that the filaments of these lamps are of tungsten operating at the low efficiency of about 7 watts per candle, and that they only require to be used for exceptional reference purposes at intervals of several years, there would seem to be every reason for expecting the unit of candle-power to be maintained constant. by means of them for an indefinite period. The authors desire to place on record their obligations to Dr. R. T. Glazebrook, C.B., F.R.S., Director of the National Physical Laboratory, and on behalf of the Laboratory to acknowledge the generosity and help of the General Electric Co. and the Osram Lamp Works for the many expensive and special standard lamps which they have made and presented in connexion with this work. eens V. The Scattering and Regular Reflexion of Light by Gas Molecules—Part Il. By C. V. Burton, D.Se.* 28. WT will now be convenient to consider further a result obtained in Part It The region 0= —Aipvr3/8r’.. . . (36) This result is true only when yu! or vA*/47? is small ; moreover it applies only to the acoustical case considered. The corresponding electromagnetic (or optical) case is re- ferred to in § 48. 33. Some practical interest attaches to the transmission of normally incident plane-waves through a lamina of finite thickness occupied by a swarm of secondary vibrators; for it is on this that certain interferometer measurements of anomalous dispersion are based. From (18), (19) we obtain, with the help of (21), (22), E=—y"'B=—y1[C, exp iué + C, exp (—ipné) ] =—2A { [(u—1) exp {—i(u—1)n}— (ut 1)] expipé —[(u—1) exp {—i((u+1)n}—(w +1)] exp ing +[—(u+ 1) exp é (u—1)n+ (u—1)] exp (—in€) —[—(u+1) exp i(u+1)9+ (a—1)] exp (—iné) | + { (w+1)? exp i(u—1)9 + (u—1)2 exp {—i(u—1)n} — (u—1)? exp {—i(u+1)n} —(u +1)? exp itu +1)n}, 90 Dr. C. V. Burton on the Scattering and and when &=y (that is, when «=L) this becomes E,=—8A iz sin | (w+ 1)? exp i(u—1) + (u— 1) exp {—i(u—1)n} —(u—1) exp {—i(ut Ln} —(wt 1)? exp (w+ 1)n | =A ol? WS expt (—y—s— om)... 2 (as we find from (27) after some reductions), where W = (my? + poe? + 14+ 2p)? exp 2m + (my? + My” + 1— 2)? exp (— 2m) —2(pyt + py + 1 + 2 py? ho? — 2py” — Opty”) cos Zurn + 89 (juy? + fy”? — 1) sin 249; cos w= W-?[ exp fen «(fr — pe” +1+ 2m) cos (u,—1)n + (21M + 22) Sin (44—1)n} + exp (— pom) . { — (Hr? — pe” + 1— 2p) cos (wy +1)y + (2 pple — 2u2) sin (my +1)y}]3 7 sin a= W-2| exp fon . {(H17 — py” +14 21) sin (u1—1)y — (244 fy +22) Cos (u;—1)7} + exp (M9) { (M1? pe? + 1-2) sin (m1 + 1)y cos 93, sin 3 = (1, M2) (oa? + Me”) -?. Since 4, M2 are given in terms of y1, v2 by (28), (29), E, is known in terms of the complex constant NE 34. The transmitted regular disturbance is thus = 4A (uy? + 2)! W-Fexp i(pt—va—S—a), . (38) which assumes a simpler form when yu? is small. For in that case (28), (29) become Py = lye, Me KU; so that va a(t ye) exp (ala) sin o = sin YL —po/p, . cos XL = sin (y,L—S), and (38) becomes ap’ = A exp (—y,L) expi(pt—vx—yL). . (39) The refractivity, in the ordinary optical sense, is thus, to our order of approximation, x2; though strictly speaking #,—1 does not vanish with y.v~' when terms in y;’v~? are retained. 35. When the secondary vibrators are Rayleigh resonators, distributed through the region 0 1... Gea) d=0, 0, Au sin (pi—vz). 2) es ee We have now to introduce the condition that the secon- dary vibrator merely scatters energy of frequency p/2zq, without changing its total amount. To de this, we may imagine a closed surface drawn enclosing the vibrator, and write down the expression for the surface-integral of the Poynting vector; the condition then is that the time-average of this surface-integral should vanish. More conveniently the closed surface is replaced by two parallel planes, c= —l1 and w=1, where J: is large. Thus of the vector-product of d+d’ and h+h’ there is only the x-component to be considered, namely v’{ A sin (pt—vwv) + Cr-3(a?+y”) sin (pt—vur—y) {Asin (pt—vz) + Cr~*e sin (pt—vr—y) } =v"| A’ sin? (pi—vax) + Cr ?x(2? + y”) sin? (pt—ur—y) +AC{r3(a?+y7) +r722} sin (pt—vz2) sin (pt—ur—y) ]. The quantity which has to vanish is the surface-integral of the last-written expression over the plane 2=/ minus the surface-integral over the plane e=—J/. Accordingly those terms may be omitted from the expression which remain unaltered when the sign of w is reversed. On averaging the remaining terms with respect to time, and taking account of Regular Reflexion of Light by Gas Molecules. 93 both planes, we get as the multiplier of dydz in the final integral vy? | LC? a(x? +?) + SAC {r-3 (a? + y?) +17 2x} cos {u(r—2) +} | =v"|[ Cr *l(P +4”) + $ACr*l {cos (u.r—l +e) + cos (U.r+l+y)} +4ACr-(P? +4°){ cos (vu. r—l+y)—cos(v.r+l+y)}]. Since /is arbitrary, let it be so chosen that ul is a multiple of 27; 1 may then be omitted from the arguments of the cosines, and dropping also the constant factor Cu7/, we have for the quantity whose surface-integral must vanish Cr-°(l? ++ y?) + Ar? cos (ur+y¥). 39. If g?=y’+2’, the above expression must be multiplied by 2mgqdq, that is by 2ardr, and integrated from r=/ to y=, the result being equated to zero. Thus of r-*(P? + y*\dr+ af r—1(cos ur cos y —sin ur sin y)dr=0. l l (45) In the first integral, for any given value of r, y? may be replaced by its average value $q?=4(r?—/*). The first term in (48) is therefore f= J DER h 2 HS jae -—-l — 2(17-1 =20l . r=l The second term in (48) may be dealt with in two portions, each of which, by successive integration by parts, is trans- formed into a series of descending powers of v/. Remem- bering that vl may be chosen as large as we please, and that it has already been designated a multiple of 27, we readily obtain for the second integral the value —Av~7'siny. Hence finally C=3Avu7!siny, and (42) becomes [type l.| a=0, 0, ae sinycos(pt—ur—y); . . (49) corresponding to the primary disturbance (45). 40. By way of example, two types of electromagnetic vibrators are considered in this paper; type I., whatever its orientation, when excited by plane-polarized waves is capable of vibrating symmetrically with respect to an axis parallel to the electric vector in those waves. Such a vibrator, placed at the origin and excited by primary waves (45), emits the secondary disturbance (49) in which the 94 Dr. C. V. Burton on the Scattering and phase-lag y depends'on the “tuning”; the response being greatest when y= = that is, whence there is resonance. The vibrator or radiator of type II. is specified in § 42. 41. Consider next a multitude of vibrators of type I., all lying in the plane of yz, and within a square whose sides are formed by the lines y= +6, z= +0. If these vibrators are distributed with complete irregularity, and are all vibrating with the same amplitude and in the saine phase, the vibra- tions of each being symmetrical about an axis parallel to the axis of z, the method of §§ 4-7 can be applied to determine the ratio of the plane-wave energy propagated (say) in the direction of #-decreasing to the energy irregularly scattered. The notation need not be changed, if a is now understood to be the amplitude of the electric vector due to a single vibrator at a point distant f from it, the direction of / being perpendicular to the z-axis. The expression o7b?a7\?/? represents as before, on an arbitrary scale, the energy-flux in the diffraction pattern ; while the average value of sin?@ enters as a new factor into the corresponding expression Arab’a?f*av .sin? 0 for the scattered energy; 0 being the co-latitude of any point on a complete spherical surface. The average value of sin’@ is 2, and hence the plane-wave energy, reckoned in one direction only, bears to the scattered energy the ratio [Type L.] SONG] Sa S1ra/ 20%.) 0) ee (50) 42, A radiator of type II. has an axis fixed in it, with respect to which its vibrations are understood to be always symmetrical. Let such a radiator be exposed to the action of a primary plane-polarized disturbance, with electric vector parallel to the z-axis, the angle ¢ between the axis of the radiator and the z-axis being in general finite. The ampli- tude of vibration will be only cos ¢ times as great as if the two directions had agreed; and we can, moreover, resolve the secondary disturbance into two components: one with electric vector parallel to the plane of xy, the other with electric vector parallel to the axis of z. The amplitude of this latter component is only cos’) times as great as if ¢ had been zero. Let the problem of §41 be now modified by substituting secondary radiators of type II. for those of type I., the axes of the radiators being oriented indiscrimi- nately ; and suppose that plane-polarized plane-waves such as those represented by (45) are incident upon the sheet of secondary radiators. Then evidently the only regular waves Regular Reflexion of Light by Gas Molecules. 95 emitted are plane-polarized, with electric vector parallel to z; all else belongs to the irregular disturbance. If a now represents the amplitude at distance f due to a single secon- dary radiator, when the axis of the radiator is parallel to the z-axis, and the distance 7 perpendicular thereto, it is evident that the resultant amplitude at any point of the diffraction pattern contains as a new factor the average value of cos’¢, where ¢ may be regarded as the co-latitude of any point on the surface of asphere. The average of cos*¢ is 4, so that at each point of the diffraction pattern the amplitude of disturbance is 4 as great and the energy-flux 4 as great as if the secondary radiators had been of type I. At the same time the expression for the scattered energy contains a new factor av.cos*f or 4+; and accordingly the regularly re- flected energy bears to the scattered energy a ratio one-third as great as that given by (50), namely, [Type II. ] gen Or) wrapper eats. (SE) 43. Now let the whole plane of yz be scattered over with secondary radiators, whether of type I. or of type IL. ; the distribution (of o radiators per unit of area) being completely irregular, while the average vector-potential due to a single radiator is given, as regards direction and magnitude, by an’ =0, 0, Ora, expi(pt—ur,—y). . . (52) Then the vector-potential in the plane-waves propagated in the direction of w increasing will be a!’=0, 0, Cr," exp i( pt—ur,—y). As in the diagram in Part I., let O be the origin, Pa point (w, 0,0) and p?=y7?+2*. If Q is any point in the yz-plane, distant p from the origin and s from the point P, the angle e made by PQ with the axis of z is given by cose=sin Ocos¢; where @ is the angle OPQ, and ¢ the angle made by OQ with the axis of «. For the moment we are concerned only with the regular waves, which depend on average values, and not with the diffuse radiation, which depends on deviations from the average: thus from symmetry the resultant vector potential at any point is always parallel to the z-axis. The contribution of an elementary area in the neighbourhood ‘ TT of Q has to be twice resolved through an angle 5—e, so a that the corresponding element of the vector-potential at P 96 Dr. C. V. Burton on the Scattering and will contain a factor sin?e = 1—sin? 6 cos? ¢; the average value of which, for any given value of @, is 2 1—4sin? @ = (1+). Thus to the z-component of the vector-potential at P the annulus 27pdp (or 27sds) contributes Cs-lexpi(pt—us—y).o.27s ds 3(14%), the integral of which is TR 9 TwaC | exp? (pt—us—y) (1455) as se bay Jo As in $11, the limits of integration are s=z and s=R, where R is very great compared with 2. 44, The first term in (53) 1s (within a constant term) —imoCu-} exp i (pt—va—y) = —moCulexp2 (pt—va—y+4r). The second term in (53) depends on the integration of s~*exp(—ivs)ds between the limits s=a2 and s=R. Its value is readily found by successive integration by parts, if we remember that vx is large and may be chosen as large as we please, provided only that R:wa is very large. The value finally obtained for (53) is —2maCu! expi(pt—vi—y+4r), and the waves propagated in either sense from the plane of yz are given by a’; a!’ = 0, 0, —2mro0Cu~ exp2 (pt fua—yt+4m). (54) 45. Suppose now that the secondary radiators, irregularly distributed in the plane of yz, are sending forth disturbances (represented on an average by (52)) owing to the incidence of primary waves a= 0,0, Aexp7 (pt—vue); . '. 0. ae and let od? be so small that each radiator is sensibly uninfluenced by the disturbances reaching it from its neighbours. If the radiators are of type I. we ean apply (49); y being the phase-lag in the case of an isolated secondary radiator. The secondary plane waves are thus [Type I.] a'’; al'’=0, 0, —3mrcAvu~’ sin y expz (pt + ue—y+4rr). (56) Regular Reflexion of Light by Gas Molecules. a If we are dealing (as in §22) with a thin lamina O0<2<6x, witb v secondary vibrators of type I. per unit of volume, c in (56) must be replaced by véz, and corresponding to the resultant incident waves (55), the secondary waves emitted are 0,0, —xdc.Aexpi(pt+uz), where [Type Ll.] y= 3mv-*siny exp (447—y). . (57a) 46. Under the primary stimulus (55), when the secondary radiators are of type II., with axes promiscuously oriented, the average radiator gives out a disturbance which differs from that for a type I. radiator only in having an additional factor 4, as explained in §42. ‘Thus [Type II.] xy = mvv~?sinyexpz (47—¥). . (57b) 47. Hither (57a) or (57b)—according to the type of secondary radiator with which we have to deal—may now be made use of in adapting some results already obtained to the circumstances of optical problems. For instance, equations (9) and (24) still hold good, with the single emendation that the value attributed to w is multipled by $ or by 4 as the case may be. In particular, when the radiators are tuned as resonators, w=37ou~” (Type I.) or w=mov ” (Type II.), and as before the validity of (9) and (24) is limited only by the conditions that the radiators must scatter without absorbing wave-energy of frequency p27, and must be distributed with perfect irregularity like gas-molecules. The expressions (50), (51) likewise hold good rigorously provided the same conditions are fulfilled. It appears from Wood’s researches that, as the density of a swarm of resonant molecules is increased, the condition that true absorption should be absent fails long before any marked departure from the laws of ideal gases has become apparent. This is so even in the case’of mereury vapour, which can be raised to a pressure of several atmospheres without destroying the property of resonance. 48. Equation (36), giving the amplitude of waves regularly reflected from an attenuated swarm of secondary vibrators, must now be replaced by [Type I.] Al" = — 3Aimvy = —3Aivr3/16 77? 58) [Type ll.] A’ =— $Aimvy = — Aivd3/1 677? } oe Phil. Mag. 8. 6. Vol. 30. No. 175. July 1915. H 98 Dr. C. V. Burton on the Scattering and Again, in place of (40) we shall have [Type L.] y.v71=32rvu~? sin 2y=3pn? sin CT a (59) Type II. gv !=Lrvu sin 2y= vas sin Qy/16r? J ’ ; and in place of (41), for the extinction-coefficient, [Type I. ] Y= 3rvu? sin? y=3n"p sin? y/4ar 60 [Type I1.] X1= Try *sin?y= Dv sin? y/4rj oy 49. The two types of molecule which have been especially considered, though far from exhausting the a prior? possi- bilities, may serve as examples ; and it should be possible to decide experimentally whether type I. or type IL. represents the more nearly the properties of the actual molecules of any given gas—or whether a type distinctly different from either is indicated. The suggestions now offered in this connexion relate only to the means of discriminating between molecules | of type I. and those of type Il. The most obvious method is to use piane-waves completely polarized as the primary disturbance, and to observe the intensity and degree of polarization of the diffuse secondary radiation emitted in definite directions. The density of the vapour should in these tests be small enough to make the tertiary etc. radiation insignificant in comparison with the secondary ; otherwise the distinctions to be observed would be partially obliterated, and the whole problem would become more complicated. From the fact that Wood * observed no trace of polarization in the radiation emitted laterally by his ‘resonance lamp,” it may be surmised that a considerable proportion of the resonance radiation was of tertiary or higher order. For it seems hardly possible to imagine a molecule such that the secondary } radiation emitted per- pendicularly to the original existing beam would not be at least partially polarized. Indeed Wood, in the papers referred to, has emphasized the prominence of the radia- tion of higher orders. To secure an approximately pure secondary radiation, it may be necessary to use mercury vapour under as low a pressure as ‘001 mm. contained in a vessel of much smaller dimensions than those hitherto used. In the following two paragraphs it is assumed that such precautions have been taken. * Phil. Mag. May 1912, p. 712. + What is referred to above as “secondary radiation” is the same as Wood’s ‘‘ primary resonance radiation,” the term ‘“ primary” being in this paper applied to the original incident beam. Regular Reflexion of Light by Gas Molecules. 99 50. As hitherto, let the wave-fronts of the primary dis- turbance be parallel to the plane of yz, with the electric vector parallel to the z-axis; and let the resonant gas- molecules be contained in a rectangular vessel with trans- parent sides parallel to the three co-ordinate planes. Then, if the molecules are of type I., each of them will be vibrating ‘symmetrically with respect to an axis parallel to the axis of <, and the following deductions can be immediately made. The secondary diffuse radiation proceeding in any assigned direc- tion will be fully polarized, and in a direction making an angle @ with the axis of z, the intensity of the secondary diffuse radiation will be proportional to sin?6. o1. Alternatively, suppose the molecules to be of type IL., each having fixed within it a definite axis, with respect to which any induced vibration will be symmetrical. Consider a moleeule whose axis is inclined @ to the z-axis, while the plane parallel to the z-axis through the axis of the molecule makes an angle @ with the plane of wz. Then the vibration of the molecule can be resolved into three rectangular com- ponents having amplitudes proportional to sin@cos¢, sin@sind, cos é. For diffuse secondary radiation emitted in the direction of the axis of a, the polarized components have intensities in the proportion av cos? 8: avsin? 7 sin? ¢ = 2:1; averag> values being denoted by the prefix av, and the stronger component being that for which the electric vector is parallel to the axis of <. The same evidently holds good for the diffuse secondary radiation emitted in any direction parallel to the plane of zy. For a direction parallel to the axis of z, the ratio is one of equality, the radiation being unpolarized. Under the conditions contemplated in this paragraph, it is easily seen that the total intensity of diffuse secondary radiation emitted parallel to the axis nie y bears to that emitted parallel to the axis of < the ratio 3: 2. 52. From his experimental study of the absorption of radiation (A 2536) in mercury-vapour at low pressure, with some considerations of a general nature, Wood has drawn the conclusion that, at any given instant, only a small pro- portion of the mercury molecules are acting as resonators. This seems to be borne out by a somewhat more detailed examination of the question. In Wood’s experiments the radiation from a quartz mercury are, restricted to X 2536, | 5 Me 100 Dr. C. V. Burton on the Scattering and was caused to pass through mercury vapour at a pressure of ‘001mm. The radiation laterally scattered was most intense where the primary beam entered, and gradually fell off as the beam penetrated further into the vapour; the intensity being reduced to one-half after a depth of 5 mm. had been traversed. The proportional reduction of amplitude in traversing a layer comparable in thickness with 2/27 is. evidently quite small, and the approximations of § 34 can be used. For the present let attention be confined to the primary (or incident) and secondary radiations; also let the possible influence of the Doppler effect and of collisions. between molecules be disregarded. 53. The coefficient of extinction, for light of any assigned! frequency, is proportional to sin*y, where y is the amount by which the secondary radiation from a molecule lags. in phase behind the resultant incident radiation: a result. derived from the sole assumption that no absorption of energy takes place. But when we want to express y as: a function of the particular frequency (p) in question, some: further assumption must be made, and one that readily presents itself is that y is related to p as it would be in a purely dynamical system ‘To express such a relation, new quantities have to be introduced, though these do not appear in the final result. If the free vibrations of a singly-free: system with co-ordinate u are conditioned by * ee e ; 2 ee, U+Kutnu = 0, so that the natural (undamped) frequency is n/2zr, then the prolonged action of a force proportional to cos pt will give rise to a vibration which lags in phase behind the force by y,. where tan y = px/(n?—p?”) fT. Since the effective range of frequency with which we are concerned is extremely narrow, no appreciable error is. involved in replacing this last equation by tan y=nk/(n?—p?), n? K? @aopyeme * ° * GD whence ey sin? y= 54. Let the energy-flux in the incident beam between the limits p?=p? and p?=p?+d(p*) be f(p)d(p?). Then since * In the present optical application, the presence of the term «xz must be attributed wholly to radiation from the molecule. + Rayleigh, ‘Theory of Sound,’ vol. i. § 46. Regular Reflexion of Light by Gas Molecules. 101 the values of » which are effective belong to a very narrow spectrum-line in the middle of a much broader incident line, we may simplify by treating f(p) as a constant. After a thickness (LL say) of the vapour has been traversed, the distribution of energy in the primary beam will be different. For any assigned value of p, the energy of the primary beam has been changed in the proportion exp(—¥x,L), and %, as we know is proportional to sin?y ; so that in place of exp(—x,L) we may put exp(—Ksin’y), where K is a constant multiple of L, and is equal to y,L cosec?y. At the same time it has to be remembered that, corresponding to any definite value of p (which determines y), the diffuse secondary radiation emitted per molecule * has an intensity proportional jointly to the intensity of the exciting radiation and to sin’y. It follows that the intensity of the diffuse secondary radiation at a depth L within the mass of vapour will be less than that where the primary beam first enters in the ratio {sin?y. exp (—K sin? y) d(p®) : J sin? y d(p”). Making use of (61), and at the same time putting M in place of (p?—n”)/n«, we can write for the ratio me K i Me een (oe Me hm |. 62 \ eri? ( we) ™ \ea™ ee) ‘Only a very small range of values (negative and positive) of M contributes sensibly to the integrals, and the limits of integration must be wide enough to include the whole of this range. 5). Using a rough graphic method, and ranging from M=—10 to M=+10 (or from 0 to 10, which amounts to the same thing), I find that the ratio (62) would be as small as 0:5 probably for K=1'5 and certainly for K=1°6. The latter figure may be compared with Wood’s experimental result, referred to in § 52 above. Thus 1'6=K=y, cosec? y x 0°5 cm. ‘On reference to (60) the values to be attributed to v (the number of resonators per c.cm.) are seen to be y=87 x 1°6/8X7= 6:7 x 10° for type I. cape (63) y=8rx1-6/2 =201x10° , IL ,, ; * The energies of the diffuse secondary radiations from the various molecules being simply additive in an attenuated vapour sensibly obeying the gas-laws (cf. §7, part 1.). 102 Dr. C. V. Burton on the Scattering and The actual density of mercury vapour was equivalent to one molecule in a volume A’, or 6°07 x 10° molecules per c.c.; and thus (63) implies that at any instant only about one in 9000 (one in 3000) of them are effective resonators: the molecules being of type I. (type I.). This agrees with Wood’s qualitative conclusion. 56. There are, however, some further considerations which should not be lost sight of, and which seem to indicate that one in 9000 (or one in 8000) is an under-estimate. In the first place the secondary diffuse radiation, though effectively restricted to an extr emely narrow range of frequency, is far from homogeneous in regard to its absorbability by mercury yapour ; at that which is excited well within the mass of vapour is due to a primary beam in which the more absorbable frequencies have already been selectively weakened. It is. thus less absorbable than that excited where the primary beam first enters. Since the scattered radiation has to traverse some thickness of mercury vapour before escaping through the quartz window at the side of the vessel, it follows that the diffuse brightness along the track of the primary beam will appear more uniform than it should, thus leading to an under-estimate of the primary absorption. At the same time the deviation of the general course of absorption from a simple exponential law will be rendered less con- spicuous; though Wood has already suspected from his measurements that such deviation exists. 57. In the next place, the energy removed from the primary beam takes the form of secondary radiations which in turn are found to excite tertiary radiations and so on; and thus along the course of the primary beam the falling off in the total radiation available for exciting the resonant mole- cules is less rapid than the extinction of the primary beam itself. _ 58. Incidentally it should be remarked that, when the highest homogeneousness in the radiation from a resonance lamp is desired, the total thickness of vapour traversed by incident and scattered radiation should be very moderate ; the scattered energy being then necessarily but a small part of the incident, even in the case of the most effective frequencies. o9. It is not easy to make satisfactory allowance for the sources of error referred to in §§ 56, 57. From the point of view of theoretical treatment the following slightly modified method seems to have some advantages. ‘A resonance lamp, fulfilling the condition referred to in the last paragraph, is used as a source of radiation, the illuminated object being a small Regular Reflexion of Light by Gas Molecules. 103 hole in an opaque screen placed before the lamp. By means of a quartz lens an image of the hole is formed on a photo- graphic plate, a parallel-sided quartz cell, successively empty and filled with low-pressure mercury vapour, being inter- posed close to the lens. The illuminated hole being small, the image formed on the plate need be but little affected by the diffuse radiation from the absorbing cell. Since the radiation from the resonance lamp over the range of frequency p? =p’ to p?>=p*+d(p?) will be approximately proportional to sin*y, it is easy to see that the mercurial absorption will reduce the brightness of the image in the ratio (62); whence by arithmetical trial and error we can find K, that is x cosec? yL (where L is the internal thickness of the cell), proceeding as in § 55. 60. From the form of the expression (62) it seems to be a not unreasonable conclusion that, when resonance radiation, due to a single resonant frequency, is emitted from an atten- nuated vapour, the law of general enfeeblement of that radiation, when passing through the same vapour, depends only on the type and number per cubic centimetre of molecular resonators. It is here to be understood that the resonance radiation is emitted without either absorption or change of frequency, and that the original excitation is due to a ‘‘line” much wider spectroscopically than that which represents the re- sonance radiation. ‘The suggestion is put forward with some reserve, because the relation (61) between 27x frequency (p) and phase-lag (vy), involving also the dissipative co- efficient x, is derived from the assumption of a formal analogy between the resonating molecule and an ordinary dynamical system. However, « disappears from (62), and it is not easy to imagine the relation between frequency and phase-lag so different in form from (61) that the ratio (62) would have to be replaced by one of a difterent order of magnitude. The conclusion that only one molecule of mercury vapour in several thousand forms an effective re- sonator for 72536 suggests the question: to how many classes, differing spectroscopically from one another, do the mercury molecules belong? And in a vapour such as that of iodine, which yields very complicated resonance spectra, is the number of spectroscopic classes greater? Determina- tions of the extinction of iodine-resonance-radiations by iodine vapour, under conditions precluding true absorption, might help to indicate how many effective resonators are present for each line examined; but here fresh complications arise. 61. In the foregoing discussions, the aim has been to 104 Scattering and Reflexion of Light by Gas Molecules. assume as little as possible regarding the processes of radia- tion; the activity of a radiating molecule has been specified merely by the disturbance produced at a distance, and it may be remarked that the validity of the results is not directly conditioned by the smallness of the molecules against the wave-length involved; but it has been frequently as- sumed that the molecules are distributed approximately as in an ideal gas, and thus many of the formule would fail if the linear dimensions of the molecules weré comparable with their mean free path. The influence of the Doppler effect and of collisions between molecules must be very small in pure mercury vapour at room temperature ; this is indicated by elementary gas-theory, and is confirmed by the experi- mental evidence for the extreme narrowness of the resonance line 2536. 62. More doubt may be felt as to the assumed absence of true absorption in the pure attenuated vapour. The assump- tion is supported on general grounds by Wood, who insists, however, on the urgent need for verification. And here I venture to suggest that, provided sufficient sensitiveness can be realized, a satisfactory way of testing for absorption of radiation is to determine the value (whether zero or finite) of some effect to which absorption would directly give rise. Absorbed radiation will, in fact, heat the vapour and any air that may be mixed with it, thus causing expansion ; and there seems to be reason for supposing that the sensitiveness obtainable on these lines would be abundant for the purpose in view. The form of apparatus proposed is indicated in plan in the diagram; it was suggested by F. W. Jordan’s paper * “ On a new type of thermo-galvanometer.” A isa little flask of fused silica, whose neck projects into, and is fused to, the bulb B. C is a small thin circular disk, pre- ferably of very thin fused silica, platinized. This is suspended * Phys. Soc. Proc. vol. xxvi. part ii. (April 1914). Number of Electrons concerned in Metallic Conduction. 105 in the position indicated by a fine quartz fibre, the bulb B being provided with a vertical tube (not shown) for carrying the suspension. If we begin heating the residual air or vapour in A, a puff will issue from the neck, and will turn the vane C in azimuth, the deflexion being read through the window D. A good notion of suitable dimensions can he gained from Jordan’s paper. Meanwhile it may be mentioned that one of that author’s galvanometers gave a deflexion of 4 mm. at a scale-distance of 64 em. for a steady heating of one micro-watt suddenly commenced; the maxi- mum excursion being reached in about 2 seconds, after which the vane gradually recovered its normal position. In this case the air was at atmospheric pressure, and general con- siderations seem to suggest that at lower pressure a greater effect should be obtained for a like rate of heat-production. 63. Suppose that, to begin with, the system contains air at ‘01 mm. pressure*, free from mercury vapour. A beam from the resonance-lamp is suddenly made to fall on A, and if the vane © executes a significant movement, the extent of this is noted. (Probably the movement of the vane would be very small, for in 2 séconds or so any slight absorption by the silica walls would hardly have time to affect the temperature of the contained air perceptibly.) A drop of mercury is now introduced, and the test repeated; in the absence of true absorption the excursion of the vane should be about the same as before. The apparatus would of course need calibrating, if only roughly. Further details would be out of place here, but I cannot help thinking that in some such manner definite results might be obtained by a skilful observer. Boar’s Hill, Oxford, 3rd May, 1915. VI. On the Number of Electrons concerned in Metallic Conduction. By G. H. Livensf. HE mathematical theory of metallic conduction has been extensively developed by various writers with a view not only to placing the theory on as firm a foundation as possible, but also with the object of determining more pre- cisely the general electrical properties of the metals. The results of certain forms of theory are certainly in fair agree- ment with experiment, but the agreement obtained, even * This air-pressure is suggested because Wood has found it to have no perceptible influence on the absorption of 2536 radiation by mercury- ‘vapour at room temperature. + Communicated by the Author. 106 Mr. G. H. Livens on the Number of under the most favourable circumstances, is perhaps not so exact as one might desire. One of the most fundamental quantities it is for many reasons desired to calculate precisely, is the density of the electrons which take part in the ordinary phenomena of conduction, and which are, therefore, more or less freely moveable in the space between the atoms. The simplest and first investigation of the number of these effective free electrons, pr oceeding from a minimum of assumptions, was made by Schuster*, whose conclusion is that the number of free electrons in a metal at ordinary temperatures is equal to the number of atoms, or exceeds that number not more than three times, More recently an elaborate investigation of the whole question concerning the number of free electrons in the metal has been undertaken by Nicholson}, with a view to discriminating between the results of the various forms of the theory which have been suggested. His conclusion is that the particular form of theory proposed by H. A. Wilsont is not only the only theory which provides results consistent with those obtained experimentally, at Jeast, within the limits of experimental errer, but also that this theory provides a very exact estimate of ie required number of electrons. Unfortunately for this conclusion, it appears that Wilson’s theory is incomplete, and the formulze he obtains and which are used by Nicholson are certainly not those which should follow from his fundamental assuinptions. In a previous paper § the present writer has attempted to: formulate a general form of the electron theory of the optical properties of metals, and results were obtained which, although of the same form as those obtained by Wilson, are, nevertheless, fundamentally different from his results. These results possess the advantage of being perfectly consistent with the more usual results for steady currents, and it is suggested that, on the usual basis of these theories, they are the only ones which can consistently be obtained. The object of the present paper is to examine the formule obtained in the previous paper with a view to their application in the calculation of the number of free electrons. On « priort grounds, it would, of course, appear to be * Phil. Mag. February 1904." qe nil, Mag. August 1911. “It is perhaps fair to add that I hear from Dr. Nicholson that he no longer credits his results with the accuracy at. first expected. ¢ Phil. Mag. November 1910. § Phil. Mag. May 1915, Electrons concerned in Metallic Conduction. 107 hardly probable that any approach to the exactness apparently expected by Nicholson can be obtained from any theory of the present type; and the more complete examination of the results of a theory which takes full account of all actions which are certainly known to be in operation only tends to substantiate this view. It is, therefore, the opinion of the present writer that precision in this direction cannot possibly be obtained with the data at present at our disposal, and we must therefore, at present at least, be content with the roughest calculations on this basis as representing the probable order of magnitude of the quantities. It is not suggested that proper and more exact estimates can never be ebtained, but it is insisted that far more knowledge is re- quired concerning the details of the behaviour of the substance in these respects before such exactness can be expected. It appears, however, at the present stage sufficiently interesting to examine how the various factors which enter in a full expression of the theory are effective in modifying the calculations usually made on the present basis. This is all that is attempted in the present paper. The general formula obtained in the previous paper, from which all the circumstances of optical dispersion andabsorption can be obtained, was written in the form ret Ae 1 A a ip’ l—aA - (1+ ax) where wp is the index of refraction, « the extinction or absorption coefficient, p the frequency of the light used, and C is a function of p and the usual electron constants of the metal which is given by pie. se. fe re OP ae Sdn pe Cee ae a ee am ih of Be U wherein e denotes the charge of an electron, m its mass, N the required total number of free electrons per unit volume of the metal, and q is a constant connected with the mean square of the velocities of the electrons (w»°) by the relation » re) Ip Qe and /,, is a constant which may be taken to be the length of the mean free path of the electrons in their undisturbed motion. 108 Mr. G. H. Livens on the Number of The quantity A which appears in this formula arises entirely trom the resonance electrons, which, for other reasons, are supposed to exist in the atoms, and which will therefore be effective in modifying the simple direct action of the electric field in the incident radiation. It consists, as in the usual theory of dispersion, in dielectrics mainly of terms of type Aas e?/m 2 TLL ag p Np — nN? —INNy the sum > being taken per unit volume over all these resonance electrons. The constant a represents an absolute constant of which an ideal estimate gives a=1/3. In any real case, however, and particularly in solid media, this number may be widely departed from; to this extent alone the theory provides a distinctly uncertain result, although it must be said that a sufficient knowledge of the optical properties of the metal over a wide range in the spectrum would enable us to obtain a precise estimate even of this constant. We shall now make the assumption that there is no appre- ciable absorption due to the resonance electrons. This as- sumption is fully justified if we are using light-waves of frequency different from those of the free oscillations of the resonance electrons. We may then take A to be real, and SO we put Sie Sat 1—aA’ and py is the index of refraction of the metal with all the freely moving electrons removed. We have, then, that 1 l1—aA We now separate C into real and imaginary parts, and write C= Cy + iCo, , 8&#Neqln gq? ( we-% du C= : iy | 3m ’ pln? - eQ : f 4 ; u N 8apN e? ql q° °° ree dy ae 7 ae mJy 4 Palm * = fe eon Pp =1+ = te (ln a Lie so that whilst Electrons concerned in Metallic Conduction. 109 and then we have Phas as - ~ Mp p(1—aA)’ a Maybe L—aA ‘zp’ vy Soe _AmTNe* gl J ih @ we~ te du Seat 3p(1—aA)m m 14 ln? a7 [eae and BAN iy 87 Ne? 37 Ne? ql? i © ret du iat a (L—aA)3m Mle? h o 14} u We now introduce the expression for the conductivity for steady currents, viz., 4Ne?I,, sea/ 2, 09 = = dm which will enable us to eliminate the unknown quantity lm from the above formule. We use also s=quv? and pe — be= Whence 9p?m?oy” =P ln?g@= “T6N2e! and then we get, Bons that ee Ne tt (weds ew WL—aA) ), sta’ an Dae pill) >» 3fmrma2 (* s3e-*ds ete ee cat 2 ae 5) 4Ne*(l—aA) J, sta or using 270 Ces) ! xX Pe a 42 el 3,/mmay? | s? eds Ws 2— pee cere Shiabecnal, a Riot, ff AN ba A) ie It is perhaps necessary to remember that the Hertz-Heaviside system of units is adopted here as in all my previous work. If the application is to light in the visible part of the spectrum where the wave-length is not too long the constant a is large, and we may w ith good approximation expand 110 Mr. G. H. Livens on the Number of each integral in powers of ; and retain the first few terms only. We have, for instance, 25—s 2 fee =: (e(1—5 +57 jena (et Sea TL Be Nt ge ieee Bons, whilst = 3/2 .—s 2 2 \s é <=7\ 8%(1- 2 +... )errds SH+a a 0 a 2 Naa Mo 10560 ) = G-ie tie ves so'that we have with sufficient approximation 2arpKe Soe (1- 3 =) a? : ho aa a and on) dee eae For metals of good conductivity, for which, therefore, Fn to eee 9amoy? =( Pease o> ) OT? ene ay : is very small, we should have a 2 ar [AKC 7 a A CESS Aloe whence inserting the value of « Zaps Se Oe % — 9p?m?oo(1—aA)> -g0 that , Sp*mearpKae.,. Noe Eu (Ley or since ps ae e pa » %9ImMemuKkc ts ee a | This result differs from Nicholson’s formula by the factor 9 9 (1—aA). RPMew Care Electrons concerned in Metallic Conduction. 111 Thus even excluding the effect of resonance electrons the present form of theory would give a value of N about twice that deduced by Nicholson. It is, however, just the re- sonance electrons which render the result too indefinite to be of any real assistance in the determination of N, unless we have sufficient information to enable us to accurately determine the factor 1—aA or, its equivalent, the reciprocal of {1+a(u,2—1)}. An error in the assumed value of this factor will result in a corresponding error in the estimation of N. More accurate formule may be obtained by successive approximation, as, for example, the next approximation to i a3 97° m2wKay Omuxc(1—aA) gyll! ey ell aida ale eget 4 N2= eat (1—aA) E a oo , so that even the correction term for the next approximation will depend on the values of a@ and A. There appears, however, very little necessity at the present stage for using anything more than the first term as representing a suitable approximation to the quantity required. The second equation can be used, after Nicholson, to determine the value of pp? when the values of N, w, and « are known; but here again the calculation is considerably modified by the presence of the resonance electrons, which introduces the factors (1—a.\), which are themselves functions of p,?. Very little weight can, therefore, be attached to the exactness of Nicholson’s calculations on this basis. The conclusion must therefore be stated at the outset. The theory is too complex for any very definite results to be obtained from it until our knowledge of the behaviour of the metals is equally complex. Until then we must content ourselves with very rough approximations to the numbers required. Of course any great precision in the results on the basis of the present theory is necessarily nullified by the fact that the basis adopted is obviously only a first order approxima- tion to the actual state of affairs ; so that even though full information were at hand, the theory would probably not fit the facts very well*. If we are prepared to regard the basis of the present theory as the correct one, there are other and apparently better and more direct means of obtaining an * It is known for instance that the assumption that the electrons and atoms are elastic spheres is not in agreement with all the facts and that a more general type of dynamical interaction is necessary. These points are discussed in detail in separate communications. 112 Mr. G. H. Livens on the Electron estimate of the number of free electrons taking part in the conduction phenomena. As is well known, the various. coefficients occurring in connexion with the Hall effect and allied phenomena contain this number as the only unknown constant in their expression in terms of the electron constants. of the metal. It is, however, very significant that just in this one particular part of the subject, where a good estimate of the number could be expected, the results obtained on the basis of the present theory are hopelessly inadequate- to express the actual experimental results determined. This fact suggests and strongly emphasizes the view that the simple theory is by no means consistent with the actual state of affairs. It is therefore held that it is useless at present to make: any attempt to obtain the accuracy which Nicholson claims. for his calculations, and also, therefore, these calculations. cannot be said to result in favour of any theory, and more: particularly of the incomplete theory due to H. A. Wilson. The University, Sheffield, December 4, 1914. VIL. On the Electron Theory of Metallic Conduction.—III. 7 By G. H. Livens *. HE researches of Riecke, ~ de. Thomson, Lorentz, and others have shown that tu ductivity of metals for electricity and heat may be satisfactorily explained on the hypothesis that a metal contains a very large number of free electrons, and that these electrons, taking part in the heat motion of the body, move to and fro with a speed depending on the temperature. Problems f relating to the motion of enumerable electrons. in a piece of metal are best treated by the statistical method _which Maxwell introduced into the kinetic theory of gases, and which may be represented in a simple geometrical form so long as we are concerned only with the motion of trans--. lation of the electrons. Indeed, it is clear that, if we construct a diagram in which the velocity of each electron is represented in direction and magnitude by a vector OP drawn from a fixed point O, the distribution of the ends P’ of these vectors, the velocity points, will give us an image of the state of the motion of the electrons. * Communicated by the Author. + Vide Lorentz, ‘The Theory of Electrons’ (pp. 266-271), from which the first few paragraphs of this paper are taken. Theory of Metallic Conduction. 113 Tf the positions of the velocity points are referred to axes of coordinates parallel to those that have been chosen in the metal itself, the coordinates of a velocity point are equal to the components (&, 7, €) of the velocity of the corresponding electron. Let dv be an element of volume in the diagram, situated - at the point (&, 7, €), so small that we may neglect the changes of (§, 7, €) from one of its points to another and yet so large that it contains a great number of velocity points. This number may then be reckoned to be proportional to dv. Representing it by FE, 0, O)dv per unit volume of the metal, we may say that, from a statistical point of view, the function 7 determines the motion of the swarm of electrons. This function f is determined by an equation that is to be regarded as the fundamental formula of the theory, and which we may proceed to establish on the general assumption that the electrons are subjected at the instant ¢ to a force which imparts to them an acceleration whose components in the three principal directions (X, Y, Z) are the same for all the electrons in one of the groups considered. Let us fix our attention on the electrons lying, at the time ¢, in an element of volume dV of the metal, and having their velocity points in the element dv of the diagram. If there were no encounters either with other electrons or with metallic atoms, those electrons would be found at the time t+ dt in an element dV’ of the metal equal to dV, and _ lying at the point (7+ &dt, y+ndt,z+€dt). At the same time their velocity points would have been displaced to an element dv' equal to dv and situated at the point (E+Xdt, n+Ydt, €+Zdt), of the diagram, so that we should have het Xdt, n+ Ydt, c+ Zdt, xt Edt, a nat, cet Edt, t+dt)dV'dv' Ih Cols Oy 25 C)a Va. The impacts which take place during the interval of time considered require us to modify this equation. The number of electrons constituting at the time ¢+dt, the group speci- fied by dV’ and dv’ is no longer equal to the number of those which, at the time t, belonged to the group (dV, dv), the latter number having to be diminished by the number of impacts which the group of electrons under consideration undergoes during the time dt, and increased by the number of the impacts by which an electron, originally not belonging Phil. Mag. 8. 6. Vol. 30. No. 175. July 1915. I 114 ) Mr. G. H. Livens on the Electron to the group, is made to enter it. Writing adVdvdt and bdVdvdt for these two numbers, we have, after division by dVdv=dV'dv’, E+ Xdt, n+ Ydt, 6+ Zdt, «+ Edt, y+ndt, z+ Cdt, t+dt) =/(E, UE) C, L,Y, &; t)+ (b—a)dt, or, since the function on the left-hand side may be re- placed by jlé, 7, ©, ®, Ys 2s t)+ (xSsv2 +290 +90 ash +68) ae Oe. Oy, Os Oy this equation is the same as Ff yo 7 Of, OF OF, OF Meet % 5, | ach ae 1 oyna Oe ee This is the general equation of which we have spoken. We have now to calculate a and 0, or at least their difference b—a). Thus far the argument is precisely the same as that given by Lorentz ; it is, in fact, verbally transcribed from his own account, which I am not inclined to attempt to improve upon. We can, however, now proceed by a slightly different line of argument which has the advantage of exhibiting the beautiful generality of Lorentz’s form of the theory. The present view of metallic conduction is that it takes place by the free electrons whose velocities are prescribed by collisions with the molecules, and so are taken as deter- mined in the initial instants after collision by the law of equality of mean energies in gas theory. This view, as Drude states, seems to require tacitly that the average velocity is in most instances restored by collision at the end of each free path, although it might appear that such a statement is not necessarily to be taken as literally true for each individual electron, but merely collectively for the whole swarm. Now in calculating bdtdVdv, which is the number of electrons which enter the specified group by collision during the time dt, we may notice that all the electrons which collide with an atom during the small interval dé will, when taken after collision, have the velocities assigned to them by Maxwell’s law, since the whole effect of the external fields has been on the average obliterated by the collisions. The number bdVdvdt would then be exactly the same as the number adVdvdt, which is the number of electrons leaving the aforesaid group in the same small interval, if there was Theory of Metallic Conduction. 115 nothing to modify the velocity distribution (specified by Maxwell’s law) during the free-path motions prior to the collisions which occur in this small interval. As a matter of fact, however, certain electrons which have been moving on a free path for a finite time since their last collision previous to any instant ¢ and which started with velocities outside the specified limits, are brought into tae group by the action of the external agencies and others are made to leave it in the same manner, so that at the instant ¢ the number of electrons in the group is different from the number as specified by Maxwell’s law. Moreover, some of the electrons added to the group by external agencies previous to the instant ¢ will be removed from it by collision during the next small interval dt, and these must therefore be included in adtdVadv ; it is, in fact, precisely these collisions that make the differ- ence between adVdvdt and bdVdvdt. We may therefore in such a case state that the total number of electrons extracted from the specified group by collisions during the interval dV, or (a—b)dVdv dt ean be specified as the number of collisions which occur during the same interval among those electrons in the specified group at the time ¢, which are over and above the number of electrons in the same group as would be specified by Maxwell’s law for the same instant. But in calculating the number of collisions in the small interval dt we need not trouble about the state of the metal varying from one point to the other, or even of the effect of the external field ; we may, in fact, simply calculate the number characteristic of the particular group of electrons, which is the number of collisions they would undergo if they started at the same instant ¢ with the same motions, but under the action of no external or internal agencies. Now if N denotes the number of electrons per unit volume at any point in the metal, and wu,” the mean square of their velocities, the number of these electrons which have their velocity components between (&, 9, ) and (£+dé, n+dn, €+d%) according to Maxwell’s law is deter- mined by Ae~% dv, wherein ) wa=E +7740, dv=dEdndé, 2 3 Aan, /%, Ig= y,2* and T2 116 Mr. G. H. Livens on the Electron If, then, we write Fé UE C v, Y a2) t)—Ae~“ = (E, UE iG vy Y <5 t), then the function ¢ is the mathematical expression for the change which an external force or difference of temperature produces in the state of motion of the system of electrons. » We have now to calculate the number of collisions which occur during the small interval of time between ¢ and t+dé in the group of electrons in which there are in all ON=ddVdr electrons with their velocity components in the specified range. oF us first of all find the part of this number 6N of electrons, which, after travelling for a time + after the instant considered have not yet struck against an atom, a number that is evidently some function of r. If we denote this part by 6N’, then during the next succeeding interval dr a certain part d ON’ of this number will be disturbed in their motion by collision, and since this part will be proportional to SN’ and to dt we may write it BON'dr, where @ is a constant, which may, however, be a function of the velocity components (£, 7, €) and the position in the metal (a, y, 2). Hence, during the small interval dr the number 6N’ changes by déN'= —BSN'dr, so that we have ON'’=6Ne-8*, because ON’=6N for r=0. We thus find that the number of the group dN of electrons which collide between the instants tT and t+ dT is BdNe7- fdr, so that the average time that elapses before collision for all the electrons in the group is TH =| aerPs an = s Hence we may say that the number of the group of electrons which collide in the interval between 7+ and 7+-dt is Fiat SN ™mdtT. Tm ‘ Theory of Metallic Conduction. 117 Whence we may conclude that the number of the same group which collide during the small interval dé is at ey (a—DyaVdoae = anf eo Tmt Tm dN dt Tm _ bdVdvdt Tm Thus ) or again 1 The differential equation satisfied by the function 7 thus assumes the form Ff yi 79 129, OF, OF, OF Tm (xay 2 4257 Feat ay a ne +5, = Ra ef In this equation 7, may be taken to have the value it would have in the absence of any external field, and may therefore simply be put equal to nt ™m=-—») wherein /,, denotes the mean free path of the undisturbed motion of the electrons, which under the usual assumptions is independent of the velocity. It will be convenient for the general purposes of this paper if we adopt the general notation of the vectorial calculus, using R for the general vector of acceleration whose components are (X, Y, Z) and w as the vector of velocity of the electron. The above differential equation satisfied by f may then be written in the form apie VD Mitvas EVI TeV its bios eM, wherein V7, and Y denote the usual Hamiltonian vector operators whose components are v= (3% 2) (W)= (Se dy de) and 118 Mr. G. H. Livens on the A slight and more suitable modification of this equation is obtained by interpreting it in terms of ¢ ; it is then very easy to see how the various methods of approximation which are usually adopted in this theory are effective in the final result. If we therefore write faotadem, the equation becomes (RV uot (UV )b + of a8 2. s= Aen? [2 Ras ue (uV)A +? (uV) ] = ~q\ A q > "ist 1 Qg(uR) — WV) AF WT Y9, the equation for ¢ is obtained in the form (RV»)6+ WV) G +98 é — yAe wm, This is a linear partial differential equation of a well-known type. To obtain the general form of the function ¢ which satisfies it, we must first try to obtain seven integrals of the equivalent differential relations gE na dg XY" 2 ae eel yaw —# We notice, however, that six of these relations are the same as dE _ dn _~ a s Ay at cia ne at 7 72 dx dy dz di. SF & dt = 1s dt aus co which are exactly the equations of motion of the typical electron. The solutions of these six equations are obtained at once, provided we know the values of (X, Y, Z) as functions of the time and position. This provides us with six integrals of the relations, and the seventh is got from dg en —qu?2 aa” where, however, the function on the left is interpreted, by Electron Theory of Metallic Conduction. 119 substitution of the values of R, vu, A, and q previously found or known, as a function of the time. The solution of this last equation is easily obtained in the form ape ek) d(t) = mf @ Tm KXyAie7 a" dt,. The suthx indicates that all the quantities affected are to be interpreted as functions of ¢, which is taken as the argument of the integration. This is the most generally suitable form of the function ¢ which is consistent with the above differential equation. We must, however, choose the constants properly in order to make it represent the more particular function which is to be the characteristic function of our specified group of electrons. To do this we must choose the constants in the integrals of the equations of motion so that the position and velocity of each electron is such that it occurs in the group specified by (dV, dv) at the instant ¢. Then, again, in order to remove entirely the effect of the initial conditions of the system under ee ae we must choose the lower limit of the integral for 6 as(—%«). This gives us the complete physical solution of our problem in the most general possible form. A more convenient form for the function , however, expresses it as a double eae in the form oo =("« Tn wef xehie™ Quy 4 ahi. This is easily verified to be equivalent to the form given above, and has an important physical significance which has been discussed at length in a previous communication. It is equivalent to the statement that the distribution of velocities at the initial instants of beginning the free-path motions being pursued at any instant ¢ is ‘precisely that specified by Maxwell’s law. This fact can be used forja direct evaluation of the function @. The general form of the law of distribution of velocities is thus obtained, and according to it the number of electrons per unit volume of the metal at any point in it with their velocity-points in the small volume dv of the diagram is fade = | Aew qu a | ies ae at yy Aono dt, |. eo tT As stated above, the function ¢ is the expression for the change which an external force or difference of temperature produces in the state of motion of the system of electrons. 120 Mr. G. H. Livens on the Now, as a matter of actual experience, this change is known to be extremely small in all real cases; so that various methods of approximation may be adopted in evaluating the integrals in the expression for 7. General rules can hardly -be given for such methods, as they will essentially depend on the conditions of the particular problem to be dealt with ; but the few particular cases worked out in detail below will illustrate the points quite clearly. It may, however, be worth noticing two conditions which the physical circum- stances of the case necessarily imply for the function 9. We know that whatever be the new siate of the motion of the swarm of electrons, we shall always have a definite number N of electrons per unit volume, and a definite value of the mean square of their velocities. These conditions imply that Jodv=0, (gvdv=0, the integrations being over the whole of the space in the velocity diagram. This implies, generally speaking, that @ is an odd function of (&, 7, €); as a matter of actual experience, a linear function is usually found sufficient. The implied non-existence of square terms in the expression shows that squares and products of direct accelerating forces are negligible, as are also the products of these forces by the gradient of the conditions in the metal. The formula thus far are perfectly general and involve probably none but very easily justifiable assumptions. A tew applications to special cases are worth noticing. 1. If the acceleration is due to a steady uniform electric field of intensity E, and the conditions of the metal at each point are also constant in time, then the integrations for are immediately effected. In this case Ro m?’ and thus 2Qa0e iL 9 x= —- (uw) — a (uV)A+wv(uV)¢q. Thus y depends on the time only through the terms involving &,, $,andu. As, however, these quantities only vary from their undisturbed values by quantities of the order of EH, we may neglect their variable parts altogether. With this ec eaton x is independent of the time, and we thus ave fone? Miers |, Electron Theory of Metallic Conduction. 121 and if we use we recognize at once the particular form obtained by Lorentz in \his treatment of steady electric and thermal currents. 2. If the acceleration is due to a steady uniform electric field of intensity E superposed on a uniform magnetic field of intensity H, the result is obtained equally readily, but it is necessarily much more complicated. For simplicity it is best to choose one of the coordinate axes, say the « axis, along the direction of the lines of magnetic force ; the equations for the free-path motion of the typical electron, whose velocities at the time ¢ are (&,, 7,, €,), are then m abt —f 1) eH a =eH,+ - . ot l id m' : =ch,— — Mt If we use eH C= — me’ it is easily verified that the last two equations are equivalent to the following equations : m=n+ (= a ¥+2) sin nt + (< —n) (1—cos nt), eH, eH, ) G= C+ + (°. Big (10 AT Baan | Breakdown Voltage. Thickness: \ |==3.- : When there is ay absorption ta = and the work done on thickness. ¢ is as before (equation (3)) Fe OMe ie — ure Bui) + ran . VA. To represent this graphically let OA (fig. 3) be the dis- placement #) caused by the external field and AB that of . Fig. 3. interattraction. Let AC be the force Fe and DE=Ce?/d?. The area OAC is then the work done in polarization when there is no absorption, and the area CDE that done through interaction but supplied by the external circuit. The curve CE is that of fig. 2, displaced vertically to pass through C. The shaded areas together represent the total work of polarization. Let F be the breakdown gradient, the areas are then respectively kv*/8mt and kav/47i, and their sum has been found to be constant. F is, however, not constant but is less in thick specimens. Let F change by an amount 6F, the displacement a, must be such that Féa,=2,6F. Since 2)/F is a constant quantity the condition is that «=, provided that the position of the are OE is not appreciably changed, which is so since OC is nearly tangential to the curve. We have, therefore, that in order that the total work of breakdown should be independent of thickness, the eatra displacement caused by interattraction. should be the same as that initially produced by the applied jield. The total displacement is therefore as a consequence proportional to the applied field at breakdown. Electric Strength of Solid Dielectrics. 133 This very simple relation may be otherwise derived from the consideration that failure must occur when the total displacement reaches some definite limit depending on the atomic structure. Let this be X, let gy=my the quantity Fig. 4. <—- —Lg- —— —x,— zed WY iS LE RIL TILIZZII which appears at once on the application of the field, and that slowly developed q,=mz,. The magnitude of the first polarization is such that the second is a minimum, the total displacement at breakdown being fixed. The external energy which must be supplied to the medium in order to produce the second displacement is w,=m(X—.2»)v, v being constant. But in the first stage w/v is constant, =a say; thus and this is a minimum when y= X/2, or 77=2%. 5. Influence of Frequency of Field. Electric breakdown occurs more easily at high frequency. In the case of glass the effect of raising the latter from 50 to 8500 is to lower the disruptive voltage in a ratio 2°5*. From equation (3) it is seen that the work of breakdown ‘must be less at higher frequencies, in order that at a given voltage the breakdown thickness should be greater. AC in fig. 3 must then be less for a given value of a, that is a smaller voltage will cause failure. The reason why breakdown occurs at a particular dis- placement peculiar to each frequency is not to be found in the system of force external to the atom. To account for it the internal restoring force must be weakened, as it would be if the atomic charges were unable to take up their full displacement corresponding to the field instantly on reversal. Let OA, OB be the amplitude of the instantaneous polari- zation wy, AC and BD the secondary polarization 2. Let the field be reversed so that A and B are interchanged. * Moscicki, 2. 7. Z. loc. cit. 134 Prof. W. M. Thornton on the If then, AC, BD take time to establish there is for a moment a distribution of charge as in ii. fig. 5. In the end C moves to C’, D to D', for the viscous polarization has first to be wiped out and then established in the opposite direction ; Fig. 5. il. but the condition at reversal is that the charges at C and D are separated by a smaller amount than if there had been no viscosity. The internal restoring force varies directly as the separation of the charges and the displacement increases, until when C and D are at their full outward positions 2% is greater than it would have been if AC and BD could have been also instantly reversed. In other words, because of a viscous displacement AC superposed upon an elastic dis- placement OA, a smaller field and less work produce the limiting displacement, and breakdown eventually occurs at a lower voltage. 6. Intensity of Internal Field. From equation (2), v= me —a, or the breakdown voltage is less by the term representing interattraction. The value of a is A/2B volts, since A=ha/ daw, Aoiel ele k | Sarw. a is not the gradient but the equivalent applied voltage, that which would cause the same polarization as is in fact produced by the interattraction. This inner field is in every case but paraffin greater than the applied field, as shown by Table IX. The external breakdown voltage, therefore, although it roughly differentiates between good and bad insulators, gives no indication of the atomic forces under which a material Electric Strength of Solid Dielectrics. 135 TasLe [X.—Values of ; and ; volts per centimetre — for the smallest thicknesses. | _ Internal field. | Applied field. | v/t. | Material. ae Ratio a/v. PEA 3742s a tndesveeas se nO 27 105 0 TLL Se | O-17.10° 32. 43 0-54 Lee i |g a orae: 3s 86 Ge, EO ler)’, 34:0. ,, 4°85 Vuleanized Rubber ...... OCS 24 4 BO 0°65 2) ete ho, 4 O23 55 2°4 1a Ee pee WIWA seSron a Na 0:96..,, 3:4 Oiled Canvas ............ Wat), 4 eaReSER alas i a ee 1:9 fails. Rupture is chiefly an internal effect, an electric ex- plosion or recombination of atomic charges of such violence that the molecular structure is permanently damaged along the line of action. ‘lhat electric breakdown is of the nature of an explosion along the line of failure is supported by the appearance of the point of passage of the spark, which has a burr at both ends, and by the fact that in many cases, unless the testing transformer or generator is large, a spark ‘through’ insulation is not followed by permanent break- down of insulation resistance. This, on the contrary, may actually be improved by the passage of a spark intended to break down an incipient fault*. 7. Amplitude of Electrical Displacement. The area OAC, fig. 3, is 4F exo, that of CDE is e2 19% (Ta) — 51 - the second term being the area CD’, fig. 3. The ratio the two areas is v/a, so ; that 765.003 | 4 Feay=a/, From Table IX. the mean value of a/v, omitting paraffin and porcelain as exceptional, is 2°2. So that with eee tO, P =a00 H.8.U., d=5. ya ee (* _ 25.1019 300. Ree eT ioe * See Beaver, I. E. E. “Cables,” 1914, discussion by Bridge. or 136 On the Electric Strength of Solid Dielectrics. According to this a very small electrical displacement relatively to the diameter of a molecule is sufficient to strain the atemic structure to breaking point. If, as in porcelain, a/v is iarge, = 17-2, 2 =0°016. Inthe ease of paraffin, N.Fe.x/2=w, where w is the work of breakdown per unit volume and N the number of molecules in unit volume. N is nearly 2/d?, so that fis EN a ele The breakdown gradient is 9000 E.S8.U. and w=8500 ergs. Therefore with d=5.1078, - —:002. 8. TLotal work of Electric Breakdown. The total work done upon a material up to the point of rupture is w=k/87B. It therefore depends upon the di- electric constant &. The energy of the first polarization at any voltage is kv?/8mt, of the second kav/47t, in each case divided by 9.10* to convert to ergs, when v is in volts and ¢ in centimetres. The ratio of these energies is given in Table LX. and the total work in Table X. TaBLE X.—HEnergy absorbed before failure. | | ‘Total energy | Energy on one Material. k. per sq.cm., | molecular : in ergs. area. ; Bt | | poe a MET a ala uses ron ganas 6:0 265 40? oS 210) Mere ancelain' acces 53 6B. 1186. ORs Wale Rubber «9 L980 ) Oke G3 LOne Gases Bk Gad 2:3 M07 Baad: cs. yee 2-3 85 oy PROG) —12 Presspahin (..0c5..24 | 49 158 . ,, 31. 10m Biled Canvas 00%, Og 1:25 .,, 25 Vim —12 Impregnated Jute ...... | 4°3 POs 2:0 .10 Glass fails under mechanical tension at about 5.10° ergs per cubic centimetre, so that it is 50,000 times stronger mechanically than electrically. The energy to break down a single chain of molecules of sectional area 2.107 sq. em. in the above typical insulators is of the mean order of 107“ erg. This may be compared with the energy required to change the charge in an atom by an amount e. ‘That required to ionize a hydrogen atom is 1°7.107" erg. gl Cie ae IX. The Coefficient of Hnd-Correction.—Part I. By P. J. Danieut, B.A., Assistant Professor in Applied Mathe- matics, The Rice Institute, Houston, Texas*. $1. [* an electrical current passes through a long cylindrical tube of conducting material, and then out into a large hemispherical volume of the same, the total resistance is proportional to the total length of the tube plus a certain multiple of the radius. This multiple is the coefficient of end-correction which we require to find. Rayleigh, in his Theory of Sound,’ found first that +785 < this coefficient k<°845. In the appendix he showed further that k <°8242, and he supposed that its true value did not differ greatly from this. The solution depends on assigning some form to the current-flow across the open end stated as a function of the distance w from the central axis. Rayleigh assumes the axial velocity to be of the form 1+ya?+p'o*, taking the radius of the tube to be 1. In a paper being published in the American Journal of Mathematics the author has assumed a general law of the form R 1 —> Kec J o(k,or) rS1 where k, is the rth root of J,(/)=0. R was taken as 7, and then the coefficient required was found to be less than *8222. The convergence was slow, and in the work the coefficients, #,, were found to be roughly the same as if the axial velocity were of a form (l—@a’)~", In the corresponding two-dimensional case the disconti- nuity at the edge of the opening is just of this type. Again, Rayleigh found that the a term was of greater importance than the w?. All these considerations have led the author to assume a form A+ B(1—o’) + C(1—3a’)- for the axial velocity, or current. This forms the substance of this paper, and it is found (i.) That B is small and its effect on & almost negligible; - (ii.) That if the total current is 7 and the radius 1 the axial velocity is very nearly }+4(1—a’)-14 ; (ii.) That 4<*82168 and is probably extremely close to this value. For a more detailed explanation of the methods employed * Communicated by the Author. 138 Prof. P. J. Daniell on the the author’s paper in the American Journal of Mathematics and Rayleigh’s ‘Theory of Sound,’ Appendix A, may be read. § 2. Let us use cylindrical coordinates aw, z, and let us take the radius of the tube to be 1, which will not affect the general nature of the problem, and its length to be L. Let us divide the whole space into two parts: firstly, the hemisphere a > 0, z> 0, and w?+2? < R? where R is large ; secondly, the cylinder O< a <1, —L1, or f(a) =0 when o> 1. So that V= FJ (kar )dk Jo(kp)pdo. \ e-™ J (ka) (ro (kp)pdp Or © 1 (V)z=0 =| J s(ke)dt | I (p)Jo(kp) pap, and i J, ¥-0(-3 )_ade=| ak (\Aorteciprodp) Further, at the other boundary w?+ 2?=R’, V is proportional il OV seal ee OV : to R and 30 toys: and the integral ies dS over this * Rayleigh, ‘ Theory of Sound,’ vol. i. Appendix A. + Lamb, ‘ Hydrodynamics,’ p. 129. OA I ET EP ea OT ee EE TEE ED ng OG he, eo - Coefficient of End-Correction. 139 boundary can be made as small as we please by choosing R sufficiently great. Then OV oo 1 : \v an = 2m dk (| Ao)So(kp)edp ) : at e -$") bas) = A+B1—a’)+CO—a’)-1%, 0z Then 1 a 1 {/ Aersolteyede = A “IuCtpiede+B( "IuCkp)(1—p*)pde 1 +C \ To(kp)(1—p?)" "dp. 0 Now * Tb) 1 { Jo(kp)(1—p?)’-!pdp = amg oy Pri t gt ie) Then | ; da(# Ja(h Joa h (HordCtrrede = 52M + 252 4 c2-wrg a, or “3 OV x - Ji(k) | gpdeolh) 1 9-1/3 Jaa(k) ]?. in| V 5,48 =| dk |At= + 2BW2S) +0 2-wr(g) 28 | But f * J(k)I,(k)dk _ K(3)0w+y) oR eeD +3) T e+ aot et) Then P(g) T'(4) 21 (3) PB) P(3) +2pCLOPor 4 ehOre rant) POHL) *° LLVOLOLCg +B? bo Ls) So — bol ~~ —~ om eee — ly Nc 8 847. (T(8)7? 9 9 4 2 pt et 3? bois ee eee | At+2ABS + BY Ane great |2AC. +2BC 55-55 Me) 104; * Schafheitlin, ‘ Bessel Functions,’ p. 31. t Nielsen, ‘ Cylinderfunktionen,’ p. 194. a 3{P)} + C? 140 Prof. P. J. Daniell on the Let us denote a by ® Then (VS yo Y dS= = ial +288 8 +B 2 AU [BN + 2BC a. Bi 32 39 +02 5 Va | J a ee N@) sles I NG XS) _ * y-8(1—y) “WPdy. fy Me eae 2 ra) “ 0 ie Sy We da —_— sa _— een eens Put gaa 3) chen 5 pel 1 l— cos¢ : lem Peel ce ya meee so that @ lies between 0 and 7. Te) IN ee dd pane X Om BM) iene AV Sly where ia - = sim hoe. by 1 33/4 a or x =a (By 2° L k?) Xu 2 will give the value of 2. In the region II. a solution of Laplace’s equation will be given by Va—24D43 ape” So(kyo). This satisfies the proper conditions; for near the end _ z= —L we have V=—2+D, while at the boundary wa=1, ON. On|? aa or J1(k,) =0- Thus the /,’s must be chosen so as to satisfy this equation. Then (- OV —— = ] — Sa,k ® )e Se ayn S aaliyah) Coefficient of End-Correction. 141 Since the current is continuous at <=0 between the regions and IT., S(@) =A+ B(1—2a’) 4. C(1—a’)-8 must =1—Sa,k J (kyo) Multiply by Jo(4,a)@ and integrate from 0 to 1. ar hen 1 1 — tp { 4S o°(kr) } = A Jo(h,o)ada +B ( J(hko)(1l—o)ada 0 eo 0 1 +C ( Jo(ka)(1— a") - ade ev 0 + 2M) 4 ENG) re Oe ) by (i.) § . : But “J1(k;) cad and Jo(k;) = 7 JiChr) —Jo(k;) = —J (k;). Then . Death) 2 pk MT@ Iuelk) gre Opa tO ae AR) Ving= D4 Sach (lia) A Sa eae Sn = (Bz ay = Et Sebel) a vet dS [over the open oe = £ =|. a [D+SaJ okra) | | - 1+ Sank Soka) = 3D + 540%, —yp+0r2-Erayp[s eee, | —2BC 2?°1(2) E ang + B4 E a ; ov V Ta dS [over the end -<=—L] — =( (L+ Dysde Sth +D). J 9 142 Prof. P. J. Daniell on the Then 1 oV | - V is dS oe Boe) =F +072 9{0@)}? [2 craters | 23 2/3(kr) 2 af pein ¢ 2/3 2 2/3 2 2BC 2 T@)[ = parC) | +B 4/37 | me The author could find no tables for Jg3(k), so that the J 2/3(Kr) value of aaa) was found by means of an asymptotic expansion. It is known ®* that : 2 Bea) ZA r ) + Q,(z) cos («— — n)\, where (4y? —1)(4v? —9) iby —1)(4v?—9)(4v?—25)(4V?—49) _ Bee) ogee th ky eee et Ay?—1 (4r°—1)(4v°—9) (40? — 25 Q, (= en ye eee FP ichwees Alot Sol IEA) Hl) = 1 so that Jo(kp) = a But ve 2 T Tie)~a/ = \ P,(2) cos («— *)- Q,(#) sin (2- 7 } : while Ree J3(x) =0~ \/ =, | P,;(2) sin («— i) + Q:(2) cos («— ) } i Therefore cos( L— i)= P,(@)A (z) sin (2 _ i)= — Qi (2) A(z), where (P,?+ Q,?)A?7=1 * Schatheitlin, ‘Bessel Functions,’ p. 50. + Schafheitlin, ‘ Bessel Functions, p. 47, Coefficient of End-Correction. 143 Then J,(x) =J,(«)Y,(2) > Jo(z) ae | PA) sin (2- ae | am Q,(2) cos (#— — 1 rr) i ~. Substitute the values of cos(e— 7) d si —7) ;) an sin(. i) then we have : [cos [P,PA+Q.QA]— sin [P.QA—PiQA] if J, (2) or ~ e085 (P;P,+ Q,Q,) + iia 5 (PQ, —Q,P,), Jo(x) v(v?—2 — 2?) v?(p? — 2?)*(p? — 4?) ie P,P, + Q,Q,=1— 2! (22)? cee ALIEN seeeee _v-l (v?— 1”) (v? —3”) P,Q,—.P, an 1— Bla)? 3 oo oe } : If we stop at the terms last written and if w2=h, 23°84 the error will be less than 1 in 3000. Putting vy=2/3 we obtain the approximate relations BW 2 See lat 3 2aalts) Fr) es ys . > [103 + 3 > 38 — 53 an | r ke aY 62 (ky) 9 8k ki = "02809, 4S 3(k, 573. 1 41 1 ea} i V3 ae et a = Ere pe” 18 5 5 8h > at ; > 7208 9 ~ j28 =*015080, a o and a 010261 Now f(s) =A+B(1—o’) +C(l—a*)-*° = 1—Shyard o(k,o). 144 Prof. P. J. Daniell on the nse Ay 1B, 3 ie rs Oo: +35t 212? then’ Ll=A+BIURC I ao ole .) Substitute these values into equation (ili.) and then, since 2-U8T'(2) = 10748, yaw. dS=3L+ B”(-04104) — 2B'C'(-02161) + a 01442). om um On Substitute them also into equation (ii.), then OV ‘AT ae odpl® weld 8) ou rn a\ Von 8=3 a [APH 2ABT, +B 75-7 +240 / aa 40 48 i) Pheer al 39 49 F? 5.34 EF ]’ oR C= where F denotes #65 ae 2—v'3) _ 10174. o° oe § 3. Then . a alae e 19 ees = [At4 2AB (1 +°06667) + B’?(1+°21905) On . 2A(Y(1—-04383) + 2B'C’ (1—-01932) + C1 + 06150) ]} and v OV as— 51+ 5 [B#(-09670) —2B'C'-05091) + 0%(-03398) J = a ee Then the required D is the minimum value of a [A?+2AB'(1 +-06667)-+B'2(1 +:31575) +2A0"(1—04383) ey | + 0'2(1—*02752) + 2B'C'(1 — 07023) |. Coefficient of End- Correction. 145 But from (iv.), A+B'+C'=1. Then $7 D—1=minimum value of 2B'(-06667) —2C0’ (04383) +B'(-18242) + C’2(-06014) — 2B’C'(:09307). Firstly let us take B’/=0, then a minimum is given by O'=°7288 and 37 =D —1= —°7288(:04383) = —°03194. Secondly, if B’ is taken into consideration ; for a miuimum B'(18242) —C’ (09307) = — 06667, B’(:09307)— C’/(:06014) = —-04383. Then B'= +:030, 9-307 1 ae : , — 7759 ,= 7288+ 030 x 6-014 17323 Then 30 Oe Oe ee El te a, 2 D—1=—-03194+ 030( 06667 04383 ) — —-03194—-000035. In the former case D="82171. or &£<'82171; in the latter D='82168 or k<°82168. The introduction of a B’ term evidently makes very little difference, and it seems probable that we are approaching the actua] value of £ very closely. In the first case, B'=0; Cl =729, then. . . A=‘271,- B=0, and C=3C'=486. In the second case, Ib) = "080, -O' "775; then A='195, C=°51T, .B=060. Phil. Mag.8. 6. Vol. 30. No. 175. July 1915. L 146 Mr. G. von Kaufmann on the Theory of Corresponding Roughly we could take Oe=aa Cl Ss 7D. 0 amd see ae so that f@)=}440 92). If the radius of the tube be a, the total current I, and the specific resistance o, the total potential difference required Ba Ne gna (L+ak), where k is about °8217. The normal current at the open end is approximately I ae [2+4(1— wa?) -18], X. On the General Theory of Corresponding States, and the Thermodynamic State-Equation. By GEORGE VON KAUFMANN *. Part I. Analysis and generalization of the Theory of Corresponding States, and a new method of obtaining the Reduced State- Equation. HE Theory of Corresponding States has its origin in the famous equation of van der Waals, which was deduced as a closer approximation to the state-equation of real gases from certain assumptions in the kinetic molecular theory of gases. Van der Waals’ equation (v4 5) (o-4)= Fe T . contains three specific constants, and three variables of different dimensions, measured in independent units. Van der Waals showed by simple algebraic calculation that by introducing the quantities p/p, =’, v/v,=¢, T/T,=0’, where Pw Vy» 1, are the critical values, a, 8, and R/M are elimi- nated, and the equation (7'+ gs) (3 —1)=86" Lo, = is obtained, which is identical for all substances. On this “Reduced Equation” was built up the whole of the extensive and far-reaching theory of Corresponding States. * Communicated by Prof. W.J. Pope, F.R.S. States, and the Thermodynamic State-Equation. 147 This theory was at first regarded largely in the light of molecular theory *, to which it owes its origin. Its con- sequences are, however, so important and so extended that the empirical side of it soon came into the forefront, just as van der Waals’ equation for many purposes is treated as a purely empirical thermodynamic state-equation. Moreover, as other equations, such as those of D. Berthelot and Clausius, were set up empirically to represent the facts more nearly than van der Waals’ equation, so it was realized by many scientists, including van der Waals himself, that the theory of corresponding states follows not only from that equation, but from any state-equation with not more than three specific constants. Meslin f first showed that this is a purely mathematical consequence. [or the general con- ditions defining the critical point, namely (dp/dv)r=0, (0?p/dv”)r=0, when applied to the equation F(p, vs de a, Bs y) =0, give p,, v,, 1, as functions of «, 8, y: 1. ¢., 4, 8, y in terms of p,, v,, T,, and hence a new equation G(p, v, I, Pe Ye T,,) =0 is obtained, which must be of the form a (2 wr) =0 Py % Ty since p, v, T are measured in entirely independent units. Furthermore, as Curie ¢ also pointed out, any “critical point” thus defined by general equations will serve the same purpose ; and Berthelot § has, in fact, set up reduced equations on the basis of three other critical points which have certain special thermodynamic properties defined analytically. For instance, experiment confirms the result deduced from van der Waals’ equation, that at each temperature there exists one finite pressure (p)o, at which the product pv has the same value as it has at zero pressure and at that temperature. Berthelot chooses as one of his critieal points the point, defined by the relation d(p))/dT=0, at which (p)o * See, for instance, K. Onnes, 4k. Wetsch. Amsterdam (2) xvi. p. 241 (1881) and Arch. Néerland. xxx. p. 101 (1897). See also Happel, Phys. Zeitschrift, vi. p. 889 (1905), where an excellent résumé and bibliography of the work on the theory of corresponding states is to be found. t C. R. exvi (a), p. 135 (1893). t Arch, Sc. Phys. Genéve, xxvi. p. 13 (1893). § Journ. de Phys. (4) ii. p. 186 (1903). 4 148 Mr. G. von Kaufmann on the Theory of Corresponding GC is a maximum, and using this as “unit point” obtains the reduced equation ("+ gin) (#"— 5) =0". My The experimental testing of the theory of corresponding states, undertaken chiefly by S. Young, depends in the first place on the possibility of measuring Pw xy T,, and the comparison of the. different functions at equal values of m' or 6’, for various substances. A still more simple method is that first suggested by Amagat *, the principle of which is most simply carried out by plotting curves with the variables log p, logv, log T, &e. (Raveau ft). It follows at once from the theory of corresponding states that these curves should be of identical shape (superimposable by parallel shifting of the axes) for the different substances. This method requires no knowledge of the critical values. Although in many cases a fairly approximate corre- spondence of states has been found to exist, the theory in its entirety has been proved without doubt inexact ; it has not been found completely true for even a single pair of sub- stances. There is therefore no general (p, v, T)-state- equation with only three specific constants. Nevertheless, a theory which is so far-reaching and fundamental as this, and which over a large range of phenomena gives a good first approximation to the facts, will not lightly be discarded ; and in the present position ‘attempts are being made to modify it in such a way as to bring it more into : agreement with the truth. Thus, Kristine Meyer t has found that a fairly accurate correspondence of states exists if the reduced variables PP,» (Osa) te oy TT) are chosen, where v,, T have certain values for each substance, being selected zero for fluorbenzene. Berthelot expresses the same facts as a correspondence with respect to p/p,, (v—v,)/(v,—2,,)> (T—T,)/4,—-T,,), and suggests a physical meaning for v , alae The following contains a re-statement of the fundamental principles of the theory of corresponding states, from a rather different and more general point of view, and a suggestion as to the direction in which the true nature of the generalities here certainly existing is to be sought. * C. R. exxiii. pp. 30, 83 (1893). t+ Journ.de Phys. (8) vi. p. 432 (1897). { Zeit. Phys. Chem. xxxil. p. 1 (1900). States, and the Thermodynamic State-Equation, 149 At the same time there is indicated an extension of the theory to phenomena involving other than the purely thermodynamic quantities p, v, T. One point revealed by the work of Meslin, Curie, Berthelot, and others has perhaps not been sufficiently emphasized, namely, the fact that the theory of corresponding states is in its foundations quite independent of any ‘critical point.” The very essence of its truth is contained herein: by as- signing to each substance special or specific units of p, v, T, three specific constants are eliminated from the state-equation. The usual method of assigning specific units is to put P,.=0,=1,=1 for each substance, where k refers to some arbitrarily selected critical point. There is, however, a directly mathematical method of choosing specific units which is more fundamental and more general. Consider the general case :—The variables a , involving 7 independent dimensions, are related to one another by the equation : Ly la ae ee fae ea pi containing 7 specific constants a, B, y, ....... Substitute for these constants certain functions of them, n of which i, P40, ...... ) are of the dimensions of z, y, 2, .:.... re- spectively, and the remainder (Aj, Ag, ...... ) of zero dimension. The equation must then be obtained in the form ea 2 a(®, 4, ae e208 aie ) As Ao, we etee == 0; aja=£€, ylb=n, 2/c=, &e., may now be called the reduced variables, and the reduced equation is written G(é, Ns g, elaine 3 M1 No, nes ens.) Oa In the case when x¢71, there are no constants (A) of zero dimension. ‘The reduced equation Se a aa )=0 is then identical for all the systems to which the original equation applied. ‘The purely algebraic process reduces the equation to a non-dimensional form, containing n specific constants less than the original one. To apply this method to van der Waals’ equation (1) above. Here a=a/§?, b= 8, c=Ma/R8 are the constants of dimensions of p, v, T respectively, and the “algebraic ” reduced equation is : ("+ p)G—-1)=8. By At dwadaccslss (AD 150 Mr. G. von Kaufmann on the Theory of Corresponding Since P,= 274, = 3), T=, the “ critical’? reduced equation (2) is obtained at once from this by putting T=pla=j;7', &c., &e. Similarly, Berthelot’s reduced equation (3) and any other “critical” reduced equations follow from (4), which is the fundamental and the simplest form. The equation of Clausius : [p+ wag] @-P= a7 ore gives the algebraic reduced equation i Ws nes aD ares — es e e s id 6 [+ agaxp| G-D=9 (6) a=(aR/B'M)3, b=8, c=—(eM/BR)*, A—oiee Clausius’ equation contains four constants. Hence there appears, in the reduced equation, one constant » of zero dimension. By putting X=0 in this equation, the reduced equation derived from Berthelot’s state-equation : (pt+a/Tv?) (v—B8)=RT/M, Here is given, This purely algebraic method of obtaining the reduced equation is thus essentially similar to the old method, and directly expresses the theory of corresponding states. The algebraic reduced equation involves the reduced equations obtained by the use of any critical points, in all their consequences: it is the simplest form and the most general. The great use of this method is that it opens the way to applying the theory of corresponding states to other phenomena which do not involve easily recognizable critical points. The physical facts corresponding to the above mathematical process are, in general, these :—The variables represent a certain set of energies and their factors (and, in certain cases, time), the relations of which are exhibited by a variety of material systems. Underlying the relations of these quantities are certain natural laws, within whose scope those relations may differ in the different systems. Now a certain degree of this difference is always expressible by choosing the units of the quantities specifically in the different systems. And if the degree of difference be not too great, the whole of itis thus expressible. In that case, the sets of corresponding quantities (#/a, y/b, ...... ) are identical in all the systems; just as the sets of quantities (, y, ...... ) would be identical if the natural laws gave no scope of difference j : { States, and the Thermodynamic State-Equation. 151 in the systems with respect to the relation of the given quantities. This may ke expressed by saying that the systems “correspond proportionally” with respect to the given quan- tities, corresponding states being defined by equal values of certain fixed, specific ratios of the quantities. As was seen, the mathematical condition for this proportional corre- spondence is that the number of specific constants in the equation connecting the quantities be not greater than the number of independent dimensions. The geometrical ex- pression is the superimposability of the logarithmic curves and surfaces, as shown by Raveau for the (7p, v, T)-state- equation. It may at first sight be objected to this generalized idea of corresponding states that it is purely theoretical, and cannot be put to the test of experiment or to empirical uses. This is, however, not true. The logarithmic test of Raveau, or a test equivalent to it, can in all cases be applied with the greatest simplicity, and with any desired accuracy. And the manifold theorems which have been set up about different functions at corresponding temperatures or pressures, in the old theory, indicate the great advantage of being able to define corresponding sets of values, in different systems, of other physical quantities. Most of the applications are cases of the general fact (pointed out by K. Onnes* for the (p, v, T)-state-equation) that any function of the given quantities which is of zero dimension has the same value in all the svstems at states corresponding with respect to the given quantities. It may safely be postulated that, for all phenomena con- sisting in the interaction of energies in several material systems, there exist and can be found factors and functions of the energies involved which are connected by a common or non-specific reduced equation—. e., with respect to which the systems correspond proportionally. Thus, considering the relation of heat- and volume-energies in the systems defined as ideal gases, there is no proportional correspond- ence with respect to the quantities Hnergy, Entropy, and Volume f ; but if the variables Pressure, Temperature, and Volume be chosen, there is correspondence of states with respect to these, as expressed in the reduced equation : mp=0. It is a fundamental problem of physical chemistry to find, for each set of phenomena, the quantities with respect to which correspondence of states exists. So, in the case of * Loc. cit. + See below, p. 153. 152 Mr. G. von Kaufmann on the Theory of Corresponding the purely thermodynamic relations of volume- and heat- energies in chemical individuals, although no accurate corre- spondence of states exists with respect to pressure, volume, and temperature (except at great dilution), the theory is not exhausted ; it remains to determine the thermodynamic functions in which there is proportional correspondence, by whose means corresponding thermodynamic states of sub- stances can be defined. Such an attempt has already been made, with some success, in the work of K. Meyer and D. Berthelot cited above. In conclusion, some examples will be given of the extension of the theory of corresponding states to the most diverse phenomena in physics and chemistry. In the first place, there are a number of natural laws expressed as a proportionality, by an equation containing only one specific constant, the proportionality factor. These all give non-specific reduced equations by the specific choice of only one of the units involved, and the logarithmic curves are superimposable by shifting only one of the axes. The proportionality factor may in these cases be regarded as a ratio of specific constants of certain dimensions, characteristic of the different systems. Thus, in the equation E/C=R, the resistance R can be looked on as the ratio of specific con- stants e and 2, of the dimensions of Potential and Current respectively. Again, in the equation qgq'/r?=K f (Coulomb’s law), and in any other equations involving K, the dielectric constant K can be regarded as a ratio of specific constants 91, %, 4, of dimensions of Electric Charge, Energy, and Length respectively : K=9,?/tly. The value of the theory is, however, more immediately evident when it is applied to rather more complicated rela- tions. Thus, in a large number of the systems defined as metallic thermo-couples in which the temperature of one Junction is kept constant at zero temperature, the E.M.F. is related to the temperature of the other junction by the equation E=at+ Be. Here, introducing the constants e=a?/8, c=e/@, of dimen- sions of H and ¢ respectively, and putting E/e=e, t/c=0, we obtain the reduced equation, common to all the thermo- couples, e= 0+ 6. In this case, there is an obvious “critical point,” the so-called neutral point, the vertex of the parabola, with the coordinates States, and the Thermodynamic State-Equation. 1538 EH, =—7e,t, =—yc. The corresponding “ critical” reduced equation is e' = 20'—9”, For another example: in all the systems defined as homo- geneous solutions in which a chemical reaction of the first order is taking place at constant temperature, the relation between concentration w of reacting substance and time ¢ is expressed by the equation kt=log (a/(a—)). The reduced equation, common to all these systems, is T=log (1/(1—&)). For reactions of the second and third orders, the initial concentrations of the reacting molecules being equal, the reduced equations are, respectively, am E/(1 8); r=£(2—£)/(1—€)?. Corresponding times and concentrations are thus defined : for 1st order reactions by equal values of kt and 2/a, ss, 2nd i 5 a akt and w/a, ard i _ ie 2a*kt and w/a. When no non-specific reduced equation exists, 7. ¢., when there are more specific constants than independent dimensions in the original equation, it may yet be of value to form the reduced equation, which is simpler in form and may shed light on the inner nature of the relation which is expressed. Thus, the (u, s, v) or Energy-Hntropy- Volume equation of ideal gases :— eo x (t—a)'. ory where ¢ is the specific heat at constant volume and m= M/R, contains four specific constants. It is reduced by putting : a=, I/m=c/A=5), b= —s,(log v,+2r. log wy). Then 2, s;, 7; have the dimensions of uw, s, v, respectively, and writing u/u;=v, s/s;=0o, v/v, =¢, the reduced equation is e7=(v—1). ¢. Here X=cm is the specific constant of zero dimension remaining in the reduced equation. 154 Mr. G. von Kaufmann on the Theory of Corresponding Again, the vapour-pressure formula of Nernst * logp=— — + Plog T— aT +log A, gives the very simple reduced equation Here A=er,/R’ is the specific constant of zero dimension. Another instance is found in the 4-constant state-equation of Clausius, whose reduced form was given above (6), and is used in the subsequent calculation. Part II. Calculation of the complete conditions of Inquid-vapour coexistence from the State-equations of van der Waals, D. Berthelot, and Clausws. The problem of calculating the equations connecting temperature, pressure, and coexisting volumes in the two- phase system, from the (p, v, T) state-equation for the homogeneous substance, has received very little attention hitherto. Theoretically, it is solved through the thermo- dynamic equation first given by Maxwell (Heat), and by Clausius : { p. Qv=p(t%y—%), together with the state-equation applied to each of the coexisting volumes : Bip, 7 TF) =0, DCP tk) —0: These three equations suffice to give any one of the quantities p, v1, 2, T in terms of any otherone. In practice, the solution of these equations offers some difficulty, and leads to somewhat complicated results, even with such simple equations as that of van der Waals. This circumstance, together with the fact that all the well-known state-equations are known to be inaccurate at temperatures below the critical point, probably accounts for the long neglect of a theoretically very interesting problem. A partial solution has been given by P. Ritter f, who, working with the critical reduced equation of van der Waals, * ¢ Applications of Thermodynamics to Chemistry ’ (1907), p. 66. t Wiener Berichte, cxi. p. 1046 (1902). States, and the Thermodynamic State-Equation. 155 obtained complicated implicit relations connecting (1, p) and (v,, p). A better method is due to Planck*, who calculated the two-phase relations for CO, from Clausius’ state-equation (5) above. The three equations here under consideration all give, by simple calculation, an implicit relation of v, and vz Planck introduces an independent variable yy, by putting 11—B=¢ . cos’ y/2, %— B= . sin? 2. Substituting these values for 7, v2, an equation is obtained which gives q explicitly in terms of W; hence v, v2, and so p, T are all obtained as functions of y. The problem is thus solved. The functions are, however, complicated, and are not worked out beyond the equation giving q in terms of w. The method here adopted was suggested by that of Planck and is essentially similar to it. Working with the algebraic reduced equations, the independent variable, or parameter, ¢ is introduced, by writing : $,—1l=q(1+1)/(2+¢), g2:—1l=q/(2+?). The rest of the calculation follows the same lines as in Planck’s method. The relations obtained are, however, comparatively speaking, very simple, and have been worked out completely for all three equations. In addition to p, », v, T, the quantities M pr, M pv, d log p M r fo? ee dlog T’ Bis ak (where r is the Latent Heat), have all been calculated as functions of ¢. It will be observed that each of these quantities is of zero dimension and should therefore have the _ same values at corresponding states in all substances: Particularly simple expressions are obtained for Mr/RT and dlp/dlT, which is of interest, since the former is the well- known ‘Trouton’s constant, divided by R=1:985, and the latter is closely related to Crafts’ correction constant for boiling-point under different atmospheric pressures. The resulting functions are all tabulated on p. 156. For purposes of abbreviation, the following symbols are used :— L=log,1+t), W,=t—L. W.=(14+t)L—-t. X=(24+2)L—2t. Y=?—-(14+4L*%. Z=t(24+t)—2(14+24)L. * Wied. Ann. xiii. p. 535 (1881). 156 Mr. G. von Kaufmann on the Theory of Corresponding | VAN DER WAALS. D. BerriExor. CLAUSIUS. | | [ti Che ee ee ame f j T t. W, WA +tW 1 oT 4 », 4 | tut be fi at d(24e— 50+ sauces =0,/0. PF Sees. (sia Wes. | NA+ tWe ; i . A (2-14 50+ Poet ) =0,/b. S—t4+ EP +... lea (s an fa a CH) ee ™ 2. WW ry ne Wi W 2A x (Berthelot) a a ie Aly ee on (1-+A)? =Pia. |x. Ge gerbes ). (1— at +...] | ne x 7a ey Hee | bo Wiehe t Wee We | | 0 Bexeiety 8 4 \ (1-FA)2 | =T/e. | 5 (1— 56% Bieta ) ae (1- 50+ kt ). | OY) 6: (1+). W,Y +0 YO = 9 6 [2 (1+X).¢ _M py Shee "i 2)+3 re Sie aia, el) ey BCL ae Pe an tsi W.Y Y O7eome ete (yer Gea ae Pee 96 (1+A). | _M. py By de at | A+3 | eR pc 8 3°t igo’ t veceeee | es eeesece haven Ley BAe dio ne Y _ Up Lae Diy ‘ aly’ 44 Bet SPs is T+ e+ Pacinie'ss aeney iy f DE awe a : Z M. - = ee ate i (t-5e+ tee ) ! | States, and the Thermodynamic State-Equation. 157 For investigating the two-phase conditions near the critical point, for which t=0, these functions are expanded’ in powers of ¢. The first three terms of these expansions are given in the Table, in small print, below the respective functions. Finally, we reiterate, for comparison, the reduced equations used :— ed i van der Waals: r= at — de 6 1 Clausius: 7= wey pan ees : $—1~ b+} To each of these equations, written for the two values 7 and v2, the equation (oe -ag=n(-4.) is added, the integration being carried out with 7 given as function of @ at constant temperature, by the respective: equations. The solutions, in terms of the parameter t=(¢i1—$2)/(¢2—1), are then as follows (p. 156). The equations of van der Waals and Berthelot give the- same expressions for ¢;, $2, 7,/9, md/8. Clausius’ equation gives for 7 and @ the expressions given by Berthelot’s equation, divided by (1+)? and (1+2)* respectively. These two equations give the same expressions for dlp/dlT, Mr/RT. It is interesting to note that the relation of the latter quantities as calculated from Clausius’ equation does not involve the specific constant X; this 4-constant equation in p, v, T gives an exact correspondence of states with respect to dlp/dlT and M7/RT. For purposes of reference, the values of the different quantities, as oblained from van der Waals’ and Berthelot’s equations, have been calculated for values of ¢ ranging from 0 to 10’, and are given as common logarithms in the following Table * (p. 160). Corresponding states in the two-. phase system are of course defined by equal values of the parameter ¢. _ * The calculation is made with the slide-rule, and there is a slight uncertainty in the last place of decimals; for any likely applications. these values will, however, be sufficiently accurate. 158 Mr. G. von Kaufmann on the Theory of Corresponding In fig. 1, the logarithms of $), $9, 7, 0, as obtained by these two equations, are plotted as functions of logy)(1+¢) ; Fig. 1. 5 6 108 (1¢t) ———> ‘The curves are all drawn to the same scale, the 6-curves being moved two units upwards, and the z-curves moved seven units upwards as indicated by the small figures. -and fig. 2 gives the curves showing log z as function of log 0. The latter curves may be compared with experimental loga- rithmic vapour-pressure curves, as a test of the equations by the method of Raveau. Clausius’ equation here gives a curve of the same shape as that given by Berthelot’s, but shifted -according to the value of A, and the resulting values of 7, and @,. According to the usual value of 7,¢,/0,=0°266 approx.: the value of A for most substances would be about 6°5. If the values of ¢;, ¢: according to Clausius’ equation -are required, they are readily obtained from the tables as follows :—Let ® be any value of ¢; or qe, as obtained from Se” cae Se retes a Ue bea: oe States, and the Thermodynamic State-Equation. 159 the tables (giving logs of $;, 2 by Waals’ and Berthelot’s equations). Then the corresponding quantity by Clausius’ equation is: ®+A(®—1). Fig. 2. /og,,9 The curves showing logarithms of ¢,, $s, 7, @ in terms of log(1+¢) give a clear idea of the relations of these quantities inter se, and their rates of change with respect to one another. ‘The very rapid decrease of pressure in comparison to temperature is especially striking, and also the shape of the curves near the critical point. The rates of change of v, and v, with respect to p and T there become equal and opposite, approaching the limit « (see the expansions). It is also interesting that dlp/dlT becomes very nearly constant as the critical point is approached; in the expansion there is no term involving t¢. With regard to the relation of dlp/diT and Mr/RT; all three equations lead, in the limit, at low reduced temperatures, to the equality dip _ Mr ie ae This is connected, through the latent heat equation, with the fact that all three equations lead at low temperatures to the limit R P(%1 — te) =pi,= NM Es a form which at once suggests the Ideal Gas Equation. 160 Mr. G. von Kaufmann on the Theory of Corresponding be 0 Ol 1 5) ad logy (1+¢)] mr Re 0:000 | 0:000 041 7 0-095 301 § 0°69 477 | 1:10 0°602 # 1°38 699 | 1°61 TiS t 79 845 | 1°95 0:903 | 2:08 954 f 2:20 1-000 | 2°30 041 | 2°40 1:322 | 3°04 491] 343 613 ] 3°71 708 | 3:93 1785 | 4:11 908 | 4°40 2:000 | 4°61 O79 | 4:79 2°146 | 4°94 204 | 5:07 255 | 5:19 344 | 5°40 2417 | 5:56 479 # 5°70 558 f 5°88 624 | 6°04 2°681 | 617 733 | 6°29 779 | 6°40. 3000 | 6°91 3301 | 7:60 4:000 | 9°21 477 {10°3 5000 711°5 5) 5477 1126 6000 F13°8 477 J 14:9 7000 7 16°1 +00 | +00 Van DER WAALS. ve ay 4-000 40007 4:005 4:04 4°12 ANT 4:21 4°25 4:28 4°32 4°35 4:39 4:64 4°80 4°94 5:05 D515 5°32 5°44 5°55 5°65 a°74 5°83 5°98 6°09 6:20 6°34 6°46 6:56 6°65 6°74 716 Ceiths. 9°27 10°3 11°5 SS log w. | log.@. log ¢,.| log g,. 2°5687| 1:4716] 0°477110°4771 5683] 47151 4911) 4636 546 | 466] 589 | 3878 510 | 458] 659 | 3403 2:479 | 1-451 10°716 |0°3127 446 | 4431 763 | 2917 417 | 486] 800 | 2755 392 | 430] 834 | 26238 2:368 | 1-424 10-865 |0-2530 343 | 418] 891 | 2437 323 | 413] 915°'| 2362 303 | 408] 938 | 2297 2-150 | 1-374 |1:099 |0-:1907 044 | 352] 204] 1715 3964 | 336] 281 | 1590 898 | 322] 3431 1502 3:844 | 1-311 11:393 |0°1434 751 | 294] 480! 1336 678 | 281] 548! 1269 616 | 270] 604!) 1219 3562 | 1:261 }1°652 |0°1176 516 | 2524 694] 1140 467 | 245] 731) 1111 403 | 232] 794] 1062 3342 | 1-222 11-848 10-1025] 1 289 | 213] 895.| 0997 223 | 203] 956 | 0962 164 | 194 |2:007 | 0930 3112 | 1-186 }2:052 |0:0909 066 | 179] 092} 0888 _025 | 173] 128] 0871 4819 | 144] 305 | 0795 4°548 |1:107 | 2°551 |0:0708 5878 |_ 029 13:141 | 0561 _ 423 | 2-981 | 558 | 0494 6:913.| 935 14:022 | 0434 6445 | 2896 |-4-451 |0-:0392]1 7-929 | 857 | 928 | 0354 _ 45 825 15°366 | 0326 8-940 | 790} 850! 0302 —o | —xo | +o |0:0000 log 9. jlog 7.| gi" | RE 1°7358] 2-8330 7357| 8326 733 | 818 729 | 781 1°725 | 2°753 722 | 724 718 | 700 715 | 677 1712 | 2-656 709 | 635 706 | 617 704 | 599 1687 | 2-463 676 | 367 668 | 296 661 | 237 1°655 | 2:189 647 | 104 641 |_ 037 635 | 3-981 1°630 | 3-932 626 | 889 622 | 851 616 | 786 1-611 |3°731 607 | 682 602 | 618 597.| 565 1°592 |3°518 589 | 477 586 | 439 572 | 253 1553 | 4:995 514 | _ 367 491 | 5-932 468 | 445 1-448 | 6:997 428 | 501 412 |_ 044 395 | 7-545 —co | —c BERTHELOT. 7000 700138 7010 708 7°28 7:34 742 7:50 T57 764 770 ove 8:28 8:60 8°88 9°10 9°30 9°64 9:38 LO Ns 103 fad at pe wy toly wo Ovco States, and the Thermodynamic State-Equation. 161 It will be of great interest to put these two relations to the test of experiment at low reduced temperatures. Under ordinary atmospheric conditions, the value of M7/RT lies between 10°0 and 11°5 for most substances ; for some, such as the alcohols, it is about 13 or 14. For these values, van der Waals’ equation already gives practically equal values of dlp/diT ; Berthelot’s and Clausius’ equations still give rather higher values of dip/diT. For most substances the value of dlp/diT is rather greater than that of Mr/RT, but there is no exact agreement with either equation. Accordingly, at atmo- spheric conditions, p(vy—v2)/T is generally less than R/M. The following examples are taken from substances which give almost equal values of Mr/RT and dip/dlT. The quantities are calculated from data in Landolt-Bornstein, Tabellen, 1912. Mr dlp T p(v,—v,) fF RT* alt” : T Water, H,0O. 131 — 1380 — 373 — 0:00443 R/M=0°00456 ft. 303 — 00456 T,,=638, 263 — 00472 253 — C0481 Ethyl ether, 0,H,,O. 10°3* \ 73 — 000091 R/M=0:00111. 110 f— 102 — 308 T= 468. 273 — 00107 252 — OO114 Carbon Disulphide, CS,. 10°1* 10°1* } 373 — 0°00100 R/M=0:00108. 104 f— 103 f— 320 T= 546. 273 — 00105 259 — 00104 These results are not conclusive. Such as they are, they indicate that at low reduced temperature p(v,—v.)/'I’ does not always tend to the limit R/M, but passes that value and becomes greater. Conclusive experiments on this point will be very interesting from a thermodynamic point of view. The following Table shows the relation of 'I'routon’s constant and Crafts’ constant as experimentally determined for about forty substances. The values are calculated from Landolt-Bornstein’s data, extreme values of different experi- menters being included. Column I. gives the experimental values of Mr/RT, Trouton’s constant divided by R=1'985, at atmospheric pressure. Column II. gives the corresponding experimental values of dlogp/dlogT. The values of dlogp/dlogT calculated from the experimental values of M7/RT by means of Clausius’ and Berthelot’s equations are given in column III.; the values in column I. are identical with the values of this constant obtained by van der Waals’ equation. * In cases where there are considerable discrepancies between the results of different experimenters, both results are given. mere we unit of energy chosen is the Litre-Atmosphere. = ‘20 ’ ite Phil. Mag. 8. 6. Vol. 30. No. 175. July 1915. M 162 Theory of Corresponding States. Waiter ge ees Go. toa MENLO hx ach eet eee Carbon disulphide ......... Ethylidine chloride......... ithylene bromide ......... Mihiy lL wodide. 2.5. eto scs. ese: Ethyl bromide............... Hthyl ether 17.5....b.-essee+s Miothiy lal): o8... coer. eee AGOTONE 51d: dons. cacesceqeoeten Benzaldebydevs-s.u.-35- cee. Acetophenone }.....0800+-0e-. Pseudocymene ......... hoe Cymene.... ek eesseees ee Benzene. «..tecnenccrmencte aaetee MPoluene,s. Sete esa, zt p= Xylene ea sees m-Kylene 32s whe cen o=Ky lene, Toes eens es Methyl formate ............ Hthylvtormiare Sc9. asec ec Propyl formate ............ Methyl acetate............... Kithylvdeetaten, vse. eee Propyl aeetater...-. ise. .e. a6 Ethyl propionate,........... Methyl z-butyrate ......... Methyl butyrate ............ Benzyl alcohol. ose Methyl! alcohols. sc... Ethyl aleoholiy, 3c... Propy! aleghol .. 22 .:.3cru n-Butyl alcohol ............ 4-Amy] alcohol,.............. Propionie’acid 223527, 2 n-Butyric acid ............... m-Valeric acid ............... The Chemical Laboratory, Cambridge University. i, 131 109 10-1 10°4 10-1 10:3 10°4 10°8 10:5 10-4 106 12° 11 et — Or pa ey Ft Or oosouncoOrs] & aK ad Ys eo SO 11:8 Pweg XI. Radiation from an Electric Source. (Second Paper.) By Li. SIBeRsteIn, Ph.D., Lecturer in Natural Philosophy at the University of Rome *. CoNTENTS, Remarks on the Method adopted. Mean Energy of the Source. Unit Permittivity. Specific-heat Curves. Large K, and molecular dimensions of source. N the first Note on this subjectt I have applied my general formula for the radiation from an electric source to the case of line spectra (which corresponds to high values of the permittivity K of the source), and more especially to the case of the hydrogen series. Having attributed there to the source only the simplest kind of dispersion, 2. é. 53 ? SindK=atsF 5... + Ga) (taken as valid only down to X=¥y, of course), I at- tempted to show that the formula in question represents fairly well at least twenty-five lines of the hydrogen series. (Cf. Tables I-JII.) Since that time I have found that the few rebellious lines, and especially H,, can be reconciled with the remaining ones by taking y = Az, i.e. Balmer’s theoretical limit, and by introducing a second “free period” or convergence point in the far infra-red. I hope to be able to report on the corresponding numerical results at a later opportunity. Meanwhile I desire to develop somewhat further the general properties of the source, and especially such as may be interesting in con- nexion with subsequent physical applications and are likely to bring into prominence the essential features of the investigation. Remarks on the Method adopted. Maxwell’s equations for an isotropic, non-magnetic insulator are KOE/dt=c. curl M, OM/dt=—c.curlE, . (a) K being the permittivity of the medium and ¢ the velocity * Communicated by the Author. + Phil. Mag. May 1915, p. 709. The symbols of the first (“ Pre- liminary ”) Note will be retained here, and the numeration of formulz will be continued. M 2 164 Dr. L. Silberstein on Radiation of light in vacuo. If w=3(KEH?+M?’) be the density, and F=c VEM the flux, of electromagnetic energy, then, where- ever these equations are valid, we have the relation dwo/dttdivF=0, ..... @) meaning that the loss of energy of each volume-element equals the amount of energy radiated outwards. Thus the energy belonging to any initial field E, M will almost instan- taneously spread out beyond all limits, leaving behind either no field at all or a purely statical one. Genuine sources of energy are usually treated us singular points of integrals of the unmodified equations (a), as for instance in the well- known case of the Hertzian dipole, and in those derivable from it by axial differentiation. All of these are ultimately point-sources, and lead essentially to the X~*-type of the law of radiation. Another method of treatment consists in adapting particular solutions of (a) to the surface of a perfect conductor, usually a sphere, which plays the part of an elec- tric oscillator ; but here again the supply of energy is not explicitly taken into account. A third method consists in treating the sources as elaborate electronic mechanisms. The latter method, which has already yielded some inter- esting results, is, undoubtedly, very tempting; but it has the disadvantage that it obliges us to enter into all the minute details of the hypothetical structure. Again, in the existing investigations of this type even the continuous supply of energy is not explicitly stated. So much to justify a return to Heaviside’s method of dealing with sources, which is free from the above objec- tions and which recommends itself by its very breadth. It seems that it has not received from modern physicists the attention it deserves. It may be useful to recall shortly Heaviside’s general concepts and equations *. Let s be the productivity of the source, per unit time and unit volume ; then, instead of (a’), s=dwfoitdivF, . . . : . @ F’ being the flux of energy in this more general case. Heaviside splits s into its electric and magnetic parts ; each of these parts is represented as the scalar product of the corresponding current into the auxiliary vectors e, m, the impressed electric and magnetic force, respectively. Thus, * For a full exposition see Oliver Heavicide’s ‘ Electromagnetic Theory,’ vol. i. 1893. from an Electric Source. 165 for a non-magnetic insulator, s=KeE+mM, and the developed form of (6') becomes K(E—e)E+ (M—m)M= — div F’. This, the equation of energy, could be satisfied in a variety of ways, according to the manner in which Maxwell’s equa- tions (a) are modified within the “sources,” i.e. within the seats ofe,m. Heaviside’s modification consists in replacing (a) by KOE/Ot=c.curl(M—m), OM/dt=c.curl(e—E). (0) The solenoidal conditions remain unchanged. The equation of energy is obviously satisfied by (b), with F’=cV(E—e)(M—m) as the energy flux. The boundary conditions are: con- tinuity of the normal components of M and (in absence of charges) of KE, precisely as in the case of no impressed forces, and continuity of the tangential components of E—e and M—m. This is all we shall require in the sequel. A peculiar feature of Heaviside’s, as compared with Max- well’s, equations is that the former contain eaplicit functions of the time, the impressed forces, so that the physical com- pleteness of the system * is given up. The introduction of such functions, far from being a methodological disadvantage, is but the manifest, and very desirable, expression of our ignorance of the intrinsic mechanism of the sources of electromagnetic energy t+. And when it is first found out which are the appropriate “impressed forces” it will always be possible to attempt to replace them by mechanisms and thus to amplify the system until it has become complete. Mean Energy of the Source. Let the source be a sphere of radius a. Impressed electric force, e=ey sin nt, homogeneous throughout the source. No magnetic impressed force; also, magnetic permeability of * That is, the property of its past and future being determined by its present state alone. + It seems also worth noticing that equations (0), at least for K=1, satisfy rigorously the principle of relativity. But this property, with some of its consequences, is better reserved for a later publication, 166 Dr. L. Silberstein on Radiation the sphere =1. Then the electromagnetic field within the sphere is, by the equations (bd), KOE/Ot=c. curl M, OM/O‘= c. curl (e~E) *, divE=div M=0, 1 Od. 2 cos 0 Of ” Kp an ae ee * | ‘ (6) J where v=cK~1? is, in the case of a real positive K, the velocity of propagation of disturbances within the sphere ; i sin 6(% ob + ee aly d= — 2A cos (nt+n). sin, 2 — C . e ° more generally, K = = (1—o)*, where o is the “ extinction index” and s=,/ —1; then K in the first differential equa- tion stands for a differential operator, as in the usual treatment. Except in the immediate neighbourhood of a convergence point the influence of o in our dispersion formule will be disregarded. Our K will therefore in practical applications he always real, and generally posi- tive ; but it may, in certain begians of the spectrum, acquire negative values; then K = sah —o*) and ¢ >13; such regions will, in fact, be epnatlle interesting in connexion with band spectra. The meaning of the remaining symbols in (6) is as follows :—R the radial, and P the meridional components of the electric force E; M the intensity of the magnetic force, the magnetic lines being circles of latitude, on each sphere concentric with a; axis coinciding with e); @ angle contained between axis and any radius vector; finally, A and » functions of the frequency n, which, together with the corresponding magnitudes A’, 7', outside the sphere, are determined by the boundary conditions (KR)=(R), P+esmo@e), QD=QW)) Saieee where () means ‘‘ for r=a,” and dashed letters refer to the * The impressed force being homogeneous, curl e vanishes everywhere (leaving only a vortex sheet on the surface of the sphere), but the presence of e is essential in as far as it leads to the boundary condition : tangential component of EH —e (not E alone) continuous. The remaining “condi- tions are as in the usual Maxwellian theory. These boundary conditions determine A, 7 in (6) and the corresponding A’, 7’ in tne field outside the sphere. from an Electric Source. 167 surrounding medium. Of. these functions only A will be needed here. It will be given presently. To abbreviate, write nr p=—, g(p)=sinp/p—cosp. . . . (7) UV Then the mean values of }KP?, 4 KR’, 4M’, taken over a period of oscillation, will be A?’ sin? @ 4A? cos? 8 A? sin? @ 9g e 2 » g 2 -\ ——___77(p sin p+ ci Kat oie atte cyt (p Pp 9) ? ort J; eur? respectively. Integrate each of these expressions through the volume of the sphere, add up, and write nalv = u. Then the mean energy of the source will be 8 A? 3 U=> kG, gid ig CURE St (8) where wu 1 2 9 2 . G(w) =) [Goer grtee +9°kdp. whe #83) 0 Develop the integrand and integrate by parts the terms containing sin? p/e* and sin 2p/p*. Then the result will be 2 cos 2u—3 sin 2u sin?u ins ee en Oe a 987 (9 ee . + 28i(2u) + a ii (9) where Si is the so-called sine-integral, G(u) =u+ sin wv . Si(e) = { St 0 & For moderate values of x, even up to v = 17, ge jd Bh mn Sb) S St yu A numerical tables, from «=0 up to e=107, and a graph, have been given many years ago by Glaisher*. The limiting value is Si(«o)=7/2. But for the present we shall require the values of the sine-integral for small or moderate values of x only. Developing the several terms in (9) into series, the reader * J. W. Glaisher, Phil. Trans. clx. (1870), pp. 367-387 ; reproduced in E. Jahnke & F. Emde’s Funktionentafeln, Teubner (1909), pp. 20-21. Si(v) = «— hat Bk Aw 3 ee nae YS Sa = IE a eS —_ -? ii i" ' ; t i i 168 Dr. L. Silberstein on Radiation will find that all the negative as well as the first and second positive, and all the even powers of wu, cancel mutually, thus giving the series G(u)=ag8 pagh st oO... which turns out to be very convenient for fractional values of uw, and even up to u=1'2. The coefficients are — 39 L6G a Ge 5) — ey et — (5+ sr) 64 1 2 2 =— 128(5, + a)+ ese aes and so on. For the first five of these I find [a;,=2/3, ds=—7/75, a;=74/(3.5.7)?, &e.], in decimal logarithms, to five figures, which will be Found convenient for purposes of numerical calculation, log as = 182391, log (—as) = 2°97004, f (11) log a; = 3°82685, log (—a,) = 447401, log ay, = 6°95402. The calculation of the following coefficients is left to the reader. What is still required is the value of A, to be substituted in (8). This coefficient, which is easily found by means of (6 aj, is given, in general, by 1+ w’ (1+ w*)A(u) = <—¢5e, ——_——— — ee A Beox 3 (GE) cos7, wtann = Diaig(D) where g(u) is as in (7), $h(u) =u? sin u+(K—1)ug(u), and —1, (12) U na ora pak apace aes aan Vale The expression for h(u) can be introduced into that of tan y, and so on, and the resulting expression for A is easily con- densed into a more elegant shape, for any permittivity and for any wave-length X. But in the present paper we shall be concerned only with two particular cases of (12), viz.: Ist, the case of moderate permittivities, which is, essentially, that of K=1; and 2nd, the case of very large permittivities, such as are required fur line-spectra. These two cases will be treated separately and, for the present, as concisely as possible, in the following two sec- tions. With regard to physical applications, it will be well to remark here, in passing, that in the former case the “source” and its functions would represent the cooperation from an Electric Source. 169 of a large number of crowded and mutually interfering atoms or molecules, and in the latter case (a being atomic), a single atom or molecule in its undisturbed individual action. But at the same time I must warn the reader that even the mere thought of entering into the structural details of the atomic ‘‘source”’ is entirely foreign to the circle of ideas here developed. Unit Permittivity. Specific-heat Curves. For K = 1 we have u = w and, by (12), A? = a®e?(1+w’)/4w®. Thus, the mean energy of the source becomes, by (8), 9 2 U = Paty 22 G(w), . . . » (18) w? where w = 27a/X, as above, and G(w) is given by (9) or (10). The mean energy emission J, per unit volume of the source, and per unit time, is, in the present case, FH gA(w), a eS a Saas ee) g being defined by (7). It is precisely the curve (14), whose first ‘‘ branch,” from w=0 down to the first root of g(w) = 0, turned out to fit closely with the black-body radiation curve, as was mentioned in the first Note. Formula (14) is the particular case, K=1, of the formula numbered (1), but uot written out in that Note; it will be given in the next section. If the absolute temperature T is introduced as a parameter determining the reduction of scale of the radiation curve (14) *, in accordance with Wien’s law, then ae,” is proportional to T, and a to T-*, say 2rra Ta hia Ta (15) a 2 —_ ‘ —— . -_—— a. A=} oO f where a and y are “ universal ” constants. * It will be understood that, from the atomistic point of view, this is legitimate only as long as the sphere a is large enough to contain quite a crowd of molecules, but not when it shrinks down to molecular dimensions. No inferences therefore will be drawn about the rdéle of “temperature ” in the case of large K and molecular a, corresponding to line spectra. 170 Dr. L. Silberstein on Radiation The mean energy of the source, (13), can now be written Uo l Pw), os i oy 6 where 21+w? , ra yee: G(w) 2 tee = 5 [ as+ (a3 + a3)w? + (a5+ a7)wt+.. y | Tt seemed interesting to consider the differential coefficient of the mean energy with respect to the temperature, at constant A,=A,, say, C= (UO). . 2 which might be called the sphere’s “ specific heat” for con- stant frequency. By (16), and writing Fi sas Co, we have ultimately Cn =F] E(w) — WF (w) |. wee F¥ being given by (17). In series form we can write, by the developed form of (17), and remembering that a,=2/3, C/Cq = 1—byw? + bywt—bew® +... . . (19a) w= TT, 0d itl: oe where T, stands for the constant 27ra/A,. The coefficients 6, which—as far as I have calculated them—are all positive, are by = dees as), bi= ~* (as+az), Og= S (a7 +g), ete. ; thus, for instance, b,=43/50, 6,=191/490; the following ones are far from claiming to be elegant numbers. For purposes of practical calculation the logarithms of these coefficients will be most convenient; these, enabling the reader to use (19a) up to w’, are log by = 1-93450)) )) log 6, = 1:°59083, (24) log by = 2°68220, log bg = 3°48187. B They will be found sufficient up to w=1:1 or 1:2, and beyond that the original form (19), with the u»per line of (17), must be used, the power series then ceasing to converge. | It may be mentioned here that a direct comparison of the radiation formula (14) with Lummer and Pringsheim’s from an Electric Source. 171 experimental isotherms 1087 and 1377 deer. absol. gives for a very nearly the value 2 = ‘20 cm. x degree. But this only by the way. The essential thing is that « plays the part of an “universal” constant (as long, of course, as a is large enough), so that T. is ultimately proportional to N=. It will be noticed that, w being dimensionless, T, is a certain temperature. If the reader is willing to treat our © as the specific heat of a concrete substance, he may call T, “the characteristic temperature” of that substance *. But as a matter of fact, T, is a certain magnitude associated with our electric source, proportional to its actual frequency n=n,, which according to the definition (18) is to be kept constant, and which from case to case can have different numerical values. It is, in short, a jived parameter of the electric source, just in the same way as T’, in general, or, still better, a itself, is 4 slowly variable parameter of the source. We could investigate a number of similar sources with different frequencies of oscillation, say from n=O up toa certain fixed value, and find the corresponding average of C in order to identify such an average with the specitic heat of a given substance. But before doing so it seemed in- teresting to examine the “specific heat”’ of a single source as given by (19) or (19a). Now, having plotted C against T for a set of different values of T.= const... I have been surprised by the fact that the resulting curves, which have C =C,, for their common asymptote, exhibit a very close resemblance with the atomic-heat curves of various elements. At first I had only the w?-term of (19 a), and correspondingly . small pieces of the curves; but with the next two or three terms the agreement proved to be much closer. Some numerical results are given in Table IV. which needs no further explanation. TaBLeE LY. o sett a oivers ‘Ts 50°12. ee . 535 331 | 200 | 100 "7 | 538 | 514 | 455 |} | —.| F cale ‘99, 98, 24, | +1805 10g) “224 47, | *45 | | al ATRER aa) ears R c Ves Tae ? St Ca Boe -| “99, | "96, | 94 | 80, 67, | ‘50, | ‘47, | 41 | | * Although there is nothing characteristic or peculiar happening just at that temperature. What I wish to say is that the curve (19) has no singularity in that neighbourhood; singularities, which, if undesirable, can be abolished, set in at temperatures much lower than Te, as will be seen later on. 172 Dr. L. Silberstein on Radiation An equally good agreement has been obtained in the case of aluminium, for which the suitable value of T, is about 92 deer. absol. Bnt since w=T,/'l’ enters in our formula always in the same way, the above example will suflice. With regard to the above numerical values of T,, it will be well to remark that they are altogether different from those of Debye’s ‘characteristic temperature.” This is due to the circumstance that our assumptions, leading to (13) and (19), have nothing in common with the theory of Debye, which is based on the concept of “quanta” and is con- structed on non-electromagnetic lines*, In fact, the series (19 a), although consisting of alternately positive and nega- tive terms in (T./T)*, (T./T)*, &e., as that of Debye, has entirely different coefficients. Unlike Debye’s, the proposed formula is in intimate connexion with that for the emission. But a somewhat detailed comparison of the two, utterly different, points of view would be out of place here. In fact we have stopped here to consider specific heats at all only for the sake of illustration of the formula (13) for the mean energy of the electric source. But before leaving this subject a few more remarks seem indispensable. ‘The above table is broken off at about 45 (2. e. a few degrees below T.) not because the practical application of (19), instead of the series, was laborious, but solely for the reason that our specific-heat curve, corre- sponding to a constant é, does not descend steadily right down to zero, but in approaching the absolute zero of temperature begins to go up and down. The amplitude of these oscillations does not even decrease indefinitely but tends to a finite and comparatively large value. Indeed, using (9) and (17) in the rigorous formula (19), we find, for very large values of w=T,/T, C/C,, = 3[1—497(w)] = 3(1—3 cos? w), 2. é. incessant oscillations between 1°5 and 3, which is utterly unlike any facts of experience. This apparently undesirable behaviour of the curve at lowest temperatures, which fully corresponds to what I have expected beforehand, is intimately connected with the peculiarity of the radiation curve (14) in consisting of an infinite number of branches, contained between the consecutive zeros of g(w). The first of these * Cf. P. Debye’s paper “Zur Theorie der spezifischen Warmen,” Annal. der Phusik, vol. xxxix. (1912), pp. 789-839, from which the “observed” values of Table IV. have been taken. Debye’s “ charac- teristic temperature ” (@) is, for Ag, 215°, and for Al, 396° absol. For a comparison of coeflicients see Debye’s formula (12'). from an Electric Source. WS branches keeps close to the experimental black-body iso- therm ; the following ones have nothing to do with experience as far as it actually goes. Now, as I shall show in detail at a later opportunity, the second branch of the radiation curve, stretching from the first zero down to the second zero of radiation, can be easily abolished as such. In fact, if e), the amplitude of the impressed force, instead of being assumed constant, as hitherto, is taken to be inversely proportional to . w—w,*, the first two branches coalesce into one continuous branch which fits with the experimental isotherms as well as the first by itself didt, thus pushing the difficulty back to the next, that is, originally the third branch. This possibility is based upon the property of g(w) of giving Lim Jf 9 (2) —_——_ - =4u, sin w iP ieee ww, U(w/w,)?—-1 ee + so that J(w,) becomes, apart from a constant factor, equal to wy," sin? wy instead of vanishing at that critical frequency. Similarly, by using the divisor w?—w,” the second zero can be abolished, and soon. In this way we might obtain the coalescence of as many branches of the radiation curve as we like, pushing the difficulty back, towards higher and higher frequencies (or lower temperatures), until it ceases to be a practical nuisance at all. And, pari passu with this procedure, the specific-heat curve will be deprived of its capricest. But in thus abolishing branch after branch we must not go so far as to forget that the seyeral maxima of the emission curve are not merely so many mathematical nuisances. For it is only due to their existence that our source is capable of emitting line-, and, generally, non-continuous spectra. And it is certainly not undesirable to have essentially one and the same conceptual radiator for all kinds of spectra. Notice that the passage from continuous (via broad-lined) to the sharpest line-spectra consists in (increasing K and) reducing the dimensions of the source until it contains but a single * Where w,=4'4954 is the first root of g(w)=0, the following ones being w.=7'7253, w3=10°9041, &e. + And even slightly closer. { In the case of silver, fur example, we should not have to go very far, for the time being, because the original formula, with e,=const., works quite well down to 45° or 43° absol., and the obseivations have. uot, in that case, been pushed below 55°. ‘ 174 Dr. L. Silberstein on Radiation atom. In connexion with this I should like to remark here briefly, that the introduction of the above divisors is by no means a purely mathematical artifice but seems to have an obvious physical meaning. To see this, notice that assuming e=é).sinnt with e) constant, 2.¢. independent of n, is equivalent to writing down the differential equation 9 mal 22 y —— + ne=(), dt® and that using a set of the divisors in question is as much as postulating, instead of this, a chain of equations d?e/dt?? +n ve=e', de'/dt? + n2e’ =e", &e., &e., ending, say, with ¢&™=e, sin nt, where e)™ is a constant. For this leads at once to va ; sin nt e=const. CME FP ER the supplementary oscillations of frequency 7, n,, ... being themselves irrelevant, since they are strictly self-contained and thus do not contribute to the radiation of the source. Now, achain of differential equations of the above form is suggestive of a linkage of entities otherwise unconnected. I shall attempt to develop further these rapid remarks in a later publication. Meanwhile let us return once more to the expressions (13) and (14) for the mean energy and the rate of emission of the electric source. Hach of these is a function of A/a(=27/w), and contains, besides, a alone and éy in its constant (or slowly variable) factor. It is obviously interesting to consider the ratio of emission to store of energy. Remember that U stands for the mean energy of the whole source, while J is the emission per (unit time and) unit volume of the sphere. The period of oscillation being 2/n, define what might be ealled the relative emissivity or the prodigality of the source by __emission per period — (2r/n) x vol.x J (23) ~ mean store of energy _ U ‘ By the definition (23), and by (13) and (14), we have, for K=1, and for any w, 2 be) OS hi c= An eae Gene oe eee from an Electric Source. 175 and since g and G contain besides w purely numerical co- efficients only, ¢ is a function of d/a alone, independent of ep, of the size of the source, and even of the choice of time- and length-units. (We shall see in the next section that this property, with a slight modification, belongs also to a source of any permittivity.) For ‘‘long” waves, 2. e. for small values of w, we have, by (7), 2 4 6 Asia: gw) = ayo — ny wet 7 shih Ta and, by (10), (1 + w?)G(w) =w*az+(ast+as5)w?+ ...] +35 wv, so that (24) becomes 2a RS o— ae a N . (x) + o . . . e my (24a) where N is a purely numerical coefficient. Thus with increasing “long” waves e¢ rapidly decreases to zero*. Formula (24a) is valid up to terms of the fourth order, since the next neglected term is in w’. Up to terms of the sixth order I find, from the above series, 2ar 33 stole c= yw (Lt 550) MaMa arg) 2.9 which can be used up to w=*90 or ‘95. Beyond this the rigorous formula (24) must be employed. Since G(w) is, by (9'), essentially positive, so is also e, as might have been expected. It attains, between each pair of consecutive zeros (w, = 4:49, w. = 7°73, &e.), a maximum. * The reader will be familiar with formul such as (24a) from the current, “elementary,” theory of the Hertzian radiator, which, by a very rough treatment of the problem, gives, for the ratio of the energy emitted per period to the original electrostatic energy of the apparatus, 1671 /?R' oa TARE Ca KR’ being the radius of each of the (larger) spherical conductors and / the distance of their centres apart. Sve, for instance, Drude’s ‘Physik des Aethers,’ 2nd ed. (1912), p. 545. Thus, b'/? takes the place of our a°, For the actual data of Hertz’s experiment (R'=15, 7=100, A=480 cm.) Drude finds (per half-period ‘352, z. e.) «="704, which is certainly a huge prodigality. The reader will notice that A is here not very “long,” being, in fact, only about Y times (/°R')!3, and 4'8 times J, z.e. less than five times the length of the axis of the “ dipole’ which has conceptually to correspond to the actual oscillator. € 176 Dr. L. Silberstein on Radiation Large K, and molecular dimensions of source. For any value of K, equations (6) give for the mean rate of emission, per unit time and per unit volume of source, 4n* u9?(u) s J= ee sae Ne Sir ah, pale ate (25°) This, intermediate, form is easily obtained by remembering that the productivity of the source is (by the very definition of Heaviside’s concept of impressed force) given by the scalar product KeE. Substituting A? from the first of (12), and developing the second of (12) for tany, we have, ultimately, i een See J= epee oe wtann+1_ 1fsinu — wet+tl ~ ulg(u) pe | rs 5) J In the same way A? can be introduced into the expression (8) for the mean energy U of the source. But, having already J, it will be more convenient to write down the expression for the more characteristic ratio, the relative emis- sivity e, as defined in the last section. For this sake use the intermediate form (25'), multiply it by the volume of the source and by the period of oscillation, and divide bv (8). Then, remembering that u= Kw, the result will be, for any K and any frequency, slit, MOO). | ut healer ‘The reader can verify at once that (14) and (24) are particular cases of (25) and of (26), respectively, viz. for K=1. The relative emissivity of the source is again independent of anything but the ratios of the wave-lengths 2, d,, outside and inside the source, to its radius*. The factor w/(1+w’) is as in (24), and in the other factor of (26) wu has simply taken the part of w. And since w is for the source precisely the same thing as w for the surrounding vacuum, we can say that the structure of ¢ is the same in the particular and the general case. This makes the intrinsic character of ¢ even more pronounced. Our present interest lies in large values of K and even of clu=K?. The reason for contemplating values of ¢/v such * w=2ra/r, u=IQra/As; As=A/K12, from an Electrie Source. Li as 10%, especially when line-spectra are aimed at, will be explained later. Meanwhile, in order to fix the ideas, sup- pose that c/v and therefore also u/w is of the order 10%, and the radius of the sphere of the order 1078 cm., 2. e. molecular or atomic. Then, for the whole extent of the physically interesting spectrum (with the exception, for the present, of the admirable X-ray domain recently discovered), w will be negligible in the presence of 1, although u will mount up to many units. Under these circumstances formula (26) becomes ug? (u) . e=47rw— Bde Met Ree aM (3) 7: : G(x) ’ oe) and (25), after some easy reductions, Ceo” Cos? ) ak wes! |] 1) bi P | 4 1 Re inh? a (25a) w tan n= ssl te ob = | gu) Cay We shall denote by ww, w, &c., the successive roots of g(u)=0, i.e. of tanu=u. ‘The values of the first three of these roots have already been quoted (footnote, p. 173). A formula, due to Euler, convenient for the calculation of the following ones, is 2 Loic pw TRG) Uy =4,—4,1— 5a — aa ?— Le; ye ely Ce U=L;—-# gt 15" ion”* ? (27) e (i . . . where v= (22+1)5*. Beginning from :=20 or so we can safely write u;=(2i+ es For w=, wv, &. we have tann=~x, and therefore, rigorously, J =0, just as in the case of unit permittivity, and, since G(w)>0 always, also e=0. The corresponding _wave-lengths will be denoted by »,, v2, &e.; these waves are strictly internal; not a trace of them escapes from the source, Since w? varies comparatively slowly, the successive * The tables of Jahnke and Emde (loc. cit. p.3) contain sixteen of these roots, to four decimal figures. Formula (27) will be found in Euler’s ‘ Introductio in anal. inf. ii. p. 319; also in Lord Rayleigh’s ‘Theory of Sound,’ 2nd ed. vol. i. p. 334. Phil. Mag. 8.6. Vol. 80. No. 175. July 1915. N 178 Radiation from an Electrie Source. maxima of radiation will, by (25a), be given very nearly by Cen re Geo ey i. a fice en Ana They will be denoted by J,, Jo, &ce., and the corresponding wave-lengths X by Ay, Ay, Ke. They corrrespond to cos’ n=1, i.e. tan 7 = +o, andare therefore roots of the transcendental equation cosu K Day as a ans. - ° . . ( as mentioned in the first Note. With the only exception of the nearest neighbourhood of these roots, we can write cos? n/w? = (w? + w? tan? 7)-1 = (w tan )~”, that is, instead of (25a), CO,” fF COSMO arn, Nees ie e J= 4 (oa +=3) Sa . ° (25d) because K/u? is (everywhere but at w, w2, &e.) large * in comparison with cosu/g(u). For the same reason, as is seen at a glance from (29), the positions of the maaima are very near the corresponding zeros of radiation, 1. e. Ay, Az, Ke. are slightly > 1, ve, Ke., the differences being almost imperceptible. Numerical examples will be given in a later note. To put it briefly, the spectrum, for large K, is practically confined to very narrow bands with Jmax. proportional to (A/a)?, the intensity of radiation elsewhere being proportional to (a/d)?. Asa rule (to which exceptions may occur in connexion with possible peculiarities of K as a function of 2), the passage from light to darkness is more abrupt towards the ultra-violet than towards the infra-red, although it is, for all purposes, abrupt enough on both sides. The position of these sharp spectrum “lines” is practically given by the roots uy, U:, Ke. themselves. The promised justification of contemplating large permit- tivities at all, and enormous ones especially, will be given in the next Note. May 5, 1915. * Even for high values of wu (viz. ui), for these occur only in the neighbourhood of a convergence-point or ‘head,’ where K is very rapidly increasing. tree] XII. On the Resistance experienced by Small Plates exposed to a Stream of Fluid. By Lord Rayueisy, O.M., F.RS.* a. a recent paper on Molian Tones f I had occasion to determine the velocity of wind from its action upon a narrow strip of mirror (10°1 cm. x1°6 cm.), the incidence being normal. But there was some doubt as to the coefficient to be employed in deducing the velocity from the density of the air and the force per unit area, Observations both by Hiffel and by Stanton had indicated that the resultant pressure (force reckoned per unit area) is Jess on small plane areas than on larger ones; and although I used provisionally a diminished value of C in the equation P=CpV? in view of the narrowness of the strip, it was not without hesitation f. I had in fact already commenced experiments which appeared to show that no variation in C was to be detected. Sub- sequently the matter was carried a little further; and I think it worth while to describe briefly the method employed. In any case I could hardly hope to attain finality, which would almost certainly require the aid of a proper wind channel, but this is now of less consequence as | learn that the matter is engaging attention at the National Physicai Laboratory. According to the principle of similitude a departure from the simple law would be most apparent when the kinematic viscosity is large and the stream velocity small. Thus, if the delicacy can be made adequate, the use of air resistance and such low speeds as can be reached by walking through a still atmosphere should be favourable. The principle ‘of the method consists in balancing the two areas to be com- pared by mounting them upon a vertical axis, situated in their common plane, and capable of turning with the mini- mum of friction. If the areas are equal, their centres must be at the same distance (on opposite sides) from the axis. When the apparatus is carried forward through the air, equality of mean pressures is witnessed by the plane of the obstacles assuming a position of perpendicularity to the line of motion. If in this position the mean pressure on one side is somewhat deficient, the plane on that side advances against the relative stream, until a stable balance is attained in an oblique position, in virtue of the displacement (forwards) of the centres of pressure from the centres of figure, * Communicated by the Author. t Phil. Mag. vol. xxix. p, 442 (1915). t See footnote on p. 445, N 2 180 Lord Rayleigh on the Resistance experienced The plates under test can be cut from thin card and of course must be accurately measured. In my experiments the axis of rotation was a sewing-needle held in a U-shaped strip of brass provided with conical indentations. The longi- tudinal pressure upon the needle, dependent upon the spring of the brass, should be no more than is necessary to obviate shift. The arms connecting the plates with the needle are as slender as possible consistent with the necessary rigidity, not merely in order to save weight but to minimise their resistance. They may be made of wood, provided it be accurately shaped, or of wire, preferably of aluminium. Regard must be paid to the proper balancing of the re- sistances of these arms, and this may require otherwise superfluous additions. It would seem that a practical solu- tion may be attained, though it must remain deficient in mathematical exactness. The junctions of the various pieces can be effected quite satisfactorily with sealing-wax used sparingly. The brass U itself is mounted at the end of a rod held horizontally in front of the observer and parallel to the direction of motion. I found it best to work indoors in a long room or gallery. Although in use the needle is approximately vertical, it is necessary to eliminate the possible effect of gravity more completely than can thus be attained. When the apparatus is otherwise complete, it is turned so as to make the needle horizontal, and small balance weights (finally of wax) adjusted behind the plates until equilibrium is neutral. In this process a good opinion can be formed respecting the freedom of movement. In an experiment, suggested by the case of the mirror above referred to, the comparison was between a rectangular plate 2 inches x 14 inches and an elongated strip °51 inch broad, the length of the strip being parallel to the needle, 2.é. vertical in use. At first this length was a little in excess, but was cut down until the resistance balance was attained. For this purpose it seemed that equal areas were required to an accuracy of about one per cent., nearly on the limit set by the delicacy of the apparatus. According to the principle of similitude the influence of linear scale (/) upon the mean pressure should enter only as a function of v/V/, wnere v is the kinematic viscosity of air and V the velocity of travel. In the present case v="1505, V (4 miles per hour) =180, and J, identified with the width of the strip, =1°27, all in c.a.s. measure. Thus v/V1="00066. by Small Plates exposed to a Stream of Fluid. 181 In view of the smallness of this quantity, it is not surprising that the influence of linear scale should fail to manifest itself. In virtue of the more complete symmetry realisable when the plates to be compared are not merely equal in area but also similar in shape, this method would be specially advan- tageous for the investigation of the possible influence of thickness and of the smoothness of the surfaces. When the areas to be compared are unequal, so that their centres need to be at different distances from the axis, the resistance balance of the auxiliary parts demands special attention. I have experimented upon circular disks whose areas are as 2:1. When there was but one smaller disk (6 cm. in diameter) the arms of the lever have to be also as 2:1 (fig.1). In another experiment two small disks Fig. 1 A (each 4 cm. in diameter) were balanced against a larger one of equal total area (fig. 2). Probably this arrangement is Fig. 2. c the better. In neither case was any difference of mean pressures detected. In the figures AA represents the needle, B and © the large and small disks respectively, D the extra attachments needed for the resistance balance of the auxiliary parts. XIII. On the Distribution of Electric Force in the Discharge at Low Pressures. By T. Harris, B.A., B.Sc., AR.CS* by eae object of the work to be described below was to determine the distribution of the electric forces in a discharge through air at low pressures, in a cylindrical discharge-tube. Sinular measurements at higher pressures have been made by many workers on this subject who have mostly used a probe method, in which a wire is inserted into the gas and assumed to take up the potential of the gas at that point. Objections have been raised against this methodt, and at low pressures it is specially unreliable, so that the method was adopted which has been used by Thomson f and Aston§, in which a fine beam of cathode rays is projected through the field of force, the magnitude of which is estimated from the deflexion of the beam. Thus if v is the velocity of the cathode rays passing through d cm. of a uniform field, X, at right angles to the direction of the field, thence proceeding to the screen upon which they excite a phosphorescent spot situated at a dis- tance / cm. from the edge of the field, then the deflexion of the spot on the screen from its position when X=0O is given by b=X. 5d ($ +4), "my?" where e/m is the ratio of the charge to the mass of one electron. This holds if the assumption is made that the deflexion of the beam is small enough for the path of the rays to be considered always as perpendicular to the field. The velocity v is given by tmv’= Ve, where V is the generating potential of the rays. Substituting this value of v xX d * Communicated by the Author. | For experimental work see Wehnelt, Deutsch. Phys. Gesell. Verh. Xill. xiv. pp. 505-510, July 1911. t J. J. Thomson, Phil. Mag. xviii. pp. 441-451, Oct. 1909. § Aston, Proc. Roy. Soc. A. 1910-1911, p. 526. cea LNstribution of Electric Force at Low Pressures. 183 The force per unit deflexion of the spot is given by ges ie ry: i a+) Therefore by multiplying the actual deflexion of the spot by X’ the value of the force in the field through which the rays have passed is obtained. If V is measured in volts, X is found in volts/em. In the present work d is taken as the diameter of the discharge-tube. | Apparatus. Two types of apparatus were used. The final form is shown in fig. 1. The discharge took place in the tube M To Wimskurst / BS oe |: @o= =: Sf a} ‘a Spark gap Ge To earth Willemite screen W To earth To earth Poles of To mercury pump, P>0 d 1 e fipiiy iM ‘¥ pump, F2 0; drying tub Wimshurst MS Leod £auge, charcoal bulb. between parallel plate electrodes which were capable of being moved up and down the tube whilst being kept at a fixed distance apart. The beam of cathode rays was produced in 8, passing from 184. Mr. T. Harris on Distribution of Electric Force the cathode c, through the earthed tube anode a, and the earthed metal tube ¢, entering the discharge-tube through the hole, A, bored in the side of the tube, emerging from the discharge-tube through a similar hole diametrically opposite, whence it proceeded through the earthed metal tube t’ to the wiilemite screen, W. Thus the cathode rays were shielded from. electrostatic disturbances by earthed metal tubes during the whole of their journey, except when passing through the field it was required to measure. The main discharge M was run froma six-plate Wimshurst, the secondary discharge S from a small Wimshurst, both machines motor-driven with both positive poles earthed. The voltage across M was measured by a Braun electroscope, that across S by measuring the equivalent spark-gap G. The application of the method for the measurement of big forces is difficult. It is necessary to use tww discharges, one running at a very much greater voltage than the other. If the gas is at the same pressure in both cases this can be accomplished by putting one cathode in a constricted space, so that the charged walls surrounding it require the use of a big voltage to cause a discharge. This necessitates the use of a small cathode placed in a narrow-bore tube, and the difficulty arose that the discharge was inclined to be very irregular for P.D. greater than 15,000 volts, ¢. g. taking place from the edges of the cathode instead of forming a steady beam of rays from the central portion. So much time was spent in surmounting this difficulty that a full description of the design of the tube 8 is not unwarranted. The cathode is shown in fig. 2. It was constructed as Fig. 2. Capillary tubing ZILIA le < Aluminium core A$ S TFicstble mata! Cathode follows. A small bulb was blown in a piece of capillary tubing and drawn out until it was of the same external diameter as the rest of the tube. This was cut through and ground down until the plane of the end was perpendicular to the axis of the tube. A piece of aluminium rod which fitted the capillary was pushed through until it projected a little past the end. The space between this and the walls was filled with pieces of fusible metal, which on warming fused and on cooling solidified and expanded, completely filling the whole of the space between the rod and the glass. to) The metal end was ground down flush with the glass and im the Discharge at Low Pressures. 185 polished. By this means a cathode was obtained with no projecting edges and the central portion from which the discharges take place of aluminium, thus getting rid of sputtering. This cathode was pushed into a glass tube, which it just fitted, in which a bulb 2 or 3 cm. in diameter was blown; the cathode was about 1 cm. from the bulb. The anode was a brass tube closed at both ends by sheet metal ; through the end nearest the cathode a hole 1 mm. in diam. was bored, and in the other a fine hole pierced with a needle- point. Fig. 3 shows the complete arrangement. This gave Fig. 3, At Sal A, es mmm A>~APD PPLE esis eerie . Anode Cathode Brass ball for eee th Secondary discharge tube- Wimshurst. a steady beam of rays up to P.D. of 30,000 to 40,000 volts*, the spot of phosphorescence on the willemite screen being sharp and about 0°5 mm. in diameter. 5 Corrections. It is assumed in applying the cathode ray method that the equipotential surfaces in the discharge are planes perpen- dicular to the axis of the tube. At very low pressures this is certainly not the case, the appearance of the discharge varies from point to point across any section near the cathode perpendicular to the axis, and at the lowest pressures used here a well-defined pencil of cathode rays passed down the centre of the tube. At the higher pressures when the boundary of the negative glow and Crookes’ dark space extends evenly across the tube, the forces in the dark space may be calculated, but at lower pressures the method gives only a rough idea of the magnitude of the forces in the discharge. For this reason it is useless to apply any cor- rections to the formula already noted. Eaperimental Procedure. The apparatus was exhausted by a water-pump and mercury-pump, and finally completed with the aid of liquid air and charcoal. The discharges were started and allowed to run for a short time to reach a steady state. The pressure * A constriction in the tube between the cathode and the bulb “hardens ” the discharge considerably. 186 =Mr. T. Harris on Distribution of Electric Force of gas and voltage across the discharge were noted, and the length of the equivalent spark-gap found for the potential across the secondary tube. The position of the electrodes of the main discharge was found from a scale fixed to the tube. These electrodes were raised until the beam from the secondary tube passed close to the anode, and the reading on a scale on the willemite screen was taken. The electrodes were lowered em. by em. until the beam of rays passed as close to the cathode as its velocity permitted, the spot of phosphorescence being at the extreme end of the scale. The effects of varying (1) the gas pressure, (2) the distances between the electrodes for aluminium and nickel cathode, (3) diameter of the discharge-tubes, were determined. Results. The results are shown in the curves, in which electrie force is plotted against distance from electrode. Fig. 41s for a tube 2°7 cm. in diameter with aluminium electrodes 12 cm. apart. Starting with a fairly high pressure when the con- ditions are such that the discharge consists simply of a well- defined dark space with negative glow extending right up to the anode, the curve A is obtained. The point q is the edge of the dark space; right through the negative glow the force was too small to measure. Aston found, under similar circumstances, a linear curve with a discontinuity at the edge of the dark space. Here there is no discontinuity, but this may be explained by the fact that the boundary of the dark space was slightly concave to the cathade, insted of a plane parallel. As the pressure is lowered the appearance of the discharge varies rapidly. The dark space begins to extend towards the anode, and the negative glow gradually becomes fainter until the boundary between it and the dark space becomes practically indetinable. The discharge at the cathode takes on the appentaae of a broad beam no longer filling the whole tube, but tending to concentrate in the central portion of the anode. Green phosphorescence begins to appear upon the walls of the tube about the anode. This state of the tube is represented by curve B. If the pressure is still decreased the negative glow fades completely away, the broad band at the Wha, rye. . Sega ee ee ay — pee eee ee Se in the Discharge at Low Pressures. 187 -eathode narrows into a pencil of cathode rays, and the green phosphorescence becomes much brighter and fills the tube from anode to cathode. Curve C represents the measure- ments under these conditions. Not much reliance must be placed upon the actual magnitudes of the forces, but it can Fig. 4, SERpSASE Ss S558 goad Sess Beseraeats Hii oe 1600 1400 a iS seetsest teat \vi200 ! ee ~~ Hi Sonatas a ie -§ 1000 pas eg esis ate G4 ete ee % He eae et ee ee cee Fe Pea Bie 8 Sea a e & iH ae ise GO § seed baal treat $seS2 002035 Bish dz i ‘HEU Farin ae Ae an be ee pot ES HE dis ; “4 600 on i A oO o i itt fel Cifia tot Hee bet He 1 Distance from cathode in c.m. be seen from the curves that at any one point in the tube near the cathode the force increases rapidly as the pressure is decreased. Hxamining the points where the curves cut the axis, it is to be noticed that the dark space for curve B, represented by pv, is much longer than for A, pq, while for the -further decrease in pressure the extension to § is slight. Similar results were found for a tube of diameter 4°7 em. using a nickel cathode, except that the total length of the dark space was about 8 cm., when the total length tended to become independent of pressure. With this tube also the effect of pushing the anode into the normal dark space, 7. e. 188 Mr. T. Harris on Distribution of Electric Force constricting the length of the dark space, was examined (fig.5). Here A and B (two measurements at different times) were taken when the negative glow was faint without a definite boundary, and the discharg ge was beginning to concentrate 1000 ela Heese ee a iS : 600 paises eaess < He ee ate S Ee & 500 ae E Ee v elas inn 2 EG a e iueee Es 400 asian is, Electric tit i Ba Les) Oo Oo Distance from cathode in c.m. into the centre of the cathode. The voltage required to drive it was 750 volts. C was taken at a lower pressure when the glow had faded away and phosphorescence was beginning to appear round the anode ; D, when the phos- phorescence had extended about half way up the tube and the visible discharge was a beam of cathode rays, voltage 6000; F, when the pressure was still lower, the tube phos- phorescing brightly between anode and cathode, visible discharge a narrow pencil of cathode rays, voltage 8500. In this case the curves all end at about the same point, and show that there is a region in front of the anode where the force is extremely small, although it is inside the normal dark space, which, as was seen above, is about 8 cm. in the Discharge at Low Pressures. 189 Fig. 6 gives the results for a tube 3°5 cm. in diameter, electrodes of aluminium 13°6 cm. apart. To curve A there corresponds a faint negative glow, a broad beam from the Fig. 6, aged noses eae est Esze; Hees id SEES Er Saath ee al HH 5 bisttrrteH paar i ee ne sh = es Pepesenaad tt eeeseberrctecttee aa, ae Seanissaystsssqusess ett a es { Hee See Ea ati: fl a ieee Distance from cathode in ¢c.m. cathode, voltage 1700 ; to B, the point when the glow had faded away, voltage 4700; CO. thin pencil of cathode rays, bright phosphorescence, voltage 7000. In spite of the decrease in pressure and the big increase in voltage, from 4700 volts for B to 8000 for C, there is only a small extension of the region in which there is any appreciable force. The length of this field is about 7°5 cm., being intermediate between the corresponding values for tubes of greater and smaller diameter. In one experiment the anode was placed in a side tube 20 or 30 cm. away from the cathode. The pressure was very low, 12,000 volts being required to drive the discharge, and the glass was phosphorescing 40 em. away from the cathode ; cathode rays were traversing the whole length of the tube. No force could be detected except in the region just in front of the cathode. The pressure-gauge and mercury-pump were cut off and evacuation carried out solely with the aid of liquid air and charcoal, thus eliminating mercury vapour from the tube ; 190 =Mr. T. Harris on Distribution of Electric Force the results were similar. Thus the effects noticed do not depend upon the position of the anode when it is beyond a certain distance from the cathode nor upon the presence of mercury vapour. Conclusion. In all cases there was a rapid increase of potential fall across the discharge when the negative glow became faint and the charges on the walls began to alter the character of the discharge. With the electrodes far enough apart, the leneth of the field corresponding to the dark space at first increased rapidly with the decrease in pressure, but as soon as the negative glow had faded from view and the visible discharge became concentrated into the centre of the cathode, the length of this field became very nearly independent of the pressure. The length of the field at this stage depends upon the diameter of the tube, being longer for ee tubes. Throughout the rest of the discharge the force was too small to be detectable, z. e. under a few volts per cm. With the electrodes close enough for the anode to con- strict the dark space, the length of the field for the range of pressures used was practically constant, and there was a small region in front of the anode where the force was extremely small. In the one case where the pressure was high enough for the negative glow to have a well-defined boundary, the curve appears approximately a straight line. Applying the equation aX Fp ATP: where -X=the electric force along the axis, «x =distance along the axis, p=the density of the excess of positive over negative electricity ; it is seen that in the dark space the positive electricity is in excess, and that by a practically constant amount throughout the space, which was shown more accurately by Aston. The same holds roughly in the other cases as the cathode is approached, but towards the anode end of the field not much information can be obtained ow ing to the probable distortion of the equipotential surfaces. Throughout the rest of the discharge the positive and negative electricity are present in equal quantities. The distortion of the equipotential surfaces does not affect the measurements of the length of the fields of force, for the in the Discharge at Low Pressures. 191 cathode beam is here acting as a detector rather than as a measuring instrument. The result that the extension of the field becomes practically independent of the pressure is rather surprising, for one would think this length would be a function of the mean free path in the gas rand therefore of the pressure. It is to be expected that the length should depend on the diameter of the tube, for with w ider tubes the pressure can be reduced further, with a consequent greater extension of the field, before the charging of the. walls changes the character of the discharge, after which the length of the field remains practically independent of the pressure. Perhaps it is possible to correlate these results with those at higher pressures. Referring to Aston’s work™ on air it can be seen that at a certain pressure and for a certain current density the length of the dark space was 1°805 cm. Decreasing the pressure in the ratio 9/12 caused the dark space to expand to 2°18U em., while increasing the current density at the lower aera to four times its previous value caused a contraction from 2°180 cm. to 1:900 em. De- creasing the pressure caused an extension of the field of force; increasing the current density caused a contraction, and the two effects may be made to balance. When the pressure is so low that the visible discharge is concentrated -into the central portions of the cathode, Wehneltt has shown that practically the whole flow of current from the cathode is concentrated in this beam, the current from the outer spaces being but a very small fraction of the total current. In the experiments deseribed in this paper the field of force began to become almost independent of pressure always when ‘the dischar ge began to be concentrated into the centre. At this point, therefore, decrease of pres- sure may increase the current density, as the cross-sectional area of the discharge rapidly becomes smaller when the beam of cathode rays is developed. The extension of the field due to decrease in pressure may be counterbalanced by the contraction due to this increase in current density, and the result of the two effects be to make the length of the field - almost independent of the pressure. The conditions of the discharge, even at higher pressures, are so complicated that no quantitative theory has ever been proposed, and at low pressures the conditions become more complicated still. It may be that the effect mentioned above # Aston, Proc. Roy. Soc. A. lxxix. 1907. p. 94. t Wehnelt, Deutsch. Phys. Gesell. Verh. xv. ii. pp. 47-52, Jin, 1913, 192 Sir J. J. Thomson on Conduction is a consequence of a change in the mechanism of the dis- charge, as the high speed cathode rays make their appearance with a consequent increase in the amount of Rontgen radia- tion in the tube, that an unknown factor which is of small importance at high pressure predominates at the lower pressure, causing the length of the field to be independent of pressure. I have much pleasure in recording my thanks to Professor Sir J. J. Thomson for suggesting this work to me. East London College. XIV. Conduction of Electricity through Metals. By Sir J. J. THomson, O.M., F.R.S.* rFINHE investigations of Kamerlingh Onnes on the resistance of metals at the temperature of liqnid helium have led to results which are of vital importance in the theory of metallic conduction ; they have shown, for example, that some metals can exist in a state where their specific resistance is less than one hundred thousandth millionth part of that at 0O°C. The transition from the state in which the resistance is diminishing normally with the temperature to the one where they possess this super-conductivity takes place abruptly at a definite temperature, and the difference in the electrical properties of the metal above and below this temperature are as well marked as the difference in elastic properties when a solid melts, or in the magnetic ones when a piece of iron passes through the temperature of recalescence. One of the most remarkable effects discovered by Kamer- lingh Onnes is that when a current was started in a small ring of lead at a temperature of about 4° absolute, by bringing a magnet close to it, the current instead of dying away, as it would have done at 0° C., as soon as the magnet was stopped, went on with practically undiminished intensity, its rate of decay being so slow that Kamerlingh Onnes estimated that it would take four days to fall to half its initial value. This power of transmitting a current for long periods when no external electromotive force acts on the metal is one that has to be accounted for by any theory of metallic conduction: any such theory must indicate that in certain metals a change of electrical state takes place at a definite temperature, that above this temperature the current dies away almost instantaneously after the electromotive force is * Communicated by the Author. Read before the Physical Society of London, June 25, 1915, of Electricity through Metals. 193 removed, while below it the current may persist for days without undergoing any considerable diminution. It seems to me that this is another, and fatal objection to the theory that metallic conduction is due to the presence in the metal of free electrons which drift under the electric force, for no permissible increase in the number of free electrons or in the mean free path would explain the difference between the ordinary and super-conducting state. In the case of the lead ring the maximum free path (equal to the longest chord that can be drawn in the ring) cannot be more than a few millimetres. It is the object of this paper to show that the effects discovered by Kamerlingh Onnes are in accordance with the theory of Metallic Conduction which I gave in ‘The Corpuscular Theory of Matter ’ (page 86), and which, with the substitution of an electron tor a charged atom, is sub- stantially the same as that given in my ‘ Applications of Dynamies to Physics and Chemistry,’ 1838. On this theory, the atoms of some substances, including the metals, contain electrical doublets, i. e. pairs of equal and opposite electrical charges at a small distance apart. In the normal state of a body the axes of the large number of doublets occurring in even a small volume are uniformly ‘distributed in all directions: when, however, an electrical force acts on the body, the axes of the doublets tend to point in the direction of the force and the moments of the doublets have a finite resultant in this direction. If the axes of the doublets were quite free to set in any direction, the smallest electrical force would be able to pull the axes of all the doublets into line and thus produce the maximum polarization. There are, however, several influences at work which limit the number of doublets which point in the direction of the electric force. ; In the case of gases, for example, there are collisions between the various molecules which tend to knock the axes of the doublets out of line as fast as they are brought into it by the electric force. Langevin has calculated from the principles of the Kinetic theory of Gases the magnitude of this effect, and has shown that if M is the moment of each doublet, N the number of doublets in unit volume, I the resultant of these moments parallel to v, and X the force on a doublet in this direction, I=NM { See ae id =| x when x=MX/RO, @ being the absolute temperature and R@ the mean kinetic Phil, Mag. 8. 6. Vol. 30. No. 175. July 1915. O +e 1 194 Sir J. J. Thomson on Conduction energy of a molecule at this temperature ; when w is very small, [=4NMa, when it is very large, I= NM. In the case of solids and liquids, though there may not be collisions between the molecules, the rotation of the molecules endows them with a quasi rigidity, making each molecule behave very much as if its axis of rotation were acted on by a restoring couple proportional to the angle through which the axis is displaced and _ proportional ‘also to the kinetic energy possessed by the body in virtue of its rotation : it behaves in fact very much like a spring whose stiffness is proportional to its kinetic energy. The value of I will bea function of the ratio of XM, the deflecting couple acting on the doublet, to the restoring couple brought into play when the axis is deflected through unit angle : ; as this couple is proportional to w, the average kinetic ener oy of the molecules, : : we have [=NMF(AMw). Thus we see that for solids and liquids, as well as for gases, I is a function of MX/w. We need not here go into the question whether the form of the function depends on whether the body is in the solid, liquid, or gaseous state. It is sufficient to notice that whatever the state, when «=0, F(«)=0, and when c=m, Ei@\e= 1, Thus F(#) will be represented by a curve of the type shown in fig. 1. The foree X which occurs in this expression Aah for # is not merely the external electric force acting on the system, the polarized doublets will themselves give rise to strong electric forces, and X is the resultant of such forces and the external electric force. We shall take the force due to the polarization of the doublets as proportional to [ and of Electricity through Metals. 195 put it equal to"kI. Thus, if Xy is the external electric force X= X,+ KI, and pox MiMot A) ays wW or OR ts] ee l= a -F° This relation between I and « is represented graphically by a straight line, and the value of I corresponding to any value of X, can be determined by finding where this line intersects the curve I=NM F(z). The effects corresponding to any finite value of I will be the same as if I doublets per unit volume pointed; in the direction of the electric force, while the axes of the rest were uniformly distributed in all directions ; and we may picture the substance as containing a number of chains of polarized atoms whose doublets all point in the direction of the electric force as in fig. 2. Fig. 2. F E D : Cc B aA x So far as we have gone there has been nothing to differ- entiate between insulators and metals ; in each of these the doublets set under the electric field and give tothe substance specific inductive capacity, the yalue of which is proportional to the value of I when X, is unity. It will be noticed that the electrons in the atoms of the substance will be under the influence of forces excited by neighbouring polarized atoms. Thus in the case represented in the figure these forces tend to make the electrons in A _ move towards B, and those in B to © and so on. On this _ theory the peculiarity of metals is that electrons, not necessarily nor probably those in the doublets, are very easily abstracted by these forces from the atoms when these are crowded together. Thus we may suppose that under these forces an electron is torn from A and goes to B, another _ from B going to C, and so on along the line,—the electrons q oe along the chain of atoms like a company in single _ file passing over a series of stepping-stones. Let us suppose 9 - if i, it 196 Sir J. J. Thomson on Conduction that p electrons pass along each of these chains per second, then if there are n of these chains passing through unit area at right angles to the electric force, the current 2 through unit area will be epn, e being the charge on an electron. If d is the distance between adjacent atoms in the chain, there will be 1/d atoms per unit length of chain, and I the number of doublets per unit volume pointing in the direction of the electric force will be equal to n/d. Thus n=Id, and therefore i=enld, The specific conductivity of the metal c is equal to i/Xo, so ae c=epd I/ Xo. The force exerted by the polarized atoms on the nearest electron in a neighbouring atom will be very large compared with that exerted by the external electric force, so that p will be determined by these inter-atomic forces and will not to an appreciable extent depend on the external electric force. The ratio of the current to this force will therefore follow the same laws as the ratio of I to the force. We have seen that the value of I is determined by the intersection of the line AOD T= Mi me (1) with the curve I=NM F@), 0. where w is the kinetic energy of a molecule ; unless the temperature is very low w=R@, where @ is the absolute temperature and R the gas constant: when the temperature falls to the stage where the specific heat diminishes with the temperature, w will be smaller than the value given by this equation. When w/MK is considerable the line (1) will be steep and will intersect the curve near the origin, where it approxi- mates to the straight line I=NMeF'(0) .... the intersection of (1) and (3) is given by NMPE'(0)X, I= D—NMPEF'(0)’ and 2 the current by ._ epdN M?F'(0)X, ~ “w—NMEF'(0) * of Electricity through Metals. 197 Thus the current is proportional to X, and Ohm’s law holds ; the specific resistance o is given by _ w—NMWLE'(0) ~ epdN M?B"(0) * Now, except at very low temperatures, w is equal to R@, so that o is expressed by an equation of the form ao=a(O—b). It is thus a linear function of the temperature, this is very approximately true for pure metals. Super Conductivity. Let us now consider what happens when the temperature is diminished, the slope of the line (1) continually decreases and the intersection of this, line with the curve gets further and further away from the origin; when the intersection comes on a part of the curve at an appreciable distance from the tangent at the origin, Ohm’s law will no longer hold. Suppose that the slope of the line (1) has fallen so that, as in fig. 3, itis less than that of the tangent at the origin to Fig. 3. the curve I=NMEF(z2), and after the application of a force X, suppose the force is gradually removed, the value of I corresponding to the diminished force will be got by drawing parallels to PQ continually getting nearer to the origin, and its value when the force has been entirely removed by drawing a parallel through the origin itself. We see from the figure that in this case the line through the origin will 198 Sir J. J. Thomson on Conduction intersect the curve again at S, showing that I retains the finite value SN after the electric force has disappeared. From the point of view of this paper, however, the part played by the electric force in metallic conduction is to polarize the metal, 7. e. to torm chains; when once these are formed the electricity is transmitted along them by the forces exerted by the atoms on the electrons in their neighbours. Thus if the polarization remains after the electric force is removed the current will remain too, just as it did in Kamer- lingh Onnes’ experiment with the lead ring. The argument is similar to that by which Weiss explained the existence of permanent magnetism below a critical temperature. We see that we shall have the current remaining after the removal of the electric force; 7. e., the metal will be in the super-conducting state as soon as the slope of the line is less than that of the tangent at the origin to the curve, 1. e. when an is less than NMF’(0), or w less than kNM?F"(0). Thus the temperature at which the metal passes into the super-conducting state is such that w=NMPLE'(0). NMzk& is the electrical force exerted by the doublets when they all point in one direction: if we denote this force by P, w=MPF'(0). If the specific heat of the metal had not commenced to diminish at this temperature, 0) the temperature of transition into this state would be given by the equation RO,=NMKF’(0). As, however, the transition takes place at very low tempe- ratures, when the specific heats are variable and w no longer equal to R@ we must use a more general expression for w in terms of @ to determine the critical temperature. _The per- sistence of the chains after the removal of the electric force is due to the disturbance due to thermal agitation being too weak to break up the chains when once they are formed. The chains are held together by the electric force due to the doublets in the chain itself as well as by the external electric force, and when we approach the critical temperature the : , —— of Electricity through Metals. 199 force due to the doublets is much greater than that due to the external field. We see this from the expression ‘GNM (0) Ry ~ w—kNMPE'(0)* If wy is the value of w at the critical temperature, oper COV at, ah! 1a yew 6 (4) so that I= - or kI i, Vee Now A&I is the part of the force on a doublet due to the other doublets, and we see from this expression that when w is nearly equal to wy) kI is very large compared with Xo, so that the removal of Xp will not appreciably weaken the coherence of the chains. On the other hand, at temperatures considerably above the critical, kI is small compared with X,, so that the external force is essential for the coherence of the chains. If the disturbing effect on the chains is entirely due to the thermal energy and if this energy vanishes at the zero of temperature, it will always be possible to find a value of w which satisfies equation (4), and there will always be a critical temperature, 7. e. the metal will be able to pass into the super-conducting state. Itis probable, however, that the action of adjacent atoms may, independently of thermal agitation, tend to make the axes of the doublet take up a definite orientation, and that the doublets, when disturbed from this alignment, come under the action of couples tending to restore them to their original positions. We can easily take this into account, all that we have to do is to replace w in the preceding equation by w+D, where D is proportional to the restoring couple for unit angular dis- placement, due to the mutually directive action of the atoms. If L is the local electric force due to the action of the adjacent atoms, D=LM. The equation to the straight line (1) is now fo wt+tD = Xo 5 = “ME ai ar . lla hfe Gk) ( ) We should expect the directive force either to be independent of the temperature or to vary but slowly with it. In this ease the slope of the line will not diminish indefinitely as 200 Sir J. J. Thomson on Conduction the temperature, but will reach a minimum value whose tangent is D/Mk. If this slope is greater than that of the tangent to the curve at the origin, w hose tan gent is NMF'(0), there will be no critical temperature, hence the condition for a critical temperature is D —. less than NMF"(0), Mk r D less than w) where we=NMPLF'(0), this is equivalent i: L less than PF’(0). The value of w at the critical tempe- rature is now w)—D. When the slope of the line is considerable, we have from equations (3) and (5) il Wo x mee Wl Dmnniyals x _. 1 w+ D—wo) i ASK z or Wo epd and o the specific resistance is equal to k (w+ D—wy) ed. jd aioe fool wahei ane (7) Unless the temperature is very low we may put w=R@, and we have k a (“ D—wp ) ~ epd Wo If o is the resistance at 0° ©., and « the temperature coefficient of the resistance o=o (1+ at), where ¢ is the centigrade temperature; comparing this with the previous expression we see that 1 2734+ D—w, | ae The condition for the existence of a critical temperature is D < wo, 2. e. that the temperature coefficient of the resistance when. the temperature is not very low should be greater than 1/273. When D is considerable the line (5) will be steep, so that at all temperatures the intersection of the curve and the line will be quite close to the origin; we may, therefore, use a= of Electricity through Metals. 201 equation (3) even for very low temperatures, so that at all temperatures k (w+ D—wy) cpd Wy The temperature coefficient of the resistance is proportional to do/d@, and this as we sce is proportional to dw/dé. This quantity, the rate of increase of the energy with the tempe- rature, is proportional to the specific heat at the temperature. As the specific heats of many substances are very much smaller at the low temperatures obtained by the use of liquid hydrogen or helium, than at normal temperatures, we see that on this theory the temperature coefficients of metals which have no critical temperature ought to be very small at low temperatures. The experiments of Kamerlingh Onnes, and Dewar and Fleming, show that this is in some cases a very well marked effect. Jig. 4 shows the variation Fig. 4. of resistance of gold and platinum suspected of not being quite pure : it will be noticed that at very low temperatures the resistance becomes almost independent of the tempera- ture. Similar effects are shown by many alloys; they would on this theory be shown at Jow temperatures by any metal or mixture which had not too small a value of D and whose specific heat fell appreciably at low temperatures. In fact the general behaviour of alloys seems to admit of a satisfactory explanation on the supposition that in them, or at any rate in those whose resistance is considerably greater than the value calculated from their percentage composition, the restoring couple D is much greater than in pure metals. This seems what we might expect when the alloy is not a mere mixture; for if it was a definite compound of the two metals, we should expect that there would be a tendency for the axes of the molecules of one metal to have definite orientation with refsrence to those of the molecules of the other. The same thing would also apply if the metals did 202 Conduction of Electricity through Metals. not form definite compounds with each other but did form mixed crystals, we should expect the local force L to be increased by anything analogous to chemical combination. We see from the preceding equations that if D were large for these alloys, they would have (1) a small temperature coefficient at normal temperatures and a very small one indeed at temperatures low enough to diminish the specific heats, (2) they would not have a critical temperature and would never pass into the super-conducting state. These are characteristic properties of the resistance of alloys Again, if there are m molecules of the metal (1), n of the other (2) per unit} volume, we should, from the expressions (7) for the specific resistance of a pure metal, expect that o the specific resistance of the alloy would be given by a formula of the type o={ oh (w+ D;— wo) D,—w) ve mikey eet [Gata ep dy Wo e pails Wo As this involves the restoring couples D, and D, it cannot be calculated from the resistances of the pure metals; we see, however, that o;—oy the, difference between the specific resistances of the alloy at the temperatures ¢ and T, is given by the equation o—op={ mhky (we —ur) rf ike (Wr—wn) Y [omen ep1d, Wo epads | ag ed the D’s have disappeared from this equation, and it is exactly the value we should have calculated on the supposi- tion that the alloy is a mechanical mixture. This is the result known as Matthiessen’s rule, which states that even when the specific resistance of the alloys cannot, the dif- ference between the resistances at two temperatures can, be calculated from its constituents; another way of stating it is that the difference between the observed and caleulated value is independent of the temperature. We have supposed that D is independent of the temperature; if it changes appreciably with it, as it might be expected to do if the nature of the compounds, or mixed crystals formed by the two metals did so, the temperature coefiicients would show anomalies such as those found in alloys which have negative temperature coefficients. I have shown (‘Corpuseular Theory of Matter,’ p. 86) that the electric and thermal conductivities will on this theory bear a nearly constant ratio to each other if the electrons which take part in the conduction are in thermal equilibrium with the metal in their neighbourhood. rr a05) 3 XV. Onan Anomalous Variation of the Rigidity of Phosphor Bronze when in the form of Strips. By HaRroLp PEALING, M.Sc., Lecturer in Physics at South African College, Cape Town, late Oliver Lodge Fellow, University of Liverpool*. gee when in the form of very fine wires or ex- cessively thin strips or blades show variations in their elastic constants. An extreme case of this behaviour is shown by phosphor-bronze strips 0°005 em. thick or less. In the Phil. Mag. for March 1913 I gave an account of the variations of torsional rigidity of phosphor-bronze strips of that thickness when subjected to different tensions. ‘hese variations I attributed to overstrains given to the strips during the process of manufacture. Now it has been urged that this variation in the restoring couple per unit angle of displacement with different tension, which I considered to be wholly due to variation in the torsional rigidity, may be due to a bifilar action of the stript. Indeed, it has been said quite explicitly that a bifilar etfect ought to have been observed in the strips on which I experimented f. Now there are no grounds, from a theoretical point of view, for supposing that we would get a bifilar action for small oscillations. Searle has given a fairly complete treat- ment of the subject§. He finds that when a blade (or strip) is twisted about its central filament that the central plane becomes a helicoid, and all filaments in the central plane of the untwisted blade remain unchanged in lenyth when the blade ts twisted. Applying this result, we see that when the couple is applied by means of a moving inertia bar, the axis of the inertia bar will be in the same horizontal plane at all phases of the oscillations, and therefore there can be no restoring couple due to bifilar action. Searle also gives experimental evidence in proof of this|]. From this it is clear that there is abundant evidence both theoretical and practical against the view that we should expect a bifilar action in phosphor- bronze strips. The object of this investigation was to try to show by experiment that there was no bifilar effect in * Communicated by the Author. + Campbell, Proc. Phys. Soc. April 1915. ft Buckley, Phil. Mag. December 1914. § Searle, ‘ Experimental Hlasticity,’ p. 63 &e. '| Searle, ‘ Experimental Elasticity, p. 132. 204. Mr. H. Pealing on an Anomalous Variation of thin phosphor-bronze strips; and to obtain further informa- tion about the effect of annealing the strips. In the experi- ments I made with strips 0°0043 cm. thick I showed that the discrepancy, which amounted in the case of one strip examined to 200 per cent., was entirely removed by annealing it. The simplest way of explaining this result is that the rigidity is constant over the range of the experiments when annealed, and that there is no bifilar action. The width of the strip was printed by mistake 0°48 em. The value stated should have been 0°048 cm. In the case of strips 0°001 em. thick I was never able entirely to remove the discrepancy, and this I attributed to imperfect annealing, but offered no proof for that statement. That is to say, the experiments did not definitely decide that there was no bifilar action in the very thin strips. Heperiments with Phosphor Bronze Strips. The dynamical method was used to determine the rigidity of the strips of which relative values only were obtained. One end of the strip used was soldered to a piece of brass which was rigidly held in a torsion head, and the other was soldered to a ght copper stirrup which supported in a horizontal position circular brass rods whose weights varied from ‘3 erm. te £5 grm.,and whose lengths were about 9 cm. The whole was enclosed in a glass cover. When a rod of given weight had been placed in the stirrup, the time of a convenieat number of swings was observed. From this, relative values of the restoring couple per unit angle of displacement could be calculated. It was found that the initial behaviour of the strips was entirely different in amount, and partly also in the kind of variation, from what had been observed before. The table gives the results obtained with four strips the dimensions of each of which were 25 x 0:024 x 0:0012 em. They were supplied to ne through the kindness of Professor Wilberforce, of Liverpool University. The thickness given of the strips is approximate only. ‘The strips were perfectly smooth and free from kinks. They were manufactured by Messrs. Johnson and Matthey. Strips I., I1., and III. were consecutive lengths of strip off a new bobbin of material, while one metre of strip separated when on the bobbin the end of the third strip from the beginning of the fourth. Rigidity of Phosphor Bronze in Strips. 205 | Torsional rigidity (relative values) for strips ae numbered J., II., IIL, & IV. Tension in grams weight. ih, eens l | EE. Lea. age eae 88 58 885 | 541 1430 S66 | 4764 9] ie = Si4 1500 et. 83() 1:26 556 «| ~=— 907 1503 431 | 1053 2-9 | AD h. ’ | 754 1497 662 1536 6-6 | 340 519 1431 540 | 1792 14-0 G95) «+ B92 1469 705 | 1970 | 25-0 1162 | 356 1472 906 2100 | | 313 1612 | 635 «| 1451 eg tpl Saree | | 45:0 eet Woe: «158% ey | | The first column gives the tension to which the strip was subjected in grams weight. The figures under the columns headed I., II., I1I., and IV. give the relative values of the torsional rigidity of strips I., IL. III., and IV. when first set up. ‘The figures under IIA. give the relative values of the torsional rigidity obtained when strip II. was annealed. Remarks on the Initial Behaviour of the Strips ; and on their use as Galvanometer Suspensions. In contradistinction to the behaviour of strips 0-004 em. thick, the rigidity for a given tension was found to vary during the time a set of readings were being taken. Ex- amining this matter a little more closely, it is seen that there is no real difference. In the case of the thicker strips it was found that the behaviour of the strips altered in the course of a few days, although nothing more had been done to them further than putting them under tension and making vibration experiments with them. Now in the case of the thinner strips, the time of ten swings for the heavier rods was often as much as forty-two minutes. In this time, much may happen to alter in a permanent manner the behaviour of the strips. The alteration in time of swing was, however, not very great; and when the time of any particular experi- ment with a given rod was reduced to ten minutes or less, very little alteration was observed. This was the case whether the experiments with the smaller tensions were 206 Mr. H. Pealing on an Anomalous Variation of made before those with the larger tensions, or vice versa. The figures in the column may therefore be considered to represent with fair accuracy the rigidity of the strips as they are before being experimented upon. The initial behaviour of strips I. and IL., it will be seen from the table, is very similar, but that of IL. is very remarkable indeed. For sinall tensions the rigidity is fairly high, then we get a sharp drop, and after that a more gradual fall until we reach a tension of twenty-five grams weight. For greater tensions than that we get a very sharp rise indeed. The behaviour of strip IIL. is searcely less remarkable. For small tensions the rigidity is very low, and as the tension is increased we get a very sharp rise throughout the range of the experi- ments. Entirely different from the behaviour of 1.) Dan ITI. is that of strip IV. In that we get a low value of the So rigidity for small tensions, then a very sharp rise followed by a sudden transition to a gradual rise to a high value. In the third strip, the rigidity for a tension of atone one gram is only 0°27 of its value for a tension of twenty-five grams weight and 0°143 of its value for a tension of forty- five grams weight, a very extraordinary variation. Now phosphor-bronze strips are very largely used for galvanometer suspensions and universally for moving-coil galvanometers. No doubt when the galvanometers have been working for some time, their behaviour will be regular. Tt is, however, no less certain that when a new phosphor- bronze suspension is fitted to a galvanometer, the re- storing couple per unit angle of displacement will gradually increase for the first few days. A second point of im- portance is the figure of merit of a galvanometer that we may expect to oot when using phosphor -bronze suspensions of given dimensions. The above results show that this cannot be predicted with any certainty at all. Take for example the restoring couple per unit angle of displacement for tension of °91 grm. In strip III. it is less than one-third of the value for strips I. and II., and only about one quarter of the value for strip IV. This discrepancy is no less for greater tension, as the values for a tension of twenty-five erms. show. Liffect of annealing the strips. It is not easy to anneal satisfactorily very thin strips. If they are annealed while they are under no ‘tension or pres- sure of any sort, they are very liable to crinkle and develop numerous kinks. The method I found most satisfactory was ee se Oe ee 4 } \ q 4 ' : ] Rigidity of Phosphor Bronze in Strips. 207 to place the strip carefully between two sqiare rods of brass whose dimensions were 12 in. x 2? in.x 3 in. ‘The whole was then heated to a suitable temperature and kept hot for about an hour. But even in this case slight ciinkles developed in the strips. ‘The easiest explanation of this is as follows. The strips during the process of manufacture are overstrained, particularly on the surface. These overstrains are of such «a nature that they make the strip perfectly uniform in thickness and free from kinks. During the process of an- nealing, the function of the outer lay er in jacketing the inner ‘Javers is removed, and the strip takes on its natural shape, which is by no means smooth and regular. If this view is correct, it is a very doubtful advantage to anneal the o strips, for while that removes one disadvantage it introduces another. By the exercise of care the last disadvantage could be very considerably minimised. All the strips on which experiments were made, when annealed showed a very small variation in the torsional rigidity under different tensions. This was particularly well brought out in the case of the second strip. The values obtained are shown in the table. For loads up to thirty grams the rigidity is practically constant. The value for a load of forty-five grams showed a slight increase. In the case of the other strips the increase in rigidity for the heayy loads was a little more pronounced, especially in the case of the fourth one. ‘This increase I consider to be due to two reasons : (1) Imperfect annealing. (2) The straightening out of the annealed strips by the heavy loads. The first was the more usual cause of the increase as further annealing reduced the increase. In all cases the increase was very small compared to the increase obtained when using unannealed strips. The experimental evidence is such that we are entitled to conclude that if there is a bifilar effect in the strips, it is excessively small and completely masked by other effects. On the other hand, the removal of the effect by annealing ean only be easily explained by the fact that during the process of manufacture the strips were considerably over- strained, and seems to justify the conclusion that it is an anomalous variation in the rigidity of the phosphor bronze, and not an apparently anomalous one. Liverpool University. 208" XVI. Notices respecting New Books. Physics, an elementary teat-book for University Classes. Third Editien. By C. G. Knorr, D.Sc., F.R.S.E. Pp. vi+292+4 370 (Iwo parts). London and Edinburgh: .W. & fi Chambers. 1913. Price 7s. 6d. al text-book is destined for the use of junior University Students, and especially medical ones, in their first year of study. The first edition was published before the discovery of the phenomena of radioactivity. The present, third, edition is therefore enlarged, in its Second Part, dedicated to “‘ Matter, Ether and Energy,” by a new chapter (pp. 336-350) on ‘‘The Hlectron Theory and Radio-Activity,” and, in connexion with this, by short indications concerning the recent advance in various lines of physical research. The new chapter contains a short description of the kathode, anode, and Rontgen rays, of the discoveries made by Becquerel and the Curies, and of the a, 3, and y rays, followed by short sections on ‘ The Emanations and Active Deposits,” the ‘““Euergy of the Radiations,’ and the ‘ Distribution of Radio- Active Bodies.” The subject of that chapter gives the author an opportunity of intercalating some of the concepts of the electron theory. Photoelectricity: The Liberation of Electrons by Light. By H. StanbtEy ALLEN, D.Sc. Pp. vut221. Longmans, Green & Co: Ws. Gd. net, Tuts book, which is one of the Monographs of Science which Messrs. Longmans, Green’& Co. are now publishing, illustrates very well the remarkable growth in electronic science during the last two decades. For although it extends to over two hundred pages it deals with only one small portion of that science, and yet perforce —owing to the enormous mass of detail to be dealt with—is com- pelled to treat it in a very summary fashion. Use made of the volume since the reviewer received it convinces him that it gives a very thorough account of the subject, and where room has been found for criticism or comment Dr. Allen has been able to supply. it with a sound judgment based upon a thorough knowledge of the facts. The subject is still growing fast and existing theories cannot be regarded as much more than tentative guesses after the truth. It is this fact which must have made the book exceedingly difficult to write. It would be futile to claim that the volume left one without any sense of confusion; but we think that Dr. Allen can hardly be blamed for this; it is simply the result of his being obliged to compress so much growing material into such a small space. It is the first detailed account of the subject in English, and should be in the hands of every student. Not the least inter- esting chapter is the last, on the photographic image. bye) a} THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. AUGUST 915. ia ees: XVII. The Relation between Uranium and Radium.—Part VI. The Life-period of Ionium. By FREDERICK Soppy, M.A., F.RS., and Miss Apa IF. R. Hrreniys, B.Sc., Carnegie Research Scholar, University of Aberdeen*. A* experimental examination of the question whether radium is produced from uranium has been in progress by one of us since 1902. A clear growth of radium ina uranium solution, initially purified from radium by precipi- tating barium sulphate in the solution, was observed in 19047, but the extreme slowness of the growth suggested the existence of a long-lived intermediate parent of radium, which was separated by Boltwood in 1907 and named ionium. The present series of experiments were started in conjunction with Mr. T. D. Mackenzie in 1905. Uranium preparations were purified as carefully as possible by methods designed to eliminate all other substances, so that neither r: adium nor the hypothetical intermediate parent of radium would initially be presentt. Accounts of the progress of the measurements on the quantities of radium in the various solutions have been published from time to time§. In 1912, * Communicated by the Authors. 1 F. Soddy, ‘ Nature, May 12, 1904; Jan. 26,1905; Phil. Mag. | 6] ix. p. 768 (1905) ; compare W. C. D. Whetham, ‘ Nature,’ May 5, 1904 ; Feb. 2, 1905. in Soddy and T. D. Mackenzie, Phil. Mag. [6] xiv. p. 272 (1907). § F. Soddy, Phil. Mag. [6] xvi. p. 632 (1908) ; xviii. p. 846 (1909) ; xx. p. 840 (1910). Phil, Mag. 8, 6. Vol, 30. No, 176. Aug. 1915, FP 210 Prof. F. Soddy and Miss A. F. R. Hitchins on the a connected account of the whole work up to that date was given in a lecture at the Royal Institution (March 15th, 1912)*, and the conclusions then drawn may be briefly restated. In the following table particulars are given of the four uranium preparations studied :— ~ Grams of Date of Method of oi ee ee Dy growth ca Uranium, | purification. | purification, icthetiaiae a rte kilogram of uranium. ge boa wey Sanayi Ether. 13x10 g. AM clas OS 14/ 8/06 Ether. 8x10) ¢. TH. ..:| 408 | 13/12/06 Ether. 35x10 g. TV. eal |}, OOO 4/ 6/09 | Recrystallization. 25 10s ae The methods of purification of the uranium adopted, ex- traction with ether in the case of the first three preparations and repeated recrystallization from water in the case of the fourth, are those generally employed to remove uranium X from uranium, and, since uranium X is now known to be isotopic with ionium, the best possible methods for removing ionium had unknowingly been employed. A very slow rate of growth was apparent in all four preparations, diminishing in order from 138X107” gram of radium per year per kilogram of uranium in the case of the first, to about one- fifth of this rate in the case of the fourth. These differences can only be due to the more successful elimination of initial ionium in the successive preparations, and prove that, in the first two preparations at least, the growth of radium is to be ascribed mainly to initial ionium. As Rutherford has pointed out, the growth of radium from uranium, if ionium is the only long-lived intermediate product, must proceed initially according to the square of the time. But in 1912 there was no evidence that the growth of radium in any of the pre- parations was proceeding other than linearly with the time. That is to say, it was certainly due, for the most part, at least in the case of the first preparations, to initial ionium that had survived the purification processes, and there was in 1912 no positive evidence that uranium was producing radium at ail. On the assumption that the whole of the radium came from the uranium, the minimum possible period of average life of ionium can be found. This was deduced * Trans. Royal Institution, 1912. EE —————— S.C Oe ee i ot > a ee ee a = Relation between Uranium and Radium. 211 to be 80,000 years from the results with the third prepara- tion, and 70,000 years from the results with the fourth. Recalculating on the same data as are used later in this paper, these periods must be increased by the factor 1:2, that is to 96,000 and 84,000 years respectively. If part of the growth of radium is derived from initial ionium the period of ionium, naturally, is increased. Since it was thought that even in these most highly purified preparations part of the growth at least must be due to initial ionium, the period of the latter was estimated as probably at least 100,000 years (now 120,000 years). In 1912, however, the direct experi- ments only fixed a lower limit to the value of the period, and gave no indication of the true period. Measurements on the rate of growth of radium in the various preparations were continued till September 1914, in Glasgow, under the same conditions and with the same instrument as previously*, and indicated a clear increase of the rate of growth in the fourth and largest uranium pre- paration, containing 3 kilograms of uranium (element). This increased rate has been confirmed by subsequent measurements with a new instrument and under slightly altered conditions of measurement in Aberdeen, whither the preparations were all successfully transported. It is now possible to say that a definite growth of ionium from uranium has been experimentally observed, and to fix the true period of ionium approximately. Method of Measurement. Until the suinmer of 1914, the method of measurement was the same as previously adopted and described*. In October 1914, the whole of the preparations were successfully transported in their sealed flasks to Aberdeen, and a new electroscope had to be set up for the measurements. Ad- vantage was taken of this necessary break in the continuity of the measurements to modify the method slightly to render the observations less time-consuming. Hitherto, all the measurements had been made with the leaf charged nega- tively, maintaining the leaf charged as it leaked away during the 3-hour interval between admission of the emanation and measurement, to avoid errors through changes in the distri- bution of the active deposit Tt. It is more convenient to charge the leaf positively, though the sensitiveness of the electroscope is thereby reduced, as then the instrument can * Phil. Mag. [6] xviii. p. 847 (1909). + Ibid. p. 850, neil - sit = nll BE rn i = ~ — SESE ai te! baat? a is ——— | : i ! al 212 Prof. F. Soddy and Miss A. F. R. Hitchins on the be kept discharged during the waiting interval without changing its constant. This is a point of considerable im- portance in measurements of radium by the emanation method, and had long ago been adopted in all other measure- ments by this method; but, in order to preserve continuity, the old method had been retained with all the measurements by the old instrument in this set of experiments. With the new electroscope the new plan was adopted. Very great care was taken to avoid any introduction of error by this change, and measurements taken with the same standards by the old and new magnifying-power methods agreed perfectly. In addition, the magnifying power of the new microscope was considerably less than that of the old, and the two changes together caused a reduction of the sensitive- ness of the new instrument to about one-third of that of the old. The accuracy of the measurements, however, was not affected by these changes. In former papers, somewhat different values have been employed for the ratio of radium to uranium in pitchblende, and all the former results have been recalculated to the same value, viz. 3°4x1077 @. radium per gram of uranium. In the present work, a large number of new standards from carefully analysed uranium minerals were prepared, as it was found that the old standards, prepared in 1909, no longer agreed among themselves. Asis well known, it is practically impossible to keep such standards indefinitely, owing to the tendency of part of the infinitesimal amount of radium present to precipitate out of solution. In the following three tables are given particulars of the calibrations of the instruments. The first refers to the original instrument, the second to the new instrument, and the third to the latter after an accident to the gold leaf. The uranium minerals used were those employed in an earlier research*, on the ratio of radium to uranium in minerals. The radium-barium chloride preparations used were some containing about 10 mg. of radium (element) per kilogram, in which the radium has been determined by y-ray measurements of spherical samples against a radium standard, according to the method described in ‘Chemistry of the Radio-Hlements,’ Part I. Second edition, 1915, p. 93, in which the absorption of the y-rays in the preparation itself is corrected for. Known weights of these were dissolved, and the solutions diluted to convenient strength. The electro- scope was thus calibrated independently on the y-ray standard * F. Soddy and Miss R, Pirret, Phil. Mag. [6] xx. p. 345 (1910) ; xxi. p. 652 (1911). Relation between Uranium and Radium. 213 and the ratio of radium to uranium required to make the : results agree with those from uranium minerals was 3 3°4%10-7. Calibration of Electroscope. I. Old Instrument. Radium | Leak | Constant Standard. Uranium | (grms, of Ba x 10-12 (grms, | (divs. per , 4 (mgs.). | mee Sa wee _ - required to give ' [< 218 Prof. F. Soddy and Miss A. F. R. Hitchins on the of radium in time ¢, and A, and Az are the radioactive con- stants of ionium and radium respectively. The ratio of the mass of radium in equilibrium with 1 gram of uranium is taken throughout this paper to be 34107". This factor agrees best with the experiments before referred to, and 1s somewhat higher than the Rutherford and Boltwood value, recalculated to the International Standard, viz. 3°23 x 1077 * and in better agreement with the value of Heimann and Marckwald, viz. 3°33 x 107‘ f. Hence Ryo=3'4x 10°? x 2587220 > where P is the number of uranium atoms experimented upon. 1f M is the mass of radium formed froma mass U of uranium, M/jU=R/P x226/238; and R/P= srpAzt? x 34 X 10-* x 238/226. Hence M/U = Sins x ot x 10 If 1/A3 is 2375 yearst, 1), = (710 x 106") UAE For Preparation [V., taking the mean of the firsttwo and last two measurements given in the Table (p. 215), when ¢ was 0:26 M was 41, and when ¢ was 5°85 M was 116°5 (x 107" g.). Hence 1/Ag= 7:16 x 107" x 3000 x (5°85? — 0°26?) /75°5 x 107” = 97,000 years. In previous calculations the factor 6 instead of 7:16, deduced above, has been used in these calculations. The old factor would make the period 81,000 years if used above. With this may be compared the previously published 70,000 years, deduced three years ago from this experiment as the minimum period of ionium. The curve drawn through the observations on Preparation IV. in fig. 1 is the theoretical curve deduced from the above equation, taking 1/d, as 100,000 years in the above equation. It agrees fairly well with the experimental observations. Of the other preparations only Preparation III. can yet give any information. In this the initial quantity of radium was excessively minute and the greater relative accuracy of the measurements, in consequence, and the greater age In part compensate for the smallness of the quantity of uranium, * Sir E. Rutherford, Phil. Mag. [6] xxviii. p. 323 (1914). + Heimann and Marckwald, Phystkal. Zettsch. xiv. p. 803 (1918), { Sir E. Rutherford, loc. cit. p. 323. Relation between Uranium and Radium. 219 compared with Preparation IV. ‘The curve in fig. 1 for this preparation is the theoretical curve drawn on the same assumption as those for the curve of Preparation IV. As before, taking the mean of the first two and last two obser- vations in the Table (p. 215), 1/Ag=7-16 x 107"! x 408 x (8°325? — 1°795?)/18°5 x 107” = 104,000 years, a value agreeing well with that given by Preparation IV., and with 96,000 years calculated three years ago for this preparation. But in this case the intermediate observations lie consistently above the theoretical curve. The departure amounts, as a rule, to less than 8x 10~” g. of radium, and it would be unwise at the present stage to lay too much stress uponit. Ifitis real, it indicates that the true period of ionium is somewhat longer than that calculated and that in both Preparations III. and IV. some ionium was initially present. From the results with the earlier Preparations I. and IT., where certainly some ionium was initially present, nothing can yet be deduced as to the period of ionium. It may be stated in conclusion that the period of average life of ionium is probably about 100,000 years on the assumption that that of radium is 2375 years. This value still partakes of the nature of a minimum period, but it is unlikely that it is very far from the true period. Summary. The continuation of the measurements on the growth of radium from purified uranium preparations has shown an unmistakable increase in the rate of growth of radium in the case of the preparation containing 3 kilograms of uranium. The growth of radium appears to be proceeding according to the square of the time, as theory requires if ionium is the only long-lived intermediate member of the series. There is thus now, for the first time, direct experimental evidence that uranium is the ultimate parent of radium. ‘The period of average life of ionium calculated from this experiment is about 100,000 years, assuming 2375 years as the period of radium. An earlier preparation containing 408 grams of uranium gives practically the same value for the period of ionium, calculated on the assumption that ionium was initially absent. The effect of any ionium initially present in the preparations would be to lengthen the period of ionium, but 100,000 years is probably not far from the actual period of average life. [ 920 4 XVIII. Note on the Excitation of y Rays by B Rays. By Javwica Szmipt (Petrograd) *. i was shown by J. A. Grayt that the @ rays from radium E falling on metallic radiators produce a certain amount of secondary radiation of a y-ray type. The penetrating power of this secondary radiation was found to vary largely, being as a rule softer the lower the atomic weight of the radiator. A similar result holds when cathode particles fall on the radiator t or when X rays of suitable penetrating power fall on the radiator, exciting its characteristic radia- tion §. It was thought of interest to determine whether the secondary y and X rays were identical by comparing their absorption in aluminium. The particular arrangement originally used by Gray not being suitable for the purpose, the following method was adopted :— The source R—radium D in equilibrium with its product radium H—was placed in a strong magnetic field and the primary f rays therefore could not strike the metallic sheet under investigation, when the latter was placed at A (fig. 1). The metallic sheet absorbed the primary y-radiation to a * Communicated by Sir Ernest Rutherford, F.R.S. + J. A. Gray, Proc. Roy. Soc., A. vol. Ixxxv. p. 181 (1911). { Beatty, Proc. Roy. Soc. A, vol. lxxxvii. p. 51] (19138). § Barkla and Sadler, Phil. Mag. vol. xvii. p. 789 (1909). ——— On the Excitation of y Rays by B Rays. 221 large extent, especially the softer constituent ; absorbing aluminium-foils were placed between sheet and electroscope close to A, and an absorption curve obtained in the usual way. The electroscope H was of aluminium and contained inethyl iodide*, since this gas is known to increase relatively the effects of the soft radiations. If now the metallic sheet is shifted to position B close to the source, a very large increase in ionization is noticed: this must be due to the excitation of secondary rays by the @ particles, which now are allowed to strike the radiator ; a simple geometrical con- sideration shows that it would be impossible to attribute more than a very small part of this secondary effect to primary y-rays striking the radiator in position B under a larger solid angle than in position A. A second absorption curve was obtained with the radiator at B and the absorbing foils in the same position as before, that is at A. By comparing the two curves and subtracting the primary effect from the effect increased by secondary rays, we find the absorption curve for the purely excited radiation. The advantages of this method are the following :-— (a) The emergent secondary radiation is examined, which is known to be much larger than the incident f, especially for thin sheets of light metals. (4) No correction has to be made for scattered radiation, as the scattering of the primary beam must be nearly the same in both cases. (c) The effect of the secondary radiation is of the same order as that of the primary rays ; the latter always have to pass through the radiator, whether it is placed at A or B, whereas the secondary rays produced in the surface facing the electroscope and in the nearest layers are not absorbed at all or only to a slight extent by the radiator itself. Iron, nickel, copper, and zine were investigated in this way, the respective sheets having a thickness of 0°043, 0°058, 0°022, and 0:025 mm. These thicknesses, out of all avail- able, proved the most suitable in order to obtain a compara- tively large amount of secondary radiation. A large number of measurements had to be made, as the results were not obtained directly but by subtraction of two values, each of which might contain an experimental error of about 1 per cent. ‘The case of copper is represented in fig. 2, where the logarithms of ionization are plotted against the thicknesses of * Rutherford and Richardson, Phil. Mag. vol. xxvi. p. 824 (1918). ft J. A. Gray, Proc, Roy. Soc., A. vol. Ixxxvi. p. 518 (1912). 292 Miss Jadwiga Szmidt on the the absorbing foils. In Table I. the values of p/p in the four cases are given together with the corresponding values for “ characteristic” X rays found by Barkla and Sadler *. Fig. 2. Lag of sonssation Vy 40 ssouyriyy / e WNIT) TABLE I. AE GERO EEE Mass absorption coefficient u/p in aluminium for ar y rays. | X rays. Tio ee p58 91-3 em.~! Al 88:5 cm. Al Nickell! cooee: 61:3 58°8 59:1 Copper ...... 63°6 468 47-7 DINO’ he vest 65°4 39°8 39°4 It is seen that there is a close resemblance between the radiations in the two cases. The difference in values is not greater than the experimental error. The amount of secondary radiation, never exceeding 5-6 divisions a minute, varied in the different cases, but was as a rule larger for the lighter radiators. In the case of elements heavier than zine—namely, silver and tin—the secondary effect was too weak to be examined with the * Loe. ett, e soe Excitation of y Rays by B Rays. 223 accuracy required. This seems to disagree with Gray’s con- clusion, for he found that the higher the atomic weight of radiator the larger the amount of secondary radiation *. The disagreement between the results is probably due to differences in the experimental arrangements: Gray mea- sured the ionizing effect of rays which had passed through a certain thickness of iron used to cut out the primary #-rays; a large amount of soft y radiation was absorbed at the same time and the effect of the lighter elements was therefore largely decreased in favour of the heavier elements, which give rise to harder rays. In addition Gray used a #-ray electroscope filled with air, whereas in the present investiga- tion a very thin sheet of mica covered the face of the electro- scope which contained the vapour of methyl] iodide. In order to examine the secondary radiation from silver and tin, a different arrangement was found more satisfactory, being simply an improvement of Gray’s method as regards sensibility. The soft primary y rays were cut out by a plate of carbon 5 mm. thick ; this thickness was also sufti- cient to stop all primary 8 particles, so that no magnetic field was necessary ; but to get rid of the electrons excited in the carbon and absorbers, it was necessary to cover the face of the electroscope with some cardboard. The inezdent secondary radiation was examined by placing the radiator immediately behind the source. Subtracting the effect with- out radiator from that with the radiator in position, when different absorbing foils are placed in the path of the rays, we obtain the values for the secondary radiation. The results are shown in Table II., and are in fairly good agree- ment with the values for the “ K” radiation from silver and tin determined by Barkla. TABLE JI. Atomic weight. y rays. X rays. | - eo a a Silver......... 107°9 26 (em.)~1 Al | 2°5(em.)~! Al ae 119:0 16 | £57 From the results of Barkla, another softer radiation might be expected belonging to the “ L” series, but this * Loe, cit, 224 Prof. H. Taylor Jones on most [fective Primary could not be detected with the experimental arrangement employed. I wish to express my sincere thanks to Sir Ernest Rutherford for his help and advice. Physical Laboratory, The University, Manchester. July 1914. Note-—While these experiments were in progress, H. Richardson (Proc. Roy. Soc. A. xe. p. 521, 1914) has examined the secondary radiation excited in different elements by the @ rays from an intense source containing radium Band radium C. The method he used was the same by which Chadwick * in 1912 had proved the existence of this radiation, but the experimental arrangement was more suitable to detect soft radiations. The mass ab- sorption coefficient for the secondary rays excited in a large number of elements has been determined in this way. The values found for eleven comparatively light elements (atomic weights 58°8—137°4) show the same striking resem- blance with the corresponding values obtained by Barkla for the “K” series of characteristic X rays, which has been pointed out in the present note. Six heavier elements (195°3-238) gave results very close to those found for the “L” series of characteristic rays. ‘he few results obtained in the present investigation by using radium D and radium H, in addition to those obtained by Richardson, who used a quite different source of @ radiation, leave no doubt as to the identity of secondary rays excited in the different substances by any stimulus, whether by @ or X rays. XIX. On the most Kfective Primary Capacity for Induction- cols and Tesla Coils. By HW. Tayvuor Joxnusi aise Professor of Physics in the University College of North Wales, Bangor f. | the theory put forward in recent papers} by the writer the induction-coil is regarded ag an oscillation trans- former, in which the two circuits are closely coupled, and in which the secondary potential arises from the superposition of two oscillations differing in amplitude, * J. Chadwick, Phil. Mag, vol. xxiv. p. 594 (1912). {| Communicated by the Author. { Phil. Mag. xxii. p. 706 (1911) [with D. E. Roberts]; xxvii. p. 565 (USTED 2 o-are. ea ee EIS NS) Capacity for Induction-coils and ‘Tesla Coils. 995 frequency, and damping coefficient. The expression for the secondary potential was worked out in certain cases from the “constants”? of the circuits, and found to agree closely with experiment both in wave-form and in maximum potential as deduced from the observed spark-length. According to this theory the secondary potential at time t after the interruption of the primary current is repre- sented, if the resistances of the circuits are neglected, by the expression | eee Arr Liattomins (nz sin 2arnyt—n, sin 2Zangt), . (1) Ng? — ny in which L,, is the coefficient of induction of the primary coil on the secondary, 2 is the primary current interrupted, and 71, nz are the two frequencies of oscillation of the system (ng >). For given values of Lg), %, n, and nz, the value of Vz is stationary at times given by dV,/dt=0, 2. e. cos 2arnyt — cos2a7nt=0, . . . . (2) or sin @(ny+7o)t. sin w(mg—m)t=0.. . . (3) The stationary values of V, therefore occur at the times 1 2 3 7=(), >> eee N+tNg Ny+Nyg Ny+Ny (4) and 1 2 3 Nyg—Ny? Ng—Ny? Nyg—Ny _ At any stationary value we have, by (2), sin 27ngt=+ sin2mnt, . . . . (5) the upper sign giving the numerical minima of V,, the lower sign the maxima. Substituting in (1) we find that the numerical maxima lie on the (¢, y) curve ae aera ¥y = 277 Linity : 2 2 e . Bite Saree Crile: etre? Oe Ng— Ny dis (6) the minima on the curve NyzNo ny + No y=2rLygii, Steere yn ie ye CD In dealing with the greatest sparking potential of an induction-coil we are only concerned with the greatest maximum of V, in the first half-period of the slower oscil- lation. Hyen though there should be a closer coincidence of maxima of the two waves in some subsequent half-period, the amplitudes are by this time so much reduced by the Phil. Mag. 8. 6. Vol. 30. No. 176. Aug. 1915. Q 226 Prof. E. Taylor Jones on most Effective Primary damping that the potential seldom, if ever, reaches a value equal to the greatest in the first half-period. The principal maximum of V, is the maximum which occurs nearest to the first summit of the curve (6), 2. e. at the time nearest to ¢=1/4n,. The first maximum occurs at the time (= , and this is the principal maximum if the frequency- Ny + Ng ratio no/n, is between Land 5, If ny=5n, the first maximum 2 My +Ng’? Ny +g If n,/n, is between 5 and 9 the second maximum is the So te Ny + Ng Nz= In, the second and third maxima are equal, and if 2,/n, is between 9 and 13 the third maximum is the principal is equal to the second, and they occur at times principal maximum, occurring at the time maximum, and it occurs at 1= ; and so on. Ny + Ng Consequently the principal maximum secondary potential 1s given by the equation 14 s n . Vem = 27 Lgiiy—~. sin o, . peel where 274 5 is 8 o= ——— if — is between 1 and 5, N+ Ny Ny Atrn ue 1 55 o= Seung aa ” oP) oy) 5) 9 9, N11 Ng 67rn 1 ¢ 99 o= a) 99 99 9 99 13, N+ No If n/n, has one of the values 3, 7,11,... the maxima of the two oscillations occur simultaneously, the principal ; : : i 7 AU i maximum occurring at the time 1/47; (@=F) and being equai to the sum of the amplitudes of the oscillations. These cases may be called the comerdences. ; If the resistances of the circuits were taken into account the value of the maximum secondary potential would be considerably less than that given by (8), chiefly owing to the damping effect of the resistances*. In certain cases * The full expression for V,, with damping factors and phase-angles, was given in the Philosophical Magazine for April 1914, pp. 565, 574. Several examples of the (Y,,¢) curve were given in the papers referred to above. Capacity for Induction-coils and Tesla Coils. 227 examined by the writer, in which an 18-inch coil was used, the damping was found to be sufficient to reduce the maximum secondary potential by about 25 per cent. It is, however, useful to consider the simpler theory in which the oscilla- tions are regarded as undamped, since it enables us to trace with sufficient accuracy the general nature of the effect of varying one or more of the induction coefficients or capacities of the system. ‘The other effects of the resistances, viz. on the frequencies, initial amplitudes, and phase relations of the oscillations, are generally small, and no serious error is in- troduced by neglecting them. ‘Thus, while the value of the maximum secondary potential given by (8) is considerably in excess of the value found by experiment, the conditions in which it occurs do not differ greatly from those deduced from the simpler theory. For example, in three cases worked out in a former paper*, in which the damping and phase- angles were taken into account, the values of the angle 2mnyt — 6, at which the maximum secondary potential occurred were 107°4, 65°°8, and 74%3 respectively. The values of o tor these cases, calculated as explained above, are 111%3, 62°°8, and 71°7. The calculation of the most effective primary capacity. One important problem connected with the induction-coil is that of determining the optimum primary capacity for a given coil, 7. e. the capacity which, connected across the interruptor, allows the greatest secondary potential to be developed, when the induction coefficients of the circuits, the secondary capacity, and the primary current interrupted are all given. In considering this problem, and other problems of this kind, it is convenient to express the sum of the amplitudes of the two oscillations in the secondary circuit in terms of =20) and (=z), We have then for the principal maximum Soy ae Vom = 21 Ligy75 .sin Ng — Ty Loxto wing een Wa) Wi dal MRI GD) where | a: Ce * Phil, Mag. xxvii. pp. 572-577 (1914). + Phil. Mag. xxvii. p. 584 (1914). Q 2 228 Prof. E. Taylor Jones on most Effective Primary The angle ¢ can be calculated from the ratio of the frequencies, which is given by the equation mf _l+ut+ VO=uPtahu ay 2 ie ees WV (1—u)?+4h7u In the present problem, in which the primary capacity Cy alone is varied, we have to determine the conditions in which U sin ¢ is a maximum, u varying while & is constant. The function U has a maximum value of 1/k at u=1—h’, and there is a series of values of & for which the maxima of U and sin ¢ occur at the same value of u. If k has one of these values the optimum primary capacity is that which makes u equal to 1—k?, The necessary values of & may be calculated from (11) by putting in this equation u=1—k’, and n/n, successively equal to 3,7, 11,... The first four values of the series were given ina recent paper” by the writer, in which it was shown that this series of adjustments has the further property that in any one of them the system has unit efficiency, 7. e. the maximum electrostatic energy in the secondary circuit is equal to the initial electrokinetic energy (3L,%*) in the primary circuit, on the assumption of negligible resistances and perfect suddenness of the inter- ruption of the primary current. If k has not one of these special values the maximum value of U sind does not occur when ¢ is exactly 7/2, 7. e. when the maxima of the two waves occur simultaneously. If, however, one of these coincidences occurs at a value of u not far from 1—k?, dU/du is small, and this coincidence determines very approximately a maximum of V,. The greatest maximum secondary potential for any value of k? is then determined by that coincidence which comes nearest to the maximum value of U. The matter may be illustrated by the curves shown in figs. 1 and 2, in which the full line represents values of U sin @, the broken line those of U, for various values of uw. The full-line curve therefore shows how the principal maximum secondary potential changes when the primary capacity alone is varied. The points of contact of the two curves determine the coincidences, cor- responding to @=7/2. Fig. 1 refers to the case in which #?, the square of the coupling coefficient, is equal to 0°768, the value for an 18-inch coil with which the writer has experimented. In this case the maximum value of U occurs at w=0°232, and the greatest maximum of U sing at u=0°11 (n/n, =6°801), 7. e. nearly * Phil. Mag. xxix. p. 3 (1915). Capacity for Induction-coils and Tesla Coiis. 229 at the 7/1 coincidence. As u diminishes from this value Using goes through a series of maxima and minima, the maxima ae approximately to the ratios n,/ny=11,15,19,... There is also another maximum of U sin ¢ at abate! u=0°46 (n/ny=4'237), and a marked minimum near w=0'25 (n/n; =5). Fig, I. rH PH peeneieest tezcssnset Pro Pitty fe -- aoe FH si irstasint it sgcaauee eeeierees oe vanke res a beeeeeeses rH = GOES Seaeeeeees BREESE sam a eee eee as ofa poae SeeESH Hace eect ntaaeee eee een EESSeey cee en cet ne eee Sine nee eee eee EERE iesrsniae: scadosuitaat pases the ft tty Ey ae HHH we fetid +44 mands 2 SeRus 2ebe82 Beles sae! ou Sees ecaee one seeeae SHEER igs sie Gene sate seceeeatesseivasi ea lei iee erent peees teserecees Ba ebeee Cgesd Suit adeess Heousoteg csestasseeietesteontfassveniit aeouas so aah em —- aren epee mRawyseana oh | 1] ae pests te sivciceiitiais iit SHE Hecho Hebe Lele See aE SC SRRRs CHRee es “Ss Hifi ee oe pt HH oe ae = Het EEE pp ER feeii ine as SSRbd Snel Canas Exbeneceed Gnceaseeas recesses Seen taste ae S5EEE HEESEIEEH HET EuGat toessesoss creszasaes cesesstese seit searee poee Hi : oS H tH H HH oH an ee eee in See EEEISEEE EERE EEE EE aan Eee eae SG Haee HSE eee EP eee Hea Sessicpsey poessenead os pot eae uuene ae aan Peed SBiSdsecence OUSHRaABeTS RAR RAAsS Eussugeusesnedeastns tsese0e PtH Hae naniinsntinii Sagee aw! aes a a2 SSSR eeeeeen wuase H 4 tosecsswes Teceeeueas Bauns Sssgecuss cosesseees teeneeeess 0:8 = Uu Curve showing the variation of maximum secondary potential of an induction-coil when the primary capacity is varied, %? being 0°768. The abscissa represents « (= L,C,/L,C,), the ordinate of the full- line curve U sin 9, of the broken-line curve U. Some of the chief features of the U sind curve may be traced experimentally by observing the spark-lengths ob- tained with various primary capacities. Thus, with the 18-inch coil, the greatest spark- length is found to occur near u=0'1 (C,=0°06 mfd.*), there is a minimum spark- length when w is about 0°25 (C,=0°15 mfd.), and a second __* These spark-length measurements were described in a former paper (Phil. Mag. April 1914, pp. 582, 583), in which 0:05 mfd. was given as the optimum capacity. At that time I was not able to try 0:06 mfd., but I have since found that this represents more accurately the optimum. U sin ¢. 230 Prof. H. Taylor Jones on most Effective Primary (smaller) maximum near u=0'46 (C,== 0°28 mfd.). The maxima corresponding to the higher ratios 11, 15,... are more difficult to demonstrate experimentally owing to the smallness of the primary capacities required and the con- sequent difficulty of obtaining good interruptions. | Fig. 2 shows the curves for the case k?=0°571, the first feeea# Jobe bu eas oseee gees bert cv seseaues eeeerebvel foezsbeees Cogzceregs seseesaeesseaszocei! poses seeueuacue 1-3 rH Srecesaretaenent 1-2 | SRST at Hy co aes Hy Be lI Il Su ecco: acini POY sssszossss cuezesatatzs sieavasdsedes eastasees fantesont iets 0 O-2 0-4 0-6 0:8 t U Curve showing the variation of maximum secondary potential of an induction-coil when the primary capacity is varied, 4? being 0'571. of the unit-efficiency values. In this case the greatest maximum of Usind (n/m=3, ¢=7/2) agrees with the maximum of U, both occurring at uw=0°429. There is a minimum of Using at w=0'l, and a series of diminishing maxima at smaller values of wu, the first of which occurs at u=0'06, near the 7/1 coincidence. In this case also some of the main features of the U sin @ curve can be followed by observing spark-lengths. The coupling coefficient of the 18-inch coil being reduced, by the addition of series in- ductance to the primary circuit, so that k? =0°571, the greatest -spark-length is found to be given by the primary capacity Capacity for Induction-coils and Tesla Coils. 231 which makes wu equal to 0°429, a minimum occurs when u=0'1, and a secondary maximum when w is about 0°06. At this last value of u, however, the primary capacity is only 0:025 mfd., and good interruptions are rare if the primary current is as great as 1 ampere. For certain values of k? there are two equal greatest maxima of Usin@, one corresponding to a value of wu greater than, the other less than, 1—A?. This happens, for example, when k? is about 0°71, the two equally effective values of n,/n, being in this case 3°767 and 67595. It also happens when k? has the approximate values 0°87 and 0°92 (see Table I.). From the point of view of efficiency these adjustments are not so advantageous as those in which the maxima of U and sin ¢ occur together. The values of w which give the greatest maxima of U sing have been calculated in the same way for other values of k? ranging from 0°92 to 0°5. The results of these calculations are given in Table I., in which the first column contains the Tabe I. Adjustments for maximum secondary potential. 2 L,C, N, ; | aoe Efficiency, - u(= po) é ny Me / Using. | 7212 gin? 9. 0:92 0.0628 15 } 1042 | 1042 | 0-999 0 1297 11 1040 | 140 | O'995 0:90 00965 11 1054 | ‘1054 | 10 0:87 0-075 10°79 | 1:067 1-067 0-990 a (0:21 7183 1066 | 1-066 0-989 0°835 0-165 7-0 1-094 1094 1-0 0°768 O11 (801 1125 ht? ae O-“7N 0-71 0:09 6595 fol >in UO bd Ds 0-919 # O44 3767 1 i (: (pt ae O918 0:70 0445 368 1183 1153 0-950 0:64 0°45 3299 1:246 | 1:288 0-981 0°60 04335 312 | 1-290 | 1-288 0996 0-571 0-429 3-0 1323 | 1:32 1:0 OD 0°41 2°752 1408 | 1-401 O-981 | values of k?, the second column the values of w required for the greatest maxima of U sin ¢, the third column the cor- responding values of the frequency-ratio n,/nj. The fourth and fifth columns contain the values of U and Using, corresponding to these values of k? and u, and the last column gives the product k?U? sin? ¢, which is equal to the efficiency of the arrangement, i. e. the ratio of the greatest maximum electrostatic energy in the secondary circuit, L “ide Was 4 OV em. Te? to the initial energy-supply $147". 232 Prof. E. Taylor Jones on most Hfective Primary For any one of these values of k? the greatest maximum secondary potential is found (on the assumption of negligible resistances and perfect interruptions) by multiplying the corresponding value of U sing (fifth column) by the factor Lintp/ VW L,Cy. If this factor is constant in all these adjust- ments the numbers in the fifth column are proportional to the greatest maximum secondary potential. ‘This case is considered in the next section. The greatest maxima of U sing are also shown in fig. 3 plotted as a carve, from which may be found the value corresponding to any degree of coupling within the same range. It will be seen from the Table that for the higher values of k? the greatest maxima of U sin ¢ correspond very closely to coincidences (d=7/2), and that this applies with con- siderable accuracy down to the value 0°71 of 42. Even at this degree of coupling the 7/1 coincidence value of Using (approximately 1°134 at w=0-078) only differs by 4 parts in 1100 from the neighbouring maximum value (1-138) given in the Table. The rule, already stated in a former paper*, that the optimum primary capacity is that which determines the coincidence nearest to the maximum value of U, applies therefore with sufficient accuracy for all values of k? between O92 and 0-71. The more exact value of the optimum capacity, or rather of the optimum ratio L,C,/L,C,, for any value of k? within these limits may be found by plotting the (4?, u) curve from the values given in the Table. This curve has two distinet segments, one covering the range k?=0°92 to 0'87, the other running from 0°87 to 0:71. At those values of k? for which there are two equal greatest maxima of Usin 9, it 1s better in practice to choose the greater of the two values of u, since the larger capacity involved does not so readily allow trouble- some sparking at the interruptor. Over the lower range of values of k’,.viz. 0°71 to 0°5, the greatest maxima of U sing do not correspond so closely with the 3/1 coincidences, except near the value k2=0°571. For this range, however, a simpler and more accurate rule may be stated. It will be noticed from Table I. that there is, within these limits, no great variation in the optimum value of uw. It should also be noted that, for such values of k?, Using varies very slowly with wu near the greatest maxi- mum: these portions of the (u, Using) curves are very flat-topped. We shall therefore make only a verv slight error if we take w as constant and equal to the mean value * Phil. Mag. xxvii. p. 584 (1914). Capacity for Induction-coils and Tesla Coils. 233 over this range. The values of U sin@ obtained in this way do not in fact differ by more than 1 in the third place of decimals from the maximum values given in the Table. Hence the rule may be stated :—If k? lies between 0°71 and 0°5 the optimum primary capacity is that which makes Poe 0-43 ibe ae Of the adjustments specified in Table I. three give unit efficiency when the resistances are neglected, and the efficiency is least at k?=0°71, one of the cases of equal greatest maxima. In this case rather over 8 per cent. of the initial energy 4 Lyi)? appears as electrostatic energy in the primary con- denser and electrokinetic energy in the primary circuit at the moment when the secondary potential reaches its greatest value. It can be shown that of this 8 per cent. about one quarter represents the energy of charge of the condenser, the remainder the energy of the primary current. How this percentage is affected in practice by the resistances may be considered on a future occasion. On the use of Series Inductance in the Primary Circuit. Y The series of adjustments given in Table I., or a portion of the series, may be effected with an induction-coil in various ways, among which the method of reducing k? by the addition of series self-inductance to the primary circuit is the most important if we keep in view the object of obtaining the greatest spark-length with a given current. Jn this process the quantities L,; and L,C, are constant, so that the values of U sin ¢ in the fifth column of the Table are proportional to the maximum secondary potential attainable at the inter- ruption of a given primary current. The optimum value of the ratio L,C,/L,C, for any value of k? is given in the second column. In fig. 3 the greatest maxima of Using are plotted as a curve the abscissa of which represents 1/k?, this quantity being proportional to the total self-inductance of the primary circuit. The full-line curve, the ordinate of which is pro- portional to the greatest maximum secondary potential at each stage, consists of three portions A, B, CU, corresponding to adjustments in which the frequency-ratio n/n, is near 3, 7, 1l respectively. The points at which these sections of the curve meet represent those cases in which there are two equal greatest maximaof V>. The broken-line continuations of the curves at these points correspond to secondary maxima in the (vu, U sing) curves. At each of these points of inter- section also the efficiency is a minimum for “optimum” adjustments. U sin dmax: kU? sin? ¢. 234 Prof. E. Taylor Jones on most Effective Primary At the points marked a, 0, ¢ on the curve the frequency- ratio has the exact values 3, 7, 11, and the efficiency in these adjustments 1s a maximum. The efficiency curve is also shown in fig. 3, by the line DEF, the ordinate of which represents k?U? Sine d. Fig. 3. Le? Curve CBA shows the increase of maximum secondary potential accom- panying the addition of series self-inductance to the primary circuit of an induction-coil. The abscissa 1/k” is proportional to the total self-inductance of the primary circuit. The ordinate represents the greatest maximum of U sin @ (see figs. 1 and 2) at each stage. Curve FED shows the variation of efficiency. Capacity for Induction-coils and Tesla Coils. 235 The curve CBA shows how the maximum secondary potential for a given primary current may be increased by adding series inductance to the primary circuit, in the case of an ideal induction-coil with negligible resistances. In practice the curve is, owing to the damping effect of the resistances, considerably below the curve CBA of fig. 3. It may some- times bend downwards near the point corresportding to a, as I found in some experiments described recently*. I wasthen under the impression that this is theoretically correct in all cases, but curve A shows that this view is erroneous ; and on repeating the experiments, using for the series inductance oply air-core coils of low resistance, I found that the spark- length continued to increase when the self-inductance was increased beyond the value corresponding to the point a+. If suitable coils are used for the series inductance, and if the interruptions are good, the experimental results therefore conform in this respect with the theory as represented by the curve in fig. 3. The fact is that as the primary selt- inductance is increased more energy is being put into the system, and, while the efficiency falls off from the point a, the maximum secondary potential continues to increase. ‘This increase becomes, however, less and less rapid as L, is increased, and the facts that it is accompanied by loss of efficiency, that under less favourable circumstances it is converted into a diminution, and that a large increase of self-inductance gives rise to troublesome sparking at the interruptor, indicate that it is inadvisable to continue the process beyond the above-mentioned point. In the Annalen der Physik, vol. v. p. 837 (1901), an induc- tion-coil is described by Klingelfuss in which the primary and secondary coils are wound on a nearly closed magnetic circuit. The instrument is said to give a very long spark with a comparatively small number of turns in the secondary coil. The advantage of such an arrangement appears to arise from the low electrical resistance of the secondary coil and the low magnetic reluctance of the core. Since Ls, is approximately proportional to the number of secondary turns, and L, to its square, the factor Ly//L, of equation (9) is not changed by a reduction of the number of turns, but the lowering of secondary resistance, due both to the reduction of the length of wire and to the use of thicker wire for the secondary coil, diminishes the damping of the oscillations and thus tends to allow a higher secondary potential to be * Phil. Mag. January 1915, p. 7. + This effect was not found when coils with iron-wire or sheet-iron cores were used. Probably Maxwell’s coil of maximum self-inductance for given length and diameter of wire is the best form for the purpose. 236 Prof. H. Taylor Jones on most Effective Primary developed. Again, L,; and L, are inversely proportional to the magnetic reluctance of the core, and the factor L,/ /L, is therefore increased by closing, or nearly closing, the magnetic circuit of the core™. On the other hand, the conditions do not seem so favourable from the point of view of the other factor of (9), viz. Using. The values of the inductances and of the coupling coefficient are not stated, but from the description it would seem that the primary and secondary coils are very closely coupled. If this is the case U sin ¢ cannot be large (see Table I. or fig. 3), and according to the present theory the instrument should give a longer spark for a given primary current, or the same spark-length at a smaller current, if suitable self-inductance were connected in series with the primary coil. If, for example, the instrument is so adjusted that k?=0°835 and w=0°165 (n,./n;=7), and if L, and ©, are increased, the former by 46 per cent. and the latter by 78 per cent., we should then have k?=0°571, u=0°429, and the maximum secondary potential for a given primary current should—apart from any effect due to change in the logarithmic decrements of the oscillations—be increased by over 20 per cent. without loss of efficiency. The Tesla Coil. In the case of the Tesla coil the difference of potential V, of the terminals of the secondary coil, at time ¢ after the beginning of the discharge in the primary circuit, is (neglect- ing resistances) 1:01 Vo i = 0s Z = = / (iC, bi Cae (cos Zn t— cos 2anzt), (12) where V, is the initial (discharge) potential of the primary condenser. The primary potential difference is given by 2, 2 22 nn 1—4r?n,? LC Vi=V) > He 3 *cos 2arnyt No — ny Ng 1 —4q7n,?L,C ——_—_} "cos 2g). >. Sen Ny Tat These results may be deduced from the solutions given in Fleming’s ‘Principles of Electric Wave Telegraphy and Telephony,’ 2nd ed. p. 264. The full expression for Vg, in * See du Bois, ‘The Magnetic Circuit,’ p. 276, 1896. Capacity for Induction-coils and Tesla Coils. Zonk which the resistances are included, was given by Drude in a well-known paper*, but the above expressions will serve for the present purpose. Drude considers the problem in which the secondary coil is given, and in which it 1s required to find whai arrangement of the primary circuit gives the highest secondary potential f. After remarking (J. ¢. p. 539) that it is necessary to dis- tinguish between the two cases in which (a) the capacity ©, is varied, and (b) the self-inductance L, is the variable quantity, he apparently comes to the conclusion (p. 540) that in either case the highest secondary potential is attained when 1,0, is equal (or nearly equal) to L,.C,, 2. e. when the periods of the primary and secondary circuits, separated from each other, are equal. This is the case of so-called “ re- sonance,” and all the subsequent calculations and conclusions given by Drude, including the tables and curves (I. «. pp- 946-551), are based on the assumption that this con- dition (L,C,;=L,C,) is satisfied. Drude finally arrives at the result that, if the damping of the oscillations is small, the secondary potential is greatest when the coupling co- efficient is 0°6 and the primary capacity is so adjusted as to bring the primary circuit into “resonance” with the secondary, this adjustment of the system giving the frequency- ratio n/n, = 2. The reasoning by which Drude arrives at the above result is, I think, not perfectly clear, and the result does not hold in case (a). Denoting the ratio L,C,/L,C, by m, the expres- sion (12) for V, becomes Vo=> 7 ot = (cos 2arn,t—cos Qarngt 14 nthe m™n,t—cos 2rnzt). . (14) If the primary capacity C, alone is varied (Ly, L,;, £?, V,, and LC, being constant), the denominator of this expression has a minimum value when Ee eae ai) ee @ If also n, and ny are suitably related, the maxima of the two waves in the secondary coil will occur simultaneously, so that at time 1/2n,, cos 27njt=—1, cos 2ant=1. This happens, for example, when k?=0°265t, m=1—2k?=0-47, which adjustment makes the frequency-ratio n/n, equal to 2. It follows from (14) that if 47=0°265 the most effective primary capacity is that which makes m=0°47, i. e. L,C,=2°128 L.C,. At this degree of coupling, therefore, * P. Drude, Ann. d. Physik, xiii. p. 512 (1904). + The primary sparking potential Vo is also supposed to be given. J} More exactly 4° =9/34, 238 Prof. KE. Taylor Jones on most Effective Primary the optimum primary capacity should be more than twice as great as the value necessary for “resonance.” The maximum value of V, in this case is numerically, by (14), Van = 2/267 =H (16) The adjustment recommended by Drude (k?=0°36, L,C,=L,C,) gives for the maximum secondary potential Le Viegas ee | Ly Even at this degree of coupling, however, a higher secondary potential may be obtained with a larger primary capacity. For example, the adjustment #?=0°36, L,C,=2L, C,, gives the trequency-ratio Ny/ny=2°197, and a maximum secondary potential, at time ¢=0°925/2n,, the value of which is , Lig, V Von= 998 ane (18) From these examples it will be seen that Drude’s rule does not in general give even approximately the correct value of the optimum primary capacity. The most effective capacity is generally greater than the ** resonance ”’ value. In some preliminary experiments made by the writer last summer, it was found that in certain cases the best effect was obtained with a primary capacity considerably greater than that required to make the periods of the two circuits equal when separated, but a more extended series of experiments is in progress in the laboratory with the object of determining the best value of m for various values of k”. The conditions are very different if the primary capacity is kept constant and Ly is varied, e. g. by means of variable series inductance in the primary circuit. In this case Ly, is constant, £? is inversely proportional to L,, and the co- efficient of (cos 2an,t—cos 2mngt) in (14) has a maximum value of Ha os when m=1. If in additionn,=2n, 1g Us (k?=0°36) the numerical maximum of Vy, is given by bet Vane= Vi n/ee Oe 5 * The adjustment 4° =0°262, m=1, gives approximately Ven=1'88 L,, V,/L,. Capacity for Induction-coils and Tesla Coils. 239 consequently Li “Ly 4. e. the maximum electrostatic energy in the secondary coil is equal to the initial energy in the primary condenser. In Drude’s adjustment, therefore, the efficiency is unity if the resistances are negligible. This result may also be seen from the expression (13) for the primary potential, L(V? 2 2 Vom — = C,V0’, which reduces, if cos 2arn,t= —1, cos 27n,t=1, to V.=v m—1 8 (m—1)? +44 m’ and therefore vanishes when m=1. At the moment when the secondary potential reaches its maximum value, there- fore, the primary condenser is uncharged, and, since aVy _ 0 dV, a? dt: that the whole of the energy exists as electrostatic energy in the secondary coil. In this kind of variation, in which L, alone is varied, the initial energy supplied to the system is constant, and the adjustment which gives maximum efficiency must also give the greatest secondary potential. But when the primary capacity is increased the energy supplied to the system becomes greater, and the secondary potential may be in- creased to a certain extent beyond the point corresponding to maximum efficiency. Both forms of the problem may present themselves in practice. If, for example, the Tesla coil is given, and ample energy is available for charging the condenser, the capacity of the condenser may always be adjusted to the “ optimum ” value, whatever be the value of k?. But if, on the other hand, the energy is limited and it is required to construct a Tesla coil which will make the fullest use of it in generating high secondary potential, the primary capacity is then con- stant, viz.: the greatest that can be charged to the required sparking potential with the energy available at each dis- charge of the induction-coil. In this case maximum efficiency and small secondary capacity should be the chief considera- tions borne in mind in the construction of the Tesla coil. Bangor, May 1915. =(, there is no current in either circuit, so [ 240 J XX. The Passage of « Particles through Hydrogen.—ll. By Prof. EH, Marsprn, D.Sc., and W. C. LanrsBerry, Wa S0L" ie a previous paper t by one of us, evidence was given that in the passage of @ particles through hydrogen, certain intimate collisions take place with the nuclei of the hydrogen atoms. These encounters result in the nuclei attaining velocities greater than those olf the « particles impinging on them, giving them a correspondingly greater: penetrating power or range. ‘The maximum value of this range was found to be about 100 cm. in hydrogen at ordinary temperature and pressure. ‘The absorption by various metals. was also studied, and relations were found to hold for the hydrogen particles similar to Bragg’s laws of absorption for a particles. As Rutherford } has pointed out, these encounters between, a particles and hydrogen nuclei are very important from the point of view of the estimation of their dimensions. If the laws of their relative directions after impact follow simple: theory, as given by Darwin 9, it can be predicted that the radii of the nucleus of the hydrogen atom and of the « par- ticle are certainly less than 107* ecm., 7. e. less than the. radius of an electron. This follows from a calculation of the distance they are apart at the moment of nearest approach. in a collision. All the experiments described in the previous paper show ed a good qualitative agreement with Darwin’s calculations. It seemed desirable, however, to make quantitative measure- ments of the number and distribution of the H particles when a parallel beam of « particles of definite velocity and intensity falls on a known thickness of hydrogen gas. The present experiments were undertaken with this aim in view,, but unfortunately we have not been able to complete them. However, certain points of interest have arisen, and it seems advisable to publish them. The formula derived by Darwin for the number vy and, distribution @ of the H particles is 1 e?H? p= QNtw . Vi Ve ° 25 sec.?0, where Hf and M are the charge and mass of the a particle, * Communicated by the Authors, + Phil. Mag. xxvii. p. 824 (1914). t+ Phil. Mag. xxvii. p. 488 (1914). § Phil. Mag. xxvii. p. 499 (1914). Passage of « Particles through Hydrogen. 241 and e the charge of the hydrogen nucleus; V is the velocity of the « particles, and @ is the angle between the direction of their initial motion and the direction of projection of the H particles considered; @ is the number of particles in the main beam, N the number of hydrogen atoms per unit volume and ¢ the thickness of the absorbing layer, and the solid angle over which the H particles are observed. It will be seen that the number of H particles projected in a direction making an angle @ with the directioa of the a particles should vary as see’ 6, a very different law from that for the scattering of a particles in single collisions with heavy atoms, where the number is found to vary as cosec* }/2. To make measurements, the same apparatus was used as in the experiments of Geiger and Marsden * on the scattering of « particles, minor alterations being introduced owing to the smallness of the effect to be expected. A pencil of « particles from an e-ray tube containing radium emanation fell on a thin sheet (10m) of wax, and the H particles projected from the hydrogen in the wax were counted in different directions by means of a zinc-sulphide screen. The a particles were eliminated by placing sheets of aluminium, sufficient to absorb them, between the wax and the zinc sulphide. The effects observed were several times greater than anticipated by formula. It was found, however, that the source itself was emitting long-range particles capable of scintillating, and that these were scattered in the wax, thus causing the disturbance. Consequently an investigation of these long-range particles had to be undertaken. Long-Range Particles emitted from the source of Radium Emanation. The «-ray tube was placed in air at a distance of 8 cm. from a zinc-sulphide screen, the path between the two being in a transverse magnetic field to get rid of disturbing luminosity due to 8 particles. Although the « particles did not penetrate more than 5°8 cm. from the a-ray tube, yet scintillations were observed on the screen. These scintilla- tions were similar in appearance to those produced by H particles. On moving the zinc-sulphide screen further trom the source, the numbers fell off in the same way as * Phil. Mag. xxv. p. 604 (1918). Phil. Mag. 8S. 6. Vol. 30. No. 176. Ang. 1915. R — es oe ees | | 1 i deer eisai <- Dia nia a a ST ———eE—OE—————e Ss = = a ee ee ee 242 Prof. Marsden and Mr. Lantsberry on the H particles might be expected to be absorbed. Further, on interposing sheets of aluminium between the source and screen, the curve of absorption was the same, within the experimental error, as the curve for H particles given in the previous paper. These results show that H particles are given off from the source, and that, unlike « particles, their velocities are not uniform, but are distributed in the same manner as those of H particles produced during the ordinary transmission of « particles through hydrogen. These results would be explained if we were to assume that there is sufficient hydrogen either in the gas inside the a-ray tube, or in the material of the glass of the tube, or in any water-vapour or hydrogen occluded in the surface of the glass. An estimate was therefore made of the amount of hydrogen which would be required to produce the number of H par- ticles observed. A film of wax of thickness about 10y (stopping power about 1:2 cm. air) was placed round the a-ray tube and comparative measurements were made of the number of H particles with and without the wax. It was found that the presence of the wax practically doubled the number of H particles, showing that about the same amount of hydrogen must be associated with the a-ray tube as with the wax. The mass per unit area of the wax was about 1:0 x 107? gm. and its formula approximately C.,Hs.. Consequently the amount of hydrogen per unit area was about 0°15x 107%. The weight of the glass of the a-ray tube per unit area was about 1°6x107*. Consequently, as the glass does not ordinarily contain as much even as 0-5 per cent. of hydrogen, we cannot look to the glass as the source of H particles. The mass of hydrogen required represents about 1°6 cm. thickness at N.T.P. This was also verified by actual comparison with the number of H particles produced in a tube containing hydrogen, the results of which showed that the hydrogen associated with the a-ray tube would be equivalent to about 2°0 cm. This amount of hydrogen cannot be contained in the «ray tube, particu- larly as the emanation was sparked with excess of oxygen before condensation. There remains only the occluded hydrogen or water vapour in the glass. unless we are to assume that the H particles arise in some way from the interior of the radioactive atoms. A strong source of radium C deposited on nickel was next used, the nickel being heated to about 150° C. to remove water-vapour. In this case also, H particles were observed, Passage of « Particles through Hydrogen. 243 in greater number even than for the same activity (y ray) from an a-ray tube. As an average of several fairly concordant experiments, the following figures were found for the number of H particles per equivalent mgrm. radium element, observed in air at a distance of 8 cm. from the source. The area of the zinc-sulphide screen observed was 1°17 sq. mm. Radium C... 0°20 per minute. Ra.Em.(--A+C in a-ray tube)... 0°07 ,, x Tt must be remarked, however, that in the case of radium C on nickel, there is not as much absorption of the lower velocity H particles as in the case of an a-ray tube, since there is the extra thickness of glass in the latter case. At low pressure the number of a particles reaching the screen from the same source of radium C would be 3:1 x10! per minute. The comparatively large number of H particles emitted from radium C on nickel seems to suggest that occluded hydrogen or water-vapour is not the cause, unless radium C forms some compound with hydrogen involving an increased chance of collision of the « particles with it. Experiments were also made with a quartz a-ray tube, and again the H particles were observed, though slightly less than half as many as from a glass a-ray tube. The numbers are not strictly comparable however, since the quartz «-ray | tube had nearly three times the thickness of the glass «-ray tube, and would therefore absorb the slower H particles. Thus there seems a strong suspicion that H particles are emitted from the radioactive atoms themselves, though not with aniform velocity. This does not necessarily disagree with the nuclear hypothesis unless it were further assumed that a particles are ejected not only with definite velocities from the radioactive atoms, but also in definite directions relative to the nuclear and electronic arrangements *. In conclusion, we desire to thank Sir Ernest Rutherford for his kind, inspiring interest in these experiments and for providing us with the necessary material. Wellington, N.Z., May 1915. * Cf. Rutherford, Phil. Mag, xxviii. p. 310 (1914). R 2 = ee si = — a oS AROS = ea ES st ie oe ae ee a ae [ 244 ] XXI. On the Probability of Ionization and Radiation of Gas Molecules due to Collision with Electrons. By K. F. NESTURCH *, AS recently shown by Franck and Hertzy, the electrons in electropositive gases (He, Ne, Ar, also mercury vapour), set in motion by the action of an electric field, collide elastically with neutral gas molecules. Consequently when passing a fall of potential, the electrons accumulate kinetic energy independently of the number of collisions on their way, 7. e. independently of the pressure of the gas. When an electron has obtained a certain amount of energy, its next collision with a gas molecule is an inelastic one; it loses its energy, which is then transferred to the molecule. The amount of energy, possessed by the electron at the moment of the inelastic collision, is a characteristic quantity of the gas, the corresponding fall of potential being the “ionizing potential” of the latter. Franck and Hertz have brought forward numerous argu- ments { in favour of the view that the energy, which an electron loses by this inelastic shock, can either ionize the molecule or force it to emit radiation of a definite wave- length, but that both processes cannot occur simultaneously. . To this it may be added that according to the model of the atom, recently designed by Dr. Bohr §, the possibility of such transformation of energy is very obvious in both cases. It is possible to suppose then that the ratio of probabilities of ionization and radiation is a characteristic property of the as. In this article the author will attempt to indicate a method for a rough determination of the numerical values of the above probabilities for electropositive gases only. In the first place a formula is to be given for the value of the electric current between the two plane electrodes in a homo- geneous field, the gradient of potential being X. The two following assumptions are made for the sake of simplicity. 1. The “ionizing distance,” which the electron must pass before it has gained the amount of energy sufficient to expe- rience an inelastic shock, must be great as compared with the mean free path of the electrons. 2.1, the distance between the plates, must be an exact multiple of the ionizing * Communicated by the Author, having been read before the Russian Physico-Chemical Society at the meeting of 22 December, 1914. t Verh. d. D. Phys. Ges. xv. (1918) ; xvi. p. 457 (1914), { Z.c. and Verh. d. D. Phys. Ges, xvi. p. 512 (1914). § Phil. Mag. xxvi. p. 1 (1918). Tomzation and Radiation of Gas Molecules. 245 distance; of course the same relation must exist between the potential difference X/, on the electrodes, and the ionizing potential V. Let y be the probability of the ionization of the gas molecule due to an inelastic collision. Then the probability of the emission of light will be equal to 1—y. Suppose that ny electrons set free by some external cause (for instance, by photoelectric effect) are made to travel from the cathode towards the anode plate. After having passed the fall of potential V they will inelastically collide with the gas molecules and produce ny new pairs of ions. Therefore in the electric field there will be mp(1+-y) electrons, and after a new series of collisions the number will be increased to n(1+)*, and so on. It is obvious that the total number of electrons reaching the positive plate will be equal to a) Te eG In Prof. Townsend’s theory* the same quantity is denoted by the expression nye", 6) Liat le : wt ee tet) cat | he (2) where « is the number of ions which an electron produces by collision in going through a centimetre of the gas. The numerical values of a were experimentally determined over a wide range of gradient X and pressure p by Prof. Townsend and his pupils. ‘These values can be easily obtained from the . a : . x curves, representing — as a function of —. By means of the expressions (1) and (2) we can write an equation for the determination of y, log nat (1+y)= Ya. eee ey Te), This equation may be considered in a slightly modified form a= lognat (L+7), . Site ie ue (4) which determines a2 as a linear function of X. This formula holds good only, when the above-mentioned limitation is made for the “ionizing distances.” With increase of X the “jonizing distances ” are diminished and become comparable with the mean free paths of the electrons. In this case after * J, 8. Townsend, “ The Theory of Ionization of Gases by Collision.” 246 Mr. K. F. Nesturch on Jonization and an electron has gained an amount of energy, sufficiently great to be able to ionize, it must sometimes pass a considerable distance before it comes into collision with a gas molecule. We can make the probable assumption that, when such an electron overcharged with energy collides with a gas molecule, the electron newly formed by the process will have the initial velocity at the expense of the excess of energy of the parent electron. Therefore it will be further moved as if it were formed right on the plane, parallel to the cathode plate, where the parent electron had acquired the critical amount of energy sufficient for ionization. The movement of the latter, however, will be slower when compared with that assumed in the simplified calculations of the expression (1). Consequently for large values of X, the charge received by the anode and therefore the values of « are less than those calculated by means of the expression (1) and formula (4). As stated by Prof. Townsend”, the highest possible value of a does not depend upon X, and is equal to the number of free paths of the electron per centimetre of its way. Therefore one might expect that the lines, representing « - as a function of X for large values of X, will become more and more concave towards the axis of abscissee. Further, since according to the formula (4) these lines ought to pass nae the origin, they will have the form represented in enili Fig. 1. Fig. 2. “BIR S12 va Experimental curves, however, have the shape given in fig. 2, and this shows that “ isa linear function only for e xX certain values of —. Such disagreement may be due to different causes. Hither ele oa me Lyir ean} : Radiation of Gas Molecules by Collision with Electrons. 247 the value of y for small * is not constant, or for small = an inelastic shock does not take place at every collision of the electron, possessing the critical amount of energy, with the gas molecule, but depends upon their relative position at ‘the moment of the shock. On the strength of the present experimental results it is impossible to explain the shape of the curve for small values of —. Whatever cause determines the form of the curve, its influence disappears with the increase of ” For the determination of values of y we Sratinly must take the straight part of the curve, prolong it as far as the intersection with the axis of abscissee, and take this point for the new origin. In this way we can determine the value of a for a given X, and further, using for V the data found by Franck and Hertz, we then can calculate the numerical value of y by means of the equation (3). The described method has as yet been applied to argon and helium only, and the following results were obtained. Argon. The point of intersection with the axis of abscissze of the prolonged straight part of the curve corresponds to cha The ionizing potential of argon, according to a and Hertz, is equal to 12 volts. Thus y= about 0-4. Helium. In the case of helium the experimental results are not sufficiently definite. Gill and Pidduck in a careful investigation* have given a characteristic curve for the gas, containing 2 percent. ofimpurities. The point of intersection for this curve is —=8. The ionizing potential of helium is 20°5 volts. The value of y, calculated by means of these data, is about 0°9. So far as pure helium is concerned, the results are notso certain, and the curve is not characteristic enough to permit the application of this method. There is no doubt, however, that the ionization increases with the purity of the gas, and the value y for the pure gas must consequently be greater than 0°9. From the difference of values of y for the two gases it is to be expected that under the impact of electrons ‘helium will be more readily ionized thanargon, while argon will be more capable of emitting light. The following observations, made by Collie and Ramsay+, may be considered as a confirmation * Phil. Mag. xvi. p. 280 (1908). + Proc. Roy. Soe. lix. pp. 257-270 (1896). 248 Prof. P. J. Daniell on the of the above results. A small quantity of argon, present in helium when a discharge is sent througha Geissler tube, may be easily detected spectroscopically; but when some helium is present in argon its quantity must be considerably greater in order to give a distinct enough spectrum. With regard to the electronegative gases, gases in which electrons lose their energy at every collision with the mole- cules, the analogous considerations are probably far more complicated. Petrograd, Physical Institute of the University. XXII. The Coefficient of End-Correction—Part II. By P. J. Dantevt, B.A., Assistant Professor in Applied Mathematics, The Rice Institute, Houston, Texas *. nls |e an electrical current passes through a long cylin- drical tube of conducting material, and then out into a large hemispherical volume of the same, the total resistance is proportional to the total length of the tube plus a certain multiple of the radius. This multiple is the coefiicient of end-correction which we require to find. Rayleigh in his ‘Theory of Sound’ found that ‘785 < this coefficient k < °845. In the previous paper with the same title the author assumed the normal current at the open end to be of a form A+ B(1—oa’) + C1—a’) “14. Thus it was found that (1) if B is neglected, the approximate value of & is *82171; (2) considering B, the approximate value of k is °82168. Then the method shows that the real value of k < °82168. In this paper the author states a method by which an approximate value of & can be found which is less than the real value. In fact *82141 < real value &. Thus & is confined between the narrow limits °82141 < k< -82168. The method by which Rayleigh found the lower limit "785 for k, was to assume a value for the potential V at the * Communicated by the Author. Coefficient of End-Correction. 249 open end. The author has amplified this and assumed that at the open end V is of the form A+BA—@e’) +C0(1—a’)?+ DA—a’)*. § 2. As before, let us use cylindrical coordinates aw, z and let us take the radius of the tube to be 1, and its length to be L. Let us divide the whole space into two parts: first, the hemisphere « =0, z=0, and #7 +2?= R’ where R is large ; secondly, the cylinder OS a1, -LXSz50. Let V be the potential at any point, then a solution is given by Dirichlet’s condition that | V =— dS isa minimum, On where Sn is an element of normal drawn outwards from the region over whose surface the integral is taken. But in this case we shall assume, not a given value of the current, but of the potentials at z= —Land at z=+n. In the region II. a solution of Laplace’s equation will be given by V=—Iz+ B+ 32,7 (hyo). This satisfies the proper conditions; for near the end <= —L we have V=—l/z+H, while at the boundary «=1, or Ji(k,) =(). Thus the k,’s must be chosen so as to satisfy this equation. Then Vi=0= E+ Ya, JS (kyo) =A -- B(1—o’) + C(1—o’)?+ Di1l—2a’)?. Multiply by Jo(4,@)@ and integrate from 0 to 1. Then Bis e(k.) = { Tish aides i Hit ee We Melee: 0 i 0 ‘1 +| C(1—2a’) J o(h,o)ada +{ D(1— x PJ o(k,.o)ada. 0 1 But * : 1 { Jo(ka)(1—a’)’ ada = 2”—'T (p) A : 0 & * Schafheitlin, ‘ Bessel Functions,’ p, 31. 250 Prof. P. J. Daniell on the Then Jo (kr) _ , SiGe) , p 2do(hr) Po aa ter But J1(ke) = then 2°J ae Lee 4 peed a 2 Ju(h,)= 7 Isle) —Iol by) = —Io (ly) Ay) “ 4. J3 (k,) — & Je (ky) ear, Ji(ky) mers z, 30's) > Isle) = 2 Isls) — Tulle) = —JIo(k) (F —1). Then — iy Jo(kr) 2B & 32C A8D / 24 1 Dee hed Ap fi ) Again, OV OZ) Aye el eetele Veep ; Therefore v3 ov ag at end z=O0 of I: ={ VS (< )_ ode Jo7(/ #) = — gE Shpate . i At the other end Ove ON = pgp pelag 2 / | V=/L+H, Ao anne so that ra vo" gg =) (PL+IE). Then = ee VOU dS = YL + Shon, De =1PT 4 (OBS Z, + 2(2B) (32055, ee (82055 +2(2B)(48D)5 S(p— Ds none i, = =1) ¢. (i.) J 24 \? +(48D)°S 7,(75 — —1). Coefficient of End-Correction. 251 Again, since V,--0=H+ Yard (kro) =A+B(1—a’) + C(1—a’)?+ D(1l—a’), if we multiply by w and integrate from 0 to 1, HB —A+SB+iC+2De 2)... (it) In the region I. we have to make V have a definite value over z=0, a<1 and V=0 when z=+o and oY. = (0 when 2=0), a 1. 02 Introduce * new variables £ € which satisfy the relations e=f 0 w= /(1—-F)(L +2). Then the solution of Laplace’s equation which makes V=0 at infinity and finite when w = 0 is V = ZAnPa(E)Qn (8) : where P,, is the first, Q, the second Legendre function, and iis the imaginary ,/—1. But when -=0, =( aol, €=0, OV and therefore DE =0 when €=0. Then only even integers n may be taken, n=2r say. When z=0, a<1, €=0, so that V2=0= DAorPo-(E) Qo,(0). Let A>,Qo-(0) = do. F Then, since when €=0, l—w’ = it follows that 2borPor(E) =V,9>=A+ BE + CE + DE. This equation will yield the coefficients bz,. When -=0, w<1 or €=0, OV __1oVv 2 Te ae iJf 7] fe) — go AaP» (E) Lye Qutit) 2=0 * Jeans, ‘ Electricity and Magnetism,’ p. 252. 252 7 Prof. P. J. Daniell on the But Qo,(26) ry (E+ V&+1 cosh u)-?"+Ydu or ee) Q,,(0) aio cosh u-@"+Ddu. 0 Also foo) $, Quit) —=4-(2r+)) (f+ VC+I1 cosh u) = CH 2 Ope 1) i x (14+ ——— cosh udu, ( Vo°+1 or when z=(0 this =i rt) — 2r— >) cosh u7@"t2)du, 0 Then (tr) ager yes Co,= (2r +1) { (cosh w)-@"+2) du / i (cosh u)-@r+Ddu 0 0 of Cee 2 24 TED S- i Ve-0(— e" )_zde= { : [ Sbe,Par( é) | | Sea-Co-Por(E) | - Ede oe Ar+1° 2 = » bor Cor r a Between the regions I. and II. the boundary of non- conducting material is continuous, that is to say there is no chance of any current escaping between the regions I. and II. except through the open end itself. The total current passing into the region I must then be the same as wl, the total current in the region II. This total current is and =7l on the other hand. Then by= 4 l. Coefficient of End-Correction. 253 Now 1= Po; / 8 5 4 Po zm) B= 5, (Pit 5Pat 3 Po) 16 9 ee. 33 |S) aes al ay pes oe f= 357(Pot 5Pit g fat ig Po). But by Pot boP 2 +).Py + boPe= At BE + Ge-+ DE". Then b= At ,B+ 50+ 2D, = t 2 1 Ae) b= gB+7C+5,D, Bit oA = gs U+ 77D, 16 oe gaa But Con eae 1 1 1 and ‘ 1 | Ae: os so that oe a | 2 3 H=7!l+ @B+ 7, C+ og. The potential V at the end z=—L is /L+E and is assumed to be fixed. 1 1 ea ; se Then U(L+ t) + g B+ is C+ 58D is fixed.,. (iti.) In the region I. at the boundary #?+<2?=R’, the potential V is proportional to = and On to ea so that the integral R On R?’ od yo as ean b d I l by i si a, e made as small as we please by increasing R sufficiently. 254 Prof. P. J. Daniell on the Thus OV Gg = =\¥ -=of -2") joe ae = Babar! Ca ie i =}.8.Qpstos Joy +3.£.8. (hosMny Combining (i.) and (iv.) and using the calculated values of the coefficients Dirichlet’s condition becomes that 2 z(t 7) 4. BY[-182024 + 226354] + 2BC[-164180-+-194017] 4 O[-171548 +-192580] + 2BD[-13972-+°16168] + 2CD[-15863 ++17442] + D?[-15483-+-16675] is minimum, J ( T 1 9 3 ; while (Ltt) gb Ot oh is fixed. § 3. The problem then is to find the minimum value of 2 a1 + 2) +B%(-408379) + 2BC(-358197) + (°(-364128) 4+2BD(-30140) + 20D(-33305) + D?(-32159), when (+ 7 +: ;B+ ces 2 D=fixed in value.. (v.) By the method of indeterminate multipliers, B(-408379) -+ ©(-358197) + D(-30140) = ah B(-358197) + O(-364128) + _D(-33305) = = B(-30140) +C(-33305) +D(32159)= 2 l ah v or ) Ns Coefficient of End-Correction. 255 Then 1 B[-408379 x °364128 — (-358197)?] =a E of 364128 — “ ae 358197 | —D[-30140 x 364128 —-33305 x -358197], or B(-020397) =r(-0129284) + D(-00955). Also C[-408379 x -364128— (-358197)?] =a E of 408379 — : of ‘358197 | —D [33305 x 408379 —-30140 x *358197], or (020397) = —a(-005249) — D(-02805), or B= (-63384) + D(-4682), C= —A(‘ 25734) — D(1°3752). Then D(°30140 x °4682 —°33305 x 1:3752 + °32159) = (53 — 30140 x 63384 +-33305 x 25734) or D(-00470) =a(:00181) or D=A(‘385). The fixed value of the linear expression fv) is equal to the potential at the end <=-- L, that is =/(L+4) where & is the coefficient required. But L+4 is proportional to (Viz-1)? hee dS ¥ Lidigt © dS that is to say, making OV OV a minimum for a fixed V,—-y corresponds to finding the maximum k. Bach term introduced will increase this maximum and therefore the k we obtain is less than the proper value, contrary to the case where we gave the current a fixed value 256 On the Coefficient of End-Correction. when the value of & obtained was greater than the proper value. Here we find a eee (L+=U(L+ *) +, B+i50+ SD, or since A= al ! (-63384) — + (-95734) Phe were {ion hee 2 Pee +5( 385)( j of -4682— = of 137524 5) a + + 035664 + 000348. Then neglecting the term D(1—a’)* we obtain k= ~ + 035664 = "821062, and introducing the term D(1—2a’)3 ky = 821062 + 000348 = "821410. Then using the result of the previous paper, 82141 < k < ‘°82168. The change produced in & by introducing a further term is about ‘00035 in this paper. In the previous paper it was about °000030. Let us assume that the actual errors are proportional to these changes. Then & will probably differ very slightly from 3 ° 56 = 2282 = *R9 82168 ap: 3 ( 82168 —-82141) or from °82166. Then certainly "82141 < k < °82168 and probably k=about °82166. coger all XXIII. Construction of the Diamond with Theoretical Carbon Atoms. By Apert ©. Crenore, Ph.D.* (From the Department of Physiology of Columbia University.) | io a former paper f a crystal of rock-salt was constructed with the theoretical atoms of sodium and chlorine, the atoms being in cubical array, alternating, sodium and chlorine, along each edge in three directions. The direc- tion of the axis of rotation of each atom was shown to be along the diagonal of the elementary cube, taking such direction along the diagonal that one plane may be brought into coincidence with any other plane by properly moving the plane. When the edge of the elementary cube has the dimensions 2°814 x 107° em. it was shown that each atom in the whole structure, due to the forces upon it of all the other atoms, is in stable equilibrium, both as to translational and rotational forces. Tig. 1 illustrates one of these planes of atoms. The arrows are not in the plane, but point upward toward the centre of a cube on the elementary square base. In the diamond it has been shown, by the study of X-ray spectra, that each carbon atom is at the centre of a regular tetrahedron. It is more difficult to represent the exact arrangement of the tetrahedral structure by plane figures in perspective than is the case with a cube. A model is of much assistance in seeing the relations. There are many possible ways in which the axes of rotation of the atoms may be placed, but only one way to make the equilibrium perfectly stable. From the following description a model may be con- structed to show the necessary positions of the axes of rotation for stable equilibrium. The whole crystal may be built up by placing planes of atoms as represented in fig. 2, one above another, and making 1 1 the distances between planes alternately Dp / 61 and 4 Ay AE where / is the side of the elementary triangle and also the edge of an elementary tetrahedron in the diamond. This amounts to dividing the vertical distance into equal spaces, IL! pe 4 $e equal to 1 /6l, and alternately using two and omitting two consecutive planes of atoms. Hach plane may be made to coincide with every other plane in all respects, including * Communicated by the Author. Tt Phil. Mag. June 1915, p. 750. Phil. Mag. 8.6. Vol. 80. No. 176. Aug. 1915. S 258 Dr. A. C. Crehore on the Construction of the the directions of the axes, by giving it the proper motion ; but one plane is not immediately above another in a similar position. Where the wide spacing between planes occurs equal to a) 61, the planes, including the directions of axes of rotation, do exactly coincide when moved vertically, but this is not the case with the pairs of nearer planes. PENNS SSS SYN NNN Pf SO NNSA SZ eS AN ANS SSS S/S SY NA LPP ue = ve ik A SANS ieee AN Ne SN SONOS A Ne RN A) LN Re oN The arrows represent the axes of rotation, such that when the observer looks from the head of the arrow toward the | centre, the rotation of the electrons is clockwise. None of the axes in fig. 2 lie in the plane of the paper, but three- fourths of them take such directions as to point toward the centres of the regular tetrahedra whose bases are the ele- mental triangles and whose centres are above the plane of the paper, as the point 0 for example. The other one-fourth Diamond canes Theoretical Carbon Atoms. 259 of the axes represented by the crosses to indicate the tails of the arrows points downward perpendicular to the plane of the paper. Designating this plane of atoms, bounded by the full marginal lines, as plane number —1, the plane of atoms next Fig. 2. ee ee ee es ee ee we a ee ar A Fat ee. ‘ee OM at ee ee rt ce ee OR Oe oN eee ww of], = re he eS See at eae DP Se Be a ge i eR Ce ae as PS Oe OSS Bee ea ee ee Ge a Se a ae a ae dl | Sigs Sue above this at a vertical distance Vv 6 will contain the point 0 and all similar centres of tetrahedra, and may be numbered plane zero. If the plane No. —1, bounded by the full lines, is moved, carrying the atoms with it, into the position indicated by the lines of dashes, the upper and lower boundaries remaining in the same position, this will be the correct position for all atoms and axes in plane No. zero, and the atom taking the position 0 will point duwnward perpen- dicular to the paper. The next plane of atoms may be called plane No. +1, at a vertical distance iV above No. zero. This coincides in all respects with plane zero if brought into coincidence by moving perpendicular to the paper. The : j tony plane No. +2 above No. +1 is at short interval pv 6, and is represented in its proper relation to plane No. —1 by moving the full line area into coincidence with the line of dots, so that the atom 0 in plane zero is directly under the centre of a triangle where the arrows point away from its ae : Bis centre. Plane +3 is ata long interval n V6l above +2 8 2 260 Dr. A. C. Crehore on the Construction of the and coincides with it in position. Plane +4 is at a short interval and is a repetition of plane —1 directly over it, and this completes the cycle. In brief, there are but three situations of similar planes, Nos. —12, —11, —6, —5, 0, +1, +6, +7 being like the plane bounded by dashes; Nos. —10, —9, —4, —3, +2, +3, +8, +9 like planes bounded by dots; Nos. —8, —7, —2, —1, +4, +5, +10, +11 lke planes bounded by the full lines. A study of the model made from these directions shows perfect symmetry in each of the four directions corre- sponding to the elementary tetrahedron, and the model may be turned so as to make any one of the four faces of a tetra- hedron horizontal, when each becomes a duplicate of the others. We will select the atom 0 in plane zero with its axis pointing downward and calculate the forces of every other atom uponit. The directions of the axes have been chosen after many trials so as to make the turning moments of all other atoms upon any given one zero and to produce Rig. 3. stable equilibrium for these moments. This may easily be shown to be true for this arrangement of axes as follows: Referring to fig. 3, let 0 be the atom selected in plane zero with axis pointing downward perpendicular to the zero plane. Diamond with Theoretical Carbon Atoms. 261 If we at first consider only those atoms in planes + 2, zero, and —2, it is always true that those atoms nearest to 0 form a regular geometrical figure with fourteen faces, six squares, and eight equilateral triangles, one carbon atom being located at each of the points 1-12 inclusive. These atoms are in pairs, each pair being on opposite sides of 0 at equal dis~ tances on some diameter through 0. An inspection shows that the axes of rotation of each atom in one pair are parallel, as land 4. The axis of 1 points away from the centre J of the tetrahedron 0123, and of 4 toward the centre of 0456, and these directions are parallel. The axes of the atoms in the hexagon 7-12 inclusive point in pairs toward the centres of the tetrahedra whose bases lie outside the hexagon, 7 and 8 toward 13, which is the centre of a tetrahedron on base 7, 8,16; 9 and 10 toward 14, the centre of a tetrahedron on base 9, 10, 17; 11 and 12 toward 15, the centre of a tetra- hedron on base 11,12,18. Therefore 7 and 10, 8 and 11, 9 and 12 are parallel to each other in pairs. It is evident that the turning moment of the triangle of atoms 1, 2, 3, on 0 is zero, for the moment of 1 on 0 may be represented by a vector perpendicular to the plane 1J0, which contains both the axes of l and 0. Similarly, the moment of 2 on 0 is represented by a vector perpendicular to plane 2J0 and equal in amount, and the moment of 3 on 0 by a vector perpendicular to plane 3J0 of equal amount. ‘These three vectors take the directions 0, 8; 0,10; and 0, 12 respec- tively, making 120° with each other in the same plane, and the sum is, therefore, zero. It is seen from this that the sum of the moments of all the atoms from 1 to 12 inclusive is zero, for these twelve atoms form three squares 1, 4, 8,11; 2,5, 7, 10; and 3, 6, 9, 12; the axes in the first square all being parallel to the axis of 1, in the second square parallel to 2, and in the third parallel to 3. The total moment of the twelve atoms is, therefore, four times the moment of the triangle 1, 2, 3, namely, zero. It may also be shown that these moments produce stability for small displacements of the axis of 0, but these are not all the atoms which must be considered. The triangle of atoms 1, 2, 3 lies in the plane we have called No. +2, the centre of the triangle being a point where the tails of the three arrows meet just to the left of 0 in fig. 2. A study of this figure shows that the total moment of all the atoms in this plane, +2, is zero. By drawing concentric circles in this figure from a centre where the tails meet it is easy to show that the sum of the moments of all atoms in each circle is zero. The next outlying atoms to 1, 2, 3, in this plane form an 262 Dr. A. C. Crehore on the Construction of the equilateral triangle, each axis being parallel to that of 0 and in the same direction, thus assisting the stability and making the moments zero. Plane —2, containing atoms 4, 5, and 6, may be similarly studied, the whole plane making for stability, giving zero moments. Similarly, each plane not yet considered in detail, particularly planes —1, zero, and +1, may be shown to give zero moments and produce stability. Hach atom lies in the centre of a tetrahedron and the axes of the four corner atoms are always in different directions from each other, parallel to one of the lines joining a centre of some tetrahedron anda corner of it. There are but four directions of axes of rotation in the whole crystal determined by these four fundamental medial lines of the tetrahedron. The tetrahedron of which 0 is the centre is omitted from fig. 3 to avoid complication, but is shown in fig. 4. The atom J, however, is in the centre of the tetrahedron 0, 1, 2,3. Since it is always true that the sum of the moments of any Diamond with Theoretical Carbon Atoms. 263 three corner atoms is zero in every tetrahedron, the axis of the central atom as determined by the tetrahedron alone must take the direction of the axis of one of the corner atoms. In this case the axis of J takes the direction of that of 0. It might, however, take the direction of the axis of any one of the osher three corner atoms as far as this one tetrahedron is con- cerned, and remain stable in each of four positions for small displacements. A large displacement would be: required before it would leave one position and go over to another. Translational Forces. The consideration of the translational forces exerted upon some selected atom 0 is not apparently capable of such simple treatment as was the case with the cubic rock-salt crystal. In rock-salt every atom has a duplicate at the same distance on opposite sides of the selected atom with axes parallel to each other. The general formule (23), (24), and (25) of the former paper * show that the translational force upon 0 is equal and opposite for each pair of such atoms, and is therefore zero for the whole crystal, irrespective of the dimensions of an edge of the elementary cube. ‘The con- ditions for stabelity of the equilibrium determine the length of the edge of this cube. In the diamond now under consideration we may eliminate all atoms in planes numbered +2 and —2, and +4 and —4. An inspection shows that each atom in plane +2 has a corre- sponding atom in plane —2 at an equal distance from 0 along a diameter with axes parallel to each other. The total of these two planes, therefore, gives zero translational force on 0. For a similar reason atoms in plane +4 cancel those in plane —4, and atoms in plane zero cancel each other. The forces due to the nearest atoms in planes +1, zero, —1, and —3 have been calculated by the formule referred to, giving the following results. The axis of z must by the formula always lie in the direction of the axis of the atom, 0, upon which the force is to be calculated. The axis of y must by the formula lie in the direction of the vector k x k’, where & is a unit vector along the axis of atom 0, and k' a unit vector along the axis of the second atom. The result is that as the formula stands the directions of the # and y axes differ when the force of different atoms upon 0 is figured. They must always lie in the plane of the hexagon, however, 7-12 inclusive, in fig. 3, the z axis being downward perpen- dicular to this plane. The directions of the w and y axes will, therefore, be specified for each atom caleulated. It will be * Loc. cit. 264 Dr. A. CU. Crehore on the Construction of the seen from symmetry that the sum of all w and y forces in the plane of the hexagon, the zero plane, due to all other atoms in the structure upon atom 0 vanish, and it is then only required to figure the z component forces ; ; for upon substi- tuting the values of the quantities given in Table I. in the equations, we must obtain the same 2 and y component forces for atoms 20, 22, and 24, because the values are identical. The « components are therefore equal and their directions are along 0—28, O—26, and 0—27 respectively, TABLE I. | | 3 | + DiGeSEIOR of Axes Beobatois _ |#Direction oF Axes| Protwenl x Ly [2 |r [x | [2 | fol sina seer PEE EET ew i O- at S S an ae 9 % N t\ i = N =(s\=I 7 al ~~ =I8 iS IN herr) | ae + N S ! &y| i In N IS S1919s 1s {9s Bh | Ri ® |S) Rie 9/9 S| ° S| HLS + x) ~~ S & S y S NS S S = ©, -) wy 9 3 +. + is SS N N G | WG GW G) w G Nh NM | Nf fh ins) Digi] A] @WINI Oo}; Un Pi ws] nt] - ~ = ~ ~~ = = Nie ~~ a|@ ~~ 0-28| 0-10 pablo making 120° with each other, and therefore evidently vanish, without makin g any further calculation. The y components lie along 0—10, 0—12, and 0 —8, and also vanish for a similar reason. The wand y components of atom J, which is on the axis of 0, both vanish. The three atoms, 29, 30, and 31, fig. 4, in the base of the tetrahedron of which 0 is the centre, similarly make the sum of the x and y components vanish, and, lastly, the lower triangle of atoms 35, 36, and a7 in plane —3 makes the w and y components vanish. Diamond with Theoretical Carbon Atoms. 265 We have, therefore, to calculate the z component forces . for five different sets of coordinates only : first, the force of J in plane +1; and, second, the force due to 20 in plane +1, which, multiplied by 6, gives the z component of the whole hexagon 20-25 inclusive ; third, the force of 29 in the base of the tetrahedron, which is to be multiplied by 3; and, fourth, the force of 33 in the same plane —1, which is to be multiplied by 3; and, lastly, the force of 36, which is to be multiplied by 3, for the lower triangle in fig. 4. The results are as follows :—~— Plane. Atoms, +1 J B= + °¢ {—10-60606 P°8 21-44 149-2229 eee, «6 «(067830 «=O, «=O + «117086 C=, eee, 6) — 197531 «=, «= + 1176348, —1 32-34 F.= , {— 059008 ,, — 24363 e Soe = 4, j— 182k 5 = 5757 : Tote. F= ,, {—15°'7225 ,, +252°83 o K is the specific inductive capacity of the medium ; a, the radius of the orbit in centimetres of the single electron in the hydrogen atom; @, the ratio of its linear velocity to that of light; w, its angular velocity; ¢ the velocity of light; P the number of electrons in the carbon atom ; Sami! the sum of the squares of the radii of the orbits of each electron in the carbon atom; and / the length of the edge of the elementary tetrahedron in the diamond, measured in a, units. a,, 8,, and w, are constants whose values have been previously + determined. ata, 0-200 4 107% & 15:0. x 19" a ee ee i = 0'00103 (7) mee S910" me 15°) R10? For the carbon atom the number of electrons is 12, and P=12. The determination given later of the distance / in diamond gives 2°528 x 10~® centimetre, which is equivalent to 122,100 a, units. That is to say, the edge of the tetra- hedron is this number of times greater than the radius of the SP et ele te «| (8) + Loc, cit. pp. 772, 778, equations (63) and (68). 266 Dr. A. ©. Crehore on the Construction of the orbit of the hydrogen electron, or 15,010 times greater than the radius of the carbon atom itself, as we will show that the carbon atom has a radius of 8°14 a,, units. The value of 1 may now be found by equating the total translational force in (6) to zero, giving Sin? SLAB P . f= 4-010 PA, dy Tamils. y es ys i To obtain / in centimetres multiply (7) by a, and obtain : >m? ae fot ALN CD : : ‘ —= Ses P centimetres. . . . (10) And numerically = Ss x 02% L0™ * Sm? =°006683 x 107 * 2m’, cm. 27) Kquating this ere value to the experimental value of J given above, namely, 2°528 x 10-8, we obtain the value of the sum of the squares of the radii 9°528 ; — ( ( « . ‘ 2 San? = O0GGS3 = 318, Lor carbon atom: |.) ame This result is in very close agreement with the theory given in the paper f referred to, which makes the carbon atom con- sist of an outside ring of eight and an inside ring of four electrons. In this theory the radius of the positive electron is ‘735 x 10-!? cm., and the radius of the carbon atom with 12 electrons °735 x 107 x 12% = 1:684 x 10-7” cm,, .or 8°14 a, units. Applying the formula (75) of a former paper f to find the radius of the outside ring of electrons on the assumption that the four inner electrons are at the centre of the atom, the radius of the outside ring is *828 of the radius of the positive sphere, or 6°735 a, units. Using the experimental deter- mination with suspended charged spheres of the relative values of the inner and outer rings as given in the table, p- 37, of the paper referred to for the combination 12=8+4, 6°73) 2°88 the radius of the inner ring. The sum of the squares of the namely, the ratio 2°88, we get =a>> = 2°338 a, units for t Loe. cit. ¢ Phil. Mag. vol. xxvi. p. 79, July 1918, ~l Diamond with Theoretical Carbon Atoms. 26 radii, =m? for the carbon atom is, therefore, a 65150" %/8:= 363 Bou 6 = 29 Sm? = ON ihe ase orie & CES) This is to be compared with this same quantity determined from the diamond in (11) above, namely, 378. The agreement is surprisingly close. To find the Length of the Edge of the Elementary Tetrahedron in Diamond. It may be of interest to give the calculation of the length of the edge of the elementary tetraliedron in diamond, as the method differs somewhat from the usual but gives sub- stantially the same value. 1 = distance between corners of tetrahedron. Pe pv 6! = f i nearest planes of atoms. a = the edge of a cube of the crystal containing one gram. No. planes of atoms in distance “ = 74/6. 2 2D : a’? 2./3 No. atoms in one plane = E37 No. atoms in cube of edge a = 4/2 0. atoms In cube of edgea = 2/2... . . . . (14) d = density. 3 1 : a° = volume per gram = qoucem. «2 eee (15) No. electrons in a gram = 6 x 10". No. carbon atoms in a gram = wiGea ee 107 (16) Replacing a? in (14) by 2 and equating to (16) we find 4,/2 p. ive ELS age og 7 eR ae a Taking d = 3°5 for diamond, we get P= 16°16x10-* cu. cm. and / = 2-528 x 10-8 em. . (18) 8) 268 Dr. A. C. Crehore on the Construction of the It may be objected that it is not sufficient to calculate the forces upon a single atom 0; but it may be shown that the example calculated is representative of all atoms, owing to the symmetry of the crystal. Referring to fig. 2, the atom 0 represents any one of the crosses, meaning that the axis points down, perpendicular to the paper. One-quarter of the total number of atoms in this plane are crosses, and an inspection shows that we must obtain the same result for any one of these crosses ; and the same remark will apply to everyone of the different planes. That is to say, our selected atom 0 really represents one-quarter of ail the atoms in the crystal. Had we started with one of the other atoms, for example, that just below 0 in fig. 2, we should have used the sets of planes perpendicular to this arrow. The crystal may be built up as well of parallel planes perpendicular to this arrow as of those perpendicular to any other arrow, each plane being an exact duplicate of fig. 2. The atom 0 is therefore representative of every atom in the whole crystal. A word of caution seems advisable lest attempts be made to apply this result for the diamond to other crystals whose atoms are arranged in a similar way. A case in point is crystal zincblende, ZnS. It has been shown by X-ray analysis* that the structure of this crystal is like the diamond, zine atoms replacing the carbon atoms in one set of tetrahedra and sulphur atoms replacing the carbon atoms in the other, interpenetrating, set of tetrahedra. The zine atom, according to the theory, has a ring of only two elec- trons at its centre and the sulphur atom a single electron. The general equations (20), (24), and (25), so far developed, do not apply to the case where each atom has one or two electrons at the centre. They do apply to give the force of the zinc on the sulphur because there is presumably no syn- chronous rotation in such a combination, but not to the case of the zine on the zine or the sulphur on the sulphur. To be more explicit, if the selected atom 0 in fig. 8 is zinc, then the atoms in planes +2 and —2 are also zine. So far as we know, therefore, we have no right to say that these planes cancel each other, as was the case with the diamond, because we do not know that the forces of these atoms cancel in pairs. The phase angles of the single electron in sulphur and the double electron in zinc come into the account, and render the solution of the problem far more complicated than in the case of the diamond, where the equilibrium is independent of * W. H. Bragg, Roy. Soc. Proc., ser. A. Ixxxix. pp. 480-438, Jan. 1, 1914. Diamond with Theoretical Carbon Atoms. 269 phase angles. If we apply the formulz here developed to ZnS we obtain ee Oe ae aly « {19} PP and the theory as given in fig. 3 of a former paper * gives eae = 2S ee ee es 10) Fig. 5. The almost exact coincidence between these theoretical and experimentai values in the diamond, where the theory applies rigorously, leads to the conclusion that the discrepancies in * Loe. cit. p. 323. 270 Mr. E. Howard Smart on the Third-Order ZnS above, as well as in NaCl in the previous paper, are due to this cause. The diamond appears to be unique in that there is only one kind of atom and that the inner ring does not contain one or two electrons. I am not aware that any other crystal has been studied experimentally where none of the atoms have, according to the theory, one or two electrons at the centre. Fig. 5 is a photograph of a model of the diamond, and may be found of assistance in supplementing figs. 3 and 4. ; Note. Since this paper was communicated a considerable number of crystals on the isometric system has been worked out, and experiment found to agree with theory. These include zine and copper. XXIV. The Third-Order Aberrations of a Symmetrical Optical Instrument. By EH. Howarp Smart, .A., Head of the Mathematical Department, Birkbeck College, London™. HE five third-order aberrations of a symmetrical optical instrument, commonly associated with the name of Von Seidel, have been frequently discussed. But the mode of presentation often leaves something to be desired from the practical optician’s point of view, direction cosines of rays and the like being to him of inferior importance com- pared with angular aperture and field of view. Sometimes even the whole subject is treated in general terms, the constants of the instrument not being considered; and occasionally there is some obscurity regarding the relations between the several errors. In this paper an attempt has been made to effect some improvement in these respects, and at the same time to indicate a method by which the investigations could be extended so as to deal with the fifth-order corrections. Let C; (fig. 1) be the centre and O; the vertex of the ith of a system of coaxial spherical surfaces, and let this surface separate media of refractive indices w,, and w,. Let r, be the radius of curvature of the surface considered positive when the surface is convex to the incident light. Take O;as origin and a pair of rectangular tangents at O,, and the axis of symmetry as coordinate axes. 7 * Communicated by the Author. Aberrations of a Symmetrical Optical Instrument. 27] Assuming a plane object perpendicular to the axis, let the successive Gauss image-planes for the several surfaces be constructed, and let s;, s,/ be the distances from O,; of those connected with the ith surface before and and after refraction Fig. 1. respectively. Let K;, K;,; be the positions on these planes of the images of a point on the object supposing there were no aberration. Then C;K;K;,; is a straight line. Leta ray from the same point on the object incident on the surface at Q.(E, §;) cut this line at P(x, + d2x;, Y,+ 8Y,, sit As,), and let the retracted ray Q;P.41 cut it at Pya(ti41 “++ 62; 41; Ysa t OY 14 si + As;,’) ’ (4, Yio Si), Ad (Wi41, Y;445 Si’) being the coordinates of K; 200g. Then we have Ee tne + b= 267, ig c, 5 (#8 hae r? ): neglecting 4th order terms . (1) Also it readily follows from the law of refraction that OF; OP.) 5 cP) (55 Ay elie (2) QPP = (wi + bay E)?+ (y, + by,—9,)? + (8, + Asi— £)2 = (wat Oui)? + (y; + 8y,)? + (s:-+As,)?—2(E,(0; + 82,) +n(y; + 8y,)) = 26(si+ As;—r). 272 Mr. E. Howard Smart on the Third-Order Also C;P,/C.P; 4, = eae + 62) /(@i4y Ss OU + 1). ao Es (3) Hence (2) gives Pei he + O2;) a Bi (@ai+ 64%:41) above expression for Q,P, corresponding expression tor Q;Pi41_ Let (4) be expanded by Taylor’s theorem. Denoting Mi-1Vi i b i; ae tye + 57 —2(E2i + ny3) — 250% — 7) ‘e y we get oF; oF; oF; As, —— F, + d2; ‘Oe; + oy Yi OY, at Sj OS; oF ra ae oe We develop this expansion in terms of = yy c Xi U4] Cc; i as &e Ta De. eae hace eae C., Sj ee of the same order of magnitude. ; (up to terms of the third order) = P14 E? me pi =| aa: Ge; +, id a Si ZS; Vy 8; Se PA making use of (1), and F;,;=a similar expression with Mi Cina Yiav 8¢ fOr py, ®y Ye Sp dw,|v,=dy,/y;= As,/(s,—7;) 62 541/541 = OY i4 /Yig =A /(5;) —7)) from the figure Also and (5') Kliminating 6z,, dy,, 6v;,,, 5Y;,, With the help of these from (5), we get, after a little reduction, as the coefticient of Si on the left-hand side of (5), ST; bi; ef Ex, + ay,+ C(s,—7,) —1 iS; } a + ye + Se =e (Eu, +;",)— 20(s, a 2 with a similar expression on the right-hand side. Aberrations of a Symmetrical Optical Instrument. 273 Approximating in (5) we have, rejecting 3rd order terms, B; -1 ee is t+1 : 3 \ PY ihe ; (6) si Si This is obviously true, for by considering the points K,, K; +; in the figure, we have Ge _ Vi+1 Sane ae) at Spay and the positions of the Gaussian image-planes are connected by the relation ine 1 eaiiadt ehnisay sn yz ots (? To the next approximation including 3rd order terms we have Bs a4 + M;_} ) ES aia oft lp E52 i v7) Si t } 8; 8 1 a?ty? Bh; Ase, YS; ate ti ee ane _ be stot S14 Peta d Ver ae 4 Stig t Mi 3 | r; PY 7 Las, + Vig ‘ “so Asia, 78; at > encamey 5) meap Me : ay) a j ae si — 1; Making use of (6) and denoting the equal expressions in (7) by the symbol Q;, we have as the simplified form of the last equation, pAs! AS; _ 1 j g;” ys: so 5a as eas a) rs 2) (FE 1 1 “S) East) (— a) “8; igh 15; ve (w $+y)Q(ss rd ~~): Now Q, G- pe a) = bh; =(; - )— Mes AG i 1 ee Ga): ° PB 1 MS! fy_ Sid 8) Phil. Mag. 8. 6. Vol. 30. No. 176. Aug. 1915. T 274 Mr. E. Howard Smart on the Third-Order So the last equation takes the final form NS! fh MS. it Be eo FONE En} 5; M18; Py_1 agi a 1 ) 7a oF 1 —5(4=) (72+ y,") ) = Si bs; a Pi: +3(") (a? ae —) salle lt area This is the recurrence formula for the longitudinal aberration of the successive images P,, P;41. To sum this throughout the system we require first relations between successive &’s and 7’s, the coordinates of the points at which the ray strikes the successive surfaces. The condition that the refracted ray passes through the point (&,,4, 4, 4;+ 41), where d, is the distance between the ith and (0+ 1)th surfaces, is Sas E, ae Dea es _ 4+ SiH S$ U, +80; ae ¥.,1 + O44 — 9; sop Ase In approximating to this we need only retain first-order quantities ; so remembering that Ox, éy, NSS nN at +1 atl : : ee . and S 9) is 9 Pires | Year $s; ‘i i are all of the second order, we get from the equality of the first and third § Making use of (6) this may be written symmetrically Mig 15541 Ms GES 4 leat FU iy Si41 S; SjSi+1 since Sear...) 5) a ae (10) Similarly i+. MYM,» Merb PY = +1 a YN ay. SY ior yey (11) Si41 S, S83 2274+ 1 Aberrations of a Symmetrical Optical Instrument. 275 For purposes of summation we shall also need the quantities hy, hg... hi... which are the heights above the axis at which a paraxial ray, starting from the axial point of the object and passing through the system, cuts the successive surfaces. It should be pointed out, however, that these are introduced merely as a convenient device for summation, as used by Seidel; the quantities h must not be confused with the &’s and 7’s which indicate the true order of approximation as regards aperture. We have ! f s: S; CC eae : pe) a a eo att". GAZ) ieee | he Be 8 GS Hence the relation (6) may be written er iy -@;h; ey ae aay, Pee ) a ee @ sintilarly 4 = he, : Si4] Si Si-1 = Motil — >» (say), aha gt) flay 1 where A is a constant for the ray throughout its passage through the instrument. Similarly MY sr; I BELEN on &e.... = MA — another constant fe. (14) i+1 Z Summing the equations (10) and (11) we have pa@ier&i4. uy Mot Ey a PS pp 1d yp tp +1 ‘ SS ee ef = ae | Si+1 8] p=1 SpSp+1 MiYit+ii+1 ot Boi + PS Poe n"n+1 s 2 Kee) <1 ar 9 Si+1 $s) a) SpSp+1 which by (13) and (14) may be written compactly Git &1 Se. ha dp x a i ' ’ a Ke — eas? ke isd hy p=1 Myliphy+1 Aiay hy p=1 Mypltnhy+1 We can now effect the summation of (9) for all the refracting surfaces to the 7th. For brevity we shall denote Pet) i die 1 1 1 > a te by > and —,— —— by A (=). p=1 Myh» Yo+] isi Bi-1%; KS 79 276 Mr. E. Howard Smart on the Third-Order Then ene equation (9) through by h?, and since Ny Sian Be ASialiey aoe — 1 5 Qh i Tine a 2 a m/) > t 1 Sy torus }a(Z)—Qanea(Z) FAM + orn —500+e)A(F +3 = Op ie By summation, since 4s,;=0, Ns ch? pirates TH Sie (Ea) = anna (=) — (EAE) 5 nea (2) + Qnay) ie -) (1+ QheSy +5 (x2 + ps Bese | (16) When A and p are each zero the object and image are on the axis. The aberration is then the central ‘spherical aberration only and is given by the first term of the right- hand side of (16). The condition for no central Hhoprationd is ee é if if > sit( = — ———) =0 e e 8 e As, 1 Q Misty Mi-1& ( ) Supposing this satisfied, let 7% and mw be now small quantities whose square may be neglected. Then the expression for the Cas aberration is (Papa Bote nels “\(14+ Qh? = 5% ). hy 1 isi! Te as ‘4 p= 5 He EiN+ mph h The evanescence of the coefficient of is the condition for the absence of coma, the balloon-shaped flare produced in the image-plane owing to the images for different values of &, and ™ being distributed along the line C;K;Ki41 in fig. 1. As in Whittaker’s tract (‘Theory of Optical Instruments,’ 1907, § 25) it may be shown that to this order of approximation this is identical with Abbe’s sine condition. Aberrations of a Symmetrical Optical Instrument. 277 We shall now consider the union of rays in the primary and secondary planes respectively. Since in equation (16) A/z;=/y, we must have in primary planes &,/N=y,/u and for secondary planes €A+nu=0. Let ,As;’ denote the aberration from the image-plane of the primary focus of a small pencil proceeding from a point distant x; from the axis of the instrument and s; from the ith surface, and whose chief ray is incident on the latter at a distance H, from the axis (where now H?=£&7?+7,7). Then, taking for convenience the primary plane as the plane of xz, the equation of the refracted ray being L- 5, ys a ek 1 3 s;| I Ns; — a Ly + bu; H; we must have for primary foci (si oa 5) (a, + da — H;) 27; \__) — 3M 3 A & (1+ Qih? =)” HR ey cy Leal =). 3 Saas 1° TY \ fc” e131 The separation of the secondary focus is easily written down from (16). Denoting this by ,As;’ we have eee 3G) 2h A 2 Toes LS ZB23)2 Be = 5(G, ) = WEA(T )—5V SAT 1+ Qi?) s jes 1 egret pa $5ME—(=— 4 e e ° e s e 20 2 1 Ti\KFi = Pi-1 ( ) Assuming, then, the errors of spherical aberration and coma to have been eliminated, it is clear from equations (19) and (20) that the primary and secondary foci can only be brought into complete coincidence if also > A(=.)4 Qn? SO. 1 bs This term occurring as the coefficient of 27, and therefore depending on z?, indicates that the corresponding astigmatism, which the evanescence of this expression removes, is due to the outer parts of the field. It may be further noted, from equations (16), (19), and (20), that when central spherical aberration, coma, and astigmatism have been corrected the image still suffers a displacement from the focal plane given by ees at (pet Spay 2 17% \KMi Mi-1 ie Sen ] oy i 1 ). Hence agi mare bg Saunas (— -—). ea pax? 41 17 \i Mi-1 \ Aberrations of a Symmetrical Optical Instrument. 279 The left-hand side of equation (21) is iF times the curvature of the image after the ith refraction. Hence for a flat image we must have oS ke Taa(y, pi pi] S if (; 1 1% the well-known Petzval condition. It remains to consider the defect of distortion due to unequal magnification in the inner and outer parts of the field. It is assumed that the image has been corrected for central spherical aberration, coma, astigmatism, and curva- ture, so that it now lies in the focal plane. We proceed to obtain the condition that it shall be a faithful copy to scale of the object. The refracted ray at the ith surface is ly, Se aR X—&; Y—n; Oe 5, Moe eh } Damages, ye one dL gs4y) eos Fer OH —7 si t Asi — 9 (E? +9;) This cuts the image plane ZL=s;' where X=, +-Az;, Y=yi + Ay; (say). ay Then we get (1+ Bets By (+ =f ‘— bx; gE; I 9 9 and similarly for the y’s. Rejecting terms of the fourth order this gives Aer; bai! As; i. 1 a = -—-— Xi Vi Si By (5') we have Aa,’ Ss" As i ghey 1. Rk; ST ee a ee We “) ra = a *7). nibs As/! Xi Siti Sj X; s Qi 5 Si a piAs;' 1 hé, i ; =" | Ot xg from (13) A similar equation follows for the incident ray. 280 Mr. E. Howard Smart on the Third-Order Therefore by (15) and (16) ae as in Whittaker’s tract, sieepigy Us pag, (1+ Qa? 3), We! Ae fi i) 4 bie 5) — On be eee a 2 se me Cig = 4 seared: (e geo ne a3 = { Or a Se TH) soiui(a EY tm + 52 (sa +0,U?2 | OF +u)= ; +20 vent Unc 1fl I 24 42 vif ev 2-2 (7 — f he: ras tO pe a) tO oe Ki Pi-1/ = Now the quantities £,, m, fi, 4, and mp are the same throughout the system: hence by summation Az;' Aa eee UBEP + 2) + QE >0,;U?2 a; hy? sp OBE grain Eryn eS ate a hy hy Mi Pi-1 #0244) 3 Vig @U2-=(— alee ios i Ki See Now correcting for the four errors above mentioned we have in order 50;=0, 50,U,;=0, 2OU2 =0, = Lp pi yee / But = must be independent of ?,and p?, since these depend on the square of the distance of the image-point from the axis. Ti \Pi — Pi-1 is the condition for freedom from distortion for full pencils to this order of approximation. The same result is obtained by considering Ay;'/yi'. This of course necessitates that Aberrations oj a Symmetrical Optical Instrument. 281 Az;'=0 Ay:'=0, as might be expected; since the image- point must now coincide with the normal magnified position given by (a;', yi) in the final image plane. he expressions for the coordinates of the point in which the ray, after refraction at the ith surface, cuts the image plane, in the general case when the errors are uncorrected, may be easily written down. Denoting | bay I 1 2 O,, Sy @:U;, s 0,;U?, eS == (~ — ), Ti \fi — Pi-1 and > U(O,U?7— boy occ Wy =) Ti \KFi Bi-1 by A, B, C, D, E respectively, and putting y;,=0 so that pw also vanishes, we have, if M be the linear magnification, a; = May and the 2-coordinate of the required point is Mar [1 — yi (AE (E? +2) + BOM) BEE +n) + (830—D) (Mhy)*E, + BM)9) | where 2 is given by (13), The y-coordinate is similarly 1 My, E ~ hs (Am(E? + m1’) + 2BEym(AA,) + (C— D)(vra)?n: | ' We shall conclude by deducing as illustrations of formule (18) and (20) the expressions for the deflexion of ihe primary and secondary foci from the focal plane in the case of a small parallel pencil incident centrically on a lens of index w at an angle @ with the axis. Here for the first refraction, since /h,= tan 4d, a) ; 7 the deflexion of the primary focus is given 4g 1 an? tb +t ae cha | 3 =F Ga 3G) =a" ae At the second refraction similarly (2) =288"# (a, L-N) or ear df 3(4— 1) =). 2 PX ery rt Ais 282 Mr. W. Gordon Brown on Hence the total deflexion of the primary focus is given by si _ tan’? But ly . bh 1 i (7)= oF (CAT ) since p= 4—1)(--;). Similarly for the secondary focus the deflexion 6( p) is LAY tan—3(2) {oto d —1 il 1 and since Ae bb ey — pb pry Bry we have 5 1 A tan? @ (ot 7 (i)=—se (“7 These formule were given by Coddington, but of late have been somewhat ignored by textbook writers, though used with success in the design of some modern photographic objectives. XXV. Note on Reflexion from a Moving Mirror. By W. Gorpon Brown ™*. Wie regard to the question of alteration of amplitude by the Doppler effect, which was raised by Mr. Edser in Octoberf, I should like to point out that the relation between the kinematics and electromagnetics of the problem is very clearly brought out by the Faraday tube theory of radiation. In this theory a tube of force lying along the direction of the ray is supposed to transmit transverse vibrations like an elastic string. The displacement of the tube from its equi- librium position alters the direction of electric intensity. If it is assumed that the magnitude of the component parallel to the ray is unaltered, then the transverse component of electric force will be equal to this unaltered intensity mul- tiplied by the tangent of the angle between the tube and the ray. Thus if r measures distance along the ray, z is the transverse displacement, and R, E are the 7-, z- components of electric intensity, Oz ee en ee The magnetic force is the product of the velocity of the * Communicated by the Author. + Phil. Mag. vol. xxviii. p. 508. Reflexion from a Moving Mirror. 283 tube, which is perpendicular to the ray, and the electric intensity perpendicular to that velocity, divided by c?, so that Let zZ=2,sin m(a cos 0+y sin @+¢ct), (3) where xcosO+ysind=r. Then B= kh = | = Rmzy cos m(x cos + y sin 6 + ct), so that the amplitude of F is Merrimac. iets g)asiies ayibay ta (4) Similarly, Hy= ~Rmmzy. 4) las raw le 9 GF) Now if the waves are obliquely reflected by a moving reflector, we have the following relations e+vecos@ _ sinO _ m ec—vcos@'’ sin@!’ Mm — (6) Further, MOEMSER ck ys ya) wae, «AD Combining the relations (4)...(7), Ho _ Bo’ mm’ _ sind _ c+vcosd (8) 1: PS m sin@'’ c—vcos@’ ~*~ ~* This result is independent of the plane of polarization, and is in complete agreement with those of both Sir Joseph Larmor and Mr. Edser. Equation (7) corresponds to the equation §+£&'=0 quoted originally by Mr. Edser, and z can be identified with the ethereal displacement & The continuity of the magnetic force normal to the reflector of course follows from (8) when the plane of polarization is such that the magnetic forces lie in the plane of the reflector. It is also apparent that & is not 284 Prof. Morton and Mr. Vint on the Paths of the time integral of magnetic force, but proportional and perpendicular to it; this seems to be the explanation of the difficulty noticed by Sir Joseph Larmor in the last complete paragraph on page 705, Phil. Mag. Nov. 1914. The con- tinuity of magnetic force normal to the reflector is due to the alteration in the angle of reflexion, so that m’ sin 6’=m sin 6. The assumption that the electric intensity along the ray is constant, and is not altered by reflexion, is essential to the argument given above. AXVI. On the Paths of the Particles in some cases of Motion of Frictionless Fluid in a Rotating Enclosure. By W. B. Morton, 1.A., and Jas. Vint, M.A., Queen’s University, Belfast *. : [Plates I. & II.] [* a paper by one of us (Proc. Roy. Soc. vol. Ixxxix. p. 106, 1913), drawings were given of the paths of the liquid particles in some cases of two-dimensional motion. The present communication contains some notes supple- mentary to the former treatment of the cases of rotating cylinders containing liquid, followed by a discussion of some three-dimensional paths which occur in the rotation of an ellipsoidal shell containing liquid. Paths in rotating cylinders—The stream-function = Cr” cos n@ gives the motion when the form of the containing cylinder and of the relative stream-lines within it are given by Cr” cos n? — 4 wr? = const. The cases n=2 and n=3 were discussed in the former paper. The case n=4 can also be treated by use of elliptic functions. It corresponds to the rotation of a cylinder having square symmetry, and has been treated by St. Venant for the elastic torsional interpretation of yy: particulars are given in Thomson & Tait, part il. p. 244. For properly chosen values of the constants we get a very close approximation to a square section, the corners being slightly rounded off: another case is that of a section bounded by hyperbolic ares. We have drawn a number of paths of particles in these cases, but it does not seem worth while to reproduce these, their general form being quite similar to those already given for the triangular prism. The angular width of the loops on the paths shrinks to zero at the centre. This will always be the * Communicated by the Authors. Particles in Motion of Frictionless Fluid. 285 case when n>2, because then the relative paths approach the circular form when r is small. For the elliptic cylinder, on the other hand, all the paths are geometrically similar, and the angular width of the loops remains the same right in to the origin. A point not noticed in the former paper is that these paths are trochoids gene- rated by the rolling of a circle of radius (a—b)?/2(a+6) outside a circle of radius 2ab/(a+b), the tracing point being at distance $(a—b) from the centre of the former circle. This follows easily from the method of generation of the curves. Paths in cylinders rotating about eccentric azxes.—lIt is obvious that the stream-lines relative to the cylinder are the same in this more general case as when the rotation takes place round the axis of symmetry. Further, the displace- ment of the particle along the relative stream-line for a given angle of rotation remains the same. Thus it is easy to calculate the paths in space for rotation round any point, once the case of rotation round the axis of symmetry has been treated. The manner in which the looped path is modified depends on the progressive change in the arrangements of the normals which can be drawn from the centre of rotation to the relative stream-line. We have worked out the case of the elliptic cylinder and the triangular prism. Some diagrams relating to the former are shown on figs. 1-8 (Pl. I.). The dotted ellipse shows the position of the cylinder relative to the axis of rotation which goes through the centre of the looped path. It will be seen that for points on the minor axis alternate loops shrink and disappear, while for points on the major axis pairs of adjacent loops coalesce as the centre of rotation moves away from the centre of figure. Paths in a rotating ellipsoidal shell.—If the ellipsoid (abc) rotates with angular velocity Q round the axis (J mn) through its centre, then, as was shown by Lord Kelvin, the particles of the contained liquid move, relatively to the ellipsoid, round ellipses lying in the planes lx my ne a2(b? +0) © b?(c? + a”) Le (a? +0) Particles on the boundary move round the intersections of their planes with the ellipsoid, other particles move round similar ellipses. This elliptical motion is retrograde and simple harmonic, having period 27/#, where wo” =( 2bel ) 4 2eam \? 2abn \? OF NG Fe Ta) * (arH) = const. 286 Paths of Particles in Motion of Frictionless Fluid. This magnitude is < 1. The motion of a particle in space is thus obtained by com! ining simple harmonic motion round an ellipse with a rotation of the ellipse itself round the axis (lmn). To examine the simplest re-entrant forms of these space- paths we take w, © in the ratio of small numbers. When the value of this ratio is assigned a certain limitation is imposed on the form of the ellipsoid. The axis of rotation (Jmn) must lie on the cone 2 o 2 he 2 (oe = Bye (A or) # + (B ge + (© o)e= where A= Zobel (b?-+ c*), ete. Therefore w/Q must lie between the greatest and least of the magnitudes A, B, C. This is equivalent to saying that the greatest and least of the axis-ratios of the ellipsoid (taken as Jess than unity) must lie on opposite sides of the magnitude (l— yl—#’)/k where k=o@/Q. The values of this are, for tees 2s 382 3 268 4 172 4 iPad These cases have been worked out numerically taking the ellipsoid (5, 2,1) for the first two and (10, 3, 1) for the second two. In each case the cone which contains the possible axes of rotation surrounds the mean axis of the ellipsoid. The particular generator selected was such that it made a considerable angle with the normal in the plane of the ellipses, so that the path in space should have solid relief. Models of the curves were made in lead wire and stereo- scopic photographs were taken of these. For k=, 1,1 the paths of particles lying on the central section only were con- structed. These are shown on figs. 9, 10, 11 (PL II.). For k==4 the paths were calculated also for a series of sections at different distances from the central plane, showing how the path is continuously altered until it becomes a circle round the axis of rotation for the particle on the surface situated at the point where the tangent plane is parallel to the planes of the paths relative to the ellipsoid. ‘The successive forms are given in figs. 12 to 16. The following is the detailed description of the cases of motion for which the curves have been constructed. Fig. 9. Hllipsoid (5 2 1) rotating about the axis whose cosines, referred to the axes-of figure, are (‘59, *52, °62). The Electron Theory of Metallic Conduction. 287 The planes in which the particles move relatively to the ellipsoid are normal to the direction (°21, °22, :95) which makes the angle 34° 7’ with the axis of rotation. Hach particle goes twice round its ellipse relative to the shell, while the latter makes three revolutions round the axis. The motion relative to the ellipsoid is, of course, retrograde. Wig. 10. Hllipsoid (10 3 1) round axis (:33, °87, °37), planes of motion normal to (‘09, °27, :96) which is inclined to the axis of rotation at angle 51° 42’, one rotation relative to the shell to three turns of the shell round the axis. Fig. 11. LEllipsoid (10 3 1) round axis (22, -96, :17) planes of motion normal to (‘12, °56, °82), angle between these directions 45° 15’, one relative rotation to four of the shell round the axis. Figs. 12-16. LEllipsoid (5 2 1) round axis (:33, °86, °38), planes of motion normal to (‘17, °53, °83), angle 34° 10/, one relative rotation to two of the shell. Fig. 12 shows the space-path for a particle on a central section, fig. 13 on a section way out towards the parallel tangent plane, figs. 14 and 15 on sections ‘90 and +98 of the way out respectively, and fig. 16 the limiting circular form. os XXVIII. On the Electron Theory of Metallic Conduction.—lIV. By G. H. Livens*. PRE general theory of metallic conduction based on the idea that it takes place by free electrons whose velocities are prescribed by collision with the molecules, has been inter- preted in its most general possible form by Drude and Thomson and by Lorentz on rather different but apparently fundamentally identical lines. According to Lorentz, the whole problem turns on the determination of the law of distribution of velocities, which in the absence of all external forces is taken to be determined by the Maxwellian law of equality of mean energies as in gas theory, and which in their presence is presumed to depart but slightly from this law. As Lorentz proves J, the most general law of distribution of velocities consistent with the more fundamental basis of the theory and determined by the specification of the number of electrons per unit volume at the point (2, y, z) in the metal and with their velocity components in the range between (&, 7, €) and (&+d&, »+dn, €+d£), depends on the * Communicated by the Author. + Vide ‘The Theory of Electrons,’ pp. 266-271. 288 Mr. G: H. Tavend on the Electron solution of a differential equation, which he writes in the form xSi pv 47.9 os +69! nhs O04 of APL Herein the function f is defined so that the number of electrons in the specified group is WS Te XY, %, t)d&.dyn.de; (X, Y, Z) are the component accelerations produced on the typical electron of the group by the external fields and the same for all of the electrons in the group; the difference (6—a) is such that (b5—a)d& dn dé represents the increase per unit volume of the number of electrons in the specified group brought about per unit time by the collisions of these electrons with the atoms. In the particular case examined by Lorentz when both the electrons and atoms are assumed to be hard elastic spheres, the atoms being rigidly fixed, the number ()—a) may be simply replaced by fom: Tm where f, is the particular value of f which expresses Max- well’s law of distribution of the electronic motions and T» 1s the mean time between two successive collisions of an elec- tron of the specified group. The equation for f is in this case OF. v OF 7S eae nn aod of, 4 Fok Raet+ Ysot Laer fa, +75) + S52 + mS of which the general solution may be 2 i written down. It was thought at the time when the previous paper of this series was written, that this equation was perfectly general and independent of the particular dynamical nature of the interaction between an electron and an atom on collision. In a certain sense this statement is true, although in the general case it is necessary to specify very carefully the appropriate value of tm. The object of the present paper is to formulate explicitly the problem for the more general case and thereby to determine the appropriate form for tm to be used with the equation. We may generally assume that the atoms are rigid spherical nuclei fixed in the metal, which act on the free Theory of Metallic Conduction. 289 electrons in such a way that the potential energy of the electrons when at a distance r from the centre is m™ / p\§ 2(F) relative to the atom; this is the most general form of law of action appropriate to the present type of problem. There is every reason to believe that in the simpler metals s=2 gives a sufficiently good approximation to the actual facts, but we shall not make this particular assumption in the following work. The particular case examined by Lorentz. is easily deduced if w is taken equal to the radius of the atom and s made infinity. In the more general problem now under review it will be necessary to take into account a factor which was neglected in the previous argument, viz., the persistence of the velocity of the electron after a collision. It is well known from the corresponding problems in gas theory that the velocity of an electron after a collision is not entirely independent of the velocity before collision; in other words, the collisions do not entirely obliterate the effects of the reorganizing forces in the external field produced previous to the collisions. This persistence of the effects may be mathematically expressed by saying that if the distribution of velocities among the electrons just before collision be such that there is the number ON of electrons per unit volume with their velocity components in the specified range over and above the number expressed by Maxwell’s law, then the number in the same group taken just after their respective collisions is eON, where e is a factor (<1) which expresses the dis- organizing effect of the collisions: this of course implies that the expectation of the particular velocity defining the group is reduced from 6N to edN by the collisions. The collisions have therefore only removed the number (1—e)dN of electrons from the group considered. Jn this way it is easily seen that Rae a—b= a (f—So) where T» is used properly for the mean time between two successive collisions of an electron. If we use Phil. Mag. 8. 6. Vol. 30. No. 176. Aug. 1915. U 290 Mr. G. H. Livens on the Electron then a—b= eras Tn and the equation for f is thus x2Fy v4 284 4 Wy Bly Wa_foh Met ay’ Mot Fac? (ona. (fer of which the general solution pee to the present type of problem is a ~ at Clee? a Aon T, X1 dt, 1 Ofo Oso Ofo , ony no = OF, net Yar Boe + o 3a yep Pees and the suffix ; indicates that the oie eae ex- plicitly as functions of the general time variable ¢ are to be taken for the time t=¢,. We now require some idea of the value of tr, in the par- ticular case under investigation: this is obtained at once from an investigation of the dynamics of a collision between an electron and an atom. We introduce polar coordinates in the plane of the motion of the electron and the centre of the atom, with the origin at this centre and the axis along the dee of the asymptote in the direction in which the. electron approaches the atom. The two first integrals of the equations of motion of the electron are then of the usual type, PO=h, P+ P= ia +0, where M1 = & where C and A are constants. If wu is the velocity of approach from infinity and p the perpendicular from the centre of the atom on to the asymptote described with the velocity u, then h=pu, C= 17) If we eliminate the time differential from the two equations of motion and substitute the value of the constants, we get the equation to the path of the electron in the form is i Bes gt (a ee ue i Theory of Metallic Conduction. 291 or introducing the new variable, and writing this equation becomes dv 2 9 Vv s 6 z (phe v\s i) \/1 —v'+ (°) The apsidal distance is obtained directly from the fact that so that there, and it therefore corresponds to the one positive root of the equation #=(2) +1. a Denoting this root by vp, we see that the angle between the asymptote and apsidal radius is 6) where The angle the direction of motion of the electron is turned through, which is the angle between the asymptotes to its path, is then 20). The important conclusion is that 6 is a function of « only. Now let us examine the change in the velocity of the electron which is brought about by this encounter : this is easily obtained because the new values of the velocity may be written down from the fact that the velocities after collision parallel and perpendicular to the apsidal radius are —u cos 24, and usin 24). U 2 292 Mr. G. H. Livens on the Electron The new component along the direction of the «-axis is therefore E’ =E—2F cos? Oy + Vu? — E sin 24) cos yp, where (£, 7, €) are used for the components of w, and yf is the azimuth angle between the plane of the motion of the electron and the plane through the polar axis parallel to the z-axis. Similar expressions hold for 7’ and €’. The question of the frequency of the collision next arises. The law of force assumed is such that the electrons are in reality influenced by the atoms at all distances, although of course the influence may only become perceptible when the electrons are near enough to the atoms. We may therefore firstly suppose that in the general case the influence is perceptible when the electrons come within a distance a from the centre of the atoms ; this will give us an approxi- mation which is the better the greater is the value of a. To obtain the actual exact result we may then in our final formule pass to the limit when a is infinite. With this definition we see that the probability of an electron with its velocity components in the specified range, and describing an orbit for which the elements p and w lie within the limits (p, yr) and (p+dp, +d) colliding with an atom in the next succeeding small interval of time df, is equal to the probability of finding an atom in a certain small cylinder of height udt and base pdpdy, which is nup dpdrrdt, wherein 7 denotes the number of atoms per unit volume in the metal. The total number of the electrons per unit volume in the group specified as having their velocity components between (E,, €) and (€+dé&, »+dn, €+d¢), which collide during the same sinall time dt, is therefore a 27 ( ( nudN dtdrpdp ~0:20 =ra’nudN dt, wherein © dN=fdédndé. Moreover, the velocity of each of these electrons has the same € component before collision, and therefore the average Theory of Metallic Conduction, 25a value of this same component after collision is f 2 fe [E(1—2 cos? A) + Jw sin 20) cos ]pdpdyp E Tn ° °2T (‘a ) \ieaav = e(1- at cos? O)p dp), and similar results hold for the other components, the factor being the same in each case. We may therefore conclude that the coefficient of per- sistence of the velocities of the electrons after the collisions is quite generally e=1- es { “cos*6opap ur a’ No We have therefore l1—e= : { “cos*@.pap. “er |: - In this case, however, the mean time between two colli- sions is | ie | Tm — Toe. ey Uu NTAU so that eee it, —— =4irnu\ cos? Oypdp, Tm 0 and the general formula is to be used with this expression. We may, however, before adopting it proceed to the limiting case, where a is infinite, so that the most general equation is to be used with hie = dren | cos’ Oypdp. Tm 0 It is for many reasons more convenient to interpret this result in terms of the constant a, introduced above instead of * Or at least the statistical effect of the collisions may be so inter- preted. The average value of the velocity component after collision is just as if the number in each partial group were reduced by this factor. The argument here is purposely stated indefinitely, as it is probably of an extremely tentative character. == =. »YyY —_ _ =. —. — 294 The Klectron Theory of Metallic Conduction. p as here; we have Bq 2 at = a ; be so that pdp= ada, he and then 1l—e_ 4arnp*u (” FR pe ml cos? Oyada. The integral in this expression is a mere numerical factor of which estimates may be found by quadrature as soon as s is given. If we write 7 lm 5 =4rnp’ ( cos? Ojada, 0 2 then the equation for the fundamental function is thus to be interpreted with The further developments of the theory now proceed along the usual lines laid down by Lorentz, Bohr, Enskog, Richardson *, and others. Before concluding this paper reference must be made to the work of Enskog + and Bohr, who have given elaborate: discussions of the theoretical basis of the present theory, but in a form which is hardly as simple or as direct as that suggested above. Bohr reduces the problem of the deter- mination of the distribution function to an integral equation whose solution can be effected only in certain special cases, but which is otherwise much too general to be of any real service. In his paper, however, Enskog proceeds ona rather more tentative method, and assumes a general solution for the fundamental function which differs from that here given by the presence of additional terms of the second order in (€, nf). The presence of these additional terms is required if, and only if, the theory is generalized to include the effect of the collisions of the electrons with one another, a factor which has been purposely neglected in the above discussion. * Phil. Mag. July 1911. t Ann. der Physik, xxxviii. p. 731 (1912), where the.reference to- Bohr’s dissertation will he found. Prof. O. W. Richardson on Metallic Conduction. 295 As most of the more important results are independent of these terms, and their retention would lead to unnecessary complications in the statement of the problem, it was thought preferable to give the present analysis in its simpler form pending a discussion of the correlative problems in gas theory. The University, Sheffield, Jan. 4th, 1915. [Note added April 1st.] Since the above was written I have, through the kindness of Dr. Bohr, had the pleasure of reading his dissertation, and I find that the reference made in the text to this work is hardly just. The method employed by him is as stated, but it is necessitated by the far more general circumstances under which he treats the problem. I had not access to his work and relied on the reference to it given by Enskog. Bohr’s analysis is, however, rather different from thai, here given, being based on the idea of the momentum distribution rather than the velocity distribution as here. XXVIII. Metallic Conduction. By O. W. Ricuarpson, F.RS., Wheatstone Professor of Physics, University of London, King’s College *. HE form of theory of metallic conduction t which has recently been revived and extended by Sir J. J. Thomson {, and has been strikingly successful in explaining the variation of metallic conduction with temperature, suggests a number of interesting questions some of which are discussed in the present note. According to the theory under consideration, the transference of electricity in metals is due to the motion of electrons from the various atoms to those immediately adjoining them. The electrons are imagined to be projected, from the atoms at which they originate, in directions which coincide with those of the axes of electric doublets which are also supposed to be present in the atoms. In the absence of an external electric field the axes of the doublets are distributed uniformly in all directions, so that there is no preponderance in favour of any particular direction of projection and therefore no * Communicated by the Author. + J.J. Thomson, ‘Corpuscular Theory of Matter,’ p. 86, New York (1907). t Phil. Mag. vol. xiii. p. 192 (1915). 296 Prof. O. W. Richardson on average electric transportation. In the presence of an electric field the axes of the doublets tend to be pulled into alignment with the field, so that on the average more electrons travel in this direction than in others, and this gives rise to an electric current. At low temperatures the tendency to alignment is enormously enhanced by the mutual action of the neighbouring doublets, and this state of affairs causes phenomena closely resembling the super-conductive state discovered by Kamerlingh Onnes. Evidently one sharply marked difference between this form and the more usual forms of electron theory is that the conductivity is due to the effect of the electric field on the atoms from which the electrons are ejected, and not to its effect on the motion of electrons in the interval between collisions. At tempera- tures sufficiently high for the specific heats of metals to have become normal, this theory leads to the following ex- pression for the electrical conductivity c at temperature T° K, namely :— el NepdM == 3 Va 5 . e e ° e . (1) where N=the number of doublets in unit volume, e=the electronic charge, p=the number of times an atom emits an electron per second, d=the distance between the centres of adjacent atoms, M=the moment of a doublet, and k= Boltzmann’s constant. It is clearly of considerable importance to determine the values of the quantities entering into equation (1). An estimate of the moment M of the doublets may be obtained by considering the emission of electrons from the surface of the hot metal. Since the electrons only pass from an atom to its nearest neighbours, the number of electrons crossing unit area in the positive direction in the interior of the substance in unit time will be }Npd in the absence of an electric field. The number which escape from unit area of the surface per second will be less than this on account of the work which an electron has to do against the attraction of the metal for. it. To calculate this number it is necessary to know the velocity of the electrons. Thomson* has shown that in * ‘Corpuscular Theory of Matter,’ Joc. cit. Metallic Conduction. 297 order to account for the law of Wiedemann and Franz, it is necessary to suppose that at the temperatures at which this law is valid, the electrons possess the kinetic energy appro- priate to the molecules of a monatomic gas at the same temperature. This suggests that the velocities are distributed amongst the electrons in accordance with Maxwell’s law. If we adopt this assumption as a working hypothesis we can calculate the number of electrons emitted from unit area of the surface in unit time in the following manner :— The number which start outwards from unit area of the surface per second is me tal PL tA!) POE RO CIA 9) Of these, the proportion which have a velocity component u normal to the surface between the limits wu and u+du is hm \% es —hmu2 (2 bamad ap olnegbiiiny Moro¢gy In order to escape it is necessary that ea ao ay kate, fern gs (4) where m is the mass of an electron and w is the work done at the surface in escaping. The number which escape in unit time is therefore Npa(") | eee debe ae ey mee, .(3) hae WAS; VT oe} Nahe ™m Since 2hw is a fairly large number, the integral on the right- hand side of (5) is very nearly equal to 4(2hw)-ie-?, and, after substituting from (1), the number which escape becomes, since 2h= (kT)7}, Hedy ey ee ee he er POM ON ee OE gti tO where 7 is the surface-density of the saturation thermionic current at temperature T. Since cT’ is constant, this makes t of the form AT?e—/?, where A and 6 are independent of T, if M is independent of T. As is well known, this form of expression agrees with the experimental results. A form which is in better accord with the thermodynamical require- ments and agrees as well with the experiments is i= AT?e~°7. 298 Prof. QO. W. Richardson on This would require M to vary as T-*?. Using equation (6) to find the value of M we get MGe wee fe a 1 esate ; ey é hee a ete et Mette (7) In the case of tungsten at 2000° K, taking K. K. Smith’s values for 7 and w, and putting 16 x 10? Kh. Iie Blee appears that M=4°65 x 10-72 E.M.U. x cm. If the moment is due to charges -re separated by a distance 6 the value of the distance is 0= 2°92 K108 This is of the order of the linear dimensions of a tungsten atom, and is therefore of the order to be expected from general considerations. Langmuir’s data for tungsten would give values of M and 6 about 10 times as large as those above. It seems doubtful, however, if the value of M can be much greater than 5x 10-28 without making the emitting particles larger than atoms, and this would conflict with the theory in other directions. A knowledge of M helps towards an estimation of the number of “tree” electrons present at any instant on this theory. The number ejected from atoms in unit volume per second is Np. If the electrons travel on the average a distance d, with velocity V before recombining, they will be free for a time ¢=d,/V. The number instantaneously present in the free state is thus eee Nod; _ 3kT dy 3vkT ne (8 “=a oe MoV TO) where pv is the number of molecules in 1 ¢.c. of a gas under standard conditions. Putting c =6 x 10) HAS P= 20007 Ke vk=3'711x10%erpdeg 4, ve=4327 E.M.U., M=4-65 x 10-8 H.M.U., and V=3 x 10’ em./sec., n! =6-2 x 1089. eee hs i (9) The number of atoms per c.c. of tungsten is 6°16 x 10”. ihe agreement of this number with the numerical factor in * Phil, Mag. vol. xxix. p. 802 (1915). Metallic Conduction. 299 (9) is quite possibly only a coincidence. The value of the numerical factor depends on that of M, which in turn has been deduced from thermionic measurements. As we have seen, there is still some uncertainty about the thermionic data, even with tungsten, so that we have no right to expect more than a rough estimate of M to be got in this way. At the same time, it seems very unlikely from what we know already about the structure of atoms that M can be very much larger than the value estimated above. Consequently it would seem that n' cannot be much less than the value given by (9). If for the purpose of discussing the question we assume (9) to be true, it would follow that the number of free electrons is to the number of tungsten atoms as d, is to d, where d, is the distance travelled by an electron in passing from one atom to another, and d is the distance between the centres of neighbouring atoms. If the atoms are similar to planetary systems one would expect d, to be comparable with d, so that nearly half the specific heat of the substance would be located in the “free” electrons, If, on the other hand, the atoms have a definite geometrical boundary and electrons only pass between two atoms in actual contact, d, would he zero and the “ free” electrons would contribute nothing to the specific heat. The view of electrical conduction under consideration contemplates the more or less continuous transference of electrons through atoms. If variations of velocity of the electrons during this motion are disregarded, the formula suggests that an atom loses half its heat energy when an electron passes out of it and that this energy is regained asa result of recombination: in other words, that the heat capacities of metals are equally divided between the electrons which carry the currents and the positive residues which form the rest of the atoms. If the true value of the somewhat doubtful numerical factor in (9) is @ times that given above, similar statements would apply to 2 times the specific heat of the substance. For the reasons already stated it is unlikely that 2 is much less than unity, so that, even on the type of theory under consideration, it is improbable that the heat capacity of the current carrying electrons can be disregarded. O°] XXIX. Experimental Determination of the Law of Reflexion of Gas Molecules. By R. W. Woop, Professor of Eaperi- mental Physics, Johns Hopkins Univer sity *. [Plate III. ] 7 ARIOUS phenomena connected with the flow of gases have led us to assume that the gas molecule, on collision with a solid wall, is reflected at an angle which is quite independent of the angle of incidence, and that the number reflected in any given direction is proportional to the cosine of the angle which this direction makes with the normal to the surface. This law is assumed by Knudsen in his recent treatment of the kinetic theory, but so far as I know it has never been verified directly by experiment. Some eight or ten years ago I made some efforts to realize what might be termed a one-dimensional gas by allowing the gas to enter a long highly exhausted tube through a very minute hole while in communication with a small bulb, which was broken after the system had been for some time in communication with the pump. It was assumed that the first molecules to reach the far end of the tube would be moving in parallel paths, and the electrical behaviour of such a gas was the phenomenon studied, the tube being non-con- ducting initially. As no very definite results were obtained, it is not worth while to record the details of the experiment. Attempts were also made to realize a one-dimensional gas by vaporizing mercury in a tube, a portion of which was cooled by solid carbonic acid, but these experiments gave also negative results. The very interesting observations of L. Dunoyer on the formation of sharply defined deposits of metallic sodium in highly exhausted tubes, due to the pro- jection of a flight of molecules down the tube, led me to repeat my earlier experiments with mercury. M. Dunoyer had pointed out to me the necessity for employ- ing a very perfect vacuum, and after one or two experiments 1 had no difficulty in observing the phenomenon with mer- cury. In fact, mercury appears to be better adapted to the production of the one-dimensional gas than sodium, for the latter contains occluded hydrogen, and the last traces of this can be removed only with great difficulty, if at all. This gas impairs the vacuum very rapidly, and gives rise to deposits with ill-defined edges as a result of molecular collisions, * Communicated by the Author. Determination of Law of Reflexion of Gas Molecules. 301 The first experiment was made with a tube of the form shown in fig. 1 A. A droplet of mercury was placed in the bulb and the tube highly exhausted, the walls being heated with a Bunsen burner. After the occluded gas had been driven from the wall, the tube was allowed to cool, and the bulb heated until all of the mercury had distilled into the cooler portion of the tube. It was then sealed off from the pump, the capillary being heated very gradually, so as to have it at a temperature just below the softening point of the glass for some minutes before it finally collapsed. The mercury was now gathered into a single drop which was brought into the narrow neck of the tubes The whole was then introduced into a flask of liquid air with the exception of the turned-over neck. At the end of twelve hours a faint circular deposit of mercury, of the same diameter as the con- stricted portion of the tube just above the bulb, was observed. In this case the distillation had taken place at room tem- perature. The tube was now wound with insulated german- silver wire from a point just above the surface of the liquid air, to the closed tip. ‘This was heated electrically to a temperature which felt uncomfortably hot to the finger. Under these conditions the phenomenon is reproduced in a minute or two. The mercury vapour enters the upper por- tion of the tube, and all of the molecules which are moving sideways are condensed on the wall, the deposit being very heavy at the top of the tube, and gradually thinning away to nothing a few centimetres below the surface of the liquid air. Below this point the molecules are moving all in the same direction, like bullets from a machine gun, and no further deposit is found until the constricted portion of the tube is reached. Here the molecular stream strikes the sloping walls of the constriction, and a heavy deposit of the metal occurs. Passing through the small opening, the gas traverses the exhausted bulb in the form of a jet which shows no tendency to spread ont laterally, and deposits on the wall in the form of a small circular patch with very sharply defined edges (fig. 1). A photograph of the deposit, formed on an oblique plate of glass within the bulb, is reproduced on PI. II. fig. 1. In this case the tube was kept in liquid air for some time before starting the vaporization, to allow the plate to take up the low temperature. In the study of the reflexion of the jet the reflecting surface must be kept at a relatively high temperature, to prevent the condensation of the vapour. A large drop of glass was ground off at an angle of 45°, and the oblique 802 Prof. R. W. Wood: Experimental Determination surface polished with rouge on pitch. The drop was mounted on a slender stem of glass which was fused to the bottom of the drawn-out portion of the bulb (Pl. III. fig. 2). In this case the deposit began to appear in about three minutes after the heat'ng-current had been started, and in ten minutes it was so opaque as to be quite impervious to light. As will be seen from the photograph, the deposit covers the of the Law of Reflexion of Gas Molecales. 303 | upper oblique half of the bulb with the exception of a narrow circular zone just above the plane of the plate. This clear zone is extremely interesting. I at first attributed it to the circumstance that the dia- meter of the bulb was small in comparison with that of the reflecting surface. To test this hypothesis the experiment was repeated with a larger bulb, and a deposit as shown in fig. 3 was obtained, in which the cosine law and the random direction of reflexion are well shown. Professor Max Mason, of the University of Wisconsin, made a calculation of the distribution of density ina bulb small in comparison with the reflecting surface, which by the way is not an easy problem, and found that the clear zone was not accounted for on this supposition. J then repeated the experiment with the large bulb, and found that, if the deposit was allowed to form for fifteen or twenty minutes, the clear zone was quite as marked as with the smaller bulb. It occurred to me that the clear zone might have something to do with the structure of the surface, irregularities of molecular size perhaps pre- venting the nearly grazing reflexion. I tried a freshly split surface of mica mounted on a glass plate, and obtained the deposit shown in fig. 4, the clear zone being quite as con- spicuous as with the glass surface. Dr. Wright, of the Geophysical Laboratory, informs me, however, that some experiments which they have made, suggested by the account of my experiments which I gave at the Spring meeting of the National Academy, have yielded quite remarkable results in the case of the reflexion from the surfaces of certain crystal sections. An account of these experiments will be published shortly. In the reflexion experiments the reflecting surface keeps the temperature which it had before the immersion of the apparatus in liquid air, as it is isolated in a non-conduct- ing vacuum, and the only escape for the heat is through the long thin stem which supports it. In some cases I heated the bulb while it was still full of air, and in this way brought the reflecting surface to a rather high temperature before the exhaustion was commenced. The tubes were usually provided with a lateral branch, joined on midway between the two constrictions and bent up. The exhaustion was effected through this tube, and it was found advantageous to keep the pump in operation throughout the course of the experiment. The results appear to show that the cosine law is approxi- mately followed, for the density is greatest on the line of the normal, falling off gradually as the angle increases. At 304 Determination of Law of Reflexion of Gas Molecules. angles greater than about 80 degrees there appears to be no reflexion, at least in the case of the two surfaces which I have examined. The jet of molecules obtained in this way is interesting in that both the temperature and pressure are vector quantities. Molecular collisions have been completely abolished and the lateral pressure is zero. ‘he same may be said of the temperature I suppose, though we must specify what we mean by temperature in such a case. An experiment was made to determine whether the jet was capable of rendering its path conducting. The lower constriction was made rather large, and a platinum wire sealed in so as to lie across the path of the jet, which played upon another electrode at the bottom of the bulb. The bulb was non-conducting, and no visible discharge occurred when the jet was started. It is possible, however, that experiments more carefully carried out, on the electrical properties of a one-dimensional gas, might yield interesting results. It is my intention to illuminate the jet with the light of the 2536 mercury line, and determine the molecular velocity by the displacement of the resonance radiation line. It may very well be that interesting results can be obtained with an apparatus in which the reflecting surface can be held at any desired temperature, for example at a temperature a hundred degrees or.so higher than the tem- perature at which the mercury vapour is liberated. Possibly some information can be obtained in this way as to the nature of the molecular agitation which we call heat. It is obvious that the reflexion of the molecules results solely from this agitation, since, if we remove the heat from the reflect- ing body, reflexion no longer occurs and the molecules remain fixed to the wall. It will also be interesting to work at a temperature intermediate between that at which com- plete reflexion occurs, and that at which complete condensa- tion takes place. Experiments along these lines will be undertaken in the autumn. I have not tried projecting the phenomenon, but see no reason why it could not be done with a form of apparatus similar to that shown in fig. 1 of the Plate. The success of the experiment would depend largely upon whether the heat rays which are absorbed by liquid air could be removed from the beam of the lantern. fe 805)" XXX. The Structure of the Spinel Group of Crystals. By W. H. Brace, D.Se., FLRS., Cavendish Professor of Physics in the University of Leeds*. | Meee spinel group of crystals is placed by crystallographers in their Class 32, the members of which are cubic and possess the highest possible number of symmetries. The composition is given by the formula R’R,'’0,, where the divalent metal R” may be Mg, Fe, Zn, or Mn, and the tri- valent metal R’’’ may be Fe, Mn, Cr, or Al. It is very interesting that a somewhat complicated composition should be associated with such complete crystalline symmetry. Magnetite, FeFe,O,, is a member of this group. Its. X-ray spectra are shown in fig. 1, the same method of Pin, : : : : (100) : e 184 : 76; ‘ : (110) ci : : : ’ j ‘43 '270 ‘8 : 13 (111) : 6 a 46 78 40 20 37 ae. 1 eee . i Fs o3 . < —>sin@ (@) 4 0°5 Spectra of magnetite representation being used as in previous cases. The heights of the vertical lines represent the intensities of the various orders of reflexion by the three most important planes, and their positions represent the sines of the glancing angles of reflexion. The X-ray used is the a#-ray of rhodium for which 4 = 0°614 A.U. We first find the spacings of the three sets of planes. According to the usual formula, \= 2d sin 6. For the (100) planes @ = 8° 30’. 0614 Hence Adin. = 01478 = 4:15 A.U. In the same way 2dj)= 9°88 A.U., and 2d,,,= 9°60 A.U. We now try to connect these values with the molecular dimensions of the crystal. * Communicated by the Author. Phil. Mag. 8. 6. Vol. 30. No. 176. Aug. 1915. BS . 306 Prof. W. H. Bragg on the Structure oj Let a be the length of the edge of the cube containing one molecule, that is to say the volume per molecule put into cubic form. The specific gravity of the crystal is 5°2 approximately, its molecular weight is 3x 56+4x 16=232, and the weight of a hydrogen atom is 1°64 x 10~*4, Hence exe? = 232% bt ae. a = 713'2 x 109, a=418A.U. The length of a face diagonal of such a cube is 5°92 ALU. and the length of the cube diagonal is 7:25 A.U. A cube containing eight molecules has twice these dimensions. There is a clear connexion between these dimensions and the spacings of the planes which we have calculated. We may state it simply in the following form :— The spacing of the (100) planes is a quarter of the edge of a cube containing eight molecules, the spacing of the (110) planes is a quarter of the face diagonal of sucha cube, and the spacing of the (111) planes isa third of the cube diagonal. | Now this is exactly what has already been found to be true of the diamond, atoms of carbon replacing molecules of Fe,0,*. We conclude, therefore, that magnetite has funda- mentally the same structure as diamond, a molecule in the magnetite corresponding to an atom in the diamond. It has been suggested by Barlow (Proc. Roy. Soe. xci. p- 1) that such arguments as this are ambiguous, and that atoms lying on any of the cubic space lattices may be shifted in a manner which he describes without the fact being betrayed by the X-ray spectrometer. This is not the case, however. ‘The moves which he describes will, amongst other things, introduce a periodicity into the spacings of the (110) planes which is twice as great as before. If such a periodicity existed, it would be detected at once by the existence of a spectrum at half the normal glancing angle of the first order. The structure described above, so long as each molecule is represented by a point, has all the full symmetries of Glacs32.. We have now to place the atoms in the molecule: in doing which we are guided by two requirements. (1) The symmetry must not be degraded. (2) The relative intensities of the spectra of different orders must be explained. * ‘X-rays and Crystal Structure’ (G. Bell and Sons, 1915), pp. 102-6. the Spinel Group of Crystals. 307 We begin with the oxygen atoms, of which there are four. In order to maintain the trigonal symmetries these must lie at the corners of a regular tetrahedron, so oriented that the lines drawn from the centre to the corners of the tetrahedron are parallel to the four diagonals of the crystal cube. Such an arrangement does not interfere with the sym- metries about the (100) and (110) planes, as may be seen in the following way. Consider first the case of the diamond, the structure of which is shown in fig. 2 (loc. cit. p. 100). The black circles Fig. 2. and white circles both represent carbon atoms, but the blacks and whites can be considered separately as each forming face-centred lattices. They can be derived from one another by a shift along a cube diagonal equal to one quarter of the length of that diagonal. The symmetry about the plane egca is not an absolute mirror symmetry; but if all that is on the right of that plane is reflected in it and then shifted by an amount and in a direction represented for example by a move from D half way te 0, the reflected right coincides absolutely with the left. White circles slip into the places of black circles, but since both represent carbon atoms no difference is made. Suppose now that the oxygen tetrahedra take the place of the carbon atoms and are arranged, as already described, so that the four perpendiculars from the corners on the faces are parallel to the four cube diagonals. There are two ways of making this arrangement. We may describe them by saying that the coordinates of the four corners referred to rectangular axes are in one case (111) (,—1,-1) mi,i,—1) (-l;—ly); m the other (—1, —1, —1) (=1,1,1) d,-1L,) (1,1,—1). . Two tetrahedra so arranged are the reflexions of each other in the planes (ay), (yz), or ae 308 © Prof. W. H. Bragg on the Structure of Let the carbon atoms of one face-centred lattice be re- placed by oxygen tetrahedra of one orientation, and the atoms of the other lattice by the other orientation. It then follows that when the process of reflexion and shifting is carried out as in the case of the diamond, we still obtain the same absolute coincidence. From the crystallographic point of view the symmetry about the (100) plane is complete. As regards the (110) planes the introduction of the oxygen tetrahedra in place of the carbon atoms does not disturb the: simple mirror symmetry which already exists. Any (110) plane passing through a tetrahedron contains some one edge and bisects the opposite edge at right angles, thus dividing it into two halves which are the reflexions of each other in the plane. A tetrahedron of one kind does not reflect into a tetra- hedron of the other kind over a (111) plane, but the crystal of course possesses no symmetry of that kind; so the introduction of tetrahedra creates no difficulty. Thus all the symmetries are maintained in full. Next consider the effects of the substitution on the spectra. As regards the (100) planes the spacing in the case of the diamond is one quarter of the edge of the cube shown in fig. 2, a cube which contains a volume associated with eight atoms of carbon. The (100) planes contain alternately alk blacks and all whites; and when both blacks and whites are carbon atoms they are necessarily exactly similar. When the blacks represent oxygen tetrahedra of the one orientation and the whites tetrahedra of the other, the planes are still effectively identical. A (100) plane divides a tetrahedron so that two oxygen atoms lie on each side of it. All four are at the same distance from the plane, though the pair on one side is not the reflexion of the pair on the other. Hach (100) plane of carbon atoms becomes, therefore, a pair of oxygen planes, and whether the original plane contained blacks or whites these sets of two are indistinguishable from an X-ray point of view. If they were not we should have a new spacing double as great as the old, and the (100) spectra would be interleaved by new ones bisecting the spaces between those of the diamond. We have nothing of this sort to account for, and the tetrahedra therefore still satisly the conditions required. ; As regards the (110) planes no difficulty arises because each (110) plane contains blacks and whites in equal numbers, and the spacing of the (110) planes does not the Spinel Group of Crystals. 309 depend on whether the blacks and whites have the same or different significance. As regards the (111) planes, the spacing in the case of the diamond (loc. cit. p. 100) is the distance from one plane containing blacks only to the next that contains blacks only; the planes containing only whites lie between those con- taining blacks and divide the spacings of the latter in the ratio 1 to3. Whether or not the blacks and whites represent the same things again makes no difference in the spacings. It makes a difference in the intensities, of course: there is no second order spectrum in the diamond because the blacks and whites represent the same carbon atoms and there is perfect interference (loc. cit. p. 103). In zinc-blende they do not, and the second (111) spectrum is only partially destroyed (loc. cit. p. 98). In the present case the blacks and whites do not represent the same thing as regards the (111) planes, and so the second (111) spectrum remains as in the case of zine-blende ; but we are coming to this point presently. Thus the replacement of the carbon atoms by oxygen tetrahedra does not interfere with the spacings of the (100) (110) and (111) planes, and we still have them of the same description as those of the diamond, which is in agreement with our experimental results. The placing of the oxygen atoms has been completed satisfactorily. The size of the tetrahedron remains undetermined as yet. Let us next consider the iron atoms. We have three, of which chemical considerations would distinguish one as divalent from the other two as trivalent. ; Symmetry considerations are in agreement with such a division. It is not possible to place three atoms round a point so as to have all the symmetries required. We may increase the three to six, provided that we so place each iron atom that it is shared equally by two oxygen tetrahedra, and that each oxygen tetrahedron has shares in four iron atoms. Jt does not seem possible to do this; nor do extensions of the number to nine, twelve, and so on seem to offer a solution. In any case, such a disposition would make no difference between the divalent and trivalent atoms. In the spinel MgAl,0, the magnesium atoms must surely be placed in some different way to the aluminium or else we should, in securing the trigonal symmetries, be obliged to assume that Mg and Al behave alike to the X-rays, which would be contrary to all our experience. Let us, therefore, take the divalent and trivalent irons separately. The most simple and obvious place for the single 310 Prof. W. H. Bragg on the Structure of divalent iron atom is the centre of the oxygen tetrahedron. It might of course be multiplied by four, provided each was placed at equal distances from four oxygen tetrahedra and each such tetrahedron had shares in four iron atoms; but this comes to the same thing as the first suggestion, since, owing to the peculiar diamond structure, a point which is equally distant from four oxygen tetrahedra is itself the centre of a tetrahedron. 7 We have two iron atoms left. These cannot be assigned entirely to a single tetrahedron so as to have the symmetries required. But if they are increased to four, each being placed on one of the four perpendiculars from the corners of the tetrahedron on the opposite faces, and halfway between two tetrahedra so as to be shared equally by them, every condition of symmetry and spacing is fulfilled, for it has already been shown that a properly oriented tetrahedron, such as that on which the trivalent atoms now lie, may be substituted for a single point centre without degrading the Symmetry or increasing the spacings. We conclude that this is the real position of the trivalent atoms, and the structure is complete, except as regards (1) a knowledge of the dimen- sions of the tetrahedron and the determining whether two neighbouring tetrahedra point towards or away from each other, (2) a choice between two alternative positions of the iron atom which we will discuss immediately. Fig. 3 a. a Hi cir ’ hans ; . : 1 4 S u es t : e . 1 x G ----29- a ‘ A ‘8 hay TT otReS -----. ° az we ioe ee. Qe ao a Arrangement of atoms on diamond. ABHC trigonal axis, AB=152A.U. BH=4:56A.U. A very good way to realize the stage we have reached is to consider a small portion of the crystal consisting principally of atoms lying along a trigonal axis. Fig. 3a shows four such atoms of carbon in the case of the diamond, viz., A, B, the Spinel Group of Crystals, alt H,and ©. J, K, and L are the three other atoms which lie at the same distance from B as H does. H, J, K, and L form a regular tetrahedron. The length of AB is 1°52 A.U., and of BH 4:6 A.U., being three times as large as AB. The (111) spacing is not shown directly in the figure, but is four thirds of AB: a glance at the photograph of the model shown in ‘X-rays and Crystal Structure,’ p. 107, will make this clear. In magnetite the distance AB becomes 3°60 A.U., and BD 10°80 A.U. In fig. 36 divalent iron atoms have replaced the carbon atoms at A, B, H, J, and K; L is not shown. The oxygen tetrahedra are shown, and it will be observed that there are two orientations, A, H, J, and K being of one description and B of the other. The former replace atoms belonging to one of the face-centred lattices of the diamond, the latter corresponds to the other. So far as our arguments have gone, the size of the tetrahedron has not entered into consideration: it might be of any size, but is drawn small enough to be clear of other atoms. The structure seems easier to grasp when this is done. We shall come presently to arguments which relate to the size, but they are based on quite new considerations. We may anticipate so far as to say that the oxygen tetrahedron is so much larger than it is drawn in the figure that it takes in five divalent iron atoms instead of one*. Probably it is not the same size in all the crystals of the spinel group. Trivalent iron atoms are shownat D,E, F,andG. It will be observed that the positions of the divalent and trivalent atoms are essentially different. JB lies inside a tetrahedron, D lies between the bases of two tetrahedra and has connexions with six oxygen atoms, whereas B has relation to four. This difference is quite independent of the size of the tetrahedron; if the latter is enlarged, other oxygen atoms than those shown move toward the iron atoms and the neighbours change. But it always remains true that the trivalent atom has three neighbours to the divalent atom’s two. Let us now take up the points we left undecided. The trivalent iron atom might be halfway between A and B or halfway between B and H, the latter being the position adopted in the figure. The reason for placing it as shown is derived from considerations of the (111) spectra, and in * Note added subsequently. I see that the disposition of the oxygen atom here described as consisting of certain large tetrahedra pointing towards each other may with equal exactness be described as consisting of smaller tetrahedra each containing only one divalent iron atom and pointing away from each other. 312 Prof. W. H. Bragg on the Structure of Fig. 36.—Diagram illustrating structure of magnetite, Oxygen tetrahedra shown correctly as regards form and position, but not as regards size. © Divalent iron atom. @)Trivalent iron atom. © Oxygen atom. AB= 3°60 A.U. BD=DH=BE=EJ=BF=FK =BG=5-40 4.U. BH, BJ, BK, BG are trigonal axes. the Spinel Group of Crystals. 313 particular the fact that the first order spectrum is so very small. We gather from this fact that two sets of planes occur alternately at half the actual (111) spacing, and nearly balance one another. If we place the trivalent atom halfway between every pair of divalent neighbours such as A and B, we obtain the two sets of planes we want, but one set is so much stronger than the other that it would be hard to explain the weakness of the first order reflexion. If we place it as shown, viz., halfway between B and H, we again obtain the two sets of planes (it is rather hard to see without a model), but one consists of a single plane with three atoms in it, and the other three planes fairly close together each with a single atom. Thus they nearly balance, and so we make the first order reflexion small. In all this we are neglecting the oxygen atoms, it is true. But they are so much lighter than the iron atoms that it seems unlikely that they can make much difference. Let us next consider the size of the oxygen tetrahedron. So far as we have gone we have made such a distribution of the atoms in the crystal that the distribution in the (111) planes will be found to be as shown in fig. 4; the distance Fig. 4. epee 30fe" oreo Fe'30 3 Fe” from 3}e'” to 3Fe’” being 4°80 A.U. The constituents of two molecules are employed to represent the arrangement. We may represent the effect of such a distribution in the manner employed by W. L. Bragg in previous papers. Take the central Fe’ plane as the zero plane. We have then (1) The central Fe’”’ plane. (2) Two Ie’ planes having phases 7/4 and —7/4 respec- tively. (3) One 3Fe’” plane having a phase 7: the other is only a repeat of the first. 314 Prof. W. H. Bragg on the Structure of (4) Two planes containing three oxygen atoms each and having phases 7/4+a and —7/4—a respectively. If a is determined the size of the tetrahedron is known. (5) Two planes containing one oxygen atom each and having phases 7/4—3e and —7/4+ 3a respectively. The combination of the whole gives the same effect as that of a single plane having a weight Nir 56+ 112 cos a + 168 cosn7+ 32 cosn (T — 32) + 96 cosn (7 = where n is the order of the spectrum. We here assume what is no doubt an approximation only, that every atom contributes to every order of reflexion in proportion to its weight. In order to find a value for « let us first examine the effects of the iron atoms alone. Putting n=1,2,3..., the first three terms give us the following series of values:— — 34, 224, —190, 112, —190, 294, —34, 336. We have to allow for the rapid decline in intensities as we proceed to higher orders, which seems an invariable effect. In the Bakerian lecture for 1915 it is ascribed to the fact that the centres scattering the X-rays are distributed through the volume of the atom and are not concentrated in one point. In general, we find that the intensity falls off in- versely as the square of the order, where the reflecting planes are similar and similarly spaced. We may approxi- mately allow for this fact by multiplying the experimental results, each by the square of its order, and then comparing them with the figures just found by calculation. The experimental figures (see fig. 1) are 6 28°53) -48 yom 256 02"S (es Aree which then become 6. 114” 432) 32007" 315,584 14 ees There is a steady rise to the 3rd or 4th followed by a steady fall and a large rise to the 8th. Comparing these with the calculated figures due to the iron alone, we see that the latter are at fault in having the 2nd and 6th too large and the 4th too small. We shall want all the influence we can get from the oxygen the Spinel Group of Crystals. 315 atoms in order to make these alterations, which we do by making the tetrahedra at A and B point towards each other and putting 4e=7. The oxygen terms in the formula then become simply 8x 16 cos or, in the different orders 0 —128 0 +128 0 —128 O +128 Combining these with the effects due to the iron atoms we get the calculated series —34 96 —190 240 —190 96 34 464, which has the proper rise and fall, though the maximum is actually at the 4th instead of between the 3rd and the 4th. The 8th is large as it should be. It is too large in fact: but that need not trouble us because the spectra of high order seem to diminish even faster than the inverse square of the order. At this stage the plus and minus signs have no significance. When a is put equal to 7 the structure becomes much simpler. The two oxygen tetrahedra round A and B in fig. 2 should be supposed to grow until the two corners on the line AB pass each other and finally lie each on a face of the other tetrahedron. The oxygen atoms then all lie on two planes. When a model is made it is seen that each trivalent iron atom lies at the centre of a regular octahedron of oxygen atoms and each divalent at the centre of a tetra- hedron of oxygen atoms. The (100) planes consist of FeO, planes having the spacing 2:07 A.U. interleaved with Fe planes. This makes the second order spectrum more intense than is normally the case and agrees with experiment (see fig. 1). The (110) planes may be looked on as consisting of Fe,0, planes of spacing 2°94 A.U. interleaved with FeO, planes. The near equality of these two kinds of plane will readily account for the fact that the first (110) reflexion is so very small (fig. 1). The results for spinel MgAl,O, work out in exactly the same way, though it looks as “iE the oxygen tetrahedron was not quite so large. But these details may be left to be discussed hereafter. eae XXXII. The Lffect of Electric and Magnetic Fields on the Emission Lines of Solids. By C. E. MENDENHALL and R. W. Woop*. ; [Plate IV.] S° little is yet known about the cause and real nature of the Stark-Lo Surdo effect, that a search for it under any experimental conditions different from those used by its discoverers is certainly worth while. ‘This is the more true since Paschen + failed to detect any electric decomposition of the Hg 2536°7 line excited by resonance according to the method used by one cf the present writers{. It is true Paschen’s method was indirect, but it should be extremely sensitive as a qualitative test, Paschen’s negative result raises the question which started the present work, namely, is the absence of any influence of an external electric field due in this case to the nature of the vibrating system giving this line, as compared with the system giving other mercury lines for which the effect has been demonstrated §, or is it due to the different method of excitation? For, with this exception, all sources so far examined for the Stark effect have been excited by canal-ray bombardment, and it seemed possible that this mode of excitation might be a necessary, though not a sufficient, condition for obtaining the effect. It was therefore proposed to look for the Stark effect in solids, both with fluorescent emission lines and with absorp- tion lines, in the hope of finding an “inverse” effect. Incidentally, some other aspects of these fluorescent and absorption lines were examined, and the results are included. It should be said at once that, as regards the Stark effect, the results have all been negative ; but it seems to us that negative results in this connexion are of sufficient interest to warrant a description of the conditions under which they were obtained. Apparatus.—Two plane grating spectrographs were used from time to time, one of 1 metre focus using about 4 inches of ruled surface, 15,000 lines per inch, the other 3 m. focus and 6 in. ruled surface, in each case in the 1st order spectrum. The definition and light efficiency of these instruments were excellent ; without exposing for more than an hour the slit could be used in every case so narrow that resolution was limited by the physical width of the lines. All but a few * Communicated by the Authors. + Paschen & Gerlach, Phys. Zeit. May 15th, 1914. t Wood, Phil. Mag. April, 1913, p. 433. § Wendt & Wetzel, Phys. Rev., Nov. 1914, p. 549 (abstr.). On the Emission Lines of Solids. 317 of the observations were photographic ; with the smaller instrument the D lines appeared about 1 mm, apart, with the larger instrument about 3 mm. The electric fields were applied by means of a motor-driven Wimshurst machine, and measured either by the equivalent spark-gap between brass balls in air, or by an electrostatic voltmeter up to 10,000: volts. The observations were carried out at liquid-air temperatures whenever possible, because of the greater sharpness of the lines. For work in liquid air, the specimens were pressed between two parallel brass plates about 12 mm. in diameter, and held apart 2 mm. or 3 nm. by the specimen. A 1 mm, hole through the centre of each plate served to transmit the light in the absorption tests. In the fluorescence tests the exciting light was focussed from the side, and the excited light taken off through one of the holes. The first line examined was the strong sharp fluorescent line X 5736, which is part of a rather complicated spectrum emitted by a certain specimen of Weardale fluorite, studied and described with great care by Morse *, through whose kindness we had a specimen available. As this line is most strongly excited by ultra-violet light, and there was no quartz Dewar flask available, observations were made at room- temperature only, in the case of the electric field. Tinfoil electrodes were waxed on the opposite faces of the crystal ; they were about 4°5 mm. in diameter and 13 mm. apart (the thickness of the crystal), so that the field was by no means uniform. However, the difficulty of insulating was con- siderable, and this arrangement seemed good enough for a qualitative test. Between these electrodes a_ potential difference of about 50,000 v./em. was maintained, the fluores- cence transverse to the field being examined. ‘The line was excited by light from a heavy spark between mg. electrodes, and appeared slightly wider than one of the D lines on the plates, taken with the same slit-width. The electric field produced no change whatever in the appearance of the line. As is usual in such cases the fluorescent light is slightly polarized ; but examination showed that the electric field produced no change in the amount of polarization. In order to still further study the sensibility of this line to external influence, it was examined for the longitudinal Zeeman { effect, at liquid-air temperatures. For this purpose the crystal was mounted in a cell in one end of a half-inch * H. W. Morse, Astrophys. Journ. xxi, p. 83 (1905). + The Weiss magnet was very kindly loaned by Professor Richards, of the Department of Physics, University of Pennsylvania. 318 Mr. Mendenhall and Prof. Wood on the Effect of copper rod, which dipped for most of its length in liquid air. Glass cones waxed into holes in this cell allowed the (transverse) entrance of the exciting light and the exit of the excited light through a polepiece. These cones were closed with quartz and glass windows respectively, far enough from the cold copper to avoid condensation of moisture. The line is very considerably sharpened at low temperatures, becoming nearly as narrow as the D lines of a flame very poor in sodium. Not the slightest trace of broadening or splitting could be observed, however, with a field of 22,000 gauss. An attempt to detect any sharply selective absorption at this same wave-length, both with and without fluorescent excitation, resulted negatively ; but as this was for only one transmission through about 13 mm. of fluorite, it cannot be taken as conclusive against the existence of a corresponding absorption line. A subsequent experiment in which the path was increased to 36 mm. by multiple reflexions gave also negative results. If the line represents a normal mode of vibration of an electronic system, absorption would cer- tainly be expected, as well as response or resonance to exciting light of the same wave-length. Since the line does not form part of any obvious group or series of lines, one’s expectation would be to find absorption and resonance in case a sufficient number of vibrators were available, 2. e. a sufficient thickness of crystal. The next lines studied were the brilliant red fluorescent pair of the ruby, % 6918, 6932, which are excited by a wide range of visible and ultra-violet wave-lengths, but most strongly by the yellow-green. These lines are much wider than the fluorite line at room-temperatures, but are sharpened more by lowering the temperature, so that in liquid air the difference is not so great, though they never become quite so narrow as the fluorite line at—185° C. A field of 45,000 volts/em. produced not the slightest change in the lines. In this case there is a sharp absorption doublet of the same wave- length, but this was also quite uninfluenced by the field. In addition to the narrowing of the lines by the low tem- perature there was a shift of 12 A.U. towards the violet, somewhat greater than that noted by Du Bois and Elias. The wave-len oths of these lines, determined from the neon comparison spectrum (neon line 6929°8) are, at—180°, 6920 and 6934. r) ' Da ; \ : * ’ . " f , : SPC tne 6a eae iy fate ne ste Wp Raa Tey —s Nt nt 7 if ve , vee tat at rt > xe lz ae SS. Sat Urwin . cee ; : ES re ; . xi ms ; a ‘ ry = $ ; - . é on k - Fi; ‘ iy * ; ‘ \ ; j £ . aoe , t aN al ie 6% o, Ser. 6, Vol. 30, Pl. IL. Phil. Mas —_* ~ Woop, Phil. Mag. Ser. 6, Vol. 30, Pl. III. Fig. 3. Fra. 4. Menpeynanrt & Woop. Phil. Mag. Ser. 6, Vol. 30, Pl. TV. 2 =} 0°: Fig. l. -/80° | Fig. 2. H=22,000 THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. Fe en Fe [SIXTH SERIES] ag \ s 'T Y + . oN SEPTEMBER Wi, LB Re “YY acct Vf XXXII. The Mobility of Negative Ions at Low Pressures. By Sir J.J. TaHomson, 0.1, £.R.S.* POM the experiments made by Franck and Pohl on the mobility of the negative ions in argon and_ nitrogen carefully freed from oxygen, and also from the very interest- ing investigation on the mobility of the negative ions in air, recently published by Wellisch+,it seems clear that the electron can traverse in a free state distances which are large compared with the mean free path. In a recent communication to the Cambridge Philosophical Society, I showed that this result would explain many of the peculiarities shown by the negative ion at low pressures, such, for example, as its abnormally large mobility and: the lack of proportionality between its velocity and the electric force acting upon it. In this paper I wish to find expressions for the magnitude of effects due to this cause and the factors on which they depend. The case considered is when the charged particles are moving under a uniform electric field through gas enclosed by two parallel plates at right angles fo the lines of electric force: this corresponds to the conditions under which the mobility of the negative ion is usually investigated. All the negative iuns are supposed to begin as electrons, though some of them ultimately unite with the molecules of the gas and become negative ions. It is probable that when * Communicated by the Author. y+ American Journal of Science, May 1915. Phil. Mag. 8. 6. Vol. 30. No. 177. Sept. 1915. bd 322 Sir J. J. Thomson on the Mobility of once the electron has become attached to a molecule it stays so for a time which is large compared with its life as a free electron ; for the sake of simplicity we shall suppose that when once the electron is attached to a molecule it reaches one of the parallel plates and is removed from the field before it gets free again. | Let X be the electric force between the plates, k, the mobility of the electron, then /,X is the velocity of drift of the electron : we assume that this is small compared with V, the velocity of the electron due to thermal agitation. We shall suppose that when an electron collides with a molecule the chance of its uniting with it and forming a negative ion is L/n. We shall first calculate the expectation of an electron traversing a distance w parallel to the electric force without becoming attached to a molecule. The time taken by the electron to traverse this distance is v/k,X, and in this time, since V is assumed to be large com- pared with 4,X, it passes over a space which is approximately VealkyX. We shall write « for V/k,X, so that this space is equal to ax. The expectation of the electron travelling over this space without a collision is e~%7/*, where is the mean free path of the electron, and this is the expectation that it should pass over this space without making a collision and without uniting with a molecule and becoming a negative ion. The expectation that it makes one collision, but no more, in passing over the space # may be found as follows. The expectation that it passes over a distance € without a collision ise“; the expectation that it should make a collision between & and &+dE is adé/X ; and the expectation that it should make the rest of the journey without a collision is e-(@—-5)/A. Hence the expectation that it makes but one collision, and that between € and €+d€, is it e~ *5/ (ad E/N)e~ 0 — S)/A or x ead E, Hence the expectation that it makes one and only one collision in the whole journey is x { c-asn ade or en yay, 0 \ The expectation that this collision does not result in recombination is (n—1)/n. Hence the expectation of the Negative Ions at Low Pressures. 323 one-collision electron traversing « without becoming a negative ion is n—l1\ ax a7 ti), %| (eae, | viele oo ( n r The expectation that the electron makes two collisions and no more is Xr for e~%°/\(a£/n) is the expectation that it should have made one, and only one, collision before reaching &; ad&/d, the expectation that it should make a collision between & and &+dé; and e~ (—5)/\, the expectation that it should not make a collision in the rest of its journey. The value of the integral is {eanat : a +e a(e—8))R, 2 Le-a2/rf 20 et ) The expectation that neither of the collisions should result in union is ((n—1)/n)’?. Hence the expectation for a two- collision electron without recombination is 1 /n—1 ax\? -—az/Xr hia 5 ( n =) Similarly the expectation for a three-collision electron without union is oe er e7ar/d + (” 1 =) ee ae and so on. Hence, since any electron must have made collisions varying in number from nought to infinity, the expectation of the electron traversing the distance w in a free state is Beet am 5("— =) "(oe ) } i { i n + 3( Re X +913 n NX ety —_-At/h (n—1)ar/nr _ --a2/mr, Thus if N electrons start from #=0 the number which traverse 2 without recombination is Ne~#/”, In experiments on the mobility of ions the quantity measured is, in many cases, the charge of electricity received by one plate during an interval T when a layer of gas close to the other plate has been ionized at the beginning of the be 324 Sir J. J. Thomson on the Mobility of interval by a flash of Rontgen rays. If d is the distance between the plates, it is evident that no charge at all will be received until T=d/k,X, and that apart from the loss by recombination all the negative ions will have reached the plate when T=d/k,.X, where kis the mobility of the negative ion, k, that of the electron, and X the electric force between the plates. When T is between these limits the charge which reaches the plates can be readily calculated by the aid of the preceding expression... Let us consider the case of a negative particle which takes the whole time T to get across, travelling through a distance 2 as an electron and d—# asa negative ion. We have Ci Oa Ay hh hy X i ks d—kXT (L—Agfhy) All the molecules which reach this distance « without combining will be in time to give up their charge to the plate, while those which combine before reaching this distance will be too late. Thus the charge Q received by the plate will be the charge carried by those electrons which travel this distance without combining, and this by the preceding expression is equal to as Ais (d—k,XT) Nee (i= hefty)’ where e is the charge on an electron and a= V/k,X. We see that QQ is a function of T, of the electric force, of the distance between the plates, of the pressure (since » and the k’s vary inversely as the pressure), and of the temperature through V. Let us consider how Q varies when any one of these quantities changes, the others remaining constant. Variation of ( with T.—Considered as a function of T, Q is zero until T=d/k,X and becomes constant when T equals d/k,X ; between these limits it varies as e", where b= Vko/nr(ky at ky). The graph representing the relation between Q and T is shown in fig. 1. By means of the points P and Q we can determine 4, and ky, while w can be determined by the equation log (Qi/Q2) = w (Ti1—Ts), where Q; and (, are the values of Q corresponding to the C= Negative Ions at Low Pressures. 325 intervals T, and T,. When “,, ko, and wu are known we can determine the value of nd. Fig. 1. Q o/h, X o/ kX Variation of Q with X.—Considered as a function of X, pd ; Q varies as pea The smallest value of X which can send any charge to the plate is d/kT, and when X=d/k,T, Q reaches the maximum value Ne. The Q and X graph has a point of inflexion when X= pd/2k,== Vd/2nr(ky— ke). The shape of the graph is shown in fig. 2. From the phd 7) Fig. 2. p o/k,T kT points P and Q the values of k, and ky can be determined, and w, and therefore nX by means of the equation log (Q/Q) = 4°(g-— x} where Q, and Q, are the values corresponding to X; and X, respectively. 326 Sir J. J. Thomson on the Mobility of Variation of Q with the pressure.—Since ky, ko, and ® are all inversely proportional to the pressure p, Q will be of the form ¢~"?(—-7), Q will therefore be very small until p diminishes so as to be comparable with B-?. Atthis stage it will rapidly increase as the pressure diminishes, and there will be be an appreciable number of ions crossing with a mobility greater than kz, and thus possessing an abnormally large mobility. Since @ is proportional to d/X, the ab- normality will set in at higher and higher pressures, as d is diminished and X increased. The apparent mobility will thus depend on the electric force, and also it will no longer be inversely proportional to the pressure. bean Considered as a function of d, Q varies ase *“4—42) and thus diminishes exponentially with d. Since V varies as 1 6: where 0 is the absolute temperature, Q will vary as « //". It is probable that the electron would more readily escape from a molecule against which it struck when its kinetic energy is large than when it is small, so that we should expect n to depend upon @; n and @ increasing together. It will be seen from the preceding results that the mobility of a negative particle, as measured by its average velocity over a given distance when acted on by unit electric force, is not a definite quantity ; some particles have one mobility, others another, and particles can be found possessing any given mobility, provided this is between k, and k,. When, however, the conditions are such that dV/nAk,X is large, as it will be unless the pressure is so low that nX is com- parable with d, the number of particles having mobilities appreciably greater than k, is exceedingly small, so that in this case the mobility of the negative particle is substantially definite. | The number of ions which have a mobility greater than some given value K can easily be calculated as follows. If the electron travels a distance at least equal to # before uniting with a molecule to form a negative ion, the time taken to traverse the distance d will not be greater than The average speed will therefore not be less than aX : ax/ky + (d—2) [ky ’ so that K the mobility will not be less than d ak, + (d—a)/[ky Negative Ions at Low Pressures. 327 The value of « which makes this expression equal to K is given by the equation mie d(1—ks/ KK) ~ (L—ko/ky) The number of particles which have a mobility not less than K is the sameas the number of electrons which travel a distance x without uniting with a molecule, and by the preceding expression is equal to _4(1—h/K)_V_ Here (1—sky/ky) kyX If we take as a measure of the mobility of the negative particles a mobility such that half the particles have mobilities greater and half less than this value, then K must: make this expression equal to N/2, and therefore aed ar ay OE 1s f Cua eek ee or K—k, ere, X(k eer) ty el ied pA ana ead K ae Vv Since Xk,/V is the ratio of the speed parallel to x to the total speed of the electron, it is never greater than unity, so that K will not differ appreciably from k, until 1A is com- parable with d. When md is large compared with d, and Xk; smaller than V, K will increase with X, and in this case the mobility of the negative ion will increase with the electric force. We have supposed that there was a flash of ionization at the beginning of the interval T. If, however, the ionization is continuous over this time and if g electrons are produced per second, the number reaching the plate will, if ‘I’ is less than d/k,X, be equal to T—d/k,X { ge7(@-*:X(T—1)) ay 0 7 (<—td—h.T) _ ¢-wi(1— B)) BH Ry din cs able) v or where v= V/k,XndA(1—ky/hy). If T is greater than d/k,X, then all the ions emitted from 328 The Mobility of Negative Ions at Low Pressures. t=0 tot=T— oe reach the plate, so that the total number ga reaching the plate is T—djk,X (t-7x)+4 ge (d—EX(E— 9) a Q+ T—d/k,X i ae ( ~- ud (1-2 (ies ro aa saith A). If we regard T as constant, then Q the quantity reaching the plate will be represented by .(@) from X=d/h, 0a X=d/koT and for greater values of X by (8). The graph representing the relation between @ and X will differ from that represented in fig. 2,in that it will start from zero when X=d/hk,T and will not reach a constant value when X=d/kJT, but will asymptotically approach the constant value gT. In comparing the preceding theory with experiment it is necessary to remember that we have supposed all the negative particles to leave one plate as electrons; we could not directly apply it to sucha case as that tried by Mr. Wellisch, when the gas was ionized behind the plate by polonium and the negative ions driven through holes in this plate by a small electric field. With this arrangement some of the electrons may unite with the molecules before they get through the plate, and so start on their journey as ions “and not as electrons; it is possible too that some methods of ionization may produce negative ions initially as well as electrons. It would be necessary to test the theory to produce the electrons at the surface of the plate, either by incandescent metal or by allowing ultra-violet light to fall on the plate. We have at present, except in the case of flames, no direct determinations of k; the mobility of an electron : these, how- ever, would be of very great value and would enable us to settle many points in connexion with the theory of the mobility of ions which are at present uncertain. The advantage possessed by the electron is that there is no uncertainty as to its nature: it is the same whatever may be the nature, temperature, or pressure of the gas. The ion, on the other hand, whether positive or negative may vary with each or all of these conditions. This uncertainty as to the size of the ion makes the evidence as to the nature of the action between ions and molecules, afforded by such pheno- mena as the variation of their mobility with temperature, pressure, aud the nature of the gas, ambiguous. [ 329 } XXXII. On the Stability of the Simple Shearing Motion of a Viscous Incompressible Fluid. By Lord Rayuricn, 0.M,, FR S.* A PRECISE formulation of the problem for free in- finitesimal disturbances was made by Orr (1907)f. It is supposed that ¢ (the vorticity) and v (the velocity perpendicular to the walls) are proportional to e’” e**, where n=pt+ig. If y?v=8, we have 2S ; Ge =P +, (et hey) }8, nie) Ae) ars ©) with the boundary conditions that v=0, dv/dy=0 at the walls where yis constant. Here v is the kinematic viscosity, and 8 is proportional to the initial constant vorticity. Orr easily shows that the period-eqnation takes the form J Si el dy. JS. e-” dy— |S, e- dy. | S.e%dy=0, (3) where §,, 8, are any two independent solutions of (1) and the integrations are extended over the interval between the walls. An equivalent equation was given a little later (1908) independently by Sommerfeld. Stability requires that for no value of & shall any of the gs determined by (3) be negative. In his discussion Orr arrives at the conclusion that this condition is satisfied. Another of Orr’s results may be mentioned. He shows that ptkBy necessarily changes sign in the interval between the walls t. In the paper quoted reference was made also to the work of v. Mises and Hopf, and it was suggested that the problem might be simplified if it could be shown that g—vhk? cannot vanish. If so, it will follow that g is always positive and indeed greater than vk’, inasmuch as this is certainly the case when B=0§. The assumption that g=vk?, by which the real part of the { } in (1) disappears, is indeed a con- siderable simplification, but my hope that it would lead to an easy solution of the stability problem has been dis- appointed. Nevertheless, a certain amount of progress has * Communicated by the Author. t+ Proc. Roy. Irish Acad. vol. xxvii. t Phil. Mag. vol. xxviii. p. 618 (1914). § Phil. Mag. vol. xxxiy. p. 69 (1892); Scientific Papers, vol. iii. p. 583. 330 Lord Rayleigh on Stability of Simple Shearing been made which it may be desirable to record, especially as: the preliminary results may have other applications. If we take a real 7 such that p+ kBy= “Qe B)im, 7. we obtain aS dn? = SinS. ° ° ° ° ° - (5): This is the equation discussed by Stokes in several papers*,,. if we take e& in his equation (18) to be the pure imaginary i. The boundary equation (3) retains the same form with: e" dn for e dy, where MS WPI BA. 8 In (5), (6) 7 and X are non-dimensional. Stokes exhibits the general solution of the equation as oes in two forms. In ascending series which are aiways con-. vergent, ae) Voge 989 S= Alt peed ee ie vt Q2 47 Q3 10 BY pi Auishens VG LL ed \i +Blet+y atsarettsaeTaiot of: The alternative semi-convergent form, suitable for calcula— tion when @ is large, is +~928=0° 0 2 UY MON 1 14443? "1.9 14a aly Dy eae D Ey, 1 1.2.3. 144829 Aas gordi [ 1.5 Toe dualal aie ve 1.14423? * 1.2. 14423 ty 11.13.47 einee toby (9) S=Ca7 e228? J 1— Xe 1.2.3. 144329/2 in which, however, the constants ( and D are liable to a * Especially Camb. Phil. Trans. vol. x. p. 106 (1857) ; Collected Papers, vol. iv. p. 77. Motion of a Viscous Incompressible Fluid. 331 discontinuity. When 2 is real—the case in which Stokes was mainly interested, or a pure imaginary, the calculations. are of course simplified. If we take as §, and 8, the two series in (8), the real and imaginary parts of each are readily separated. ‘Thus if S,=s, + 7t, 2 Sotto, ° . Signs (10), we have on introduction of 2 n=l 555+ 5356 So are) = ae elias eg te ot £2) Pe atta t beam sag + eT 9 10.18.13 — aay hoa in which it will be seen that s,, s, are even in , while ¢,, ty are odd. When <2 these ascending series are suitable. When n>2,it is better to use the descending series, but for this purpose it is necessary to know the connexion between the constants A, B and C, D. For «=in these are (Stokes) A=n-it(5){ O+De#}, B=3nn(5){ -C+Dems} : j Thus for the first series 8; (A=1, B=0 in (8)) log D= 15820516, C=De*; . . (16) snd tor 8, (A=0, B=1) log D'=1-4012366, —C'=D'e-i®,. (17) so that if the two functions in (9) be called =, and &g, S=Clee De, SH Os,4+DS, -. . (18) These values may be confirmed by a comparison of results calculated first from the ascending series and secondly from the descending series when »=2. Much of the necessary (15) 332 Lord Rayleigh on Stability of Simple Shearing arithmetic has been given already by Stokes*. Thus from the ascendiny series $;(2) = —13°33010, 4, (2) =11°62838 ; $9(2) = — 2°25237, t,(2) = —11°44664. In calculating from the descending series the more important part is 4, since __ 93/2 __9;3/2,,3/2 3/211 —7 + veh alae AS ; 20 “yn i N 2.3/2(1 i). For »=2 Stokes finds J = —14-98520 + 43°81046 j, of which the log. modulus is 1:°6656036, and the phase + 108° 52’ 58'"99. When the multiplier C or ©’ is intro- duced, there will be an addition of +30° to this phase. Towards the value of 8, I find —13°32487+11°630962 ; and towards that of S, — 2°24892—11°44495 2. For the other part involving D or D!' we get in like manner —°00523 —:00258 2, and —°00345—-001702. It appears that with the values of C, D, C', D’ defined by (16), (17) the calculations from the ascending and descending series lead to the same results when n=2. What is more, and it is for this reason principally that I have detailed the numbers, the second part involving &, loses its importance when 7 exceeds 2. Beyond this point the numbers given in the table are calculated from >, only. Thus (n>2) s;+it;=Dn-te N29? | —i( 2.9°?4 2 /8—7/6) re 1p) ,. oe . {aerate 1.2.1447(in)? op oe : 1 3/2 _ 372s) Sotity= —D'n-te V2" eT N20" + 77/84 77/6) 1... 1 sD.2de tam x _ aoa 20 {1 1.144 (im)52* 1.2. 142 (in) h, ee) * Loc. cit. Appendix. It was to take advantage of this that the “9” was introduced in (5). . Motion of a Viscous Incompressible Fluid. 333. the only difference being the change from D to —D’ and the reversal of sign in 7/6, equivalent to the introduction of a constant (complex) factor. TABLE I, q). Si. | te Sy. ee 0:0 + 1-:0000 — -0000 + 0000 + -0000 Ol + 1:0000 = 0015 + 0001 + +1000 0:2 + 1:0000 = “120 + -0012 4+ +2000 0:3 + -9997 — +0405 + :0061 + +3000 0-4 4+ +9982 —~ 0960 + -0192 4+ 3997 0°5 + +9930 1874: + -0469 4. 4987 06 + :9790 — 3234 + -0971 4+ 5955 0:7 + 9393 = h5aea | | 4b) 1969 4+. +6845 0:8 + 8825 — 7605 + 3055 + +7663 0:9 + -7619 — 1:0717 + +4865 + +8234 1:0 + 554 — 1-444 | + 784 “i “B40 11 4+ ‘215 — 2-007 + 1-057 sey 7 oO) 12 et F310 — 2°304 | + 1:456 4+ 634 1:3 =— 1-083 — 2707 | + 1-923 4+. +320 i |) — 2173 — 2-979 | + 2-494 — 22) Py oi 8-685 — 2-972 | 4. 2-893 ~ 1-067 16 — 5-493 | — 2-466 + 3-212 —. 2-308 17 — 7694 — 1161 + 3191 -- 3:998 1°8 —10-057 4+ 1°825 + 2-550 — 6173 1:9 —12:177 4+ 5441 ue Se — 8745 2-0 —13'330 +11-628 — 2-252 —11:447 21 — 12°34 +20°19 — 7°46 —13°70 2-9 — 7-49 +3101 — 15°24 —14-50 2:3 + 3°54 +43:20 | —v5-84 — 12-22 2-4 +23°55 454-54 — 38-90 — 453 25 455-20 +60°44 — 52-70 +11°59 | When 7 exceeds 2°5, the second term of the series within { } in &, is less than 10~*, so that for rough purposes the { } may be omitted altogether. We then have . V2 3/2 3/2 mee wo Pde yy erat) a) es. C2Y) »NQn@? “ss t{=—Dn-te*" sin(,/2.” —7/24),. . . . (22) so al pint 3/2 Sg== D'n-* e sin(,/2.» —7/24—7/6),. . . (23) 9 3/2 3/2 ts=D'n-te 2” cos (\/2.n —m/24—m/6).. . . (24) Here D and D’ are both positive—the logarithms have already been given—and we see that s,, f are somewhat approximately in the same phase, and ¢;, s, in approximately opposite phases. When 7 exceeds a small integer, the 034 Lord Rayleigh on Stability of Simple Shearing functions fluctuate with great rapidity and with correspond- ingly increasing maxima and minima. When in one period / 2.7 increases by 27r, the exponential factor is multiplied by e?” viz. 535°4. From the approximate expressions appli- cable when 7 exceeds a small integer it appears that sj, t, are in quadrature, as also 8p, fp. For some purposes it may be more convenient to take 1, 42, or (expressed more correctly) the functions which identify themselves with %,, 2. when 7 is great, rather than S,, S2, as fundamental solutions. When 7 is small, these functions must be calculated from the ascending series ihusby (15) (C= Do) S,=0T(5)8,~ 30a (5) 8.) eee mane (C= 0), D1) sey DU UN ae Lor (3) e—*/6S, 4 30720" (5) e'™/68., (26) Some general properties of the solutions of (5) are worthy ot notice. If S=s-+it, we have d*s/dn?=Int, d°*t/dn?= —Yns. Let R=34(s?+27) ; then aR _ ds, jae dn "dn dn’ ‘cand aR ds \? MONAie, ds at ae = (an) + (aq) tage tage ‘of which the two last terms cancel, su that d?R/dn? is always positive. In the case of 8,, when »=0, s,(0)=1, t,(0)=0, s;'(0) =0, so that R(O)=4, R’(0)=0. Again, when 7=0, -6.(0)=0, ¢,(0) =0, so that R(O)=0, R’(0}=0. In neither case can R vanish for a finite (real) value of 7, and the same is true of S, and S,. Since (5) is a differential equation of the second order, its solutions are connected in a well known manner. Thus dS, CS Si dr? —S, dn? =i) e e « ° (27) and on integration dS ds Sa —f, de =constant=t, Mian enn ( 3) Motion of a Viscous Incompressible Fluid. 330 ‘as appears from the value assumed when 7=0. Thus See ti dn S, =21 Sy? st W ier th Pei Weare le (he (29) 20 which defines S, in terms of Sj. A simiJar relation holds for any two particular solutions. For example, em ME WN 80) ee The difficulty of the stability problem lies in the treatment of the boundary condition N2 r No oe 2 pe , { S,e-"dn | Soe igo : Sie Man { q Se edn =0. i] y ii Ny in which 7, 7, and % are arbitrary, except that we may suppose 7, and X to be positive, and 4; negative. In (31) we may replace e\1, e—™ by cosh Xn, sinh Xn respectively, and the substitution is especially useful when the limits of integration are such that 7,=—7,. For in this case No Ne i S cosh An dn =2 ( s cosh An dn, nh 9 t F Cia | 6 \ S sinh An dn = 22 | t sinh An dy ; Ny a9 and the equation reduces to n Ma 4 * 5, cosh An dy . ( “ty sinh An dy 0 e/0 -{ sy cosh An dn A t; sinh An dn=0, (32) 0 0 thus assuming a real form, derived, however, from the imaginary term in (31). In general with separation of real * Rather to my surprise I find this condition already laid down in ‘private papers of Jan. 18938. 336 Lord Rayleigh on Stability of Simple Shearing and imaginary parts we have by (31) from the real part fs edn ; (.. oe "dy -{ ty edn ft edn —(s edn As edn + fe e—ndn ; fo edn =(); (33) and from the imaginary part i? {s edn fo edn + \% e "dn Ae edn $e v » a) ES -(s e X11 In { to Mn —| Sp edn te "da=l0: x 2 (34) If we introduce the notation of double integrals, these equations become {{simn (n—7!) { s(n) . 52(79') —#1(9) « to(7’) } dy dy'=0, (35) {\sim M(n—7') | suln) .ty(9') —s9(n) . in!) bay dn' =0, (36) the limits for » and 7’ being in both cases y, and y,. In these we see that the parts for which 7» and 7’ are nearly equal contribute little to the result. A case admitting of comparatively simple treatment occurs. when A is so large that the exponential terms eh eh” dominate the integrals. As we may see by integration by parts, (31) then reduces to Si (42) - 89071) Sim) -So(no) = 0, ee or with use of (29) S; (nn) See Ole en. 8°) We have already seen that 8,(7) cannot vanish; and it only remains to prove that neither can the integral do so. Owing to the character of 8,, only moderate values of » contribute sensibly to its value. For further examination it conduces to clearness to write 7»=a,7,=—b, where a and Motion of a Viscous Incompressible Fluid. 337 b are ae Thus ies dn + Ce ae » (it it)? + a (s,—it)? au os! — ty’) dy i °(sP —t”)dn -2i {* Syt,dn ta 0 (sy? oF belt 0 (s\ ar ty)” 0 (s) + t,”)? CE elroy Bol / : and it suffices to show that { ( enh DLL cannot vanish. 0 (s)’ a t;*)? A short table makes this apparent. TaBLeE II, | alte, wy iGt o ints 8,7—7,7 Sums of 7: §°— ty". (s)°+7,")". (87 0,2)? fourth column. A + 1:000 1:000 +1:000 1-000 3 + 0-997 1-002 + ‘995 1995 3) + 0951 1-042 + 913 2°908 ‘7 + 0°581 1-399 + ‘415 3323 ‘9 — 0°569 2°989 — ‘191 3'132 rl — 3982 16°60 — ‘240 2°892 1:3 — 6155 72°25 — ‘085 2°807 15 + 4:38 485'8 + -009 2816 i oe + 57°9 3660°0 + ‘016 2°832 19 + 119:0 31700°0 + ‘004 2°836 21 — 255:0 314000°0 — ‘001 2°835 2°3 —1854°0 353 x 10-4 — ‘001 2834 2°5 — 616°0 45x 10-6 _ ‘000 2°834 The fifth column represents the sums 7 to various values (s; Toy *) dn (sf +t)? 2x 2°834 or ‘567. The true vals of this integral is (D'/D) sin 60° or *571, as we see from (30) and (19), (20). We conclude that (37) cannot be satisfied with any values of ym, and 7. When the value of X is not sufficiently great to justify the substitution of (37) for (31) in the general case, we may still apply the argument in a rough manner to the special case (72 +71=9) “of (32), at any rate when 7 is moderately great. For, although capable of evanescence, the functions 51, t1, So, tg increase in amplitude so rapidly with 7 that the extreme value of 7 may be said to dominate the integrals. Phil. Mag. 8. 6. Vol. 30. No. 177. Sept. 1915. Z of 7. The approximate value of if is thus 338 Sumple Shearing Motion of Viscous Incompressible Fluid. The hyperbolic functions then disappear and the equation reduces ™ to 81(2) + to(2) — So(M2) - t1(72) =0, . . ~~. (40) which cannot be satisfied by a moderately large value of mp. For it appears from the appropriate expressions (21) ... (24) that the left-hand member of (40) is then DD! ¢2¥2-° cos (17/6), a positive and rapidly increasing quantity. Again, it is evident from Table I. that the left-hand member of (32) remains positive for all values of 4, from zero up to some value which must exceed 1-1, since up to that point the functions s,, s,, tg are positive while t, is negative. Hven without further examination it seems fairly safe to conclude that (32) cannot be satisfied by any values of 7, and X. Another case admitting of simple treatment occurs when m2 and 7, are both small, although X% may be great. We have approximately ii {= -- 3n’, Sg= 3m", t2=7, the next terms being in each case of 6 higher degrees in . Thus with omission of terms in 7’ under the integral sign, (31) becomes (ora An e hn ay—(e™ dn fn e dn=0, (41) or on effecting the integrations M2—M1) sinh A(j2—m) +2—2 cosh A(n2—m)=0. (42) It is easy to show that (42) cannot be satisfied. For, writing A(7,—m) = 2, ee xe esinh gaa Tore + 73 457° wz 6 OMe x ee A Ge every term of the first series exceeding the corresponding term of the second series. The left-hand member of (42) is accordingly always positive. This disposes of the whole question when 4, and 7 are small enough (numerically), say distinctly less than unity. 2(cosha—1)=a?+ * Regard being paid to the character of the functions. Needless to say, it is no general proposition that the value of an integral is deter- mined by the greatest value, however excessive, of the integrand. [ 339 |] XXXIV. Maximum Frequency of the X Rays from a Coolidge Tube for Different Voltages. By Sir ERNEST RUTHERFORD, F.R.S., Protessor J. Barnes, Ph.D., and H. RicHarpson,, M.Sc.* a the course of last year, Mr. C. G. Darwin began an investigation in the University of Manchester to examine the relation between the velocity of cathode rays and the frequency of X rays excited by them in different radiators. ‘The cathode rays were generated by the electric discharge in a suitable vacuum-tube, and by means of an adjustable magnetic field rays of definite speed were allowed to fall on a radiator. It was the intention of Mr. Darwin to examine the frequency of the X rays initially by measuring their absorption in aluminium, and if possible by direct reflexion from crystals. Some difficulty was experienced in obtaining a sutiiciently good and constant vacuum, and measurements were interrupted by the departure of Mr. Darwin to the seat of war. The experiments were continued by Mr. H. Richardson, but in the complicated apparatus employed it was found difficult to keep the vacuum sufficiently constant when a discharge was passed. As soon as the Coolidge tube was put on the market, it was recognized that it afforded a much more convenient method of attacking a part of the main problem and over a much wider range of voltage. As is well known, the discharge in the very perfectly exhausted Coolidge tube is mainly carried by the negative electrons liberated from a tungsten spiral heated to a high temperature by means of the electric current. The anticathode is of tungsten, and the exhaustion in the tube employed in the present experi- ments was so perfect that the tube maintained, with suitable precautions, 175,000 volts across its terminals without any obvious breakdown of the vacuum. Since tungsten is of atomic weight 184, and atomic number 74, the X radiation from the Coolidge tube corresponds to that emitted from a heavy element ; but it is to be expected that the frequency of the X radiation for a high voltage should be somewhat less than from the ordinary X-ray tube with a platinum or platinum-iridium anticathode, since the atomic weights of iridium and platinum are 193 and 195, and atomic numbers 77 and 78 respectively f. * Communicated by the Authors. Tt See Moseley, Phil. Mag. xxvii. p. 703 (1914). Z 2 3840 Sir E. Rutherford, Prof. Barnes, and Mr. Richardson: General method of the Hxperiment. The primary object of the experiment was to determine the maximum freguency of the X rays emitted from a - Coolidge tube excited by different constant voltages. Since the penetrating power of X rays in a light element like aluminium increases regularly and rapidly with the fre- quency, an estimate of the maximum frequency of the radiation can be made by determining the absorption in that metal of the “end” radiation from the tube, 7. e. the ab- sorption of the most penetrating rays present when the rays of smaller frequency have been almost completely absorbed. It has been known for some time that the absorption ccoefiicient « in aluminium of the characteristic X radiation from different radiators is given approximately by »=hA*? where A is the average wave-length of the radiation *. This relation has been recently examined in detail by W. H. Bragg and Piercet by determining the value of mw in aluminium for individual spectrum lines of definite fre- quency, and found to hold fairly accurately over the limited range employed, viz. for wave-lengths between °49 x 107° and ‘615x 107° em., 2. e. for radiations which are reflected from rock-salt between angles of 5° and 6°. There seems to be no doubt that this relation will hold very approximately for the much shorter wave-lengths contained in the more penetrating y radiations emitted by radium B and radium C. Té can be deduced from Bragg’s results as a mean of the measurements on the silver 6 and palladium @ lines that a radiation of wave-length A=5x107° em. hus an absorption coefiicient #=5°6 (cm.)~*> in aluminium. If the above relation between absorption and frequency for aluminium holds, the y radiation from radium C, which has the value #=07115 in aluminium, corresponds to a radiation of X=1:06 x 10-® em., which should be reflected from rock- salt at an angle of 1° 5’. By far the strongest line in the y-ray spectrum of radium C was found by Rutherford and Andrade ¢ to be reflected at an angle of 1° from rock-salt. There is thus a very fair accord between experiment and calculation when the relation is extrapolated over a wide range in the value of w, viz. nearly 50 times. We may consequently assume with confidence that the relation p=kd?? holds very approximately over the whole range #* See Owen, Proc. Roy. Soc. lxxxvi. p. 426 (1912). + ‘X rays and Crystal Structure, by W. H. & W. L. Bragg, pp. 180, 181. a: { Rutherford and Andrade, Phil. Mag. xxviii. p. 268 (1914). Maximum Frequency of X Rays from a Coolidge Tube. 341 of wave-lengths employed in the experiment, viz. from A=14x10-*% cm. toA=1x107° cm. It was hoped at the same time to make a systematic examination of the X-ray spectrum of the radiations, and to determine if possible the voltage at which the spectrum-lines appeared. Barnes in an accompanying paper™ has given the wave-lengths of the spectrum-lines observed with the Coolidge tube. On account of the thickness of the glass of the vacuum-tube, the intensity of the issuing “L” characteristic radiation was weak when examined electrically or photographically, but permitted of the determination of the wave-lengths of the stronger lines. No evidence of other well marked lines was noted in the region of higher frequencies, but the main part of the spectrum appeared to be continuous with the crystals employed. As will be seen later, the experiments on absorption of the radiation showed that the “end” frequency increased regularly with increase of voltage. No evidence was obtained that the maximum frequency for different voltages varied by jumps, such as might be expected if the issuing radiations were mainly confined to a few waves of definite frequency. Arrangement of the Experiment. The energy of the electrons striking the anticathode in the high vacuum of the Coolidge tube depends only on the voltage applied. In order to investigate the effect of elec- trons at definite speed, it was necessary to employ a constant voltage delivered by an influence-machine in place of the variable voltage due to an induction-coil or transformer. For this purpose, the only machine available was a large Wimshurst of 12 plates of diameter 71 cm. ‘This had been presented to the Department about 15 years previously, and was in some respects not nearly so well suited for the experiment as one of the more modern types of high-speed machines. The Wimshurst, of which 10 plates had survived, was run by a motor, and after a month’s fairly continuous running, four more of the plates cracked and were removed. This proved fortunate, for it was found that the machine with six plates gave nearly the same maximum voltage and delivered nearly the same current as in the beginning, and in addition ran much more steadily. In order to reduce the losses to a minimum, all the con- ductors consisted of light metal tubing 2°7 cm. in diameter with rounded ends. These passed through large paraffin insulators to the Coolidge tube T, which was placed in a box * Infra, p. 368. i See ee) AAAANUANAARN lou} ANNI WN AN fs ~ Co} | = W a AUWRRAWAARYE’ Se = ANNA NNN ANANNANNANT 342 Sir E. Rutherford, Prof. Barnes, and Mr. Richardson: covered with sheet-lead 3 mm. thick. The general arrange- ment of the apparatus is shown in fig. 1. Special precautions were taken to prevent losses from the external electrodes of the bulb and to prevent discharges over its surface. For this purpose, the dust collecting on its surface was regularly removed and the surface washed with alcohol. ; Fig. 1. B ANN As the machine after stoppage occasionally reversed its voltage, suitable cross connexions (see fig. 1) were arranged to rectify rapidly the direction of the current. The acces- sories for the Coolidge tube, viz., the battery, adjustable resistance, etc., were placed on an insulated stand and com- pletely covered with a rounded metal case. The current could be controlled by an insulating handle coming through a small opening. The conductors for the heating current passed inside the hollow metal tubes. Measurement of Voltage and Current. In order to determine the absorption curve of the radiation with accuracy, it was necessary to keep the voltage very constant, and to have some method of knowing the voltage at any moment. Tor this purpose, the conductors to the bulb were shunted through a high resistance in series with a galvanometer. ‘The resistance consisted of two capillary tubes, R; and Re, filled with xylol and thoroughly insulated by a thick layer of paraffin. The moving-coil galvanometer G with suitable shunts was placed between the two resist- ances, so that its potential was never far from zero. In Maximum Frequency of X Rays froma Coolidge Tube. 343 order to prevent electrostatic disturbances and electrical losses, the galvanometer was placed on a rounded metal base, insulated on paraffin blocks and covered with a round metal screen, BB. The resistance in series in the galvanometer was about 5000 megohms, and was such that only about 30 per cent. of the total current from the machine passed through it even at the highest voltage employed, viz. 115,000 volts. It was at first intended to measure the voltage across the tube directly by determination of the current through the known high resistance. On account of the charging up of the xylol tubes after the current had been passed for some time, it was not found possible to determine with the requisite accuracy the resistance of the xylol tube under the conditions of temperature etc. when in actual use. The deflexions of the galvanometer were instead standard- ized by the spark method. For this purpose, we made use of a spark-gap composed of large copper spheres,. SS, 20°5 cm. in diameter, constructed by Dr. Makower some years ago. Over the range of voltage employed, the voltage required to produce a spark was practically the same as for parallel plates. The tables employed were those given by C. Miiller, Ann. d. Phys. xxviii. p. 612 (1907). The method of standardization was as follows :—The heating current was adjusted to the required value and the machine was run for five or six minutes at about the voltage required, so that the resistance of the xylol tubes should reach a steady state. The length of the spark-gap was then carefully adjusted, and a number of observations made of the deflexion of the galvanometer at the moment the spark passed, care being taken that the voltage rose slowly to the sparking-point. In a similar way, observations were made at the end of a series of experiments, but no certain change in the deflexion was ever observed over the interval of a few hours. A gradual increase of the xylol resistance was, however, observed over the interval of several months re- quired for the experiments. The deflexion of the galvano- meter was found to be nearly proportional to the voltage over a considerable range. For voltages greater than 30,000, the spark-gap method was very suitable for cali- brating the galvanometer directly. For voltages below this, it was found more convenient and accurate to take the deflexion of the galvanometer as proportional to the voltage. The correctness of this was confirmed on several occasions by determination of the deflexion of the galvanometer for a voltage read directly on a Kelvin astatic voltmeter. 344 Sir H. Rutherford, Prof. Barnes, and Mr. Richardson : As it was very important to keep the voltage constant during an absorption experiment, it was necessary to control the voltage within small limits by means of an adjustable point discharger D placed near one of the high potential conductors. The arrangement for this purpose is seen clearly in fig. 1. The voltage galvanometer worked through- out in a very satisfactory manner, and it was possible by its aid to keep the potential steady within about one per cent. over the interval of an hour or more required for a complete absorption experiment. Since the galvanometer was highly damped, it would not indicate any rapid surges in the voltage. These surges, however, occasionally made themselves manifest at the extreme end of the absorption curve, where the intensity of the radiation had been reduced to about 1/1000 of its initial value. Even when the voltage appeared quite steady by the galvanometer, the presence of surges could be detected by the irregular rate of movement of the electrometer. Measurement of Current. A moving-coil galvanometer E was placed in the main circuit to measure directly the current delivered by the machine, and was protected against electrostatic disturbances by a metal shield FI’, as in the other case. The actual current passing through the bulb was measured in the following way. The deflexion was first observed under the experimental conditions of excitation of the radiation. The heating circuit was broken, and the voltage retained at the same value by means of the point discharge and by altering the speed of the machine. The deflexion rapidly dropped to a constant value, which was due mainly to the current taken by the voltage galvanometer, but partly also to conduction over the surface of the heated bulb. The difference between these two readings thus gave a definite measure of the current passing through the bulb, quite independently of all other losses in the circuits. The current passing through the bulb in the various experiments at different voltages varied from 1/100 to 6/100 of a milli- ampere. It was found in all experiments that the ionization for a given voltage applied to the tube was directly pro- portional to the current through the bulb—in other words, the intensity of the radiation was directly proportional to the number of electrons incident on the anticathode, Maximum Frequency of X Rays from a Coolidge Tube. 345. Determination of the Absorption Curves. The general arrangement of the apparatus for this purpose is shown in fig. 2. The X rays, passing through a rect- angular opening (6x6 cm.) in the lead box, entered the Fig. 2. k iiss} ionization chamber A. This consisted of a lead box (15x 15x15 cm.) 3 mm. thick, divided into two equal partitions by the electrode B. The two insulated plates C and D were generally connected together and charged to a P.D. of 1000 volts, which was found to be sufficient to. produce saturation under the experimental conditions. Two equal openings were cut in the front face, which were closed with a thin sheet of mica to make the vessel air-tight, and covered with thin aluminium foil to make it conducting. Two thick lead slides, SS, were constructed to control the width of the beam of rays entering either half of the ioniza- tion vessel. The current supplied to the electrode B was measured by a Lutz string electrometer, which proved very suitable in all respects for this purpose. The instrument was very easily set up, and sufficient sensibility was obtained when the plates were charged to +15 volts, while the zero. was very steady and the deflexions read with ease. The quartz fibre was broken during the course of the experi- ments, and the later measurements were made with a Kaye- Wilson electroscope. The main difficulty in these experiments lay in the fluctuations of the current through the bulb, and consequent variation in the intensity of the radiation. To correct for this, the radiation before entering the ionization chamber A passed through a “ standardizing vessel” V of thin parallel 346 Sir E. Rutherford, Prof. Barnes, and Mr. Richardson: aluminium sheets. The plates 1, 3 were connected to the high potential battery and the plate 4 earthed. The plate 2 was connected with a mica condenser M of suitable capacity, which could be connected at will through the insulating key K to the string electrometer. The method of conducting an experiment was as follows. By means of the key K, the standardizing vessel was disconnected from the electro- meter and the connexion of the latter with earth broken. A thick lead slide L, which completely stopped the radia- tion, was rapidly drawn aside and the radiation allowed to pass into the ionization vessel for a period varying from 10 to 60 seconds in different experiments. The slide was then closed, and the steady deflexion of the electrometer read. After discharge of the latter, the standardizing vessel was connected, and the deflexion again read. The corrected deflexions for the two vessels should be proportional to each other provided the fluctuations are in the current and not in the voltage ; for variations of the latter alter the pene- trating power of the radiation as well as the current through the bulb. In this way, provided the voltage was kept steady, it was found possible 1o correct for any changes in the intensity of the radiation over the long interval of an hour -er more required to obtain a complete absorption curve. It was necessary to measure the absorption curve, espe- cially for high voltages, over a very wide range of thickness of aluminium where the current in the ionization vessel was reduced in some cases to 1/10000 of its initial value. This was done as follows. An air-condenser F of capacity about 400 e.s. units was placed parallel with the electrometer, and the width of the opening of the ionization vessel adjusted till there was a convenient deflexion of the electrometer in 10 seconds. Successive screens of aluminium were intro- duced, and the currents in the testing and standardizing vessels compared as the current diminished. The air con- denser was removed, and before the current became too small to measure with accuracy, the opening of the ionization vessel was widened until again a convenient deflexion was obtained when the condenser was in the circuit. This process was continued two or three times, depending on the variation in range of the ionization current to be measured. For the end part of the curve, the testing vessel was always com- pletely open to the radiation. In this way it was possible to determine the complete absorption curve over a very wide range without introducing any uncertainty in regard to saturation. The capacities of the circuits were carefully : a Log of /onisation Ger, Maximum Frequency of X Rays from a Coolidge Tube. 347 determined, and the readings corrected for change of capacity of the systems and for inequalities in the electro- meter scale. In the later experiments, an additional method was used in order to correct for changes in intensity of the radiation. This depended on the observed fact that the intensity of the radiation for a given voltage was directly proportional to the current through the bulb. The value of the latter was read by means of the galvanometer during the course of each observation by the methods already described. The values obtained by the two standardizing methods were in good agreement, and each served as a useful check on the other. Haperimental Results. Some of the absorption curves in aluminium of the radia- tion at different voltages are shown in fig. 3, where the Fig. 3. rent ue Pe eee et 15 2 Thickness of Aluminium in mms. logarithm of the current is ordinate, and the abscissx the thickness of aluminium. If the absorption were exponential, the curve should be a straight line, but it is seen that this is not the case for any of the curves. The radiation at first rapidly diminishes owing to the absorption of the softer radiations, and on the average gradually becomes more 348 Sir E. Rutherford, Prof. Barnes, and Mr. Richardson : penetrating with increase of thickness of absorber *. Finally, for the higher voltages, when the current is reduced to about: 1/500 of its value, the absorption curve becomes very nearly a straight line until the radiation is completely absorbed. This shows that the end radiation is approximately homo-. geneous, and it is the absorption coefficient of this end radiation that was carefully determined. From 40,000 volts upwards, the end radiation was ab-. sorbed exponentially over a considerable range of intensity, but below this voltage the penetrating power of the radia-- tion appeared to slightly increase until it was completely absorbed. The analysis of the radiation by reflexion of crystals, as given in the following paper, p. 361, shows that. the ‘“‘ L” characteristic radiation of tungsten escapes from the bulb. Its intensity, however, is greatly reduced by passing through the glass of the bulb, which was found by direct measurement to be about *5 mm. thick. Careful observations were made of the voltage for which the ioniza- tion in the testing-vessel was first measurable. In order to increase the electrical effect, the testing-vessel was filled with sulphur dioxide. No ionization was observed below 10,300 volts, and it then increased very rapidly with the voltage. The absorption of the main radiation by aluminium was examined at the lowest possible voltage, and was found to be w=69 cm.~! in aluminium. This coefficient no doubt corresponds to that of the characteristic “L” radiation of tungsten under experimental conditions. The absorption coefficient of this radiation measured by Chapmanf was. found to be 81, but as in passing through the bulb the softer components were relatively cut out, the issuing radiation would be expected to be more penetrating than that observed under normal conditions with no absorber. The range over which the ionization could be measured increased rapidly with the voltage above 10,000 volts, but on account of the presence of a large proportion of softer radiations, it was. difficult to determine with certainty the absorption coefficient of the end radiation for the lower voltages. In every experi-- ment it was found, however, that the penetrating. power of the radiation increased rapidly and regularly with increase of the voltage. * Preliminary measurements on the absorption of the radiation from a Coolidge tube have been made by S. Russ (Journ. Réntgen Soe. April 1915), using an induction-coil to excite the rays. He observed that the radiations were not homogeneous, but tended to become so with increase of thickness of absorber. t+ Proc. Roy. Soc. A. lxxxvi. p. 489 (1912). Maximum Frequency of X Rays from a Coolidge Tube. 349 Attempts were made to test whether the absorption curves ‘could be analysed into components corresponding to definite characteristic radiations. Kaye* has made numerous ex- periments of this kind to analyse the radiation from different anticathodes, and obtained indications that by the use of suitable absorbers the radiation could be analysed into definite components. While it was not difficult to express the absorption curve for any particular voltage with con- ‘siderable accuracy by the sum of three exponentials, the values of the constants changed with voltaze, and it was concluded that such an analysis, apart from showing the presence of the “L” characteristic, had no physical meaning. It may yet be possible, as Kaye has suggested, to analyse the radiation as a mixture of two or more characteristic radiations, but before this can be accomplished it will be necessary to carry out a large number of experiments on the absorption of the radiation by different materials. It appears to us, however, unlikely that the absorption curves can be completely expressed as the result of a small number of characteristic radiations each of which are absorbed ‘exponentially. Heperiments with Induction-coil, It was not found feasible to examine the absorption curves of the radiation excited by the Wimshurst above 115,000 volts. In order to carry the experiments still ‘further, a large induction-coil which gave a 20-inch spark, operated by a Sanax break, was used. The gap between the sparking spheres was set to the potential required, and ‘the current through the coil carefully adjusted so that an -occasional spark passed. In many cases, a spark-gap ending in fine needle-points was used in parallel with the sparking ‘spheres, to test whether there was any sensible alteration of the potential required to spark across the spheres, owing to possible alteration of their surface by the passage of the preliminary sparks. The current through the heating-coil was adjusted so that the intensity of the radiation was about the same as that ‘excited by the Wimshurst, and the absorption curves in aluminium determined as before. It was anticipated that the radiation excited by the variable voltage of the induction-coil would on the average contain a larger proportion of softer radiation than the radiation ‘excited by the Wimshurst. 1'o our surprise, however, we found that the curves for equal intensities of radiation were * Kaye, ‘X Rays,’ pp. 121-123, Longmaas, Green & Co., 1914. 350 Sir E. Rutherford, Prof. Barnes, and Mr. Richardson: very similar, and it was difficult to distinguish one from the other. Russ (loc. cit.) had observed that the proportion of pene- trating radiation increased with increase of current through the bulb. This, if true, is a very important observation, for it would indicate that the quality of the rays depends not only on the velocity of the electrons but also on their number. In our preliminary examination of this point, results were obtained in general agreement with those of Russ, but the difference between the absorption curves was finally traced to another cause. When working with very intense radiations, it was necessary to nearly close the opening in the ionization vessel by means of the lead slides. Some radiation was found to enter the vessel by scattering from one lead plate to another or by the excitation of charac- teristic radiations. When the front of the ionization vessel was covered with a thick lead sheet and the rays allowed to enter through a small opening, the disturbance was elimi- nated, and the absorption curves were found to be indepen- dent of the current through the bulb over a wide range.., Since a much greater intensity of radiation could be excited by the use of the coil, the absorption curve could be obtained over a greater range of thickness of aluminium. In such cases, the penetrating power of the “end ” radiation appeared to be slightly greater than that observed with the Wimshurst. On account, however, of the uncertainty as to. the equality of the maximum potential in the two cases, not much stress can be laid on this difference ; for it is probable that the minimum voltage required to produce a given length of spark is greater for the coil than for the Wims- hurst, on account of the rapid variations in the potential of the former. Since for the “end” radiation, the coil gave about the same value as the Wimshurst, the former was employed to. determine the penetrating power of the radiation between 110,000 and 175,000 volts. For such high voltages, the end radiation is absorbed nearly exponentially over a wide range. On account of the danger that the bulb might break down under such high voltages, the experiments were confined to an examination of the end radiation alone. In order to detect sinall variations in the absorption of the rays, the experiments for 125,000, 145,000, and 175,000 volts were made in the following way. Two sheets. of lead each of thickness °62 mm. were placed in front of the ionization vessel, and the current determined with the fixed capacity in parallel. A thickness of aluminium 3°24 cm. was Maximum Frequency of X Rays froma Coolidge Tube. 351 added, and the current measured with the capacity removed. An exactly similar process was carried out at 145,000 and 175,000 volts, and the variation of the ratios of the two currents allowed of an accurate measure of the small. change in absorption coefiicient in the three cases. Variation of Absorption of End Radiation with Voltage. The curve showing the values of w in aluminium for the. end radiation at different voltages is shown graphically in fig. 4, where the abscissze represent volts, and the ordinates log w. The experimental points are indicated in the figure, Fig. 4. Absorption of end radiation at different voltages pf} | Se eae Mine Se a Trsecal of Voie. and from the smoothed curve the values of yu Sir different voltages are given in the following table. Taking X=ky?", the values of the wave-length » are calculated, the value of k being deduced from the data of Bragg previously quoted, viz. =5°6 (em.)~? for X=5°0 x 107% cm. It will be seen that the value of uw decreases rapidly at first with voltage, but decreases slowly after 100,000 volts, and very slowly after 125,000 volts. There is no measur- able change in w between 145, 000 and 175,000 volts. During recent years, experimental evidence of various kinds has indicated that the energy of radiation is emitted in definite quanta, expressed in the simplest case by Planck’s relation E=fv, when E is the energy of the ray and y its frequency, and A Planck’s fundamental constant, which 352 Sir HE. Rutherford, Prof. Barnes.and Mr. Richardson : has a value h=6°55x10-*" erg. It is of great interest to see how far such a relation holds for the excitation of X rays by electrons. If all the energy E of the electron can be converted into a single X ray of definite frequency vy, then E=h/y. Assuming that such a relation holds for excitation, we can at once calculate the values of w and the wave-length X to be expected for each voltage. The calculated tables for poand ® are given in columns III. and V. of Table I. The value of e/h=7:27 x10" is taken to calculate the value of X. This is deduced from the results given by Warburg, Leithauser, Hupka, and Miiller (Annal. d. Physik, xl. p. 609, 1913) without any assumption of the value of e. TABLE I. al. LT. ITT. VE We Calculated p Calculated Voltage in| Observed p | in aluminium Observed wave-length thousands.|in aluminium.| on quantum | wave-length. on quantum theory. theory. —1 =i! -9 -9 13:2 33 (cm.) 26°3 (cm.) 102x10 cm. | 94x10 © cm. 20 12 Die as, 6°8 % Oto ee 3 4°7 ae Sh er 4°66 ‘ op Seale 40 2°46 166.0... DOO. 3102 ae 50 1°53 105) c., BOSE igs ZAG lies 60 1:07 “60! 2°58 i 2:06 ue 70 ‘81 aI, 231, Dim): 80 66 29, Ziloh) 3, LSD) 7G: 90 ‘54 Dai ies 1:96 5 138 es 100 ‘48 lige 1:87 0 1:24 +, 110 “45 153) ;, 1-82 if Piste 125 "42 085 ,, 17 9 oo: ae 145 23) 066 ,, LOO ‘OD ne 175 "39 Onis. 1°72 + Se iow The relation between calculated and observed frequency is simply shown in fig. 5, where the frequency is plotted as ordinate and the voltage as abscisse. Since the energy E of the electron is proportional to the voltage, the theoretical ‘curve on Planck’s relation is a straight line. For the lowest voltages, the experimental curve is seen to fall below _ the theoretical, the observed frequencies being about 10 per cent. less than the calculated. The departure between theory and experiment becomes more and more marked with in- crease of voltage, and at about 142,000 volts the frequency reaches a maximum which is not altered by increase to 175,000 volts. 3 We shall discuss later the probable reason why the theory Maximum Frequency of X Rays from a Coolidge Tube. 353 is so widely departed from in excitation of X-rays by swift electrons, but at present we shall consider whether there is any simple relation between frequency and voltage. The frequency curve is approximately parabolic in shape and can Fig. 5. (Frequency of end racti2tion X10 fe 60 80. 100 Voltage in thousands. be expressed by the relation v=aV—OV’, where a and 6 are constants and V is the voltage. This relation can be put into a simpler form, viz., ficial See SC Aime a hs i an ei da Q) where E is the energy of the electron moving through a difference of potential H and ¢ a constant. If Planck’s relation held for excitation; H=hv, where vy, is the caleu- lated frequency. It is seen that the relation reduces to the simple form v/v, =1—kV. By differentiating equation (1), it is seen that the frequency y reaches a maximum when H=1/2c or V=1/2k. If V is expressed in volts, the value of 1/k which fits the results best is 285,000 volts. The frequency reaches a maximum at half this voltage, or at 142,500 volts. Phil. Mag. S. 6. Vol. 30. No. 177. Sept.1915, 2A 354 Sir E. Rutherford, Prof. Barnes, and Mr. Richardson : The following table shows how closely the frequencies up to the maximum as given by this empirical relation agree with the observed values. Tasie IT. | Volts in Calculated Observed | thousands. frequency. | frequency. | 132 3°07 x 1018 | 2°94 x 1018 A he sip 454, Air 30 655. 6-44 40 839015, 834, 50 10:06 |. IGN ce 60 es See Lica Oe 70 12 S80K5 1 0one 80 1404, RAVE ie 90 19302), | 1 Nae Sea 100 LDN ee | 16022 | 110 16°47 ,, eS eee 125 LF awe TG Oa} 142°5 se I fe eee With the exception of the values for low voltages, where the frequency is difficult to determine accurately, the agree- ment between the calculated and observed values is remark- ably good and within experimental error. Absorption in Lead. Several experiments were made to find the absorption in lead of the radiation excited by the higher voltages. Pre- liminary experiments showed that the “end” radiation was absorbed nearly exponentially in that substance as in aluminium. The values of the absorption coefficient mw for lead were found to be 36, 29, 23, 23, 23 for the end radiation excited by 96,000, 110,000, 125,000, 155,000, 175,000 volts respectively. Since the absorption coefficient for lead on the average decreases rapidly with increased frequency over the range considered, it was thought that the small change of frequency observed by measuring the absorption of the radiation in aluminium would show up more promi- nently when lead was the absorber. No certain difference in the absorption was, however, observed from 125,000 to 175,000 volts. The final measurement of .the end radiation was made after the rays had passed through 2°49 mm. of lead. The ionization observable through 3 mm. of lead is certainly less than 1/10000 of the initial value. Hven with very intense Maximum Frequency of X Rays from a Coolidge Tube. 355 radiation, a thickness of 3 mm. of lead affords practically complete protection against the rays. Through a thickness of 4 mm. it would be difficult to detect the ionization even when the bulb was strongly excited at 125,000 volts. Discussion of the Results. It has been shown that the penetrating power in aluminium of the X rays from a Coolidge tube reaches a maximum at 142,000 volts, and that no sensible alteration has been ob- served when the voltage is raised to 175,000. The maximum value of the absorption coefficient w in aluminium is ‘39. Remembering that the value of w for the penetrating rays from radium C is ‘115 in aluminium, it is seen that the rays from the Coolidge tube have only about 3/10 of the pene- trating power of the gamma rays from radium C. The radiation from the Coolidge tube is, however, slightly more penetrating than some of the rays from radium B, for which » has been found to be *51. From the variation of frequency with voltage, it would appear that the frequency of the radiation reaches a natural limit, probably controlled by the frequency of the “K” characteristic radiation of that element. This and other points are very clearly brought out by comparison of the radiations from the Coolidge tube at different voltages with the y rays emitted by radium B. The latter has been carefully analysed by H. Richardson *. By examining the absorption in aluminium, the rays were found to consist of two components, for which »=40 and ‘5; the former radiation, which is easily absorbed, undoubtedly corresponds to the * L.” radiation of radium B. The analysis of the radiation was carried still further by determining the absorption curve for lead. In addition to the “LL” characteristic, the rays were found to consist of three components, for which yu in lead was 46, 6, and 1:5(cm.)~*. Since the most penetrating radiation of the Coolidge tube gives a value w=23 for lead, it is clear that the radiation from the Coolidge tube is more penetrating than one of the main components of the radiation from radium B, but is far less penetrating than two other com- ponents. In the following table are given the wave-lengths of the chief lines observed in the radium B spectrum by Rutherford and Andrade f, and the absorption coefficients in aluminium and in lead to be probably ascribed to the * H. Richardson, Proc. Roy. Soc. A. xci. p. 396 (1915); Rutherford & Richardson, Phil. Mag. xxv. p. 722 (1918). + Rutherford & Andrade, Phil. Mag. xxviii. p. 263 (1914). 2A2 356 Sir E. Rutherford, Prof. Barnes, and Mr. Richardson : radiations, and also some comparative results obtained with | the Coolidge tube. Tasue III. Wave-length of | Absorption coeffi- | Wave-length of | Absorption coeffi- strong lines cient of radiations | radiation from | cient of radiations from Radium B. | from Radium B. Coolidge tube. | from Coolidge tube. WAS | eve etn. 1°37 x 10 = 1-5 (em.)\ | in. Pb. IE a a aa 169, ond 1:72x10 em. |u—23(em.) in Pb. ama a LETS les | 191x107" em. | =86° aie ; SDL Alay in Al.,| 4570 (0 aimee: PASO A tk) a | | Dro ah, 3 “TL” yadiation. | 10°82 ‘‘ 7,” radiation. ce bO to _9 KE eae ae i =—40)in Aly 14°77X10 > cm. wa in Al. The line in radium B,A=1°96 x 10~°, appears to be mainly responsible for the radiation which has an absorption w="51 in aluminium and w=46 in lead. This is clear from a comparison of the results with the Coolidge tube. Taking Moseley’s observation that the frequency of the correspond- ing lines in the K radiations of different elements is pro- portional to (N—1)? where N is the atomic number, it follows that the line in the radium B spectrum for which X=1:37x10-* should have a value A=1:'69x10>-" for tungsten, since the latter has an atomic number 74 and radium B, 82. This calculated value is in good agreement with the maximum wave-length X=1°73 x 10~° found for tungsten. It thus appears probable that the radiation from tungsten is analogous to the radiationfrom radium B. Since the speed of the beta rays issuing from radium B corresponds to a fall of potential of at least 400,000 volts, and from radium © of 2,000,000 volts, it seems clear that we cannot expect to obtain a more penetrating radiation from tungsten unless possibly a voltage of the order of 1,000,000 volts is applied. Hven with electrons corresponding in energy to over 2,000,000 volts, the wave-length of the strong line due to the penetrating gamma rays from radium Q, viz. A='99X10-%, is only 6/10 of the shortest wave from the Coolidge tube. The comparison of the results with the Coolidge tube with the gamma rays thus leads to the conclu- sion that there is a definite limit to the maximum frequency to be obtained from an element bombarded by swift electrons. Maximum Frequency of X Rays from a Coolidge Tube. 357 This limit is probabiy determined by the characteristic radiation of highest frequency which exists in the atom. Since radium C has an atomic number 83 and uranium—the heaviest known element—92, we should anticipate from Moseley’s relation that the shortest wave-length to be obtained with a uranium anticathode in a vacuum-tube is rA=1:40x10-°. The penetrating power of this radiation in aluminium should be »=°23 instead of =*39 from the Coolidge tube. Under possible laboratory conditions, it thus appears very improbable that we can obtain X rays as pene- trating as the gamma rays from radium CU. The Excitation of X rays and the Quantum Theory. We have seen that for the Coolidge tube the connexion between the maximum frequency v and the energy E ot the exciting electron is given by. hv=HK—ch” v/vp»=1—kV, or where v, is the frequency to be expected on Planck’s theory 1f the whole energy of the electron is transferred into radia- tion; V the voltage ; ¢ and & are constants. These formule do not hold beyond the maximum frequency given by H= 5,” 1 Vics 2h 1/285000. If this formula holds for lower voltages than those actually examined, it is seen that the value of v becomes more nearly equal to v, the lower the voltage. The formula suggests that, for a heavy atom like tungsten, the frequency excited by low voltages should very closely agree with that expected on the quantum theory, supposing that the whole energy of the electron is transformed into that of the Xray. It thus appears probable that the simple quantum theory holds tor excitation if the voltage is sufficiently small, but that a large correction is required for high voltages. It will be of very great interest to examine the correspond- ing relations between frequency and voltage for lighter atoms, and to test whether such a simple relation as that found for tungsten holds in such cases. It is to be anti- cipated that the maximum frequency would be reached for a voltage which diminishes in value as the atomic weight or atomic number decreases. There is one point, however, in this connexion that should be mentioned. We have seen that the maximum frequency obtainable from tungsten is about If V is expressed in ‘volts, the value of & is 358 Sir HE. Rutherford, Prof. Barnes, and Mr. Richardson: that to be expected for the component of the shortest wave- length for the ‘“‘ K” characteristic radiation of that element. On the other hand, the experiments of Kaye have shown that a radiation can be obtained from aluminium at 30,000 volts, which is much more penetrating than the “K” characteristic of that element. Results of a similar kind have been obtained by Rawlinson *, who found that at about 50,000 volts he obtained a radiation from nickel for which y=6°9, while the value of » for the “K ” characteristic is 148(cm.)~*. Such results show that the highest frequency to be obtained from these atoms is much greater than that of the K radiation. It is possible, however, that the radiation of higher frequency may represent the octave or still higher harmonic of the fundamental mode of radiation which is represented by the ““K” characteristic. A close analysis of the frequency-voltage curves of the radiation from different elements should throw much light on this question. Arrangements have been made to continue experiments, such as have been made for tungsten, for a number of other elements. We have already drawn attention to the fact that, even with the very high velocity of projection of the beta rays from radium B and radium ©, no frequencies have been observed much higher than those to be expected for the “K ” radiation. Some years ago, Whiddingtont made a number of experiments on the minimum voltage required to excite the “ K” characteristic of a number of light elements. Assuming, as seems probable, that it is necessary to excite the shorter wave-length of the two main components of the *“K ” radiation before the characteristic appears, the voltage required on the simple quantum theory to excite the radia- tions is given in the following table, taking the values for the beta component of the “‘ K ” radiation found by Moseley. It is seen that for all metals except aluminium the voltage required to excite the “‘ K ”’ radiation is distinctly higher than that expected on the simple quantum theory. Such a result is to be anticipated if the relation between frequency and voltage is of the same general form as that observed for tungsten. We have seen that the radiation from the Coolidge tube was first detected at 10,300 volts. It seems probable that this corresponds to the voltage required to excite the “[,” characteristic of tungsten. Assuming that the line * Rawlinson, Phil. Mag. xxviii. p. 274 (1914). + Whiddington, Proc. Roy. Soc. A. Ixxxv. p. 328 (1911). Maximum Frequency of X Rays from a Coolidge Tube. 359 X=1:277x 10-8 cm.* must be excited for the appearance of the “‘L” radiation, the theoretical voltage is 9600 volts, TABLE LV. Atomic | A for the Calculated | Observed | Volts cale. Element. number. B line. | voltage. | voltage. | Volts obs. a hig volts. | volts. Aluminium ... 13 791x10 foi@ |. S208 is Chromium...... 24 |2098x10*| 5900 | 7820 | - 80 igi. ...>..... eee ittoxio: |. 7000 |. 94600,,.|> | 78 Nickel ......... | 28 |1:506x10°°} 8200 | 10750 | 76 Copper ......... | 29 |1-402x1078) ss00 | 11080 | 79 PHN) eons: | 30 1306 x 107° | 9500 | 11280 83 while the voltage deduced from the equation v/vy,=1—£V is 9800. ‘The difference in this case between the observed and computed voltage is about 2 per cent., and is not beyond the experimental error. In order for the quantum theory to hold for excitation, it is necessary that the whole of the energy of the electron should be directly converted into energy of radiation. It seems probable that if an electron makes an “end on” collision with another, the whole of the energy of the one is transferred to the other, but we have no definite evidence that radiation is emitted in such a collision. Taking, for example, the point of view proposed by Bohr f, that the radiation arises from the fall of an electron from one ring of electrons to the next, the essential antecedent to the emission of radiation is the removal of an electron from one of the rings. If the whole of the energy of the incident electron is expended in ejecting the electron from the ring, it is to be anticipated that the energy of the X radiation should equal the energy of the incident electron. This condition should approximately be fulfilled in the relatively sparse distribution of electrons in the outer rings. In order, however, to excite the higher frequencies, the electron must penetrate deep into the atom where the electrons are more closely packed, and part of its energy will be used up in setting neighbouring electrons in motion, and only a fraction will be available to eject an electron from its ring. Quite apart from any special theory of the mechanism of radiation, such a factor must * See Barnes, infra, p. 368. Tt Bohr, Phil. Mag. July 1913. 360 Maximum Frequency of X Rays from a Coolidge Tube. always enter into the energy of the electron finally available to excite a characteristic radiation. This point of view offers a simple and probable explanation of the reason why the quantum theory holds closely for excitation of low frequencies by slow speed electrons, but fails for high frequencies. The relation found experimentally between v/v, for tungsten suggests that the correction for this effect increases rapidly with the frequency. Summary. (1) The absorption curves in aluminium of the X radia- tion from a Coolidge tube have been examined over a wide range of constant voltages supplied by a Wimshurst machine. While the main radiation is complex, the ‘‘ end” radiation is found to be absorbed exponentially. (2) The absorption curves for different voltages obtained with an induction-coil are nearly the same as for the Wims- hurst machine, and the penetrating power of the end radia- tion is nearly the same. The maximum frequency of the “end” radiation for voltages between 13,000 and 175,000 volts has been deduced by examining its absorption in aluminium. ‘The frequency and penetrating power reach a maximum value at 145,000 volts, and are not altered by increase of the voltage to 175,000. (3) The shortest wave-length emitted by the Coolidge tube is 1°71 x107-° em. or :17 A.U. The absorption coefti- cient of this radiation in aluminium is °39 (cm.)~1, and in lead 23(cm.)~1. The penetrating power of this radiation is about 3/10 of the penetrating gamma rays from radium C. (4) The relation between the frequency and the voltage is expressed by the formula hky=H—cH’, where HE is the energy of the electron, c a constant, and A Planck’s fundamental constant. This relation holds up to 142,000 volts when the radiation has its maximum frequency. Evidence is given that the quantum theory is directly applicable for the exci- tation of waves of low frequency, but for higher frequencies requires a correcting factor, the value of which increases rapidly with the frequency. (5) A comparison is given of the radiation froma Coolidge tube with the gamma radiations emitted by radium B and radium C. University of Manchester, July 1915. XXAV. Lficiency of Production of X Rays from a Coolidge Tube. By Sir Ernest Rutuerrorp, £.A.S., and Prof. J. BArgnus, Phil). | oy the preceding paper we have given an account of experiments which have been made to determine the maximum frequency of the X radiation excited in a Coolidge tube for different constant voltages supplied by a Wimshurst machine. It was thought desirable to extend the experiments to determine the efficiency of the production of X rays in the Coolidge tube for comparatively high voltages. The question of the efficiency of the production of X rays, i. e. the ratio of the energy of the generated X rays to that of the exciting catho‘le rays, has been the subject of several investigations. Wien} determined the energy of the X rays enerated in a platinum anticathode for a potential difference of 58,700 volts, by measuring the heating effect of the radiation. Similar experiments have been made by Angerert and Carter §. ‘The iatter observed that the efficiency increased with the exciting voltage. In general, it was found that the efficiency was of the order of 1/1000. Recently the question has been attacked under more definite conditions by R.T. Beatty||. Cathode rays of definite velocity sorted out by a magnetic field fell ona radiator. The generated X rays passed out through a very thin window, and were then completely absorbed in a cylinder 150 em. long filled with the vapour of methyl iodide. The total ionization of the rays was thus measured, and the corresponding energy deduced from general data. When characteristic radiation was not excited, the energy of the X rays for an equal number of exciting electrons was found to be proportional to the fourth power of the velocity of the cathode rays. Kaye] had previously observed that the energy of X rays under constant condition of excitation was approximately proportional to the atomic weight A of the radiator. Beatty finally concluded that the efficiency of production of X rays by matter in general was given by the formula X-ray energy b 2s = 2°54 x 104A", cathode-ray energy * Communicated by the Authors, + Wien, Ann. d. Phys. xviii. p. 991 (1905). f Angerer, Ann. d. Phys. xxi. p. 87 (1906). § Carter, Ann. d. Phys, xxi. p. 955 (1906). || Beatty, Proc. Roy. Soc, A. lxxxix. p. 314 (1913). {| Kaye, Phil. Trans. Roy. Soc. A. cciv. p. 123 (1908) ; Proc. Roy. Soc. A. Ixxxi. p. 337 (1908). 362 Sir E. Rutherford and Prof. J. Barnes on Lfficency where 8 is the velocity of the cathode as a fraction of the velocity of light. From the curves given by Beatty, it would appear that the maximum speed of the cathode rays employed by him corresponded to 23,000 volts. In the experiments with the Coolidge tube we have exa- mined the efficiency of production of the radiation escaping from the bulb for voltages 48,000, 64,000, and 96,000 volts. The rays were excited by a Wimshurst machine, and the current through the bulb measured by a galvanometer. The general arrangements for controlling and calibrating the voltage were the same as those described in the accom- panying paper. For the high voltages employed, the radiation is too penetrating for complete absorption in a reasonable length of air or other gas. In order to overcome this difficulty, we have measured the ionization due to a definite length of a beam of X rays in air, and deduced the absorption in air indirectly by determining the absorption curve of the radiation in water. It is known that the absorption of X rays by complex molecules is additive. The absorption of the radiation by water is mainly due to the oxygen atoms, whose atomic weight differs only slightly from that of the average atomic weight of air. For the relatively penetrating rays employed, we can assume with very little error that the absorption of the X rays by water is very nearly equal to that of a column of air of the same thickness compressed to the same density. The absorption by 1 em. thickness of water is thus equivalent to that by 8-2 metres of air at laboratory temperature. The ionization due to the X rays was measured in air without the X rays impinging on the electrodes in order to avoid the introduction of surface effects due to scattering or Fig. 1. Aa Battery “70 electrometer Earth % excitation of characteristic radiations. The general arrange- ment of the apparatus is shown in fig. J. The ionization vessel consisted of a rectangular box (20 x 12 x 12 cm.) lined of Production of X Rays from a Coolidge Tube. 363 with aluminium plate. The ionization was measured between the aluminium plate A and the central plate B. The former was connected to a 1000-volt battery and the latter to the electrometer. Two additional plates C, D connected with earth extended on both sides of the plate B. The cone of rays entering the ionization vessel was fixed by the circular opening O, 2 cm. in diameter, in alead plate, the size of opening being adjusted so that no radiation fell on the electrodes. The disturbing effect due to the X rays impinging on the aluminium end of the box was small, and could be neglected. In order to determine the absorption curve for water, a vessel was constructed consisting of two closely fitting brass tubes H and F. Openings, G, H, cut in the ends of these tubes, were covered with a thin sheet of mica. The length of the column of water in the path of the rays could be simply adjusted, and care was taken that the radiation entering the ionization vessel did noi strike the brass ends of the vessel. The maximum length of the column of water in the experiment was 5cm. It is seen from fig. 2 that this is sufficient to reduce the intensity of the radiation to a small fraction of its initial value. Fig. 2. lorisation Current 20 Millimetres of water In order to determine the absorption of the end part of the radiation by water, the vessel EF was removed, and a sut- ficient thickness of aluminium introduced in the path of the rays to reduce the ionization to an equal degree. The water column was again introduced, and the experiments continued up to a thickness of 15 cm. of water. The capacity of the circuit and of the condensers in parallel was carefully determined. Changes in the intensity of the 364 Sir HE. Rutherford and Prof. J. Barnes on Efficiency radiation were controlled by means of the “ standardizing vessel *’ described in a previous paper, and also by the current, passing through the Coolidge tube. This current was deter- mined by the method described in a previous paper. The deflexion of the galvanometer in the circuit was observed ; the current through the tungsten spiral broken, and the voltage kept at the same value by varying the speed of the machine and by means of adjustable point discharge. The deflexion observed under the latter conditions was due to the current through the xylol resistances and voltage galvano- meter in parallel with the Coolidge tube, and to electrical losses in the leads or over the surface of the bulb. The difference between the two readings served as a measure of the actual current conveyed by the electrons from the heated spiral. The earlier parts of the absorption curves in water for 64,000 and 96,000 volts are shown in fig. 2. The absorption is not exponential, but decreases steadily with increase of thickness of water. Knowing the initial saturation current through the air with no absorber, the total ionization current due to complete absorption of the radiation can be at once deduced by determining the area included between the curve and the two axes, assuming that 1 cm. of water is equi- valent in absorbing power to 8°2 metres of air at 15° C. Supposing that the X radiation from the tungsten anti- cathode is emitted equally in all directions, the fr action of the total radiation entering the ionization vessel was 8°6 x 107°. The intensity of the radiation was found to be directly proportional to the current through the bulb. In the following table the total ionization current in air due to complete absorption of the whole radiation is expressed for each voltage in terms of 100 divisions of the current galvanometer which correspond to 2°92 x 107° amp. | ‘Total Ionization | | | ws . Current 7, for | Voltage. | oad Pee Complete Absorp- Sty fte.! ee en eee eens coe tion of the | | Radiation. | | | L E | | | 48,000 | 292x10-F amp. 25x10~5amp.| 86 | “406 | 32 | | 64,000), vy des a ibe) |» U4 ily laine | | | 96,000 | We Site eo) BOL Saree amass Beatty ice. cit.) concluded from his experiments that X='58 AB, where A is the atomic weight of radiator and 6 of Production of X Rays from a Coolidge Tube. 365 the velocity of the cathode rays as a fraction of the velocity of light. It is seen from the last column that X/* is sensibly constant. Substituting the value of A=184 for tungsten, the values of X to be expected from this equation are 2°9, 4°8, 9:1 for 48,000, 64,000, and 96,000 volts respectively in place of 86, 1:4, 3:0 observed experimentally. The observed vaiues are about one third of the values calculated on Beatty’s relation. Correction for Absorption in Bulb. In our calculations, however, we have measured the total ionization produced outside the bulb, and have not corrected for the absorption of the rays by the wall of the bulb and by the air and other absorbers in the path of the rays. Special measurements showed that the wall of the bulb where the rays issued was *5 mm. in thickness. The absorption in glass for equal thicknesses is about the same as for aluminium, and there will not be much error in taking the absorption of the rays before entering the ionization vessel as equivalent to 6 mm. of aluminium. Comparing the relative absorption of air and aluminium for soft radiations, *6 mm. of aluminium is equivalent to about 8 mm. of water. Until experiments are made of the total radiation from tungsten under conditions that the absorption of radiation in escaping from the tube is a minimum, it is difficult to make more than a rough estimate of this correcting factor for absorption. From a consideration of the absorption curves in water, it seems probable that the correcting factor for the total ionization is at least 2 for 48,000 volts, and may be somewhat greater. It is to be expected that this factor would be somewhat less for the higher voltages. Making this correction, it is seen that the results for tungsten are in very fair agreement with Beatty’s relation, even though it is extrapolated over a wide range of atomic weight and voltage. Energy of the X Rays. Knowing the current due to complete absorption of the radiation, the energy of the radiation can at once be deduced if the average energy required to produce a pair of ions in airis known. The value of this important quantity can best be deduced from the total ionization current produced by the absorption of a single alpha particle of known energy. Geiger found that asingle alpha particle from radium C gave rise to 2°37 x 10° ions in air each of charge 4°65x 107?" es, units, 7. e. a quantity of electricity 3°67 x 10-© e.m. units. a a Se eB. Sn 366 Sir E. Rutherford and Prof. J. Barnes on Efficiency On the latest data *, the initial energy of the alpha particle from radium C is 7°66 x 10!e, where e is the charge on the ions in electromagnetic units, From this it can be deduced that the energy required to produce a pair of ions in air is equal to the energy acquired by the unit charge in moving freely through a potential difference of 33 volts. If 2, is the total ionization current and 7, the electronic current Me ears ee es cathode-ray energy %,V’ where v=33 volts and V=the voltage applied to the tube. Voltage. ae Efficiency. Ge. Efficiency/?. hes 48,000 "86 59x 1073 165 36 64,000 1:44 s(t ‘211 35 96,000 3-01 1:04 ,, 992 3:6 The efficiency deduced in the above table is for ordinary working conditions when no correction is made for absorption in the glass walls, &c. Under these limitations, the efficiency is seen to be proportional to B?. No doubt the closeness of the agreement is accidental, for the numbers would be changed ~ if corrections were made for absorption of the radiation. We have seen earlier that about half the energy of the radiation is probably absorbed in the bulb. We should consequently expect the efficiency under ideal conditions to be about 1/500 for 96,000 volts, and 1/800 for 48,000 volts. It is of interest to note that Wien (doc. cit.) found an efficiency of 1:09 x 10-? for a platinum anticathode, using a bolometer method to measure the energy of the X rays for a potential of 58,700 volts supplied by an induction-coil. Since the average potential of the discharge due to an induction-coil is less than the maximum, it is to be expected that the efficiency of the coil would be somewhat higher than for an equal steady voltage supplied by a machine. Beatty (loc. ct.) found the efficiency to be given by H=2°54x 107-*AQ?. The value of the numerical factor involves the average energy required to produce a pair of ions. This was deduced by a very indirect method by a combination of distinct in- vestigations by Glasson and Whiddington. From the data * Rutherford and Robinson, Phil. Mag. xxviii. p. 551 (1914). of Production of X Rays from a Coolidge Tube. 367 given by Beatty, itcan be calculated that the energy assumed to produce a pair of ions in methyl iodide corresponds to 110 volts. ‘This is undoubtedly more than three times too large, and the value of E=7°6 x 10-°A? is nearer the truth, assuming the correctness of the other data involved. The formula gives an efficiency for 96,000 volts of 4:1 x 10-°, which is about twice as high as that to be expected from our experiments with a Coolidge tube after the probable cor- rection for absorption has been made”. In these calculations no correction has been made for reflexion or scattering of cathode rays by the tungsten. No doubt the correction for this varies with the speed of the electrons, and must be considerable for very high voltages. In the absence of any definite data on this question, it seems desirable, however, to give the efficiency of the conversion of cathode rays into X rays under actual working conditions. The relations given by Beatty only apply to the ‘* general” or “independent ” radiation from an X-ray tube. As pointed out by Beatty, the efficiency rises rapidly when a characteristic radiation is strongly excited. The low percentage value obtained for the efficiency of a Coolidge tube for high voltages is thus an indication that the radiation is mainly of the “independent”? type, and that the “K” characteristic radiation is not so strongly excited as in the case of metals of lower atomic weight. This is borne out by the difficulty of detecting the presence of the ‘‘ K ” characteristic of tungsten by absorption experiments, or by reflexion from crystals. It is only in the case of radioactive substances that the characteristic radiations of high frequency are strongly excited. No doubt this is due to the ideal conditions of excitation in this case, for the exciting electrons all come from the nucleus of the atom. In the preceding paper we have shown that the voltage required to excite the most penetrating rays in tungsten is about twice that to be expected on the quantum theory, indicating that about half of the energy of the exciting electron can be transformed into radiation. From the low value of the efticiency at high voltages, viz. about 1/500, it is clear that, on the average, 1 electron only in 200 is efficient in producing radiation. University of Manchester, July 1915, * Dr. Beatty has drawn my attention to the recent measurements of Barkla (Phil. Mag. xxv. p. 838, 1913), in which he finds that the total ionization in methyl iodide for cathode rays is 1°48 times that in air. Using the amended data, Beatty’s relation becomes E='51 x 107 *Ap?, which is in fair accord with our measurements, PY Bash a XXXVI. The High-Frequency Spectrum of Tungsten. By James Barnes, Ph.D., Research Fellow, University of Manchester *. TYNE investigation of the high-frequency spectrum of the d X rays coming from a Coolidge tube with a tungsten anticathode was thought to be of interest in connexion with the accompanying paper t. The method used for obtaining and photographing the spectrum was the same as that employed by Moseley ft. The crystal was placed on a goniometer table and adjusted so that the face upon which the rays were incident passed through the axis of rotation. The beam of rays coming through the slit of about 0°5 mm. width was intersected by the axis and fell approximately on the centre of the face of the crystal. The perpendicular distance from the axis to the photographic plate was 20°12 cm., and the distance from the slit to the axis was made as near as possible the same. The crystal was kept throughout the exposure in slow rotation by the method used by Professor Rutherford and Dr. Andrade § in their determination of the spectrum of the soft gamma rays from radium B. The lines were first photographed on one side of the central image of the slit, then the crystal was turned through 180° and the lines obtained on the other side. The so-called reflexion angle @ can then be calculated from the expression lee The 20=tan 40-94? where a is the distance between like lines on the same plate. The longest time of exposure employed was six hours. Satisfactory photographs of the strongest lines in the “ L” series could easily be got in half an hour. The glass wall of the bulb being 0°5 mm. thick, at least one-fiftieth of the intensity of this soft radiation is absorbed by the glass. With a bulb having a side tube with its end covered with thin aluminium, good photographs of this series would probably be obtained in a few minutes under similar conditions. The following table contains the results obtained with a rock-salt crystal, 1°6 cm. thick, whose face was 3°5 x 1:9 em. * Communicated by Sir Ernest Rutherford, F.R.S. + Phil. Mag. supra, p. 339. t Phil. Mas. xxvil. p. 703 (1914). g Phil. Mag. XXVll. p. 858 (1914). The High-Frequency Spectrum of Tungsten. 369 The reflexion angles are believed to be accurate within +3’. The wave-lengths are calculated from the formula 7 r= 2d sin 8, where d is taken as 2°814 x 107° cm. Reflexion angle. Wave-lengths x 10° cm. Intensity. 15° 13’ 1477 Strong. 13° 29’ 1-312 Weak. 18° 19 1-296 Strong. 13° 07' 1-277 Weak. 12° 55! 1-258 Strong. 11° 24’ | 1/113 | Strong. 11° 05' 1:082 Weak. It will be noticed from this table that the first line corresponds to the only line in the spectrum of tungsten referred to by Moseley, which he designates the « line of the “L” series. The lines 11°296 and X1:258 fill exactly the gaps in the curves as given by the beta and gamma lines of this series tor the elements from zirconium to gold. X1:277 is probably the ¢ radiation. It will be interesting to investigate if these elements have also radiations analogous to 7. 1'312, X1°113, and 21-082. Extrapolating Moseley’s* results as expressed by the relation y=((N—1), where v is the frequency, © a constant, and N the atomic number, which for tungsten is 74, we get 0°22 x 10-8 cm. for the wave-length of the « line of the “K” series. This wave-length corresponds to a reflexion angle of 2° 18’ from rock-salt. The photographs obtained with the thick crystal showed no signs of this line, but a continuous spectrum extended from 4° 54’ almost to the edge of the central image. Beyond this band towards the longer wave-lengths was another band much less uniform in intensity, and ex- tending almost to the lines of the “L” series. De Broglie t found similar results for tungsten and platinum (atomic number 78); and Malmer { observed in the case of cerium * Phil. Mag. xxvii. p. 710 (1914). + Comptes Rendus, elviii. p. 177 (1914). t Phil. Mag. xxviii. p. 792 (1914). Phil. Mag. 8. 6. Vol. 80. No. 177. Sept. 1915. 2B 370 Dr. L. Silberstein on (atomic number 58) a continuous spectrum in the place where the lines of the “K” series should appear. It was thought that possibly the spectrum of this series might consist of a large number of lines which blended into one another, giving the appearance of a continuous spectrum and due to the layers of the crystal not being accurately enough spaced for these short radiations. The penetrating power of these radiations being so great, every layer in the thick crystal is almost equally active, while for the “lL” series, most of the reflexion comes from the planes in the first millimetre. A number of thin rock-salt crystals about one millimetre thick were then used, with the result that the bands were not nearly so uniform in intensity as in the photographs with the thick crystal. The lines were broad and diffuse, and many were wavy, so that measurements of any accuracy were impossible. In any case, there were no clearly marked characteristics of the ‘“‘ K”’ series, even when the maximum potential on the bulb was 100,000 volts. According to Planck’s relation, H=hyv, the @ line should appear when the electrons have the energy of approximately 58,000 volts, while according to the observations in the accompanying paper, this characteristic if present should appear at 74,000 volts. These spectroscopic observations, therefore, agree with the absorption results, the curves of which showed no evidence of being able to be resolved into a few exponentials representing characteristic radiations. In conclusion, the author wishes to express his best thanks to Sir Ernest Rutherford for many suggestions in the course of the above observations, and also for the privilege of working in his laboratory. XXXVI. On Mutual Electromagnetic Mass. By L. SILBER- STEIN, Ph.D., Lecturer in Natural Philosophy at the University of Rome*. HE electromagnetic mass of a system of two charges, é; and és, is of the form My, + Me+ M49, where m, mp are determined by the shape and size of, and the distribution of electricity in, 1 and 2, respectively; whereas mj, depends on the mutual relations of the two charged regions to one another. The first two terms of the above sum being called the masses of 1 and 2, respectively, * Communicated by the Author. Mutual Electromagnetic Mass. ont the appropriate name for m=, which I have employed since 1909, is the mutual electromagnetic mass of 1 and 2. Let 1, 2 be spheres of radii aj, a2,and let 7 be the dint: ance apart of their centres. Let each of these spheres have a homogeneous volume charge. It will be enough, for the purposes of the present Note, to consider the limiting values of the masses for small velocities, or the so-called rest-masses (although, as I have shown in papers to be quoted presently, there is no difficulty in treating any velocity common to the two spheres and directed along the central line), Vhen, in Heaviside’s rational units, 2/2 2/2 m= eee 1a = UL) Oy aaa ee BS OTT, OT A, where c is the velocity of lightin empty space. The expression for the mutual mass mj. of these spherical charges assumes different forms according as— (1) the spheres exclude one another (7 > u,+ 4g), (2) one is entirely contained in the other, or (3) the spheres are only partly overlapping. In a communication to the LIV. Congress of the Societa Italiana per il Progresso delle Scienze, Naples, December, 1910, L developed the complete and general formule for the first two cases*, the third being more complicated and of little physical interest. Quite recently I have been surprised to see this very problem treated by Prof. Nicholson +, without any mention of my previous work. It seems, in fact, that Prof. Nicholson has had no knowledge of my Italian communication, although it has appeared also, with some more det tails, in the more widely circulated Phys. Zeitschrift (vol. xii. p. 87, 1911), and wa; shortly afterwards reviewed in Journal de Physique. Now, if it were only for the sake of claiming priority, the matter would hardly be worth mentioning , especially as the mathematical side of the problem is of utter simplicity. But it so happens that Prof. Nicholson’s solution—formula (12), loc. cit.—is wrong in its numerical coefiicient, and, what is more important, “would (if correct) apply only to the case * Cf. the Atti of the said Meeting. t J. W. Nicholson, Proceedings of the Phys. Soe. Lond. vol. xxvii. Part 3, April 15, 1915, pp. 217- 228, 2B3 372 On Mutual Electromagnetic Mass. of comparatively large distances*. In fact, Prof. Nicholson himself states (p. 227, loc. cit.) that ‘under this condition ” (rv of the order of a) his formula “ ceases to give the mutual mass etc.””? Why, then, not calculate the rigorous expression, which, if the problem is appropriately treated, is as easily obtained as the approximate one ? In view of these circumstances it may be useful to recall here, without repeating details of the calculation (which was based on the electromagnetic momentum of the system of the two charges), my final formule for the mutual mass of a pair of charges as specified above, for the cases 1 and 2. They are :— When the spheres exclude one another. eel | a a7 +a,” pies pine TIA a nee 2TH gun atn 7 CA) “mis= or, in virtue of (1), TOES 2 2 151A ay +a (Al). mp aA : 21 le cieae a Zr G where + corresponds to charges of equal, and — to charges of opposite signs. When the sphere 1 is entirely contained in the sphere 2. €1€,/¢2 ay +r (B) Mi9= a [i al *) Qeras Day” or, again, by (1), als Fe 2 a ad (B,) Mj5= ta/mymy ? oe [ pee st | a 2 The signs being interpreted as above. It is worth noticing that B follows from (A) by a cyclic permutation of ay, 7, ag. These formule are rigorously valid for any value of r. aos illustrating them will be found in my papers quoted above. London, July 1915. * J. e, small values of (a,2+a,")/?°. gra). XXXVIII. On the Temperature Coefficient of Young’s Modulus for Electrically Heated Iron Wire. By H. P. Harrison, PhD., F.RSL., Professor of Physics, Presidency College. Calcutta, and Susir K. Caaxgravarii, I.Sc., Bengal Government Research Scholar *. (* recent years several observers + of the effect of tem- perature on Young’s Modulus claim that for loads below a certain magnitude the modulus for metal wires reaches a maximum in the neighbourhood of 100° C. The circumstances in which this peculiarity is said to occur are somewhat restricted; the total load must be rela- tively small, and heating must be effected by the direct passage of an electric current. When other methods of heating are employed, such as vapour-baths or the radiation from a surrounding helix conveying a current, no maximum is observed, the coefficient of elasticity decreasing uniformly as the temperature rises. Shakespearf alone appears to have noticed an increase in the modulus during ordinary heating (steam), but he found that the effect ceased after several heatings and coolings, whereby the wire was reduced toa condition in which it behaved normally and possessed a negative temperature coefficient of elasticity. One of the present writers §, in a series of experiments recently pub- lished, failed to detect any sign of a maximum in Young’s modulus for nickel during electric heating ; but the research being more particularly concerned with values of the elas- ticity near the Curie point at 400°, did not afford very strong evidence either for or against the existence of a maximum below 100°. Consequently the experiments described below were undertaken as an additional attempt to settle what appears to be a somewhat doubtful issue. Present Huperiments. lt was decided to use iron wire, electrically heated, and to carry out two separate tests, one ona portion of wire annealed at 400° C., and the other on an unannealed portion straightened by stretching when cold. Both portions were taken from the same hank of wire. The apparatus and method used for finding the modulus * Communicated by Prof. A. W. Porter, F.R.S. + Noyes, Physical Review, vol. ii. & vol. iii. (1895), vol. iii. (1896). H. Walker, Proc. R. S. Edinburgh, vol. xxvii. Part 4, No. 34 (1907) ; vol. xxviii. Part 8, No. 40 (1908) ; vol. xxxi. Part 1, No. 10 (1911). { Shakespear, Phil. Mag. vol. xlvii. (1899). § Harrison, Proc. Phys. Soc. London, vol. xxvii. Dec. 15 (1914). “ , ° . . (1) where fh is Planck’s constant and A, and A, are the energies of the system in the two stationary states. . That the dynamical equilibrium of the systems in the stationary states is governed by the ordinary laws of mechanics, while these laws do not hold fee the transition from one state to another. D. That the various possible stationary states of a system consisting of an electron rotating round a positive nucleus are determined by the relation DPR 0). |. a ny im where T is the mean value of the kinetic energy of the system, o the frequency of rotation, and n a whole number. Ii will be seen that these assumptions are closely analogous to those geet used by Planck about the emission of radiation in quanta, and about the relation between the frequency of an atomic resonator (of constant frequency) and its energy. It can be shown that, for any system con- taining one electron rotating in a closed orbit, the assumption CG and the relation (2) will secure a connexion between the frequency calculated by (1) and that to be expected from ordinary electrodynamics, in the limit where the difference between the frequency of the rotation of the electron in successive stationary states is very small compared with the absolute value of the frequency (see [V. p. 310). On the nucleus theory this occurs in the region of very slow vibrations. If the orbit ef the electron is circular, the assumption D is equivalent to the condition that the angular momentum of the system in the stationary states is an integral multiple of h/27. The possible importance of the Radiation and the Structure of the Atom. 397 angular momentum in the discussion of atomic systems in relation to Planck’s theory was first pointed out by J. W. Nicholson *. In paper I. it was shown that the above assumptions lead to an interpretation of the Balmer formula for the hydrogen spectrum, and to a determination of the Rydberg constant which was in close agreement with the measurements. In these considerations it is not necessary to make any assump- tion about the degree of excentricity of the orbit of the electron, and we shall see in the next section that it cannot be assumed that the orbit is always circular. So far we have considered systems which contain only one electron, but the general validity of the assumptions A and B seems strongly supported by the fact that they offer a simple interpreiation of the general principle of combination of spectral lines (see LV. p.507). This principle was originally discovered by Ritz to hold for the ordinary series spectra of the elements. It has recently acquired increased interest by Fowler’s work on the series spectra of enhanced lines emitted from many elements when subject to a powerful electric discharge. Fowler showed that. the principle of combination holds for these spectra although the laws governing the numerical relation between the lines at an important point (see section 3) differed from those of the ordinary series spectra. There is also, as we shall see in section 4, some indication that the principle holds for the high frequency spectra revealed by interference in crystals. In this connexion it may also be remarked that the assump- tion A recently has obtained direct support by experiments of A. Hinstein and J. W. de Haas f, who have succeeded in detecting and measuring a rotational mechanical effect pro- duced when an iron bar is magnetized. Their results agree very closely with those to be expected on the assumption that the magnetism of iron is due to rotating electrons, and as pointed out by Hinstein and Haas, these experiments therefore indicate very strongly that electrons can rotate in atoms without emission of energy radiation. When we try to apply assumptions, analogous with C and D, to systems containing more than one electron, we meet with difficulties, since in this case the application of ordinary * Nicholson, Month. Not. Roy. Astr. Soc. Ixxii. p. 679 (1912). + Einstein and Haas, Verh. d. D. Phys. Ges. xvii. p. 152 (1915). That such a mechanical rotational effect was to be expected on the electron theory of magnetism was pointed out several years ago by O. W. Richardson, Phys. Review, xxvi. p. 248 (1908). Richardson tried to detect this effect but without decisive results. 398 Di. N. Bohr on the Quantum Theory of mechanics in general does not lead to periodic orbits. An exception to this, however, occurs if the electrons are arranged in rings and rotate in circular orbits, and from simple considerations of analogy the following assumption was proposed (see I. p. 24). EH. In any atomic or molecular system consisting of positive nuclei and electrons in which the nuclei are at rest relative to each other, and the electrons move in circular orbits, the angular momentum of each electron round the centre of its orbit will be equal to h/2a7 in the “normal” state of the system, 2.e. the state in which the total energy is a minimum. It was shown that in a number of different cases this assumption led to results in approximate agreement with ex- perimental facts. In general, no stable configuration in which the electrons rotate in circular orbits can exist if the problem of stability is discussed on ordinary mechanics. This is no objection, however, since it is assumed already that the mechanics do not hold for the transition between two stationary states. Simple considerations led to the following condition of stability. IF. A configuration satisfying the condition E is stable if the total energy of the system is less than in any neighbouring configuration satisfying the same condition of angular momentum of the electrons. As already mentioned, the foundation for the hypothesis E was sought in analogy with the simple system consisting of one electron and one nucleus. Additional support, however, was obtained from a closer consideration of the formation of the systems. It was shown how simple processes could be imagined by which the confluence of different rings of elec- trons could be effected without any change in the angular momentum of the electrons, if the angular momentum of each electron before the process was the same. Such considerations led to a theory of formation of molecules. It must be emphasized that only in the case of circular orbits has the angular momentum any connexion with the principles of the Quantum theory. If, therefore, the applica- tion of ordinary mechanics to the stationary ‘states of the system does not lead to strictly circular orbits, the assumption E cannot be applied. This case occurs if we’ consider con- figurations in which the electrons are arranged in different rings which do not rotate with the same frequency. Such configurations, however, are apparently necessary in order Radiation and the Structure oj the Atom. 399 to explain many characteristic properties of the atoms. In my previous papers an attempt was made in certain cases to overcome this difficulty by assuming, that if a very small alteration of the forces would make circular orbits possible on ordinary mechanics, the configuration and energy of the actual system would only differ very little from that calcu- lated for the altered system. It will be seen that this assumption is most intimately connected with the hypo- thesis F on the stability of the configurations. Such considerations were used to explain the general appearance of the Rydberg constant in the spectra of the elements, and were also applied in discussing possible configurations of the electrons in the atoms suggested by the observed chemical properties. These calculations have been criticised by Nicholson *, who has attempted to show that the configura- tions chosen for the electrons in the atoms are inconsistent with the majn principles of the theory, and has also attempted to prove the impossibility of accounting for other spectra by help of assumptions similar to those used in the interpreta- tion of the hydrogen spectrum. Although I am quite ready to admit that these points involve great and unsolved difficulties, I am unable to agree with Nicholson’s conclusions. In the first place, his caleu- lations rest upon a particular application to non-circular orbits of the principle of constancy of angular momentum for each electron, which it does not seem possible to justify cither on the Quantum theory or on the ordinary mechanics, and which has no direct connexion with the assumptions used in my papers. It has not been proved that the configura- tions proposed are inconsistent with the assumption C. But even if it were possible to prove that the unrestricted use of ordinary mechanics to the stationary states is inconsistent with the configurations of the electrons, apparently necessary to explain the observed properties of the elements, this would not constitute a serious objection to the deductions in my papers. It must be remarked that all the applica- tions of ordinary mechanics are essentially connected with the assumption of periodic orbits. As far as the applications are concerned, the first part of the assumption C might just as well have been given the following more cautious form :— “The relation between the frequency and energy of the particles in the stationary states can be determined by means of the ordinary laws of mechanics if these laws lead to periodic orbits.” * Nicholson, Phil. Mag. xxvii. p. 541 and xxviii. p. 90 (1914). 400 Dr. N. Bohr on the Quantum Theory of The possible necessity for an alteration of this kind in assumption C may perhaps not seem unlikely when it is remembered that the laws of mechanics are only known to hold for certain mean values of the motion of the electrons. In this connexion it should also be remarked that when considering periodic orbits only mean values are essential (comp. I. p.7). The preliminary and tentative character of the formulation of the general assumptions cannot be too strongly emphasized, and admittedly they are made to suit certain simple applications. Hor example, it has been already shown in paper LV. that the assumption B needs modification in order to account for the effect of a magnetic field on spectral lines. In the following sections some of the recent experimental evidence on line spectra and characteristic Rontgen rays will be considered, and I shall endeavour to show that it seems to give strong support to the main principles of the theory. 2. Spectra emitted from systems containing only one electron. In the former papers it was shown that the general assumptions led to the following formula for the spectrum emitted by an electron rotating round a positive nucleus 277%e'Mm [.1 iz h?(M +m) ( ) ) Ne, —e, M, m are the electric charges and the masses of the nucleus and the electron respectively. The frequency of rotation and the major axis of the relative orbit of the particles in the stationary states are given by AnetMm 1 1 1?(M+m) iinre TE Es annem he 9 ic h?(M +m) n®” GN: Qar2e Nios ao ae (4) The energy necessary to remove the electron to infinite distance from the nucleus is Wie Ne p= N? ni" ie 2777e4@Mm un “ f(y om), 1? gs Oh eae ea (9) This expression is also equal to the mean value of the kinetic energy of the system. Since —W, is equal to the total energy A, of the system we get from (4) and (5) = hwy. ° ' e e ° . . (6) If we compare (6) with the relation (1), we see that the Radiation and the Structure of the Atom. 4Q1 connexion with ordinary mechanics in the region of slow vibration, mentioned in the former section, is satisfied. Putting N=1 in (3) we get the ordinary series spectrum of hydrogen. Putting N=2 we get a spectrum which, on the theory, should be expected to be emitted by an electron rotating round a helium nucleus. The formula is found very closely to represent some series of lines observed by Fowler * and Hvanst+. ‘These series correspond to ny=3 and n,;=4 f. The theoretical value for the ratio between the second factor in (3) for this spectrum and for the hydrogen spectrum is 1:000409 ; the value calculated from Fowler’s measurements is 1:000408§. Some of the lines under consideration have been observed earlier in star spectra, and have been ascribed to hydrogen not only on account of the close numerical rela- tion with the lines of the Balmer series, but also on account of the fact that the lines observed, together with the lines of the Balmer series, constitutes a spectrum which shows a marked analogy with the spectra of the alkali metals. This analogy, however, has been completely disturbed by Fowler’s and Evans’ observations, that the two new series contain twice as many lines as is to be expected on this analogy. In addition, vans has succeeded in obtaining the lines in such pure helium that no trace of the ordinary hydrogen lines could be observed ||. ‘The great difference between the con- ditions for the production of the Balmer series and the series under consideration is also brought out very strikingly by some recent experiments of Rau{[ on the minimum voltage necessary for the production of spectral lines. While about 13 volts was sufficient to excite the lines of the Balmer series, about 80 volts was found necessary to excite the other series. These values agree closely with the values calculated from the assumption EH for the energies necessary to remove the electron from the hydrogen atom and to remove both electrons from the helium atom, viz. 13°6 and 81:3 volts respectively. It has recently been argued ** that the lines are not so sharp as should be expecied trom the atomic weight of helium on Lord Rayleigh’s theory of the width of spectral lines. This might, however, be explained by the fact that the systems * Fowler, Month. Not. Roy. Astr. Soc. lxxiii. Dec. 1912. f Evans, Nature, xcii. p. 5 (1913) ; Phil. Mag. xxix. p. 284 (1915). { For n,=2 we get a series in the extreme ultraviolet of which some lines have recently been observed by Lyman (Nature, xev. p. 343, 1915). § See Nature, xcii. p. 231 (1913). || See also Stark, Verh. d. D. Phys. Ges. xvi. p. 468 (1914). §| Rau, Sttz. Ber. d. Phys. Med. Ges. Wiirzbwry (1914). ** Merton, Nature, xcy. p, 65 (1915); Proc. Roy. Soc. A. xci. p. 889 (1915). Phil. Mag. 8.6. Vol, 830. No. 177. Sept. 1915. 2D 402 Dr. N. Bohr on the Quantum Theory of emitting the spectrum, in contrast to those emitting the hydrogen spectrum, are supposed to carry an excess posi- tive charge, and therefore must be expected to acquire great. velocities in the electric field in the discharge-tube. In paper IV. an attempt was made on the basis of the present theory to explain the characteristic effect of an electric field on the hydrogen spectrum recently discovered by Stark. This author observed that if luminous hydrogen is placed in an intense electric field, each of the lines of the Balmer series is split up into a number of homogeneous components. These components are situated symmetrically with regard to the original lines, and their distance apart is. proportional to the intensity of the external electric field. By spectroscopic observation in a direction perpendicular to: _ the field, the components are linearly polarized, some parallel and some perpendicular to the field. Further experiments. have shown that the phenomenon is even more complex than was at first expected. By applying greater dispersion, the number of components observed has been greatly increased, and the numbers as well as the intensities of the components are found to vary in a complex manner from line to line*. Although the present development of the theory does not allow us to account in detail for the observations, it seems that the considerations in paper IV. offer a simple interpretation of several characteristic features of the phenomenon. The calculation can be made considerably simpler than in the former paper by an application of Hamilton’s principle. Consider a particle moving in a closed orbit in a stationary field. Let w be the frequency of revolution, T the mean value of the kinetic energy during the revolution, and W the mean value of the sum of the kinetic energy and the potential energy of the particle relative to the stationary field. We- have then for a small arbitrary variation of the orbit sW= —208(*). eee @ This equation was used in paper IV. to prove the equivalence. of the formule (2) and (6) for any system governed by ordinary mechanics. The equation (7) further shows that if the relations (2) and (6) hold for a system of orbits, they will hold also for any small variation of these orbits for which the value of W is unaltered. Ifa hydrogen atom in one of its stationary states is placed in an external electric field and the electron rotates in a closed orbit, we shall therefore expect that W is not altered by the introduction of the atom in * Stark, Elektrische Spektralanalyse chemischer Atome, Leipzig, 1914. Radiation and the Structure of the Atom. 403 the field, and that the only variation of the total energy of the system will be due to the variation of the mean value of the potential energy relative to the external field. In the former paper it was pointed out that the orbit of the electron will be deformed by the external field. This defor- mation will in course of time be considerable even if the external electric force is very small compared with the force of attraction between the particles. The orbit of the electron may at any moment be considered as an ellipse with the nucleus in the focus, and the length of the major axis will approximately remain constant, but the effect of the field wiil consist in a gradual variation of the direction of the major axis as well as the excentricity of the orbit. A detailed investigation of the very complicated motion of the electron was not attempted, but it was simply pointed out that the pro- blem allows of two stationary orbits of the electron, and that these may be taken as representing two possible stationary states. In these orbits the excentricity is equal to 1, and the major axis parallel to the external force ; the orbits simply consisting of a straight line through the nucleus parallel to: the axis of the field, one on each side of it. It can very simply be shown that the mean value of the potential energy relative to the field for these rectilinear orbits is equal to +3/2 aeH, where EH is the external electric force and 2a the major axis of the orbit, and the two signs correspond to orbits in which the direction of the major axis from the nucleus is the same or’ opposite to that of the electric force respectively. Using the formulee (4) and (5) and neglecting the mass of the electron compared with that of the nucleus, we get, therefore, for the energy of the system in the two states 27cm 1 — ae ea ee + eNGh eae wn) EP respectively. This expression is the same as that deduced in paper LV. by an application of (6) to the expressions for the energy and frequency of the system. Applying the relation (1) and using the same arguments asin paper LV. p.515, we are therefore led to expect that the hydrogen spectrum in an electric field will contain two components polarized parallel to the field and of a frequency given by Ls 2ar2e*m ( 1 ‘ie get 3h y= -(A,, —A,.) =N? — = —. } 3. } —_____(n,?—n,”), (9) i! m—An,) ae Ts" Sa?Nem ‘ 2 v) (9) The table below contains Stark’s recent measurements of the frequency difference between the two strong outer com- ponents polarized parallel to the field for the five first lines 2D2 A,=—N? 404 Dr. N. Bohr on the Quantum Theory of in the Balmer series *. The first column gives the values for the numbers n,and m;._ The second and fourth columns give the frequency difference Av corresponding to a field of 28500 and 74000 volts per em. respectively. The third and fifth columns give the values of Aarem a= Ap: 3 Hh ( 192 —n,")’ where « should be a constant for all the lines and equal to unity. 28500 volts. per cm. 74000 volts. per cm. | N, Ny Any ie a | Av. Oko a 2 0:46 $33 | Ki | 9 4 1-04 0:79 386 | 088 ao 2:06 0:89 541 | 0:90 Zt 0 3°16 0:90 781 | 0°85 eed £47 0:90 | | Considering the difficulties of accurate measurement of the quantities involved, it will be seen that the agreement with regard to the variation of the frequency difterences from line to line is very good. The fact that all the observed values are a little smaller than the calculated may be due to a slight over-estimate of the intensity of the fields used in the experiments (see Stark, loc. cit. pp. 38 and 118). Besides the two strong outer components polarized parallel to the field, Stark’s experiments have revealed a large number of inner weaker components polarized in the same way, and also a number of components polarized perpendicular to the field. This complexity of the phenomenon, however, cannot be con- sidered as inconsistent with the theory. The above simple calculations deal only with the two extreme cases, and we may expect to find a number of stationary states correspond- ing to orbits of smaller excentricity. In a discussion of such non-periodic orbits, however, the general principles applied are no longer sufficient guidance. Apart from the agreement with the calculations, Stark’s experiments seem to give strong support to the interpretation of the origin of the two outer components. It was found . that the two outer components have not always equal inten- sities; when the spectrum is produced by positive rays, it * Stark, Joc. cit. pp. 51, 54, 55, & 56. = ei) + - - —s oe 7, ——_ Theory of the Optical Properties of Metals. 435 complete specification of the effect of resonance electrons which is omitted by all the above authors *. The results of the present investigation seem to require special emphasis as completing in the present respect the work of Thomson and Jeans in the general theory of radia- tion. These two authors have examined in great detail the emission of light from a metal of the type under considera- tiont. They agree in the conclusion that the most probable law of force is the inverse cube law, this alone leading to results which are consistent with the special as well as the general laws of radiation. Reference may also be made to an elaborate investigation carried out by Prof. McLaren tf, who attempts to obtain absolute generality in the problem. His main result is that the absorption of energy from radiation in the metal must ultimately be determined by a function of exactly the same type as that which deter- mines the emission, 7. e. it must fall off exponentially as the wave-length decreases. It appears, however, that McLaren’s analysis, which is not so general as at first sight appears, will not stand the test of application in particular cases and must therefore ultimately involve some oversight. McLaren states that his analysis involves no assumption whatever, which is equivalent to neglecting the Doppler effect, pres- sure of radiation or the square of the velocity of an electron to that of radiation (varying mass of an electron) ; but, as I understand his analysis, it appears that his dynamical equa- tions, which are the particular ones usually associated with Hamilton’s name, and the analysis he bases on them, do ip fact involve a neglect of all these factors. It is well known that Hamilton’s equations, in the particular form in which McLaren uses them, are restricted for application to systems in which the kinetic energy is expressible as a homogeneous quadratic function of the generalised momenta, and this is certainly not the case unless the velocity of the electrons is small compared with that of light. Secondly, the analysis of the emission of the light as given by McLaren is appropriate only so far as the effect of the collisions does not impress itself on the radiation formula: a con- dition which necessarily restricts it for application to long wave radiation only as in the simple case examined by Lorentz. * Inskog does attempt to include this effect, but only partially, as he neglects entirely the action of the internal local field on the conduction electrons. + Thomson, Phil. Mag. Aug. 1907; Jeans, Phil. Mag. June-July, 1909. ¢ Phil. Mag. July, 1911. » = 2 436 Mr. G. H. Livens on the Electron In any case it would appear quite hopeless under present circumstances to attempt to obtain the great generality which Mclaren claims for his theory, because if we are prepared to disregard the restriction as to smallness of velocity of the electron we must equally disregard altogether the whole concept of the electron as a definite entity, and -we must then include in its specification not only the position of its nucleus but also a definition of the condition of the ‘whole of the surrounding ethereal strain field, which but loosely follows the nucleus in its motion, unless that motion is comparatively small or at least of the quasi-stationary type. oho advantage can therefore be gained by attempting such generality of procedure, and no excuse is therefore required for offering the analysis for the more special case, which is, after all, sufficiently appropriate for our purposes. 2. The Basis of the Theory. The whole theory turns on the evaluation of a function f which determines the statistical distribution of the motions among the electrons at any point in the metal. This function is such that, at any point in the metal whose coordinates referred to a definitely chosen system of rectangular axes are (x, y, z), the number of electrons per unit volume with their velocity components (£, 7, €) in the limits between (&, 7, €) and (&€+dé, n+dn, €+d€) is fag dy db, and it is, of course, a function of (£, n, €) (a, y, z) and ¢, the time. If the electrons in the metal are subject to action by external fields, the results of which may be specified by the acceleration (X, Y, Z) which they impose on the typical electron, then the function 7 is shown to satisfy the dif- ferential equation of Of Sip Oi Pe OLT til Oia NOME weet ae art on tA oe - die 1 =e ana b—a, where (b—a)dé dy af dt denotes the increase of the specified group of electrons during the next succeeding small interval dt, owing to the collisions taking place in this interval. The present writer has shown in another place that under the assumption that the dynamical character of the collision between an electron and an atom is similar to that between a fixed centre of force, repelling the particle in such a -way Theory of the Optical Properties of Metals. 437 that the potential energy of the particle at a distance r from the centre of the force is then f—to Sep oF nm b—-a=-— wherein, if denotes the number of atoms per unit volume: in the metal Tn = E i anpe—l where Wim +a +e and i= 4{ cos? 6, a dz. 0 The relation between @) and « being defined by the equation where v% is the only real positive root of the equation 1 riated =|, pe A We have also used “ght an to — na /S, e*, wherein N denotes the total number of electrons per unit volume, and g a constant connected with the mean square ot their velocities (w,”) by the relation I£ we now use 438 Mr. G. H. Livens on the Electron so that l m — 49 Vian then the differential equation for f is Of , vy Of ..7 Oh) eo! 1 sO) Ojon Mae ty 5, oe on ae cil el a na on Throughout the present paper we may assume that the thermal conditions are uniform throughout the metal, so that the function / will not depend on the coordinates (a, y, 2). We may also restrict our analysis to the simple case when Y=Z=0, so that the differential equation for f is of the form of foe T—ho ie Fe ee es o€ oF ot T. me The general solution of this equation appropriate to the present type of problem is easily obtained and can be written in the form Tv t,=t Me Gee oe al: ( Ofo pany ae m x ——— at “ P=Jo i) i A Oe: : The suffix 1 indicating that the integrand, interpreted explicitly as a function of the time ¢, is to be taken for the time ¢,. Using the value of fy) given above, this becomes th ee! dt Ch= fan /%, eg E a 2q ( é ik Be ( EX,dt, | ‘ «0 =t—r Tmt, — ig | We may now, as in all these problems, neglect squares of the accelerations produced by the external fields; this means that we can ignore the variable part of €, and thus we may write > ae ES i d t,=t j= N n/ S en”. E + 2ue\ e “m = Xie, | 3 0 m 60 ee Theory of the Optical Properties of Metals. 439 The applications of this function in the ea paper are to be made with X periodic—say, of ek _" ; we may ¢hen use X =X e?". It is then easily found that t=ho ee ex | ~—] 3 ie or, since T' =/ ? mn 4 = ; 2Qql_u? X we have t=ho Ce sll \ 1+zpl_u* The’current of electricity in the metal in this case is directed entirely along the axis of x and is of a density +a ("+0 (?+0 =| { i} e&f dé dn dE Ae iy 3 ay ye ne x 1+<¢pr,, This integral may be transformed by the usual spherical polar transformation, and thereby pee at once to i ie hae Ee q° ee wy 8S Tae cE 1+ ipl nu” If we now use cag’, po ?= Pm this integral becomes eas ee ie a eh ‘pe Gi eae Days and it is not hee to reduce it further, except perhaps to another irreducible integral of the integral- logarithmic type. 440 Mr. G. H. Livens on the Electron 3. The Optical Equations. The general optical equations for the metal are identical with those contained in the former paper, and need not therefore be developed in detail in the present instance. We may consider the general propagation of a plane homogeneous wave train in the positive direction of the axis of z, so that all functions specifying the propagation are dependent on the time and space variables only through - the factor 6 ia gint—q), We may also assume that the waves are polarized so that the electric force vector E is in the direction of the z-axis, so that in the metal xa =: The whole of the circumstances of the propagation are then determined by the generalized quasi-index of refrac- tion, which is in general a complex quantity (w—7k), the real part (u) of which determines the true refractive index and the imaginary part (k) the absorption. From the optical equations it is then easily deduced that : C 1 A Bay Ry Dial Ni NG Ieee) ea ip aI a Ke wherein (eo) 2 anne (eae ,_. 4re7l,,N vi, q° o ee C= awe 7 201 omg ne 9 1+(8c)? ‘ 2 and Ae Sey rte tpn, — PP which is a sum arising from the presence of the resonance electrons bound by quasi-elastic forces in the interior of the atoms, and is in tact taken over all of these electrons, each with a proper frequency n, and resistance coefficient (mn,). The constant a which occurs in this relation is a numerical constant of which an ideal estimate is 1/3. If we write C — OF + iCo, A Wp 2 ———— and (p,—1k,) te aN Theory of the Optical Properties of Metals. A4} so that oe 2 PREP A Abts Avre?l,,N g3 o i Pete. ik = nee See AL one TRIP URE MT TAN) ee ee as : 3mq 1+ (Be) con eels 44 Amr pel, N g a 8 %de re fie 2 a3 a ee 3mgs * 9 1+ (Ba)° and 1—aA then it is easily verified that N we— he = pr — he? — (1 + apr? — k,? —1)) Cs + ta es P 4 ph=p,k,+(1+alp— k,—1)) . + Zap ‘ If we are dealing with light of a period somewhat removed from the proper periods of the free vibratiuns of the resonance electrons, there will be no absorption due to these, and thus k,=0 and the formule reduce to ; —— 0 we—h=p,— (1+ ap,” — 1) i Y— “ay Arel,N (1 +ap ih tad oa 72 3mqs 1+ (Bo)s eg he, a ee eel @ 2 He 2are?l,,N (1 + apy’ — 1) gs | at tee 0 3mpq AM These are the general results of the more detailed theory. If we use o, for the conductivity for steady currents, and pr, the value of w, when p=0, then the last equation may be written in the form 2 wh= 1 +a (pr? —1) , 70 oF ¥e "de 1+ a(u,—1) 2 i” ro pl (5 +2) | Rage Ss 1+ (8c) s It is obvious that the present investigation does not materially affect the general comparison of the theoretical ihe Die o® *2¢ "de 73 4 bi A492 Mr. G. H. Livens on the Electron and practical results *, which is possible in the present state of the subject, as our knowledge in this branch of work is hardly complex enough for a discrimination between the different forms of the theory. One or two general remarks bearing on the application of these results may, however, not be out of place. The formule here obtained exhibit very clearly the exact limitations of the simpler Maxwellian theory, which is applicable so long as the period of the light-waves is large compared with the mean time of description of a free path. This has been experimentally verified by the experiments of Hagen and Rubens, who found that complete metallic ‘conduction is fully established in a small fraction of the period of ultra-red radiation, up to which limit all radiation is reflected from all metals in proportions determined by their ohmic conductivities alone. The simple relation pavel my required by Maxwell’s theory is modified, firstly, by the factor nk 1+a(u,—1) L+a(uz—1)’ which arises from the resonance electrons, and, secondly, by the factor 2 ae 2 d EG P28) 74 ¢es): which is a function of the period of the light used. The theory thus contains an effective account of the observed departures from the simpler relation. For long-wave radiation the square of the quasi-index of refraction is a purely imaginary quantity ; for the opposite extreme case of very short waves it appears that, in the absence of absorption due to the resonance electrons, the square of the quasi-index must be a real negative but small quantity. The optical determinations of Drude indicate that this is not very far from being the case with light- waves for some of the nobler metals, although, as the more >) —1 * See Enskog’s paper previously referred to, where a comparison is attempted. His results, however, require the modification necessitated by the more complete theory, which includes the full effect of the resonance electrons. Theory of the Optical Properties of Metals. 443 general theory indicates, the presence of the resonance electrons may considerabiy modify this statement *. 4. The Emission of Energy and the Complete Radiation Formula. Although I have no intention of entering into an analysis of the radiation from a-metal on the basis of the present form of the theory, as this side of the problem has already been amply dealt with in the work of Jeans and Thomson, I should, nevertheless, like to take this opportunity of making a few remarks bearing on this subject, with a view to emphasizing some aspects of the theory which appear to have been overlooked in the discussions of the more general problem. In a previous paper on the subject of the emission of light from a metal, I have ventured to give in full detaila dis- cussion of Lorentz’s theory, which regards the radiation from a metal as arising from the motions of the free electrons inside the metal, basing myself mainly on the general assumptions of the simpler form of the theory, which regards the electrons and atoms as elastic spheres. The final result of this analysis is that if it is possible to assume that the time of duration of an encounter of an electron with a molecule is always negligibly small, then the emission is of such a type that the Rayleigh-Jeans formula for the com- plete radiation is generally applicable all along the spectrum. This absurd conclusion arises, however, as before stated, from the fact that the assumption on which the analysis is based virtually implies that the total amount of energy radiated from the moving electrons, depending essentially on their accelerations, is of definite amount. It is therefore suggested that the theory and the Rayleigh-Jeans formula which is derived from it are considerably restricted in their applicability to the actual state of affairs, and are valid only so far as the general nature of the collisions between the molecules and electrons are not of primary importance in determining the radiation from the electrons. Ultimately, of course, the complete radiation formula must contain some account of the general dynamical nature of these collisions, although it is to be expected that any very special character- istics would be eliminated by the usnal statistical methods of analysis, and such a conclusion is amply borne out by the researches of Thomson and Jeans. * Vide J. Larmor, “ On the Range of Freedom of Electrons in Metals,” Phil. Mag. August 1907. 444 Electron Theory of the Optical Properties of Metals. The calculations of Thomson and Jeans prove decisively that the energy emission in a particular wave-length is a function of that wave-length which falls off exponentially as the wave-length is increased. The above calculations. show that the absorption diminishes as a function of the wave-length like r atn” The complete radiation formula determined by the ratio of the emissivity to the absorption is therefore precisely of the general form required by experience, although the difficulties of the analysis preclude the deduction of any simple relation. It must therefore be insisted that although the theory in its present state may be too indefinite in its details to give anything beyond a tentative account of the general phe- nomena of the radiation from the metal, the general conclusions which can be drawn from it are in themselves: not materially inconsistent with the known facts as contvined in the experimental results ; and that therefore there seems: to be no imperative need for resorting to a wholesale and phantastical modification of the general basis of physical theories, on the grounds that such a modification appears at: first sight to be required in one aspect of the general problem. In his attack on the theory McLaren comes to a conclusion which is just the opposite of that just stated, and he attempts. to support his result by a general assertion that on alk dynamical principles the absorption is a function of the same type as the emission. There does not, however, appear to be any legitimate ground for such an assertion, which, as. experience and theory prove, is only true so long as the wave-length is long compared with the mean time of the free molecular motions in matter. It is to be insisted that. the two phenomena of absorption and emission are essentially ditferent in their mean aspects, and this difference even extends. to the dynamical characteristics of the phenomena. Absorp- tion is essentially a free path phenomenon, whilst the emission must ultimately depend in its main features on the collisions at the ends of the free paths, and must therefore contain a much more intimate account of these collisions than any other phenomenon, which is, practically speaking, in no way concerned with them, except, perhaps, in so far as they temporarily remove the electrons concerned from con- sideration in the treatment of the phenomenon. Whilst AVotices respecting New Books. 445 the absorption is mainly concerned with the organizing effect of the electric force in the incident radiation fields, the emission is more concerned with the disorganizing col- lisions, and it is only in the case of very slow alternations that a balance between the two has time to be established before the one or the other is modified by the altering field or by the collisions. The University, Sheffield, Jan. 4th, 1915. XLVI. Notices respecting New Books. Calculus Made Easy: Being a very-simplest Introduction to those beautiful Methods of Reckoning which are generally called by the terrifying Names of the Differential and the Integral Calculus. By F.R.S. Second Edition, enlarged. Macmillan & Co., Ltd.,1914. Pp. xi+265. Price 2s. net. HE motto of this little book is : ‘‘What one fool can do, another can.” It makes, in fact, some portions of the Calculus “easy,” and, putting aside all claims of modern rigour, it can be said that the facilitation attained has been achieved without serious sacrifices of correctness. Also there is no doubt that the bovk has been and will be useful to some classes of beginners. But one thing is to be regret'ed : the author has diligently inter- ealated at every step such words and notions as ‘“‘ dodges,” “tricks,” ‘« pitfalls,” ‘‘ triumphs,” &c., and used throughout a corresponding style of exposition, to such a degree, that the reader, having gone through all the little chapters, will most likely experience the feeling of having conquered some hostile camp (where “ trade- secrets” have been kept in custody by the caste of “ professional mathematicians ”) instead of enjoying the acquired knowledge of areally beautiful domain of human thought. Faraday, Tyndall,and Clifford more especially, have taught us that almost everything ean be popularized in the good sense of the word, 7. e. without being unjust to the more subtle sides of the subject and without leaving on it the brutal stamp of practical human concerns. Frenchmen express this requirement shortly by distinguishing * popularization ” from “ vulgarization.” In spite of the author’s warning (cf. “‘ Epilogue and Apologue ” there are in the book not many “ most grievous and deplorable errors,” and the few that the reviewer has remarked will certainly do no harm to the beginner. They will simply pass unobserved. But such a statement as one reads on p. 204, viz.:— “Tf the figure of the solid be expressed by the function F (x, y, z), then the whole solid will have the volume-integral, volume = is (x, y,2). dz. dy. dz,” will certainly require a careful correction in a future edition. Again, remarks such as that concerning \a-2 dx, on p. 200, seem rather misleading and, to say the least, unnecessary. [ 446 J XLVI. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from vol. xxix. p. 848.] February 3rd, 1915.—Dr. A. Smith Woodward, F.R.S., President, : in the Chair. (pede following communications were read :— 1. ‘On the Gravels of East Anglia.’ By Prof. T. McKenny Hughes, M.A., F.R.S., F.G.S. The author discusses the sources from which the subangular gravels that cover such large areas in Hast Anglia can have been derived. He points out that their great variety of fracture, colour, ete. proves that they cannot have come directly from the Chalk, nor from Boulder Clay derived directly from the Chalk, nor from the Lower London Tertiaries, none of which contain subangular gravels. but only beds of pebbles, and those mostly of small size. The character of the flints in the gravels indicates that they have been derived from surface-soils which have been winnowed and shifted by soil-creep, rain, and streams, until arrested on the terraces and flats of the valleys. The dry land of Miocene age was the first over which the flints. of our gravel-beds could have received that subaérial treatment which they all seem to have undergone. There was then no Boulder Clay to protect and obscure the Chalk-with-Flints. Then came the submergence which let in the Crag Sea. This. rapidly invaded the land, not giving time to reduce the flints to pebbles but burying remains of animals and plants in coarse sub- angular gravel. In time the subsidence affected more distant shores, and compensating rises of mountain-regions far away began to modify chmatal conditions; ice floated southwards, stranding at. various depths according to size, ploughed up and crumpled the shore-deposits, and dropped masses of far-transported material. It is not difficult to distinguish the old shore-deposits, even whem thev have been crumpled up, from the foreign material introduced by the floating ice. When the land had sunk so low that the wind- and tide-waves. could not sort the material, it remained, as brought, a boulder-clay, which is therefore widely spread over the peneplain of the East. Anglian heights, and is generally above the gravel and sand of the advancing sea. Where the sea was able to work longer at pounding and rolling the flints, immense beds of pebbly shingle or of sand are the result. The Plateau Gravel is traced from section to section across the country, and the characteristics by which it can be recognized are- pointed out. Geological Society. AAT Since an irregular land-surface was thus depressed beneath the sea, one might expect that in some of the deeper valleys deposits. older than the submergence might be detected. Also, seeing that the land has slowly risen again many hundred feet, we ought to have evidence of the subaérial denudation which has been going on since the land began to rise. Thus the author starts with the definition of three important ages of long duration, and proceeds to refer some of the best exposed. gravel-deposits of East Anglia to one or other of them. They are, in descending order :— (A) The stage of which the Barnwell Gravel is taken as a type. (B) The stage of which the Plateau Gravel is the most important representative. (C) The stage to which he suggests that the Barrington Beds may belong. The rest of the paper consists of descriptions of sections, and. discussion of evidence derived from fossil remains. 2. ‘The Pitchstones of Mull and their Genesis.’ By Ernest Masson Anderson, B.Sc., M.A., F.G.S., and E. G. Radley. The pitchstones here discussed occur with extraordinary frequency, intruded into the Tertiary plateau-lavas of the eastern portion of the Ross of Mull, as well as in less number in other parts of the island. They fall into two main divisions, distinguished by the absence or by the presence of porphyritic felspars. ‘Those of the non- porphyritic class are the most prevalent, and usually form the central portion of sills or inclined sheets. The marginal portion of these intrusions is crystalline or stony. The petrological characters of these pitchstones, and their more crystalline margins, are such that they seem to warrant the grouping of the rocks under a new type-name, and the name leidleite has been chosen. The por- phyritic pitchstones occur as flat or gently-inclined sheets ; they also are associated with a more crystalline phase, and have been grouped under the type-name inninmorite. The relation of the stony margins of the pitchstone-intrusions to their glassy centres is usually seen clearly. A typical leidleite may have 5 feet of pitchstone in the centre, with margins of stony matter 3 feet thick on each side. Occasionally, the central glassy portion may be split up by stony partings. A feature that occurs very frequently in these rocks is what has been termed ‘sheath-and-core’ structure. In this case, the stony base and top of an intrusion send off narrow sheets of stony character which traverse the glassy portion in a branching and sinuous manner. The glassy nature of the cores is clearly not due to a greater rapidity of cooling; but, with the object of ascer- taining the reason for the devitrification, a chemical investigation of both the glassy and the stony portions was undertaken. 448 Geological Society. It has been found that there is a much greater percentage of water given off from the rock at 105° C. in the case of the glassy variety, and the authors suggest that the escape of this excess of water, soon after the consolidation of the rock, has resulted in the devitrification of the sheaths and margins. February 24th.—Dr. A. Smith Woodward, F.R.S., President, in the Chair. The following communications were read :— 1. ‘The Ashgillian Succession in the Tract to the West of Coniston Lake.’ By John Edward Marr, Sc.D., F.R.S., F.G:S. 2. ‘The Radioactive Methods of Determining Geological Time.’ By H. 8. Shelton, B.Sc. The radioactive method of determining geological time, while of great interest, is not of such certainty as to be independent of con- firmation from other lines of investigation. The various radio- active methods, helium ratios, lead ratios, and pleochroic haloes are severally examined, and the various sources of uncertainty, general and particular, are pointed out. ‘The most important general ‘cause of uncertainty is to be found in the fact that mechanical and chemical changes of composition in minerals are the rule rather than the exception; and, in instances where constancy of com- position throughout long periods of geological time is asserted, the burden of proof les with those who make the assumption. The attempt to assess exact, or even approximate times by means ‘of lead ratios is premature and entirely invalid. At the same time, the weight of the evidence is such as to render it exceedingly probable, so far as radioactive evidence goes, that geological time must be reckoned at least in hundreds of millions of years. There is a high degree of improbability that the errors in the radioactive methods should always be errors of overestimation. 'The next step in the investigation of the time problem is to be found in a rever- sion to other lines of reasoning. ‘The sea-salt methods, and those based on the thickness of the sedimentary rocks in particular, need. careful reconsideration. Reference is made to a number of papers which show that the first of these is worthless, and the second based on a misapprehension of the nature of deposition. The argument from tidal retardation is still of value, as also is that from the evolution of carbonate of lime. To the author radioactive experi- ments come as a confirmation of views held on other grounds, but are not sufficiently important in themselves to be authoritative against the balance of the evidence derived from other lines of investigation. McLennan & Murray. Phil. Mag. Ser. 6, Vol. 30, Pl. V. | RG oe Frq. 4. Freeros ‘ ay ™ 7 4 = oe . + ; ty e > oo c : ; _ wv Lode! . 5 ; : ) = ‘ r ¥ ™ Fool ae ebm aM caride can tee a S a ere Mrs doe mre late DO a rate in sah oOo peotumeenny umn os navn anteynd nV ees 7s Pale t : Ai aie . : igs ii SRS LAVA y, iD tear tke us re ri x THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. Fae ae ee ie [SIXTH SERIES‘] WOrOnER Ainge ver)" | ke / ~~ —. _ herpes XLVI. A Further Study of the Fluorescence ~produced by Ultra-Schumann Rays. By C. F. Meyer and R. W. Woop*. (Plate VI.] N 1910 one of the writers ¢ described experiments showing the existence of a radiant emission from the spark which had not been previously detected. The subject was more fully investigated in collaboration with G. A. Hemsalech f, and many interesting phenomena observed. All experiments indicated that the emission which was being studied con- sisted of ultra-Schumann waves. No method was found, however, of determining their wave-length, and the experi- mental difficulties throughout the work were so great that many phenomena were only incompletely studied, and many points were left uncertain. The present authors have there- fore attacked the problem anew, and the results they have obtained will be discussed in this paper. The radiant emission in question cannot be directly observed or photographed, but its existence is shown by the fluorescence which it causes in certain gases. The essential parts of the apparatus used in its study were the sume as in the above-mentioned investigations, except that the box forming the jet-chamber this time consisted of metal instead of wood. * Communicated by the Authors, + Wood, Phil. Mag. [6] xx. p. 707 (1910). t+ Wood and Hemsalech, Phil. Mag. [6] xxvii. p. 899 (1914). Phil. Mag. 8.6. Vol. 0. No. 178. Oct. 1915. 2G 450 Mr. C. F. Meyer and Prof. R. W. Wood on Apparatus. Referring to fig. 1, A is a circular copper plate 3 mm. thick and ¢°5 cm. in diameter, part of the plate being represented in the diagram as cut away. ‘The plate rested on and was sealed to a short piece of brass tubing 7 cm. in Fig. 1. diameter and 1 em. long, which in turn was soldered over an opening of nearly as great diameter in the bottom of the metal box J. Through a hole in the centre of this plate a truncated portion of a copper rivet had been driven. The rivet had a vertical slit S cut into it, 2mm. long and -2 mm. wide. B isa piece of heavy copper wire. The copper rivet in the centre of plate A and the wire B served as terminals between which a spark froma transformer was passed. The radiation from the spark passed up through the slit 8 into the jet- chamber J, together with the ordinary visible and ultra-violet light. The special radiation, which we may call ultra- Schumann radiation, causes a short jet of ultra-violet fluor- escence in the air or other gas above the slit S in the jet-chamber J. This fluorescence was photographed through the quartz window W, by means of the quartz spectroscope represented diagrammatically by the lenses L, the prism P, and the photographie plate Q@. ‘The spectroscope had had its slit-tube removed, the jet itself servingas slit. The auxiliary chamber K is added to the apparatus in order to ensure a dark background. The chambers J and K (13x9x9 em.), which consisted of metal boxes blackened inside, served as a dark enclosure, and also as a container for the parlicular gas in which it might be desired to study the jet. The tube T served for the introduction of the desired gas, Fluorescence produced by Ultra-Schumann Rays. 451 and was so placed that the stream delivered by it passed directly over the slits, 7. e. through the fluorescent jet. The arrangement of the plate A, slit S, lower electrode B, and the tube T can also be seen in the small sectional diagram of tig. 1, which is drawn as seen by an observer looking from the direction of the spectroscope. The glass window W, was introduced for the purpose of determining whether the radiations from the spark were coming up through the slit properly. During some of the experimental work, all lines along which parts of the apparatus joined were sealed up with wax, and the bottom of the box J was sealed with mercury. During other parts of the work the lines of juncture were left unsealed. The spark was produced bya 3 kilowatt, 110 volt, 60 cycle transformer. Across the spark-gap was placed a condenser consisting of 36 copper plates, 15x20 cm., insulated by ordinary window-glass, the whole being immersed in oil. No influence of the nature of the spark upon the nature of the fluorescent spectrum of the jet was ever ascertained, except that a strong spark gives a more intense fluorescence than a weak one. The authors do not feel able to state, however, that there is no influence at all upon the nature of the fluorescence, the difficulty of ascertaining such an in- fluence lying in the fact that it is difficult to vary the spark and keep all other conditions constant. The Fluorescent Spectrum in Various Gases. In the work of Wood and Hemsalech above referred to, it was found that the strongest fluorescence of the jet was obtained when nitrogen was used in the jet-chamber, the spectrum then consisting of the water band 23064 A.U., and under favourable circumstances the nitrogen bands. Their nitrogen, however, contained impurities, especially OXY gen ; and the present authors thought it to be of interest to determine the effect of remov ing the last trace of oxygen. Nitrogen obtained from a bomb was accordingly cleared of the oxygen it contained by the method described by ©. Van Brunt*, dried, and passed into the jet-chamber. Plate VI. fig. 1 shows the spectrum with commercial nitrogen, containing about 1 per cent. oxygen. Fig. 2 shows the effect of removing the oxygen. ‘he horizontal Jine in these photographs is the continuous spectrum due to the light from the spark diffusely reflected by the upper * Journ. American Chem, Soc, July 1914. 452 Mr, C, F. Meyer and Prof. R. W. Wood on edges of the slit S. The spectrum appears asa line since the slit is presented to the spectroscope “end-on.” The rather diffuse vertical streamers are the monochromatic images of the jet, dispersed by the spectroscope. Long wave-leneths are toward the left. The streamer marked with a dot is the water band which is interrupted where the current of nitrogen delivered by the tube T crosses the jet, for reasons which will be further discussed. The position of the tube is indicated. To the left of the water band are the nitrogen bands. In the commercial nitrogen only the first and second, counting from the water band, are present ; while in the purified nitrogen the first (3369) is faint, and the second and third (3556 and 3778) are prominent. Some time after these photographs were obtained, when the appa- ratus had all been taken down, cleaned, and set up again, a different type of spectrum was obtained with purified nitrogen, in which the first and second nitrogen bands were faint and the third strong (fig. 3). The type of spectrum in the commercial nitrogen remained the same. The cause of the difference between fig. 2 and fig. 3 was not discovered even after making a number of exposures under various conditions for the purpose of ascertaining it. Some little time was spent in repeating the more important parts of the work of Wood and Hemsalech, done in Paris. With entirely new apparatus, and different sources of supply of our gases, these attempts at repetition often resulted at first in very perplexing failures. Investigation into the. cause of these preliminary failures, however, resulted in most cases in throwing much light on the phenomena themselves. For example, the first attempts to duplicate the photographs (1. c. figs. 5, 6, 7) obtained with moving and stationary gases. were unsuccessful. In these a stream of nitrogen was blown across the fluorescent jet, the stream causing an interruption of the water band, and the appearance of the nitrogen bands. in the moving stream. Later attempts showed that in order to obtain the interruption of the water band it was best to. bring the mouth of the tube T close up to the slit S (say within *> cm.). Hxperiments made with smoke indicated, however, that the stream of gas delivered by the tube at the velocities which we were using maintained approximately its form and area of cross-section, and consequently its velocity, for several centimetres beyond the mouth of the tube. This suggested that the appearance of the water band in the stationary portion of the gas, and its disappearance in the moving stream, might be due, not primarily to the rest or motion of the gas with reference to the fluorescent jet, but. Fluorescence produced by Ultra-Schumann Rays. 453 to a difference in the constitution of the gas just leaving the tube T, and that immediately around the stream, even though the jet-chamber was entirely closed, and had been washed out with a stream of nitrogen delivering a litre and a half per minute for four minutes before beginning the exposure. Figs. 1, 2, and 3 (Pl. VI.) show the type of spectrum obtained with the tube close up to the slit and the jet-chamber washed out in advance. To increase the difference in constitution between the gas in the stream and that surrounding it, an opening was made in the jet-chamber to admit air. Purified nitrogen was blown across the slit and an exposure made. The result is shown in fig. 4. It is seen that the water band is again interrupted where the nitrogen crosses it, and that the nitrogen bands show only in the stream. An attempt was also made to explain the interruption of the water band found when a stream of air was blown over the slit S, and the jet-chamber was filled with air (J. c. p. 905, fig. 13), as being due to a residual difference between the air in the stream and that in the jet-chamber. When air from the room, which of course also filled our apparatus, was blown by means of a bellows through the tube T over the slit, no interruption of the water band was shown. When the air was moistened by passing through wet cotton still no interruption was shown; but when the air from the bellows was dried and blown over the slit, and a source of moisture was provided in the jet-chamber, so that the stagnant air around the air current might take up moisture, the photograph of fig. 5 was obtained. Moreover, when a strong current of dried air (about 2°5 litres per minute) was blown into the apparatus for five minutes before exposure was begun, and no source of moisture was provided in the jet-chamber, the fluorescence was so faint that it could not be photographed in fifteen minutes, which is about the time of exposure of the other photographs. It thus appears that water vapour is necessary to obtain the fluorescence of the water band, while the presence of oxygen mixed with nitrogen, either in large or small quantities, will not give it. An exposure was also made in which nitrogen purified and subsequently moistened was blown across the slit S. This plate shows the water band uninterrupted where the current of nitrogen crosses the fluorescent jet; shows faintly two of the three nitrogen bands 3369 and 3556 on the long wave-length side of the water band, and a fourth band or line on the short wave-length side, probably the fainter water band 2811. The fact that the water band is interrupted when the stream 454 Mr. C. F. Meyer and Prof. R. W. Wood on of nitrogen delivered bythe tube T is perfectly dry, but is no longer interrupted if the nitrogen is moistened, would lead us to believe that if the stagnant gas in the jet-chamber consisted of entirely pure dry nitrogen, then the fluorescent spectrum would consist of the three nitrogen bands only. Experiments tried with gases other than nitr ogen and air were not carried far enough to lead to results of sufficient interest and certainty to warrant discussion, except in the case of iodine vapour. Some crystals of iodine were placed in a glass tube through which nitrogen was passed, and the resulting mixture of iodine vapour and nitrogen was blown from the tube T across the slit. The iodine fluoresces in the visible region with a bluish-green light as was noted by Wood and Hemsalech (le espe Goi: and in the ultra-violet. A spectro-photograph of the streamer in ultra-violet is repro- duced in fig. 6. This fluorescence was so bright that it could be easily photographed by throwing an image of the fluorescent jet upon the slit of a spectr oscope with a quartz lens, thus obtaining its spectrum -in very much greater detail. This was very y kindly done for us by Mr. Voss of this laboratory, and the result is shown by the photograph of fig. 7, which was made with an improvised quartz spectrograph of the Littrow type, furnished with two Cornu prisms and a lens of about 2 metres focus. The comparison spectrum is that. of the iron are. The band is highly monochromatic, as will appear from the wave-lengths of the principal lines which make up the band: they are as follows :— aore'd d434°8 3388'8 3418°5 3400°0 3423°9 3406°3 3426°4 3408°8 b431°4 3413°3 0435°3 The spectrum of the visible bluish-green light has not yet been photographed, as it is very much fainter than the ultra. violet band. Transmission, Refraction, and Reflexion. In the study of the transmission of the rays exciting the fluorescent Jet an entirely new jet-apparatus was used, the spectroscope and electrical equipment remaining the same. This new apparatus was based on the same general priaciples. as the old one, but differed in being smaller; in providing for the delivery of the fresh gas from the rear through a large co) tube (1°5 cm. in diameter), which at the same time ‘formed a Fluorescence produced by Ultra-Schumann Rays. 455 dark background for the jet; in having a special mounting for the truncated copper rivet with the slit forming the one electrode, and the copper wire forming the other, so that these could be readily removed and replaced to allow for cleaning of the plate the transmission of which was being tested. Moreover, the entire jet-apparatus was mounted ina position inverted with respect to that shown in fig. 1, so that the spark was above, while the jet pointed down. The new form and mounting are represented diagrammatically in fig. 2 in the text. J is again the jet-chamber, T the tube delivering the gas, also serving as dark background, W, the window through which the jet is photographed, R the rivet having a slit in it in the plane of the drawing, B the wire forming the other electrode. R is driven through a thin copper plate A; A and B are both fastened to an arm which rotates about a horizontal axis in the plane of the drawing, so that both may be readily raised, by raising this arm, and then replaced. E is the plate or lamina whose transparency is to be tested ; A’ is a thin brass plate upon which Hi rests, and serves also to prevent light scattered by the edges and surfaces of Ik from reaching the spectroscope. The spectroscope is on tie left as in fig. 1. The first experiments in transmission were made with a small disk of clear colourless fluorite °57 mm. thick. Fig. 8 (P1. VI.) shows the transmission of the rays exciting the fluor- escence of the water band. Long wave-lengths are now on the right. It will be noticed that the vertical streamer is much 456 Mr. C. F. Meyer and Prof. R. W. Wood on wider in this than in the previous photographs, this being due to the use of a very much wider slit (45 mm.) in this part of the work. The gas passing into the jet-chamber through the tube T consisted of nitrogen taken directly from the bomb, and the fluorescent spectrum when the fluorite plate was not in the apparatus consisted of one very broad streamer indicating the water band and the one or two nitrogen bands nearest it all merging into each other. When the fluorite plate was inserted only the water band appeared, showing that the fluorite was more transparent to the radiation ex- citing the water band than to that exciting the nitrogen bands. Three exposures made under approximately the same conditions indicated that an exposure of from fifty to seventy times as long was necessary to obtain an image of the same intensity for the water band when the fluorite plate was in as when it was absent. This indicates that the radiation lies outside of the so-called Schumann region, for which fluorite is quite transparent. The plate reproduced in fig. 8 was exposed for 50 minutes. In making these exposures it was necessary to clean the fluorite plate frequently. It was asa rule wiped with dry cotton every two minutes, and every fifteen or twenty minutes cleaned with nitric acid, water, and alcohol. The time taken for wiping and cleaning is of course not included in the time of exposure. In these ex- posures the amount of nitrogen delivered into the jet-chamber was about a quarter of a litre a minute, the apparatus being washed out for from five to ten minutes before the exposure was begun. When purified nitrogen was used the fluorescent spectrum showed the nitrogen band of longest wave-length predominant. Inserting the fluorite plate into the apparatus showed that the radiation exciting this band was also transmitted, but very much less readily than the radiation exciting the water band—perhaps a third or fifth as well. The study of the transmission of quartz was attended by more difficulties, and yielded less definite results than that of fluorite. Using nitrogen directly from the bomb, in the spectrum of which the water band predominated, no trans- mission of the radiation exciting the water band has ever been detected, though the conditions under which the ex- posures were made were not entirely favourable. Using purified nitrogen, in which the nitrogen band of longest wave-length came out most strongly, it was found that after transmission through quartz the region of most intense fluorescence was displaced toward the short wave-length side of the spectrum, indicating that it was probably the Fluorescence produced by Ultra-Schumann Rays. 457 radiation exciting one of the nitrogen bands of shorter wave- length which was being transmitted. In all the photographs of long exposure when purified nitrogen was used, there is also a fluorescence at about 2300 A.U. The radiation exciting this fluorescence is not very strongly absorbed by either quartz or fluorite. Exposures made to detect a possible transmission of any of the radiation exciting fluorescence through laminz of fused quariz and thin films of mica gave entirely negative results. If the radiation from the spark which has been studied in this paper is really of the nature of light, then it should be possible to refract it by passing it through the thin edge of a prism of quartz or fluorite. A quartz prism ground toa razor edge by Petitdidier was accordingly placed in the appa- ratus in such a way that the radiations passed through the very edge of the prism. The edge of the prism was pressed against a piece of black paper to prevent any of the radiation from passing by without going through the quartz. Tor this experiment the original jet-apparatus (fig. 1) was used, mounted in inverted fashion and adapted to the present work. A tube was provided to feed the nitrogen from the rear (similar to the tube T, fig.2). The arrangement of the spark, slit S, prism Pr, bedplate A’, and tube T viewed end-on, is shown in the sectional diagram fig. 3 in the text, which is drawn as seen from the direction of the spectroscope. Passing a current of purified nitrogen of about one-third litre per minute into the jet-chamber and exposing for an hour and a quarter, cleaning the prism frequently, the ex- posure of fig. 9 was obtained. This photograph shows in the first place that the radiations here being studied are refracted. Further, since the fluorescent streamer is fairly broad, it indicates that for an extremely thin refracting edge 458 JS luorescence produced by Ultra-Schumann Rays. as was here used, the rays exciting at least two, and possibly three of the nitr ogen bands, and possibly also those exciting the water band, are transmitted thr ough the quartz. In the original photoet ‘aph the fluorescent jet “of wave-len eth about 2300 A.U., mentioned above as being present in ‘long eX- posures when purified nitrogen is used, is also to be seen. Its position is marked with a short oblique line in the repro- duction. Fig. 9 cannot be used to determine the index of refraction of quartz for the rays exciting the fluorescence, as the photographic plate, lying in the focal plane of the spectroscope, was inclined to the axis of the focus tube, and consequently greatly exaggerated the deviation of the jet. To obtain a value of the deviation of the rays by the prism in the jet-apparatus several exposures were made, using the spectroscope merely as a camera (2. e. with the prism of the spectroscope removed). An exposure was then made with smoke in the jet-chamber. The light scattered by the smoke was \filtered through a silver film, and thus the deviation produced by the prism in the jet-apparatus upon light of wave-leneth 3000 A.U. (the transmission band of silver) was obtained. The index of refraction of quartz for this wave-length is given in tables as 1°57. [From this value and the relative deviation for this wave-length and for the hight exciting the fluorescence, we calculate for the latter an index of 1:75-+:08, that 1s, a probable value somewhat greater than the greatest index in the ordinary transmission 1 region of quartz. The object of finding the index of refraction of quartz for these rays was to see if it might not be less than the index for ordinary ultra-violet hight. If this had proven to be so, it would have indicated ihat the rays exciting the fluorescence were on the short wave-length side of the ultra- violet absorption band of quartz. But as this has not been found to be true, we must conclude that they are still on the long wave-length side of the absorption region, unless indeed they should be on the long wave-length side of an entirely hypothetical second quartz absorption band, still further out in the ultra-violet. Hxperiments under way show that the radiation exciting the water band can also be reflected. ‘The reflecting surface used was a cathode deposit of silicon on glass made by Dr. E.O. Hulburt. At 45° incidence the amount of reflexion lies between ten and twenty per cent. Summary. The fluorescent spectra excited by ultra-Schumann radiation in nitrogen containing some oxygen, in nitrogen free from oxygen, in moistened nitrogen, in dry and moist air, and Polarization of Light scattered by Metal Particles. 459 in nitrogen with an admixture of iodine vapour, have been studied. An explanation for the effects obtained with moving streams of gas is given. Fluorite and crystalline quartz are slightly transparent for the rays studied. Those exciting the fluorescence of the water band pass more easily through fluorite than do those exciting the nitrogen bands. Jor crystalline quartz the reverse 1S probably true, Rays exciting a very faint fluor- escence at about 2300 A.U. in purified nitrogen pass readily through fluorite and crystalline quartz. Fused quartz and mica are found opaque for all the rays exciting fluorescence. The ultra-Schumann radiations have been refracted. The index of refraction of quartz for them is 1°75 +°08, indicating that they lie on the long wave-length side of the quartz absorption band. The radiation exciting the water band has been reflected. XLIX. On the Polarization of Light scattered by Spherical Metal Particles of Dimensions comparable with the Wave- length. By HK. Tausor Paris, B.Sc., Physics Research Student at University College, London”. A SOLUTION of the problem of the scattering of electro- magnetic waves by a spherical obstacle has been given by Professor A. E. H. Lovet. Lord Rayleight has shown that Professor Love’s results admit of considerable simplifi- cation, and has used the simplified forms to calculate the polarization of the scattered wave when light is incident on a transparent sphere having a refractive index 1° d, and of dimensions comparable with the wav e-length. In the present paper similar calculations have been effected for the cases (1) when the Leen is supposed to be perfectly electrically con- ducting, and (2) when the sphere is composed of silver. In the latter case the incident light is taken to have a waye- length of 550 up. In both cases the calculations have been made for spheres having circumferences not less than one wave-length, and not greater than two wave-lengths. With a view to verifying the numerical results obtained, observations have been made on the polarization of the light scattered by uniform suspensions vf silver. These experiments are described in the last part of the paper. * Communicated by Prof. A. W. Porter, F.R.S. + Proc. Math. Soc. Lond. vol. xxx. p. 308 (1899). t Proc. Roy. Soe., A, vol. lxxxiv. pp. 25-46 (1910). 460 Mr. E. Talbot Paris on the Polarization of 1. Theoretical. Let a beam of unpolarized light, travelling in the negative direction along the axis of z, be incident on a spherical obstacle placed with its centre at the origin (fig. 1). Sup- pose that observations on the polarization of the scattered Fig. 1. light are made in the w, 2 plane at a distance r from the origin. Let 6 be the angle between the direction of obser- vation and the incident beam. ‘The simplest thing to measure experimentally is the ratio of the intensity of the vertical component of the scattered light, parallel to Oy, to the in- tensity of the horizontal component in the «, z plane, per- pendicular to the direction of observation. If X, Y, Z denote the electric forces parallel to the three axes in the scattered wave, these two components are respec- ; wL—2X tively Y and : Se solution leads to the following expressions for these two components :— Rayleigh has shown that Love’s ra an+1 = _ \r+1 a et Y > ae mae n(n+1)Py} ek(ct—7) +NaP,’ | —> +. @ aL—2X r 2 pean AE erate Bs Na Ed ea pi heat) = > i ne a [| Natu? Cee wall age eh(ct—r) seal Ht G “MP. at ss 1 Laght scattered by Spherical Metal Particles. 461 In these equations k= a r being the wave-length of the weident licht, ~=cos8, and P,, or P,(z), is a zonal harmonic of degree n, whose axis is the axis of z. The quantities M, and N, depend on the boundary conditions to be satisfied at the surface of thesphere. Expressions similar to (1) and (2) have been used by G. Mie* to calculate the intensity and polarization of the light scattered by small gold particles. It may be nuted that Mie’s solution of the problem of the scattering of light by a sphere is identical with Love’s, though obtained independently and expressed somewhat differently. The quantities { 2n+1 pan ol Meee n(n+ 1) ‘3 n(n+1)” are functions of m and yw only, and depend on the direction of observation. Their logarithmic values for various values of » have been tabulated by Rayleight. M, and N, are functions of the size (relatively to the wave-length) and optical properties of the spherical obstacle. Mod Y and aoe * give the amplitudes of the two components, BP,’ —(2n4+1)P,, § and and their squares give the intensities. The complete ex- pression for N,, is{ : ! Ryan) - {KY g 5 +B eS eri a ers ( anieere sur 1(m) ( ae aetT a wrn(n ) n(7) K is the dielectric constant of the material composing the sphere, that of the surrounding medium being supposed equal to unity. The expression for M,, is obtained by writing #, the magnetic permeability, instead of K. In optical problems we may take ~=1, when M,, becomes _ Yay i a Wr (7') Ya) ~E, a(n) + Y=) Bam) vn(n') 462 Mr. E. Talbot Paris on the Polarization of K may be replaced by m?, m being the refractive index of the material composing the sphere relatively to the sur- rounding medium, and then n'=mn. The functions yr, and E, are defined by bso) (— 1-8. Fee ay (- d yes 0 (3) ndn) 9 iad Ne aed Tey) (8h oo ee ed (Go) hor (7) ( ) \ Mite n (4) —tX,(n) is the imaginary part of E,(7). So that, sepa- rating the real and imaginary parts, E,(n) =WVi(7)—yal). Tables of WY and wW have been given by Rayleigh* for abe ~ = > if Ro PIS! arguments ranging from 1 to Og 2. Numerical Results for a Perfect Conductor. If the sphere is supposed to be perfectly conducting the expressions for M, and N,, become much simpler, and the numerical work is consequently much lighter. In order to effect this calculation we have only to find the appropriate values of M, and N, and substitute them in equations (1) and (2) given above. ‘'o this end we may make use of the expressions given by Sir Joseph Thomson in ‘ Recent Researches in Electricity and Magnetism,’ page 445. These expressions give the electric forces in the wave scattered by a sphere having the character of a perfect conductor, but they are applicable to ordinary conducting materials pro- vided that the distance which the alternating currents (set up by the incident vibration) penetrate into the sphere is small compared with the diameter of the sphere. ‘Ihis dis- eae , (27pp\-? | tance is given by d= (“) > where « = the magnetic permeability, p=27 x the frequency of the incident wave, and p= the specific resistance of the material composing the sphere. There is no reason to suppose that this condition could be fulfilled by any known metal when we are dealing with oscillations of the frequency of light. Nevertheless, the calculation is not without interest, for the above conditions would be strictly fulfilled if we were dealing with Hertzian waves and spheres (or spherical shells) composed of a metal such as copper. Moreover, the curves exhibiting the polari- zation of the scattered light, obtained by assuming the sphere to be perfectly conducting, do not differ very materially from # Proc. Roy. Soc., A, vol. Ixxxiv. p. 39 (1910). Light scattered by Spherical Metal Particles. 463 those obtained by putting in the appropriate values of the optical constants of silver, except in the case when the spheres are very small compared with the wave-length. In this case, as is well known, the perfectly conducting particles should give rise to plane-polarized light in a direction making 120° with the incident beam, while the full theory, taking account of the particular optical properties of the metal composing the particles, shows that there should be plane-polarized light in a direction making 90° with the incident beam, a result which is confirmed by experiment. In fact, the polarization of the light scattered by small imperfectly conducting particles is the same as if the par- ticles were non-conducting. Indeed, it would be impossible to tell by observation on the polarization of the scattered light alone whether we were dealing with particles of a heavily absorbing metallic substance like silver, or of an almost transparent substance like sulphur. The intensity of the scattered light is, however, very different, the metal exhibiting ‘optical resonance,’ a phenomenon which has been fully discussed by Mie. When suitable alterations in the notation have been made, the expressions given by Sir Joseph Thomson for the electric forces in the scattered wave correspond exactly with those given by Professor Love, the sole difference being in the quantities written in place of M, and N,. We find that the equivalents of M, and N, in Thomson’s calculation for the perfect conductor are : PDs (7) M,.= Lat eae ae ae | . . . . . . . . n" TAn) (5) d and N Re Ree: Ia 1% Sn) \ ——F () CRE oe {a +h f(7) } In this, as before, k= “7 and a is the radius of the sphere. Also nfl “d \"sin Sn = (+ paso ff 7 . . . Wie \G a) (7) and =(‘ yo . hin= n dn 2 . . « e e e (8) It is convenient to have M, and N, expressed in terms of the functions Wn» and E,, equations (3) and (4), so that the 464 Mr. E. Talbot Paris on the Polarization of numerical values of these functions tabulated by Rayleigh can be used. From equations (3) and (4) we see that 8.0) = sag Ree ae - d i ee i) = (3.5 ne) hbo ea ye Mey lt ED } d d For N,, since n=ka and ia ee we have Differentiating n— d _ mt), (2) +9 Ta Wn(7) d (x +1) E,(m) + dy bo) Now ,(m) satisfies the equation 1 gg Vol) =n 41) jYual—vier)}. . 2) and H.,(m) satisfies an equation of the same form. There- fore, ria (2n a Daley (7) — ns, (1) “eS @i-eDBa(a) aig) 7 By separating E, into its real and imaginary parts, the values of M, and N, given by (11) and (138) can be caleu- lated without further trouble for any value of » for which y and WV have been tabulated by Rayleigh. Alternatively, the values of M, and N,, for the case of a perfect conductor can be found by putting K=o, and, in order that, Ky, may remain finite, ~=0 in the general expres- ions for M, and N,. This is analogous to the method used Light scattered by Spherical Metal Particles. 465 by Rayleigh to obtain a solution for the case of a perfectly conducting particle, small compared with the wave-length”. The values found in this way are the same as those given above. The numerical values of M, and N, have been calculated for n=1, 1:5, 1:75, and 2. They are given in Table J. As the perfect conductor is a standard case, the numerical results are given in some detail. TABLE I. | Nd. M,,- | i Deas ah, Pie. —0°20807—10°04535 | 0°45465—1x 029193 ah. —0-01720 —10-00030 0-03036—1 x 0-00092 Bo icee —0-00054 | 000076 n= 15, is —0-42978—1x0:24448 | 0-49960—.x0-48012 i. —0-09380—1x0°00888 | 0:19780—1x0-04079 ae —0-00747 —«X0-00005 | 001134—10-00013 : —0°00030 | 000089 n= 4d 0, he —0°49244—1x0°41339 | 0:49054—10-40320 ss. —0°16420—1x 002773 | 0°34143—-1 x 0:13473 "ae —0:01903—1x0-00036 | 0-08083—1x«0-00095 Bs. —0 00107 | 000144 5. ...| —0-00004 ' 0.00004 n=2. | 0:44722—1x 027641 ‘ikea _0-48850—1 x0-60663 | —0:25195—1x0-06812 | 0-49571—1<0'43468 icv —0°04084—1 x 0:00167 0:07147 —.x0:00514 4......, —0°00316—1 x 0-00001 0:00438—1 x 0:00002 5...) —0°00014 | 0:00018 These values of M, and N, may be checked in the follow- ing way. When light passes through a turbid medium, the particles distributed through which are of metal or other absorbing material, there is loss of light from the original beam owing to two causes: (1) the loss due to scattering, and (2) the loss due to the absorption of light by the material composing the particles—the surrounding medium being * Phil. Mag. vol. xliv. p. 48 (1897); Scientific Papers, vol. iv. p. 322. Phil. Mag. 8. 6. Vol. 30. No. 178. Oct. 1915. 2H 466 Mr. HE. Talbot Paris on the Polarization of supposed perfectly transparent. Let « be the coefficient of “absorption ” of the turbid medium, taking into account the losses of light due to both the above causes ; and let «’ be the coefficient which represents the loss due to scattering alone. Then («—x«') may be called the coefficient of “ pure absorption.” Expressions for « and «’ have been found by G. Mie*. If there are N particles per cubic centimetre, these expressions, when translated into the present notation, may be written «=the real part of «x eee (2n+1)(M,+N,), n=1 e'=N72"S" (2n+1) (mod M,2+ mod Ny). n=1 If the particles are perfectly conducting, then since a perfect conductor cannot absorb energy, the coefficient of “ pure absorption,’ K—K’, must be zero, and M, and N, must satisfy the equation mod M,?+ mod N,’?=the real part of (.M,+Nn). (14) And the same relation must be satisfied if the particles are transparent. For example, when n=1, we find from Table L., mod M,?+ mod N,?=0°33728, and real part of 1(M, +N,)=0°33728T. The logarithmic values of (aera ee { (Qn+1)yP,’ 7 n(n + 1) an n(n+1) (2n+1)P, b have been tabulated by Rayleigh for the values of w yiven * Annalen der Physik, 4 Folge, Bd. xxv. p. 486 (1908). t+ A quantity a,, corresponding to N,, has been tabulated by Mie for 3 the case of a perfect conductor. a, and N, are connected by N,= = ay. The only point at which the values of a, overlap those of N,, given above, is when n (or ain Mie’s notation) =1. In this case the value given for a, is 0°638—. 0-487, whence N, =0:425—:X0:291 instead of 0:-455—.X0°292 as in Table I. It can, however, be asserted, without recalculation, that the value of N, given in Table I. is correct, since it satisfies equation (14). Light scattered by Spherical Metal Particles. 467 below. Their actual values, as far as they are required in the present calculation, are given in the following tables:— z ! Wablaco (2n+I1)P, n(n+1) N. =|) e— i. pai. | p—. 1...| +41'5000 +1-5000 +1°5000 +1°5000 +1°5000 Ee 0 +0:62500 +1:2500 +1°8750 +2°5000 3...| —0°87500 —0°60156 +0°21875 +1°3859 +3°5000 acs 0 —0°72070 —0°70313 +0°79102 +4:5000 5...| +0°68750 +0°14234 —0°81641 —0°15845 +5°5000 2n+1)uP Table of iD 2n+1)P,. Cane (2n+1)P, n p=0. =F p=p w= h. w=. J 0 —0°37500 —0°75000 —1+1250 — 15000 2 +2°5000 +2°1875 +-1:2500 —0°31250 — 2°5000 ras 0 +2°2012 +3:1719 +1°6816 — 3°5000 4 —3:3750 — 1:5996 +2°2500 +3°7441 — 45000 5 0 —3'7014 —1°3965 +4-4614 — 55000 : } 2n+1)P,’ : : When vp is negative, ee changes sign if n is (2n+1)uP,’ : : aig fms on 4 1)P. bch even, ‘ ‘A ( sig ) ny changes sign if n 4 wL—2z2X , is odd. Y and - ae and their moduli, can now be calculated by means of equations (1) and (2). [Table, p. 468. ] The values of Y and ~ when 7=2 were also found, but less accurately than in the table. In fig. 2 ; I,-I, . the quantity P=100 ion 8 is plotted against 0. I, and I, are the intensities of the vertical and horizontal components respectively, and are given by (mod Y)? and (mod eo Lae 2H 2 468 Mr. E. Talbot Paris on the Polarization of p. se 2 Modulus. Me Modulus. Ie —1...| 0:4035—1x0:5093; 0°6498 || 0-4035—1x0°5093 | 0°6498 —3...| 0:1745—-1x0°3973| 0:4339 || 0°5011—.x0-4907 | 07013 —2...|—0-:0331 -1x0'2862| 02881 || 0°5873—.x0'4727 | 0:7539 —3,..|—0:2201—.X0°1757| 02816 || 0-6613—1x0-4548 | 0:8027 0...) —0°3875—1 x 0:0657| 0°3980 || 0°7243—.x 04371 | 0:8460 +2...|—0°5363-41x 00443} 0°5380 || 0:°7770—1x0°4197 | 08831 42. |—~0:6672+1X0:1524| 0°6844 || 0°8200—1 x 0-4024 | 0-9134 +3...|—0°7814-+1x0:2607| 0:8238 || 0°8541—. x 0°3885 | 09383 +1...|-0°8797-+:0°3683| 0:9537 || 0°8797—«.X0°3683 | 0°9537 nabs —i...) O-3787—«x1-2117| 1:2695 | 0:3787—-.x1-2117 | 12695 —3.../—0-2519—1x0°9362| 09694 | 0:6394—1x1:0746 | 12504 —4.../—0°6727 —.x0°6865| 0°9611 | 0:8181—.x0'9433 | 1:2486 —4.,.]—0:9683—10-4628| 1:07382 | 0°9259-.x0°8177 | 12353 0.../—1:1813—1x0-2647| 11619 | 0-9730—.x0-6979 | 1-1974 +2,,.;—1:1762—cx0-0921| 11798 | 0:9687—.x0°5836 | 11309 -+-4...,-1°1161+200550| 11175 | 0-9215—.x0-4750 | 1:0367 +2,..|—0:9630++x01772| 0:9792 || 0:8390—.%0-3717 | 0-9177 +1...;—0°7278+1x0:2739| 07776 | 0°7278—.x0-2739 | 0-7776 4=1°75. —1...| 0°4882—.1°6356| 1:7055 || 0-4832—1«1-6356 | 1:7055 —3,..|—0-4764—1 X 11668, 1:2608 || 0°8569—1 x 1:3320 | 1:5838 —3,,.|—1-1071—ex07858} 13576 || 1-:0668—.x«1-0477 | 1°4952 —1,,.—1-4574—-1 04916} 15381 || 1:1442-.x0-7820 | 1:3859 0.../—1:5708—« x 02829) 1:5961 11158105316 | 1:2373 +41,..|—1:4862—.0°1587| 1:4946 1-0053—t x 03051 | 10506 +4.../—1:2884—.x01177| 12440 || 0°8334—.0-0930 | 0-8386 +3...|—0-8586—1X0:1588| 0:8732 || 06179. x0-1033 | 0:6265 +1...|—0°3741—1x0:2807| 04677 || 0°3741-1x0-2807 | 0-4677 The reason for plotting this quantity is that it is the most convenient for comparison with experiments. Measurements on the polarization of light scattered from a turbid medium are best made with a Cornu polarimeter, consisting of a small rectangular opening through which the light to be examined passes, a double-image prism, and a nicol capable of being rotated about its own axis and attached to a divided circle. The double-image prism is fixed at such a distance from the rectangular opening that the two images, one polarized vertically and the other horizontally, are seen side by side. These are viewed through the nicol, which is then set so that the two images appear equally bright. If the zero position of the nicol is taken to be that position in which it transmits only the vertical component, and the angle Light scattered by Spherical Metal Particles. 469 through which it is moved is called w, then P= —100 cos 20. In practice it is better to make two settings of the nicol, one on each side of the zero position; the angle between the settings is then 2w. The polarization curve for the case when the particles are very small compared with the wave-length (7-0), taken from Mie’s paper, is reproduced in fig. 2 for the sake of Fig. 2. + 100 +680 +60 +40 +20 -20 -60 L. 1B0 150° 120° 30° 60° 30° 0' a Theortical polarization curves for perfectly conducting particles. comparison. It shows the scattered wave to be plane- polarized in a direction making 120° with the incident beam. When 7=1 the scattered wave is no longer plane-polarized in any direction, but there is a direction of maximum polari- zation at about @=110°. As 7 increases this maximum moves back again through the 120° position, for when »=1°5 it is at about 130° with the incident beam. At the same time a neutral point (P=0) appears, which when 7=1°5 is 470 Mr. E. Talbot Paris on the Polarization of at @=85°. Between the neutral point and @=0 the polari- zation is ‘‘ reversed,” that is, the horizontal component I, is greater than the vertical component I;. When 7=1°75 the neutral point is at about 110°, and the reversal has become more pronounced. The curve for 7=2 shows a remarkable change in the character of the polarization. There are now two maxima, two neutral points, and a strong reversal. If the dimensions of the sphere were large compared with the wave-length ordinary reflexion would occur, and the examination of the “scattered” light would consist in examining light which had been reflected from the equatorial region of the sphere. Since light could not be polarized by reflexion from a perfectly conducting surface, we should have when 7>o, P=0 for all values of 0, provided the incident beam is unpolarized. 3. Numerical Results for Silver (A=550 wp). In order to calculate the polarization of the light scattered by silver it is necessary to make use of the general expres- sions for M, and N, already quoted, giving 7’ its appropriate value. In the experiments that have been carried out the silver particles were suspended in water and the observations were made with nearly monochromatic light of wave-length 550 uu. For the purposes of this calculation the value of the complex refractive index of silver for X=550 wu given by R. 8. Minor* has been used, viz. m=0°176—ex 3°305. This must be divided by the refractive index of water, taken to be 4, giving 1! =m'n=(0°132—4 x 2°479)>m. The first step towards obtaining M, and N, is to calculate the values of ,(n') from the series 4 19 Ve hae Diy n abenlay )==d 7 oes OL ao, + Bee - (15) The convergence of this series is best when 7 is large, and we could, as explained by Rayleigh, use (15) to obtain the values of n(n’) for moderate values of n, and then use the sequence-equation 12 — Fg vey) =(2nt 1) (Yaa!) —ala')}, (26) to obtain the values of Warm’) of lower order. The work * Annalen der Physik, 4 Folge, Bd. x. p. 617 (1903). Light scattered by Spherical Metal Particles. 471 can be shortened a little in the following manner. Jn the general expression for My and N, the w-functions with 7’ as argument only occur in the form Yn) Dividing (16) Yn’) by wn(m') and rearranging the terms we have DOE Piare ore : 17 hae 2n-+l.2n43 Dep a ! where ®, is written for aca - Thus two values of Wn(7’) can first be found from (15), for example >; and Wg when n=1. These give the value of D,, and ®;, ®,, Kc. can be obtained by using (17). It is convenient to tabulate Bn and Ke caer et ae vf | Pa={ oy (m at er 2 We then have, omitting the arguments, numerator of Ma=Pbry,-1—Pa . Wn, denominator of Mz=«x numerator —Wyz_1+ Pn. Vn. And numerator of Na=m'*bry,-1—T a. rn, denominator of Nn=cxX numerator — m?V,_,4+T,. Va. The following values of Vy, and y are required, in addition to those given by Rayleigh, to enable the polarization to be calculated for 7=1°25, which was very near to some of the values of 7 obtained experimentally. n Vn. Vn n=1°25 0. 0°25226 0°75919 e. 2 3064 0-85222 | D) 19-720 0:89312 | ts: 390°06 0:91622 | | 4 ' 14932 0:93108 | et 0-94145 | | 6 | an 094910 | 472 Mr. E. Talbot Paris on the Polarization of The values of Mn and N, are given in the following table :— TABLE LI. ANY tia ae | N. Mn: | iN; n=l. | aaa —0:072—1 x 0-062 0:185—: x0°888 Dd eas —0:003 0:059—« x 0008 uaeee 0-001 Linea —0°1638—10°038 0-081—1 x 0963 PE —0:012—.x0°001 0°197 —.x0:057 Bens 0-006 n=1'5 een —0:265 —1 x 0:009 0:233 —1 x 0'910 BAN, —0:032—1 «0003 0:-410—1. x 0°338 Daeane —0:002 0:022—1. x 0°002 Ali: 0001 n7=2. EAN —0°463—10°345 0'404—1x0°748 Qa —0°129—1« 0:022 0:105—.x0°918 SA ee —0:0138-+¢x 0°001 0:173 —.x 0048 Ae ais —0:001 0:008—:x0:001 r, hd From these the values of Y and wh —2X are obtained by means of equations (1) and (2). The most useful quantity for comparison with experimental results is P, the values of which are given below. Values of P=100 t Bu Ip 1,+1, pe n=l == 2d n=1d. | n=2. n—->0. Shane ean 0 0) ie 0 0) LEER 26 32 Tide \Winee 28 SES 55 5d BB i GN 60 SSO 82 62 22 6 88 Oe 96 58 15 ae 100 SUN AON 91 52 —35 — 58 838 Sy, 60 38 pe. Aan Mea (| 60 EEE NN 28 22 SEO A 28 PS Meio 0 0 0 0 Light scattered by Spherical Metal Particles. 473 In fig. 3 P is plotted against @= cos". When 7>0 the direction of maximum polarization is at 90° to the incident beam, as for transparent particles, in which direction the light is plane-polarized. The curve for »=1 follows very Fig. 3. +100 AG VA +80] +60 +40 +20 oO \\ ho #20 -40 “60 ° ° ° © o ° axe 180 150 120 90° 60 30 om ) Theoretical polarization curves for silver particles (A=650 py). closely that for 7>0, but the maximum has shifted a little from the 90° position back towards the incident beam. In this respect it follows the procedure in the case of transparent particles, and the calculation agrees with the conclusion of G. W. Walker* that, for these small sizes, the polarization of light scattered by particles possessing a certain amount of conductivity should not differ from that scattered by dielectric particles. It may also be recalled that in the case of per- tectly conducting particles the maximum polarization first shifts from the 120° position back towards the incident beam. So that in all cases, as we pass from the case of * Quart. Journ. of Pure and Applied Maths., vol. xxx. p. 217 (1899). ATA Mr. E. Talbot Paris on the Polarization of infinitely small particles to those of dimensions comparable with the wave-length, the direction of maximum polarization first moves back towards the incident beam, the movement being less marked in the case of conducting particles than of dielectric particles. When 7=1°25 the maximum has moved back through the 90° position to about 0=108°, so that there must be two eases when the scattered light would have its maximum polarization at 90° to the incident beam, one when 7-0 and again when 7 is between 1 and 1°25. The curves for 7=1'5 and 7=2 show the same general feature as those for the perfect conductor, though in the latter case the effects are much less marked. If the sphere is supposed large compared with the wave- length the light would be regularly reflected and P would show a low maximum at about 6=146°. 4, Heperimental. 1. Preparation of the Silver Suspensions.—The suspensions of silver were prepared by mixing a solution of silver nitrate and pyrogallolin pyridine with water. The pyridine solution Was pogees by ye seeaes 1:436 gms. of silver nitrate in 20 cm.* of pyridine, and 0°228 om. of pyrogallol in 10 cm.? of pyridine; these two solutions were mixed®, For the pre- paration of small particles it is convenient to dilute this solution down to |'5 of its original strength. Jairly uniform suspensions of silver can be prepared by. mixing this solution with water; the greater the proportion of water the smaller are the resultin @ particles. 2. Preparation of Uniform Suspensions.—In order to obtain more uniform suspensions use was made of the process of fractional centrifuging. The process, as described by Pro- fessor Perrin}, consists in centrifuging a suspension con- taining particles of various sizes, for a certain time, say 7}. Let a, represent the radius of a particle which, under the influence of the centrifugal force, would travel from the surface of the liquid in one of the test-tubes carried by the centrifuge to the bottem of the tube in the time 7. Then the sediment which collects at the bottom of the tube contains most of the particles of radius greater than a, together with smaller particles which started from positions nearer the bottom of the tube. The sediment is mixed with water and centrifuged again, and the process is repeated * Pieroni, G'azetia, 43, vol. i. pp. 197-200 (1918). t+ Perrin, Ann. d. Chim. et de Phys. 8me séries, Sept. 1909; or Perrin, ‘Brownian Movement and Molecular R eality’ (English translation, Taylor & Francis). Light scattered by Spherical Metal Particles. 475 until no particles are left behind in suspension. All the particles are then of radius greater than a, and the suspen- sion obtained by mixing these with water is now centrifuged for a time 72, less than 7,, by which means particles of radius greater than a, are removed. TF inally, a suspension is obtained containing only particles having radii between a, and ap. In order to obtain strictly uniform suspensions by this method it is necessary that a large quantity of raw material be available. Professor Perrin in some of his experiments used several kilograms of mastic, the centrifuging extending over a period of several months. Such a procedure is im- practicable for the preparation of suspensions of silver, and it is therefore desirable to obtain some idea of the degree cf uniformity possessed by suspensions which have been prepared in a less exacting manner. The following simple theory, which is piety applicable only to a hypothetical centrifuge, working at a uniform speed, and which neglects the effect of convection currents in the test-tubes, changes in temperature, and errors due to speeding up and slowing down, shows in more detail the kind of processes that occur during fractional centrifuging. Let m=the effective mass of a particle, a=the radius of a particle, w=the angular velocity of the centrifuge, and r=the distance of the particle from the axis of the centrifuge. Then by Stokes’s law—assuming that the velocity at each position has the limiting value cor responding to that position*— mo*r= brnav, where 7 is the viscosity of water and v is the velocity of the particle. Replacing v by = and m by its equivalent 477a?(p—p')—p being the density of the particle and p’ that of water—we fin whence 9 o . ha eke, 2a(p—p')a® 8 7, and i) T, 2 = ' . aa , long 2 ; : 27(p—p)w Tr) * This is justifiable as long as 3rnu/m for the particles dealt with is large compared with w. 476 Mr. E. Talbot Paris on the Polarization of From this, if 7, and 7, are the distances from the axis of the centrifuge to the surface of the liquid in the tube, and to the bottom of the tube, respectively, when the centrifuge is working, then a, and a, corresponding to the times 7, and 7, can be determined. In general the average radius of the particles in the re- sulting suspension will not be the arithmetic mean of a, and az, and may not even be between them. Suppose that initially the suspension contains an equal number of particles of all sizes within reasonable limits— say N particles of each kind per cubic centimetre. In practice it is immaterial whether or not the condition is strictly fulfilled provided it obtains in the neighbourhood of those particles which it is desired to separate into a uniform suspension. Let this suspension be centrifuged for a time rt. Then, if the centrifuge works perfectly, all particles having radii greater than = 4 log as ~ Lor(p—p)e? * 7, will be removed. To find the number of particles removed having radii equal to a, less than a’, we must find 7, the distance from the axis from which the last particle of radius a to reach the bottom of the tube, at 7, started. Tf C= 20*(p — p') oY TENT ops: r ‘ and r= log, {log.r2—C .a’r}. The number removed is (72—7) N, supposing the tube to have unit cross-sectional area ; while the number originally present is (r2—7)N. The proportion removed is therefore Mg” *g— log, “{logerg—C.a?r} ee Hm N77 NEI CT: enwan oes and the proportion remaining is (1—p). Some calculations have been made, based on the dimen- sions and speed of the centrifuge used in the present ex- periments. The results are shown in fig. 4, where the percentage of particles is plotted against their diameter. | ee Light scattered by Spherical Metal Particles. 477 The curve I. indicates the kind of suspension that would be obtained by centrifuging once for 30 mins. and once for 20 mins. The most numerous particles are those having diameters of about 100 wy. If the suspension were centri- fuged six times for 30 mins. and then once for 20 mins., the Fig. 4. Pipe be Rn: 50 ticles p Qa fa Cie ra yal id re) Percents fe nm re 6) 40 60 iz0 1€0 200 Diameter of particles in pip. Results of centrifuging. distribution shown by the curve II. would be obtained. The maximum is now for those particles of diameter about 125 ww, and is much sharper. The result of centrifuging the suspension six times for 30 mins. and six times for 20 mins. is shown by the curve III., where the ordinates have been multiplied by 100. The most numerous particles now have diameters less than 100 wy, and the maximum is not so sharp. Moreover, the suspension has been much impoverished, there being less than 0°25 per cent. of the original particles left even at the maximum. It is therefore important to centrifuge the suspension a large number of times for the longer period in order to free it effectively from small particles, and then to centrifuge it just a few times for the shorter period in order to remove large particles. The suspensions used for the observations on the polariza- tion of the scattered light were centrifuged from five toa dozen times for the longer period. 478 Mr. E. Talbot Paris on the Polarization of 3. Determination of the Dimensions of the Particles—The following methods were used to determine the size of the particles :— (1) By observing the velocity of fall of the particles in water under the influence of gravity, applying Stokes’s law. This method was used for the largest particles. (2) The number of particles per cubic centimetre was determined by observing the average number in a known volume illuminated in the ultramicroscope. The whole of the silver was then collected by centrifuging and dissolved in nitric acid, the solution being made up to a certain volume. The silver having been precipitated by the addition of hydro- chloric acid, the turbidity of the resulting suspension was matched by precipitating silver chloride from a standard solution of silver nitrate, under as nearly as possible similar conditions. (3) The size was calculated from the speed and dimensions of the centrifuge. 4, Observations on the Polarization of the Scattered Light. The observations were made with a polarimeter, the essential parts of which were a rectangular opening, a double-image prism, and a nicol attached to a divided circle. The arrangement of the apparatus is shown in fig. 5. z.. Apparatus for observing the polarization of the scattered light. The suspension to be examined was placed in the small beaker B on the table of a spectrometer. Light from the are C is focussed by the lens L so that an almost parallel beam of light passes through the liquid in the beaker. The width of the beam was adjusted to about 1 em. by means of | ; | | : Light scattered by Spherical Metal Particles. 479 the iris diaphragm I. The scattered light was examined through the telescope tube of the spectrometer, which was adapted to carry the rectangular opening A (1 cm. x 0°5 cm.), the double-image prism P, and the nicol N. Two images of the rectangular openings are seen side by side through the nicol and double-image prism, and the nicol is set in two positions successively for which the intensities of the two halves of the field appear equal. If the readings on the divided circle are a, and ag, then = aise 100= —100 cos (w,—@.). : I,+1, The angle between the direction of observation and the beam passing through the beaker is read off from the scale on the spectrometer table, a preliminary setting being made to find the reading for @=7. Readings were taken every 10°, and intermediately in special cases. Reflexions from the side of the beaker obscured the readings for 2=0° to 30° and 150° to 180°. To obtain light of the required wave-length the incident light from the are passed through a Wratten mercury-green- line filter. This filter transmits a bright beam at about A950 and a much less intense beam at the extreme red end of the spectrum. The following sources of error may be noted :-— (a) The presence of a small quantity of red light trans- mitted by the filter. This, though small, would tend to diminish the polarization of the scattered light, for the polarization phenomena were observed to be less marked with red than green light, tor a given size of the particles. (6) The scattering of light by dust and other particles which were always found to be present in the distilled water used for preparing the suspensions. Observations made on distilled water alone showed the intensity of this light to be extremely feeble, but exhibiting a direction of maximum polarization at 90° to the incident beam. (c) The reflexion of light from the walls of the beaker. This was very small except in the direction noted above. (d) The polarization of the light by its passage through the walls of the beaker. It is doubtful if any of these sources of error would make any perceptible difference in the polarization of the scattered light. “The results of the observations are shown in fig. 6, where the experimentally determined values of P for particles of -. 480 Mr. E. Talbot Paris on the Polarization of different diameters are plotted against 6. 1 is, of course, calculated for the wave-length in water, viz. 2 x 550 wp. Fig. 6. 0,———— ee +20 2: 0 ~20 -40 160° i50° i20° @ Experimental polarization curves for silver particles in water (A=690 pp). Ne Comparison of Theoretical with Heperimental Results. On comparing the curves in fig. 6 with those in fig. 4, it will be seen that, in a general way, as the particles increase in size, the polarization of the scattered light follows the lines indicated by theory. Thus, when the particles are very small, the maximum is at 90° to the incident beam; as the particles increase in size the maximum moves away, making an angle greater than 90° with the incident beam ; for still larger particles a neutral point andareversal appear. Now, in figs. 2 and 3, when 7=2, there are two neutral points, two maxima and a minimum. Nothing corresponding to this was observed in any of the experiments. In fig. 6 the curve for 7=1°96 is more like what would have been expected ———— a SS Light scattered by Spherical Metal Particles. 481 for 7<1°5 and >1°:25. The reason for the discrepancy is not clear. The following points may be noted :— (1) The polarization is calculated for a single isolated particle. In the experiments, we are dealing with a mass of particles, there being usually about 10° particles per cubic centimetre. (2) A different selection of the value of mia DE) Mo might have brought the theoretical curves nearer the experi- mental ones. The values of » and nx chosen for the caleu- lation are those given by R. 8. Minor, viz. n=0°176, nk =3°305. Other values that might have been used are _those given by Hagen and Rubens, nx=3°78 for chemically prepared silver, with a reflecting power R=0°927, giving n=0°31; Shea gives n=0°20 for 486, and n=0°27 for 589. There is obviously very great uncertainty in regard to these data. (3) In the theory it is assumed that all the particles are strictly spherical. With very small particles this condition would no doubt be fulfilled, the forces of surface tension completely overcoming forces tending to produce crystalline structure. But as the particles increase in size a point must be reached at which the surface-tension forces are no longer capable of keeping the particle in a spherical form. 6. Summary. The polarization of the light scattered by spherical par- ticles whose circumferences measure between one and two wave-lengths has been calculated. First, on the supposition that the particles are perfectly electrical conductors ; and, secondly, for the case of silver particles immersed in water, the incident light having a wave-length of 550 up. Obser- vations have been made on the polarization of the light scattered by silver suspensions containing particles of known size. The behaviour indicated by the theory is followed except when the circumference of the particles approaches two wave-lengths. In this case the theory indicates some remarkable changes in the polarization, which have not been observed in any experiment. I am deeply indebted to Professor A. W. Porter for suggesting the subject of this investigation, and for the unfailing interest which he has taken in its progress. University College, London, May 1915. Phil. Mag. 8. 6. Vol. 30. No. 178. Oct. 1915. 25 Pi ABBr | L. On the Ultraviolet Spectrum of Hlementary Silicon. By Professor J. C. McLennan, £.RS., and Hvan Hpwarps, J.A., University of Toronto ™. | N the course of some work by the writers with a fluorite spectograph on the ultraviolet spectrum of mercury, cadmium, and zinc, their attention was. drawn to a paper recently published by Sir William Crookes — on “ The Spectrum of Klementary Silicon.” In this paper the wave- lengths in the spark spectrum of this element are given down to X= 212463 A.U. As the only wave-length as yet published for this element, shorter than this limiting one, is one recorded by Eder and Valentat at 7X=1929-0 A.U., it was thought that it might be well to examine the spectrum in the region below 7=2125 A.U. This has now been done, and the present paper contains a list of the lines which have been observed down to X=1842°2 A.U. Sir William Crookes in his paper records that the elementary silicon of the highest purity which he was able to obtain was supplied to him by the Carborundum Company at Niagara Falls. He also records that the three samples supplied by them to him gave an analysis 99°56, 99°86, and 99:98 per cent. of silicon, the impurities being titanium, iron, and aluminium. On account of these samples being so pure the writers applied to the Manager of the above-mentioned company for a few small pieces, and these were sent to us with the accom- panying statement that they were presumably of as high purity as those with which the observations of Sir William Crookes were made. Our experiments were made with these pieces of silicon, and we desire here to acknowledge our indebtedness to the Carborundum Company for their kindness in the matter. In making the experiments the spark was obtained from the discharge of two one-gallon leyden-jars charged with a 10-inch induction-coil. A great many plates were taken with different pieces of the silicon, and seven of the best of * Communicated by the Authors. Read before the Royal Society of Canada, May 26th, 1915. { Sir William Crookes, Proc. Roy. Soc. No. A. 621, vol. xc. Aug. 1914, p. 512. { Eder and Valenta, Atlas Typischer Spectren, Wien, 1911. On the Ultraviolet Spectrum of Elementary Silicon. 483 these were carefully measured up to arrive at the wave- lengths of the different lines which came out. In taking the photographs the spark spectrum of zinc was ' first taken on the plate, then the spark spectrum of silicon, and finally the spark spectrum of aluminium. The prominent zine and aluminium lines in the region examined were :— Zine lines *. Aluminium lines f. A=2138-66 A.U. \=1990 57 A.U. 02:35 ,, 35:9, 00:08 __,, 3115 ,, 206432 ,, 186281 _,, 62:08, 532, aol 5, 54°8 and these were used as standards when determining the wave-lengths of the silicon lines. In measuring up a plate the distances of the various zine, silicon, and aluminium lines from the edge of the plate were carefully measured with a Hilger comparator. The distances of the above- mentioned zine and aluminium lines from the edge of the plate were used for the abscissz of a calibration curve, and the wave-lengths of the lines as ordinates. ‘This calibration curve was then used to determine the wave-lengths of the silicon lines. Photographs were also taken of the spark spectra of iron and aluminium to make certain that no lines of these elements were included in those obtained with the samples of silicon, but as we had no samples of titanium at hand we were not able to take the precaution of making absolutely certain that no lines of that element were included in those ascribed to silicon. However, the high purity of the silicon used, and the fact that the same spectrum came out on the plates when different pieces of the metal and different points on the same pieces were used as sparking terminals, would seem to guarantee the purity of the spectrum observed. The mean values of the measurements made on the wave- lengths of all the lines observed in the spark spectrum of silicon are :— * Eder and Valenta, Atlas Typischer Spectren, Wien. + Handke, Inaugural-Dissertation, Berlin, 1909, p. 18. 912 as and 434 Prof. McLennan and Mr. Keys on the Wave-lengths in Relative Intensity. Angstrom Units. Arbitrary scale. A= 2124-4 10 2073'1 2 65:2 i 61:4 2 54:9 2 24°9 1 1990°1 4 86:2 1 217 1 09:3 1 01:0 5 1885°5 10 58-4 Faint 539 il 50-0 Faint 45°5 2 49-2 1 The Physical Laboratory, University of Toronto, May Ist, 1915. LI. On the Mobilities of Ions in Air at High Pressures. By Professor J. ©. McLennan, F.R.S., and Davip A. Keys, B.A., University of Toronto *. I. Introduction. | ie a paper by the writers “ On the Electrical Conductivity imparted to Liquid Air by Alpha Rays,” attention was called to the exceedingly high insulating properties possessed by liquid air. The paper also included some measurements on the saturation currents in liquid air and in air at high pressures when these were ionized by alpha rays. In the discussion of some phenomena connected with these currents attention was drawn to the necessity for making measure- ments on the mobilities of ions, both in liquid air and in air at very high pressures. Since the publication of that paper we have on several occasions made attempts to measure the mobilities of ions produced in liquid air, but up to the present have not succeeded in getting any trustworthy results. Convection currents due to the motion of air-bubbles formed in the liquid air, and the contamination of the liquid air * Communicated by the Authors. Read before the Royal Society of Canada, May 26th, 1915. . ' J » } , : i 4 : j : 7 | : Mobilities of Ions in Air at High Pressures. 485 by ice-crystals formed from condensed atmospheric water- vapour, have been two disturbing factors which we have not as yet been able satisfactorily to eliminate. It has been difficult, too, to reduce the size of the ionization chamber of the measuring apparatus to dimensions small enough to permit of its use in a mass of liquid air small enough to be jacketed and kept at a low temperature by an outside vessel of liquid air maintained at a low temperature by rapid evaporation. With regard to measurements on the mobilities of ions in air at high pressures, however, it has been quite different, for it has been found easy to make measurements on the mobilities at all pressures up to as high as approximately one hundred and ninety atmospheres, for such high pressures were obtained quite readily by the use of a liquid-air com- pressor. The only experiments which have been made hitherto on the mobilities of ions in air at high pressures appear to be the ones made by Dempster* and those made by Kovarik fT. In his work Dempster used pressures as high as 100 atmospheres, and he found that over the range from 1 atmo- sphere to this limit the mobility of the positive ion made in air, by alpha rays varied inversely as the pressure. He found, however, that the mobility of the negative ion at the higher part of the range did not appear to vary inversely as the pressure, but it decreased less rapidly with the pressure than it should have done it the inverse-pressure law had been valid. Kovarik in his experiments, on the other hand, worked with pressures from 13°3 to 74°6 atmo- spheres, and over the whole of this range he found that the mobilities of both positive and negative ions made in air by alpha rays followed the inverse-pressure law. In the present investigation the mobilities of the two kinds of ions were measured in air over a range of pressures commencing at 66°86 atmospheres and extending to 181°5 atmospheres. At the lower pressures of this range the mobilities obtained agreed with the results of Kovarik, but at the higher pressures it was found that the mobilities of the two kinds of ions began to approach each other in value, and both decreased less rapidly with increases in pressure than they should according to the inverse-pressure law. * Dempster, Phys. Review, vol. xxiv. No. 1, Jan. 1912, p. 53. t Kovarik, Proc. Roy. Soc. A. vol. Ixxxvi. p. 154 (1912). 486 Prof. McLennan and Mr. Keys on the Il. Apparatus. In making the measurements the appsratus shown in fic. 1 was used. AB was a thick circular plate of brass about 8 cm. in diameter into which a polonium-coated Fig. 1. TO ELECTROMETER UY ERT Yy YY copper plate CD was inserted. GH was a circular plate of brass 2 cm. in diameter, and EFKL was a circular guard- plate surrounding GH. The plate GH was held firmly in position with ebonite supports with its lower face flush with that of the guard-plate EL. The upper face of CD, which _ was the one coated with polonium, was also flush with the upper face of the plate AB into which it was inserted. The plate CD was square and had an area of 16 sq. em. The plates GH «and EL were kept at a distance of 1 em. from the upper face of AB by means of ebonite supports. The clearance between GH and the guard-plate EL was less than one-half a millimetre. When this ionization chamber was in use it was placed in a strong steel cylinder which had a capacity of about 1°5 litres. The guard-plate was electrically connected to the steel chamber, which was itself kept joined to earth. One terminal of a battery of small storage-cells was joined to earth, and the other terminal was joined by a wire, which passed through an insulating plug of ebcnite in the walls of the steel cylinder, to the plate AB. An insulated wire PR also passed out through the walls of the steel cylinder and was joined up to a pair of quadrants of a Dolezalek electro- meter. With this arrangement any desired uniform electric field, positive or negative, could be established and main- tained between the polonium-coated copper plate CD and — ee ee ee ee ey 6 ee Mobilities of Ions in Air at High Pressures. 487 the electrode GH. As the range of the alpha rays from polonium is only about 3°8 cm. in air at atmospheric pressure, it will be seen that, at a pressure of about 70 atinospheres and higher, the ionized portion of the air between GH and CD was confined to a very thin layer close to the latter piate. The experiment consisted in measuring the current between CD and GH with various positive and negative voltages applied to AB at the different pressures. The formula for determining the mobilities which is applicable to the present case is that given by Rutherford* and Child f. Iixpressed in electrostatic units the mobility of an ion is given by or et = 32 —ayg em. a second per 300 voltsacm.,. (1) where 7 is the current between ©D and GH in e.s.u. per square cm. cross-section, d the distance in em. between GH and CD, and V the potential difference between them in electrostatic units. Expressed in practical electromagnetic units, ie = 32007 - em. a second per voltacm., . (2) where d is in cm., V is in volts, and 7 is in electrostatic units, and is the current per square cm. cross-section between CD and GH. As d was 1 em. in the apparatus used by us the relation (2) reduces to 3200.7 .2 ; ik= CRE a a (3) From equation (3) it will be seen that for a selected pressure the current 7 should be proportional to V°. At all the pressures used this law of proportionality between 2 and V? was tested by giving different values to V, and in all cases it was found to hold. One of the different sets of readings obtained at a pressure of 145°35 atmo- spheres will serve to illustrate this point. The voltages applied, together with their squares and the corresponding currents per square cm. cross-section, are given in Table I. They are represented graphically in fig. 2, and it is clear from the diagram that the relation between i and V’ is a linear one. * Rutherford, Phys. Review, vol. xiii. (6) p. 321 (1901). + Child, Phys. Review, vol. xii. (3) p. 1387 (1901). 7 IN £3.u.K107° REN 488 Prof. McLennan and Mr. Keys on the TABLE I, Air. Pressure = 145°35 atmospheres. P.D. in volts, V. | Square of P.D. | Current in e.s.u. per cm.? Positive. 5 4-11 16°89 4:-51x10- 6°16 37°95 WES)! 8:21 67°3 21°51 10°26 105-27 33°87 12°35 150°69 48°71 14°35 206:27 67°24 16°4 268°0 84°89 Negative. We 4°] 168 709x10 °° 6°16 37°95 16°42 8:2 67:0 29°83 10°26 105-27 45°32 12:3 150°69 66°25 14°36 206°2 91°56 16°4 268°0 1229 eH PEPE EEE Ey EEE EEE EEE EEE EEE EEE EEE SeSSEEeSeaeteres SEERE SHORE BERNE CEES PERE SRE Ree BGS BEES OEREs DUESs DSeeeeeeen SSS ES aan Bee eect acae Ss roofHaE sn rane oo nO FH si |} EH pexpual cay aueauae Bre i Sooo siintocteteasttcazt BA sus sopee sone a 50000 BEn>. sees aa anew bee [| |. Fe - ae, Ea caeeeSeeeeeeteccevereaeete EEE eee BBERE SPREE GROSS RSARE | SES008 SEBS SEBSE BERES ASSBBSESSD VOLTAGES SQUARED=V* Mobilities of Ions in Air at High Pressures. 489 Ill. Results. Diagrams similar to that in fig. 2 were plotted from the readings taken with different alas at all the pressures selected. From them values for [V2 were calculated for each of the pressures, and on substituting these values in the relation given by equation (3) the mobilities for both positive and negative ions were deduced for the corre- sponding pressures. These mobilities are all collected in Table LL., and they are plotted in fig. 3 against the pressures as abscissze. > $EESES ssau0 gaeu0 s00u0aeue0 seues seus runssse00 55550555! 3s Sra cones peees Fnsestecs cuosstenes eeess estes taaes asa: & ede: coeed sotes peeteceuse Genes ewees soeesecet rer=rea==s = SERS REaRRE BREET ARE BEE ae ee sauce ceescoarss a ee sues Sescdeueueceuaasecas cua PH ++} a PELE CEE Li peaeeene 6<« cesar sissisreateessel = auaes Heteeses = sRGnGauaneee oH Ess BeGnau ae Poo eet Bilis eee esas SHaee Ree eee Sous cESESEEtosEGudOuEt roast casi aves tos fesitoseee saposinant SRIERE HETGEEet | ne Sesstasnstaninsiinsenestesstt ; Suuosst soustossetanieosntoussesasseas teas pecceses soporte ats pecereees Rte {JaGnSeuGnN" Gunes caeSe A BR Su aseeee ae ee ee anaJe BGs! (FROGGER eRe Ee SER SARE eee eenenn oe a eee e suites ae EEE ay Hl SS ogee ——- Seeeeeeoen isa asoae ; HEH ne SSSEEEETEE tnenstesssereed auaee HH sana negaaceees aaeeeese 1 suneeueaue suuee see8 BEE cEeeeee casmueecemae SOBSEGGEEC tppeaosse anteedeoat suite sites seesteoststtasttes SHH iirstitiniist Scoamsmana Sl Be BBS Ba Be SEGEecsess Soaueauenseeue Se iSSH Sret itz Ree ° ; 20 40 69 80 100 120 140 ] PRESSURE JN ATMOS. Mobilities calculated according to the inverse-pressure . law on the basis of the mobilities for positive and negative : ions at atmospheric pressure being respectively 1: 34 and 1°89 cm. a second per volt a cm. are also given in Table II. ; The dotted curves represent the calculated mobilities and the smooth curves the mobilities determined in the present in- vestigation. As both the table and the figure show, the mobilities did not decrease as the pressure rose so rapidly as was demanded by the inverse-pressure law. Moreover, 0 ee ee SS a eee ee = 490 On the Mobilities of ons in Air at High Pressures. AWN; 08 OE Mobilities of ions at various pressures in air. Pressure in | Atmospheres. Experimental Results. 66°86 87°21 96°9 108°53 116°28 123°1 132775 145°35 153-04 164:73 1754 181°5 Calculated on basis of pxk =constant. | Mobility in em./sec. per volt/em. | Positive=/,. 19°70x10-° 16°13 14:92 13-65 12:87 12°31 12:03 10-98 10°82 10°36 9:19 9-11 67:3x10°° 449 33:5 29-4 16'8 13:4 11-2 9°6 8-4 75 6°7 Negative=/, 2830x107 ° 21:37 19°46 18:83 17-83 16°69 15°38 15:20 14-24 14-08 12-46 11:97 945x107? 63 0 47-3 Mobility x Pressure. it will also be seen from the table and the diagram that mobility of the positive ion approached that of the negative ion as the higher pressures were reached, the ratio of the mobility of the negative ion to the positive ion dropping from 1°43 at 66°86 atmospheres to 1°31 at 181°5 atmospheres. The departure from the inverse-pressure law, however, was not very gr eat. Ratio of mobilities 4 ee: .| Positive. | Negative. 1°32 1:89 1°43 1°41 1°86 Mae 1:46 1°89 1:30 1°48 2°04 1°38 150 2:07 1°39 P52 2°05 1:36 1:60 2°04 1:28 1:60 age 1:38 1°68 221 1°32 Lega 2°32 1:36 161 2°19 1:36 1°65 yeh 131 1:34 1:89 1-41 the It will be recalled that Greinacher *, in his experiments on the ionization of paraffin oil and of petrol ether by alpha rays, found that the mobilities of the positive and negative ions produced in these liquids were practically identical. * Greinacher, Phys. Ze.t. 10 Jahr. No. 25, p. 986. On the Delta Radiation emitted by Zine. AQT In this connexion it is interesting to see that our results indicate that very probably the same equality would apply to the mobilities of positive and negative ions in liquid air. Measurements on the mobilities of ions in air at pressures still higher than those used in this investigation would be required, however, to show whether this surmise were correct or not. . In closing, we desire to express our appreciation of the services of Mr. P. Blackman, who assisted us in taking many of the readings in this investigation. The Physical Laboratory, University of Toronto, May Ist, 1915. LIT. On the Delta Radiation emitted by Zine when bombarded by Alpha Rays. By Professor J. C. McLennan, /.A.S., and ©. G. Founn, At University of Toronto * [Plate VIL] I. Jntroduction. N some experiments by V. BE. Pound tT and described by him in a paper: “On the Secondary Rays excited by Alpha Rays,” he found that the delta radiation emitted by carbon when bombarded by the alpha rays from polonium increased very considerably when the temperature of the earbon was lowered from room temperature to the temper- ature of liquid air. He also showed that this increase in the delta radiation from carbon as its temperature was lowered was due to an increase in the amount of air occluded in the surface of the carbon. Numerous observers have also found that the amount of a gas occluded in the surface of metals determines to a very considerable extent the intensity of the photo-electric effect exhibited by such metals when stimulated by ultra-violet light. Indeed, it was shown by Kiistner f that no photo- electric effect was exhibited by zine even with wave-lengths as short as NX=1850 A.U. when the metal was scraped in a vacuum alter extraordinary precautions had been taken to exclude gases, particularly the active ones. Wiedmann and * Communicated by the Authors. Read before the Royal Society of Canada, May 26th, 1915. t Pound, Phil. Mag., November 1912. + Kiistner , Phys. Zeit. p. 68 (1914). — ———————— 492. Prof. McLennan and Mr. Found on Delta Radiation Hallwachs * have shown, too, that the removal of occluded gases from potassium by repeated distillation in a very high vacuum caused its photo-electric effect to disappear com- pletely with light which included wave-lengths down to »=3400 A.U. The results of Kiistner and Wiedmann and Hallwachs have also been confirmed by Fredenhagen J. In addition, Hughes= has shown that the contact difference of potential between zinc or bismuth, both distilled wn vacuo, and platinum is exceedingly small when the surfaces of the zinc or bismuth consist of fresh deposit of the distilled metals. If traces of air, however, be admitted into the evacuated chamber containing the metals a great increase takes place in the contact difference of potential between the metals. In view of all these experiments it was thought well to investigate what the effect would be on the intensity of the delta radiation from zine under bombardment by alpha rays when care was taken to remove as far as possible all gases from the surface of the zinc bombarded. The following paper contains an account of this investigation, and from what follows it will be seen that with freshly prepared zinc surfaces the delta-ray effect is exceedingly small, but that when air is permitted to be occluded in such surfaces a very great increase takes place in the magnitude of the effect. Il. Apparatus. The apparatus used in conducting the experiments is similar to that used by Hughes § in his investigations on the photo-electric effect and is shown in fig. 1. It consisted of a glass tube about 3 cm. in diameter and about 60 cm. in length. This tube carried at its upper end a tap windlass W and at its lower end it was provided with a ground-joint for fitting it into the glass heating-chamber shown in the diagram. ‘The tube was lined with a thin-walled brass tube which was kept joined to earth through a connexion at EH. B and D were two guiding-rods of brass attached to the inner lining brass tube, and MN was a strip of brass which was supported by a cord from the windlass W and had loops on its ends about the guiding-rods Band D. An insulated brass rod H was rigidly attached to MN through the inter- mediary of a short cylinder of amber A. It carried at its i SO aia and Hallwachs, Verh. d. Deutsch. Phys. Ges. p. 107 + ee Verh. d. Deutsch. Phys. Ges. p. 201 (1914). t Hughes, Phil. Mag., Sept. 1914, p. 337. § Hughes, Phil. Trans, A. cexii. p. 205 (1912). emitted by Zine bombarded by Alpha Rays. 493 lower end a small plate of zinc X with a projecting piece S which came into contact with the cup C when the rod H was raised by the windlass. A slender brass rod connected the cup © to a sensitive electrometer. LP was a circular | | E Earth |__ fo Pump Pi To Battery To Electrometer plate of copper 2. cm. in diameter, with a deposit of polonium on its anterior face. As shown in the figure it could be connected as desired to either terminal of a battery of small storage-cells. The tube B’ was filled with coconut charcoal which was used for the purpose of improving the vacuum made with a Gaede rotary mercury-pump. F was a small fused quartz furnace-tube and was provided with platinum heating-coils as shown. It was held in an upright position by means of a short glass rod sealed into the base of the heating-chamber. When making zine deposits on the surface of the zine plate X the apparatus was first of all evacuated as highly as possible with the Gaede pump in A394 Prof. McLennan and Mr. Found on Delta Radiation conjunction with the coconut charcoal cooled with liquid air. Metallic zinc placed in F was brought to the boiling point with the heating coils, and the rod H was lowered so that the zinc plate X was directly above the opening in F and immersed in the issuing vapour. With this arrange- ment the zine plate could be readily coated with a fresh surface when desired. In studying the delta radiation from this plate the rod H was raised with the windlass W until the projection S was in electrical contact with the cup C. Under these conditions the zine plate X was directly in front of the polonium-coated plate P and was subjected to bombardment by the alpha rays which were emitted by the latter. It should also be mentioned that when in operation the tube was set up with that portion about P in the field and between the poles of an electromagnet. III. Huperiments. EXPERIMENT I.—In commencing the investigation two experiments were carried out similar to those described by Logeman * in his paper on the emission of electrons from metals bombarded by alpha rays. In the first experiment a heating-jacket was placed about the tube containing the charcoal so as to drive the air out of the latter, and the apparatus was exhausted as highly as possible with a Gaede rotary mercury-pump. After this was done the heating- jacket was removed from B’, and when the latter had dropped to room temperature it was surrounded with a Dewar flask and cooled with liquid air. When this was done a McLeod gauge attached to the apparatus showed that the pressure in the vessel had been reduced to considerably below 001 mm. of mercury. ‘The zine disk X, whose surface had been carefully scraped before it was inserted in the apparatus, was raised until contact was made between §$ and C. The polonium plate P was then charged to various positive potentials by means of the storage-battery, and the corresponding currents between P and X were measured with the Dolezalek quadrant electrometer joined to C. The capacity of the quadrants and the attached electrical system was found to be 140 e.s.u. The values of the applied potentials and the currents they produced are given in Table I., and a curve representing them is given in Pl. VII. fig. 2. * Logeman, Proc. Roy. Soc., Series A, vol. xxviii. Sept. 6, 1907. emitted by Zine bombarded by Alpha Rays. 495 TaBie I. Sensibility of Electrometer, 8, =220 mm. per volt. Capacity of Electrometer, C, =140 e.s.u. D=deflexion in mm, scale-divisions per minute. The current, 7, =C.D./300.8.60 e.s.u. Voltage on Defiexion in Current x 10° Polonium. mm. per min. e.s.u. Volts. in, 0 —3d —127 are —22 — 80 oa +o kh + 6 ual + 40 +10 16 + 58 aac 20 ee +22 20 + 73 +28 21°5 + 78 -+40 21 + 76 +50 23 + 84 +60 23 + 84 +80 23 + 84 | From the numbers in the Table and from the curve in fig. 2 it will be seen that, although the terminal P was always either at zero or at a positive potential relative to X, the current was initially negative and remained so until a potential of about 4 volts was reached, when it passed through zero and became positive, gradually increasing to a maximum with an applied positive potential of about £0 volts. In this experiment it will be noted that the current between P and X consisted of : (1) a very small positive current in the residual gas due to ionization ; (2) a positive current consisting of the stream of alpha particles emitted by the polonium ; (3) a positive current consisting of a stream of recoil atoms from the polonium; (4) a positive current due to electrons passing from X to P arising from the bombardment of X by the alpha particles; and (5) a negative current due to electrons passing from P to X which accompanied the alpha particles and had their origin either in the polonium or in the copper surfaces on which the polonium was deposited. With zero or low positive voltages the stream of electrons mentioned in (5) it will be seen completely masked the other four constituents of the current. As the positive applied voltages, however, were increased this stream of electrons was more and more prevented from leaving the electrode P, and finally when a potential 496 Prof. McLennan and Mr. Found on Delta Radiation of 40 volts was reached, none escaped from P at all, and the current became constant and consisted of the first four constituents mentioned above. This result was exactly in accordance with what Logeman had previously observed, and it showed that the apparatus was working satisfactorily. EixpEriment [].—In the second experiment the polonium- coated disk, P, was kept joined to the positive terminal of the batiery at a steady potential of 80 volts. This ensured that no electrons escaped from P. The electromagnet was then excited with currents of different intensities, and readings were taken on the corre- sponding currents through the chamber to the zine plate X. The results in one of the experiments of this type are given in Table IJ. and are represented graphically in Pl. VII. fig. 3. From these it will be seen that the current gradually fell off as the field was increased and ultimately reached a steady state with a field of approximately 1000 gauss. TasE II. Voltage on polonium plate=80 volts, positive. Ghoouils fio, | Value of the | Electrometer | Current x10? Bie ets SEE magnetic field. |deflexion per min. oat amps. gauss. mm. Zero zero 23°0 84 | 5 240 22:3 72 | 1:0 410 18-0 58:3 Lar 690 15:0 49 2°5 905 14-0 45 30 1020 13-2 43 33 1100 13°5 44 37 1190 11°5 3 4:4 1275 14-0 45 4-7 1300 12°7 41 5:0 1330 13-7 44 6:0 1400 13°7 44 From the known properties of the ionization currents of the alpha rays and of recoil atoms it is clear that a field of this intensity was not sufficient to modify to any appreciable extent the current carried by them to X, and it follows therefore that the decrease in the current observed was due to the action of the magnetic field in curling the electrons emitted by the zine plate under bombardment by emitted by Zinc bombarded by Alpha Rays. 497 the alpha rays back again into that plate. This experiment, therefore, showed that a field of 1000 gauss was sufficient when the applied potential difference was 80 volts to entirely cut off the stream of electrons. The problem before us, then, was to apply the procedure just described to the investi- gation of the intensity of the electronic stream from the zine plate X when the surface of this plate was made to undergo various modifications. Before leaving this experiment it may be pointed out that the results obtained go to show that approximately three electrons were emitted by the bombarded zine plate for every alpha particle which struck it. From the table it will be seen that the current under the electric field, combined with the maximum magnetic field, was approximately 44x10-°e.s.u. This current consisted of (1) alpha particles, (2) recoil atoms, and (3) the ionization current. As the gas-pressure in the apparatus was ex- ceedingly low the ionization current must have been negli- gible. Taking it to be so the current must have been carried by the alpha particles and the recoil atoms. If, now, we assume that as many alpha particles were shot back into the polonium plate as were projected forward from it, it follows that the number of recoil atoms taking part in the current was very closely equal to the number of alpha particles which contributed to it. Taking the charge on the alpha particles to be 2e, and that on the recoil atom to be e, we have then, since the current carried by the electrons emitted by the zine plate must have been 40x 107° e.s.u., the number of alpha particles striking X given by 44x10-°/3e. Since the number of electrons emitted py the zinc plate was 40 x 10~°/e, it follows that 2°73 electrons were emitted by the zine plate per alpha particle which struck it. EXPERIMENT IIJ.—The next experiment which was per- formed served to illustrate the fatigue of the delta-ray effect. In this case the apparatus was continuously evacuated for two days after the measurements made in experiments I. and II. were taken. At the end of this time readings were taken by applying various positive potentials up to 80 volts, and when this was reached the magnetic field was turned on and readings were taken with fields up to 1400 gauss. These are all recorded in Table TiI., and are shown graphically i in fig. 4 together with the results of experiments [. and II. From the results it will be seen that while the maximum current under 80 volts was 84x 107° e.s.u. at the beginning of the experiment, it was only 63x 107° e.s.u. after two Phil. Mag. 8. 6. Vol. 30. No. 178. Oct. 1915. 2 i 498 Prof. McLennan and Mr. Found on Delta Radiation days’ evacuation. With an applied potential difference of 80 volts and a magnetic field of 1400 gauss, however, the readings obtained on the two occasions were practically the same. This showed that the current carried by the alpha particles and the recoil atoms remained the same for the two days, but that in the interval the electronic current from the zinc plate under bombardment by the alpha rays had dropped from 40 x 10° e.s.u. to 18 x 107° e.s.u. TABLE ITT. | Voltage on Current Magnetic Defiexion Current X10° | Polonium. |through coils. field. per min. e.8.U. | Volts. amps. gauss. mm. zero zero zero —10 —29°5 2 5 ; — 6 —17°0 4 ‘3 ; it 30 8 » 19 06 14 35 AF 21 62 20 ‘i 9 21 62 40 if 3 21°7 64 60 f si 20°5 60 80 * i 21-2 62°6 80 8 240 19-5 57°6 80 10 410 17°8 53 80 VG 690 16°5 49 80 2°5 905 148 44 80 3°5 1140 14°5 43 80 4°4 1275 15 44 80 50 1830 15 44 80 o7 1390 15 44 This result made it evident that the electronic stream from the zine plate was determined to a considerable extent by the amount of air occluded in its surface. Jor it is clear that under the continuous evacuation for two days there must have been a gradual diniinution in the amount of air occluded in the metal, and as everything else in the experiment re- mained the same this diminution must have been the cause of the decrease in the stream of delta radiation. EXPERIMENT IV.—In this experiment a freshly cleaned plate of zinc was attached to the rod H at X, and the appa- ratus was left full of air at atmospheric pressure for six days. It was then exhausted as highly as possible with the Gaede pump and the coconut charcoal surrounded with liquid air. Readings were first taken with positive potentials applied to P up to 80 volts, and then keeping the potential of P at emitted by Zine bombarded by Alpha Rays. 499 S0jvolts positive readings were taken with increasing mag- netic fields up to 1245 gauss. These readings are given in Table IV., and the curve representing them is shown in Pr VIL. fig. 5. Tasie IV. | _ Zine surface first scraped clean and then exposed to air | at atmospheric pressure for six days. | Current | Magnetic : Current | eee ee through field. aaee «10° esse coils. Gauss. P : e.8.U. | Volts. aps. min, zero zero zero — 8 —25 ; +2 , : 5 16a eG : : 22 69 10 é ‘4 24°5 Ei Par: 4 ‘ 25 wa | bie 20 26 82 ie) 26 x pi 26 ae. 4) 30 3 He 26 Z ees - a “6 82 60 9 is 26 82 80 99 5 2¢ 82 80 “5 240 23 72 80 "95 405 23 72 80 13 570 22°5 Tr 80 1°65 680 20 63 80 28 860 17 53 80 2°8 995 15 47 80 ts 1100 14°5 456 80 36 1160 l4 44 80 39 1205 14 44 80 | 4-9 1245 14 44 From these it will be seen that the maximum current obtained without any magnetic field was 82 x 107° e.s.u., but that with the magnetic field applied the current fell to 44x 10~° e.s.u., and this current, as was pointed out before, consisted of (1) the residual ionization current, (2) the stream of alpha particles from P, and (3) the stream of recoil atoms from the same source. After this set of readings had been taken the liquid air was taken from about the charcoal which was then allowed to rise to room temperature. The heating-jacket was then placed round it and its temperature gently raised so as to drive off as much of the occluded air as possible. While this was being done the Gaede pump was kept constantly in action. Meanwhile the rod H was lowered with the wind- lass W until the plate X was directly over the furnace F 2K 2 500 Prof. McLennan and Mr. Found on Delta Radiation and about 2 cm. above it. When the pressure had been reduced to below ‘001 mm. of mercury a current of 10 amperes was passed through the platinum wire of the furnace for 15 minutes. This sufficed to vapourize the zine in the furnace and to deposit a good coating on the surface of the zinc plate X. The furnace current was then cut off and the rod H was raised as quickly as possible until contact was made with 8 at C. A positive potential of 80 volts was then applied to P, and readings were iaken with increasing magnetic fields at intervals of a few minutes for half an hour. The magnetic field was then cut off, and at the end of 55 minutes readings were again taken at intervals for half an hour with electric fields ranging from zero to 80 volts. At the end of 90 minutes a reading was taken with an applied field of 80 volts anda magnetic field of 1200 gauss, and at the end of 190 minutes a reading was again ‘taken under the same conditions. All these readings are recorded in Tables V. and VI., and curves drawn from them are shown in Pl. VII. figs. 6 and 7. Tapum V. Zine surface deposited from zine vapour in a high vacuum. WiSitcelen Current | Magnetic |Timesince}| Deflexion | Current | Poloni through field. surface per x 10° -olonium. : coils. Gauss. made. minute. €.S.U. Volts. amps. min. mm, +80 zero | zero 5 53 166 80 5 240 lel 52 163 80 “95 405 15 53 166 t 80 1°65 630 18 52 163 80 2°3 880 | 23 53 166 80 2°8 995 28 53 166 80 36 1160 30 53 166 Zer0 zero | zero 5d EA | — 66 2 ; ‘ 58 0 0 6 ; 3 60 19 60 10 . Me 62 23 72 14 “ | ‘i 65 22 69 20 i | 67 25 78 26 - 70 25 z 30 ; 53 2 25 78 40 a mn 15 25 7 60 % (0 25 78 80 * H 85 25 78 ee ee a a ee eee e ee ne ee a a ee ee ee Sa a a a ng BS Ps emitted by Zinc bombarded by Alpha Rays. 501 Taste VI. Zine surface deposited from zine vapour in a high vacuum, Electric field alone. Electric and magnetic field. Time since a alo Defiexion Saturation Deflexion Saturation na cereal anit EEEneu nnn. ay. Hye 10 53 166 x 10-° e.s.u. 53 166X107? e.s.u. 30 43 | 195 A 43 | 185 ai 90 25 79 y 22, 69 44 | | 190 19 60 se 15 | 47 | : One point which is brought out very prominently by these readings is that for the first half hour the current between -P and X, under a potential difference of 80 volts was the same whether a magnetic field, as high even as 1160 gauss, was applied or not. From this it was manifest that during this interval there was practically no emission of electrons from the newly deposited zine surface, under bombardment by the alpha rays. It will be noted, too, that during the interval the saturation current was about 165 x 107° e.s.u., which was about twice as great as that saturation current obtained in the previous experiment with the ordinary zine plate without the fresh deposit. This was very probably due to the air-pressure in the apparatus being somewhat higher immediately after the deposit had been than it was when the observations were made with the zine plate in its original condition. Hven with the Gaede pump in action the effect of heating the furnace would be to drive off a con- siderable quantity of air from the walls of the vessel into the apparatus, and as the volume of the apparatus was consider- able it would take time to remove this air again. That this interpretation was the correct one is shown by the readings taken in the second period extending from 55 minutes after the deposit had been made up to 85 minutes after that time. These, it will be seen, show that with increasing positive potentials the current increased and finally reached a maxi- mum of only 78 x 10~° electrostatic unit. This would indicate that during the first half hour the ionization constituent of the current was very considerable, as it should have been on account of the higher air-pressure, while at the end of 502 On Delta Radiation emitted by Zine. 85 minutes after the deposit had been made it was much less, on account of the removal of the air from the apparatus. The numbers given in Table VI. and the curves in fig. 7 are also of interest in this connexion, for they show not only that the current gradually diminished with the lapse of time owing to the diminution of the ionization current constituent arising from the gradual reduction of the air-pressure, but also that there was a gradual increase in the electronic stream from the zinc plate with the lapse of time under the bombardment by the alpha rays. From what has gone before it is evident that this develop- ment of a delta radiation from the zinc plate arose from the gradual occlusion of air into the surface of the zine. For, as the vapour was deposited on the zinc plate in a high vacuum the surface would not contain any air first. Tt would not, however, in this state be in an equilibrium condition, and a tendency towards absorption would exist. The result of this would be that so long as air was present in the apparatus, absorption would take place at least until an equilibrium was established between the air occluded in the surface and that within the apparatus. This gradual occlusion of the air by the zine surface would therefore appear to account for, and to be the cause of, the gradual development of the electronic current. IV. Summary of Results. I. In the present investigation it has been shown that when a plate of zinc with a freshly scraped surface is placed in a highly exhausted chamber and bombarded by alpha rays, there is an emission of slow-moving electrons or delta rays from it at the rate of three electrons for each alpha particle impact. II. It has also been shown that the emission of electrons from such a plate of zinc under bombardment by alpha rays diminished with the lapse of time from the moment when it was placed in the high vacuum. IT]. It has also been shown that initially there is no emission of electrons under bombardment by alpha rays from a surface of zinc deposited from zinc vapour in a high vacuum, but that as time elapses an electronic emission is gradually developed under the gradual absorption of air by the surface of the zine deposit. The Physical Laboratory, University of Toronto. May 1st, 1915. 7 BAS OSi | LIII. Lonice Mobilities in Hydrogen and Nitrogen. By W.B. Harnes, B.Sc., 1851 Scholar, University College, London*. | [ has been recorded by Franckt that when nitrogen, argon, and helium are carefully purified, there can exist in them, even at atmospheric pressure, negatively electrified particles having such high mobilities as leaves no doubt that they are free electrons. The values recorded are, 120 cm./sec. for nitrogen, 206 cm./sec. for argon, and 500 cm./sec. for helium, in unit field. These particles were found to be extremely sensitive to any impurity in the gases. The research which the present paper describes was undertaken to investigate in greater detail the effect of impurities in nitrogen, and the work has extended to some interesting results for hydrogen as well. Method. The method adopted was the modification of Rutherford’s method described by Franckt, depending on the use of an Fig. 1. SSS SSSR N BN *, ZANE NS i ° eo ai epee Fa eae ce yas pe aa Sos A) tree eee ener RC wey i Wy ES \ EN \\ alternating electric field. Fig. 1 is a diagram of the * Communicated by the Author. + Franck, Bericht d. Deut. Physik. Ges. 1909. ¢ Franck, Bericht. d. Deut. Physik. Ges. 1907. 504 Mr. W. B. Haines on Jonic Mobilities experimental chamber. Two parallel circular brass plates of 10 cm. diameter are separated by a glass cylinder which is fitted with taps for the inlet and outlet of the gas. The central part of the upper plate is of copper gauze, giving communication between the main chamber and a small side chamber. In this latter the ions are generated by the a-rays from a deposit of polonium on the centre of a small piece of copper situated at A. By maintaining a small potential difference between the brass disk B and the upper plate, the ions of the sign required can be drawn to the gauze, where they diffuse into the main chamber and come under the influence of an alternating field applied between the plates. The use of sealing-wax for assembling the parts and of ebonite for insulation are indicated in the drawing. Fig. 2 Fig. 2. = Elec err = y pane a ee L£arth Earth shows the simple electrical connexions, omitting keys. The central part of the lower plate is insulated from the rest, and is connected to a carefully-screened Dolezalek electrometer, so serving to measure the current carried by the ions. An induction-coil supplied the alternating voltage between the plates, the primary circuit containing an alternating machine and an adjustable lamp-resistance. The voltage between the plates was indicated by a Kelvin multicellular voltmeter. If d be the distance between the plates, and the voltage applied be of the form a sin pé, then the ions travel a distance 2ua/pd from the upper plate before the field reverses; u being the velocity in unit field. Then a current will begin to pass between the plates when Qua 7 =. in Hydrogen and Nitrogen. d05 If Hi be the reading of the voltmeter at this point, a=,/2H, and w==const./EH, where the constant is mnd?/,/2, n being the frequency of the alternator. The critical value EK is found by plotting readings of current against voltage. The arrangements for handling the nitrogen are shown in fig. 3. It was found that the degree of purity required could only be maintaine| by using a continuous flow of gas through the apparatus. The nitrogen was contained in two aspirators of capacity 15 litres each, where all traces of oxygen were removed by leaving for many hours in contact with phosphorus. After passing through a drying-tube the gas traversed a U-tube immersed in liquid air, where the phosphorus vapour and other remaining impurities were removed. A capillary by-pass ensured a slow and regular Fig. 3. Experimental Chamber flow of gas, which was often maintained without interruption for days. The gases to be tested as impurities were intro- duced into the nitrogen stream froma graduated side tube through a second capillary. Any desired proportion of ad- mixture could be obtained by regulating the head of mercury. A filter-pump was used to aid in washing out the tubes and chamber. Results. Nitrogen.—tThe first experiments at once revealed the presence of highly mobile negative carriers in the nitrogen, and emphasized their extreme sensitiveness to traces of impurity. After the stream of pure gas had been passed for some hours, the contamination from the walls of the chamber was reduced to a minimum, but could never be ee 506 Mr. W. B. Haines on Ionic Mobilities entirely got rid of. Even when the chamber had been thoroughly desiccated by many days’ use, this contamination would make itself felt on letting the gas stagnate for a short time. The greatest care was exercised in assembling the parts to have clean interior surfaces, the ebonite parts being covered with paraffin-wax. When the arrangements were complete, the mean value obtained for the mobility was u=367 cm./sec. per volt/em. This value is the mean of nine independent readings varying between 337 and 378 (d=8'45 cm.; n=50 aiternations per sec.). This value re- presents the practical limit reached to the purity of the gas. In one series of experiments, where the conditions were exceptionally favourable, the values u=438 and u=509 were obtained, but these were not repeated. Fig. 4 shows some specimen curves. Fig. 4. i Bh CURRENT | | ea SQ SS ance ah | | f | | | | J | i Wi | 7) 1-0 | / ) | 10 20 30 40 50 60 70 80 90 100 ite) 120: VOLTS In testing the effect of impurities it was not practicable to carry the purification beyond the point corresponding to a value of w about 200, owing to the time required to remove the very last traces of a foreign gas from the chamber. The results show that almost all ‘oases and vapours when added to the nitrogen in small proportions, reduce both the number of electrons and also their mobility. Some gases are much more effective than others, and in all cases the effect of the first addition is greater in proportion to its amount than are subsequent additions. in Hydrogen and Nitrogen. | 50T A few substances have such an immediate effect that, after the first traces have been added, no electronic carriers at all can be detected. Among these are iodine, chlorine, sulphur dioxide, nitrous oxide, carbon bisulphide, and chloroform. When once they have been admitted to the apparatus it is extremely difficult to remove their influence. The method used was to pump dry air through for hours in succession. With the other substances tested the influence was not so great, and quantitative estimates could be made. The following are the figures for oxygen:— Initial mobility (nitrogen not intentionally SMM EMERUUEIALOEL 2 re ie) ge oie vices vin s evan wince 158 em./sec. (05 per cent. oxygen ............ 88 ; 35 Oh Gear 30 a re “ap cops Beane 2 11 With carbon dioxide the addition of 2 per cent. reduced u to 100, and with 5 per cent.a mobility a little higher than the normal ion could be detected. The accompanying list of substances follows the one given above in approximate order of activity: oxygen, sulphuretted hydrogen, ammonia, acetylene, methane, ether, carbon dioxide. The vapour of naphthalene and of camphor had only a small effect, it being not greater than the contamination from unavoidable sources. But it must be remembered that the saturated vapour pres- sure of these substances is very small. There is every reason to suppose that they would considerably reduce the value of u, if present in larger quantity. When the nitrogen was saturated with phosphorus vapour, the value of uw was 90. The following figures were obtained on adding hydrogen, showing that its effect is at most very small :— jai ie Ui: call oer u254 5 per cent. hydrogen ......... 218 10 - Lt) ote 183 14 ae ‘yaaa 28 fey) 20 148 Further results showed that this small effect varied according to the care taken to first purify the hydrogen used. This indicated that the hydrogen itself might be neutral, and led to the following experiments with pure hydrogen. Franck records a negative result from an attempt at finding free electrons in hydrogen. Hydrogen.—The gas was generated as required from dilute sulphuric acid and zine in an improvised Kipp’s apparatus. After being dried it was thoroughly purified by passage 508 Mr. W. B. Haines on Lonic Mobilaties through a charcoal bulb cooled in liquid air. Using the same method of continuous flow, the experiments were at once successful in indicating negative ions of high mobilities in hydrogen. The same results were obtained from hydrogen prepared by electrolysis. A careful repetition and analysis of the curves obtained with varying degrees of purity show the following results. For hydrogen dried but not otherwise purified, the values were: negative u=7'4 cm./sec., positive w=6'0 cm./sec. These are in agreement with the measurements of Franck and Pohl, who give negative w=7'68, positive u=6°02. When the gas is purified, curves of several different types are obtained, specimens being shown in fig. 5. The normal Fie. 5. CURRENT pb fo) 32 24he p 100 Ee 200 B 300 400 A 500 VOLTS 609 negative ion has an increased mobility of about 9, and is shown in all the curves(A). The curves numbered 4, 5, and 6 show the free electron at D, giving values of u between 97 and 170 cm./sec. Intermediate between A and D there wn Eydrogen and Nitrogen. 509 are two other kinds of ion shown at B and U by curves 2 and 3 respectively. The results are grouped together in the following table :— Hydrogen. d=6°45 cm. n=650. Mean temp. 19° C. Gas not specially purified. Negative ion ........ u= 7:4 cm./sec. eeitive 10. sido. ss as y= OO Gas purified. Nevative ions...... A u= 94 (mean nine values). B u=189 (mean two values). C w=28'l (mean five values). D w%=170 (electrons). PGive iON... .... 5.) u= 6-0 (unaltered). It will be seen that the ions appear in different propor- tions in different cases. What the conditions are that deter- mine this variation remain obscure, for the degree of purity required is one beyond adequate control or definition. It can be said generally that the higher the mobility of the ion, the more sensitive it is to impurities in the gas. The ion B is the exception, as it appears in some cases but is quite absent in others seemingly as favourable. It is hoped that some further work will throw light on this point, and provide data for a theoretical discussion of the results. Summary. The mobilities of free electrons in pure nitrogen at atmo- spheric pressure have been measured, and the effect on them of various impurities has been tested. In some cases only a minute trace of the foreign gas is required to prevent com- pletely the appearance of the free electrons. In nearly all cases their life in the free condition, as measured by the mobility, is much reduced by small proportions of the impurity. Hydrogen has been found to belong to the list of gases in which tree electrons can exist at atmospheric pressure. ‘The hydrogen must be purified with great care to obtain these results. Under these conditions other negative ions of high mobility appear, for which measurements have been made. It is interesting to note that, according to results recently published by Wellisch*, the electrons appear in dry air at pressures below 10 cm. It is a pleasure to record my thanks for constant encourage- ment and advice to Professor R. J. Strutt, of the Imperial College of Science and Technology, where the work was carried out. * Wellisch, Amer. Journal of Science, May 1915. [510+ a LIV. Light Absorption and Fluorescence—Part III. By H.C. C. Baty, W.Se., F.A.S., Grant Professor of Inorganic Chemistry in the University of Liverpool”. i ies the second paper of this series} it was shown that the Bjerrum conception of the structure of the short-wave infra-red absorption bands can also be applied to the ultra-violet and visible absorption and fluorescence bands of organic compounds. Bjerrum} showed that if v, be the frequency of the central line of an absorption-band group in the short-wave region of the infra-red, there will be sym- metrically distributed on each side of this line, absorption lines the frequency of which is given by Vy al Vry where v, stands for the rotational frequency of all the mole- cules corresponding to the frequencies of the absorption bands in the long-wave infra-red region. He further showed that on the basis of the energy quantum theory the rotational frequencies must have well defined values given by the expression phn = Bal? where I is the moment of inertia, A the Planck constant, and na series of whole numbers, 1, 2,3. &c. As stated above, it has been found that the relation also holds good with the ultra-violet and visible absorption and fluorescence bands of ‘substances, for if v be the frequency of the central line of one of these bands, there are found to be symmetrically dis- tributed on each side of this line pairs of absorption or fluorescent lines, as the case may be, the frequencies of which are given by v-tv,, where v, stands for the observed frequencies of the absorption bands in the short-wave infra- red region. It was pointed out that although this relation- ship undoubtedly exists, by no means all the component lines of a given absorption-band group are thus accounted for. ‘he question therefore arises as to whether the complete Bjerrum relation holds, and whether the whole of the com- ponent lines in any one ultra-violet or visible band group can be calculated from the expression hn Dt may * Communicated by the Author. + Phil. Mag. xxix. p. 223 (1915). t Nernst, Festschrift, 1912, p. 90. Light Absorption and Fluorescence. 511 If this be proved to be correct, then it would follow that the results given in my last paper represent a portion of the complete relation, for the frequencies of the absorption bands in the short-wave region of the infra-red must themselves represent certain definite values of hn Qn? 1 It would seem to be established from further investigation that the complete Bjerrum relation does hold good, and that the whole of the absorption-band groups in the short-wave infra-red, visible, and ultra-violet regions can be calculated on the basis of Bjerrum’s theory. The first point to be considered is whether it is possible to find the values of hn 2771 for any compounds simply from the measurements of their infra-red absorption bands. In dealing with oscillation frequencies the values are inconveniently large, and where a number of calculations have to be made, it is simpler to use the reciprocals of the wave-lengths, that is to say the oscillation frequencies divided by the velocity of light. This practice will be adopted in what follows, and the reciprocals employed represent the number of waves contained in a length of 1 mm., and in order to convert them into true oscillation frequencies they must be multiplied by 3 x 10°. Now Eucken* showed that the infra-red absorption bands of water vapour of longer wave-length than about 10 w can be expressed by the Bjerrum formula; that is to say, their frequencies form consecutive multiples of two constants (he having assumed two degrees of freedom) which, if wave- length reciprocals be used, are 5°78 and 2:5 respectively. EKucken, however, entirely failed to explain the very remark- able variations in the relative intensities of the infra-red absorption bands of water. He only extended his second series to 10°0y, and offered no explanation of the extra- ordinary intensity of the great absorption bands of 6:0 p, 3, 2m, and 1:5. The whole essence of Bjerrum’s theory is that the frequencies of the centres of the infra-red hands are consecutive multiples of a fundamental constant or basis. His theory in no way accounts for the fact that certain select * Deutsch. Phys. Ges., Verh, xv. p. 1159 (1913). 512 Prof. HE. C. C. Baly on multiples of the constants give rise to absorption bands which are far more intense than the neighbouring multiples on each side. On the other hand, if there are two constants or bases, then there must exist a convergence-frequency of the two as it were, which is the least common multiple of the two bases. It is to be expected that such a frequency would necessarily be especially active since it is keyed with both series. I suggest, therefore, that this is the reason why an infra-red absorption band is especially pronounced in intensity, namely, that its frequency is either an even multiple of the two bases which are active, or that it is the least common multiple of the two bases. If this principle be accepted, it seems entirely to solve the difficulty connected with the intensity of the absorption bands, and makes the calculation of the values of the bases a simple matter. Whatever may be the relative intensities of the longer- wave infra-red absorption bands of water vapour, the most intense bands in the short-wave region are those at 6°25, 6:0, 3:0).2°0, and dese. othe reciprocals of these are 160, 166°6, 333°3, 500°0, and 666°6 respectively. Hucken has already pointed out that 2°5 is one of the basis constants of water, and the above values at once suggest that there is a second basis constant of 6°6. The first wave number 160=64 x 2°5=24 x 6°6, while the second one 166°6 is ihe least common multiple of the two bases. The last three frequencies are the least common multiple multiplied by 2, 3, and 4 respectively. It would seem probable that the intensities of the absorp- tion bands dne to the several multiples of the bases would decrease as the value of n increases in the expression hn Qa’ and consequently in the short-wave infra-red region where n is large, the bands due to these multiples acting alone will be very taint indeed. When, however, that frequency is reached which is the least common multiple of the series, then a very strong absorption band is evidenced. It follows further from this, that the only possible regions of still shorter wave-length at which absorption bands of water can occur will be frequencies which are multiples of 166°6. Since all such absorption bands are multiples of 166-6, there must naturally exist a constant difference of 166°6 between their frequencies, and thus a physical explanation is found for Light Absorption and Fluorescence. 513 the relationship between the absorption bands in the ultra- violet and visible regions and one band in the infra-red dealt with in the previous paper. Similarly it explains the existence of harmonics in the infra-red noted by Coblentz and others. Again, due regard being paid to the fact that the multiples of the least common multiple can only form the central line of the absorption bands in the short-wave regions, it would follow that the complete system of absorption lines in any one ultra-violet band group can be calculated. The central line of the group must have a frequency which is a multiple of the least common multiple of the basis constants, and in the case of water this must be some multiple of 166°6. The complete system of absorption lines in any one band group of water will be given by the expressions 166°62+nK, and 166°62+nK,, where x is some whole number, K, and K, are the basis constants 2°5 and 6°6, and n is 1, 2,3,4,.... &c. This follows naturally from an application of Bjerrum’s principle to any absorption band whether in the short-wave infra-red, the visible, or the ultra-violet region. Nothing, however, is known about the absorption bands of water in the ultra-violet region, and therefore the calculation in this case cannot be put to the test. Attention may be drawn here to the influence of tempe- rature on the breadth of absorption bands, it being a well- known fact that they tend to become narrower with fall of temperature. At the boiling-point of hydrogen it has been shown that the bands appear only as fine lines. In all pro- bability the effect of temperature is to change the molecular rotational energy, and as the temperature falls the effective values of 7 in the Bjerrum formula hn 271 will become smaller in number, and indeed at very low temperatures only the lowest multiples of h 2771 will be active. If the breadth of an absorption band group is due to hn YS oT Phil. Mag. 8. 6. Vol. 30. No. 178. Oct. 1915. 21L 514 Prof. E. C. C. Baly on it is obvious that the band will become very narrow at very low temperatures. Although nothing is known about the ultra-violet absorp- tion band of water, there are, on the other hand, several substances of which the complete systems of absorption lines in the characteristic ultra-violet band groups have been measured with considerable accuracy. It may be pointed out that the conception may in such cases be put to a more rigid test than is possible in the case of the infra-red bands calculated by Fraulein v. Bahr and by Eucken, since the measurements of the fine-line absorption composing a single band group in the ultra-violet are capable of far greater accuracy than those comprising a band group in the infra-red. We are indebted to Coblentz for a series of very accurate measurements of the infra-red bands of a large number of organic compounds between the limits of 1 and 13 pw, and one outstanding conclusion can be drawn from this werk*. The characteristic infra-red bands of a substance are also found to be in the spectra of its derivatives. Thus benzene possesses a very pronounced band at 4°26 yw, and this band is also found in a considerable number of benzene compounds. Again, the characteristic bands of water are shown by salts with water of crystallization, and moreover at least one of these is shown by substances containing the hydroxyl group. These facts at once suggest that in the spectrum of a com- pound there is an additive function of the spectra of its constituents. For example, in the case of phenol the cha- racteristic basis constants of both water and benzene are functionally active and their values are not altered in the compound. It is only in this way that it is possible to explain the occurrence of the principal bands of the constituents in the spectrum of a given compound. Before this question can be discussed it is necessary to deal with the infra-red spectra of simple substances, and find out whether the absorption bands can be expressed in terms of multiples of one or more basis constants. Benzene may be taken as the first example, and in the case of this substance there are twenty absorption bands recorded by Coblentz between the limits 14 and 134. These vary very much between themselves in intensity, and the most pronounced bands are at 9°78, 6°75, 3:25, 2°48, 2°18, and 1°68 uw. Now if the wave numbers of all the infra-red benzene bands be considered, it will at once be seen that they are nearly all multiples of 4, and indeed it appeared at * Publications of the Carnegie Institution, Washington, No. 35 (1905), | ‘i ¢ ~~ a ee ee ee, a, ee we ee eee ae ee Light Absorption and Fluorescence. 515 first sight as if they all could be expressed with very fair accuracy as multiples of 4. This, however, gives no expla- nation of the remarkable intensity differences in the bands, and, further, all the consecutive multiples of 4 do not appear as bands, even between 10 and 13, where perhaps they might be expected to evidence themselves. The two most characteristic and outstanding bands are those at 3°26 w and 6°75 uw, and the wave numbers of these are almost exactly 308 and 148, and on the least common multiple principle these may be looked upon as the least common multiples of A and 7:6, and of 4 and 3°7 respectively. On this view, therefore, we have three basis constants of benzene, namely, 3:7, 4:0, and 7:6, which explain the two most characteristic bands of benzene. Other combinations of these basis con- stants can be made as follows :— 24x 7:6 =46 x 4=184, which is the wave-number corre- sponding to X=5'43 w, a value exceedingly close to 5°5 at which Coblentz found a strong band. Again we have 60x 7:6=115 x4=460, which is the wave-number of X=2:174 and clearly corresponds to Coblentz’s measure- ment of 2:18m. Further, 2x 148 (the wave-number of A=6:75) =296 corresponding to 3°37 w, which in all pro- bability is hidden in the great band at 3°26 w; 3x 148=444 corresponds to 2°25, which is hidden in the great band at 218 p, but 4x 148=592, which corresponds to A=1°68 yp, the value obtained by Coblentz. ‘hese three basis constants therefore explain all the important infra-red bands of benzene except those at X=9°78 and A= 2°48 yw, the remainder being exhibited either at the lowest common multiple of two of the basis constants, or some multiple of it, or at some frequency which is a multiple of two of the constants. As regards the band at X=2°48y, it is somewhat remarkable that the narrow absorption lines composing the ultra-violet band group of benzene can all be arranged symmetrically round the wave-number 4050 as centre. This number is 10 x 405, which latter is very near the reciprocal of 2°48 p. Now no combination of the above three basis constants gives a number near this value; and since this value appears to be a fundamental one for benzene, it is in all probability due to there being a fourth basis constant, 10°125. The lowest common multiple of 4 and 10°125 which is 405 gives a wave- length of 2°47 p, which is exceedingly near to Coblentz’s value of 2°48. We are therefore left with the band at X=9'78 yw, which, however, does not seem to be a specially characteristic band of benzene in spite of its intensity, since 212 516 Prof. KE. C. C. Baly on it does not evidence itself with any definiteness in benzene derivatives. It is most probable that this band and the re- mainder of less intensity are due to multiples of the basis constants which happen to fall near together, with the result that their effectiveness as absorbers is enhanced. Thus 10x 10°125 gives 7~=9°88, and 28x 3:7=9°65y, and the mean of these is 9°76 pw. Added to this, the other two bases give bands in the imme- diate neighbourhood which tend still further to increase the intensity. All the remaining bands can be accounted for at once in the same way by the fact that multiples of two of the bases happen to be near together, and it is interesting to note that when such multiples do not happen to be very close,a weak and very broad band is shown. It is evident from the above that no further absorption bands of any importance can be exhibited by benzene of shorter wave-length than 1 w, except at some multiple of the least common multiple of any two of the basis constants. If, therefore, the approximate position of the ultra-violet band group of benzene be known, it should now be possible to calculate every single absorption line in such group on Bjerrum’s principle, for all the necessary data are at hand. It is peculiarly simple in this case because from previous knowledge it is clear that the central line of the ultra-violet band group is 10x 405 (the lowest common multiple of 4:0 and 10°125). The wave-numbers therefore of the absorption lines constituting the group can be calculated from 4050+nx3°7, 4050+nx4:0, 4050+nx 7-6, and 4050 +n x 10°125. From the wave-numbers obtained in this way the wave- leneths can be calculated, but in order to compare them with observed measurements they must be corrected for the refractive index of air. I have calculated the wave-lengths of the absorption lines of benzene from the above formule, and there are about 600 between X=2761 and 7X=2226 which are the limits experimentally observed by Hartley. Hartley measured the wave-lengths of about 300 lines in the band, and the agree- ment between the calculated and observed values is very good. In Table I. are shown the results for the first few lines on both sides of the central line together with the values observed by Hartley * and by Grebef. * Phil. Trans. cviii. A. p. 475 (1908). + Zevt. wiss. Phot. ix. p. 180 (1911). Laight Absorption and Fluorescence. 517 TABLE I. Ultra-violet Absorption Band of Benzene. eo, Kh 4°0,) Ko 76, K,== 10125. Red Side. Blue Side. A observed. A cale. | m,.| 2. Nas | Ngo r eale. dX observed. Grebe. Hartley. Hartley. Grebe. 2468°3 2469 24684! O 0 (8) Q | 246874 2469 24683 2470°7|} 1 2466°2 2466 2466°5 2171 2470°9 ii 2466'0 2465 2472 2472:9| 2 2463'9 2464 24731 i 2463°7 2473°4 2 2463'6 243 P474°5 2474 2474°7 1 | 2462°3 2462 2462°9 2475°2| 8 2461-7 2476 2476 2173°8 = 24611 2461 2477 2477'°5| 4 2459°4 2460 2477°8 9 2459°1 2459 2478°6 2479 2478°2 4 2458°7 2480 2480 2479°7| 5 2457°2 2457 ee is ( 2480°7 5 (2456°3 ons 2481-1 2481 |{ S750 r pie 94565 2456 2482 2482'0| 6 24550 2455 2482°5 3 9454-5 2483 2483 2483'°2 6 9453°5 2454 2454-0 2484-7 2484 24843] 7 9452°8 2453 2485 2485°6 7 (24515 2452 2486 24866! 8 | 2450°4 2451 2487 2487'1 3 | 24501 2450 2487°3 4 | 2449°9 2488°3 2488'1 8 9449°1 2489 2488'9| 9 | 94483 2448 2448: 2490 2490°6 9 | 2446°7 2447 2191 2491 2491°2| 10 244671 2446 2492 2492°0 5 24453 2493 7493"1 10 (94443 ae 2493-4 4 2444-0 | 2493°7 2494 {5 eat is 3443-9 2444 2443 2495 2495'6 11 2441-9 | 2496°3 2436 2495'8| 12 9441°7 2497 2496'8 6 2440°7 2440 2440°6 | 2498 | 2498-0| 13 | 12 2439°5 | 2439 2499°1 2499 2499°5 5 2438:0 2487°8 | 2500-0| 14 2437°3 | 2501 2501 2500°6 13 243771 2437 | 2502 | 25016 7 (2436-2 } 2502°7| 14 | 2435-0 94385 | 2503 2503'0 14 (24348 2434 | 2504-4 2505 | 2505-1] 16 124328 | 2433 2505°6 15 2432°4 2506'3 2506 2506°0 6 | 2482°0 2432 2506°4 8 2431°6 518 Prof. E. C. C. Baly on The agreement between calculated and observed values throughout appears thoroughly to justify the general con- ception here put forward. All the absorption lines calculated have not been measured by Hartley as separate lines, but it is evident from his paper that more exist than those which he definitely gives. Hartley records a great number of narrow absorption bands which are resolved into fine lines. on some of his negatives and not on others. The general conclusion may be drawn from the values of these bands given by Hartley that other fine lines exist beyond those that he specifically mentions. Grebe in his paper gives two series of measurements, one of which is more accurate than the other. Those wave-lengths given with five significant figures belong to the former series. Atiention has already been drawn to the very great intensity of certain bands in the short-wave infra-red spectrum of benzene, and it would be expected that the lines in the ultra-violet corresponding to those infra-red bands should be relatively stronger. The fact that the ultra-violet absorption-band group of benzene consists of a number of closely situated sub-groups is well known, and Hartley has given the wave-lengths of the heads of these sub-groups. It is very interesting to note that these heads correspond to the observed intense infra-red bands in each case. Messrs. A. Williams and F. G. Tryborn have very kindly measured the absorptive power of an alcoholic solution of benzene with the new Hilger spectrophotometer, and the measurements of the band heads are compared in the following table with those calculated from the observed infra-red bands. It must be remembered that a correction must be made for the effect of the solvent, this being about 18 units in the value of 1/r. The general agreement shown in Table I]. makes it very clear that the sub-groups of the bands within the whole ultra-violet band are due to the principal infra-red bands and the combination of their frequencies with that of the central line. Since the basis constants of benzene and of water are now known, it is possible to test whether the additive relation holds good in a compound of the two. For example, the test may be applied to phenol in order to see whether the basis constants of benzene and water are acting independently in this compound. The absorption of phenol has not been examined by Coblentz over the whole region from 1p to 13y, but three very strong bands were measured by him at 2°97p, 6°25, and 6°75. It is fairly obvious that the last of these Light Absorption and Fluorescence. 519, TaBLe II. Sub-groups in the Ultra-violet Band Group of Benzene. Wave-lengths of Sub-groups. | IR. Bands. y;,. ue —4050-+y,.|1/—18.| Acale. ae | d obs. ane. | | PU aS | | eee | 325m 30s | 3742 | 3724 | 2685 | 2685 | 2682 | 44 AN 3823 | 8805 | 2682 | 2630 | 2630 49 204 | 3846 py B28» | SBIR | scaees 2614 543 | 184 3866 | 3848 2599 2600 | 2600 | 9-78 102 3948 | 3930 | 2545 | 2540 | 2539 Centre 0 4050 | 4032 2480 2482 2480 11:8 8 | 4135 | 4117 | 2429-| 2429 | 2496:5 5°45 | he, ae 4231 | 4216 | 2372 | 2376 | 2376 4-4 | 227 4283 | 4265 | 2345 | 2344 325 | 308 | 4358 | 4340 | 2304 | 2300 2-18 | 460 | 4510 | 4492 | 2398 | 2930 is the benzene band due to the basis constants 3°7 and 4:0, as already discussed under benzene. Now, as was shown above, water exhibits two absorption bands at 6°25 and 64 due to the basis constants 2°5 and 6°6, and of these bands the latter is by far the stronger. On the other hand, the wave- number of 6°25, 160,is a whole multiple of each of the three basis constants 4, 2°5, and 6-6, for 160=40 x 4=64 x 2-5=24 x 6°6, Obviously, therefore, this band in the case of phenol will be enormously enhanced, since its frequency is a multiple of the basis constants of both water and benzene. It is thus clear that the three phenol bands, of which the one at 6°25, is the strongest, can be explained by the basis constants of water and benzene, for the band at 2°97 is no doubt the same as the water band at 3° Ow already dealt with. Of the three great infra-red bands of phenol, therefore, one is due to the basis constants of benzene alone, one due to the basis constants of water alone, while the third is due to those of water and benzene combined. From these facts the ultra-violet absorption lines of phenol can at once be calculated, sinee the central line of the ultra-violet band group must be a multiple of the frequency of the band at 30m or that at 6°25, more probably the latter since it is due to the basis constants of both water and benzene. Now in neutral solution, phenol shows two ultra-violet band groups and one fluorescence maximum. In alkaline solution it also 520 Prof. E. C. C. Baly on shows two ultra-violet band groups and one fluorescence maximum, all three of which are much nearer to the red than those in neutral solution. There exist differences between the wave-numbers of all these six bands which are all multiples of 160. Itfollows from this that the ultra- violet band groups must possess central wave-numbers which are multiples of 160. Now Purvis* has measured the wave-lengths of the absorption lines composing one of the ultra-violet band groups ot phenol, and they lie between the limits X=2812 and 2500 tenth-metres. The only even multiple of 160 which lies near the centre of this band is 3840, which = 160 x 24 f. The wave-numbers of the absorption lines therefore may be calcu- lated from 3840-tn x 2°5, 3840-+n x 3°7, and 3840+n”~x 4:0; and the values obtained when corrected for the refractive index of air show very good agreement with those measured by Purvis. Two of the basis constants of benzene were used and one of those of water, because phenol is undoubtedly an aromatic compound, that is to say its benzenoid properties are far more pronounced than those of a derivative of water. In Table III. are shown the results for the first few lines on both sides of the central line together with Purvis’s observations. Itis somewhat remarkable that the calculated wave-lengths of the absorption lines over the whole band group themselves together, and that Purvis has observed in most cases those iines which mark the heads of these groups. Owing to the fact that the infra-red spectrum of phenol has only partially been investigated, it is not possible to adduce any collateral evidence, as was done in the case of benzene, as regards the coicindence between the frequencies of the principal lines in the absorption band with those calculated from the infra-red bands. The number of lines calculated is so large that the agreement between these and the observed values might be criticised as being no better than the theory of probability would give. Itmust, however, be remembered that the additive principle is clearly estab- lished as far as the infra-red spectrum of phenol is concerned. Many other examples might be given in support of this principle, but it is only in a very few that the theory can - be tested upon the ultra-violet absorption band group, since * Trans. Chem. Soe. ciii. p. 1088 (19183). + The even multiple of 160 is taken because the frequency-difference between the two absorption bands of pbenol in neutral or alkaline solution is 2x 160. Light Absorption and Fluorescence. ec eos Red Side. 2611 2613 2618 r eale. 2603°4 2605°1 2605°9 2606'1 2606°7 2608°5 2608'8 2610°2 2610°9 2611°6 2611°8 2613°5 { 2613°6 2614-7 2615'3 2616°0 2617°0 2618°5 2618°7 2619°8 2620°5 2621°0 2622-2 2622°5 2623°6 26239 26252 2625°5 2626°1 2627°3 2028°0 2628°7 2629-0 2630°7 2631°3 2632°5 2633'6 26339 2684:3 2656'0 26363 °6386'°4 2037 °7 2639°C Ny - Hm Cob Or | | | Ns bo 10 ll 13 14 TasueE III. Ultra-violet Absorption Band of Phenol. Ku=37 K,=4-°0. | | | j | | Nese 0 bo io) 10 12 | Blue Side. r eale. 2603°4 2601-7 2600°9 2600°7 2600-0 2598°4 2598°0 2596'6 2595°9 2595°3 25950 2593-4 2593°3 25926 2591°4 2590-9 2589°9 2588'4 2588°2 2587°2 2586'5 tok lololo bd bot ‘ ~ 25711 2569-9 2568-7 X obs. 2603 2593 Or == 522 Prof. E. C. C. Baly on relatively few substances exhibit fine-line absorption in that region. In the case of toluene and aniline the absorption bands in the ultra-violet are resolved into fine lines, and in both cases the principle holds good; for the ultra-violet band system of toluene can be calculated from the basis constants. of benzene and an aliphatic hydrocarbon, while those of aniline can be calculated from the constants of ammonia and benzene. The infra-red spectra of a number of saturated aliphatic hydrocarbons have been measured by Coblentz, and they are all strikingly similar. The compound which was examined in a state of most probable purity was normal hexane. ‘The infra-red absorption of this substance is characterized by three very strong bands at X\=13'8y, 6°86 yu, and 3°43. The wave-numbers of these are very nearly multiples of 4, and the most probable-values are 1/A=72, 144, and 288, and the corresponding wave-lengths 13°88, 6°94, and 3°47u. Since these three bands are so intense in relation to all the others. of the aliphatic hydrocarbons, it may be concluded that a second basis constant of 7:2 is present, of which the above wave-numbers are multiples as well as of 4. The existence of these two basis constants explains all the strong bands of the saturated aliphatic hydrocarbons; and it is very interesting to note that one of them (4) is the same as in the case of benzene, which suggests that itis characteristic of the carbon chain. That the basis constants act independently and that the infra-red spectrum of a compound is due to the basis constants of its radicles, can readily be seen from the following examples. Myricyl alcohol shows very strong bands at N=2°97, 3°43, 6°86, and 13°88. Of these there is no doubt that the last three are due to the basis constants of the hydrocarbon chain, while the first is due to those of water (30m). Aguin, triethylamine shows the hydrocarbon bands at 3°43 and 6°86 and also the ammonia bands at 6:1 and 9°3 py. The infra-red spectrum of aniline shows strong bands at 2°97 and 6*ly, which are clearly due to the basis constants. of ammonia. It also shows the characteristic band of benzene at 3°25; but the 6°75 benzene band is now recorded at 6°68 w, a somewhat remarkable shift. Owing to the complexity of the infra-red spectrum of ammonia, it becomes very difficult directly to determine the fundamental frequency of aniline, of which the central frequencies of the ultra-violet bands are multiples. On the Light Absorption and Fluorescence. 523 other hand, aniline in solution shows two absorption-band groups and at very small concentration*™ (N/84,000) the wave-lengths of the centres of these bands are X\=2859-0 -and 7=2349°5. The values of 1/A are 3496°7 and 4355 respectively. From these two measurements it is possible- to calculate the most probable values of the fundamental infra-red frequency, v,, of aniline, and of the factor for each band, for we have # Xv,=3496°7 (1), yx v,=4355 (2), and (y—w)v,=758°3 (3), where a and y are integers and y—zissmall. It would seem obvious that the only possible values for # and y are 23 and 28 respectively, for the value of v, is then found to be 152°03, 151°97, and 151°65 from (1),. (2), and (3) respectively. Purvis+ has measured the wave-lengths of the com- ponent absorption lines of the less refrangible ultra-violet band group of aniline, and has recorded a strong absorption line at X=2859°5, for which 1/A=3496°1. Itis obvious that the accuracy of measurement of the centre of an absorption- band group is far greater when the centre appears as a single narrow line than in the case of a broad band such as appears in the aniline solution. The value 1/A=3496'1 may be taken therefore as a more accurate measurement of the central frequency than that obtained from the solution, namely 3496°7. The true value of v, or the fundamental infra- red frequency for aniline is therefore 3496°1+23=152°0. The wave-length corresponding to this is 6°58, and it would therefore seem that this gives an explanation of the infra-red band of aniline at 6°68u referred to above. The benzene band at 6°75 and the fundamental aniline band would seem to be merged into one band with a mean calculated wave-length of 6°665,p. As has already been pointed out, the infra-red absorption spectrum of aniline shows bands due both to ammonia and to benzene; and it follows that the basis constants of both these substances are active in aniline. Now the principal basis constant of benzene is 4, and 88x 4=152; and it would seem therefore that 3°8 must be one of the basis constants of ammonia. If this be so, then many of the infra-red absorption bands of ammonia will occur at frequencies which are mul- tiples of 3°8. This is shown to be the case in Table IV., for eleven out of the sixteen ammonia bands between 3 and 14 are thus accounted for. * The values for a very dilute solution are the same as for the vapour, a fact that will be dealt with in a further paper. + Trans. Chem. Soe. xcvii. p. 1546 (1910). 524 Prof, H. C. C. Baly on TaBueE LV. Factors. 1/A. X eale. X obs. (Coblentz). 19x38 72°2 13°85 13°7 22 3°8 83°6 11:96 11-98 23 x 3°8 87°4 11°44 11-48 25 X3°8 95:0 10°53 10°4 27 X3°8 102°6 9°75 9°9 28X3°8 106°4 9°40 93 30X35 1140 8717 8-9 40x38 152°C 6°58 6°51 43X38 163°4 6°12 61 45x38 171-0 5°85 58 88 X3'8 3344 2°99 2:98 | The agreement shown in this table would certainly justify the conclusion that 3°8 is the principal basis constant of ammonia. We are now able to calculate the wave-lengths of the component lines of the ultra-violet absorption band group of aniline, for the central frequency is known, as are the basis constants; for, as in the case of phenol, the two benzene constants may be used, 3:7 and 4:0, together with that now found for ammonia, namely 3°8. I have calculated the whole of the wave-lengths of the absorption lines, and a portion is givenin Table V. The observed values are published by Purvis* and by Kochf. In his paper Purvis gives three series of measurements made at 14°, 30°, and 45°; and the values for the same line differ in some cases by as much as one Angstrém. This cannot be due to any temperature effect, as the differences are not in the same direction with a given temperature change. The variable values must either be due to expe- rimental error or to the fact that there exist more lines than Purvis actually records. While disposed to favour the latter explanation, I have bracketed together the values which would seem from Purvis’s paper to refer to the same line. The agreement between observed and calculated values is very good; and it is also to be remarked that with very few exceptions each line on one side of the centre has its counterpart on the other side. + Locker: T Zewt. wiss. Phot. ix. p. 401 (1910). | 2900°6 | Light Absorption and Fluorescence. 525 Taste V3 Ultra-violet Band Group of Aniline. Kore C= a, Ko 4-0. Red Side. | Blue Side. Koch. Purvis. | A cakes, |. Wei | Acale. | Purvis. Koch. 2860 2859°5 2859°5| 0 0 0 |2859:5 | 2859-5 2860 : 28626) 1 2356'5 2 2861 | 2862 Sea ‘ Sea eel 2863 | 2864 2862°9 H 2563 2856 2856 2865°6| 2 2853°) ) | [ 2854-5 2866 | 2866 2865'8 2 2853°3 2852 28661 2 | 3853-0 J | | 28535 28687! 3 12850°311 oora.r 2869 | 2868 ae 4 Soo 28515 | 2850 2870 28694 3 | 2849'8 { 2849 2871 Vaan t 2871'7| 4 8475 f 28485 f a8 2871 28720. 4 28471 | 2847 2872 2872°7 | 4 |28465 | 2846 2874 ; cpa dee Y ana 28747| 5 eget {2644 5 \ ak 2875 fee | (eeu 5 2844-13 | Loga4 2876 [| | 2876 2876'0 5 | 2843°3 avr t 2877'8| 6 meet {28415 2878-5 1 | [ 28783 6 2841-0 J | bogai 2879 28793 6 | 2839-9 2840 28809; 7 2838°5 2881°5 7 2837°9 | 2838 2882'6 7 |2836'8 2837 28340} 8 28355 | 2836 2885 2884" 8 28349 | 9834-5 2886'0 8 | 2833'6 2833 2887:0| 9 | 2832 6 { 2832'5 2887°8 9 2831's S| | 2832 \ a32 2889 2889'3 9 | 2880°4 ene \ oB5p 2390'1| 10 2829°6 J’ | 2830 2891 }| Seas a. 2892 | 2892-6 10 | 2827-2 oso, | J 2893 28933) 11 98266 | 28265 | 2826 " 2894 2894-2 11 2825°7 2896 28960 11 | 2824-0 2897 2396 3| 12 2823°7 2823 2897°5 [| 1 2897-4 12 2822'1 2900 2899°3 | 12 | 2820°8 \ enor 28 99 | | 9899°4| 13 pare | et | 2901 13 | 2819°7 526 Mr. G. H. Livens on the Electron Theory Summary and Conclusions. ~1. On the Bjerrum principle as extended by Eucken that ‘the frequencies of the infra-red bands exhibited by a compound are given by the expression ee the basis constants of water, ammonia, benzene, and the _aliphatic hydrocarbons have been calculated. 2. The abnormal intensity of certain bands in the infra-red ‘spectrum of these substances has been shown to be due to -convergence-frequencies of two or more series nh nh = V,» = . Varela: (he Dera. 3. The Bjerrum principle of combination of the central frequency of an absorption-band system with those of the infra-red bands has been applied to the ultra-violet band group of benzene and the wave-lengths of the whole of the component absorption lines calculated. The values agree very closely with those observed. In addition to this, the conception gives an explanation of the division of the main absorption- band group into sub-groups. 4. The infra-red spectrum of a compound is shown to be an additive function of the spectra of its constituent radicles. ‘This is proved in the case of phenol, aniline, myricyl alcohol, and triethylamine. 5. The wave-lengths of the absorption lines composing the ultra-violet band groups of phenol and aniline have been calculated from the infra-red spectra of benzene and water, benzene and ammonia respectively, by making use of the basis constants of each. The agreement between calculated and observed wave-lengths is again very good, and well within the limits of experimental error. The University, Liverpool. V LY. The Electron Theory of the Hall Effect and Allied Phenomena. By G. H. Livens *. 1. Introduction. ie 1879 Hall found that the lines of flow of an electric ! current through a metallic conductor are distorted when the conductor is placed in a magnetic field, the distortion being of the character of that produced by a slight additional * Communicated by the Author. of the Hall Lffect and Allied Phenomena. 527 electromotive force directed at right angles to the current and to the magnetic force. Thus, if a metal bar, along which a current is flowing, be placed in a magnetic field perpendicular to the direction of the current, the current will tend to be deflected from its course and to move towards the one or the other of the edges of the bar. ‘This cross flux of electricity will in fact initially exist, but it cannot be permanent since it will give rise to a slight accumulation of charge on the onter edges of the bar, the additional electric field thus set up tending to oppose the transverse current. A final steady state of equilibrium will be attained in which there is no cross flow of electricity, the additional electric field being just sufficient to balance the transverse electro- motive force arising from the magnetic field. The additional electric field thus created is capable of precise determination by connecting the opposite edges of the bar to the terminals of an electrometer. It is in fact this potential ditterence which was actually measured by Hall and subsequent workers in this subject *. In 1886 Nernst and von Ettingshausen found that a similar potential difference can be measured when the electric current is replaced by a current of heat in the same direction along the bar, and then in 1887 von Ettingshausen found that in the arrangement adopted by Hall a temperature difference analogous to the potential difference in Hall’s experiments can be obtained. Finally, and also in the year 1887, Righi and Ledue found a similar temperature difference if the electric current is replaced as in Nernst’s experiments by a current of heat. A simple explanation of these phenomena was soon forth- coming as soon as the electron theory of metallic conduction began to be developed, and was, in fact, provided by Riecke f, Drude t, and J. J. Thomson§. An electric conduction current consists essentially in a flux of negative electrons which possess, as the result of the action of the electric driving field, a finite average velocity in the direction opposite to that of the electric force If we suppose this average velocity of drift of the electrons to be represented by the vector v, then the action of a magnetic field of force of intensity H would be such that each electron will on the average be acted on by a force specified in magnitude and * See K. Baedeker, ‘ Die elektrische Erscheinungen in metallischen Leitern’ (Braunschweig, 1911), Ch. IV. + Wied. Ann. Ixvi. (1898). t Ann. der Physik, i. p. 566; iii. p. 369 (1900), § Rapp. Congr. Physique, iii. p. 143 (Paris, 1900), 528 Mr. G. H. Livens on the Electron Theory direction by the vector product of v by H, reduced, however, by the usual radiation velocity constant c, if the Hertz- Heaviside system of units is adopted. The statistical effect of these forces on all the electrons comprising the current is exactly the same as that of a general impressed electromotive force in the same direction. Again, when the ends of a metal bar are kept at different temperatures so that there is no flow of heat along the bar, there will on the average be no drift of the electrons from one end of the bar to the other, but the electrons which are travelling from the hot end to the cold end possess a greater kinetic energy and greater velocity on the average than those which are travelling in the opposite direction. And since the force on an electron travelling in a magnetic field tending to deflect it at right angles to its direction of motion is proportional to its velocity, the force tending to deflect the electrons travelling from the hot end to the cold end will be greater than that tending to deflect (in the opposite direction) those travelling from the cold end to the hot end. There will thus, on the whole, be a differential drift of the electrons in the direction in which the electrons moving from the hot end to the cold are deflected by the magnetic field. If this explanation or its more detailed equivalent to be given in the following pages were a complete representation of the action of the magnetic field on the current, the various effects would be of the same sign in all metals and would always be proportional to the magnetic force, whereas neither of these statements is rigorously true in all cases. In all ferromagnetic metals it appears, for instance, that the effects are proportional to the magnetization produced in the metal rather than to the actual magnetic force. This and the other discrepancies which have been found will be discussed in their proper places in a future paragraph with the expla- nations which have been suggested by them. The main object of the present paper is to provide a re- discussion and generalization of the detailed analysis of the explanation offered by Thomson and Drude and roughly sketched above. The method to be followed is essentially the same as that introduced by Lorentz but with the genera- lization, which seems to be necessitated by the facts, that the electrons and atoms act on one another like point centres of force, the latter being of such comparatively large mass that their motion and energy may be neglected. A detailed discussion of the present theory for the case in which the electrons and atoms are hard elastic spheres has been given of the Hall Effect and Allied Phenomena. 529 by R. Gans *, and some of the results for the more general case have been given by Bohr fF in his very general treatment of the subject of the general electronic motions in a metal. The method adopted by Bohr is essentially different from that adopted in the following pages, depending on the establish- ment of equations to determine the changes in the components of the aggregate momentum of the electronic motions, and is to a certain extent more general. It is possible, however, to generalise the present mode of treatment to the extent assumed by Bohr, but as the results are really only workable in the particular type of problem here examined, it was decided to confine the analysis to this type. 2. General basis of the Theory. The general basis of the present theory is the one now usually adopted to explain the electrical and thermal pro- perties of metals ; its physical bearing is elaborately discussed in Dr. Bohr’s dissertation, whilst certain aspects of the analytical side have been discussed in several recent com- munications of the present author. In this theory it is usually assumed that the whole of the electrical and thermal properties of the metals arise entirely in the average motion impressed by the external circumstances on the swarm of electrons which are otherwise moving about quite irregularly and freely in the space between the atoms or atomic com- plexes. In the absence of actions from any external agency the electrons are presumed to be moving about in such a manner that the distribution of velocities among them at any instant is precisely that specified by Maxwell’s law, so that if N is the total number of free electrons per unit volume the number in the same volume with their velocity com- ponents between (&, 7, &) and (&E+d£, n+dn, €+ df) is given as usual] by the formula g —qu? SN=Ny /% e dednade. WEP +P +e and q is a constant connected with the mean value w,,2 of w? for all the electrons by the relation ee a Bly * Ann. d. Physik, xx. (1906). 7 ‘Studier over Metallernes Elektrontheori’ (Dissertation; Copen- hagen, 1911). I am greatly indebted to Dr. Bohr for the loan of a translation of this paper. It contains a wonderfully lucid exposition of the underlying assumptions and restrictions involved in the general theory. Phil. Mag. S. 6. Vol. 30. No. 178. Oct. 1915. 2M wherein | 530 Mr. G. H. Livens on the Electron Theory When, however, external electric and magnetic fields are imposed throughout the interior of the metal, all this alters because then the velocity of each electron while on its free path is modified by the forces in the external fields. The main problem is now to determine the new law of distribu- tion of the velocities which is to replace the above Maxwellian law when the effect of the fields is taken into account. Now Lorentz * proves that if in the new state of the motion TE, N, 6 2, Ys &s t) dé dn dé denotes the number of electrons per unit of volume round the point (#, y, z) with their velocity components between (€, n, €) and (€+dé, n+dn, €+d€) at the time ¢, then the function 7, the fundamental function of the theory, must satisfy a differential equation of the form ee eer es +697 49 oF or ae BE antag oy '° deem wherein (X, Y, Z) denotes the acceleration components impressed on the typical electron of the group by the external fields and (b—a) d& dn dé dt denotes the increase in the specified group of molecules during the time dé brought about by the collisions which occur during this time. Now as the present writer has recently emphasized, and as Prof. Richardson f had previously proved in special case, when the electrons are repelled from an atom in such a way that its potential energy relative to that atom when at a distance 7 from it is we may put where ] pee se — =47np7u ¢| cos? Cade, Tin P 6 being defined as a function of « by the equation oe dx =| /1-#-(2) 0 a * Vide ‘The Theory of Electrons,’ p. 266. + See Phil. Mag. April 1912. Richardson works out the case for steady electric fields only by the method of Lorentz. of the Hall Effect and Allied Phenomena. 531 where 2 is the only positive root of the equation l—2?— (2) =0. ; a Thus if we now write ¢ A=na/%, See we find that the solution for the fundamental function /7 is of the form ee) =e ic-\ erm “4 xXidty, 0 m tj=t-t where x ofo vy Ofo 4 7 Ofo gOfo, Of , -Ofo , Ofo X=R5e ean * Be ie hike Feat ag’ and the suffix 1 denotes that the functions affected are to be taken for the time t=¢,. ‘To effect this interpretation it is first necessary to integrate the six differential relations ese ies Oy egy 2 es ey) By m1 O - expressed generally as function of the time ¢,, introducing as the necessary constants of integration the values of the variables at the time ¢;=?¢. Substituting these values in y we obtain its explicit expression as a function of ¢). On substituting the value for 7, we find 1/3, DA, 2A, BA are Pas Sere = yale ee gy OF.» OF 4 0ag @ GS dy OY aa 02 i: Nic The formula for f is thus obtained directly under the most general circumstances, and itis in a form which is directly suitable for application in any special case. If we are pre- pared to put a very general interpretation on Tp, the generality of the formula will transcend that for the particular cases to be investigated. We now proceed to the reduction of this formula in the particular case when the accelerations (X, Y, Z) are produced by the combined action of an electric and magnetic field ; both fields will for the gre purposes be assumed to be 2M 2 u 532 Mr. G. H. Livens on the Electron Theory stationary, the more general case being reserved for separate treatment in connexion with a discussion of the magneto- optical properties of metals which is in course of preparation. 3. The instantaneous velocity distribution under the combined action of an electric and magnetic field. For the sake of simplicity we shall first assume that the magnetic force is directed along the z-axis of coordinates in a definitely chosen rectangular system of reference, and is of uniform intensity H throughout the metal. The electric field is assumed also to be of uniform intensity E, but for the sake of generality we need not specify its direction beyond stating that its components in the three principal directions are (H,, H,, H,). The motion of the typical electron while traversing one of its free paths is thus given by the following equations: dg, Re =eH. , d eH Mg pa ac d eH mor =A te = wie where m is the mass of an electron and e the charge on it. The last two of these equations may be written in the form 2 d?n, ie ial evi, dt? : m ants = —y?f,— evil. | ae y m? where we have used eH y= —_, me and the solutions are easily obtained in the form : e 1=7 Cos vty —§ sin vty + i ' K, sin vt, + HE (1—cos vt) i, : e ; €=C cos vty +7 sin vip + i ‘ Bid cos vé,) —H, sin vty : ‘ n, € being the respective values of 7, and & at the time t,=0, which in reality corresponds to the instant t. _ The accelerations of (£, , €) are then directly obtained by differentiation in a form suitable for substitution in the function y. It must, however, be noticed that the part of of the Hall Effect and Allied Phenomena. 533 this acceleration depending on the magnetic force may be omitted as it contributes nothing to the scalar product (EX+nY+£Z). Moreover, the parts of (&, n; ,) depending on the electric force may also be omitted, as their retention would merely introduce the properly negligible squares and products of the field and condition gradients. We may thus in the general form for x use Ge; n,=7 cos vt; —€ sin vt}, 6;=C cos vty +7 sin vty, and Ba Go Bee a ym Oe m m m~ Inserting these values and performing the integrations with respect to ¢; and 7 and noticing that Need T é “a, OT VT m sin vTe —_ = = Gere The 1+ y°7,,7 0 fo ¢) T =~ 9 9 r Tm dtr VT mn” (l1—cosvt)e ~ — ise y NL + V Tm we find that ae 2qeH,E _ EQA 2 OY papel ttre 4 ASS THES f Tih 2ge Lf) QA A oa 14,7," “46 (8, + £8.) cay a("5, 78h ) ee eee VTm> 2qeHK, 10A Od) aan al m AOz ” Oz via eae as) | This expresses the most general form of the required law for the distribution of velocities among the electrons in the metal. It might equally well have been deduced by a method exactly analogous to that employed by Gans in the more special case, or by the alternative method suggested by the 534 Mr. G. H. Livens on the Electron Theory present writer in another connexion in a previous communi- cation. The present method of deduction, however, appears to bring out more clearly the essential character of the assumption on which the results are based. If we adopt the notations of vector analysis and use u, H, H to denote the vectors of electronic velocity, electric force, and magnetic force respectively, this expression may be more concisely written in the form ™ 2 1 fa ho [+ yee Ve OD gOVAt HC} Beta l 2 On Bwome YT, { 2008 Reged ae We eae me(l+v?tm) | 4 wherein V7 denotes the usual Hamiltonian vector operator whose components are (0 fo) 0 4. The general expressions for the currents of electricity and heat. The various constituents of the complete flux of electricity and heat are now easily obtained, being calculated along exactly the same lines as developed by Lorentz for the simpler example of the same principles and by Gans in his analysis of the present problem. It will not, therefore, be necessary for me to enter into a detailed examination of each calculation. According to Lorentz, if fH Ae w+ o(&, m, 6), then the current of electricity is a vector whose components are +o (J., Jy, 1)=e I ( 1, C)o(&, > O)dE dn dé, whilst the current of heat is a vector with components ny (We, Wy, Wd (Vl (9, HCE n, Ode dn a { 22% (u, [HE])—% ( [HV]A) +H] @) } | — of the Hall Effect and Allied Phenomena. 535 and each integral in the two cases is directly evaluated after substitution of the appropriate value of ¢@ by the usual spherical polar transformation which reduces them at once to single integrals with respect to wu. In this way we find that the z-components of the electric and thermal currents are exactly the same as would be obtained in the absence ot the magnetic field, whilst the other two components of the electric flux are identical with the corresponding components of the flux vector ae ky 4 noe a a i) (“= sa & grad A+12 prad 0) Taare a 3 : m ere me 1+v?7,,” The same components of the heat flux are determined similarly by the vector QarmA ” (2geH wh on hee ae W= A = qAgrad A+w grad q ie 1 3 1+ V°t ee: (= [HE]— [HV]JA+w[HV]q) u®=+,,7e-9 | rth Et at Esch 1l+v"r,7 These expressions are quite general; by using the par- ticular value of Tt, given above we can deduce the results for the particular cases to be investigated. It will not, however, be necessary for us to retain complete generality as the limitations of the theory are probably not such as to warrant any such rigour in the analysis. We shall therefore neglect all powers of H beyond the second. Thus using A 1 Army | cos? Gada 0 so that Ag Tm lmu® and introducing the ordinary expression for the conductivity _in these cases, we find that the y and z components of the electric flux are ro) Q0¢ i ‘ 4 on —qud Al (“4 (HE) —4 [HV]A+e[HV])“7 "I. 536 Mr. G. H. Livens on the Electron Theory determined by the corresponding components of the vector J where 6 2qeJ rhe r(1 ) 2geH at mao Bs 2 | m Ne dA) ge r(2+=) eg r(2+°) 9 aad V bin” S + ot a a 5 “grad q), qs r(2+ =) eee ee r(5 a -) 2qeH + (“2 [Hb] - ;[HVIA) ME * Z LTt ey C+), + 7 [THY ]g ., La a J whilst the vector W is determined by | 6 Ag? W ts iy yl? r(2+) 2qgeH 1 s- =|2+-—-—-—{— we! e — grad A) mo 8 eM. 2\1\ m A ge V(2+- | (2+7) 6 » 1(3+ - ) 6 972 at (2+=)(3+° vim aaa (grad 7) 5) ela Wi I rec élmg? © & ) 2ge 1 ama Mins. i LE) a lHv]A Gt r(2+7)! These expressions are quite general to the order of approximation adopted, and by applying them to the four LEV ]¢ of the Hall Effect and Allied Phenomena. 537 special cases examined by Hall, Nernst, von Httingshausen, Righi, and Leduc we obtain at once a detailed account of their respective results. We now proceed at once to this examination. 5. The Hall Effect. In Hall’s experiments the induced potential gradient in a direction perpendicular to the magnetic force and current flux which is necessary to maintain the current in its un- disturbed path is determined by a statical measurement. lf the magnetic field is in the direction of the z-axis and the current is flowing along the y-axis, then the conditions parallel to the z-axis are to be such that there is no electric flux in that direction. There are of course no temperature effects to be reckoned with in this case, and the slight accumulation of charge necessary to ensure the steady conditions is entirely at the surface of the metal, so that in the interior grad A=grad.qg=0. The condition that there is no current along the z-axis is then oie irs Eg 3) lng? 8 ite ML. Cains... r(2+=) AG If we use J, to denote the component of the main current in the direction of the y-axis, then Jy=cl,, where o is as above. We thus have a LD ] ~3vn0(5 +=) fe (Nec) (I(2+ = )) This is exactly the form of the law which is usually adopted to express the magnitude of this effect: the constant in this equation which is defined to be the constant of the Hall effect thus turns out to be HE y ay ee a val (5 + -) (T (2+ *)). en R=— eS — peeermmie oot — oe Se - Sec Si SS oe a: aes 938 Mr. G. H. Livens on the Electron Theory This is the result obtained by Bohr and agrees with that obtained by Gans in the particular case he examines (which corresponds to s=oo in the present analysis). There is a second order effect which is directly associated with the Hall effect and which can easily be discussed at the present stage. In the absence of the magnetic field the current parallel to the y-axis has a density oHy, if E, is the driving force. When the magnetic field is on and the arrangement is that adopted by Hall, the current in the same direction has a density r(i+.2). ot\ veaigis ag oBy|1 m | — Sen" _* __~_“LHE, ; —- “3.5 eh r(24+ 2) ge" me P(242) s s or using the form for E; obtained above, it is 6 3). 2\Vie ve ) (FG ha) el; oHy Fae 5 9 Say 9 My: ripe ’ r(2+ = ) (T(2+)) mergs Ss a result which is usually expressed by saying that the mag- netic field decreases the conductivity of the metal for currents flowing in the direction specified and in the ratio 6 3 a Bos r(i+ 4) r( i s I ele * eae Pe CHF oa ( a ie oe a Pe Ur (2+ :) r(2+ -) mcg For s=o this reduces to which is the result obtained by Gans in this particular case. In most of the cases investigated it appears that the con- ductivity for transverse currents does decrease when the magnetic field is put on, although for the ferromagnetic metals in particular the reverse takes place. * Notice that Ac=0. for s=4, which corresponds to the case in which the molecules of the metals act on the electrons like little magnetic doublets (see Bohr, J. c.). of the Hall Effect and Allied Phenomena. 539 6. The Nernst and von Ettingshausen Effect. In the experiments of Nernst and von Httingshausen, the electric current in Hall’s experiments was replaced by a current of heat, which we shall also assume to be directed along the y-axis. Im this case, as soon as the steady state has been attained, there will be no flow of electricity along the direction of the z-axis, which is perpendicular to the direction of the magnetic force and lines of flow, the tendency to flow in this direction being counterbalanced by a potential gradient established by an initial redistribution of charge. The thermal conditions in the metal are uniform in every direction at right angles to the direction of flow of the heat, conditions which imply that The condition that there is no flux of electricity along the z-axis then becomes Ph, Hel,g?~ * ; r(5 + Ms +0) (ev _ 1 s) nm rG Ri 4 1 09g me r(24 4) A Oy r(2+ 2) Yy oy There is also to a first approximation no flux of electricity along the y-axis, so that We have therefore pa {Tet )TOrs) | JP s) P(e +3) | wisg)t ae ORE) TCR) Oe i 540 Mr. G. H. Livens on the Electron Theory Or since, if 6 is the absolute temperature 2 amily: where R is the usual absolute constant, we have _1dgq_ 2k d0 G dy m dy’ so that oe a 5) r(5 7 3) Rag? Rng? * 7,40. 2 & (2+ = S) NLC dy The coefficient of this result thus turns out to be ie r(5+-) BG, 1 s} Ring? s r (2 mn *) me The importance of this result lies in the fact that it has a different sign according as s is less or greater than 4. The first example of this type of result was given by Bohr and will be mentioned in the next paragraph, where its import will be discussed. There is also a second order effect connected with this phenomenon analogous to the effect of the magnetic field on the electrical resistance in Hall’s experiments. In fact, in the absence of the magnetic field the current of heat parallel to the y-axis has a density dé Vy? where y is the coefficient of the thermal conductivity. When the magnetic field is on and the arrangement is that adopted above, the thermal current in the same direction has a density 2 6 (1+ =) P(2+ ;) ly? \d0 ee — : — —— of the Hall Efect and Allied Phenomena. 541 or substituting the value of E, found above, 3+3)\ ii ; y?]? { (eG ts) : JG + ae 1) 9 is iho < Ud We q a a -( Dracah |B The coefficient of thermal conductivity is thus altered in the ratio ga go Ay py?) 2 r(25 7) ae” ie (1+) 5) gs r(3+ 4 rer )(s +2) (PGs 242 r (2+ =) This effect always has a positive sign whatever the value of s. It shows that the magnetic field increases the thermal con- ductivity for transverse currents. There appears to be very little evidence of an effect of this kind; the only metal for which the effect is known with certainty is bismuth, and here the conductivity is decreased instead of increased as the theory predicts: but see below, § 9. 7. The von Ettingshausen Effect. In his second series of experiments von HKttingshausen measures, with the same arrangement as in Hall’s experi- ments, the transverse difference of temperature which it is necessary to create to balance the transverse flux of heat which exists in the original experiments. There is in these experiments no flux of heat in the direction of the z-axis if the main current and magnetic force are disposed as in the previous paragraphs. A definite temperature gradient is, however, established in this direction. The condition for no 542 Mr. G. H. Livens on the Electron Theory current of heat in the direction stated is to a first approxi- mation 2geBs _ ae A (34 ye m gq Oz ee 18 r(5+5) Stel 3 292lmq? * as ; Mot = _. HE,=0, r(3+) the thermal conditions in the direction of x and y axes being uniform. It follows, therefore, by using again Siam VORe: {2.3.2 em De N dé 6 dz ot a 148 _ 2ge no) a ao : me HE [ B+ (3+ ) mc y|- Tr These is also under;the usual conditions of the experiment no flux of electricity along the direction of the z-axis, so that a 1_2 P (5 ‘i 3) élmg” F r(3 i =) Me s 1 Wd Meaty @ dN) 1dé@ =16+2)+" 5 Bae It follows therefore that that 2qe ] aa | Be+ . HE, 2 do _ ote pl feval Gt dz Mc "le e2}r(e 42) or, if H, is the potential difference observed with the arrangement for Hall’s experiment, of the Hall Effect and Allied Phenomena. 543 The importance of this result is again the difference of sign which may occur with the different values of s. This was, I believe, first pointed out by Dr. Bohr, whose result agrees with that here determined. Now it appears as a matter of actual experience that it is just this effect, and the one discussed in the previous paragraph (§ 6), whose signs do not agree with those experimentally determined unless s<4, a fact which provides important evidence for the form of the law of interaction between the electrons and molecules, as Bohr points out. Even for the cases in which all the signs are reversed it still seems necessary to assume that s<4 in order to obtain consistency in the relative signs of the various phenomena. This fact is of course in full accordance with the other branches of this theory where similar requirements are necessitated by the facts of experience. It seems very necessary to emphasize this important point, as it appears to have been overlooked in subsequent discussions bearing on the experimental work in this branch of the subject *. 8. The Right Leduc Effect. In these experiments the transverse force tending to deflect the current of heat in Nernst’s experiments is balanced by a transverse temperature gradient. In such a case there is no transverse an of heat, so that tah 3 (002): A 02 ag -Fy [T (5+ r+ 3) en, ) r(5+-) re) mT r(242) nes and there is also no flow of electricity in the same, direction, so that Pe x oe +( 2\1 dq 3.4 Gi | T(3* 3) 2, a ie a a) ] 2 r(2+°) * See for example J. Keenigsberger u. C. Gottstein, Physikalische Zeitschrift, xiv. (1913) p. 232, where an excellent summary of certain tentative suggestions Be to this theory is given. ra kee gegen 1 O94 m A oy r(2+ n(242)! 27] d44 Mr. G. H. Livens on the Electron Theory There is also to a first approximation no flow of electricity along the y-axis, so that 2qeH, 10A lio a tata TAD alg? some DR SW) KS m Aor (2 eae It follows therefore that 2geH, ,1 OA Me fe) ge en Og m Ade’ (3+5) q Oz Dhwe: & 1 2 x r(5+3) ? i \elng? *H 99 r(3 +) s 2) me g Oy’ Ss whilst age.) l OA fare hoe m A Oz +(2+ aE 02 ne) Voie "G+ a 1) lag? +H Bg BRR: me. -¢ OY r(2+= “) Whence therefore r Sd: 1 Ae Og WStaeet ee aa) aa "H dg g Oz 8s(s + 1) r(2 +=) me @ dy’ or introducing the temperature r(?.4 12 a0. Ts? F856 4 a 7) elm” * WH? dee 8s(st1) — me) | ae 2.) =) which is the usual form in which the results of these experi- ments are interpreted. The sign, which cannot change with s, is in general agreement with that usually determined, at least relative to the other coefficients. 9. Discussion of the results obtained. Hyen if full allowance be made for the possibilities of the present theory, it appears in the simple form quite incapable of being verified even qualitatively by the experiments. It is, however, possible by the introduction of certain plausible of the Hall Effect and Allied Phenomena. 545 hypotheses to render an absolutely full account of the whole set of the phenomena“. According to our theory each of the effects should have the same sign (apart from variations of s trom one metal to another) in all metals, viz. those which they have in bismuth or copper. It appears, however, and not infrequently, that the opposite sign is obtained in many cases and that there is no definite regularity in the occurrence of either sign. It is, however, of importance to notice in this connexion that the signs of all the effects are found to be reversed if one of them is, or in other words, the etfects always have the same relative direction. This observation is important as implying that the cause of the reversal is common to them all. Now, this occurrence of tle opposite sign in all the phenomena to that predicted by the theory is capable of a very simple explanation, which although perhaps of an apparently tentative and uncertain character, is nevertheless supported by several independent classes of observation. It is to be remembered that the effective magnetic field for the pheno- menon is not the simple continuation of the magnetic field examined from outside the metal but is the field at a point right inside the metal where the above “internal” field will in general be augmented by the addition of a purely local field due to the polarization in the immediately surrounding molecules or molecular groups. It is thus assumed to be probable that in the regions accessible to the freely moving electrons this local part, which will on the average be pro- portional to the intensity of the magnetization, may com- pletely reverse the sign of the total effective field. If this explanation were a valid one we should, of course, expect to find that in the strongly magnetic materials the local part of the field would greatly preponderate over the external field, and might in fact be taken simply as the effective field itself. This is, as a matter of fact, exactly what is found to be the case ; for it is found that if magnetization is substi- tuted for magnetic force in the expression of the effects in the ferromagnetic metals, then the coefficients at once assume normal values, and remain reasonably constant as the field is varied. The assumption of this local field would also provide an effective explanation of the behaviour of the phenomena in metals of more complex structure; such as bismuth. In fact, the local field is essentially constitutional in character and would thus probably be different in different directions. * For detailed information respecting the experimental work, see Baedeker, Die elektrische Erscheinungen in metallischen Leitern, Ch. iv. Phil. Mag. 8. 6. Vol. 30. No. 178. Oct.1915. 2N 546 Mr. G. H. Livens on the Electron Theory Of course, in many of the very simple non-ferromagnetic metals the local field might be extremely small and the theory in the above form would then be very exact. There is, for instance, every reason to believe that this is the case for such metals as copper, silver, and gold. If this exactness is to be expected in any case, then an exact knowledge of the constants of the various effects would, of course, provide valuable information for a determination of the electron constants of the metal; not only the constant s of the force exerted’ by the molecules on the electrons, but also of the number N of the electrons per unit volume and the quantity lL, which determines the strength of the molecular fields of force. It hardly appears, however, that the subject is in a position, at present, for this point to be pushed except, perhaps, in a rough descriptive sort of way. A striking confirmation not only of the general theory but also of the hypothetical amplification just suggested, is provided in the results of an examination into the effect of temperature on the various coefficients, more particularly that of the Hall effect. In the simpler metals like copper and silver there is practically no variation of the coefficient of the Hall effect with the temperature ; a fact which proves almost conclusively not only that the law of force for the action of a molecule on an electron is of the simple type assumed above, but also that it does not vary with the temperature. The slight variation actually observed is probably due to the variation of N, the number of free electrons per unit volume. Of course, in the ferromagnetic metals or in other cases where the local field is at all comparable with the applied field, we should expect considerable variations with the temperature of the coefficients as usually defined relative to the magnetic force, and this again is precisely what is observed. The behaviour of the ferromagnetic metals in this respect is interesting ; in these cases as the temperature is increased the Hall effect coefficient increases, exactly parallel with the magnetic permeability, until the critical temperature is reached, when it decreases very rapidly to a value more akin to that found in the simpler metals. This would appear to be almost conclusive evidence of the appro- priateness of the explanation of these irregularities suggested above. There is a difficulty of another type connected with the present theory. In fact it fails entirely to explain any effect of the magnetic field on the aggregate motion of the electrons in a direction parallel to the lines of force. This is of course | | : of the Hall Effect and Allied Phenomena. 547 hardly surprising, seeing that any such action by the magnetic field is an essentially excluded factor in the main hypothesis on which the present type of theory is based. According to this assumption, whatever action the molecules may exert on the electrons during a collision, it igs presumed that this action has on the average no reference to direction in the metal, so that any fields of the type of the magnetic fields in question, which act on the electrons in a transverse manner, cannot affect their aggregate motion parallel to the direction of the field. If we drop this assumption of isotropy the theory at once becomes much more complicated, although it is possible to predict the type of result which might be expected from it. The one constant of the above theory which depends essentially on the character of the collisions and constitution of the molecules is /,,, and mathematically speaking the assumption of isotropy merely implies that l,, is not a function of direction in the metal. In all cases, however, and more particularly when the metal is magnetic, the assumption of isotropy under the action of a magnetic field is hardly justifiable, and we ought therefore to assume that J,, is then modified so as to be a function of direction in the metal with reference to the magnetic force, the coefficients being in general even functions of the magnetic force intensity in the external field. This explanation would of course not only account for the magnetic effect on the longitudinal conductivity, but would also modify the trans- verse effect of the same type which has been effectively explained on the above theory, and probably in such a manner as to account for the irregular behaviour of this particular effect. There is of course an additional modifying action of a similar type to that already considered, which might also in certain cases provide an effective explanation of the effect of the magnetic field on the electrical conductivity. Let us first consider the ordinary problem of conduction. According to the ordinary theory, if the electric force in the external field is E the current density at any point in the metal is oH, o being the usually accepted expression for the conductivity. This result, however, neglects entirely the presence of the electrons bound in the metallic atoms ; we know, however, from independent evidence that such electrons do exist and the application of an electric field will displace them relative to the atom, so that each atom will become polarized, with, however, alsoa residual charge in some cases. But then the 2N 2 548 Electron Theory of Hall Effect and Allied Phenomena. field at any point in the interior cf the metal will not simply be the electric force in the external field, but will, as in the case of the magnetic field, be increased by a local part which, if we assume again isotropy, may be written in the form ali P defining the intensity of the polarization induced in the metallic atoms around the point. The driving field for the current is therefore E+aP. If we assume that K is the specific inductive capacity of the metal, P=D—HE=(K—1), D denoting the total electric displacement in Maxwell’s sense. The driving field is thus Hl 1+a(K—1)], so that the current is o(1+aK—1)K, or the conductivity as ordinarily defined must be taken as co = o(1 +aK—1). The application of the magnetic field will now have the. additional effect of destroying the isotropy of the local electric field, so that the constant a will become a function of direction in the metal, again with the coefficients as even functions of the magnetic force intensity. In conclusion, it would seem that there appears to be ample scope for adapting the theory propounded in the previous pages to provide an effective explanation of all the regularities and irregularities observed in connexion with the present phenomena, and although the precise mathe-. matical theory necessarily suffers from the essential incap- ability of taking a full account of the constitutional irregu- larities of the phenomena, it is nevertheless capable of interpretation in a manner which provides for an explanation of these irregularities. In this respect therefore the theory must be regarded as complete as is possible in the present. stage of our knowledge. The University, Shefiield, January 28th, 1915. [ 549. 1 LVI. The ilectron Theory of Metallic Conduction.—V. Bg G. H. Livens*. 1. Jntroduction. ° ee general theories of the conduction of heat and electricity in metals based on the idea that they take place by the free electrons only, bas been developed in various different ways. In the original theories of Drude f, Riecke f, and Thomson § the statistical motion of the electrons is practically ignored except in so far as to assume that it departs but slivhtly from the distribution of motion in the undisturbed steady state. In the theory of Lorentz|| the complete specification of the statistical motion of the electrons is fundamental. Whereas in the more recent generalization of this theory given by Jeans4, Wilson**, Bohrtt, and others, a specification not of the velocity distribution but of the momentum distribution is made the basis of all the calculations of the theory. In the original theories of Lorentz, T homson, Drude, and Wilson, the electrons and atoms of the metal are regarded as hard elastic spheres, at Jeast as far as their interaction in collision is concerned ; but in the more modern extensions developed by Jeans and Bohr it has been found possible to dispose of this restricting assumption and to formulate the theory equally well for a more general type of metal. The object of the present note is to discuss certain aspects mainly of the two earlier forms of the theory on the same more general basis as that used by Jeans and Wilson. Certain preliminary suggestions have already been made independently concerning some of the points to be here discussed, but their full analytical import has hardly been realized: they will, therefore, in the main be repeated together with their extension to the more general type of theory. * Communicated by the Author. + Ann. d. Phys. Bd. i. p. 566, and Bd. iii. p. 369 (1900). Tt Wied. Ann. Bd. lxvi. pp. 353, 545, & 1199 (1898), § Rapp. d. Congres d. Physique, ’Paris, 1900, tom. 111. p. 188. || Proce. Acad. Amsterdam, vol. vil. pp. 438, 585, & 684 (1905). “| Phil. Mag. June, July 1909. ** Phil. Mag. Noy. 1910. ++ ‘Studier over Metallernes Elektrontheori,’ Dissertation. Copen- hagen, 1911. SS Me ea em 550 Mr. G. H. Livens on the Electron 2. The bases of Lorentz’s theory. According to Lorentz* the whole theory turns on the evaluation of a function 7 which determines the statistical distribution of the motions among the electrons in the metal. This function is such that in a small volume element round the point whose coordinates referred to a definitely chosen system of rectangular axes are (w, y, z), the number of electrons per unit volume at the time ¢ with their velocity components between (&, 7, €) and (€+dE&, n+dn, €+d6) is T(E, 0, &, #, Y, 2, t) dV, where dV =dEé dy dé. If the electrons at the point under consideration are subject to action by external fields, the results of which may be specified by the accelerations (X, Y, Z) which they impose on the typical molecule, the function / is shown to satisfy a differential equation of the type OF 7 OF bon Omens cree Ast 55, 406 1 Oe of ce wherein (b—a)dVdt denotes the increase in the specified group of electrons during the next succeeding small interval dt brought about by the collisions taking place in this interval. The main difficulty experienced in the application of this equation lies in the determination of (b—a). The following remarks may, however, simplify the general problem in this respect. If, as we shall first assume, we may neglect the persistence of the velocity of an electron after collision, it is clear that the velocities of any electron at two instants between which it has encountered an atom are wholly independent of one another. This means that the distribution of the velocities among any group of electrons taken each immediately after its next collision succeeding the instant ¢, is entirely inde- pendent of the distribution at the instant ¢, and will in general be different from it unless indeed the initial distri- bution is that specified by Maxwell’s law, which is specially chosen so as to be unaltered by the collisions. A direct inference from this point of view is that the distribution of velocities among any group of electrons each taken imme- diately after its next collision after the instant ¢, will in fact be precisely that specified by Maxwell’s law, and is there- fore the same independently of the state of the motion that * See, for example, ‘The Theory of Electrons,’ pp. 266-273. Theory of Metallic Conduction. d51 may exist at the time ¢: in other words, the collisions com- pletely obliterate any regularity which existed in the elec- tronic motions previous to them. It follows again that the number (bdVdt) of electrons which enter the specified group during the small interval dt is precisely the same as the number which would enter the same group if Maxwell’s law specified the distribution both before and after the collisions, and it might therefore be calculated on this basis. The number (adVdt) of electrons leaving the group in the same time would then be exactly the same as (bdVdt) if there were no external forces or condition gradients in the meta] to modify the distribution established by the collisions. In the more general case, however, it is at once obvious that the number (a—b) dV dt can be calculated as the number of electrons removed by collision during the time dé from among the partial group of electrons contained in the specified group at the instant ¢, which is the excess of the number in this group at the instant ¢, over and above the number in the same group as specified in Maxwell’s law. The distribution of velocities which is expressed by Max- well’s law may, for the present, be taken to be specified as a particular case of the distribution described above, in which T , 1); G &,Y,; t) =o (&, UE ‘e vy, Y, 2, t), fo being a previously assigned function of known type. The distribution of electronic motions expressed by this function fo is the only perfectly chaotic distribution consistent with the general dynamical assumptions regarding the collisions between the electrons and atoms, any departure from it being the result of external forces tending to organize the irregularity in the motions. The partial group of electrons described above has then the number per unit volume with their velocity components in the small range dV,and the number of them which is removed by collision during the small interval d¢ is easily calculated by a well-known argument, and is Iho gy at. Tn where tT, is the mean time of duration of the free path ae Mr. G. H. Livens on the Electron motions of the electrons of the specified group. In this case, therefore, the differential equation above is to be inter- preted with ; il , a—b= my iets If, however, we wish to include the full effect of the persistence of the velocities, we must modify this equation in some such manner as the following argument indicates. It is to be noticed that if we neglect the persistence of velocities, the average or expected value of any electron moving at any time ¢ with velocity (&, 7, €) when taken immediately after its next collision will be zero; if we do not neglect the per- sistence, however, the expected value of the velocity will be e(&, n, €), « being a factor (a function in general of the resultant velocity) which measures the persistence. In other words, if there were ON electrons moving with this velocity before collision there will still be e6N moving with the same velocity after the collisions, one for each electron, have taken place. The collisions have thus reduced the number of electrons in the group by the factor (l—e) only. It follows, then, by the help of the argument used above that = (f-f. T a—b= The argument for this can be put in a slightly more direct and perhaps more rigorous form*. If the typical collision is onein which p is the perpendicular on the initial asymptote of the relative path and ¥ is the azimuth angle of this path, then it is easily shown that b—a= || mapa, 0 Jo where =i Maes and (£', 7', £') denotes the velocity of the electron after the typical collision. Now if in the most general case we can write f=fo (1+ 9), and if also the general dynamical nature of the collision is such as to leave the resultant velocity of an electron unaltered, then we shall have To os | * ($!=4) pdp dp. v so that then b—a=uj,\ * See, for example, O, W. Richardson, Phil. Mag. July 1912. Wake 0 Theory of Metallie Conduction. 553 Now the result obtained above in the more restricted case and the general form of the function on the left-hand side of the differential equation suggest that we may take p= $1 (et) b2 (Em &), where @, isja linear function of (€, », €) and ¢, a function of the resultant velocity. If we do this we find at once that = — uri) hy (u) An v(p)p dp dy, whence by comparison with the left-hand side of the differential equation, which is to be equalled to (b—a), the general form of ¢, and ¢, are easily deduced in the form which verifies the statement tentatively made above. It follows then that in the most general possible case if Tm 18 properly interpreted so as to include an account of the factor (l—e) expressing the persistence of the velocities, we may use NR Lal a= 3 Tm so that the above differential equation may be written in the form Yi oF Bat ae. 3 us + +6o! BF Ol 4 Ol a iy This is the fundamental differential equation of the theory in a form suitable for application in the most general case where it is possible to calculate tm. A good deal of infor- mation can of course be gained in certain general types of problem by a mere consideration of dimensions, but as a general rule a full specification of the dynamical character of the collision between an electron and an atom is necessary to determine this quantity. We canin many cases make a good deal of progress without any special knowledge of T,,. The main restrictions tacitly assumed in the above argument are firstly that the metal is isotropic in constitution but not necessarily in condition, and, secondly, that the atoms are of such comparatively large mass that the collisions of the electrons with them are not effective in altering either their velocities or the direction of their motion ; this latter condi- tion underlies the assumption made above, that the energy of an electron is unaltered bycollision. Mutual collisions between 554 Mr. G. H. Livens on the Electron the electrons themselves are also presumed to be too in- frequent to warrant consideration. The actual bearing and import of these assumptions have been elaborately discussed by Dr. Bohr in his dissertation already quoted, and the reader may be referred to this beautiful exposition of the subject for further information on this point. The solution of the above differential equation for the function 7 is easily obtained, and in the most general possible Soa of the present type of theory it may be written in the orm nae -— dr (a=! =fo- e ea x dt, 0 mt =t—r wherein yy Ofo., vy Ofo , 7 Ofo ap OF0 ... Olns, Olu amer ae +i. te omy Oey i +s7> and the suffix is taken to indicate the values of the variable functions expressed explicitly in terms of the auxiliary time variable ¢,, which is introduced to denote the integrations. To effect this interpretation it is first necessary to integrate the six auxiliary differential relations expressed generally in terms of the auxiliary time variable ¢,, introducing as the necessary constants of integration the values of the variables at the time ¢,. On substitution of these integrals in y, we obtain its complete expression as an explicit function of ¢,; the integration for 7 is then directly effected, maintaining throughout (&, 7, € «, y, 2, t) all constant. There is perhaps one remark necessary respecting the form in which the above solution is written. It is implied in it that 7, is not a function of the time, although it is essentially a function of the varying quantity w, velocity of the electron. The slight variation of +, on this account is, however, negligibly small and may safely be left out of account, especially considering that, as the final result verifies, Tt», merely appears as a factor in the already small correction term in jf. This assumption implies of course that the external forces have no direct influence on the collisions themselves, which are presumed to be of a comparatively short duration. ee a ee Theory of Metallic Conduction. D55 3. The bases of the Drude-Thomson theory. The form of the general differential equation satisfied by the fundamental function and the form of the function itself deduced from this equation are susceptible of a very simple. interpretation in a form which will be of immediate use to us. It has already been remarked by Jeans* that the effect of the persistence of velocities may simply be interpreted as implying that on the average the free path motion of the electrons is longer in the ratio l: 1—e. We might, therefore, obtain a sufficiently effective picture of the electronic motions, applicable in all cases where the actual dynamical charac- teristics of the collisions are of small importance (that is, in all so-called free path phenomena) by imagining that the free paths of all the electrons are lengthened in the corre- sponding ratio ae (which may of course be a function of the velocity), and that the collisions take place as with elastic spheres at the ends of these extended paths, so that there is no further persistence of the velocity. It follows, therefore, that any discussion given in terms of the simpler theory, which assumes the electrons and atoms to be elastic spheres, can be immediately generalized to the previous extent by the proper choice of 7,,, the mean time of description of a free path for an electron with the typical velocity. We shall, therefore, confine our remarks to this simpler case. We now turn to a consideration of the alternative but necessarily equivalent method of attack in these problems which is suggested by Thomson and Drude. This method is equally effective in giving in a simple manner the most general form for the velocity distribution, although, so far as I am aware, it has never previously been used for that purpose. In terms of the theory which regards the electrons and atoms as hard elastic spheres, the fundamental assumption underlying the theories of Thomson and Drude is that the collisions of the electrons with the atoms completely oblite- rate any regularity existing in the statistical motion of the electrons, a steady state being attained when the organizing effect of the external circurnstances is just balanced by the disorganizing collisions. The calculations on the basis of this theory, therefore, depend essentially on the fact that the distribution of velocities among any number of electrons, when taken each immediately after its last collision just preceding a certain instant, is precisely that specified by * “Dynamical Theory of Gases.’ (Cambridge, 1904.) 556 Mr. G. H. Livens on the Electron Maxwell’s law. Thus, if we take any group of electrons at the instant ¢ and trace each one back for the cor responding time T to its last collision, the distribution of motion among them will be precisely that specified by Maxwell’s law. This fact combined with a knowledge of the motion of the electrons during the times 7, as affected by the external field, may be used to obtain a form of the instantaneous velocity distri- bution, and in the following manner. We know that if we take any group of electrons, SN in number, all with the same velocity, then the number of them which have travelled for a time between 7 and 7+dr since their last collision is JaN COM ee mm Now the number of electrons per unit volume which according to Maxwell’s law have their velocity components between (&, 7; C) and (E+ dé, n+dn, t+dt) a TORS Use Coes LGN a formula which it will for the present purpose be more convenient to write in the equivalent form Hy Ca? aap eta V2 OF these electrons the number as above which have travelled for the time between +t and +t+dr since their last collision previous to the instant ¢ is ar Tm Now let us examine the motion of one of these electrons. It starts with the velocity (&, 7, €) and moves with accelera- tion (X, Y, Z) for the time tr: if we denote by (&, m, &) the components of its velocity at the time ¢, and (a, yz, 24) the coordinates of its position at this same instant, and if also we introduce, as in the last paragraph, the auxiliary time variable ¢,, we shall have Deg ee dnt, dEt, bas. dt, ae dt, = Y,, Ti, Ly, dat, dyt dz; —=71, | a ees . dt, = &t, di ae 43 Ort Orv ~j Theory of Metallic Conduction. whence if ufp=EPt ne + FP, we see that at, —t uz>=ui+2) (XE, + Yn + Ze) dey, a t,=t—T whilst Cy =i a=ut+ \ &, dt, *t=t-+t with similar expressions for y; and z;: (#y z) are of course the coordinates of the point from which the electron started. We shall write these expressions in the form ug au’? +1, MaHtt+a, YeqYtt, Y%HlOrls, denoting the integrals for the increments of the respective quantities by 2. But now we may interpret the number of the electrons under consideration (6N) in terms of their position and velocity coordinates at the time ¢, so that a ON, =dN=F (ui? —2,, Ile, Yi—ly, Zitz, t —-T) AE ne It is now a fundamental assumption in this theory that the effect of the accelerations produced by the external fields is so small that the increments z of the various quantities concerned are all so very small compared with these quantities themselves, that a sufficient approximation is obtained by neglecting second and higher degree terms in these quantities. We may therefore write the above expression for dN; in the form et as Oe eo) ea oO (uz) ony ov One a ie wherein Bo = Four”, tts Yes ot, t), AV = dE; dy: dh. If now we keep (&:, m2, && tt, Yt, 2t, ¢) constant, and inte- erate this expression for SN; over all values of 7, we shall obtain the complete group of electrons which per unit volume round the point (2, yt, z:) have velocities between (E,, ne, &) and (&+d&, mn +d, €+d6,) at the timet. Thus, dropping the suffix ¢, we have “(i Ol ghee Obi OF) OF get \e oe j= Fy “3 (ue) is a cl , ‘YY Oy cine ae OM rhe “0 Tne 558 Mr. G. H. Livens on the Electron Remembering the proper significance to be attached to Fo, we see at once that this result is, to the order of approxi- mation adopted, identical with that given above and obtained by Lorentz’s method. The two methods thus consistently provide the same results in a simple and straightforward manner in the most general possible case of the type under consideration. A, The basis of the Jeans- Wilson-Bohr theory. Before concluding this account it may serve some purpose to give a short discussion of the fundamental basis of the modern extensions of the theory originated by Jeans and developed by Wilson and Bohr, at least so far as concerns its connexion with the methods of the foregoing paragraphs. The new method of attack proposed by Wilson and corrected and completed by Bohr, consists in the determination of an equation specifying the rate of variation of the aggregate momentum of the group of all the electrons per unit volume which have their absolute velocities (uw) within a specified small range between u and u+du. The equation obtained by Bohr can easily be deduced from the results obtained above andin the following manner. The differentia] equation satisfied by the function fis of vy Of 7 OF OF | OF Of © OF ine Xan t Ys, h4ge t Speer tae al We may now remark that the function / differs but slightly from /) and by a term which is linear in (X, Y, Z) and the condition gradients in the metal, which quantities are assumed to be such that squares and products of them are negligible under all circumstances. It is, therefore, quite indifferent whether we use f or fy in the first six terms of the left-hand side of this equation. Now if for simplicity we assume, for instance, that fake =F tq' +O), 06. gon eon) “OGnwen Rap ais, toate ae 17 ay Peas then fers 0 Ow = fo [e(—20X + - on =u $2) nt 1 0A 0g +a( =2qY + 5 57 =a) Theory of Metallic Conduction. ag _” We have therefore LT foal Bie Tn _ | E( -29X+ - oo —w 92) +7(...)+... ] ene, Now multiply both sides of this equation by & and then integrate the resulting equation over all values of (&, n, £) subject to the condition wc +7? +0? < (u+du)’, and using, after Bohr, wutdu m | f EdEdndf=G,(u) du, the integration being over the same extension, we get meme) Guy) 4aarmA/ , . lod | .d¢ ee: aia ak ms elena aa)te ie, which is precisely the equation obtained by Bohr in this special case of the more general principles. The theory given by Bohr is slightly more general than that discussed above, but is apparently tractable only in the restricted cases which are included in the present treatment. In any case we may assert that most of the assumptions on which this simpler form of the theory is based are sufficiently verified in actual practice to render the theory as good an approximation to the actual facts as it is possible to obtain under the present circumstances. In conclusion we may thus state that under the usually accepted restrictions on which the general theory of the electronic motions in a metal is based, the apparently dif- ferent modes of treatment which have been suggested for dealing with the various problems under review are con- sistent with each other, and are in fact reducible all to ) method suggested by Lorentz, which is apparently the more : fundamental of the three. The University, Sheffield, January 28, 1915. OT RY B60.) LVII. Note on the Relation between the Life of Radioactive Substances and the Range of the Rays emitted. By ¥. A. LINDEMANN *. ee from the extraordinarily simple law which deter- mines the chemical properties of each successive radioactive element in terms of its predecessor and the type of ray emitted, there are only two well-established quantita- tive relations in radioactivity. One, which holds absolutely, is that all simple radioactive substinces disintegrate according to the formula xX = OY ca a X being the quantity remaining after ¢ seconds, if X) was the quantity present at time 0. The other. relation, which has not been tested so rigidly, connects the radivactive constant 7% in the above formula with the range R of the a particle emitted by the equation logA=A+BlogR. In the following paper it will be shown that both the above relations may be derived from certain fairly simple assump- tions. This theory leads to a definite physical meaning for the constants A and B, which enables one to calculate nttmerical values for certain atomic constants from them ;: as will appear, these values harmonize well with other measurements and theories. Though at present necessarily confined to substances emitting « rays, similar considerations may probably be applied to @ ray products. The first assumption is that the nucleus of a radioactive element, supposed to contain particles in movement, becomes unstable when N independent particles all pass through some unknown critical position within a short time t. ‘This view would appear quite plausible if Nutherford’s atomic model is accepted. The second assumption is that the above short time 7 is of the order of the time taken by a strain to traverse the nucleus. This hypothesis may perhaps be vindicated best by an analogy. If a number of small impulses from one side are applied to a pendulum, they will lead to the maxi- mum amplitude of swing if they are all applied within the time during which the pendulum is moving in one direction. It is sug vested that each particle passing through the critical position gives rise toa strain in the nucleus and that the atom breaks up when all these strains follow one another * Communicated by Sir Ernest Rutherford, F.R.S. Life of Radioactive Substances. 561 within a sufficiently short period. There are of course other possible assumptions, but the above seems the simplest. The particles whose position determines the atom’s stability may be taken to rotate or oscillate with the mean energy H=hy, the energy of the « particle emitted. Hach particle passes through the critical region v times per second, so that the probability of its being there within the time 7 is tv. As the particles are considered to be independent, the probability of N particles’ traversing the critical position within the time 7 is (rv)X. If X atoms are considered, the number which become unstable and explode in the time dt is : dX = — X(rv) dt, whence x — Xye- et, The radioactive constant > is thus equal to (tv)®. The 1D) Poe ; cA equation v= 7 leads to r= (7) EX, or introducing the empirical formula for the range R, R= 1°35. 10°E32, one finds log A=N(log + log 3°80. 10-°) +2/3N log R or log A= N (log 5°76 . 10° + log 7) + 2/3N log R. If, as was explained above, 7 is to be regarded as the time taken by a strain to traverse the nucleus, it is possible to determine its order of magnitude in terms of the charge on the nucleus ne, n being the atomic number, the mass of the nucleus M equal to the mass of the atom, and finally of the radius of the nucleus vr. The first essential is to calculate the velocity of propagation of an elastic wave (sound wave) in the nucleus. For this purpose we may obviously as a first approximation treat the nucleus as a homogeneous, positive volume charge of ne electrostatic units and density M p=-;——. The error introduced by neglecting to take the discontinuous structure into account cannot change the order of magnitude, as the nucleus consists of such a large number of particles. The velocity of an elastic wave may 1 ; thus be roughly taken as WA a being the compressibility of the nucleus Vv 75 V being the volume —a PER CTL ae EC REG 2 Aq ° ne? nie?’ : 2 : so that the velocity g=ne A TBM" The time 7 taken by the strain to traverse the nucleus is T= aut = pa : Inserting this value logA=N (log 5°76. 10?°+ log) +2/3N log R or rss logA=N (30°819 + log ve) +2/3N log R. If Geiger and Nuttall’s formula logX=A-+B log R were strictly true, one would be forced to the conclusion that N, the number of particles which determine the instability, is Mats : constant, and that —}~ is constant, n of course being the n atomic number. These consequences are not improbable, though a more searching test of the formule, using the data now available, might be advisable. In the meantime it is of considerable interest to see to what values of N and r the formule lead. In all the three families B=53°3, i. e. N= 80. This in conjunction with the fact that the atomic numbers of all the radioactive elements lie between 80 and 90 would seem to show that the moving particles whose position determines the stability of the atoms are the positive particles, as was to be expected ; further, that nearly all the free charges must conspire to bring about the explosion of the nucleus. The first constant A varies for the three radioactive series, the formula being : logy) A= —36'9 + 53°3 logy) R for the uranium radium family logy) A= — 38°44 53°3 logy, R for the thorium series, and logy A= —39°6 + 53'3 logy) R for the actinium series. It will suffice to evaluate the radius of the nucleus for one element, say radium. ‘Taking 226 as the atomic weight and 88 as the atomic number, one finds r=3°85.10-¥. This is in tair agreement with the radius of the nucleus of On the Calculation of Series in Spectra. 563 gold atoms deduced by Sir Ernest Rutherford from the scattering of the « rays. The difficulties of this hypothesis of the mechanism of radioactive transformations are obvious. The chief of them is probably the question as to where the energy of the 80 positive particles comes from, which changes from element to element as the substance disintegrates. This difficulty arises of course, perhaps in a slightly milder form, in any theory of radioactive transformations. On the other hand, the 80 independent factors seem almost unavoidable for X is undoubtedly determined by a question of probability, and experiments show it to be of the form E**. In view of the further numerical agreement with Sir Ernest Rutherford’s measurements of the diameter of the nucleus, it would seem most desirable to re-examine the original formule of Geiger and Nuttall, and to attempt to extend them to elements which emit 8 particles. Conclusions. It is shown that the well known formula connecting the life and the range of radioactive elements may be developed {rom a few simple and plausible assumptions. It follows that the explosion of an atom necessitates the fortuitous coincidence of some 80 independent events. A calculation of the radius of the nuclei of radioactive atoms on the above assumptions leads to a value of the order 3°9.10-*, in fair agreement with the accepted dimensions. Farnborough, July 4th, 1915. LVIII. On the Calculation of Series in Spectra. By A. G. Savipee and Prof. J. W. NicHoLson*. CERIES in spectra are usually isolated by the spectro- 4 scopist as the result of a preliminary search throughout the spectra for pairs and trios of lines whose distances apart, when expressed in wave numbers, are constant. If such trios or pairs can be found, there is usuallya probability that they are successive members of two interlacing series, one of the diffuse and one of the sharp type, whose calculated limits should be identical. Butin practice, spurious doublets and triplets—the result of chance—find their way into the list, and cause the necessary calculations to be very long and tedious. For in order to determine whether three lines are -* Communicated by the Authors. 20 2 564 Mr. Savidge and Prof. Nicholson on the successive members of a series, the numerical solution of a troublesome biquadratic equation, to several significant figures, is essential; and when the process must be repeated with various selections of three lines from a list, the labour is sometimes almost prohibitive. When the components of a series form doublets, moreover, one member of the doublet is usually very much weaker than the other; and unless the spectrum is developed very strongly —this development being controlled sometimes by circum- stances not sufficiently understood to enable it to be performed at will,—the weaker member cannot be seen and measured. One of us has shown recently that this is the underlying cause of the apparent non-existence of series in the spectra of an important group of chemical elements, which actually contain series of the usual type, so that the laws of spectra are in all probability of a universal application. This work has not yet been published, but it is mentioned here on account of the fact that the existence of the Table published in this paper alone rendered such a conclusion possible. When, in a complicated spectrum, the type of regularity typified by the recurrence of doublets is in this manner apparently absent, there is no real guide to the probable series lines, and a beginning towards their detection can only be made by the use of a graph or Table, with which any two lines can be compared, and an immediate determination made of the third member of their possible series. Graphs are used frequently in this manner by spectroscopists; but their use is somewhat limited, for they cannot give the necessary degree of accuracy. When a graph leads to the conclusion that three lines may belong to a series, the ensuing verification by a tedious calculation is always necessary, the utility of the graph being limited to the rejection of unlikely cases. The need for a Table has always been urgent, and the Table has now been calculated by Mr. Savidge. It has the advantage of being applicable equally to the ordinary are series with Rydberg’s constant N, and to the ‘‘ enhanced line” type of series recently discovered by Prof. Fowler *, in which this constant is 4N. It can also be applied to series in which this constant may be any other multiple of N. In series spectra developed in the are, the wave number n of a line is given very closely by the formula N Bc! (m+n) * Bakerian Lecture, Roy. Soc. Phil. Trans. 1914. r= Calculation of Series in Spectra. 565 where A is the limit of the series, N is Rydberg’s constant 109679°2 *, uw is Rydberg’s parameter, and m takes integral values. If n, and n, are the wave numbers of two successive lines, 1 1 m—m=N | os art Let w denote the value of (m+ ) for the first line. Then Ng—l, = N {a-z Ps (w+1)? or ee b? oN ee Se ae a Thisis a biquadratic equation, from which, on the supposition that the two lines are successive members of a series, the value of z or m+y for the first line can be obtained. The next member is then ty 1 m=mtN(o 75 eee which can be calculated at once and compared with observation. Mr. Savidge’s Table gives the solution of this biquadratic for values of «e=N/(n.—n,) ranging from 1°5 to 80, and it is suitable for direct interpolation. TABLE]. a«=1'3 to a=4-0. a. a. v. | a. x, | 1:32 1:70 1:2360 315 | 14841 | 1°34 1°72 1:2477 || 320 | 1-4434 1:36 1-74 1:2592 || 3:25 | 1:4526 | 1:38 1-76 1:2705 || 330 | 14617 1:40 1-78 12817 | 3°35 | 14707 142 1:80 1:2927 || 340 | 1:4796 | 1:44 1-82 13036 || 3:45 | 14885 1:46 1:84 13144 || 350 | 1-4978 | 1:48 1:86 1:3250 || 355 | 1:5060 1:50 1:88 13355 || 360 | 15146 1:52 1-90 1:3459 || 365 | 15231 1:54 1:92 13561 |} 3°70 | 1:5316 1:56 1-94 13662 || 375 | 1:5400 1:58 1-06 13762 || 380 | 1:5483 1:60 1:98 1:3861 || 385 | 1:5565 162 2:00 13959 || 390 | 15647 164 2:05 14056 || 3:95 | 1:5728 1:66 2°10 14152 | 4:00 | 15809 | 168 2:15 14247 | | * Curtis, Roy. Soc. Proce. vol. xe. p. 605 (1914). 566 Mr. Savidge and Prof. Nicholson on the TABLE II. a=4'0C to 2=21°0. a by Beet op a i a as . 40 | 1:58] 55 | «1797 70 1978 || 145 | 2626 4] 1597 56 1-810 75 | 2032 | 150 | 2-660 | 4:9 1-612 BT | 1-823 80 | 2085 || 155 | 2694 43 1628 58 «| «1:836 85 | 2135 || 160 | 2-727 4-4 1-643 59 «| 1-848 90 | 2183 || 165 | 9-759 4:5 1°658 6-0 1861 9-5 2-230 || 17:0 | 2:790 46 | 1-673 61 1873 || 100 | 2275 || 175 | 2:821 4°7 1687 62 | £88 || 105 | 2318 || 180 | 9852 4:8 1702 63 | 1897 || 110 | 2361 || 185 | 2-882 4:9 1716 64 | 1909 || 115 | 2402 || 190 | 9911 | 5-0 1:730 6°5 1-921 12:0 2-441 || 195 2-940 | 51 1744 66 | 1932 | 125 | 2480 || 290-0 | 2-968 | Be | 1-757 67 | 1-944 || 130 | 2518 || 205 | 2-996 | 5:3 1-771 6s | 1955 || 135 | 2555 || 21-0 | 3-02 54 1784 6-9 | 1:967 || 14-0 9-591 | TABLE II]. a=21°0 to 2a=80-0. | i | \ a. a, | a. as a. 21 3-023 || 36 3700 || 51 4:08 | 66 4-694 22 3-077 37 3-738 | 52 4-238 67 4650 93 3:129 || 38 8775 || 58 4-268 68 4-675 24 3180 39 3811 || 54 4-297 69 4-700 25 3-229 40 3:847 26 3-277 4] 3:83 | 56 4355 ral 4-749 4: | | 28 3°369 43 3°952 || 58 4-411 73 4797 29 3414 44 3°985 59 4439 74 4:82] 30 3°457 45 4018 | 60 4466 75 4°845 31 3°500 46 4051, 61 4°493 76 4:868 32 3541 47 4083 | 62 4°520 77 4-291 33 3°582 48 4115 | 63 4°546 78 4-914 34 3°622 49 4146 | 64 4°573 79 4-937 3D 3661 50 SG Minoo 4°599 80 4959 | 27 3323 42 3917 || 57 4383 72 | | | When « is greater than 80, the formula w= (2a)?-4+4+4(Qa)-3 may be used with a sufficient degree of accuracy. As an illustration of the use of the Tables, we may take three successive lines in one of the triplet series of oxygen, whose wave-lengths are Nl Lao: Ag=0000'83, A; = 4968°94, and wave numbers n= L6O200'0, oa 8(IO'O, ng = 20125°0 Calculation of Series in Speetra. 567 uncorrected to vacuo and the International scale. Taking the first two as members of a series, a=N/(ny—n,) =43°510. From the Tables, z=3°9688 for the first line, and therefore N 1 With 1 ‘ate { (4°9688)? (5°9688)? which is very close to the real value, and sufficient to indicate precisely the next line of the series, if there is a series. The accuracy of the Tables is, in fact, entirely determined by the accuracy of the Rydberg formula in any individual series. Another procedure is as follows:—If£ two lines are sus- pected to be alternate members of a series, the intermediate one may be found at once. Jor if n, and nz are the two wave numbers, } = 20122°7, 1 mmn=N 1a ai} (e+2)? where 2 is the value of m-+- for the first. If e=2y, yer ye 2 30 N 2y+1 4(ng—n,) Writing N/4(n3— n,;)=«a, y is read from the Tables, and then x=2y. The intermediate member is then 1 il ! No =N3— N (@+1)? (w+2)2 ° Similarly, if n; and n, are the wave numbers of two suc- cessive members of a “4N ” series of enhanced or spark- lines, I 1 Ng—ny=4N ee 4 eon wet)? 4N 2v+1 Ny— Ny Taking 4N/(ny.—n,)=a, wv is determined at once. The limit of the series is and 4N A=n1,+ F? and the third member is 1 1 ; ; 1g (ean pA Jets eee Other modes of using the Tables will at once suggest 568 Mr. C. E. Weatherburn: Problems un themselves to the reader. The Tables given, with the formula replacing them when «> 80, are sufficient to cover the whole range of necessary calculations. A spectrum which contains even so many lines as 150 can be explored for series in this manner in a very short time, especially when regard is paid to the fact that the intensities of the lines, as measured experimentally on any system, must diminish as we proceed towards the limit of the series. LIX. Problems in Llectrostatics and the Steady Flow of Elec- tricity under the exponential potential e~*"/r. By C. H. WEATHERBURN, M.A. (Cantab.), M.A., B.Sc. (Sydney) ; Lecturer in Mathematics and Physics, Ormond College, University of Melbourne*. Introduction. ‘Ole of the difficulties with which the present day physicist is confronted, is that of assigning a definite law of action at infinitesimal distances between two elements of electricity. Though Newtons law is universally admitted for finite distances, there is considerable doubt as to its validity for small ones. In other branches of physics, too, there is evidence that the action between particles very close together does not conform to the inverse square law. We are therefore justified in investigating laws of attrac- tion with potentials different from that of the inverse distance. In particular, the exponential potential e-*’/r, in which & is a positive constant, has peculiar claims upon our attention, for it was shown by Neumann f at the close of the last century that only under a potential of the form er = A; k r is the electric equilibrium possible within a conductor. The exponential potential includes the Newtonian as a particular case, and Green’s potential f Sisco oes p a etl ’ (Oz p: = 1) * Communicated by the Author. + Allgemeine Untersuchungen tuber das Newton’sche Princip der Fernwirkungen, Cap. 2. Teubner, Leipzig, 1896. + Green, “ Math. investigations concerning the laws of the equilibrium of fluids, &c.”” Trans. Camb. Phil. Soc. 1833. Electrostatics and the Steady Flow of Electricity. 569 as a limiting case, as is clear from the relation 7 Pigg aks tari) (Gite Oxr. Our problem is to find the potential in the outer region. This will be accomplished if we can determine the surface distribution y(t) over ©, the volume distribution y(p) through the conductors, and a further distribution which will give ‘the required discontinuity at . Asa discontinuous normal derivative is characteristic of a simple stratum, we shall -endeavour to satisfy the requirements by introducing a ‘simple stratum of density LO over &, assuming then the field to be a homogeneous dielectric. As in the first problem »(p) must be constant for each conductor and equal to $4°P;. Let f(t) represent the normal derivative at the surface © of the potential due to the volume distribution 7 and the fixed external charges. The simple stratum over } has at © a potential equal to i gta )o(a)deo -and a normal derivative | i (to) b(o)de. ‘Then, since the potential within the conductor is constant, the total normal derivative is zero, so that — p(t) + A(tS) w(S)dS+ f (6) + h(ta)¢(o)do=0. © @e a>) * Cf. Weber-Riemann, ‘Die partiellen Differentialgleichungen der smathematischen Physik,’ Bd. i. 8. 328-4 (5th ed.). Braunschweig, 1910. Also Plemelj, Monatshefte fii Math. und Physik, Bd. xviii. S. 188-194 (1907), where a similar problem is treated for the ordinary potential. ; 4 ; 4 Ta i a. ee Flectrostaties and the Steady Flow of Electricity. 575 The solution of this Fredholm equation is m=/O+| Meo) ee)de A H(ts) i (3) +f h(Sc) d(o) dads, which in virtue of (5a) reduces to ae) =f + | Hts)p(a)do, . . 2... (16) where we have written AO=KO+) HUsyf(sys. . Read B) Turning our attention to the discontinuity at } we notice that the potential due to 7 and the fixed external charges has a continuous normal derivative here equal to a known function F(s). The simple stratum p(t) over © has a potential whose norinal derivative is ie h(s3)u(3)d3. Hence by the properties of simple strata we have for the total potential V(p) Lin 0-4" 4) ] = 40) LV iV : 5 (ae (s-) +7. (s*) | 28 h(sc)d (c)do+F(s) +f hi s3) p(S)ds. On substitution of the value of (3) from (16) the second of these equations becomes [art Nt =o) ]= a(s) ah E G(se) ] (0) do where a@(s)=F(s) + | “ G(sS) f(5)ds. If the values of oe and Wy derived from these are substituted in (15), we find that the required function $(s) satisties the equation (e- +e*)b(s) + (€7 —)\ 3, 7 (sa) (a) do = 2p(s)— (e“ -e* )a(s). (18) 576 Mr. C. BE. Weatherburn : Problems in This is a Fredholm equation from which ¢(s) may be deter- mined by the usual formula. To show that it admits a unique solution it is sufficient to show that the corresponding homogeneous equation with second member zero does not admit a non-zero solution. If it did we should have a solution to the problem when both y(s) and a(s) are zero; that is, when there are no fixed external charges and no volume distribution, but only the two strata w(t) and ¢(s). The condition (15) would then read dV AN yas —e7 mee aks ae G0: Multiplying by V and integrating over & we have «| —V oN (s~)ds +e* ive! Gage 5 dn - dn Now since 7 is supposed zero the potential is zero through- out the conductors. Hence in the last equation the integra- tion may be made over & and © without affecting the equation. But each of the integrals is then an essentially positive quantity*, and their sum equal to zero. This is impossible, so that $(s), V(p), and w(t) are all zero. Since then the homogeneous integral equation does not admit a solution, (18) gives a unique value for ¢(s). This value substituted in (16) gives u(t). The required potential V(p), at any external point p is then V(n)=(9Cor)pOrrar+}aCrayn(aeg +f oro neoa+| a esaeas ) > =r) being the density of the fixed external charge at the 27 point *. § 5. Lhird Problem.—Suppose next that there is only a single conductor and that we are given not its potential but MA Meal: ; its total charget+. The volume density Te n(p) is unknown as well as the surface density a p(t). The conductor is under the influence of the same fixed external charges as before, and there is the same surface > of discontinuity. * Weatherbnrn, Joc. cit. First paper, § 3 (14). + Picard has treated the simple case in which there is no surface = of discontinuity in the field. Cf. Ann. Ecole Normale, t. 25, p. 585 (1908). (e- tet )p(s) + (€° -)( (is Gi sa)$(a)do— EHlectrostatics and the Steady Flow of Hlectricity. 577 Although 7 is unknown it is constant in virtue of (6). The functions f(t), 7, (¢), and w(s) may each be divided into two parts, the one due to the external charges, and the other due to 7 and proportional to it; so that we may write AW=R ) +nY¥ (t) V4. @ (s)=N'(s) +7 Y'(s) NT Nai ian (19) R, R’, Y, and Y’ being known functions of the points ¢ ands. The solution proceeds exactly asin the preceding case up to and including the equation (18); but now the function a(s) of the second member of (18) involves 7 which itself depends on the unknown function ¢(s). Given, however, the total charge E on the conductor, we have Rete ig a Heya.) (20) D being the volume of the conductor. On substitution of the value of w(t) from (16) this becomes nj Y@dt+ D3 + J Rd + {f Aes) 6(s)ds dt=B, (20') so that i peep) as at Pen eats where H’ and B are constants depending only on the form of the surface ©. Substituting this value of n in (19), and thence the value of a(s) in (18), we have for determining $(s) the integral equation Hoe) {RO+ EVOL. . Our problem is then reduced to the solution of the Fredholm equation b(s) +A0 NGso)d(c)do=Vi(s),. . .. (22') = whose kernel N(sc) is given by TN A AE N(so) = ng 2 (8) _ = ? | H(o)at evade while pees —> € +e" and W(s) is the second member of (22) divided by (e~ + €*). Phil. Mag. 8. 6. Vol. 30. No. 178. Oct. 1915. PAN re (ts)b(s)ds dt | (22) 578 Mr. C. E. Weatherburn: Problems in Having found ¢(s) from this equation, we determine 7 by (21) and then w(t) from (16). The final potential is expressible in terms of these as in the preceding section. The same method may be extended to the case in which there are any number n of conductors whose individual charges are given. Instead of the single equation (20) there now appears a system of n equations w(t) dt+D;=H; Ca 2, wettiae e 0; lle (Gee z oni (9). From these equations the ed n are ond as expres- sions in which the functions Le H(ts)$(s) dsdt occur linearly, and the kernel of the integral equation cor- responding to (22) becomes £ (so) + > M;(s) H (tc) dt, Ns == e; where the functions M;(s) are known functions of the points sof = Part I1.—Steapy FiLow or ELECTRICITY. § 6. General Problem+.—The general problem of the steady flow of electricity under the exponential potential requires the determination of a solution of the equation DOV eV =0,. | \.~) (oS ee which is continuous along with its first derivative except in the following particulars :— (A) There are certain surfaces, S, where the potential experiences a sudden jump 27. These “electromotive ”’ surfaces may, for instance, be the surfaces of separation of two different metals such as copper and zinc. At S then we may represent the behaviour of the potential by the equation V(st)—V(s-)=2n. aur dt denotes integration extended over the surface of the ith conductor. + Cf. Weber-Riemann, loc. cit. 8. 487-488. Our solution of the present problem follows Plemelj’s treatment of the corresponding problem for the ordinary potential. Cf Plemelj, loc. cit. S. 194-199. Electrostatics and the Steady Flow of Electricity. 579 (B) At the surface of separation, =, of conductors of different conductivity » the normal derivative is discontinuous according to the formula dV dV Na Waa) ae (or j=0. The conductivity X is positive and will be assumed constant for each conductor. (C) At the surface of separation, ©, between a conductor and a non-conductor the potential is continuous, while its normal derivative vanishes on the side of the conductor. The first of these conditions, (A), is clearly satisfied by the potential u,(p) of a double stratum of moment ~ a(s) over 8, expressed by aa uy (p)= J n(s)h(sp) ds. In order to satisfy both (A) and (C) we observe that w,(p) is continuous at the surface © separating conductor and non-conductor, so that it has a definite normal derivative there. We may then determine a solution w(p) of (2) having at © on the side of the conductor a normal derivative equal and opposite to this. Such a function u,(p) is, in virtue of (3) and (4), expressed by 7 / up) = { Gas (p3) 5" (8) ds. The sum u( p) =u (p) + u2(p) then satisfies both (A) and (C). In order to make provision for (B) express the final solution in the form V(p) =u(p) + U(p). Then U(p) must be a solution of (2) which is continuous at the surface S, while it satisfies (C), for both V and wu do so. Substituting this value of V in the boundary condition (B) we find, since u(p) is continuous at &, that eee dU _. du . Te ag (Co) =F wha} (a);.' (23) u(o) being a known function of the point oc. As a discontinuous normal derivative is characteristic of a simple stratum let us endeavour to satisfy (23) by a simple stratum potential of density 77 H(o) over &, that is v(p)= (y(po) u(e) do. A a 580 Problems in Electrostatics. This potential, however, does not satisfy (C) as U(p) must do. But we can determine another, v,{p), having at ©, on the side of the conductor, a normal derivative equal and opposite to that of v(p), and continuous elsewhere. This function v,(p) is given by I %4(p)= fel, (pS) = (3) ds =\\@., (pS)h(S2) wo) do ds. Put U(p) equal to the sum of these two potentials, 2 ¢. U(p)=0(p) +21 (Pp) ={ \ (pe) a G1(p3) h(3a)d3} u(a)do- = \ Gii(po) u(o) deo. Pv This sum satisfies both (A) and (C). It will be the required potential to supplement u(p) if we can find u(c) so that (23) holds. Now at ¥ the potential v;(p) has a continuous normal derivative, while that of v(p) is discontinuous. It follows. then that (5 [5-9-2 4) |= LrdU,. el ee sl ac Coe a (s+) | =| dn, FS) He) do s and o denoting points on the surface }. On substituting : a is aura in (23) the values of a: ) and = (st) derived from these, we find for w{s) the integral equation 7 AT d At —Dr du : mt et Ln, Hs0) Jalal do= Fyn) CD This is a Fredholm equation from which p(s) is uniquely determined. Though the kernel becomes infinite at s=o GARE | J a like — the second iterated kernel is finite and Fredhoim’s 7 ha method is applicable. That (24) admits a unique finite and continuous solution will follow if we can show that the corresponding homo- geneous equation does not. That such is the case is easily Velocity of Swiftly Moving Electrified Particles. 581 shown by noting that if the homogeneous equation admitted a solution the boundary relation (23) would read dU dU oT earn CS sia a aad A hl Multiplying by U(s) and integrating over } we have -| = UST )dse at | =e (st)ds=0. = 52 In this equation the integration may be extended over © also, seeing that on the conductor side of those surfaces the normal derivative is zero. But the potential U and its derivative are continuous in the regions bounded by =} and ©, so that the last equation makes the sum of two essentially positive expressions equal to zero. U(p) must therefore vanish identically and hence p(s) also. Thus since the homogeneous equation has no solution except zero (24) admits a unique finite and continuous solution u(s). From this the potentials v(p) and v,(p) are determined, and therefore their sum U(p). The final solution to our problem is then | V(p)=U(p) + u(p). December 24, 1914. LX. On the Decrease of Velocity of Swiftly Moving Elec- trified Particles in passing through Matter. By N. Bour, Dy. Phil. Copenhagen; p. t. Reader in Mathematical Physics, University of Manchester™. HE object of the present paper is to continue some calculations on the decrease of velocity of « and B rays published by the writer in a previous paper in this magazine. This paper was concerned only with the mean value of the rate of decrease of velocity of the swiftly moving particles, but from a closer comparison with the measurements it appears necessary, especially for 8 rays, to consider the probability distribution of the loss of velocity suffered by the single particles. This problem has been discussed briefly by K. Herzfeld, but on assumptions as to the mechanism of decrease of velocity essentially different * Communicated by Sir Ernest Rutherford, F.R.S, + Phil. Mag. xxv. p. 10 (1913). (This paper will be referred to as I.) t Phys. Zeitschr. 1912, p. 547. 982 Dr. N. Bohr on the Decrease of from those used in the following*. Another question which will be considered more fully in the present paper is the effect of the velocity of @ rays being comparable with the velocity of light. These calculations are contained in the first three sections. In the two next sections the theory is com- pared with the measurements. It will be shown that the approximate agreement obtained in the former paper is im- proved by the closer theoretical discussion, as well as by using the recent more accurate measurements. Section 6 contains some considerations on the ionization produced by aand 8 rays. - 2 < = (2) Consider next an & or @ particle penetrating through a sheet of some substance of thickness Av, and let the number of atoms in unit volume be N, each atom containing n electrons. The mean value of the number of collisions in which p has a value between p and p+dp is given by mes SOON POEs at mane Teds © wb 58) If we now could neglect the effect of the interatomic forces on the electrons, the average value of the loss of kinetic energy of the swiftly moving particle in penetrating through the sheet of matter would consequently be _ Ame? H?NnAw ( pdp Ns mV? { pte : (4) where the integration is to be performed over all the values for p, from p=0 to p=x. The value of this integral, however, is infinite. We therefore see that in order to obtain agreement with experiments it is necessary to take the effect of the interatomic forces into consideration. Let us assume, as in the electron theory of dispersion, that the electrons normally are kept in positions of stable equilibrium and, if slightly displaced, they will execute vibrations around these positions with a frequency v charac- teristic for the different electrons. In estimating the effect of the interatomic furces it is convenient to introduce the conception of the “ time of collision,” 7. e. a time interval of the same order of magnitude as that which the « or 8 par- ticle will take in travelling through a distance of length p. If this time interval is very short compared with the time of vibration of the- electron, the interatomic forces will not have time to act before the « or 8 particle has escaped again from — — ee ga 5 name ln SE a ona ES ’ : on 584 Dr. N. Bohr on the Decrease 0) the atom, and the energy transferred to the electron will therefore be very nearly the same as if the electron were free. If, on the other hand, the time of collision is long compared with the time of vibration, the electron will behave almost as if it were rigidly bound, and the energy transferred will be exceedingly small. The effect of the interatomic forces is therefore equivalent to the introduction of an upper limit for p in the integral (4), of the same order of magnitude us V/v. The rigorous consideration of the general case would involve complicated mathematical calculations, and would hardly be adequate in view of our very scanty knowledge as to the mechanism of the forces which keep the electrons in their positions in the atom. However, it is possible over a considerable range of experimental application to introduce great simplifications and to obtain results which toa high degree of approximation are independent of special assump- tions as to the action of the interatomic forces. The calculation of the total loss of energy suffered by the a or 8 particle is very much simplified if we assume that, for all collisions in which the interatomic forces have an appre- ciable influence on the transfer of energy, the displacement of the electron during the collision is small compared with p as wellas with the maximum displacement from which it will return to its original position. It can be simply shown that the displacement of the electron during the collision if it were free would be of the same order of magnitude as the above quantity a. The first assumption is therefore equi- valent to the condition that V/v is great compared with a. The second assumption is equivalent to the condition that the value for Q which we obtain by putting p=V/v in (1) is small compared with the energy W necessary to remove the electron from the atom. Under these conditions we get by a simple calculation, the detail of which was given in the former paper, that the effective upper limit p, for p in the integral (4) is equal to kV Po ony? where k=1'123. Introducing this, we get for the integral in (4), performing the integration from p=0 to p=p, and neglecting a? in comparison with p,?, ae (= me ( EV?Mm ) °8 pe) = °8 2rrvHKe(M+m)/° ge ee . “ =" —— Velocity of Swiftly Moving Electrified Particles. 585 From (4) we now get, noticing that v has different values Vjv2... vn for the ditferent electrons in the atom, dre" NAw S kV?Mm 5 a tae 2, 198 Ces rane In the above we have assumed, as in the ordinary theory of dispersion, that the electrons in the atoms normally are at rest. Qn the theory of the nucleus atom it seems, however, necessary to assume that normally the electrons rotate in closed orbits round the central nucleus. In this case it is a further condition fer the validity of the above calculations that the velocity of rotation of the electrons in their orbits is small compared with the velocity of the a or 8 particle and that the dimensions of the orbits are small compared with V/v. Ina previous paper the writer + has attempted to apply the quantum theory of radiation to the theory of the nucleus atom. It was pointed out that there appears to be strong evidence for the assumption that for every electron in the atom the energy W will be of the same order of magnitude as hy, where h is Planck’s constant. On this assumption it was deduced that in an atom containing n electrons the highest characteristic frequency of an electron will be of the same order of magnitude as 2mm , pee? the corresponding values for the velocity of rotation, for the diameter of the orbit, and for W will be of the same order of magnitude as (9) Dare? 2 a ee ae h 2) a2 4, Wa 6) TE: |g — Te. ) , and W=- j2 2 respectively. From these expressions it will be seen that the conditions underlying the above calculation will be the better satisfied the smaller the number x of the electrons in the atom. Introducing the numerical values for e, m, and A, it can be shown that all the conditions will be fulfilled, in case of a particles (V=2.10°, B=2e, M=10*m) if n< 10, and in case of 8 particles CV = 2.10", Base, Mim) » if n<100. Now according to Rutherford’s theory the number of electrons in the atom is approximately equal to half the atomic weight in terms of the atomic weight of hydrogen as unity. If, therefore, the main assumptions as to the + 1 be. | Phil. Mag. xxvi. p. 476 (1913), 2rr7e’m n —_ +1 en ee ee i etiam i TN — a i. ee er LL. —— —— 586 Dr. N. Bohr on the Decrease of mechanism of transfer of energy from the a or £ particle to the electrons are correct, we should expect that the formula (5) will hold for absorption of « rays in the lightest elements, and for 8 raysalso for the absorption in the heavier elements. In case of 8 rays it must, however, be remembered that the formula (1) is deduced under the assumption that V is small compared with the velocity of light. We shall return to this question in Section 3, when we have considered the probability variation in the loss of energy suffered by the | single particles. § 2. The probability distribution of the losses of energy suffered by the single a or B particles. The questions to be discussed in this section are intimately connected with the probability of the presence of a given number of particles at a given moment in a small limited part of a large space, in which a large number of the particles are distributed at random. ‘This problem has been investigated by M. v. Smoluchowski*, who has shown that the probability for the presence of n particles is given by W(n)=—e~*, - ek: where ¢ is the basis for the natural logarithm and @ is the mean value of the number of particles to be expected in the part of the space under consideration. If @ is very large this probability distribution is tu a high degree of approxi- mation represented by the formula W(s)ds=a / 2 e282 ds re 27 where s is defined by n=w(1+s), and W(s)ds denotes the probability that s has a value between s and s+ds. In the paper cited i. Herzfeld uses the formula (7) in calculating the probability distribution of the distance R which an « particle of a given initial velocity will penetrate through a gas before itis stopped. Herzfeld makes the simple assumption that a certain number of collisions with the gas molecules is necessary to stop the particle, and he takes this number A to be equal to the total number of ions formed by the particle in the gas. Now the number of collisions suffered by an a particle in penetrating a given distance through the gas is the same as the number of molecules * Boltzmann-Festschrift, 1904, p. 626; see also H. Bateman, Phil. Mag, xxi. p. 746 (1911). Velocity of Swiftly Moving Electrified Particles. 587 present in a tubular space round the path of the particle. The probability distribution of the number of collisions can therefore be obtained from the above formule, if for wm we introduce the mean value of the number of collisions. Since A is supposed-to be very great the variation in the ranges R of the single particle will be very small. The probability that R has a value between R,(1+s) and Ro(l+s-+ds), where R, is the mean value of the ranges, will therefore, on Herzteld’s assumption, be simply given by (7) if we put w=A. On the present theory the calculations cannot be performed quite so simply. The total number of collisions is not supposed to be sharply limited, but it is supposed that the amount of energy lost by the « or 8 particle in collisions with the electrons will depend on the distance of the electron from the path of the particle, and will decrease continuously for an increase of this distance. In order to apply con- siderations similar to Herzfeld’s, it is therefore necessary to divide the collisions up into groups in such a way that the amount of energy lost by the particles will be very nearly equal for all the collisions inside each group. Consider an « or 8 particle penetrating through a thin sheet of some substance of thickness Av, and let us divide the number of collisions of the particle with the electrons into a number of groups in such a way that the distance p has a value between p, and p,41 for the collision in the rth group. Let us now for the present assume that it is possible in this way to divide the collisions into groups so that the number in each group is large at the same time as the dif- ference between any two values for the energy Q lost by a collision in the same group is small. Let the value for Q corresponding to the 7th group be Q, and let the mean value of the number of collisions in this group be A,, and the actual number of collisions in this group suffered by the given « or f particle be A,(1+s,). The total energy lost by the particle in passing through the sheet in question is then given by A= 3 Q,A,(1 + s,). From this we get, denoting the mean value of AT by A,T, AT— Aol =e, Q,A,Ssr. Since the A’s are large numbers, we get from (7) for the probability that s, has a value between s, and s,+ds,, —2Ars V(e)ds—a / Ae mr dsr. oo x ae a D88 Dr. N. Bohr on the Decrease of Now similarly denoting the probability that AT has a value between AT and AT+dT by W(T)dT, we get by help of a fundamental theorem in the theory of probability, _ (AT-a, 1? W(AT)dT=(2nPAz)-ie 74". ay tee where Aces = (Q.Anji= 3 A,02. On the above assumptions this can simply be written PAc= | Q’dA. Introducing in this expression the values for Q and dA given by (1) and (3), and integrating for every kind of electron from p=0 to p=p, we get Ame*HAN S/1 1 Pees oe a Assuming, as in the former section, that p, is large compared with a, we get, neglecting the last term under the = and introducing in the first the value of a from (2), Ame? K?M? (M+ m)?~ T's p “ - ; . (9) It will be noticed that this expression is very simple. It depends only on the total number of electrons in unit volume, but neither on the velocity of the a or 8 particle nor on the interatomic forces. From (8) and (9) we can simply deduce the probability distribution of the thickness of the layers of matter through which particles of given initial velocity will penetrate before they have losi all their energy. Putting AT=A,l(1+s), we get for the probability that s has a value between s and P= (s+ds), | W(s)ds=4 / 3 ds, J oa wlhiere Pa Ba ON i= Ge far, ¢ being the mean value of — lf we now suppose that the straggling of the rays is small —this assumption is already indirectly involved in the assumptions used in the deduction of (8)—the formula (10) Velocity of Swiftly Moving Electrified Particles. 589 will express also the probability that a particle in order to lose the energy Aol will penetrate through a layer of thick- ness between Av=Apzr(1+s) and Az-+ dxr=Aja(1+s+4+ds),. where Ajye=AjT/d. In order to find the probability W(R)dR, that a particle in order to lose all its energy will penetrate through a layer of thickness between R and R+dR, let us now divide the interval from 0 to T in a great number of small steps A,T, Ao|l’... and let us for the rth step denote the quantities corresponding to Aw, u, d, and s by A,#, uy, by, and s, The distance through which a given particle will penetrate is equal to A,T Pr From this we get, denoting the mean value of the ranges of the particles by Ro, h=>Az= = (1+s,). A,T Pr In exactly the same manner as that used in obtaining (8) we now get R—R)= > S re _ (R-R,)” W(R)dR=(27U)-e 79 @R,. . (12) IN oe ela a where or simply TT AT \ -—3 U=P| (‘ =i dT, . . . > (13) where the differential coefficient’ stands for the mean value pe at A ee - | The equations (8) and (9) and consequently also (12) and (13) are deduced under the assumption that the collisions suffered by the swittly moving particle in penetrating a thin sheet can be divided into groups in such a way that the variation of Q for each group is small, while at the same time the number of collisions in the group is large. The condition for this is that the quantity X=dA “ is large compared with unity. Substituting from (1) and (3) we get A=mNnAa(p?+a*), 2. 2... (14) 590 Dr. N. Bohr on the Decrease 0) We see that X is equal to the average number of electrons inside a cylinder of radius ,/p?+a?. Since » decreases for decreasing p, we shall only have to consider its value for p=0. Substituting for a we get ea ee ie M?2m?V4 : If we consider a gas at ordinary temperature and pressure and introduce the numerical values for e, m, H, M, and N, we obtain both for « and 8 rays approximately nAx Va This expression varies very rapidly with V, and gives quite different results for a and for #8 particles. For a rays from radium C we have V=1°'9. 10°, this gives Ap=1l7.nAx. Now the range of a rays from radium C in hydrogen and helium is about 30 cm., and according to Rutherford’s theory, the number n of electrons in a molecule of these gases is equal to 2. We therefore see that Ap will be large compared with unity, provided the sheet of matter be not exceedingly thin compared with the range. For other gases Ay will be even greater, since the product of the number of electrons in the molecule and the range of rays is greater than for hydrogen and helium. In case of a rays we may therefore expect that the formulz deduced above should give a close approximation. In order to get an idea of the order of magnitude of the variation to be expected in the loss of energy suffered by an @ particle, consider for instance a beam of « rays penetrating a sheet of hydrogen gas 5 cm. thick. Using the experimental values for the constants, we get from (11) w=3.10° approximately. Introducing this in (10) we see that the probability variation is very small, Thus about half the particles will sutfer a loss of energy which differs less than 1 per cent. from the mean value, and less than 1 per cent. of the particles will suffer a loss which differs more than 5 per cent. In section 4 we shall return to this question and compare the formula (12) with the measurements. For 8 rays of velocity about 2.10'°, we get for a sheet of aluminium 0:01 gr. per cm.?—a thickness corresponding to that used in the experiments discussed in section 5— Aozz1'6.10-%. Since this is very small compared with unity, it is clear that the assumptions used in deducing the formule (8) and (12) are in no way satisfied. Still, it Xo pee e 10°? ° Velocity of Swiftly Moving Electrified Particles. 591 appears that it is possible from the calculations to draw some conclusions of importance for the comparison of the theory with the measurements. Consider a £ particle passing through a sheet of matter, and let us for a moment assume that no collision occurs for which 2» is smaller than a certain value tr. Let the value for p determined from (14) by putting X=7 be pr. If rt is not small compared with unity the probability distribution of the loss of energy will with considerable approximation be given by (8), if in the expression for P the integral is verformed from p=pr instead of from p=0. According to the above pr will be great compared with a, and we get instead of the expression (9) for P 1 47r7e! KAN Zn? Aw 4 P-= a (Rd A - eyo ee a (15) Introducing this in (11) we find for a sheet of aluminium 0°01 gr. per cm.” for uw approximately ur=2507. If + is not small compared with unity, we therefore see that we obtain a probability distribution of the loss of energy which is of the same character as that for « rays. The mean value for the loss of energy for the collisions in question is simply obtained from the formula (5) in the former section by re- placing a by pr. This gives 272 ee Ade? H?N Aw (”). (16) menage 208 (8 In the applications the logarithmic term in this formula will be large and A;T will depend very little upon the exact value of tr. Thus for an aluminium sheet A,T will vary only 4 per cent., if r varies from 1 to 2. Let us now consider the probability distribution of the loss of energy due to the collisions for which p is smaller than pr. Since pr is large compared with a, it follows from (14) that the average number of these collisions is very nearly equal to 7. If now 7 is a small number, e. g. T=1, it is evident that the probability distribution of the loss of energy due to the collisions will be of a type quite different from that considered above. In the first place, there is a certain probability that there will be no loss of energy at all; from (6) we get that this probability is equal to e~’. Next, if Q;is the value given by (1) if we put p=pr, no loss of energy greater than zero and smaller than Q, is possible. At Q; the probability curve suddenly rises and falls off for increasing values of Q approximately as Q-*. For the 1 ag2 Dr. N. Bohr on the Decrease of aluminium sheet considered above we have approximately A,T/Qr=16r. From these considerations it will appear that the proba- bility distribution of the loss of energy suffered by a 8 par- ticle of given initial velocity in penetrating through a thin sheet of matter will show a sharp maximum at a value very close to Arl, if r=1, and fall rapidly off on both sides. The value for the decrease of energy measured in the ex- periments is evidently this maximum, and not the mean value for AT given by the formula (5), such as was supposed in my former paper. The considerable difference between the two values is due to a very small number of very violent. collisions left out in deducing the formula (16) but included in (j). Putting 7=1 and introducing for p, and p,, we get from (16) 2 2oreP NN i (hk?V?NnAw SAL == Ge > log ( ree ) : 4 (1 7) In section 5 we shall consider the question of the loss of energy suffered by a beam of 8 rays when penetrating through a sheet of matter of greater thickness. § 3. Hifect of the velocity of B particles being comparable with the velocity of light. The calculations in the former sections are based on the formula (1) for the energy transferred to an electron by a collision with an a or 8 particle. In the deduction of this formula it is assumed that the velocity V is small compared with the velocity of light ¢. This condition is not fulfilled in case of high speed @ particles. If V is of the same order of magnitude as ¢, the calculation of the amount of energy transferred by a collision involves complicated considerations. for the general case. The problem, however, with which we. are concerned is very much simplified by the circumstance considered in the former section, that the value for the loss. of energy of @’ particles, measured in the experiments, will depend only on collisions in which the energy transferred is very small compared with the total energy of the @ par- ticle, 2. e. collisions in which u is small compared with p. Considering such collisions and calculating the force exerted on the electron by the @ particle, we can neglect the displacement of the electron during the collision as well as its reaction on the 8 particle. We need, therefore, only consider the way in which this force is influenced by the velocity of the @ particle itself. Velocity of Swiftly Moving Electrified Particles. 593 In the electron theory it is shown that the electric force, exerted on an electron at rest by a particle of charge E and uniform velocity V= ce, will be directed along the radius vector from the particle to the electron and given by* 0b REL toe OO ~ 7? (1— 6? sin? @)3? where r is the distance apart and w the angle between the radius vector and the path of the particle. Let the shortest distance from the path to the electron be p, and let o= a at the timet=0. We have thensin » =< and r?=(Vt)?+p. For the components of the force perpendicular and parallel to the path of the swiftly moving particle we now get oe ning pis Veg Tf Hi respectively. Introducing for 7, and putting (1—f?)-?=y, we get aa fy (ye We Ae E=(@vorse 7 VE We see from these expressions that the force at any moment is equal to that calculated on simple electrostatics, if we everywhere replace the velocity V of the swiftly moving particle by yV, and, in calculating the component perpen- dicular to the path, replace the charge E of the particle by E, while leaving it unaltered in calculating the component parallel to the path. In the calculation of the correction due to the high speed of the @ rays we shall, therefore, have to consider the effects of the two components separately. If the electron is free it will be simply seen that the velocity of the electron, after a collision in which a is small compared with p, will be very nearly perpendicular to the path of the B particle. In calculating the energy transferred in this case we need therefore consider only the component of the force perpendicular to the path. If V issmall compared with c we get trom (1), neglecting a compared with p, Ze? Hi? i m Vp?" If in this expression we introduce yV for V and yE for H, * See, for instance, O. W. Richardson, ‘The Electron Theory of Matter,’ p. 249, Cambridge 1914. Phil. Mag.8. 6. Vol. 30. No. 178. Oct. 1918. 2Q 594 Dr. N. Bohr on the Decrease of we see that it is unaltered. If the electrons were free there would thus be no correction to introduce in the calculation due to the effect of the velocity of the @ particle being of the same order as c. If, however, we take the eftect of the interatomic forces into account, the problem is a little more complicated. In this case it is necessary to introduce a correction in the expression for p,. In addition the effect of the interatomic forces will involve a certain transfer of energy due to the component of the force parallel to the path of the 8 particle; this is due toa sort of re- sonance effect which comes into play when the “time of collision” is of the same order of magnitude as the time of vibration of the electrons. In the former paper it was shown that the contribution to AT due to the component parallel to the path is given by* yin 2rre°?H*NnAx i mV? From (17) it therefore follows that the contribution to AT), due to the component perpendicular to the path of the 8 particle, is given by 22 n 2V2 7 Ya Ayt Zn PRPINASS (jog (PVN 22) _ 1), 1 mV2 Aqry If we now in the expression for Y replace V and E by yV and yH, and in the expression for Z replace V by yV but leave H unaltered, we get, by adding the two expressions together and substituting for y, the following corrected formula for A,T: D2 W2 n 22 2. 2 2 A,T= 2Q7re? EK NAzs [log k?V a) log(1—-q )— ee mun lie Any” * I. p. 17. The expression deduced in this paper was Are’? EK? NnAr An) aie where coal by fe 2 ie GOs ae iL), =(f (2)) dx and F (x)= |, daa L formed part ot a complicated expression, used in determining p, and evaluated by numerical calculation. The value of L, however, can be simply obtained by noticing that il f'(e)— aft) -f (2) =0. This gives ines ° f(a f'(@—f ee i |" Co 3 Sie) | he )( 6 (2 ns | z)) —( ) : Now f(0)=1 and /'(0)=f (a )=/'(~@ )=0; consequently L=3. . C3F Velocity of Swiftly Moving Electrified Particles. 595 It will be seen that the correction is very small unless V is very near to the velocity of light, since in other cases the two last terms will approximately cancel each other out. § 4. Comparison with measurements on « rays. In the former paper it was shown that the formula (5) in section 1 gives values which are in close agreement with the measurements on absorption of « rays for the light elements hydrogen and helium, if we assume that the atoms of these elements contain 1 and 2 electrons respectively, and if for the characteristic frequencies we introduce the frequencies determined by experiments on dispersion. It was also shown that an approximate agreement with the measurements of the absorption in heavier elements could be obtained by assuining that these elements, in addition to a few electrons of optical frequencies, contain a number of electrons more rigidly bound and of frequencies of the same order of mag- nitude as those determined in experiments on characteristic Roéntgen rays ; the values deduced for the number of electrons were in approximate agreement with those calculated on Sir E. Rutherford’s theory of scattering of « rays. In this section we shall therefore only consider the new evidence obtained by later more accurate measurements. Since the velocity of « particles is small compared with the velocity of light, we have T=}MV*. From (5) we therefore get d a = K, yi(log Ve— 29 log y+ K3), 2 US) where fe Aare? H?N eae — kMm a a eis: ( seaneMra) This expression depends on two quantities characteristic for the different substances, 2. e. the number of electrons in the molecule n, and the mean value of the logarithm of the cha- . . . I racteristic frequencies of the electrons ~S logy. The latter n quantity determines the characteristic differences in the “velocity curve,” 7.e. the curve connecting corresponding points ina (#, V) diagram. In the former paper formula (19) was compared with values for dV/dw deduced from the measirements. Since the quantity directly observed is the value of V corresponding to different values of x, itis simpler 2 Q 2 596 Dr. N. Bohr on the Decrease of first to integrate formula (14). This gives gi Sul 2, dz poe oW a (* de 8nK, z2,—2,!, logs’ where 4 BO ‘ log z= ° (log V3—-> log v+ Kg). ‘ ‘ 7 ? P A table for the logarithm integral in (20) is given by Glaisher *, Considering a gas at 15° and 760 mm. pressure we have Ne=1:224.10" Putting e=4:78.10-% H=2e, “ = 5°31. 10", and E/M =1:448.10", we get K,=1-131. 10% mo and —_ —21°80. In most measurements. nek from radium Care used. This corresponds to Vo=1°922.10° f. Assuming that the hydrogen atom contains one electron, we get for the hydrogen molecule n=2. If we further assume that the characteristic frequency of both electrons in the hydrogen molecule is equal to the frequency determined by experiments on dispersion in hydrogen, we get ¢ y, = 8'52.10% and > log y=35'78. 2 Using these values and the above values for Vo, K, and Ky, we get log2z9=8'75. Introducing this in formula (20) we vet for the distance travelled in hydrogen gas by @ rays from radium © before their velocity has decreased to half of its original value, @,=24°0 em, The first column of the table below contains values for 2/2, corresponding to different values for V/Vo. No accurate measurements on the velocity curve in hydrogen have been made. Such measurements would form a very desirable test of the theory since the assumptions underlying the calculations may be expected to be closely fulfilled in case of this gas. T. a. Taylor § has recently determined the range of a rays from radium CO in hydrogen. He found 30°9 em, at 15° and 760 em. Using the theoretical value #,=24°0 em., we should expect from the table that the range would be close to 27 em. This is not far from the range observed. At present it seems difficult to decide whether the small deviation may be ascribed to experimental errors in the constants involved, & Phil. Trans. Roy. Soe, elx, p, 3807 (1870). E. Rutherford and H, Robinson, Phil, Mag, xxviii, p, 552 (1914). ©. & M, Cuthbertson, Proe, Roy. Soc. A. INXXViii. p. 166 (1909), Phil. Mag, xxvi. p. 402 (1918 Gh++ — Velocity of Swijtly Moving Lleetrijied Particles. 597 V/V, ty ihe 8 | 1 V 1-0 0 0 0 0 0 09 0338 | 0315 | 0300 | O18 | 0-289 08 0502 | 0561 | 05389 | 0560 | 0520 07 v:780 | 0751 | 0730 | O710 | U729 06 o911 | 0394 | O879 | 0889 | 0:82 Od 1-000 1000 1-000 1-000 1000. | 0-4 1-055 | 1-080 03 1-087 02 1104 According to Rutherford’s theory the helium atom contains two electrons. Since helium is a monatomic gas this gives =2as for hydrogen. Experiments on dispersion in helium rive v=5'92.10", Introducing these values for n and vy in (20) we get values for w which are a little greater than those for hydrogen. ‘lhe theoretical ratio between the ranges in helium and in hydrogen is 1:09. The measurements of Ki. P. Adams’ *, discussed in the former paper, were in disagree- ment with the calculation that the range in helium was shorter than in hydrogen; the ratio between the ranges observed being only 0°87. ‘Taylor’s recent measurements, however, give for this ratio 1°05, in close agreement with the theoretical value. For air Marsden and Taylorf have recently made an accurate determination of the velocity curve. They found that a rays from radium © will travel through 5°95 em. of air at 15° and 760 mm. pressure before their velocity is reduced to}V . If we assume that the nitrogen atom con- tains 7 electrons and the oxygen atom 8 electrons, we get for the air molecule in mean n=14°4. Introducing this in the formula (20) and putting a= 5°95 for V=4Vo, we find ~ I ~ ry log 29=5°37 and = log v=38'32. The values for w/a cor- responding to this value for log zy are given in the column II, of the table. Not so many values are given as for hydrogen, since the fulfilment of the conditions mentioned in section 1, on account of the higher frequencies, claims greater values for V for air than for hydrogen. Column IY. contains the values for «w/a, observed by Marsden and Taylor. The agreement between the calculated and the observed values is very close. At the same time it will be seen that the values in column IV. differ considerably from those in * Physical Review, xxiv. p. 118 (1907). + Proc. Roy, Soc, A. Ixxxvili, p. 448 (1915), 598 Dr. N. Bohr on the Decrease of columns I. and III. The values in these columns are calcu- lated by putting log z=8'75 (see above) and log z)=4°44 (see below) respectively. If instead of log z»=5°37 we had used one of the latter values, we should instead of 14°4 have to put n=81 or n=22'5 respectively, in order to obtain the observed value for z,. It will therefore be seen that the considerable difference between the values in the columns I., II., and III. offers a method of determining n, even in cases 1 e where —> log v is not known beforehand. n Marsden and Taylor could not observe any e particle with a velocity smaller than 0-42 V>. When the velocity had decreased to this value the particles apparently disappeared suddenly. This peculiar effect is in striking contrast to what should be expected on the theory. It appears, however, that it may possibly be explained by a statistical effect due to a small want of homogeneity in the «-ray pencils used. In the first part of the velocity curve the slope varies gradually, and a possible small want of homogeneity will have only a very small effect on the mean value of the velocity. But near the end of the range the slope of the curve is very steep, and if the pencil for some reason is not quite homogeneous, the effect will be that, as we recede from the source, more and more of the particles will so to speak suddenly fall out of the beam. In this way the velocity will not start to decrease rapidly until almost all the particles are stopped ; but then the beam will contain so few particles that the final descent may be very difficult to detect. The values in column VY. correspond to Marsden and Taylor’s results for the velocity curve of rays from radium C in aluminium. The value for 2, corresponding to V=3V9 was 9°64.10~*, measured in gr. percm.? The value for K, in aluminium if # is measured in gr. per cm.? is 9°81 . 10%°. If for aluminium we assume n=13, and in (20) intro- duce #,=9°64.10° for V=4Vo, we get loga=4-44 and *¥ log v=39°02. As mentioned above this corresponds to the values in column III. It will be seen that the values in column V are much closer to those in III. than to those in I. and II., but the agreement is not nearly so good as for air. This may partly be due to the difficulty in obtaining homogeneous aluminium sheets, but it may also be due to the fact that the assumptions underlying the calculations cannot be expected to be strictly fulfilled for all the electrons in the aluminium atom (see page 586). For elements of Velocity of Swiftly Moving Electrified Particles. 599 higher atomic weight, the assumptions used in the calcula- tions are satisfied to a still smaller degree than for aluminium, and accurate agreement with the measurements cannot be obtained, although the theory offers an approximate expla- nation of the way in which the stopping power of an element and the shape of the velocity curve vary with increasing atomic weight. In section 2 we considered the probability variation in the ranges of the single particles of an initially homogeneous beam of a rays. Denoting the mean value of the ranges by Ro, we get from (12) and (13) for the probability that the range R has a value between Ry'1+s) and Ro(l+s+ds) 7) Bee Seweal bit. (BB) W(s)ds= ——= (s) 5 p/ tr € where . Piicee t2 all wae. 2 eg ee ae Parapet n) i) eee p Ry? Ro? J, (5..) a ad This expression is much simplified if we use an approxi- mate formula for dT/dx. Putting r=CT’, we get erected tees: aes 198 = a es [Teese i Br See) Introducing this in (22) we get * To See. 2 Tah : > — ' . . . 23 wie 2Pidz © st) 7? * Note added in proof. For =3, this expression is equivalent to the expression deduced by L. Flamm (/oc. cit. formula (25)) for the variation in the ranges of a particles due to collisions with the electrons. This author has considered also the collisions with the central nuclei and concluded that, although the effect of these collisions on the mean value of the rate of decrease of velocity of the « particles is very small com- pared with that due to the collisions with the electrons, their effect on the variation in the ranges is not negligible but will be given by an ex- pression of the type (21) for a value of p of the same order of magnitude as that given by (23). From considerations analogous with those applied in section 2 in the case of 8 rays it appears, however, that the collisions between the a particles and the nuclei will produce a variation in the ranges of a type different from (21). In these collisions only very few of the particles suffer a considerable diminution of their ranges, while the ereater part of the particles suffer diminutions which are very small even compared with the average differences in the ranges produced by the collisions with the electrons. It seems therefore that the effect of the collisions with the nuclei may be neglected in a comparison with the measurements. 600 Dr. N. Bohr on the Decrease of Geiger has shown that we obtain a close approximation to : TARE 3 the velocity curve in air if we put r= 5, For hydrogen we ad . e . . . ° 5 obtain a similar approximation by putting r= 3° The exact 9 . ° ° . CON 2 value for r, however, is of only little inportance since 5 e/ of is very nearly constant for values of r between 1 and 2. Putting T=3MV? and introducing the theoretical expres- sions (5) and (9) for dT/dw and P, we get Ren? SURE Oe ( kV?Mm )=4 M,, MA p? 3r—2~ 4mn > ‘i 27veH(M+m)/~ 16 m oa For x rays from radium C we now get, using the same values for log zy as above, for hydrogen and air p=0°86.107? and p=1'16.10~? respectively. For @ rays from polonium, assuming the initial velocity of the rays to be equal to 0°82 that for radium ©, we get for hydrogen and air p=0°91.10~? and p=1-20.10~? respectively. Geiger* and later '‘laylorf have made experiments in order to measure the distribution of the ranges in hydrogen and air of « rays from polonium and radium OU. They counted the number of scintillations on a zine-sulphide screen kept at a fixed distance from the radioactive source and varied the pressure of the gas between screen and source. The results do not agree with those to be expected from the theory. The straggling observed was several times larger than that to be expected and did not show the symmetry claimed by the formula (21). These results, if correct, would constitute a serious difficulty for the theory ; they seem, however, inconsistent with the results of some recent ex- periments by F. Friedmann{. The latter experiments were made in order to test Herzfeld’s theory, which also gave a straggling much smaller than that observed by Geiger and Taylor. Friedmann found a distribution of the ranges in air of « rays from polonium which coincides approximately with that given by (21), if p=1:0.10~%. As seen, this value is even a little smaller than that calculated from the theory. Further experiments on this point would be very desirable. § 5. Comparison with the measurements on B rays. The experimental evidence as to the rate of loss of energy by § particles in penetrating through matter has until * Proc. Roy. Soc. Ixxxili. p. 505 (1910). + Phil. Mag. xxvi. p. 402 (1913). t Sitzb. d. K. Akad. d. Wiss. Wien, Mat.-nat. Kl. exxii. Ila, p. 1269 (1918). ’ Velocity of Swiftly Moving Electrified Particles. 601 recently been very limited on account of the great difficulties in the measurements. Much light, however, is brought upon this question by the study of the homogeneous groups of B rays emitted from certain radioactive substances. 0. v. Baeyer*™ observed that the lines in the ‘* 8 ray spectrum,” produced when the rays are bent in a magnetic field, were shifted to the side of smaller velocities when the radioactive scurce was covered by a thin metal foil. The question has recently been more closely investigated by Danyszt, who extended the investigation to a oreat number of the groups of homogeneous rays emitted from radium Band C. The first two columns in the table below headed by Hpand A(Hp) contain the values given by Danysz for the product of the magnetic force H and the radius of curvature p for a number of groups of homogeneous f rays, and the corresponding values for the alteration in this product observed when the rays have passed through an aluminium sheet of 0-01 gr. per em.? The limit of error in the values for A(Hp) is stated to be about 15 per cent. } | p. a [8A(Hp). 5880 | 32 | 0-960 | Hop. | A(Hop). | | 1891 a ae 635 31 | 1681 9 | 0-704 3 1748 9 | O-718 33 1918 66 0750 28 i988 | 61 0-760 27 A ae ae 0770 265 2224 | 87 0795 23 i ie 0802 a3 | 2939 37 0-867 4 CO 9997 | 48 0-885 33 4789 ) 39 0°942 on ey LO Ae | / The values for Hp are connected with the velocity of the 8 particles through the equation 72 2\ —3 op . m( 1 sy : Cc deduced on the expression for the momentum of an electron which follows from the theory of relativity. Denoting V/c by 8, we get Cc =m Hp= —— BU — §?)* Bi eis ict CBU * Phys. Zeitschr. xiii. p. 485 (1912). Tt Journ. de Physique, ii. p. 949 (1915). 602 Dr. N. Bohr on the Decrease of This gives em A(Hp) = —- 1-8")? AB. On the theory of relativity we have further T=c?m((1—f?)-?7—1); from this we get AT=c’mB(.—8?)-2 AB. . = > See We have consequently AT=eBA(ip). \:.. > ae From (18) we thus have, putting E=e and V/c=8, 2re®NAa 2 PeNnAe 1— mee! A aie ea one Heecae o (Hp) mc?B? > [log 4qrv” Ve lo AG }e —# |, (27) Except for very high velocities the variation of the last factor will be very small, and we shall therefore, according to the theory, expect A(Hp) ) t» be approximately proportional to 8~*. The third column of the table contains the values for 8, and the fourth column the values for 6?A(Hp). It will be seen that the values in this column are constant within the limit of experimental errors. Putting n=13 and using the value ~ ¥ log p=a00 calculated from experiments on # rays, we get from (27) for an aluminium sheet 0-01 gr. per cm.? B= 06 OF O8 O09 0:95 BA(Hp)=. 40.) 41), 42 aa ae Considering the great difficulty in the experiments and the great difference in mass and velocity for a and § rays, it appears that the approximate agreement may be considered as satisfactory. The mean values for A(Hp), calculated from the formula (5) in section 1, would be about 1°3 times larger for the slowest velocities and would increase far more rapidly with the velocity of the 6 rays. Measurements of the decrease of velocity of 8 rays in sheets of metals of higher atomic weight are more difficult than with aluminium on account of the ‘greater effect of the scattering of rays. Danysz found that the rate of decrease of velocity was approximately proportional to the weight per cm.” of the absorbing sheet. Since the number of electrons in any substance is approximately proportional to Velocity of Swiftly Moving Electrified Particles. 603 the weight, and since the differences in the characteristic frequencies will have a very much smaller influence for fast 8 rays than for « rays, results of this kind should be expected on the theory. If we assume that the formula (18) holds also for the loss of energy suffered by @ rays in penetrating a layer of matter of greater thickness, we obtain for the “range” of the 8 particles R TmeZAt ai \ aN =| Qare!NS, where = denotes the last factor in (18) and (27). Con- sidering = as constant, and using the above formula for AT, we get CR ea "8 BdB met 0% noe R=s7ans) =A Teas (l-ON+ ay 1—3] (28) Recently R. W. Varder* has made some interesting ex- periments on the absorption of hemogeneous ® rays. He measured the variation in the ionization produced by the rays in a shallow ionization chamber wlien sheets of different thicknesses were introduced in the beam before striking the chamber. Using aluminium sheets, he found that the ioniza- tion varied very nearly linearly with the thickness of the sheets, and his diagrams give a strong indication of the existence of a “range” of the @ particles. Varder compared the ranges observed with the last factor § in the formula (28), and found that the ratio between the ranges and 8, though nearly independent of the initial velocity of the rays, de- creased slowly with this velocity. This should be expected from the above calculations, as = will increase slowly with the velocity. Measuring R in gr. per cm.? Varder found R/S=0°35 for B=0°8 and R/S=0°30 for B=0°95. The first factor in the theoretical formula is equal to 0°42 for 8=0°8 and 0°38 for B=0°95. We see that the agreement may be considered as very satisfactory. The distribution of the losses of energy, suffered by the individual particles of a bean of initially homogeneous 8 rays in penetrating through a sheet of matter of considerable thickness, cannot be represented by the formula (12) used in the former section, since—see section 2—already the dis- tribution of the loss of energy suffered in penetrating through a thin sheet differs essentially from that given by formula (8). * Phil, Mag. xxix. p. 725 (1915). 0 604 Dr. N. Bohr on the Decrease of In addition, the transverse scattering of the rays due to deflexions suffered in collisions with the electrons as well as with the positive nuclei must be taken into account. ‘This scattering will cause the mean value of the actual distances travelled by the particles in the matter to be greater than the thickness of the sheet. If, however, we for a moment negiect all collisions in which the particles suffer either abnormally big losses in their energy or big deflexions, we may, as in section 2, expect that the rest of the rays will behave in a similar way toa beam of « rays and that they will show a range of a similar degree of sharpness. There- fore the distribution of the energy of a beam of initially homogeneous £ rays emenonle from a thick layer of matter must, as for a thin sheet, be expected to exhibit a well- defined peak sharply limited on the side of the greater velocities, but falling more slowly off towards the smaller velocities. The further the rays pass through the matter the greater the chance that the particles will suffer a violent collision, and the smaller will be the number of particles present at the peak of the distribution. A simple calculation shows that by far the greater part of this effect is due to the deflexions suffered in collisions with the positive nuclei. An estimate of the effect of these collisions may be obtained in the following way. The orbit of a high speed @ particle colliding with a positive nucleus has been discussed by C.G. Darwin*. From his calculations it follows that the angle of deflexion ¢ of a B particle of velocity V =e is given by *y)!)=p(1- Bye) where ne (i= Bi)? om — pB em ne is the charge on the nucleus and p is the length of the perpendicular ‘from the nucleus to the path of the "8 particle before the collision. Let pr be the value of the p corre- sponding to =r. The probability that a @ particle will pass through a sheet of matter of thickness Aw without suffering a collision for which w>7 is equal to 1—wAa, where T= cot a pane he a JEN TB 4¢! mn Since wAw is small, this probability can be written ¢—A4z, * Phil. Mag. xxv. p. 201 (1913). Velocity of Swiftly Moving Electrified Particles. 605 and the probability that the 8 particle will penetrate through a sheet of greater thickness without suffering a single de- flexion for which w>7T is consequently given by W =e-', where \= | oAr. Substituting for Aw from the formula (18) and using the same notation as above, we get qe ie — 8?) AT 2S ems, Considering > as a constant we get from this, using the expression (25) for AT, NE dB n pal) n2 Cae ees | ai— A? 4023 8 ( 1+ (1—838)]— gs!°8( gap where S as above denotes the last factor in the expression (28) for the range R. We have consequently n? W=K(g—3)°", Whe Ac aaa 2) i J where S is approximately proportional to the range of the emergent rays, and K a constant. The formula (29) gives an estimate of the number of 8 particles left in the peak of the velocity distribution of the emergent rays, and may be compared with the ionization measured in Varder’s experiments. It will be seen that W depends to a very high degree on n, and therefore on the atomic weight of the absorbing substance. As mentioned above, & is for these fast rays approximately proportional to n, and the exponent in (29) is therefore proportional to n. If aluminium was used as absorbing substance Varder found that the ionization was approximately proportional to the range of the emergent rays, while for paper it decreased more slowly, and far more rapidly for silver and platinum. For aluminium we have n=13 and “y=18 or Uo. Putting the exponent in the expression for W equal to 1, we get in this case r=0'30 and ¢ approximately equal to 30°; this is an angle of the right order of magnitude. For paper the exponent in (29) will be halved and for platinum will be more than five times larger than for aluminium, for the same values of 7 and ¢. In connexion with the calculations in this section, it may be of interest to remark that the approximate agreement obtained between the theory and the measurements seems to 606 Dr. N. Bohr on the Decrease of give strong support to the expressions used for the momentum and the energy of a high speed electron. Let us for a moment suppose that the ordinary expressions for the momentum and the energy of slowly moving electrons could be used without alteration. This should not alter the equa- tions (26) and (27), but the values for V deduced from the values for Hp would be (1—”)~? times greater. Introducing this in the formula (27) we should have found a value for A(Hp) which for the swiftest rays would be about 30 times smaller than that observed by Danysz, and the values in the last column of the table on p. 601 would, instead of being nearly constant, be more than 20 times smaller for the slowest rays than for the swiftest. If, on the other hand, we had supposed that the expressions for the momentum were correct, but that the “longitudinal” mass of the electron was equal to the “‘ transversal ”” mass, we should have obtained the same values for V as in the table, but the equations (26) and (27) would have been altered bya factor (1L—§?)~1. In this case the value calculated for A(Hp) for the swiftest rays would have been about 15 times larger than that ob- served, and the values in the last column, instead of being nearly constant as observed, should have been expected to be 10 times greater for the fastest than for the slowest rays. It thus appears that measurements on the decrease of velocity of $8 rays in passing through matter may afford a very effective means of testing the formula for the momentum and the energy of a high speed electron. §6. The wnization produced by a and B rays. A theory of the ionization produced by « and 8 rays ina gas has been given by Sir J. J. Thomson*. In this theory it is assumed that the swiftly moving particles penetrate through the atoms of the gas and suffer collisions with the electrons contained in them. The number of pairs of ions produced is supposed to be equal to the number of collisions in which the energy transferred from the particle to the electron is greater than a certain energy W necessary to remove the latter from the atom. If we neglect the inter- atomic forces this number can be simply deduced. By dif- ferentiating (1) with regard to » and introducing for pdp in (3) we get | | 2r7re?H?NnAw dQ as mV? Q?° * Phil. Mag. xxiii. p. 449 (1912). (30) Ol ee ee ee ee Velocity of Swiftly Moving Electrified Particles. 607 Denoting by Q, the value for Q obtained by putting p=0 in (1), we get, integrating (30) from Q= W to Q=Q,, 27reHk*NnAx a Whi mee Ca (Qo ao Ou where _ 2mM?V? ; Q0= Ga)? C2) If we consider a substance in which the different electrons correspond to different values for W, we get instead of (31) simply Ire*7E2NAw & (1 T Sir J. J. Thomson showed that the formula (31) with close approximation can explain the relative number of ions produced by @ and B rays. If, however, in (31) we in- troduce the values for W calculated from the observed ionization potentials, and the values for the number of elec- trons in the atoms which were found to agree with the calculations in section 4, we obtain absolute values for Aw which are several times smaller than the ionization observed. It appears, however, that this disagreement may be explained by considering the secondary ionization produced by the electrons expelled from the atoms in the direct collisions with the « and @ particles. In Sir J. J. Thomson’s paper it is argued that this secondary ionization seems to be very small compared with the direct ionization, since the tracks of a and 8 particles on C. T. R. Wilson’s photographs show only very few branches. A calculation, however, indicates that the ranges of the great number of the secondary 1 rays able to ionize are so short that they probably would escape observation. The rays in question wiil be the electrons ex- pelled with an energy greater than W, and will be due to collisions in which the « or 8 particle loses an amount of energy greater than 2W. The number of such collisions is given by (31) if W is replaced by 2W. Let this number be PAly- The total energy lost by the particle in the col- lisions in question is equal to Pa. 2re?7HPNnAx, Qy QdA = —_.,, — log Qo oF ody mV? aw aides 2 2W Aow, approximately. The mean value of the energy of the elec- trons expelled is therefore P= W (2 log ( (Qo/2W) — 2) 2) or hydrogen and @ rays from radium C this gives Nes 608 Dr. N. Bolir on the Decrease of P=10W ; corresponding to a velocity of 6.10% and a range of about 10-*em. in hydrogen at ordinary pressure. The number of ions produced by the secondary rays cannot be calculated in the same simple way as the number produced ~ by the direct collisions with the e or 8 particles, for in the case of the secondary rays we cannot neglect the effect of the interatomic forces. From the considerations in section 1 it is seen that the conditions for the neglect of the inter- atomic forces is that the value of p corresponding to Q= W is very small compared with V/v. By help of the expression (1) for Q and the expressions for W and v on p. 585, it can be simply shown that this condition is equivalent to the con- dition that the energy 4mV? of the rays is very great compared with W. This condition is fulfilled for «and @ rays in light gases, but is not fulfilled for rays as slow as the seoondary rays. Recently J. Franck and G. Hertz* have made some very interesting experiments which throw much light on the question of ionization by slowly moving electrons. HExperi- menting with mercury vapour and helium gas, they found that an electron will rebound from the atom without loss of energy if ils velocity is less than a certain value. As soon, however, as the velocity is greater than this value the electron will be able to ionize the atom, and it was shown that the probability that ionization will occur at the first impact is considerable. For other gases the results were somewhat different, but in all cases a sharply defined limiting value for the velocity of the ionizing electrons was observed. These experiments indicate that slowly moving electrons are very effective ionizers. We may, therefore, obtain an approximate estimate of the number of ions produced by the secondary rays, if we simply assume that each of these rays will produce sions if their energy has a value between sW and (s+1)W. This would give for the total number of ions formed 2rre? HW? NnAa [/ 1 1 it 1 T=AwtAew+ ee oo (Gy = +( sav — a,)+ a: T£ Q is very great compared with W this gives approximately 2re'HNnAx 1, Q Q e Lae w oS = Aw logy. - ° (34) This formula applies only to substances for which W has the same value for all the electrons in the atom. For other substances we must take into account that an electron ex- pelled may produce ions, not only in collisions with electrons * Verh. d. Deutsch. Phys. Ges. xvi. p. 457 (1914). Velocity of Swiftly Moving Electrified Particles. 609 corresponding to the same value for W, but also in collisions with other electrons in the atom. Considering, however, the rapid decrease in the chance of ionization with increasing W , we may in a simple way obtain an approximate estimate, if we assume that all the ionization produced by the secondary rays is due to collisions with electrons corresponding to the sinallest value for W. This value will be the one which is determined in experiments on ionization potentials; let it be denoted by W,. In the same way as above we now get 27e°7K>7NAa (Aw+ Aw+w: + Awsaw, + ...)= — mV2 > IG rr =) +(wew.- @,)+ a, If Qo is big compared with all the W’s, we get approximately Q7re? E*N Aw z Qo ge l= W, og Gy): : bare tS) On account of the simplifying assumptions used, the formule (34) and (35) can only be expected to indicate an upper limit for the ionization. The minimum fall of potential P necessary to produce ionization in hydrogen, helium, nitrogen, and oxygen was measured by Franck and Hertz*. They found 11, 20°5, 7°5, Pe 300’ meecu irom this W equal to 1°75.10-", 3°25.10-", 1:20.10°", and 1°45. 10~™* respectively. The absolute number of ions produced by @ rays in air is determined by H. Geigerf. He found that every & particle from radium C in passing through 1 cm. of air at ordinary pressure and temperature produced 2°25 .10* pairs of ions From this we get, using T. 8. Taylor’st measurements of the relative ionizations in air, hydrogen, and helium, that the number of pair of ions produced by an « particle from radium C in passing through 1 cm. of one of the two latter gases is very nearly the same and equal to 4°6 . 10°. If now in (31) we introduce the above value for W for hydrogen and use the same va ues for N, n, e, E, m, and V as in section 4, we get for # rays from radium C * Verh. d. Deutsch. Phys. Ges. xv. p. 34 (19138). ft Proc. Roy. Soc. A. 1xxxii. p. 486 (1909). t Phil. Mag. xxvi. p, 402 (1913). Pll. May. S. 6. Vol. 30. No. 178. Oct. 1915. 2R e 2 and 9 volts respectively. By help of the relation W= 610 Dr. N. Bohr on the Decrease of in hydrogen Aw=1:15.10%. The value given by (34) is 5-9 Aw. The first value is 4 times smaller than the ioniza- tion observed. The latter value is of the right order of magnitude, but is a little larger than the experimental value. For helium W is nearly twice as great as for hydrogen. From (31) and (34) we should therefore expect a value for the ionization only half of that in hydrogen. Taylor, how- ever, found the same ionization in hydrogen as in helium. Since in this case the value observed is greater than that calculated from (34), the disagreement is difficult to explain, unless the high value observed by Taylor possibly may be due to the presence of a small amount of impurities in the -helium used. This seems to be supported by experiments of W. Kossel* on the ionization produced by cathode rays. This author found that the ionization in helium was only half as large as that in hydrogen—in agreement with the theory. The cathode rays used had a velocity of 1°88. 10°, corresponding to a fall of potential of 1000 volts, and the number of ions produced in passing through 1 em. of hydrogen at a pressure of 1 mm. Hg was equal to 0°882. This corresponds to 670 pairs of ions at atmospheric pressure. Putting V=1'88.10° and E=e, and using the same values for W, e, m, N, and nas above, we get from (31) Aw=300. From (34) we get T=4°5 Aw. If we consider a substance such as air, which contains a greater number of electrons in the atoms, we do not know the value of W for the different electrons. A sufficiently close approximation may, however, be obtained, if in the logarithmic term of (35) we put W=hv, where h is Planck’s constant. This gives, if we at the same time introduce the value for Qo from (32), 272 mute , 9V2 2 _ 2re?’ Eh = log 2V?mM \ l hy(M+m)?7> 7 * (28) a mV7W, If now in this formula we introduce the values for n and Ss aiee used in section 4 in calculating the absorption of n a rays in air, and put W,;=1°25 .10-",, we get I=3°6. 104. This is the same order of magnitude as the value 2°25 . 10* observed by Geiger, but somewhat larger ; this would be expected from the nature of the calculation. The value to be expected from the formula (33) cannot be stated accurately on account of the uncertainty as to the magnitude of the W’s, * Ann. d. Physik, xxxvii. p. 393 (1912). Velocity of Swiftly Moving Electrified Particles. 611 but an estimate indicates that it would be less than a fifth of the observed value. While the formule (31) and (33) give values which vary simply as the inverse square of the velocity, the formula (36) gives a variation of I with V which is similar to the variation of AT given by the formula (5). Using the same value for *S logy as above, we thus get for « rays in air, that the ratio between the values for I, given by (36), for V=1°8 .10° and for V=1'2 .10° is equal to 1°65. ‘The corresponding ratio for AT given by (5) is 1°54. This is in agreement with Geiger’s* measurements, which gave that the ionization produced by an « particle in air, at any point of the path, was nearly proportional to the loss of energy suffered by the particle; both quantities being approximately proportional to the inverse first power of the velocity. The number of ions produced by cathode rays in air has been measured by W. Kossel+ and J. L. Glassonf. For a velocity of 1°88. 10° Kossel found 3°28 pairs of ions per cm. at 1 mm. pressure. Under the same conditions Glasson found 2°01 and 0°99 pairs of ions for the velocities 4°08 . 10° and 6°12.10°% respectively. At atmospheric pressure this gives for the same three velocities 2°49. 10%, 1°53 . 10°, and 0:75.10? pairs of ions respectively ; or 9:0, 14°7, and 30-0 times smaller than Geiger’s value for a rays from radium C. The values calculated from (36) for cathode rays of the three velocities in question are 7°1, 17°4, and 41°2 times smaller than the value calculated for « rays from radium C. The calculations in this section cannot be immediately compared with measurements of the ionization produced hy high speed 8 rays, since we have made use of the formula (1) which is valid only if V is small compared with the velocity of light. In a manner analogous to the considerations in section 3,it can, however, be simply shown that the cor- rection to be introduced in the formula (36) is very small and will only affect the logarithmic term. For high speed 8 rays, the variation of this term with the velocity V will further—as in the calculations in section 5—be very small compared with the variation of the first factor. From (36) we shall therefore expect that the ionization produced by these rays will be approximately proportional to the inverse square of the velocity. This isin agreement with W. Wilsou’s§ measurements. * Proc. Roy. Soc. A. 1xxxiii. p. 505 (1910). + + Loe. cit. t Phil. Mag. xxii. p. 647 (1911). § Proc. Roy. Soc. A. Ixxxv. p. 240 (1911). 2K 2 612 Velocity of Swiftly Moving Electrified Particles. Summary. According to the theory discussed in this paper, the decrease of velocity of @ and @ rays in passing through matter depends essentially on the characteristic frequencies of the electron in the atoms, in a similar way as the pheno- mena of refraction and dispersion. In a previous paper it was shown that the theory leads to results which are in close agreement with experiments on absorption of « rays in hydrogen and helium, if we assume that the atoms of these elements contain 1 and 2 electrons respectively, and if the frequencies of these electrons are put equal to the frequencies calculated from experiments on dispersion. It was also shown that an approximate explana- tion of the absorption of « rays in heavier substances can be obtained, if we assume that the atoms of such elements, in addition to a few electrons of optical frequencies, contain a number of electrons more rigidly bound and of frequencies of the same order of magnitude as characteristic Réntgen rays. The number of electrons deduced was in approximate agreement with those calculated in Sir E. Rutherford’s theory of scattering of a rays. These conclusions have been verified by using the later more accurate measurements. In my former paper, very few data were available on the decrease of velocity of @ rays in traversing matter and the agreement between theory and experiment was not very close. The agreement between theory and experiment is improved materially, partly by using new measurements and partly by taking the probability variations in the loss of energy suffered by the individual 8 particles into account. In this connexion it is pointed out that it appears that measurements on the decrease of velocity of @ rays atford an effective test of the formule for the energy and momentum of a high | speed electron deduced on the theory of relativity. In connexion with the calculations of the absorption of a and 8 rays, the ionization produced by such rays is considered. It is shown that the theory of Sir J. J. Thomson gives results in approximate agreement with the measurements if the secondary ionization produced by the electron expelled by the direct impact of the « and B rays is taken into account. I wish to express my best thanks to Sir Ernest Rutherford for the kind interest he has taken in this work. University of Manchester, July 1915, a 61s. | LXI. Construction of Compound Molecules with Theoretical Atoms, especially the Systems of Growth of the Organic Compounds of Carbon and Hydrogen. By AuBert C. CrenoreE, Ph.D.* (From the Department of Physiology of Columbia University.) [Plates VILI.-XIT.] AVING obtained equationst for the mechanical force with which one atom acts upon another, and found that they enable us to construct crystals having properties which agree with those found in nature, a further test of this theory of the atom has been found in the building of the molecules of chemical compounds with the same atoms employed in crystal building. The present paper is a preli- minary account of the results obtained in an attempt to explain some of the complex phenomena connected with the hydrocarbon compounds of organic chemistry. This line of investigation is chosen first because a very complete description of both the hydrogen and the carbon atoms has been obtained ; and if we are successful in getting any light upon the possible ways in which these two kinds of atoms may unite into compound molecules, the results must be typical of all compound-forming processes. The results described are, so far as they go, in agreement with the well-known systems of organic chemistry. The Distinction between a Crystal and a Compound Molecule. In a crystal containing a large number of atoms it has been shown in former papers that the axes of revolution of the atoms are not parallel to each other, but that they take such positions that the turning moment of the force due to all the other atoms in the crystal upon each and every atom is zero, and in stable equilibrium for small displacements. When the number of atoms in the structure is comparatively small, however, as is the case with single molecules of chemical compounds, it may be shown that there is no position of the axes that can satisfy this required condition for the equilibrium of the turning moments except when the axes are all parallel to each other. On this account I have come to the con- clusion that the chemical molecules in distinction from a erystal are formed of atoms whose axes of revolution are all parallel to each other, and the obtaining of the different * Communicated by the Author, J Phil. Mag., June 1915, p. 750. 614 Dr. A. ©. Crehore on the Construction of systems of growth from the fields of force surrounding the atoms when their axes are parallel, which agrees with those of organic chemistry, seems to justify this view. Another important difference between a crystal and a chemical molecule is the order of magnitude of the distances between the atoms composing them. The edge of the tetra- hedron forming the basis of the diamond has a length of about 2°5x 10-8 em., whereas 107!%cm. is the order of magnitude of the distances between atoms in most of the compound molecules. Roughly the distances are 100 times greater in crystals, or we may say in crystal molecules, thinking of a ery stalas asin gle compound molecule built upon a separate and distinctive system. Not only does the distance differ, but the forces which hold the molecule together are vastly greater in the case of the chemical compound molecule due to this reduced distance. This latter fact has an important bearing upon crystal formation. It becomes possible, on account of the small distances between the atoms in a chemical compound, for a molecule composed of a group of atoms to behave as though it were a single atom; for example, in a cubic crystal like sal ammoniac NH,Cl, the group NH, may act as a single molecule taking the place, say, of Nain the compound NaCl, the atoms of which are known to be in cubical array, alternating in three directions. Graphical Method of Constructing Molecules. In some of the simpler molecules with a very small number of atoms the equilibrium positions of the atoms may be found by means of the equations of the field, as las been done in several cases; but this method very soon becomes impracticable because it requires the solution of simultaneous equations of a high order with two or more unknown quantities necessitating approximation to roots by trial, a very laborious process. Were it not possible to get approxi- mate results in any other way little progress could be made. By means of graphical charts of the fields of force between two atoms, as measured by one of them, it has proved to be possible to obtain approximate solutions for a sufficiently large number cf atoms to ascertain the different systems of growth that are possible with such fields. On the theory ef probabilities it is very probable, on account of the com- plexity and peculiarity of the known phenomena of organic chemistry, that any theory which explains these things in detail must contain premises that have certain elements of the truth as to the theory of the atoms. ——— Compound Molecules with Theoretical Atoms. 615 For the study of the compounds of hydrogen and carbon with which we are now concerned three charts are required. A chart showing both the z- and the z-component forces is required for each of the three combinations, the force of one hydrogen atom upon another, the force of a carbon atom on a hydrogen atom, and the force of a carbon atom upon a carbon atom. Since the axes of all atoms in the molecule are parallel, the components of the force in two directions only are required, the z-axis coinciding in direction with the axis of rotation of every atom, and the w-axis coinciding in direction with the equatorial planes of the atoms. The making of these charts is a laborious process. It was not known at first to what distances from the atom to extend them. At certain distances they are used much more than at others. In speaking of distances we shall use the small a,. unit explained in former papers, which may be converted into centimetres by multiplying by *207 x 10-”. PI.VILI. fig. 1 shows both the v-and the z-component forces when one hydrogen atom acts upon another. Pl. IX. fig. 2 shows them when a carbon atom acts upon a hydrogen atom, or when a hydrogen atom acts upon a carbon atom. The chart for the case of a carbon atom acting upon a carbon atom is not given, because at the small distances concerned in most of the molecules this force may be found by using the chart for the action of carbon on hydrogen. ‘The field is the same in character, 2nd the force may be obtained with sufficient accuracy up to distances of 400 or 500 by multiplying the carbon-hydrogen force by the constant 378. This constant is the sum of the squares of the radii of the orbits of the twelve electrons in the carbon atom, as determined in a former paper. In explanation of the charts, one atom is supposed to be located at the origin in the lower left corner with the direction of its axis of rotation along the vertical line, the axis of 2, and another atom to be placed anywhere upon the chart having its axis parallel to the first atom. The z-force and the ¢-force between the two atoms in this position may then be read directly by the numbers en the curves passing nearest to it. If it lies between two of the curves the force may be estimated by proportional parts. If the second atom is moved along one of the curves repre- senting the x-force the force is constant along this curve, that is the curves are lines of equal force, the one set for v and the otlier for z. In fig. 1 there are two curves marked zero, the upper curve being that where the ¢-force is zero, and the lower * Phil. Mag., Aug. 1915, equation (12), p. 266. 616 Dr. A. C. Crehore on the Construction of where the x-force is zero. The systems of z-force curves are all tangent at the origin to either the «x-axis or to the curve for F,=0. The force changes sign in crossing the curve of zero force, being negative above and positive below this curve. A positive z-force means that the second atom urges the atom at the origin upward in the positive direction. If the second hydrogen atom is on the z-axis above the origin where the z-force is negative the atoms repel each other. The zero curve for z beyond the limits of the chart forms a loop coming around and intersecting the z-axis at a distance of 1682 units. Here the force changes sign from a repulsion to an attraction, and there is a position of stable equilibrium for the two atoms. The a-force is zero along this z-axis. Along the a-axis, on the other hand, the z-force is zero and the z-force always positive, indicating an attraction between two hydrogen atoms at any distance when one is on the equator of the other. The force in this direction is not zero at any distance. The rate of diminution of the force near the origin in all directions is as the inverse fourth power of the distance. Ata great distance from the origin the law changes, so that the force is inversely as the square of the distance. The character of the field between carbon and hydrogen, shown in fig. 2, is strikingly different from that of fig. 1. The must noticeable difference is that there are two angles where the z-force is zero in the first quadrant, and the only region where the z-force is positive 1s between these two zero curves. It is negative in the region near the z-axis and the ~ a-axis. The z-force on the z-axis is zero and is a maximum on the z-axis. On the other hand, the xv-force is zero on the z-axis and a maximum on the z-axis. In great contrast to fig. 1 the w-force is negative on the equator, showing a strong repulsion between hydrogen and carbon or between carbon and carbon in this situation, whereas the hydrogen attracts hydrogen. This fact has a very significant bearing upon the compound-forming processes. Another striking difference is in the rate of diminution of the forces with the distance. At small distances, which includes all distances on this chart, the force decreases as the inverse sixth power of the distance, and at great distances as the inverse fourth power of the distance. These figures show only the forces in one quadrant, but a section of the field on a much reduced scale for the four quadrants is shown in Pl. X. fig. 3 for the case of hydrogen acting on hydrogen, and in Pl. XI. fig. 4 for the case of carbon on hydrogen. The complete fields in space may Compound Molecules with Theoretical Atoms. 617 be represented by the surfaces of revolution obtained by revolving the whole figure about the z-axis. The Forms of Compound Molecules. In giving an account of the results obtained by the use of these charts in forming molecules, much of the interesting detail will have to be omitted on account of its length; and we will give chiefly the resulting forms of stable molecules without specifying the various forces acting upon each atom. These are interesting as showing which atoms are most responsible for the equilibrium of any one atom. In all the systems it is noticeable that the hydrogen atoms perform the function of binder to hold the system together, and the carbon atoms tend to separate from each other. They are prevented from so doing by the hydrogen atoms when properly located. It was shown in a former paper that two atoms of hydrogen may unite to form a molecule, the axes of rotation of each being in the same line, and the distance between the atoms 1682a, units. The chart Pl. VIII. fig. 1 shows that three atoms of hydrogen may unite into a molecule as shown in Pl. XII. fig. 5, where the atoms H, and H; have a common axis, the distance between them being 860. The third hydrogen atom is situated at equal distances of 794 from the other two, or at a distance of 669 from the mid-point between them, making an angle of 32° 42! for the latitude of H, with respect to H,. In this position the «-force of H, on H, is zero, because this point lies on the zero curve for the z-force in fig. 1. ‘The same remark applies also to the effect of H; on H,. ‘The z-force of H, on H, evidently exactly opposes that of H; on H,, making the resultant zero. Each Phil. Mag. te Vili. p. 357 (1904). ? Phil. Mag. [5] xliii. p. 241 (1897). § Phil. Mag. [6] viii. p. 610 (1904). 3 Proc. Camb. Phil. Soc. x. p. 10 (1897). case of hydrogen the net gain is a much larger proportion of the whole ionization than in the case of air, so that it seems probable that the variations in the above table are due not merely to the different types of rays employed, but, to a much greater extent, to the type of ionization chamber used in the investigation. Crowther * made a more systematic series of measurements, : using rays of various penetrating powers. His results are | given in Table II. TABLE II. Equivalent Spark- Ionization of Hy _ gap In mm. Ionization of Air 8 0-010 12 0-013 if 0:021 16 0-068 18 0107 20 0-135 24 0:152 28 0180 * Proc. Roy. Soc. Ixxxii. p. 103 (1909). 646 Mr. G. Shearer on the In this work Crowther measured the ionization produced first in hydrogen and then in air at the same pressure, and calculated the ratio directly. He took precautions to prevent the rays from striking the electrodes so that no electrons would be emitted by ~ them, ‘but there must have been a certain gain of electrons at the ends of the chamber. The total effect of these electrons would increase with an increase in penetrating power of the exciting beam, and we should thus be led to ae that vane an increase in, ati account for the bee rise in Ass ionization between rays corresponding to spark-gaps of 16 and 20mm. The form of the curve which represents the variation of ionization with hardness of the rays, suggests that there was present in the gas some impurity whose characteristic radiation was excited by rays corresponding to a spark-gap greater than 16 mm. The latest, and probably the most reliable, work on this subject is that of Beatty *, who measured the relative ioniza- tion of hydrogen for various types of X-rays. In this work he employed homogeneous X-rays characteristic of Fe, Cx, Zn, As, and Sn, and used a method which entirely eliminated the effect of the electrons emitted from the walls or the electrodes. The method consisted in obtaining pressure- ionization curves for air and hydrogen. If there were neither a gain nor a loss of electrons at the ends of the chamber the curves would be straight lines, and the ratio of their gradients would give the relative ionization by X-raysT. The effect of the loss or gain of electrons at the ends of the chamber, although important at lower pressures, does not vary with the pressure after that point at which there is enough gas present to absorb completely the energy of the electrons. The admission of more gas is not accompanied by any extra ionization due to these electrons, and any addi- tional ionization observed must in the absence of penetrating radiations from the ends be due to processes taking place * Phil. Mag. [6] xx. p. 320 (1910). + By “ionization by X-rays” is here meant the ionization produced. directly or by easily absorbed secondary radiations from the gas itself. It is in great part due to the high-speed electrons emitted from the gas under the action of the X-rays; some ionization may be the result of the action on the gas of soft characteristic X-rays excited in the gas by the original beam of X-rays, these, of course, producing their effect through the agency of the slowly moving electrons which they eject. Tonization of Hydrogen by X-Rays. 647 in the gas itself. Thus, by measuring the slopes of the two curves after they have become straight lines, Beatty obtained a result which gave the relative ionization of hydrogen and air. His results are given in Table ITI. Tase II]. Jonization of H, Rays used. Tonization of Air WE COGEICS IE) yon sccns. | 00571 | Cu EA Wank caine neee ! 00573 FRSA NG ATEN AI ‘00570 | As PPS CRUE CSR Sen | 00573 2S SO SRE tae | 0400 For the rays from Fe to As the value of the relative ioni- zation is constant, and is much lower than that obtained by previous workers. There is, however, a sudden increase as we pass from arsenic rays to tin rays, when the value of the ionization increases sevenfold. The most natural interpre- tation of this rise would be that there exists a radiation characteristic of hydrogen, and of such a wave-length that rays of the wave-length of arsenic rays or greater are unable to excite it, while rays of shorter wave-length such as tin ravscan. If this were true, then it would be a result of extreme importance, as no evidence of characteristic X-radia- tion from the lightest gases has been otherwise obtained, On the other hand, the rise occurs at a very significant place, for if there were present in the gas a small amount of arseniuretted hydrogen, it would produce just such an effect as was observed. We have no data as to the ionization of AsH; by X-rays, but Beatty* has measured the ionization of SeH,, and from an X-ray point of view the behaviour of the two gases should be very similar. If we plot the mass absorption coefficient, A/p in Al, for the rays used, against the ionization observed for both H, and SeH,, making the ionization scales comparable, we get two curves which prac- tically coincide, as is shown in fig. 1. The rise observed occurs in very nearly the same region of the spectrum. * Proc. Roy. Soc. lxxxv. p. 578 (1911). 648 Mr. G. Shearer on the Table IV. compares the ionization in H, with that in SeH,. When we consider that values for Selly, and not for AsH;, have been used, the agreement between the two entries in the last column indicates that Beatty’s results could be Fig. 1, lonisation 0) 20 60 Mp in in Al. TABLE IV. r H H H Rays. ai in Al. Tee (Beatty). x = (Beatty). Soke ; Fe ...... 88:5 00571 30:3) Gig os 47-7 00573 | Mean | 29-2 ees YANGON: 39°4 00570 | "00572 r 20-0 ‘00019 Wavecss: 22:5 00573 / Seikie 189 30°6 } Sp res 16 ‘0400 250 ‘00016 explained completely by the presence of a small quantity of arseniuretted hydrogen. The amount of the impurity would require to be about one part in 6000. It would seem, then, that the sudden rise in the relative ionization of hydrogen between arsenic rays and tin rays was more probably due to the presence of a small quantity of AsHs, and this is all the more likely when we consider that the hydrogen used by Beatty was prepared from zine and sulphuric acid, in both of which arsenic is a common impurity which it is difficult to remove. In the preparation of his hydrogen Beatty took precautions to obtain the gas in a pure state, but beyond passing the gas through liquid air, took no special steps to rn Pe Lonization of Hydrogen by X-Rays. 649 avoid AsH;. In view of the difficulty of obtaining these chemicals in an arsenic free condition, it is quite probable that the sudden rise in ionization observed by Beatty was due to the presence of AsH;. It is possible, also, that the steep rise in Crowther’s results was due in part at least to some such impurity. Method and Apparatus. The method adopted in this investigation was, generally speaking, that used by Beatty. The manner in which the ionization in the chamber varied with the pressure was observed, and from the curves obtained for hydrogen and air the relative ionizations of the two gases were deduced. The ionization pressure curves obtained are of the form I. in fig. 2, and it is seen that only beyond a certain pressure does Fig. 2. T . oe I c etlwcce ew we eee og owe = ni pt) = wo ‘i 2 Pressure the curve becomea straight line. At this pressure the amount of gas present was sufficient to absorb completely all the electrons emitted by the walls and electrodes The admission of more gas was not accompanied by an increase of ioniza- tion due to end electrons, so that any increase in ionization observed beyond this point must be due to the action, direct or indirect, of the X-rays on the gas itself. Curve II. shows the ionization pressure curve for the electrons from the walls, or rather for the gain of electrons resulting from the inter- change of electrons between the walls and the gas. Curve III. gives the ionization produced by the action of the X-rays on the gas. This last curve is a straight line, and the slopes of 650 Mr. G. Shearer on the lines obtained in this way measure the relative ionizations of the gases in the chamber. Beatty observed the variation of ionization for pressures up to one atmosphere ; in these experiments, however, pre- liminary observations ‘showed that the ionization pressure. curves did not become straight until a pressure over half an atmosphere was reached ; it was, therefore, finally decided to observe from one to two atmospheres. A much longer range of pressures from which to estimate the gradient was thus obtained, and by this means the accuracy possible was increased. In addition, any slight leak of the apparatus at pressures less than atmospheric meant that air was entering the chamber, and the admission of even a very small quantity of air had a very appreciable effect on the resulting measure- ments. On the other hand, by using pressures greater than atmospheric, a small leak was of much smaller importance, as the tendency was for hydrogen to escape and not for air to enter. Since the ionization of hydrogen is very small, an ordinary electroscope was too insensitive for its measurement, and in- stead a Dolezalek electrometer was used with a sensitiveness of about 700 divisions to the volt. When using the electro- meter for measurements of the ionization of air, the voltage in the needle was reduced so as to give a convenient rate of leak, and the sensitiveness of the electrometer was compared with that under the previous conditions in order to make the readings of the two ionizations comparable. Fig. 3 gives a diagram of the apparatus used. The ioni- zation chamber A was a brass cylinder of length 12 em. and diameter 10 em., with a circular window 5 ecm. in diameter. The chamber was lined with thick aluminium to absorb the characteristic X-radiations from the brass walls, while the corpuscular emission was reduced and the soft secondary radiation from the aluminium absorbed by covering the whole of the inside with filter-paper. The window through which the rays entered was at first of thin aluminium, but later of mica, thin mica being found more capable of withstanding the effect of the difference of pressure on its sides than thin aluminium. As it had to stand pressures in both directions the mica window was screwed down between two brass rings, the whole being made gas-tight by the use of wax.’ An electrode of fine aluminium wire entered the chamber through an ebonite plug provided with an earthed guard-ring, while wires in earthed brass tubes connected the electrode with the electrometer. In order to allow for variations in the radiation from the bulb a standardizing Lonization of Hydrogen by X-Rays. 651 ionization chamber B was also used. It was situated so as to receive some of the rays from the radiator, while its electrode was connected by shielded wires to an ordinary Fig. 3. To Efectroscope, To Pressure Gauge. \ To Hydrogen Supply. Ay? aa Radiator jw ae electroscope placed for convenience beside the scale of the electrometer, ‘'he X-ray bulb was placed in a lead-lined box, and the rays emerged in a vertical direction through an aperture in the top of the box and fell on a radiator inclined at about 45° to the vertical. Characteristic X-rays from this radiator passed into the two ionization chambers. In order to be able to increase the pressure, another vessel C was connected to the ionization chamber, and at the start the two were filled with hydrogen from the generating apparatus. After being filled the chamber was exhausted by means of an air-pump, this process being repeated several 652 Mr. G. Shearer on the times to ensure that the air had been completely removed. In order to raise the pressure, inercury was introduced into this second chamber from a reservoir, thus displacing the gas into the ionization chamber and increasing the pressure, the value of which was measured by the pressure gauge. Later on in the course of the experiments, when the hydrogen was being prepared electrolytically, another and more convenient method of increasing the pressure was used. The voltameter used consisted of three glass tubes with platinum electrodes, the two outer ones being anodes while the central one was the cathode at which the hydrogen was produced. These were provided with stopcocks, and the central one was connected directly to the ionization chamber. When it was desired to raise the pressure, the two external tubes were closed and the current turned on, the hydrogen produced passed into the ionization chamber and the pressure increased. It was of course necessary from time to time to allow some of the oxygen which collected in the other tubes to escape. This method of varying the pressure was found to work very well, and it had the advantage that there was no possibility of any air entering along with the hydrogen, whereas in the other method there was always the danger of bubbles being carried in with the mercury. Preparation of Hydrogen. In the first series the hydrogen was prepared in a Kipp generator by the use of zine and sulphuric acid. The chemicals used were the purest obtainable and were supplied by various firms, those used first being Merck’s or Kahlbaum’s, while later some products of English firms were used. The water used was special “‘ conductivity ” water, and was kindly supplied to me by the Chemistry Department. In order to get the action to start, a little pure copper sulphate was introduced into the Kipp. The gas produced was passed through soda-lime and then through acidified potassium permanganate, the latter being inserted to remove any arsenic. Subsequently it passed through a long tube with cotton-wool, up a calcium-chloride drying-tower, finally entering the chamber through a tube containing phosphorus pentoxide. In the later experiments the hydrogen was prepared by the electrolysis of dilute pure sulphuric acid, the water employed being again “conductivity” water. The gas entered the chamber through calcium chloride and phos- phorus pentoxide drying-tubes. The hydrogen was subjected on two occasions to Marsh’s Lonization of Hydrogen by X-Rays. 653 mirror test for arsenic, the results showing that the amount of arsenic present was certainly very small. Results. In fig. 4 are shown two typical curves for hydrogen and air. The observed points were found in the case of hydrogen lonisation 310 2-0 70 90 10 Pressure (Cm) to lie within the limits of experimental error on straight lines. In the case of air it was necessary to correct for the absorption by the air of the rays in passing through the chamber; after this correction had been applied the curves for air became straight lines over the range of pressures employed. Since air absorbs the rays, especially the soft copper rays, much more than hydrogen, the mean intensity of the beam traversing the chamber in the two cases was different. From a knowledge of the absorption coefficient of air for the rays used it is easy to correct for this absorption. If I, is the intensity of the beam just after entering the chamber, and if dis the length of the chamber, and X the absorption coefficient for the gas, then the mean value of the intensity throughout the chamber d = : ( I, e~** dx 20 654 Mr. G. Shearer on the In the case of hydrogen, as the absorption is negligible, the mean value is Ip. If ¢ be the measured ionization, 2’ the corrected ionization for a beam of mean intensity Io, then 4! rAd pr (Serr or wee 2 AO ee ee The value of » for air at N.T.P. has been measured by Barkla and Collier*, and the value of the absorption co- efficient at any pressure p cm. and any temperature T° Cent. is eve wena 76 * 273+T Hence we can correct the curves obtained in the case of air and obtain a value for the relative ionizations of air and hydrogen under the same conditions. Avs TABLE V. Tonization of H, Relative Ionization by Cu Rays Tonization of Air Relative Ionization by Sn Rays — Cu Rays. Sn Rays. "0008 (?) ‘0063 “0010 0040 0024 “A 0016 aie: 0010 ae 0016 ae ue "0026 ‘0017 ‘0069 0:28 ‘0014 00380 0-47 0025 | -0046 0:54 70025—| a 0019. | 0036 0°53 0031 | 0029 1:07 0023 0034 0°68 0023 "0039 OS See ‘0009 ae kat : ‘00138 4s 0031 : 0033 ee 0036 0009 Mean="0018 Mean=:0035 Mean=0°59 The rays used were those characteristic of copper and tin, and the results obtained are given in Table V. On some * Phil. Mag. June 1912. Tonization of Hydrogen by X-Rays. 659 occasions one radiator alone was used for the series, while on others both the radiators were used, and thus the ioniza- tions were measured for the two types of rays for the same sample of gas. For these cases the ratio of the two is given in the third column of the table. Discussion of Results. It will be seen that the ionization of hydrogen relative to that of air is very small. It is also evident that the several determinations vary considerably among themselves, much more than can be accounted for by observational errors, or by slight variations in the ionizing radiation. This indicates that the gas itself is the cause of the greater part of the variation, and this is hardly to be wondered at when we remember that the effect of even avery small trace of im- purity is very important, and also that it is very unlikely that the same impurity would be present to the same extent in two different samples of gas. In support of this is the fact that when the gas was prepared electrolytically the ionizations observed were lower. Also the preparation of hydrogen in any quantity free from impurities is a task of such difficulty that we cannot perhaps expect to settle ex- perimentally whether this residual effect is really due to the gas itself or merely to the presence of foreign elements. The important part played by impurities is shown by the fact that about one part in 40,000 of a heavy substance such as AsH; would be sufficient to account for the observed ionization. In view of these facts, we are led to the conclusion that the values given in the above table are upper limits to the value of the tonization of hydrogen relative to that of air. Even if we take the mean value in the above experiments, we obtain an estimate which is smaller than that given by Beatty, while in the light of the above considerations it is clear that the mean value is an overestimate of the relative ionization. In the case of copper rays several values have been obtained which average about ‘001, while one measure- ment was also made which gave *0003 as the result; on this determination, however, too much reliance is not to be placed. In the case of tin rays, the lowest reliable estimate was 0009, a value which agrees with the lowest determination for copper rays. If, in order to get a measure of the ioniza- tion of hydrogen relative to air for copper and tin rays, we take the mean of the three lowest values obtained (excepting the value ‘0003 in the case of copper as not sufficiently 656 Mr. G. Shearer on the reliable) we get, as values for the upper limit of the ionization of hydrogen by these rays, Ionization of isi Ionization of Air (Upper limit). Cu Rays. | Sn Rays. 0010 | -0016 These values are much lower than any hitherto published, and there is, in addition, a very strong probability that they are overestimates. The lowest reliable values for copper and tin rays are practically equal, although the average for all the determi- nations for copper is smaller than that for tin. Also, although on each occasion except one when the same sample of gas was used, the value obtained for tin rays was larger than that for copper rays, we are not entitled to assume that this is a true effect in hydrogen. On one occasion the opposite was the case, while the two values were within 7 per cent. of each other. This apparent increase of relative ionization with decrease of wave-length is much more pro- bably due to impurities in the gas, these impurities being of greater importance in the case of tin rays than in the case of the softer copper rays. Beatty obtained in the interval between Cu rays and Sn rays a sevenfold increase in the ionization, while Crowther also obtained a sudden rise for rays whose wave-length was probably between that of copper and that of tin. In view of the above experiments, it seems that these effects were spurious, and there ts no conclusive evidence that the relative ionization of hydrogen increases with a decrease in wave-length. All the evidence goes to suggest that if such an increase exists it is small. In an attempt to seek out the various sources of impurity, various chemicals were used, but on the whole the results obtained were always of the same order. Also at a certain stage all the rubber tubing connecting the various parts of the apparatus was removed, and metal or glass substituted without producing any change in the magnitude of the quantities obtained. Theoretical Considerations. The fact that hydrogen seems practically incapable of ionization by the action of X-rays raises interesting questions lonization of Hydrogen by X-Rays. 657 from the theoretical point of view. The most natural sug- gestion might be that since the hydrogen atom consists of only one electron and a nucleus, these “might be so tightly joined together that the separation might be difficult, and X-rays might not have the necessary energy. This is, however, negatived by observations on the ionization of hydrogen by electrons, for it has been shown that the electrons emitted by X-rays can ionize hydrogen, so that it cannot be any lack of energy in the X-rays which makes the ionization difficult. Also observations on the ionization potential of the various gases do not place hydrogen in the place which the above results might suggest, for the value of 11 volts obtained by Thomson from observations on posi- tive rays is smaller than the ionization potential of many gases which are readily ionized. The only data as to the absorption of hydrogen by X-rays are contained in a paper by Gowdy*, who measured the absorption of Fe X-rays by hydrogen, and obtained the value ‘0003 for >A. This value is about gy of that of air, so that if ionization is a measure of absorption the ionization of hydrogen for Ke rays should only be about J, that of air. This result is certainly not supported by the work described in this paper, but Gowdy states that his determination in the case of hydrogen is only approximate. He was unable to determine the absorption coefficient for Cu rays, and it would seem that the value for Fe rays is a very large over- estimate. The work of Bohrf on the hydrogen spectrum suggests that the greatest frequency possible is 3°26x 10, which corresponds to the extreme ultra-violet, and is about ¢}, of the frequency of copper X-rays (which has a value 1°94 x 10") and abont soy, of the frequency of tin rays. On Bohr’s theory we should not expect to find a radiation for hydrogen of shorter wave-length than this line in the extreme ultra-violet, so that it is unlikely that there would be a characteristic radiation of wave- length intermediate between that of arsenic and that of tin. A gain, the impro- bability of such a radiation is strengthened by the fact that if we extrapolate from experimental results on the K-radia- tions of the elements, we find that the K-radiation of hydrogen would have a wave-length of the order of magnitude of that which Bohr’s theory suggests. Also, if we assume “that the energy absorbed when an * Journal de Physique, ser. 5, p. 622 (1913). + Phil. Mag. [6] xxvi. p. 1 (1913). Phil. Mag. 8. 6. Vol. 30. No. 178. Oct. 1915. 2U 658 On the Ionization of Hydrogen by X-Rays. atom is just ionized is re-emitted as a characteristic radiation when the displaced electron returns*, we can calculate what the characteristic radiation of hydrogen is from the observed value of its ionization potential. For, there being only one electron in the hydrogen atom, the eledizan | ahiehie dis- placed in the experiments which measure the ionization potential must be the same as that which takes part in the emission of any characteristic radiation. Thus if n is its frequency, then hn= Ve, where V is the ionization potential and h Planck’s constant. Substituting the values of h, V, ¢ tor hydrogen, we obtain n= or TO, a value which agrees with that obtained from other lines of reasoning. ‘here is thus no other evidence in favour of characteristic radiation near to that of tin, and the natural conclusion, especially in view of the above experiments, would be that it is really non-existent. Looked at from this stand- point, the position of hydrogen is not so anomalous as at first sight it appears; for it is an experimentally observed fact that as the difference between the wave-length of the exciting radiation and that of the characteristic radiation which is being excited increases, the smaller is the observed ionization. The characteristic radiation of copper has a wave-length goo and of tin soq of the value of that of hydrogen. The characteristic X-radiations of air, on the other hand, will probably have wave-lengths about 50 times that of tin, and are thus much nearer the radiations of copper and tin than that of hydrogen. ‘This, together with the fact that hydrogen contains only one electron per atom, seems capable of explaining, in terms of discovered empirical laws, the small observed value of the ionization ‘of hydrogen relative to that of air. In conclusion, I have much pleasure in thanking Professor C. G. Barkla, E.RS., who suggested the research, for his kind interest and help throughout the work. (Part of the expense of the apparatus used in the above experiments was met by a grant from the Carnegie Trust.) The Physical Laboratory, University of Edinburgh. * Barkla, ‘Nature,’ March 4, 1916. Pi 659" (J -LXVIL. Ahutual Electromagnetic Mass. To the Editors of the Philosophical Magazine. GENTLEMEN,— i the September number of the Philosophical Magazine> Dr. L. Silberstein draws attention to the problem of the mutual mass of two electrical charges placed close together. In a recent paper of my own, ‘ Electromagnetic Inertia and Atomic Weight,” I had overlooked the fact that Dr. Silberstein had already calculated formule giving this mutual mass precisely, and I had obtained an approximate formula for one of the special cases. While very willing to concede all priority to Dr. Silberstein in this connexion, perhaps I may be per- mitted to make one or two remarks about my paper. Its object was not the calculation of mutual mass, but the application of the idea to some points of nuclear structure in the atom, as in fact its title indicates. Being unaware of Dr. Silberstein’s calculation, 1 made an approximate one myself, by very obvious and simple mathematics. This approximation was all that I required, and its deduction forms an inconsiderable part of my paper. A numerical error in a coefficient, due, as I find from my manuscripts, to the transposition at one stage of a factor 2 in a fraction, does not affect my reasoning, which only dealt with orders of magnitude. I had no necessity for the exact formule in the applications I was making. It would be interesting to trace the first appreciation of the existence of mutual mass,—the term employed both by Dr. Silberstein and myself. I am sure it is not so recent as Dr. Silberstein’s note seems to imply. For example, many years ago Dr. Oliver Heaviside determined approximately the effect of an external field on the mass of a charged par- ticle,—the calculation being reproduced in his collected papers, —anid evidently at that time fully appreciated the mutual effect of two charged particles. But he did not, however, make the precise calculation which Dr. Silberstein has since given us. Believe me, Gentlemen, Yours very truly, J. W. NicHotson. King’s College, Strand, W.C. P6009 LXVIL. A Comparison of Radium Standard Solutions. By J. Moray, B.Sc., McGill University *. Section 1. History.—(a) Solid Standards. YHE fundamental Radium Standard of the present time is the International Radium Standard at Sevres, France. It was prepared by Mme. Curie in 1912, on the recom- mendation of the Congress of Radiology and Electricity which met at Brussels in 19107. It consists of 21°99 milli- grams of pure radium chloride, sealed up in a thin glass tube. This standard, before being accepted, was compared with three other purified amounts of radium chloride, pre- pared by Honigschmidt for atomic weight determinations. These all agreed with Mme Curie’s standard to within one part in three hundred. One of Hoénigsclimidt’s preparations has been preserved at the Radium Institute of Vienna, as a secondary standard, and termed the Vienna Standard. Before this, however, in 1903, the first radium standard was prepared at McGill University, Montreal, and termed the Rutherford-Boltwood Standard. A quantity of pure radium bromide was bought from Dr. Giesel of Germany, and generously presented to McGill University by Sir William Macdonald. Of this amount, 3°69 milligrams were weighed out and sealed up in a tube by Professor Eve and Dr. “Levin, and thereafter constituted a primary laboratory standard. Jt is now at Manchester University , England. Various secondary national standards exist, examples of which are the English standard at the National Physical Laboratory and the Washington standard in U.S.A. ‘These have all been accurately compared with the International and Vienna standards. After the preparation of the International standard, the Rutherford-Boltwood standard was carefully compared with it, and also with the Vienna standard. It was compared indirectly with these by means of the secondary standard at the National Physical Laboratory. These investigations showed that the Rutherford-Boltwood standard consisted of 3°O1 instead of 3°69 milligrams of radium bromide. It is therefore 4:9 per cent. low on the International. It is known that radium bromide on exposure to air gives up bromine, and changes over to the rarbonate, while water of crystal- lization is also formed. This is suggested by Sir Ernest Rutherford as an explanation of the increase of weight. (b) Solution Standards. For laboratory purposes a solution standard is prepared. * Communicated by Prof. A. S. Eve, D.Sc. + “Radioactive Substances and their Radiations,” 1913 ed. By Sir Ernest Rutherford. See also Phil. Mag. Sept. 1914 (Sir E. Rutherford). A Comparison of Radium Standard Solutions. 661 This is obtained by comparing a small quantity of the radium salt with a solid standard by the y-ray method. It is then dis- solved in distilled water with a little HCI, to keep it in solution. Such a solution, but with no acid added, was made up by Eve at about the same time as the preparation of the Rutherford-Boltwood solid standard, using about one quarter of a milligram of radium br omide. Determinations by Boltwood a few years later at New Haven with some of this solution led to results which conflicted with those obtained by Eve in similar work, where the solid standard was used. It was shown by the latier that the original solution had weakened by the deposit of radium on the walls of the flask, as no hydrochloric acid had been used in the preparation. Two new solutions were then prepared by Boltwood from a known amount of radium bromide, determined by Eve by the y-ray method. These solutions were of strength in the ratio of 100:1, one containing 1°57 x 107‘, and the other 1°57 x 10~° gram of radium perc.c. This time a little HCl was added as a precaution, to keep the radium in solution. Since the Rutherford-Boltwood solid standard is known to be 4°9 per cent. low on the International, and the Ruther- ford-Boltwood solution standards were compared with the former, considerable importance attaches to finding whether its accepted value may not also be in error, and to what extent. The more so, since determinations by Boltwood on the relative amounts of uranium and radium in rocks, and results obtained by Hive and others for the amount of eman- ation in air, and also of radium in rocks and water, are based upon these solution standards at their present accepted value. The investigation was carried out by comparing the weak solution of the Rutherford-Boltwood standard with a solution of the Washington standard. A litre of the latter was obtained by Hve in Sept. 1914, from Satterly of Toronto University, containing 9°15x107!! gram of the radium element per c.c. It was certified as follows :— “€100 c.c. of acid solution of the Washington Standard of strength 12:2 x 107° om. of radium pere.e. 10 c.c. of this were diluted to one litre. 250 c.c. of this were removed, and distilled water added to the remaining 750 ¢.c. to make up one litre, the strength of which was now three-fourths the strength of the previous litre, or 9°15x10-" om. of radium perc.c. Radium contents declared right with the International Standard to an accuracy of at least one-third per cent. Density of solvent HCl 1:08. Combined errors searcely one per cent.” Experiments were carried out by the writer with the strong and the weak solutions of the Rutherford-Boltwood standard. These served to compare the Rutherford-Boltwood standard 662 Mr. J. Moran on a Comparison of solution with the Washington, and thus also indirectly with the International. In addition they served to show if the strong and the weak solutions had remained unchanged throughout the eleven years which have elapsed since their preparation, These solutions having been prepared in the ratio of 100 : 1,if any deposits of radium had occurred during this interval the amounts deposited would probably be ina different ratio, and there would then bea discrepancy in the results, Section 2. Method. The emanation method was employed in making the above comparisons. The required volumes of radium solution were drawn off with a clean pipette, and accurately weighed. Distilled water anda little HCl were then added, and the solution was sealed up air-tight in a 500 e.c. flask, the solution occupying about half the volume of the flask. The flask was then put aside for the emanation to collect, and boiled off at intervals of about a week. The apparatus consisted of a sensitive, air-tight, gold-leaf electroscope, carefully silvered on the inside, and well earthed. It was exhausted by a water-pump, a sufficient exhaust usually being obtained in a couple of minutes. The air was deprived of moisture by phosphorus pentoxide con- tained in a U-tube. The gold-leaves were protected from a sudden inrush of air by using a capillary tube. The emanation was admitted by means of a three-way tap, and alr allowed to enter afterwards until the pressure inside the electroscope was atmospheric. The solutions were well boiled in order to drive off all the emanation, which was collected with the air over water in a bell-jar at room-temperature. Care was taken through- out to minimise possible errors, by having the solutions of nearly equal strength, by boiling each the same length of time, and by observing the maximum value of the ionization current the same interval of time after passing in the emanation. In the earlier part of the work, readings were taken every few minutes, and the growth of the ionization current traced. The practical maximum was reached in about five hours. Here the ionization current appeared to be constant, as the time-measurement of the movement of the gold-leaf showed a slightly oscillating value of less than one per cent., which was thus the probable error in taking the time of a ‘single reading. The theory of the work is well known. The natural leak was taken before introducing the emanation, and deducted from the value obtained when the ionization current had reached a maximum. This value was corrected for atmo- spheric pressure, reducing all results to standard pressure. a f q ; 4 g Radium Standard Solutions. 663 Finally, the number of divisions per minute for 10-° curie of radium emanation could be calculated by using the value of 1—e-* directly from the tables given by Rutherford, interpolating for a fraction of a day: this value we call a “fioure of merit.” By comparing the “figures of merit,” and finally reducing to percentages, we can obtain the relative values of the two standards. Section 3. Difficulties and their Solution.— Results. A number of sets of results were obtained, and practically all showed a very close agreement between the two solutions compared. As the work proceeded, a deterioration in values of the “ figure of merit ” occurred, as if boiling had a weak- ening effect. Hach value was less than the preceding one. This suggested that some of the radium might have been deposited with boiling, and so a qualitative experiment was carried out with a solution which had been boiled a number of times, by adding some HCl to see if it would dissolve the supposed radium deposit, and hence cause a rise in the value of the “figure of merit.” This is precisely what occurred. A higher value, however, was now obtained than any pre- vious one. This suggested the probability of radioactive matter in the HCl which had been added. To test this, a “blank” solution was prepared, consisting of 50 c.c. of approximately 19 per cent. strength HCl, obtained by dilu- ting 38 per cent. commercial HCl to half strength. After eight days it was tested, and on deducting the natural leak, which was *083, it gave 2°71 divisions per minute. Two c.c. of this same preparation had been added previously to the Rutherford solution standard, which had been boiled fourteen times. A reading was taken previous to adding the HCl, and it showed the steady decline due to boiling. On next boiling, however, a rise of 12-2 per cent. occurred in the “figure of merit.” By calculation, 2 per cent. of this was due to the radioactive matter in the acid which was added : the remainder must therefore be due to the re-dissolved radium, which had been deposited by boiling or otherwise rendered “ de-emanating.” ‘This quantitative experiment amplifies and confirms the behaviour of the solution in the qualitative experiment mentioned in the first part of this section. In the first trial when acid was added, the “ figure of merit’ went above the normal, due obviously to an excess of acid added, which had sufficient radioactive matter in it to account for the abnormal rise. In the second case, it did not reach the normal value when 2 per cent. was deducted for the radioactive matter in the acid added. Apparently all the radium had not been re-dissolved in the second case. These sets of experiments also established the fact that a 664 A Comparison of Radium Standard Solutions. radium bromide solution deteriorates with boiling, and that because of this, too great reliance cannot be placed on the result of a single test. Every solution tested showed a steady decline, even after adding the acid. In all, about 70 tests were made. A similar solution was prepared and used by Dr. Eve in Nov. 1908. This was also tested, and showed only 45 per cent. of the strength obtained for the Washington standard. Itthus appears that such solutions, which have been boiled a number of times and then allowed to stand very long, are not reliable, and a new preparation should be used for each set of experiments. This work showed that the weak solution of the Rutherford- Boltwood standard is to the Washington standard solution in the ratio of 98: 100. The results obtained in the comparison of the strong and the weak solutions of the Rutherford- Boltwood standard showed the strong 2 per cent. lower than the weak, or strong: weak ::98:100. This may be con- sidered fair agreement, and indicates but slight deterioration with time. We thus have Rutherford-Boltwood, strong solution: Washington = 96: 100 Rutherford-Boltwood, weak solution : Washington= 98: 100 A more complete set of experiments will be carried out later, and the exact effect of successive boilings further determined. In closing, the writer has much pleasure in expressing his appreciation for suggestions given by Professor Eve, and also by Dr. McIntosh, in the course of this work. Summary. 1. The object of this work was to determine the accurate value of the Rutherford-Boltwood standard solution at MeGill University, by comparison with the Washington standard solution. The former was compared initially with the Rutherford-Boltwood solid standard, which is known to be 4-9 per cent. too low on the International. If no errors were made in their preparation, we should expect the solu- tion standards to come out similarly. This work shows fair agreement—about 3 per cent. low, whereas the solid standard is 4°9 per cent. low. 2. The strong and the weak solutions of the Rutherford- Boltwood standard, prepared in the ratio of 100 : 1, turn out to be in fairly good agreement—probably within 2 per cent. This shows that no serious deterioration has occurred with time. 3. It has been found that a radium bromide solution deteriorates with each boiling. Also that if a sufficient amount of hydrochloric acid, free from radioactive matter, is added, the solution will recover its normal emanating power. The cause of this has not yet been ascertained. Montreal, May 22, 1915. Meyer & Woop. Phil. Mag. Ser. 6, Vol. 30, Pl. VI. Fig. 1.—Commercial Fie. 2.—Purified nitrogen. nitrogen. First type of spectrum. Fie. 3.—Puritied nitrogen. Fie. 4.—Purified nitrogen, Second type of spectrum. Jet-chamber open. Fre. 5.—Stream of dried air. Source of moisture 1n Ira. 6.—lodine vapour jet-chambez, in nitrogen. Dh OO ORION MARR Ri tree Ge bor Fie, 7.—Iodine vapour in nitrogen by Littrow Fig. 8.—Transmission spectrograph, with iron comparison spectrum. through fluorite. I'ra, 9.—Refraction by quartz prism. i‘ a 80 . Ser. 6, Vol. 30, Pl. VII. 40 ZN) ee = fas} aN > = > wy 20 - Ww 20 BS i = eS oe P4 4 & Zero 9 Zero ss . an - rH u : Hl Ss S -20 as = ~20 K ° : . E is 7% = -40 Pe 40 we fe e > BS As ~~ -~60 -100 -100 H @82G APRS g| Ses paces bones FERED Exacs AROSE Deseaceses eocecceeea Eel a _ 0 200 400 600 800 1000 1200 1400 10. 20 30 40 50 60 70 80 MAGNETIC FIELD . VOLTAGE. ON POLONIUM (POS.) (GAUSS.) $ (E.S.U,) CURRENT TO ZINC x 10 4 McLennan & Foun. Phil. Mag. Ser. 6, Vol. 80, Pl, VII. Fig. 2. Fie. 3. = ft) eo E : 60 - : 60 + i 40 40 : ' > = Si Q Zero : x Peet & : r + Se aa aa : Z ze t + He ea : =F = = = -a0 AH : & s = .) + ff : =; ; : = r) 200 1400 VOLTAGE ON POLONIUM ( POS.) Fie. 4, ; == 90 Ft ee feo Beate: Hf saa 80 = rae preted 160 eeeageas tees ie H seasosesassszeas: 60 ; : 140 + 40 east Catt + Ss iH 120 : G EEE wi 20 H a eae ry W100 -- —-- g fie papa ) Hie x ZerofH 4 Ht ly : + ‘S$ » + = > 80 - ~ 1 feet iS) N 4 + Et x 9-20 HH K t I = 3 60 K w et s : Ht « ‘ 2 = g-40 SHeeate Sra niet 2 Ee 3 see gece : oo = : : fs : -~60 ereuararet SHEE : H 20 fa (o} fs : -8i : Hike Ht HH aye He ‘ fun Ha = : 100} = pe. ee : HH it [oy 60 60 400 600 1200 ) 40 60 400 800 1200 1600 VOLTAGE ON POLONIUM MAGNETIC FIELD VOLTAGE ON POLONIUM MAGNETIC FIELD Fra. 6. 180 At T : ; Ht Eeeeeeeeeet i otf Sk tt PEPE > i + GROWTH OF DELTA RAY EFFECT 160 Pee Peet af rH 160 a Hr Feet : ere 4 : rete He treet dees eteed pe eeeeeeee jaaaa ae i Rt sas ceeees! + Ht ; Se i HH in : i 140 : f 140 : a : H i i isiseey| : HE inenaes iibses HH i fata HH = i : HH : 120 + 120 inoue f : i oe : H : i H ath : : ett estesenigii aosat nezenenasetoee = = ioaugeatatiast gaze + : Ht 5 H+ a TES 100 : G 100 ~ : EC : 4 : wi : Hy Ee = Hn: o faa a 80 Porter 80 t it : i H x : i ot : Ht iS = ~ : : ; re 60 seesaee & 60 S ieeet isi 4 Sop S : { et Ep + ++} oO 40 seseee 40 : iH H 20 20 Zero Zero fo) 20 60 80 120 140 160 180 TIME SINCE NEW SURFACE WAS MADE (MINS.) as 60 [=to} 400 6800 1200 NOLTAGE ON POLONIUM MAGNETIC FIELD ep cm — _ ei CREHORE. Phil. Mag. Ser. 6, Vol. 30, Pl. VIII. on ce Anas rs ns a: Phil. Mag. Ser. 6, Vol. 30, Pl. VIII. Y A\\ So Oe EX = ese To convert distances into centimetres multiply by 207 x 10-2, and to conyert forces into dynes multiply the figures on th Fe 19 SO 10-581 «ee gures on the curves by 75 x = 207 x 10-™ 5 | peti i“ aE J = — Phil. Mag. Ser. 6, Vol. 30, Pl. IX. into dynes multiply the figures und to convert forces LOS = 5°31x 10-5. 0 a0 UO a4 we a = - ———— Crenone. Phil. Mag. Ser. 6, Vol. 30. Pl. EX. 7] 40 OS \ Ce Sa ae als RS H)) eee) ee B ee eT ered Matern pine en! _ ih | Crenore. Phil. Mag. Ser. 6, Vol. 30, Pl. X. Bierce TPES Fra. 3. Yo convert distances into centimetres multiply by +207 x 10-", and to convert forces into dynes multiply the figures cl 2 —20 x 10 OF FE. 10-4 = 5:31 x 10-*. é —EEE Kaz = 207 x 10" on the curves by ‘ ) = -OTXcL0G> _ x Po Aq SeA.nd oy} UO sorn’s o-Ol x T6-¢ =<1-O1 x en xX LLP er-OT 2 q q4 Inoy oy} Apdiyjnur soudp oyu seor0z 410A 07 pur ‘,, OT X 20. Aq A[dyyjnM sorqouT}UEO OUT SeOULySIP J10ATOO OF, ‘p DLT ‘AUOHAU) — CreHORE. Phil. Mag. Ser. 6, Vol. 30, Pl. XI. ao REO Vs WAY Lyd | fp), f iit Fie. 4, To convert distances into centimetres multiply by -207 x 10-™, and to convert forces into dynes multiply the x 19-2 = BIE x10™ Y 10-7 = 5:31 x 10-. figures on the curves by | é = 207 = 10- : 2 Kaz Phil. Mag. Ser. 6, Vol. 30, P]. XII. I he CREHORE. C Hy METHANE FigZ C6 He BENZENE Fig. 14. Phil. Mag. Ser. 6, Vol. 30, Pl. XII. THE LONDON, EDINBURGH, anv DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. P a’ ta ¢ “ " ) nd o> i [SIXTH SE RIES. a 32 NOVEMBER {945: LXIX. The Electron Theory in Organic Chemistry. By N. P. McCuetanp, J0.A., Pembroke College, Cambridge ; 3rd Battalion, The Queen's Own, Royal West Kent Regiment *. | ear paper has been written under conditions which have rendered it impossible to quote any references or to carry the mathematical investigation of the points examined to any complexity. It should be looked on as a preliminary investigation of some interesting points which deserve more close examination; and the author would be very glad if anyone would take tliis in hand. 1. In a previous paper (Phil. Mag. xxix. p. 192) the author endeavoured to explain the absorption of light by organic substances by means of a certain model of the atom. It was surmised that an atom consists of a positively charged nucleus round which valency electrons move in circular orbits. ‘The electrons in the outer ring were called primary, those in the next secondary, and so on. When valencies become saturated in the formation of molecules, the corresponding electrons are, it is suggested, withdrawn from their original orbits, one from each atom, probably into one of greater radius about the line joining the nuclei of the two combining atoms. These electrons, moving in the same orbit, form the link between the residues of the * Communicated by Sir J. J. Thomson, O.M., F.R.S. Phil. Mag. 8. 6. Vol. 30. No. 179. Nov. 1915. 2X 666 Mr. N. P. McCleland on the two atoms. The model adopted is therefore the one suggested by Professor Rutherford. It was further supposed that while atoms are unsaturated as to (say) one primary valency, there is still one electron in the outer ring of those revolving round that atom (though the radius of the orbit may be changed); consequently a molecule of ethylene may be represented as in fig. 1. Fig. 1. “ff vA a ory Cc ® : C y y fa Ais - 6 ‘ ‘- where a, a represent the orbits of electrons uniting C and H. 8 represents ie Y C and C. 93 Y, y represent the orbits of unsaturated electrons *, This, of course, represents only the mean position about which the system oscillates. . There are possibly other rings of electrons round C and H representing higher valencies. Since these would be nearer the central nucleus, they would be more difficult to remove than those in the outer rings, and perhaps for this reason do not enter into chemical combination. They do, however, enter into physical chemistry as residual affinity. There are some points of interest in connexion with the above which must be mentioned before passing on to the main subject of this paper, namely, an investigation whether the above model is consistent with chemical facts. In the first place, the interactions between various groups round an atom will bring the groups into definite positions. * When the conditions of equilibrium are written down it is found that there is one degree of freedom, namely, the “scale” of the system or the actual distance between two chosen atoms. Since some elements of the system are fixed, namely the electric charges, the conditions of stability are likely to put limits to this scale; and the conditions such as temperature, solvent, &c., will limit them still more; therefore pro- perties such as absorption spectra will depend on the conditions. Electron Theory in Organic Chemistry. 667 Thus, for example, in a substance of the type abcC . Cdef, there will not be free rotation about the bond joining the two carbon atoms, and the experimental evidence shows that there can be at the most two positions of stable equilibrium about such an atom, and these are mirror images of each other. In connexion with this it is important to observe that the original position of a valency electron need not affect its position when saturated. For example, nitrogen appears to have three primary valencies and two secondary ones. On reducing the primary valencies we obtain ammonia, NH. On treating this with HCl we obtain ammonium chloride, NH,Cl. It would be wrong to suppose that one of the hydrogens in this substance is united to the nitrogen in a different manner to the other three; nor can we say that on splitting off the HCl again the H last added on is the one split off. Similarly, if a quaternary salt like R;R,R3;R,NClis formed, the arrangement of the R groups round the N is independent of the order in which they are linked to it. Lastly, in the previous paper the absorption of light in the visible and nearer ultraviolet regions was attributed to the vibrations of the electrons of the type marked yin fig. 1 It is now suggested that the vibrations of the a and B type cause absorption i in the infra-red region; for itis known that these bands are characteristic of certain pairs of atoms, é.g.all substances containing a C—H linkage seem to show a very similar set of bands in this region. 2. The first point which appears to need investigation is the existence of ethylene, as contrasted with the non-existence of free methyl or methylene. Now Bohr has shown that an atom consisting of a nucleus with one electron round it is incapable of free existence in the presence of similar atoms, but that such atoms combine in pairs; in other words, the stability of the equilibrium of such a system is small*. It would appear that the fact that attempts to produce free methyl always lead to the production of ethane may be due to a similar cause. Now in the previous paper (loc. cit.) it has been shown that electrons moving in separate orbits have an eftect analogous to mutual induction of currents on one another. It is therefore to be expected that forces will arise tending to * The stability of a system in stable equilibrium may be measured by the finite impulse which is required to carry it so far from its post that it will not return to it, or, in other words, the impulse which will just carry it to the nearest ‘position of unstable equilibrium. 2X2 668 Mr. N. P. McCleland on the resist any disturbance of an electron from its orbit; in other words, a system containing two eleetrons or more in separate orbits will be more stable than a system containing only one, and the stability of ethylene is to be explained as due to this effect. It is convenient to examine what this effect is in the case of currents. 3 Let there be two circuits in which constant currents u, v flow. let them be slightly displaced a distance 6@ in the coordinate 0 during a time 6¢. Let u+ &, v+7 be the currents at time ¢. We shall then obtain the following equations:— : : 6M . LE+ Mon +RE+ 5g O(vt 0) =9, Sane 6M . M,é+Nn+8n+ 59 Aut &)=9, M, being the initial value of M. And if initially &, », &, n are all zero, i. e. if the currents are not changed impulsively, we get at the end of time o¢ 6M Nu—Mu = — 39 LN We dM Lu—Mv 30. Dae gb0 ala Cee Therefore, at the end of the motion, and before the currents die away on account of resistance, the force between the e e e e 6M e circuits, instead of being wv Soe UV oM sk tre safe NL "30 50 LN—M? \crO0 od Manure Thus in addition to the ordinary force between the circuits. a force is called in to check the motion. We will call this new force —S60, where § is essentially positive so long as LN >M?*. * The following example may make the above clearer. Imagine a shallow cup filled with liquid and a particle resting at the bottom. The position of equilibrium is independent of the viscosity of the liquid, but the stability, 7.e. the impulse necessary to carry the particle out of the cup, is increased with increase of viscosity. Electron Theory in Organic Chemistry. 669 In the case where there are several circuits (1, 2,... 1) in a field it is easy to prove that S, is given by meee, Mis....Mig.... = —| 9, 2Un sy Ww) are = 6M Miz, Lz, 0, 0 Sen a eo), aise thy f M,,, 6Mi2 ie, 0) Ls, 0 1 sq? Miz, Le...... 0 \ : i Min, 0, 0, ie aga = eat Min» 0) Sane ier la 3 L,, 2 S is therefore of the order ce \ : Now since M is always positive, if M is very small 2 is very small and {at still smaller; and since, if the coefficient of mutual induction between two vibration centres ina molecule is very small, the bands of the molecule are practically the bands of the two centres superimposed on one another, with little or no change in position, we deduce that “if two vibration centres or groups occur in a molecule in such a way as not to influence one another’s absorption bands they will not stabilize one another.” Hence their chemical pro- perties will also be uninfluenced by one another. For example, the absorption band of benzylamine is in the same position as the @ series of benzene, and quite different from the two bands of aniline. It is not surprising, therefore, to find that whereas aniline differs markedly from aliphatic amines, benzylamine resembles them. This is most noticeable in the well-known fact that the intermediate product in the action of nitrous acid on aniline is stable enough to exist, but not that in the action of the same reagent on benzylaraine. Also, since the value of M between them has been found to be small, we would not expect to find any stabilizing effect between two unsaturated carbons separated by a methylene group, and there is no reason to expect a sub- stance —CH?—CH?—CH?— to exist. Further, since the effect on the absorption spectrum of the introduction of a methyl group is usually slight, it is not expected that the stability of the unsaturated group or of the C—H linkages of the methyl group will be increased. The latter is in fact diminished. This is due to another cause which wiil be dealt with later. 670 Mr. N. P. McCleland on the 3. If in the case of two electrons w=v, L=N, we find 5M ) ? B02 | Cae AALS 50 L+M If pis the frequency of the least refrangible absorption band of the molecule p?= = where ¢ is the rigidity 2 2 coefficient, and consequently S=2u? { 30 } me From this it may be deduced that in the symmetrical, and probably in not too asymmetrical cases, if in substances of the same kind we find one where the less refrangible band has an unexpectedly small frequency, the “ vibration group” will be © more reactive than is usual in that substance. For example, the absorption bands of the aliphatic aldehydes are markedly on the less-refrangible side of those of the ketones (and this is most marked in the case of formaldehyde), although the absorption is in each case due to the carbonyl group, and the methyl group usually shifts the band in the opposite direction: the ease with which the carbonyl group of aldehydes, particularly formaldehyde, reacts (in particular, polymerizes) is well known. 4, Thiele’s theory is easily deduced from the above con- siderations. In a substance of the type Ry R; ‘o=CH—CH=(( Bere oh ee ee it is easy to see that from the considerations given above the stabilizing influence on the unsaturated carbons 2, 3 is greater than on 1 and 4, provided none of the R; groups are themselves unsaturated. Therefore addition will take place more readily on to carbons 1 and 4 than on to 2 or 3, which is Thiele’s theory, It may be remarked that substances like CASEN | /CoHs C=CH—CH=C We abe, RYO EE are exceptions to the theory, which is just what one would expect from the proviso stated above. 4 Also the above does not apply to a substance like R R C= 0H 0H On=0Ci* Re] 9 3 4 5 Ry Electron Theory in Organic Chemistry. 671 for the influence of 2 on 4 is negligible, as has been pointed out. 5. We will now consider the effect of unsaturation on neighbouring saturated groups. As has been pointed out, the stabilizing effect of unsaturation on neighbouring satu- rated groups, e. g. ethyl, is negligible. There is, however, another way in which they may influ- ence the stability of neighbouring saturated groups. Taking as exainple the —CH, group, the configuration of two hydrogens and the carbon (omitting the third and reducing the system for clearness to one plane) is as shown in fig. 2, the notation being as in fig. 1. Fig. 2. Hf Cc y) ey. 8 fy The stabilizing influence on & is made up of that due to a! and that due to £. Now an unsaturated electron, while not adding anything appreciable to the stability of this part of the system for reasons given above, may nevertheless alter the positions of the orbits a, «', 8, and so their stabilizing effects on one another, for the stabilizing factor, depending as it does on (Ge) is of a lower order of magnitude than the force (due to electromagnetic effect) between the orbits, which ae depends on 3a" To discuss the change in stability due to this is at present impossible, as it requires a considerable knowledge of the velocities of the electrons and directions of their orbits. The above, however, does show that such a change should exist. Experimental evidence shows that the carbon-halogen 672 Mr. N. P. McCleland on the linkage is rendered more stable, 7. e., the halogen is less reactive when the carbon to which the halogen is attached is less stable when an adjacent carbon is unsaturated, 29-5 CH,=CHBr, CH;—CH,Br, CH;=CH—CH.Br, are in ascending order of reactivity. Similarly the C—H linkage is in general rendered more reactive, i.e. less stable, by the presence of unsaturation; and this effect is increased when it is between two unsaturated groups. Thus the hydrogens marked with a star in the following are abnormally reactive:— CE SCO 2018s — OO CEL. acetylacetone, COOC,H;—CH = CH—CH 3 —COOC,H;,glutaconic ester, OH--CH. —COOC He phenylacetic ester, C,H;—CH SELON, benzyl cyanide, 6 NH | indene, CH WSUS CH. HC—CH | | eB Olal cyclopentadiene. N i y 2 It is immaterial to the theory whether the hydrogen atoms react in the positions they occupy in the formule, or whether an isomeric change first takes place. 6. As to the stereochemical effects of the ethylene linkage, the experimental results may be summed up in the two “laws” that in a substance in stable equilibrium : (1) All the R groups lie in a plane ; (2) There are two stable positions in the plane. Electron Theory in Organic Chemistry. 673 Now in our model of the molecule it can easily be seen that if all electrons are replaced by an electron at the centre of their orbit, 7. e.in their mean position, there will be a position of equilibrium with all the atoms in a plane. We have not the knowledge which would enable us to say whether this position is stable or not in all cases, but in a symmetrical case it can be shown that stable equilibrium in the plane is possible provided the centres of the orbits marked « in fig. 1 are not very near one or other of the nuclei. Therefore, although we are not at present in a position to deduce the stereochemical laws from the model of the mole- cule, there is nothing in the model which is inconsistent with them. On this theory, the stereochemical phenomena are due to carbon atoms temporarily functioning as trivalent. Possibly, therefore, it may be found that substances of the type R, Rg exist in two forms. Some cases of isomerism possibly due to this cause are cited by Stewart, ‘ Stereochemistry,’ p. 265 seq. A change from a cis to a trans position, or vice versa, is regarded as due to a violent disturbance which ends in one part of the molecule swinging round relative to the other. Such a disturbance could be caused by setting up vibrations in the molecule by means of light or heat, or by the approach of another molecule; and it is to be expected that molecules which can add on to the free valency will be most effective, even if their approach does not happen to lead to a formation of the addition product. 7. With regard to optical activity, with our model the groups Ry, &., in a substance CR, RRR, lie in the direction ot the angular points of a tetrahedron. If the R’s are all different, this tetrahedron will be asymmetric, as in the ordinary theory. In substances of the type there does not appear to be any reason to expect optical activity unless one or more of the R groups is not restricted to one plane. Even then optical activity could not be found 674 Mr. N. P. McCleland on the unless the group is asymmetrical with respect to the plane, and is sufficiently unsaturated to make for stability. 8. A few remarks of an entirely speculative nature as to the structure of benzene may be added. Many structural formule have been proposed for the benzene ring, a critical summary of which has been given by Dr. Stewart in his book on Stereochemistry. The chief conditions to be fulfilled may be summarized as follows :— . The 6 carbon atoms are equivalent. . The 6 hydrogen atoms likewise. . The laws of position isomerism must be complied with. . The existence of derived substances such as naphthalene must be allowed for. . A reason must exist for the law of substitution, which appears to demand that the carbon atoms exist in two. sets of three, those in the same group being in the meta position to each other. 6. The hydrogens attached to ortho carbons are nearer to one another than those attached to carbons in meta or para positions. 7. The reduction of benzene to cyclohexane must be possible in a simple manner. The carbon atoms in the latter substance must be in the same cyclic order as they were in benzene. On H= OO bo Re Further, cyclohexane is very similar to other naphthenes, and it would appear that the carbon atoms in this substance are in a plane. A structural formula which appears to comply with all necessary conditions is the following, based on the “ Kekule ring’ and the octahedron formula, suggested first, I believe, by “Professor Collie. The carbon atoms are arranged at the corners of an uctahedron, formed by alternately raising and depressing the 6 carbons of the Kekule ring from the plane of the ring. The electrons forming the bonds between carbons are supposed to revolve about centres in the mean plane; though it is not possible to state that the orbits are entirely in this plane, it appears that they are very nearly so. The electrons forming the free valency of each — carbon revolve in planes (almost) | parallel to the main plane, and the hydrogen atoms are disposed outside the carbons. This model closely resembles that suggested by Barlow and Pope on crystallographic grounds. Electron Theory in Organic Chemistry. 675: Thus carbons in the meta positions are in one group and those in the ortho and para in another. The distance between the planes must depend on the con-- ditions of stable equilibrium, which are extremely hard to work out. The following points are worthy of notice :— 1. The general assumptions made in the paper on absorp- tion spectra are generally consistent with the above mode], but possibly it was not accurate to neglect the values of the mutual induction coefficients between the electrons of a group attached to the ring and the ortho carbons. Nor can we say that the assumption that m,, and m, are nearly equal is correct: this, however, can only be settled by direct observation of the y band. An alteration in these values will not affect the value of my greatly. 2. Since we have found m)=*275 (C=1) and tor ethylene m="13, it follows that the ortho carbons in benzene are eloser than those carbons of ethylene, and so the free valency electrons are more difficult to displace. 3. It will be seen that the directions of the line joining a carbon atom to the 3 atoms to which it is linked and the line perpendicular to the plane of the free electron are directed almost to the four corners of a tetrahedron. ‘This will make for stability. 4, A substance C;H; or C;H, could not be analogous to benzene. In conclusion it may be remarked that the system deve- loped by Noyes in the Journal of the American Chemical Society, where one atom is supposed to be linked to another by the passage of an electron from one to the other, leaving them with opposite charges, does not seem to be consistent with the existence of ring systems consisting of an odd number of atoms, many of which are known and some of which, as cyclopentane and its derivatives, are extremely stable. March, 1915. b676. 4 LXX. On the Ionization Tracks of Alpha Rays in Hydrogen. By Professor J. C. McLennan, F.R.S., and H. V. MERCER, M.A., B.Sc., University of Toronto*. (Plate XIIL.] I. Introduction. ib a paper by Marsden in the Philosophical Magazine of May 1914, experiments are described in which it was found that when alpha particles from radium are projected into hydrogen, velocities are given to particles of the gas which enable them to pass through a thickness of aluminium foil capable of stopping the fastest alpha particles. These ““H” particles, as Marsden designated them, have been shown by him to be capable of producing visible scintilla- tions on a zinc-sulphide screen when this is placed at a distance away from their source more than three times the distance at which the alpha particles themselves produce scintillations. | On account of the high velocities possessed by these “ H ” particles, and for various other reasons, it is believed that they are the positively charged nuclei of hydrogen atoms, just as alpha particles are the charged nuclei of helium atoms, and that they are expelled from the atoms of hydrogen when these are subjected to a direct impact by the alpha rays. Moreover, since these ‘“‘ H”’ particles are capable of pro- ducing scintillations in a zinc-sulphide screen, and since scintillations are now considered to be the results of ioniza- tion, it would appear that these ““H” particles are capable of ionizing the atoms in the zine sulphide. The question, then, which naturally arises is: Are these “ H” particles capable of ionizing the atoms of a gas through which they may be projected? In order to throw some light on this matter it was thought well by the writers to make an ex- perimental study of the ionization tracks of alpha rays in hydrogen, and the following paper contains an account of this investigation. Il. Apparatus. The method followed was precisely the same as_ that devised by C. T. R. Wilson, and used by him in obtaining * Communicated by the Authors. Read before the Royal Society of Canada, May 26th, 1915. + C. T. R. Wilson, Proe. Roy. Soc. A. vol. lxxxvii. pp. 277--292 (1912). Lomzation Tracks of Alpha Rays in Flydrogen. 677 his exceedingly beautiful ionization tracks of alpha, beta, and X-ray tracks in air. The apparatus, which was made by the Cambridge Scientific Instrument Co. under his direc- tion, is shown in fig. 1 (Pl. XIII.), and in diagram in fig, 2. SSE a m For complete details of the construction and operation of this apparatus the reader is referred to C. T. R. Wilson’s original paper *, or to the descriptive catalogue of the Cambridge Scientific Instrument Co. It will suffice here to say that the cylindrical cloud-chamber A (fig. 2), was about 16°5 cm. in diameter and 3°4 cm. in height. The walls and roof of this cloud-chamber A were coated inside with gelatine con- taining a trace of copper sulphate, and the plate-glass roof, B, of the piston or plunger, C, was coated with gelatine blackened with ink and containing 2 per cent. of copper sulphate. The expansion was effected by opening the valve H, and so connecting up the air space below the plunger with * Loe. cit. 678 Prof. McLennan and Mr. Mercer on Jonization the evacuated chamber F. In all the experiments the appa- ratus was arranged so that the ratio of the final to the initial volume V./V, of the gas in the expansion-chamber above the plunger was about 1°36: 1. | As the plunger was always at the bottom of the chamber when the expansion was completed, the value of the ratio V./V, depended upon the height to which the plunger was raised initially. It was raised therefore to any height desired by simply removing the stopper K from the expansion- chamber and blowing into the tube H. The reading taken on the gauge depicted in fig. 1 after expansion showed when the proper ratio for V./V, was secured. In order to obtain distinct ionization tracks it was necessary to have all ions in the cloud-chamber removed before the -expansion took place, and this was effected by establishing .an electric field between the top plate of the plunger and the roof-plate of the cloud-chamber: to do this the gelatine layer under the roof-plate was connected by means of a ring of tinfoil cemented between the glass cylindrical wall and the roof-plate to one terminal of a battery of storage-cells whose other terminal was connected to the metallic base of the expansion-chamber, and through the brass cylindrical portion of the plunger to the roof-plate of the latter. In taking the photographs the method followed was also ‘that adopted by C. T. R. Wilson, and consisted in passing the discharge from a set of leyden-jars charged with a Holtz machine through some vaporized mercury, in a set of four quartz tubes joined in series and placed close to the walls of the expansion-chamber. The light from these dis- -charge-tubes was projected into the expansion-chamber in beams of parallel rays by a set of four plano-convex strips of glass. The camera was directed vertically down upon the roof-plate of the cloudchamber. In setting up the apparatus it was found convenient to separate the vacuum-chamber ‘somewhat from the expansion-chamber, and the manner in which this was done is shown in the upper part of fig. 3. In the lower part of the figure the connexions are shown for producing the expansion and exciting the spark. The Holtz machine was joined to the insides of two sets of leyden-jars at P, NN’ was a wet string short-circuiting the outsides of the jars, and S represents the discharge-tubes in which the -secondary spark occurred. In taking the photographs the string joined to T (fig. 3) was cut, and the valve B was opened by the falling weight W. When this weight reached the limit of its fall a second weight, in the form of a brass ball and supported by a slender cord, broke away and fell — Oe = me —_ Tracks of Alpha Rays in Hydrogen. 679 between the terminals at P,and brought on both the primary spark at P and the secondary discharge at 8. The height of the brass ball from the spark-gap, P, was adjusted so as to bring on the discharge at the instant desired after the expansion took place. Fig. 3. Se ee ee eo | In the experiments carried out by us the cloud-chamber above the plunger was filled either with air or hydrogen or with mixtures of air and hydrogen, and the cloud tracks were formed by the condensation of either water vapour or the vapour of absolute alcohol. The source of the alpha rays was a layer of polonium on the anterior face of a small sheet of copper of area about 1sq.mm. ‘lhe plate of copper was attached to a short stout copper wire which projected into the cloud-chamber, and was held in position by being fastened with wax to the 680 Prof. McLennan and Mr. Mercer on Jonization stopper, K (fig. 2). When hydrogen was used the ionization tracks in most cases extended completely across the cloud- chamber. In the present investigation, however, it was the ends of the trails which were to be specially examined, and so the tracks in hydrogen were cut down by covering the layer of polonium with sheets of very thin aluminium leaf. This was done in all the experiments in which the photo- yraphs of tracks in hydrogen were taken which are described in this paper. The lengths of the reduced tracks were generally about 4 cm. or less. III. Experiments. Although the investigation was primarily directed to obtaining, if possible, evidence of the production and of the ionizing power of the “ H”’ particles of hydrogen, and many photographs were taken with that end in view, not one of the photographs showed any trace of cloud tracks pointing to their production or to ionization by them. Many of the tracks showed abrupt bends similar to those obtained by Wilson, and occasionally very short spurs were obtained at these bends, but no spurs were obtained such as one should expect to get with ‘““H” particles travelling with velocities such as those Marsden found they possessed. This absence of ““H” particle cloud tracks in our experi- ments cannot be taken, however, to mean that “ H” particles are not produced in hydrogen, or that they do not possess the power to ionize a gas, but it goes to confirm, rather, what has. been already surmised, that when alpha rays traverse hydrogen or hydrogen containing traces of air exceedingly few of them collide with the hydrogen atoms in such a way as to expel the “ H”’ particles. Although the experiments were disappointing in this regard, they served to bring out some points of minor interest, and a few of the photographs taken are reproduced in the present communication to illustrate these points. The pho- tograph reproduced in fig. 4 (Pl. XITT.) is one taken of water- cloud tracks of alpha rays in hydrogen, and enlargements of portions of this photograph are shown in figs. 5 and 6. In fig. 5 there is shown a sharp bend near the end of one of the tracks and a less abrupt one near the middle of a second track. Tig. 6 also shows a very sharp bend near the end of one track, but as will be seen there is no sign of a spur of any appreciable length associated with it. The tracks shown in these photographs are typical of many which we obtained, but as mentioned above none of them showed spurs such as we expected to get with expelled Tracks of Alpha Rays in Hydrogen. 681 ““H” particles. In taking the photograph shown in fig. 4a field of 70 volts was applied in the cloud-chamber, and a Zeiss planar lens F 4°5, 5 cm. focus, was used. The photo- graph was taken on an Ilford Monarch plate. The reproduction shown in fig. 7 also shows a sharp bend in one of the tracks, but there is no indication of any spur at the bend. It was obtained with hydrogen and condensed water vapour. The field applied to the cloud-chamber was 8 volts. Figs. 8, 9, and 10 are reproductions of enlargements of the ends of some alpha-ray tracks in hydrogen, the clouds of which were formed from water vapour. The field applied to the cloud-chamber when they were taken was 8 volts. The interest in these photographs lies in the fact that the ends of the tracks and the curves they exhibit are somewhat more extended than those obtained by Wilson with air. In some cases it will be seen they resemble the curve at the end of a hockey stick, while in others the bends are more like the ends of a shepherd’s crook. Some of the tracks shown in fig. 8, as will be seen, appear to be double, while others again are single and very distinct. The double tracks are clearly out of focus and more or less indistinct. From their general appearance the doubling would appear to be entirely an optical effect, but it is just possible that it represents an electrical separation of the ions in those tracks which were produced a short time before the more dis- tinct ones. Against this latter view, however, there is the fact that the separation shown on the photograph appears to be lateral, while if it had been produced by the electrical field the displacement in all probability would have been in a vertical plane or very close to it. The track in fig. 11 and its enlargement in fig. 12 was also obtained with hydrogen and water vapour, the field applied being 30 volts. In this one, the crook at the end is brought out very clearly as well as the abrupt bend at the middle. At this bend there is a slight protuberance or thickening on the convex side which looks something like a shortened spur, but it is only just noticeable. Although photographs of many alpha tracks showing abrupt bends were taken, this one represents practically the maximum indication of a spur which was obtained. This shows that if the “H” particles really do ionize a gas, the kind of collision which results in the liberation of such particles is of exceedingly rare occurrence. If the reproduction shown in fig. 12 be closely examined, it will be seen that in addition to the abrupt bend near the Phil. Mag. 8. 6. Vol. 30. No. 179. Nov. 1915. 2Y 682 Ionization Tracks of Alpha Rays in Hydrogen. middle and the crook at the end, there is a slight bend of double curvature between these two. Many of the tracks photographed showed this gradual change in curvature, and it was thought at first that they constituted evidence opposed to the view advanced by Rutherford, that the scattering of alpha particles of large amount is the result of single de- flexions through considerable angles, and not to a cumulative effect due to a very large number of minute deviations. Continued investigation, however, showed that the bending mentioned above was mechanical in its origin, and was due to a distortion impressed upon the track of ions in the gas by the irregular movement of the gas in the course of expansion after the alpha-ray particle had passed, but before the con- densation actually took place. In some of the photographs, the distortion of the alpha-ray tracks was very considerable, and an enlargement of one of a number which exhibited this ina very marked way is shown in fig. 13. Here it will be seen that the curvature commences practically at the beginning of the tracks and extends for a considerable distance over their length. In this photograph the tracks were in hydrogen, but the cloud was formed from the condensation of alcohol vapour. The field applied was 30 volts. Fig. 14 exhibits another feature which characterized a number of the photographs. In this one it will be seen that a considerable space intervenes between the polonium plate and the commencement of the alpha-ray tracks. In this particular photograph, the ionization was in hydrogen and the cloud was formed from alcohol vapour. Similar results were also obtained when the clouds were formed from water vapour. This absence of condensation at the beginning of the tracks was taken to mean that, owing to the proximity of the copper plate carrying the polonium, the gas in this region was kept sufficiently warm during the expansion to prevent condensation of the vapour. IV. Summary of Results. 1. It has been shown that although alpha-ray tracks in hydrogen are of greater length than those in air, they exhibit similar characteristics to those obtained in air. 2. It has been shown that alpha-ray ionization tracks can be exhibited by the condensation of alcohol vapour equally as well as with the condensation of water vapour. 3. The photographs obtained confirm the observation of C. T. R. Wilson, and support the contention of Rutherford, Infra-red Emission Spectrum of the Mercury Arc. 683 that apart from the crooks at the ends of the alpha-ray tracks, the “scattering” is due to single deflexions through considerable angles. 4. The experiments failed to bring out evidence in support of ionization by the nuclei of hydrogen atoms projected from the latter when alpha particles collide with the atoms. 5. The experiments go to show that when alpha particles are projected through hydrogen or through hydrogen con- taining traces of air exceedingly few collisions oceur which result in the ejection of a hydrogen nucleus. The Physical Laboratory, University of Toronto. May Ist, 1915. ~LXXI. On the Infra-red Hmission Spectrum of the Mercury Arc. By Professor J. C. McLennan, F.R.S., and Ray- MOND C. Drearuz, M.A., The University of Toronto * | Plates XIV. & XV.] I. Introduction. A the present time when efforts are being directed towards the establishment of relationships between the atomic structure of an element and special features of its spectra, it is desirable to ascertain as fully as possible the frequencies which are associated with the atoms of the element in definite and determinate physical states. The frequencies associated with mercury atoms in the neutral, or supposed neutral, state have been carefully investigated by R. W. Wood f, McLennan and Edwardst, and others in the region between 7~=6000 A.U. and »=1800 A.U. In the experiments in which this was done, it has been found that if light of wave-lengths lying within the limits mentioned be passed through non-luminous mercury vapour, there is a strong sy mmetrical absorption band at 7=1849 A. Un ig? moderately strong non-symmetrical one at X=2536°72 A. U., and one still less marked, and consisting of four narrow bands, at A=2338 A.U. From this it has been concluded that within the limits mentioned there are three groups of frequencies which cha- racterize the atoms or groups of atoms present in the vapour * Communicated by the Authors. Read before the Royal Society of Oanada, May 26th, 1915. + R. W. Wood, ‘ Physical Optics.’ p. 431. t McLennan and Edwards, Proc. Roy. Soc. of Canada, 1915. 2 Y 2 684 Prof. McLennan and Mr. Dearle on the Infra-red of mercury in the non-luminous state. It is desirable, how- ever, that a wider range of frequencies should be investigated, especially on the side of the infra-red where but little work on absorption appears to have been done as yet. With a view to proceeding in this direction, some preli- minary work has been done by the writers in that region on the emission lines in the spectrum of the mercury are. It is evident that a knowledge of the lines which characterize this spectrum in the infra-red region, as well as of their exact wave-lengths, would be of great assistance in deciding where to look for absorption by mercury vapour. It was found, on examining the work of those who have already investigated the emission spectrum of the mercury arc in the infra-red region, that considerable divergence exists in their results. The first recorded investigations were somewhat cursory attempts by Snow and by Drew, and - it was not until 1903 that we have any results in which con- fidence can be placed. These are due to Coblentz and Geer*, who worked with a rock-salt prism spectrometer and a radiometer, and found three definite lines between 0°97 uw and 1°:285 wp. In addition to these they were able to identify six lines in the neigh- bourhood of 5:0, and possibly one other near 3:0 p. Coblentz t repeated this work a couple of years later, and announced that there are no important lines beyond 1°34 except those near D0 yw. W.dJ.H. Mollt somewhat later, using a rock-salt spectrometer and thermopile in connexion with an automatic recording device, identified five lines between 1:0 wand 1*7y. In direct opposition to the results. of Coblentz and Geer, Moll states that there is no measurable - emission above 1°7 uw. Probably the most accurate measure- ments on the infra-red spectrum of the mercury are are those made by Paschen§ with a concave grating and a Rubens thermopile. By means of the better definition and the higher dispersion afforded by the grating, Paschen was able to separate maxima which had previously been recorded as single lines. In all he identified fourteen lines between 1:0 w and 1:7, and he confirmed the statement by Moll that there are no lines beyond 1:7y. Hel| subsequently repeated his measurements and found a maximum at 4°0 p,. but inasmuch as this maximum came out in the are spectrum, * W. Coblentz and W. C. Geer, Phys. Rev. xvi. pp. 279-286 (1903)... +t Coblentz, Phys. Rev. xx. pp. 122-124 (1905). ¢ Moll, Kon. Akad, Wet. Amsterdam, Proc. ix. pp. 544-548 (1907). § Paschen, Ann. d. Phys. xxvil. 5. pp. 587-570 (1907). || Paschen, Ann. d. Phys. xxxiil. 4. pp. 717-7388 (1910). Emission Spectrum of the Mercury Are. 685 of a number of the elements, he concluded that it was due to the presence of hydrogen. In these later measurements a bolometer was used in combination with a grating. More recently still H. Rubens and O. von Baeyer™ have succeeded in showing that the mercury are emits a radiation of wave- length about 313. They succeeded in isolating this radia- tion by the method of focal separation previously used by Rubens and Wood f, and in measuring its wave-length by means of a Fabry and Perot interferometer of a special type in combination with a Rubens microradiometer. Subsequent measurements{t by them on this radiation showed that it consisted of two wave-lengths, the one at about 218 w and the other in the neighbourhood of 343 wu. The only noteworthy measurements by photographie methods in the infra-red spectrum of the mercury are were made in 1912 by H. Lehmann§. He used the phosphoro- photographic method of Bergmann||, and found four lines corresponding to those of Paschen and in addition a new line at 1°46 yp. In the present investigation a careful survey was made of oD the infra-red spectrum of the mercury are in the region beyond 1:0 with the object of confirming, if possible, the existence of the lines identified by Paschen, and of seeing whether the lines found by Coblentz and Geer in the neigh- bourhood of 34 had a real existence. In this examination a number of the lines noted by Paschen were identified, the existence of a line near 3:0 w was confirmed, and in addition a number of new lines were observed. Il. Apparatus. In this work a number of different forms of mercury arc- Jamps were used as sources of the radiation, but a quartz mercury-lamp constructed by W. C. Heraeus was found to be the most satisfactory. This lamp gave a very powerful and quite concentrated arc. When in operation it was driven with a direct current of from 3:0 to 3°2 amperes with a striking potential of 110 volts. A suitable resistance was of course inserted in series with the lamp. It was found * H. Rubens and O. von Baeyer, Phil. Mag. xxi. pp. 689-695 (1911). + H. Rubens and R. W. Wood, Preuss. Akad. Wiss. Berlin, Sitz.- Ber. lii. pp. 1122-1137 (1910). { H. Rubens and O yon Baeyer, Preuss. Akad. Wiss. Berlin, Sitz.- Ber. xxx. pp. 666-667 (1911). § H. Lehmann, Ann. d. Phys. xxxix. 1. pp. 76-77 (1912). || Bergmann, Zettschr. Wiss. Phot. vi. pp. 113-1380 and pp. 145-169 (1908). 686 Prof. McLennan and Mr. Dearle on the Infra-red that even with this lamp the current through it steadily decreased for the first ten minutes after the arc was struck. This was brought about by an increase which took place in the resistance of the mercury vapour when the temperature of the lamp was rising. The following table and the curve in fig. 1 (Pl. XV.) show the variation of the current with the elapse of time in a typical case :— Time | Current. | G0. > —|— wh 4 m 0 min 6-4 amps. O min 50 secs 61 Al | ya BOe aks 59 % 4 ” 20 ” Se) 99 | 3 ” 10 ’ 5°45 ” | rks 45 ,, 5h a 4 ,, 45 ,, Wy 2a aie SUN 40 ,, 4-4 Ae Gaile Boek A113"), joe 5) 400 __,, / Dia. Opes ai RO) ieee | TOM es Ons ST eee Tass _- ot e | In all experiments care was taken to see that the lamp was in the steady state before measurements were made on the intensity of the radiation. The form of spectrometer used was one designed and con- structed by the Adam Hilger Co. It is shown in fig. 2 and in diagram in fig.38 (Pl. XIV.). The energy measurements were made with a sensitive Rubens thermopile shown in fig. 4, in conjunction with a very delicate Paschen galvanometer made by the Cambridge Scientific Instrument Co. The radiation from the lamp was allowed to fall upon a large concave mirror having a diameter of 19 cm. and a focal length of 30 cm., which brought it to a focus on the slit 8,. From this slit the rays passed to the nickel-steel concave mirror M,, thence through the rock-salt prism P to the plane nickel- steel mirror M,. From this they were reflected to the con- cave nickel-steel mirror M3, and by it they were brought to a focus on the linear junctions of the thermopile at T, which was placed immediately bebind the slit S.. The prism and plane mirror were mounted on a table which rotated about the point A. By turning this table through a small angle, any desired part of the spectrum could be brought to a focus at S,. The rotation was produced by the motion of a helical drum attached at D, which was calibrated in wave-lengths fi , : { b f F , ue i i 4, ¥, ; it i — rae ere aE ee Se ee ee a = — oe - e o , Sa ~ Emission Spectrum of the Mercury Are.. 687 up to 10 w from data on the dispersion of rock-salt as given by Paschen®* and others. An eyepiece H was attachable behind the slit S, for the purpose of focnssing lines in the visible part of the spectrum on the thermopile, ‘and of adjusting the prism so that the radiation brought to a focus at So was in agreement with the reading on the drum. ‘he prism had faces 3:2 cm. by 4-2 cm., and was ground to an angle of approximately 455°. Judging by the visible spectrum, there was very little curva- ture in the spectral lines produced by this prism. The thermopile (fig. 4) consisted of 10 junctions of bismuth- silver joined by silver solder and flattened out into rectangular plates at the exposed junctions, which were blackened. The sensitive area was 20 mm. long and 1°5 mm. wide, or a total of 30 sq. mm. _ As the slit-width used in all the experiments was only 1 mm., the effective area of the exposed junctions was only 20 sq.mm. The galvanometer used was a modified form of the Thomson galvanometer, and was specially designed by Paschen+ for radiometric measurements. ‘lhe magnet system consisted of two groups of thirteen magnets arran ged alternately on opposite sides of a fine elass stem and supported by a fine quartz fibre. The coils were elliptical in shape, and were wound with six different sizes of wire with the object of producing a maximum field for a given resistance of copper. ‘The period could be controlled hy means of a magnet, and it was adjusted to have a full period of 5 secs. It was found that while a longer period did not materially increase the sensibility, it made the zero drift consider ably greater. The resistance of the thermopile was 2°93 ohms and that of the galvanometer, with the coils connected in multiple series, which was the arrangement always adopted, was 3:0 olims. The sensitiveness of the instrument was such that a deflexion of 1mm. on a scale at the distance of one metre was produced by a current of 00025 microampere. One of the greatest difficulties met. with in the work was the variation produced by temperature changes and by stray air currents. ‘To overcome these the thermopile and slit were enclosed in a nickelled box, shown at B in fig. 3, which was both packed inside and surrounded outside with cotton- waste. The whole spectrometer was enclosed in a wooden box lined with absorbent cotton, and all the free space between * Paschen, Ann. d. Phys. xxvi. 1. pp. 120-188; 5. pp. 1029-1030 (1908). + Paschen, Ann. d. Phys. xxvii. 3. pp. 587-570 (1908). 688 Prof..McLennan and Mr. Dearle on the Infra-red the spectrometer-stand and the box was also filled with cotton- waste. The box had a window at &, covered by a shutter, and a second window as well, through which to read the wave-lengths on the drum. Asan additional precaution an asbestos screen was always placed between the lamp and the spectrometer. ‘The lamp itself gave rise to certain errors due to variations in the current, and to the occasional deposit of a drop of mercury on the face of the tube from which the radiation was taken. ‘These latter errors were sufficient at times to produce false maxima of considerable magnitude. In taking all readings, the drum was set at the desired wave- lengths and the shutter was opened until the galvanometer reached its maximum deflexion, when it was again closed. This was repeated from six to ten times, and the mean value of the deflexions was taken as a measure of the energy in the particular wave-length selected. Zero drift was always con- siderable, on some days amounting to as much as 140 mm.in readings extending over the space of an hour. To eliminate the effect of this drift the amount of deflexion on opening the shutter was read, and also the distance which the spot of light returned on closing the shutter. The mean of these two was then taken as the correct reading. When every imaginable precaution was taken, it was still found that maxima appeared in the energy curves which apparently did not represent spectral lines. However, it was possible, by repeating the readings over any given portion of the spectrum on different days, to differentiate between true and false maxima and so to identify the spectral lines. In taking the measurements, it was necessary to distinguish between the energy which was contributed by a spectral line and that which was contributed by the continuous spectrum due to the radiation from the heated quartz of the lamp itself. To do this, a circuit breaker was connected in series with the lamp, the drum was adjusted to give the wave-length of the desired line, and the shutter was opened immediately after the circuit was broken. The ensuing deflexion was read and the time noted on a stop-watch. The shutter was again closed and after an interval reopened. The deflexion pro- duced and the time corresponding to it were again noted. In this way several readings were taken, and a cooling curve was plotted from them. This curve was then extended back- wards to zero time, and from the point where it cut the ordinate axis the energy contributed by the radiation from the hot quartz of the wave-length under investigation was ascertained. This reading was subtracted from the reading taken when the lamp was in operation, and the difference Emission Spectrum of the Mercury Arc. 689 gave the energy contributed by the spectral line. A cooling curve of the type just mentioned is shown in fig. 5 per XY.). : The intensity of the radiation of any particular wave-length as measured by the spectrometer was found to vary with small displacements of the lamp or scale, and so all measure- ments were compared with the deflexion produced by the radiation from the green line X=5461 A.U. This line, which was of strong intensity, possessed the advantage of being practically outside the region of the hot quartz radiation. III. Accuracy of Measurements. In work on infra-red spectra the means generally employed to produce the spectrum are the prism and the grating. ‘The latter has the great advantage that it absorbs but little radia- tion and that it affords good definition in all parts of the spectrum. Against this, however, is the fact that the energy is divided up into several orders, that these orders often overlap in the infra-red, and that the distribution of energy in any one order does not always correspond with the true distribution in the spectrum. The prism, on the other hand, gives but one order, so that the maximum of energy is found in each and every wave-length. It limits the measurements, however, to the region where the radiation is transmitted without absorption. With prisms of rock-salt the radiation is transmitted up to wave-lengths of 60,000 A.U.,but it is difficult to secure good definition in the longer wave-lengths. This difficulty, moreover, is enhanced by the fact that in order to secure sufficient energy in the weaker lines, it is necessary to work with a fairly wide slit. On this account, one cannot ex- pect in working with a rock-salt prism to reach the precision of wave-length measurement attainable with a grating spectro- scope, or to differentiate between lines very close together with the same facility as with a grating. The prism, how- ever, enables one to obtain a reliable register of the maxima in the energy spectrum, and these can be used as a guide for finer measurements with an instrument such as a grating. The following table shows the width of spectrum covered in the different ranges by the thermopile slit for a slit-width of 1 mm. These results are also shown graphically in fig.6. Although from this table there seems to be a considerable width of spectrum covered by the slit, still it must be remembered that since in all our measurements readings were taken at intervals of Oly, the energy curves could be filled in 690 Prof. McLennan and Mr. Dearle on the Infra-red Wave-length. Width of spectrum. | 0°54 wu 0:02 » 0°58 ,, 0°03 ,, 0:82 An 0:08 ” Loo 0:28 ,, 2.00 ,, 0°62 ,, 3°00 ,, OG), | 4:00 ., 068 ,, 5°00 ., | 0°54 ,, | 6:00 ,, 0-46 ,, 7°00 ,, | 0°40 ,, 8 00 ., | 036.) | 10°00 ,, 0:32 ,, between these readings, and the wave-lengths could be assigned to the various lines with an accuracy of probably +0°01 p. IV. Observations. After the apparatus was carefully set up and found to be in good working order, a set of readings was taken on the energy spectrum of an Arons™ amalgam lamp. The amalgam in this lamp consisted of about 60 per cent. Hg, 20 per cent. Pb, 20 per cent. Bi, 4 per cent. Zn, and 4 per cent. Cd. The lamp was run on the 110 volt directcurcent oirouis qatiee suitable resistance in series. The energy curve is shown in fig. 7 (Pl. XV.). Thirty-six distinct maxima were observed in all between wave-leneths 0°70 and 3:0. Possibly as many more could have been distinguished by repeated obser- vations, but as there was no way of distinguishing which were mercury lines and which were lines of the other metals, further work with this lamp was abandoned. Readings were then taken, as stated above, with a lamp containing only pure mercury. The maxima from a large number of sets of readings were compared, and the regions which contained constantly recurring maxima were carefully examined so as to establish the exact position of each maxi- mum as closely as possible. In the same region as that investigated by Paschen+, the positions of nine lines were determined, and these accorded fairly well with the wave- lengths given by him. As was expected, in regions w here the latter gives two or three lines in close proximity, only one maximum, corresponding to the mean wave-length, was recorded in our measurements. Corresponding to the new * Arons, Ann. d. Phys. Band xxiii. (1906). + Paschen, loc. cit. ad eB oe 4 4 , , ¥ 4 A i “4 >, : 7 : \ Emission Spectrum of the Mercury Arc. 691 line observed by H. Lehmann™®™ at 1°464 w, a maximum was. found which appeared to vary slightly from 1:46 w to 1°50 p. The mean wave-length of all the readings taken on this line was 1°483. In addition to these, four other lines were recorded, two of which had already been located by W. Coblentz and W. C. Geer, with slightly higher values for the wave-lengths. In their work a line is given at 1°045 py, while in ours it was found to be at 1:038y. Close to this line we also found two others at 1067 w and 1°090 pw respec- tively. The presence of these lines was evidently suspected by Goblentz and Geert, since they state that certain obser- vations were repeated “to learn whether another line exists between 1:06 w and 1:12 ~ where the curve is very asym- metrical.” ‘Their instruments, however, were not sufticiently sensitive to detect these lines. Bey ond 1°7 7» only one line was found, which was at 3°2. This confirmed the doubtful indications recorded by the above-named pair of investigators. This part of the spectrum was not as thoroughly examined as might be desired, and it is just possible that a closer examination might have revealed other maxima, particularly in the region of 3°7 w, where indications, as will be shown below, point to the possible existence of a new line. In Table I., the wave-lengths of the lines isolated in the present investigation are given in the first column and their relative intensities in the second. Accompanying these are given in order the wave-lengths of the lines determined by other investigators, namely, Paschen, Lehmann, Moll, and Joblentz and Geer. The intensities are given for the first two, but in the case of the others no intensities are recorded in their communications. In the last column tlie frequencies of all the lines are given reduced to a vacuum. V. Discussion of Results. If the measurements made by the writers be compared with the others given in ‘lable I., it will be seen that the line observed by us at 3: 02 is probably the same as that given by Coblentz and Geer at 3:00, and the maximum at 1-72 m doubtless corresponds to the group of four lines given by Paschen between 1°692y and 1:71ly. There is no line given by the others near 1°483 unless it be that given by Paschen at 1°529q and the one given by Moll at 1°52 yp. The maximum noted at 1°377 w represents probably a com- bination of Paschen’s two lines at 1°395 w and 1°367 w, and * Lehmann, loc, cit. + Coblentz and Geer, Phys. Rev. xvi. pp. 284, 903. 692 Prof. McLennan and Mr. Dearle on the Infra-red TaBLe I. | Authors: I. |Paschen.| I. | Lehmann.| I. | Moll. Coblentz | Frequency and Geer. | in vacuo. Bi ye B be feos SiOz IG a bs sie st Bas _ 33103 Pees eal praised /ieces . Reh gay 5:00 | 3332:4 17 a ee fe nen es 5812-4 ct ie i a arf Uae fi il hl | | eee ; Bh ahs | §856°9 1:70 be | 5880°8 a fe ' 5900°8 | §908°6 ee sae | 6914:°4 1:52 se |, Goa7d bee oe | 6668°3 6828°5 71660 7260°6 7300°2 (alte 7352°2 | 7866°7 1-464 sears co Steuer saree Lo: 1-369 1'359 : NS: | 7522-4 its 285 T7784 | STIRS 82822 | 8286-6 8316°4 8410°3 85449 — ww =~J (=) Be {NOFA ONS oryo mic, eres a ls 170 | bs a) aN | 8826°5 i: 128 | 3 me ae 8863:0 sh at 9171:7 9369-7 fe 045 | 95672 9631:4 9851:2 9863°5 | it 128, 1090 | 1-067 | TUONO: AT: @p) & 1-038 races Bese an 1-015 1-014 | 30 | 1-014 Be ee ee ee rr e. sheets 8s the lines nearest to 1°329 py are that noted by Paschen at 1°351 w and that given by Coblentz and Geer at 1:285. The line at 1270 w appears to be a new one, unless it be the same as that given by Coblentz and Geer at 1:285 w. That at 1:205 w very likely represents a combination of the lines given by Paschen at 1:207 and 1:202y. The line at 1:170 « probably represents the same one as Paschen’s at 1:188 ». The maximum observed at 1:128 « was also found by Paschen and Lehmann, but the maxima at 1°090 uw and 1:067 w are new. As mentioned before, the maxima at 1:038 w and 1°045 uw probably refer to the same line. The line at 1:014 4 was also observed by Paschen and by Lehmann at L015 iu. An energy curve showing the maximum at 3°02 pw 1s given — Emission Spectrum o7 the Mercury Are. 693 in fig. 8, and others showing the new lines at 1:27 », 1:090,, and 1:067 are given in figs.9 and 10. It will be noted that the maxima at 1°27 w and 3:02 w are fairly sharp, while those shown in fig. 10 are somewhat broader. A comparison of the intensities assigned to the various lines by the different investigators affords an interesting study. Paschen says in his paper that he found the line at 1-014 » was the most intense in the whole mercury spectrum, and that on the same relative scale as that recorded in Table I., he found the mercury green line at \=5461 A.U. was represented by an intensity of 42, 7. e., rather more than ~ one-half of the intensity of the line 1-014 4. In the measure- ments of the writers it was found that if the energy in the line at 1-014 be represented by 30, that in the line at >A= 5461 A.U. was found to be represented by 40, or one- third more than that of the line 1°014y. On looking at Lehmann’s results it will be seen that though there is no value given for the intensity of the green line, that of the line 1015» is given by him as the weakest in the infra-red spectrum. .This was probably due to the method of registering the spectrum used by Lehmann, for it will be remembered that by it the lines were recorded by their inverting action on a phosphorescent screen. It is quite possible that the line 1°015 w, being nearer to the wave-lengths of the visible spectrum than any of the others recorded, would not have so strong an effect as waves of longer length. As regards the relative intensities of the lines 1:014 yu and 5461 A.U., it was noted by Paschen* that if the vapour-pressure in his lamp was increased, the relative intensity of the line 1:014 u came out still higher, while with a low vapour-pressure the intensity of the two lines was about equal. This may explain the values of the intensities found for these lines in the present investigation. In looking for series relationships among the lines given in the first column of Table I., it was seen that the frequency differences for the lines 1:038 p, 1°27 wu, and 3:02 yu are prac- tically the same as those which characterize the subordinate series triplets in the mercury spectrum given by v=2, p—m, d, and v=2,p—m,s. This will be evident from the numbers given in Table II. It will be noted, too, that the frequency difference between the line given in our list at 1:09 w and the one given by Paschen at 1:367 is equal to 1860°4, which approximates, as the table shows, to the frequency difference between the second and third members of the triplets of the two subordinate series mentioned above. * Paschen, Ann. d, Phys. xxvii. 3. p. 559 (1908). 694 Infra-red Emission Spectrum of the Mercury Are. If these two lines should turn out to be the second and third members of a triplet similar to the one already noted, there would need to be a line at about 3°70. A line in this TABLE IT. Ware-length. | Frequency. Difference. First subordinate triplet series, v=2,p—m,d. m=8. fo) 3663°05 A.U. 2729206 313166, 31922-98 ETE 2967°37_,, 33690°34 Second subordinate triplet series. v=2,p—m,s. m=1°5, Oo 5460°97 A.U. 1830673 21. 4358-66, 22937-04 Remar 4046-78 ,, 2470423 ce Triplet 1:038 p, 1:270 p, 3°02 p. | io) 380200 A.U. 33103 , 12700, 7871°8 Toe 10380 ,, 9631-4 ; Suggested triplet 1:09p, | 1°367 p, 3°70 p. Oo 37000 A.U. 2702'7 4608-6 13670 ” Tob 3 1860°4 10900 | 9171°7 position, however, was not observed in our investigation, but as already mentioned this portion of the infra-red spectrum was not examined so closely as the region of somewhat shorter wave-lengths. VI. Summary of Results. I. Thirteen lines have been recorded in the infra-red spectrum of the mercury arc between the wave-lengths 1:00 4 and 3°02 p. II. The existence of the line near 1:04 4, which was shown to exist by Coblentz and Geer but which was not found by later investigators, was confirmed by the finding of a line at 1°038 p. | Ill. The existence of at least one line with a wave-length longer than 1°70 w was proved by the discovery of a line at 3°02 4, which was also in confirmation of the work by Coblentz and Geer. | Spectra of Mercury, Cadmium, and Zine Vapours. 695 LIV. Three new lines were discovered in the infra-red spectrum at 1:067 pw, 1:090 w, and 1:270 p. V. It has been pointed out that the frequency differences for the lines 1:038 pu, 1:°270y, and 3°02 are the same as those which characterize the triplets in the subordinate series for the mercury arc spectrum given by v=2, p—m, d and v=2, p—m, s. It has also been suggested that possibly the lines at 1:09 u and 1°367 yp are the third and second members of a similar triplet with its first member in the neighbourhood of 3°70 yp. The Physical Laboratory, University of Toronto. May 1st, 1915. LXXII. On the Absorption Spectra of Mercury, Cadmium, and Zine Vapours. By Professor J. C. McLennan, F.R.S., and Evan Epwarps, M.A., B.Sc., University of Toronto *. (Plate XVI.] Il. Introduction. | fe 1907 it was pointed out by Wood f that in the absorption spectrum of non-luminous mercury vapour there is a heavy band at X= 2536-72 A.U., and a less sharply-defined one at) = 2350 A.U. Ina later paper by Wood and Guthrie } dealing with the same subject no mention is made of the absorption band at A=2350 Fa A ae but it is stated that with dense mercury vapour there is a fairly strong band at }=2338 A.U., and another very broad one at \X=2140A.U. From the work of Kirschbaum § and others it is known that light of wave-length A=1849°6 A.U. is strongly absorbed by mercury vapour. n The absorption band at X=2536°72 A.U. has been shown by Wood to be asymmetrical. It is sharply defined on the shorter wave-length side, but with increasing vapour density it gradually spreads out towards the red end of the spectrum. ' With low vapour densities it consists of two ,bands, the one at X= 2536 A.U. and the other at X=2539A.U. The band at X=2338 A.U., which is probably the same one as that originally given by Wood at A=2350A.U., does not appear * Communicated by the Authors. + Wood, Astr. Phys, Journ. vol. xxvi. p. 41 (1907). t Wood and Guthrie, Astr. Phys. Journ. vol. xxxix. No.1, p. 211 (1909). § Kirschbaum, Electrician, vol. Ixxii. p. 1074 (1914). -- eee EA alee =o oo Ss Se S ee SEES —_—— SS — — 696 Prof. McLennan and Mr. Edwards on the Absorption to have been examined in, detail. In regard to the band noted by Wood at A=2140 A.U., especially as it was obtained with high vapour densities, it appeared to the writers that it might be connected with the absorption observed by Kirschbaum at »=1849°6A.U. Some experiments were made by us to test this view, and also,to study the character of the absorption band at 7=2338 A.U., and these will be described in what follows. II. Absorption Spectrum of Mercury. In the first experiments the light from a quartz mercury arc-lamp was projected through an evacuated clear fused quartz tube containing a little mercury. The mercury was gradually heated, anda series of photographs was taken with a small Hilger quartz spectrograph. A reproduction of one of these photographs is shown in fig. 1, Plate XVI. The upper spectrum is that of the mercury arc alone, the second is that obtained when the quartz tube was moderately heated, and the third is that obtained when the mercury vapour density was considerably higher. The asymmetrical character of the absorption band is clearly brought out by the photograph. In the second experiments a photograph was first taken of the spark spectrum of mercury in air in a manner already described in a previous communication by one of us*. Photographs were also taken of the spectrum of the light from the spark between terminals of cadmium in air after it passed through the mercury vapour in the exhausted quartz tube mentioned above. These were taken with gradually in- creasing vapour density, and are shown in fig. 2. In this photo- graph the mercury spectrum is shown at the top well down into the ultraviolet and the strong lines at A=1942°1 ALU. and A= 1849-6 A.U. are clear and distinct. The succeeding four spectra show that even with small vapour density the absorption was such as to,cut off the light of wave-lengths in the region of AX=1942°1 A.U. and 7=1849°6 A, U. In the second last spectrum, absorption at A=2536°72 A.U. can just be detected, but in the last ,one it is well marked. The absorption band at X=2338 A.U., also comes out in this spectrum, and that at 7»=1849 A.U. has widened out so that on the side of longer wave-lengths it has reached N= 2144-0 A.U. From the general appearance of the photo- graph it will be seen that the absorption at N=1849°6 A.U. develops symmetrically with increasing vapour density. This photograph also shows that light of wave-lengths near to * McLennan, Proc. Roy. Soc. A. vol. xci. p. 26 (1914). Spectrum of Mercury, Cadmium, and Zine Vapours. 697 4=1849-6 A.U. was the most strongly absorbed by, mercury vapour. That in the neighbourhood of \= 2536°72 A.U. came next, while high vapour densities were required to bring out the absorption at A=2338 A.U. In the third experimenta large Hilger quartz spectrograph was used. With this instrument the arc spectrum of mercury from a quartz lamp was first taken, then the spark spectrum from aluminium terminals in water after the manner devised by Henri*, and then the spectrum of the light from the spark between these aluminium terminals in water, after it had passed through a heated clear fused quartz evacuated tube containing mercury vapour of high density. These three photographs are shown in fig. 3. The spark from aluminium terminals in water, as will be seen from the second spectrum in the figure, gives a continuous spectrum of remarkable extent. It can be obtained with ease down to A= 2150 A.U. The arrangement for producing the Henri spark is shown in fig. 4. The terminals of the induction-coil AB were Fig. 4. joined to the spark-gap at CD and to the inside coatings of two one-gallon leyden-jars HF’. The outside coatings of these jars were joined to two rods of aluminium MN. ‘hese rods constituted the terminals of the spark-gap which was the light source, and as shown they were short-circuited by a coil of small self-induction GH. The aluminium terminals MN were rods about 1 cm.in diameter. They were conically pointed, and were held clamped in a vertical plane inclined at 45° to each other. The clamps in which they were held were provided with threads which enabled one to readily alter the distance between the sparking-points. When the spark was in action the terminals MN were immersed to a depth of about 5 cm. in a vessel of water. The light from the spark passed through the water and out of a quartz window sealed into the side of the vessel. It was then * Henri, Phys. Zeit. No. 12, p. 516, June 15th, 1913. Phil. Mag. 8. 6. Vol. 30. No. 179. Nov. 1915. 2Z — _——s 7 698 Prof. McLennan and Mr. Edwards on the Absorption focussed with a cylindrical quartz lens upon the slit of the spectrograph. The third spectrum in fig. 3 is the mercury vapour absorption spectrum taken with the light from the Henri spark. In the region below 7=2150 A.U. it will be seen there is complete absorption. . The asymmetrical nature of a absorption at A=2536°72 A.U. is also brought out. At = 2338 A.U. it will be seen that the absorption is complex, a consists of four bands, one extending from A=28134.U. to X= 2320 A.U., one at A= 2322 A. U., another at N=2326 A.U.,and a wider one between X=2330 A.U. and 7=2338 A.U. The absorption moreover is strong and sharply edged on the longer wave side, but it weakens out and is less clearly defined on the side of the shorter waves. It will be noted that the third spectrum in fig. 3 also shows a narrow absorption band at ~¥=2288A.U. As cadmium vapour has an absorption band at this point, the occurrence in the photograph of such a band was ascribed to the presence of a trace of cadmiuin in the mercury. From these results it will be clear that between 1 =6000 A.U, and 71=1800 A.U. there are but three regions of absorption in the absorption spectrum of non-iuminous mercury vapour, viz. in the neighbourhood of A=1849°6A.U., of X=2338A.U., and of X=2536:72A.U. It should be noted that 7X=1849°6 A.U. 1s the first line in the series of the mercury-are spectrum given by v=1°5, S—m, P, and X=2536°72 A.U. is the first line in the series n=2, p.—m, S* of the same spectrum. The line A= 2338 A.U. has not been shown as yet to belong to any series. III. Absorption Spectra of Cadmium and Zine Vapours. It is interesting to note that in their paper on the absorp- tion spectra of metallic vapours, Wood and Guthriet found absorption with cadmium vapour at A= 2288 A.U. and at ~=3260'17 A.U. As these lines are the first members in the two series of lines in the cadmium are spectrum given by y=1'5, S—m, P, and v=2, p,—m,8,tf there isa close analogy between the absorption spectrum of non-luminous mercury vapour and that of cadmium vapour. As zinc is chemically * Dunz, Inaugural Dissertation, Tiibingen, 1911, pp. 67 & 68. } Wood and Guthrie, Ast. Phys. Journ. vol. xxix. No. 1, p. 211 (1909). (The experiments of the writers confirm the findings ‘of Ww ood and Guthrie in ,regard to the absorption bands at A\=2288 A.U. and \=326017 AU. being the only ones observable in the absorption spectrum of cadmium vapour. | t Dunz, loc. cit. Spectrum of Mercury, Cadmium, and Zine Vapours. 699 closely related to mercury and cadmium, one would expect to find absorption bands with zine vapour in positions analogous to these, where absorption was obtained with the vapours of these two metals. In the zine are spectrum the line A=21393A.U. is the first member of the series v=1°5, S—m, P, and X¥=3075'99 A.U. is tne first member of the series y=2, po—m, 8S. One should therefore expect to find absorption with zinc vapour at A=2139°3 A.U. and at 2=3075°99A.U. Inapaper by one of us, recently published*, an account was given of experiments with zinc vapour, in which it was found that there was strong and symmetrically spaced absorption at X=2139°33 A.U. No absorption, how- ever, was found at X=3075°99 A.U. These experiments have now been repeated by the writers with a column of zine vapour about 20 cm. in length, with the result that a sharp, narrow absorption band has been found at A=3075°99 A.U. In fig. 5 the upper spectrum is that of the zine spark in air taken directly, and the lower one is that of the light from the same spark after it had passed through the zine vapour. The absorption corresponding to X=2139°3 A.U., as will be seen, is extensive and clearly defined. It will also be seen that the reproduction shows a narrow band at X=3075°99 A.U. In the emission spectrum of the spark taken directly there are two lines close together at X=3075°99 A.U., while in the lower photograph only one line is seen. In order to bring out the absorption at this point more clearly a series of photo- graphs was taken with gradually increasing vapour density. One of these is shown in fig. 6. The upper spectrum shows the line at) =3075°99 A.U.to be double. Inthe second and third spectra the line is single, and in the fourth and fifth spectra a narrow dark band is seen close to and to the left of the single line. An enlargement was taken of this portion of the absorption spectrum, and it is shown in fig. 7. This photograph, it will be seen, brings out very clearly the absorption band at X=3075'99 A.U. If the absorption spectrum in fig. 5 be examined it will be seen that absorption bands are also shown at A=2288 A.U. and X= 2536°72 A.U. These were no doubt due to the presence of mereury and cadmium vapours in the tube containing the zine vapour. As this tube was a new one and had not been used previously, it would seem that the mercury and the cadmium must have been present in the zinc as impurities. It is of interest to note this, for the zinc had been purchased as being doubly distilled and specially pure. With regard to the mercury, it is just possible that when the zinc tube * McLennan, Phil. Mag. a ae Sept. 1914, p. 360. 22Z 2 a. es... oe Se aoe ————— 700 = Spectra of Mercury, Cadmium, and Zinc Vapours. was being exhausted some mercury vapour diffused back into it from the Gaede mer cury-pump. It will be noted that the mercury absorption band is also shown in fig. 6, From these results it will be seen that there is a com- plete analogy, in so far as the first members of the series v=1'5, S—m, P and v=2, pp , in the absorption spectra of mercury, cadmium, and zinc vapours. In the absorption spectra of cadmium and zinc vapours no absorption was observed corresponding to that obtained with mercury at A=2338 A.U. It will be remembered, however, that with mercury this absorption band required high vapour density to bring it out clearly. It may very well be that with cadmium and zine vapours the densities used were not sufficiently high to produce noticeable absorption at points in these spectra corresponding to the band at X=2338 A.U. in the mercury spectrum. IV. Summary of Results. 1. In the absorption spectrum of non-luminous mercury vapour there has been shown to be a strong symmetricaily spaced band at X=1849°6 A.U., a diffuse complex band at A= 2338 A.U., andan asymmetrical band at A=2536°72 A.U. The complex band at %=2338 A.U. consists of a band extending from A=2313 AOU. te V= 2320 ALU. .» one at A= 2322 AU. ., another at A= 2326 A. U., and a wider one between 1=2330 A.U. and A=2338 A.U. 2. The experiments of the writers confirm the findings of Wood and Guthrie in regard to the absorption spectrum of non-luminous cadmium vapour. There is a strong symme- trically spaced absorption band ath = 2288 A.U. and a narrow sharply defined one at X=3260°17 A.U. 3. In theabsorption spectrum of non-luminous zine vapour there is a strong symmetrically spaced absorption band at = 2139°3 A.U., and a very narrow sharply defined one at N= 3075°99 A.U. 4. With the exception of the absorption band at A= 2338 A.U., all the absorption bands found for the vapours of mercury, cadmium, and zinc are the first members of either the series represented by y=1°5, S—m, P, or that represented by v=2, p.—m, 8 We desire to acknowledge our indebtedness to Mr. P. Blackman, who assisted us in taking the photographs. The Physical Laboratory, University of Toronto. May Ist, 1915. rons | LXXIII. On Intermittent Vision. By ©. V. Raman, M.A.* NE of the most curious and interesting of the pheno- mena met with in the borderland between the physics and the physiology of vision, is the occasional appearance of “intermittency ’ seen by an observer who watches a rapidly revolving object, e. g.a disk with alternate white and black sectors revolving in its own plane. The phenomenon has been investigated by Mr. Mallock, who, in his paper on the subjectt, puts forward the somewhat startling hypothesis that a slight mechanical shock to the head or body of the observer produces a periodic but rapidly extinguished para- lysis of the perception of sight, and that the nerves on which seeing depends cannot bear more than a certain amount of mechanical acceleration without loss of sensibility. Mr. Mallock’s conclusions have been criticised in a recent paper f by Prof. Silvanus P. Thompson, who, as the result of his experiments, comes to the conclusion that Mr. Mallock’s hypothesis is unnecessary, and adds that the explanation of the phenomenon appears to be that “* when the moving images of the white sectors on the retina are suddenly shifted by a minute displacement, they fall upon some of the rods and cones which are relatively unfatigued, and which for the instant are therefore of greater sensitiveness.” I have recently had occasion to examine this subject, and I find that Prof. 8. P. Thompson’s suggestion that retinal fatigue is the cause of the effect is apparently also untenable, as it is inconsistent with the observed phenomena. ‘There is no difficulty in testing the hypothesis that the fatigue of the rods and cones in the retina is the cause of the effect. If the images of the white sectors on the retina are suddenly moved, by some means, to positions of greater sensitiveness, namely, to the portions which the dark sectors had just passed over, we should, according to Prof. 8. P. ‘Thompson, expect the white sectors to flash out bright on a dark field. By observing the revolving disk in a mirror which is sud- denly tilted by a small amount, the necessary conditions may be secured experimentally with a stationary retina, but as a matter of fact the expected effect fails to manifest itself. Paradoxically enough, it is by a sudden apparent displace- ment of the white sectors in the opposite direction, 7. e. which brings them to positions on the retina already fatigued by * Communicated by the Author. + Proc. Roy. Soc. A. vol. lxxxix. p. 407. } Proc. Roy. Soc. A. vol. xe. p. 448. 702 Mr. A. B. Wood on the Velocities of the light, that we secure the desired result, namely, a sudden brightening and appearance of white sectors on a dark field. As the two ends of the diameter of the disk move in opposite directions, these two cases may be simultaneously observed and compared. It is thus seen that the explanation in terms of retinal fatigue fails to account for the facts, and it seems unnecessary to postulate either it or else the nerve “ paralysis” suggested by Mr. Mallock as the principal factor at work. The follow- ing explanation seems more in accordance with the facts :— So long as the retina is absolutely at rest, and the white and dark sectors follow one another at intervals short compared with the period of persistence of vision, the disk appears uniformly illuminated. But if the retina is set in motion even for asmall fraction ofa second, say by a slight mechanical shock, or by the eye involuntarily following the motion of the sectors, and if the direction of this motion is such that the white sectors remain on any given portion of the retina for a longer interval than they otherwise would, the impression of light over the areas occupied by the dark sectors has time enough to die away appreciably, and we thus get the illusion of stationary white sectors on a dark ground. A movement of the retina in the opposite direction should, however, pro- duce little or no perceptible effect, provided the rotation of the disk is sufficiently rapid. This is exactly what is found in experiment. Calcutta, May 10, 1915. —_—_— LXXIV. The Velocities of the a Particles from Thorium Active Deposit. By A. B. Woop, M.Sc., Oliver Lodge Fellow and Assistant Lecturer in Physics, University of Taverpool*. (Plate XVIL] cc oe and Rosriyson, in the Philosophical Magazine of October 1914, give an account of an extremely careful and accurate determination of the mass and velocity of the a-particles from radium C. Assuming the value of the velocity thus obtained and applying it in Geiger’s formula (v?=kR) connecting ranges and velocities of a-particles, a list of the initial velocities of expulsion of a-particles from the various a-ray products is tabulated. More recently, Tunstall and Makowerft have made a direct * Communicated by the Author. t Phil. Mag. xxix. Feb. 1915. a Particles from Thorium Active Deposit. 703 comparison between the velocities of the a-particles emitted from radium A and radium C, by measuring carefully the deflexion of the path of these «-particles in strong magnetic fields. Using the value given by Rutherford and Robinson for the velocity of «-particles from radium C, the value obtained for the velocity of e-rays from radium A is in close agreement with that calculated by Geiger’s formula. The work described in the present paper is incidental to a research, in conjunction with Dr. W. Makower, on the photographic effect and magnetic deflexion of recoil atoms from radium C and thorium C. In these experiments the magnetic fields used to deflect the recoil atoms are sufficiently strong to produce a fairly wide separation of the two groups of a-particles emitted from thorium active deposit. Accurate measurements of the deflexions produced by these magnetic fields lead to a very consistent value for the ratio of the velocities of the a-particles from thorium C, (a-particles of 8°6 em. range) and thorium C, («#-particles of 4°8 cm. range). By using a combined source of thorium aud radium active deposits a direct comparison of the velocities of the a-par- ticles from thorium ©, and thorium ©, has been made with the standard velocity of a-particles emitted from radium C. 1. Determination of the ratio of the velocities of a-particles from thorium active deposit. The apparatus used was similar in principle to that employed in previous determinations of this character. A Fig. 1. pp. Section of pole-face. dd, Slotted bar (graduated in mm.). a. Active wire (0136 mm. diam.). e. Light-tight tube leading to b. Slit (0-067 mm. wide). vacuum pump and pres- ce. Photographic plate. sure gauge. 704 Mr. A. B. Wood on the Velocities of the thin platinum wire was coated with thorium active deposit (activity equivalent to 0°15 m.gm. of radium) by exposing overnight to a strongly emanating source of radio-thoriam. The active wire, slit, and photographic plate were suitably mounted on separate brass blocks, which fitted rigidly into a slotted brass bar graduated in millimetres (see fig. 1). With this arrangement the respective distances apart of the wire, slit, and plate could be easily varied and accurately measured. The whole was then placed in a rectangular brass light-tight box which could be quickly evacuated to a pressure of the order 0°001 mm. This box was introduced between the pole-pieces of a large electromagnet capable of producing fields of 10,000 gauss over a pole-face 10 cm. by 3 em. with a gap of 1°35 cm. To obtain sharp lines and good resolution a very thin wire (0136 mm. diameter) and narrow slit (0°067 mm.) were -used in the experiments. The decrease of intensity of the lines entailed in this way was amply compensated by their increased sharpness. With the arrangement just outlined exposures were made for (a) one hour with the field on, (2) one hour with the field off, and (c) one hour with the field reversed. On developing the plate after such an experiment two pairs of lines symmetrically situated with regard to a single central line were observed (see figs. 3 & 4, Pl. XVII.). With the object of testing the uniformity of the field a series of experiments were made with varying distances source-slit (/,) and slit-plate (/,). Using the relation given by Rutherford and Robinson for the deflexion (y) of the a-particles, Bid 1) Io(l, + Lg) Eo or ee we see that when J,==/, and H is constant, the deflexion (y) is proportional to the square of the distance between source and slit or between slit and plate, 2. e. Occ. is Fig. 2 shows graphically the results obtained by using a constant field (about 10,000 gauss), the distances source-slit and slit—plate being equal. The two straight lines A and B correspond to the two sets of a-particles of ranges 8°6 cm. and 4°83 cm. respectively. It will be seen that any slight deviations from the straight line have the same direction in both cases, indicating that the error is not due to inaccuracy of measurement of the deflexions (y). The deviations are a a Particles from Thorium Active Deposit. 705 due (a) to small errors in the measurement of /—this error being increased in /?, (0) to slight differences in the mag- netizing currents, or to the iron being in a non-cyclic state in some of the experiments. —> ° af in cms) Deflection a2 For the calculation of the ratio of the velocities, only the photographs showing a deflexion greater than 0°5 cm. with reversed fields have been used. All deflexions were measured by means of a travelling microscope reading directly to 0005 mm. and by estimation to 0°001 mm. Measurements have been made from ten of the best of thirty photographs, from 20 to 50 sets of obser- vations being taken in each case. The results of typical measurements are given in Table I. The magnetic fields used in these cases varied between the limits 9,400-10,500 gauss, the variations of deflexion being produced by varying /, the distance from source to slit or from slit to photographic plate. 706 Mr. A. B. Wood on the Velocities of the TaBeE LI. | l. | The d.. | | Distance in | Distance in| Distance in Ratio (0,/0,) of cm. source to| cm. between | cm. between , Ratio | the radii of cur- | slit or slitto|ThC, lines|ThC, lines d,/d,. | vature of paths of plate. (range of rays) (range of | the a-particles. 8'6 cm.). rays 4°8). | ———S | | 30cm. | 0:4900cm. | 0-5901cem. | 0:830 0-831 1 2... 850m. | 06464em.| 0-7782cm. | 0831 0°833 3.... 35em. | 0:6550cm. | 0:7893cm. | 0:830 0:832 4...| 40cm, | 08458cem.| 1:0224em. 0828 | 0-831 5... 40cm. | 0:85600cm. | 1:0322cm. | 0-829 | 0-832 p, and ps, the radii of curvature of the two streams of a-particles, are calculated from the relation given by Rutherford and Robinson, P= pa (+a Ar +a), where d is the deflexion of the a-particle at the end of its path, 7. e. half the distance between the lines obtained on the. photographic plate when the magnetic field is reversed. The mean value obtained for the ratio of the radii of curvature of the paths of the a-particles was 0°832. Now the values of the velocities of these «-particles, as calculated in the paper by Rutherford and Robinson, are 1°70 x 10° and 2°06 x 10° cm./sec., giving a ratio 0°826. These velo- cities have been calculated from a knowledge of the ranges of the a-particles. Comparing the value 0°826, the ratio of the velocities (assuming Geiger’s relation), with the value 0832 given above for the ratio of the curvatures of the paths, it will be seen that the difference is a little less than 1 per cent. It is hardly possible for such an error to occur in the measurements tabulated above, a more likely source being the probable error in the values accepted for the ranges of the e-particles. _ Now Geiger’s formula has been confirmed in so many different ways that it seems safe to accept it as a well- established fact. Consequently, instead of offering the results of this investigation in further support of Geiger’s law, if we assume this law, these results would indicate that the agreement of the velocities with known ranges shows that the two sets of a-particles are identical in mass. Over the range of velocities studied—1'70 x 10° to 2:06 x 10° cm. per sec.—no certain change in the mass of the «-particle has a Particles from Thorium Active Deposit. 707 been observed, the deviations of the measured velocities from those calculated from Geiger’s relation falling well within the limits of error involved in the measurement of the ranges. of the e«-particles, and hence within the limits of error in the calculation of the velocities. Copies of original photographs are shown in figs. 3 and 4, BEV iT. “In fig. 3 the originals are reproduced actual size. (by direct contact printing): “fig. 4 is an enlargement of (a) and (c) in fig. 3. Photographs (a), (b), and (c), fio. 3, were: obtained using approximately the same magnetic field in each case but different distances (/) from source to slit. Thus in (a) /=3°0 em., (b) l= =3°5 cm., (c) /=4:0 em. It will be seen that, although the intensity of the lines decreases with increasing distance, the sharpness remains good in all cases. An examination of these photographs reveals another in-. teresting feature. In addition to the sharpness of the lines, their relative intensity is quite noticeable. Rough observa- tions of the two deflected lines, on either side of the central line, show that the intensity of the least deflected line (due to the swift «particles of 8-6 cm. range) is at least double the intensity of the more deflected line (due to slower a-particles of 4°83 cm.range). Many observers* have shown, direetly and indirectly, that the number of particles emitted from thorium active deposit with a range of 8°6 cm. is approximately double the number emitted with a range of 4°8 cm. Thus Marsden and Barrattt showed this to be the case by both ionization and scintillation methods. The results given in the present paper are interesting as being the first attempt to illustrate this point by a photographic method. With regard to this question, however, no final statement can be made until definite information is obtained as to the relative photographic effect of «-particles of dif- ferent penetrating powers. Thus it seems probable that an a-particle of range 8°6 cm. will produce a greater blackening of the film than an @-particle of range 48cm, This question is at present under investigation. 2. Direct comparison of the velocities of the a-particles from thorium active deposit with. the velocity of the a-particles from radium C. At the suggestion of Prof. Sir E. Rutherford a number of experiments have been made to obtain an absolute value for the velocities of the «-particles, using the standard value * e.g. Bronson, Phil. Mag. xvi. p. 291 (1908). Geiger and Marsden, Phys. Zeit. xi. Jan. 1, 1910. + Proc. Phys. Soc. pp. 50-61, Dec. 1911. = > — >». —— a ee if ts 708 Mr. A. B. Wood on the Velocities of the 1°922 x 10° em./see. as the velocity of the «-particles from radium C. For this purpose it is necessary to analyse the a-particles emitted from thorium active deposit and from radium © under identical conditions. This was aceomplished by introducing a slight modification into the experimental procedure outlined in the previous section. The platinum wire, after a long exposure to thorium ema- nation, was introduced for about 15 minutes into a small chamber containing a few millicuries of radium emanation *. After an interval of 20 minutes, when the radium A on the wire had practically disappeared, the wire, now coated with thorium active deposit and radium C, was introduced as before into the deflexion apparatus. Henceforward the ex- perimental procedure was the same as in previous experi- ments. On developing the plate one observed, instead of two lines only in the deflected positions, three lines—the third, intermediate, line being obviously due to the «-particles from the radium C. The best photographs were obtained in the experiments where the magnetic field was not reversed. Fig. 5, Pl. XVII. shows the appearance of a plate obtained in such an experiment. In the deflected position 3 lines are shown, the outermost line being due to the a-particles of range 4°8 cm. (from thorium C,), the intermediate line to a-particles of range 6°94 cm. (from radium C), and the innermost line due to e-particles of range 8°6 cm. (from thorium ©,). Table II. gives a few of the accurate measure- ments made on plates of this character—-the deflexions cor- responding to a magnetic field in one direction only, measurements being made from the central undeflected line. TaBueE IT. l=4:0 cm. H varies between 9,000 and 10,000 gauss. d,. ae ds. Deflexion inem.| Deflexion in | Deflexion in | Ratio of radii of of a-particles | cm. of particles) cm. of particles} curvature of from Thorium C,.|from Radium C.) from paths. Thorium C,. Pi. Pompe Ae 0°4085 | 04385 | 0:4935 1 : 0°934 : 0°832 Ds. 0°4070 0:4380 0:4921 1 : 0:932 : 0831 3... 03914 0-4210 (0720 1 : 0'932 : 0°831 P1y Px, and p3 are calculated as in Section I. * Supplied to me throngh the kindness of Sir E. Rutherford. — | > aaa | i | a Particles from Thorium Active Deposit. 709 The mean value of the ratios Pi: pz: p3 13 1: 0°932 : 0°832. If we compare the values of the velocities as tabulated by Rutherford and Robinson we get By Ve: Ve 2: 1 : 0932 = 0-826. We may thus conclude that the value of the veloeity (v7) of the a-particles of range 8°6 cm. given in the table is correct, assuming the accuracy of the velocity for radium C, whilst the value given for the velocity of a-particles of range 4*8 cm. is slightly low, 7. e. the range 4°8 cm. is a little too small. Assuming the correct ranges of the a-particles from ThC, and RaC to be 8°60 cm. and 6:94 cm. respectively, and using the value of the ratio of the velocities 0°832 obtained in the present experiments, we deduce from Geiger’s relation that the range of the «-particles from thorium C is 4°95 cm. Taking the velocity 1°922 x 10° cm. per sec. as the velocity of the «-particles from radium ( we obtain, as the mean of all observations (a) Velocity of a-particles (of range 8°6 cm.) emitted from thorium C,, = 2-060 x 10° em./sec. (b) Velocity of a-particles (of range 4°8 cm.) emitted from thorium (C,, =1°714 x 10° cm./sec. 3 Summary of Results. (1) By measuring their relative deflexions in a magnetic field, an accurate comparison has been made, by the photo- oraphic method, between the velocities of the two groups of a-particles (ranges 4°8 cm. and 8°6 cm.) expelled from thorium active deposit. (2) The mean value of the ratio of these velocities is found to be 1: 0°832, whereas the value calculated from Geiger’s relation (v°=/R) gives 1: 0°826. To explain this difference it is suggested that the value 4°8 cm., previously accepted as the range of a-particles from thorium C, should be increased to 4°95 cm. 110 Velocities of a Particles from Thorium Active Deposit. (3) A direct comparison has been made between the velocities of the a-particles expelled from thorium active deposit and those expelled from radium ©. ‘Taking the velocity of the «particles of 8°6 cm. range (from thoriam C. 2) 4s unity, we have a-particle. Observed | Velocity calculated from CN Geiger’s relation Source. | Range. y: (v3 =FR). The, 86 cm. 1 1 RaC | 694em.| 0982 0-932 The, 4°38 cm. 0-832 0°826 (4) Using the standard value 1:922 x 10° cm./sec. as the velocity of the a-particles from radium C, it is found (a) Velocity of «-particles from Th C,= 2-060 x 10° em./sec. (b) Velocity of a-particles from Th C,=1°714 x 10° em./see. (5) The relative intensities of the two a-ray lines shown in the photographs are discussed with regard to the question of the relative numbers of «-particles expelled with ranges of 8°6 cm. and 4°8 cm. respectively. (6) No evidence has been obtained as indicating a differ ence of mass of the two sets of a-particles. In conclusion, I should like to express my thanks to Prof. L. R. Wilberforce for his kind interest and for pro- viding me with the apparatus necessary for these experiments. I must take this opportunity also to express my indebtedness and thanks to Prof. Sir H. Rutherford, F'.R.S., for suggesting the second part of this research and for supplying me with the radium emanation used in the later experiments. Holt Physics Laboratories, University of Liverpool. July 1915. acme] LXXV. The Application of Van der Waals’ Equation of State to Magnetism. By J. R. AsHwortsu, D.Sc.* INDEX TO PARAGRAPHS. . Curie’s experiments and views. . Extension of the paramagnetic equation to include ferromagnetism. . The ferromagnetic equation. . The critical constants and the fundamental magnetic constants. . Corresponding states and the ferromagnetic equation. . The application of the ferromagnetic equation to the intensity of magnetization as a function of the tempe- rature, 7. Magnitude of the intrinsic field. 8. The kinetic theory and the ferromagnetic equation. Table of constants. 9. hs 1895 P. Curie published his paper on the “ Magnetic Properties of Substances,’ and in that paper he reached two important conclusions. The first was that feebly magnetic substances have in general an intensity of magnetization in simple inverse ratio to the absolute tempe- rature in a constant field of force t. When to this is added the law that the intensity of magnetization of feebly magnetic or paramagnetic substances is directly proportional to the strength of the field, we have a complete and simple account of the behaviour of paramagnetic substances in relation to field strength and temperature. If I is the intensity of magnetization, calculated as magnetic moment per unit volume, H the strength of the field, and T the absolute temperature, we can express the two laws in the equation I=Ay or Ht where R’ is a constant the reciprocal of which is Curie’s constant A. The manner in which the intensity of mag- netization varies in relation to field strength and temperature is thus analogous to the manner in which the density of a gas varies under the influence of pressure and temperature. The intensity of magnetization I corresponds to the density p of a gas, and the strength of the magnetic field H to the pressure P applied to the gas; the absolute temperature plays the same part in both cases. We have then the analogous paramagnetic and gas equations pea, pope 1 p — bo > Or He Co =R'T, * Communicated by the Author. + Curie, Guvres, p. 330. 712 Dr. J. R. Ashworth on the Application of The constant R’ in the paramagnetic equaiion corresponds to the constant R in the gas equation. The second important conclusion to which Curie was led was that ferromagnetic substances are transformed pro- gressively when they are heated, and tend to assume the properties of substances feebly magnetic as the temperature passes and increases beyond the point of transformation ™*. Curie pointed out that there is a close analogy between curves of intensity of magnetization ot a ferromagnetic as it changes to the paramagnetic state in a constant field, and curves of density of a liquid as it passes to the gaseous state at con- stant pressures when the temperature is raised. The analogy between the curves of [=/(T) and p=f(T) prompted him to ask the question: Are there critical constants for magnetism as there are for fluids? The parallel which exists between fluids and magnetism suggests that the change of state which takes place in each at the critical temperature is due to causes of the same kind. From the point of view of molecular theories the passage of a fluid trom the liquid to the gaseous state takes place when the molecules of the fluid are released from reaction one on the other and become independent, and in like manner we may imagine that the passage from the ferromagnetic to the paramagnetic state takes place when the molecular magnets pass from reaction one upon the other to freedom from mutual control. Further molecular theories of fluids show that there must be a limit to the density under the action of high pressures and low tempe- ratures, and correspondingly molecular theories of mag- netism show that there must be a limit to the intensity of magnetization under the influence of strong fields and low temperatures. Thus we may regard the ferromagnetic and paramagnetic states in magnetism as analogous to the liquid and gaseous states in fluids. 2. The correspondence between the laws of gases and the laws of paramagnetics, the continuity of the gaseous and liquid states established by Andrews, and the like continuity of the paramagnetic and ferromagnetic states established by Curie, together with the fact that the density of a liquid and the intensity of magnetization of a ferromagnetic both tend to a limit, suggest that a general equation applicable to fluids may be applicable to magnetism, and lead one to try if the extension which Van der Waals made to the gas * Curie, Zuvres, p. 329. Van der Waals’ Equation of State to Magnetism. 713 equation to include the liquid as well as the gaseous state can be applied to the paramagnetic equation to include the ferromagnetic as well as the paramagnetic state. The salient facts which Van der Waals embodies in his equation are the reaction of the gaseous molecules one upon the other, giving rise to an intrinsic pressure and the density limit. Written in terms of density his equation is 2 Sia +m)(~—<-) =RI, RAD ao 5 where p is the density, treated as the reciprocal of the volume, pp is the limiting density, and mw is the intrinsic pressure. According to Van der Waals this intrinsic pres- sure is a function of the density, and may be put equal to ap’, where a is a constant. This expression for the intrinsic pressure makes the equation a cubic in p, and gives to the calculated isothermals and isopiestics of a fluid an appro- priate shape. The equation to a ferromagnetic will be an analogous extension of the paramagnetic equation 1 ep H; =R'T, thus ; must be replaced by (; -;) in order to make 0 allowance for the effects of a limiting intensity of magneti- zation I), and H must be replaced by (H+H,;) in order to take account of the effect of an intrinsic field H; set up by the interaction of the molecular magnets one upon another. This intrinsic field will be some function of the intensity I and may be put H;=/(I). The general magnetic equation can then be written (H+/())(;--) =. 1 The verification of this equation is embarrassed by the effects of magnetic hysteresis, but I have shown in a former paper in this Magazine * that when hysteresis is eliminated there is evidence for the constancy of R’ in the ferromagnetic state, that the form of the isothermals is obtained when a limiting intensity of magnetization is inserted as above, and that it is necessary to introduce an intrinsic field of force into the equation if the shape of the isodynamic curves is to be appropriately represented and if R’ is to have a unique value in both states. In another paper f I have shown that * Phil. Mag. xxvii. p. 357, Feb. 1914. + Phil. Mag. xxiii. p. 36, Jan. 1912. Phil. Mag. 8. 6. Vol. 30. No. 179. Nov. 1915. 3A 714 Dr. J. R. Ashworth on the Application of the ferromagnetic metals behave correspondingly when the intensity of magnetization is treated as a function of the absolute temperature, and this result leads to the conclusion that there is a molecular magnetic mechanism common to the ferromagnetic elements, and consequently that there is a general magnetic equation applicable to all of them. The equation written above may be taken as a first step towards an expression which shall represent the behaviour of any ferromagnetic body the magnetism of which is a function of the field and the temperature. 3. The investigation of the magnitude of /{I) and the form it assumes becomes now of importance in applying the equation to ferromagnetic properties. If the continuity of states is accepted then R’ must be equal to the reciprocal of COurie’s constant in the ferro- magnetic as well as in the paramagnetic state. The value of this constant for iron, nickel, and cobalt is known, and this allows an estimate of the magnitude of the intrinsic field to be made. When I and H are expressed in the usual units and Tis in absolute degrees then H;=/ (I) is easily found to be of the order of 107 gausses. For example, R’=3°7 for iron and the product R’T at ordinary tempera- tures is nearly 10*; at higher intensities ¢ — r) is of the to order 10-4, and therefore, inserting these values in the equa- tion, we have (H+ H,) x 107-*=10°, hence H; is of the order 10’ gausses. ‘The intrinsic field for nickel and cobalt can be shown to be of the same order of magnitude from a knowledge of R’ for each of these metals. It might be thought that the form which /f(I) assumes would be determined by supposing the intrinsic field to be derived from the intensity of magnetization in the same way as the external field of a magnet is derived from its magnetization. This would make the intrinsic field directly proportional to the first power of the intensity of magneti- zation. The experimentally determined shape of the curves of I[=/f(T), however, does not support this supposition, and moreover, an equation in which the intrinsic field is treated as proportional to the first power of the magnetic intensity would be incapable of representing the double inflexion which curves of [=f (T) exhibit on passing through the critical temperature. In place of this hypothesis we may try if the intrinsic field is proportional to the second power of I, and put H;=a'l’, where a’ is a constant. This makes the ferro- magnetic equation of the same form as Van der Waals’ Va Van der Waals’ Equation of State to Magnetism. 715 equation to fluids, and is consistent with the analogy between the form of the curves of I[=f(T) and p=/(T) to which “Curie drew attention. Suppose, then, leaving for future consideration the manner in which the intrinsic field is set up, we write H;=a'l’, the equation to magnetism takes the definite form aT Ste See (H+a'P)(;—p)=R'T. This may be called briefly the ferromagnetic equation and the constants a’, Ip, R’ the fundamental magnetic constants. Since this equation is precisely analogous to Van der Waals’ equation it ought to yield similar deductions, and it will be a test of the general truth of the equation if these deductions are in the main confirmed by experiment. Nevertheless, as Van der Waals’ equation of state is only an approximate representation of the properties of a fluid, it is not likely that an analogous equation to a phenomenon so different as magnetism will express the results of experiments with equal success. In what follows the application of this equation will be considered so far as it relates to l=/f(T), leaving [=¢(H) for future treatment. 4, It is at once obvious that this equation is a cubic in I, and consequently that there may be critical constants as Curie surmised. They can be written in terms of the magnetic constants after the manner of a fluid, substituting a’ for a, R’ for R, and Ip for py. Thus we have for the critical tem- perature is 8 aI, l= 57 Rr: This equation is useful in allowing a value to be assigned to a’ when R’ and I) are known, and it is now possible to determine the magnitudes of the three magnetic constants in the ferromagnetic equation in the following way. As mentioned above, from the principle of the continuity of states, R’ may be taken as the reciprocal of Curie’s constant A. Horiron .A=-O0:281*) Byes 5:56 For nickel A=0°048 7 .-. R’=20°8 For cobalt A=0°166{ .. R’= 6:0 * Curie, Giwures, p. 327. Weissand Foex, Arch. des Sc. 4 ser. t. XxXi. pp. 4, 89 (1911). + Weiss and Bloch, Arch. des Sc. t. xxxiii. p. 298 (1912). t Weiss, Arch. des Se, 4 ser. t. xxxi. pp. 5-19 & 89-117 (1911). 3A 2 Sad . Ja 4 716 Dr. J. R. Ashworth on the Application of The maximum intensity of magnetization I, has been the subject of experiments by Ewing, Du Bois, and others, so that it is known with considerable certainty for the ferro- magnetic metals. We may put For ‘iron’ | 1>=1685 For nickel I,= 510 For cobalt I,=1300 The value of a’ cannot be directly determined, but it may be estimated from a knowledge of the other constants. If T,=785°+273° for iron, then from the equation to the critical temperature we have aT Sy lames whee Nana ae Another estimate of the value of this constant can be made as follows. By differentiation of the ferromagnetic equation with respect to the temperature we have Leia eas ae ae | ; I ) ; Paty) | The temperature coefficient a, of a magnet 1s Lae so that R! at Ho eee Oe mee ee at U1 ae omitting , which may be made negligibly small. If [=m], this becomes R’ mt almlg(1 — 2m) ° Now ae; for iron when m is chosen to be 0°92 is —0:00037 *, and therefore Ba hy, Hse 3°56 oh ~ —eamIy)(1—2m) ~ 0:00037 x 0°92 x 1685 x 0°84 Calculations made in the same way for nickel give a (eS a'=92 from the critical temperature, a'=90 from the temperature coefficient, when m=0°92 and a;=0°00059 *. * Ashworth, Phil. Mag. xxi. p. 86, Jan, 1912. Van der Waals’ Equation of State to Magnetism. 717 Also for cobalt a'=21 from the critical temperature, a' = 20 from the temperature coefficient, when m=0°92 and a,=0°00031 *. _A table of the estimated values of some of the principal constants for the ferromagnetic elements is given at the end of this paper. 5. The equations in the last paragraph show that corre- sponding states ought to hold in the relations of magnetic intensity to temperature. For any ferromagnetic substance nf a'I 8 dP R! prey R/ 1 Tt a'Ty m(1—2m)? and by multiplication we get 8 1 te 97 ma — Bm)’ a relation in which a’, Ip, and R’ do not appear. Thus if m, the ratio of the given intensity of magnetization to the maximum, is the same for any ferromagnetic metal the pro- duct aI, is the same. In other words, in corresponding states the temperature coefficients of the ferromagnetic elements are inversely proportional to their absolute critical temperatures. This deduction has been confirmed by ex- periments published in this Magazinet. The fact that the product aT, is the same for the same m= 7 for all ferromagnetics does not amount to a proof of 0 the correctness of the ferromagnetic equation; it would, however, be of more importance in this respect if it could be shown that aT is numerically equal to Buigt chuap 27 m(1—2m)’ or that 8 m(1—2m)a;T,= 27° * Ashworth, doc. cit. + Ibid, BSL SS Sse = st = Laas — ES oS = — ae a ae +S al > ~~ = a Fo - a .- es “Ee. oe. SS —_——— Ee Ee 718 Dr. J. R. Ashworth on the Application of The products on the left side of this equation are 0°30 (for iron), 0°30 (for nickel), and 0°32 (for cobalt) when m=0°92, and 0°30 (for iron), 0°32 (for nickel) *, and 0°37 + (for cobalt) when m=0°60, numbers which are in fair agreement with 0°30, the approxi- mate decimal value of the fraction - . The agreement, however, is not so good at values of m intermediate to these. : 8 : : : The theoretical number 97 with which the experimental values are compared is obtained on the assumption that the ferromagnetic equation is a cubic in I. 6. The manner in which the intensity of magnetization changes with rise of temperature can be traced by considering the differential coefficient of I with respect to the temperature. As before, we have dl R’ ang a tae wa -~Tte (1 me 7) H being supposed constant. Now when [I is large and always greater than $I, the first term in the denominator is negligible, and the second term, which is then negative, measures the rate of change of intensity. Thus the in- tensity falls at an increasing rate until [=3I). But just below this value of I the second term becomes positive, a being still negligible, and therefore = is positive, which indicates that the magnetic intensity has reached a state of instability, and a rapid fall takes place until = becomes again negative, which occurs at some very low value of I. When I becomes exceedingly small the first term in the denominator becomes the important one and, when large enough to allow a’ to be neglected, Curie’s law of the inverse proportionality of intensity to temperature then comes into operation. Thus the equation represents broadly the chief features accompanying the loss of ferromagnetism, namely, at first a region of progressive and increasingly rapid loss up * Phil. Mag. xxiii. p. 36 (1912). t Honda and Shimizu, Phil. Mag. x. p. 548 (1905); and Stifler, Phys. Rev. xxxii. No. 4, p. 268. Van der Waals’ Equation of State to Magnetism. 719 to the temperature of transformation, secondly, an abrupt transition through a region of instability, and, thirdly, a region in which the magnetic intensity declines less and Koss quickly and approaches the temperature axis asymptotically. Curie’s researches showed that the curve tracing the change of state loses its abruptness and becomies less steeply-inclined to the temperature axis as the field becomes stronger*. This result is immediately deducible from the equation under discussion, where it is seen that when H is extremely large the expression. would then always be negative and the region of instability would disappear. We should expect to find a critical field where the change occurred from abrupt to rapid continuous loss. Although there is this general agreement between the ferromagnetic equation and experimental results, the re- duced curve of [=f(T) traced from the equation does not fit very well the experimental curve at all temperatures. Van der Waals’ equation, however, seems to be faulty in the same way when it is applied to the determination of the variation of liquid density with change of temperature, and there is no doubt that an improved equation of state which would fit the experimental curve of p=/(T) would at the same time more correctly represent the experimental magnetic curve of [=/(T) for the following reason. The critical temperatures of nickel and water are nearly alike, and it is possible, therefore, conveniently to compare the behaviour under variations of temperature of the magnetic intensity of nickel as a typical ferromagnetic with the density of water asa typical liquid. When values of these quantities, treated as fractions of the maxima, are plotted against a scale of reduced temperatures, the curve for nickel is almost in- distinguishable from the curve for water?. Thus an equation which is correct for one would be correct for the other. 7. From a knowledge of the value of a’ it is possible to calculate the magnitude of the intrinsic field for any intensity of magnetization. At the maximum intensity I) we have H;=a'1,”, and for iron we get Fp TO (1685)4= 272 x 10". Similarly for nickel we get H;= (92) x (510)7=2°4 x 10%, and for cobalt H;=21 x (1300)?=3°6 x 10’. An intrinsic field of this magnitude was shown to be * Curie, Quvres, p. 332. + Phil. Mag. xxiii. p. 36 (1912). Oe —————— a a Fae eS a oa: a Ce Ae =e s 720 Dr. J. R. Ashworth on the Application of necessary if R! is to be treated as a constant for both states and equal to the reciprocal of Curie’s constant. It is generally conceded that an intrinsic or molecular field exists within a magnet, but it is the magnitude of this field required by the theory which presents a difficulty. Prof. P. Weiss has developed a kinetie theory of magnetism which demands molecular fields equally large, but he has not been able to account satisfactorily for their origin*. The experiments of Prof. Weiss + which seem to confirm with least objection the large magnitude of the molecular field are those relating to the change of specific heat with rise of temperature which the ferromagnetic. metals exhibit. The specific heats of these metals augment as the tempera- ture increases up to the critical temperature, and at this point there is an almost sudden reduction to the normal values. Calculation shows that the energy required to destroy the molecular magnetic field, if it is of the order of 10’ gausses, accounts for the increase of the specific heat. The agreement between calculation and experiment is very close. There is another experiment which may be made to bear on this problem to which I have briefly drawn atten- tion t. When two pieces of the same ferromagnetic metal are dipped in a dilute acid, such as acetic or oxalic acid, and a strong magnetic field is applied to one of them, an electro- motive force is set up between them in a direction such that the ferromagnetic ions tend to travel through the electrolyte from the unmagnetized to the magnetized electrode. The magnitude of this H.M.I’. has been determined by Hurmu- zescu § and by Paillot ||, and from their experiments a cal- culation can be made of the electrical energy required to transfer an element. of volume of the magnetized material from one electrode to the other, and hence of the magnitude of the intrinsic field from which it was removed. The result of this calculation is to show that the intrinsic field is of the order of 10" gausses. Another argument in favour of a large intrinsic field is the fact that the curve of [=/(T) is not altered in a high degree by the application of a powerful exterior field. This shows that even strong exterior fields are nearly negligible compared to the intrinsic field of a ferromagnetic substance. * Weiss, Ann. de Phys. t. i. pp. 184-162. T Weiss and Beck, J. de Phys. 4 ser. t. vii. p. 249 (1908). t Ashworth, Mem. & Proc. Manchester Lit. & Phil. Soc. vol. viii. part 11. (1914). § J. de Phys. 3 ser, t. iv. p. 118. | C. R. exxxi. pp. 1194-5. . Van der Waals’ Equation of State to Magnetism. 721 The calculated magnitude of the intrinsic field depends on the value assigned to a’, and this may be derived either from the expression for the critical temperature or from the ex- pression for the temperature coefficient. It is important, however, to notice that neither of these expressions deter- mines a’ without a knowledge of R’, as it is the ratio of a’ to R' which appears in both. Thus, even if the magnitude of the intrinsic field is denied, so long as the ratio of a’ to R' remains constant, the relations of intensity of magnetization to temperature which have been derived from the ferro- magnetic equation remain true. The progressive transfor- mation from the ferromagnetic to the paramagnetic state demonstrated by Curie is, however, evidence in favour of the view that R’ is an unique constant, and if its magnitude is that which it has in the paramagnetic state then a’ is determined, and the intrinsic field must be of the order 10’ gausses. 8. Curie’s work was followed by a kinetic theory of para- magnetism developed mathematically by Langevin*, and Langevin’s theory has been extended by Prof. P. Weiss f to include ferromagnetism by making the intrinsic field the subject of an hypothesis. He assumes that the magnitude of the intrinsic field is directly proportional to the first power of the intensity of magnetization, and writes H;=NI, where N is a numerical constant. From this hypothesis he derives three important formule which it is interesting to compare with corresponding ones derived from the ferromagnetic equation. Tirstly, for the critical temperature he writes eee ee ee where A is Curie’s constant referred to magnetic moment per unit volume, and N is the constant mentioned above. The ferromagnetic equation gives aes Reins! Gapae iP Oe oe A | ese expressions agree N= 974'lo, since A=. Secondly, for corresponding states Prof. Weiss gives the equation ee hd aca T. = (eh Fi ) ee eeSIOE fol AS. | et. (B) NIn res : where c= as » being the magnetic moment of the * Langevin, Ann. Chim. Phys. 8 ser. t. v. p. 70 (1905). +t Weiss, C. R. t. cxliii. No. 26, p. 1186. 722 Application of Van der Waals’ Equation to Magnetism. molecule, and I, and I,,, being saturated values of the intensity for a given temperature and for absolute zero: respectively. The ferromagnetic equation gives all I Teir(1-z) when H is negligible, and at the critical temperature aly 8 OT tT 1m(, 1 E7E elo g) an expression which principally differs from the one above by exhibiting a region of instability below I=]. Thirdly, according to the kinetic theory, the addition of an external field to the molecular field leads to the formula A=k(T, -T,), ef eed, cechaah he) goes (y): where k is the susceptibility and T, is a temperature above the critical temperature. , The ferromagnetic equation can be made to yield the same relation, by similar reasoning. Let the equation be written _ RTI i yeauomee Lut | when H is zero or is small enough to be neglected. If a strong field is applied then T must be increased to T, to preserve I at its former value, thus | T= and therefore a'T? Real. IT2 1+0 H+a'l =e By subtraction we get il _ Pp! iia sees H=R Toy 7). If T=T,, I will be small compared with I,, and so we may. write 7 =R(L—1), and hence 1 R! =k (Tt, — T.) ’ Energy Relations involved in the Formation of Atoms, 723 a result which is the same as that given by Prof, Weiss, since aN Thus the application of Van der Waals” R equation to magnetism leads to results which are not in conflict with the kinetic theories of Langevin and Weiss. 9. In the table which follows I have collected together the values referred to in this paper of the fundamental magnetic constanis, the critical temperatures, and the maximum intrinsic fields of the three ferromagnetic elements. The value J, is the maximum intensity observed at low temperatures. Like Van der Waals’ - appear to be constant, but diminishes at higher temperatures. to which it is analogous, it does not Symbol. Iron. | Nickel. Cobalt. Curie’s Constant VEE ol yap 0-281 | 0048 0°166 } Paramagnetic Constant ... 5 =k 3:56 | 20°8 6:0 Maximum Intensity ......... is 1685 510 1300 Intrinsic Field Constant...) a’ 76 92 | 21 Critical Temperature ...,.. | Te |785°+278° 388°+4278° | 1075°+273° Maxiinum Intrinsic Field a'l,? 2-2 X 107 2°4 x 107 3°6 x 107 July 31, 1915. LXXVI. Energy Relations involved in the Formation of Com- plex Atoms. By Witiam D. Harkrys, Ph.D., Associate Professor of Chemistry in the University of Chicago, and Ernest D, Witson, 8.B., Assistant in Chemistry in the University of Chicago *. sa tendency, which seems to have always existed, to - consider that all matter is composed of some one primordial substance is now particularly strong. The idea that energy, or one form of it, electricity, might be this primordial substance is also not new, but until quite recently proof has been entirely lacking. ‘That the mass of the negative electrons is electromagnetic seems to have been established by the experiments of Kaufmann and Bucherer, and the recent work of Moseley on the X-ray spectra of the * Communicated by the Authors. 724 Prof. Harkins and Mr. Wilson on Energy Relations elements points to the fact that positive electricity must also have a discrete structure. The mass of a particle of positive electricity, which may be called the positive electron, must be greater than that of a negative electron, but the positive electron is smaller in size. Two theories exist concerning the positive electron, one due to Rutherford and one to Nicholson. Rutherford has concluded from his work on the passage of alpha rays through matter that the hydrogen nucleus is the positive electron. On the other hand, Nicholson, from his work on the spectra of the nebulex, has come to the conclusion that they contain elements with simple nuclei of charge le, 2e, 3e, &e., and that the atomic weight of the element with nucleus le, which he has called protohydrogen, is about 0°081. He considers that the nucleus of hydrogen contains a certain number of these positive electrons, and one less negative electrons, so that the resultant charge on the nucleus is le, as experiments show. Hydrogen is con- sidered to be an evolution product of the system le, helium an evolution product of the system 2e, and the other complex elements evolution products of the other simple elements with higher charges on their nuclei. It must be remem- bered that the atomic weights of Nicholson’s elements of simple nuclei are dependent on the work of Bourget, Buisson and Fabry, and were obtained using their inter- ference method for the examination of some of the lines of the nebule. It has been shown by Michelson, and Bourget, Buisson and Fabry, that the widening of certain spectral lines varies as the square of the molecular velocity. For this to hold, however, the line must be of a certain simple type, and it is perfectly possible to obtain widely differing values for the atomic weight by using different lines in the spectrum of the same element. Moreover, the accurate measuring of the faint nebular lines is a difficult matter. The accuracy of the weights as given by Nicholson would seem, then, to be open to question. Nicholson’s system for the structure of the elements found on the earth fails to account for several very important facts. Calculations show that the probability that the sum of the deviations of the atomic weights of the elements from whole numbers should accidentally be as small as it is, is only one in fifteen million. Also, there must be some reason for the fact that in radioactive disintegrations only helium is evolved, and not hydrogen. From the standpoint of Nichol- scn’s theory of the structure of the elements, the regularity found in the atomic weights remains as much of a mystery involved inthe Formation of Complex Atoms. 725 as before, and the production of helium in radioactive changes also remains unexplained. In a paper presented to the National Academy of Sciences on February 25, 1915*, the authors gave the outline of a system for the structure of the elements which explains these facts, and which is based upon the atomic weights of the elements and the well established electromagnetic theories of mass and energy. If the atomic weight of hydrogen is taken as 1:0078, and this number is multiplied by 4 for helium, 12 for carbon, 16 for oxygen, and so on for the first 27 elements in the periodic table, the average percentage deviation of the determined atomic weights from these numbers is —0°77. Of these 27 elements, six show a difference of exactly 0°77 per cent. To account for the closeness of the atomic weights to whole numbers, the simplest procedure is to consider that they are made up of hydrogen, but it is evident that this deviation of — 0:77 per cent. must be accounted for in some way. That is, if helium is made up of four hydrogen atoms, in its formation there must have been a decrease in mass equal to 0°77 per cent. If mass is considered to be electromagnetic, it may be expressed as a function of the total energy of the system. Any change in the energy of a system would cause a change in its mass. Changes in energy might be due to either a change in the kinetic energy of the system, or in the potential energy. While there might be changes in the velocity of the various particles in the formation of one helium atom from four of hydrogen, it seems unlikely that the resultant change in energy would be very great, since practically all of the mass is in the nuclei; and while in all of the recent theories of the structure of the elements it has been considered that the negative electrons are moving with high velocities, the nuclei are supposed to either be stationary with respect to the atom as a whole, or at least to have only a relatively small motion. In regard to the potential energy, it has been known for some time that the total energy of a system of charged particles is not equal to the sum of the energies of its parts if the particles are close enough together so that their fields overlap. The magnitude and sign of this “packing effect” were calculated in the paper mentioned above. The following is a brief summary of the method, * Proc. Nat. Acad. Sciences, i. p. 276; J. Amer. Chem. Soc. xxxvii. p. 1867. 726 Prof. Harkins and Mr. Wilson on Energy Relations where the analysis is given in vector notation, and Heaviside units are used. The electromagnetic momentum, G, due to a system of ‘charges is [EH] [(@E)(H)] ou LB aude EN >| E.H,; | > [E iH; | ¢= a (27) C C c C where the summation & is the vector product of each 2 with each 7. Ww) The first summation gives the electromagnetic momentum which would be due to the particles if their fields did not overlap, and the second term gives the effect of the over- lapping of the fields. This may be called the “ mutual electromagnetic momentum,” and is designated by G. For point charges, ie (1—w)ey darr?( 1 —u? sin? 6,) 9?" Let (1L—wu?) =k, : and, (l—w sin? 0.) Be. The transverse component of E due to the two particles 1 and 2 is OePen(reraed sims yn mere Aa tae + “aes S where the sign is positive if the charges have the same sign, and negative if they are of opposite sign. As only the longi- tudinal component of the vector @ is desired, only the transverse component of E is needed. Then H=wEsin ¢/c, where @ = the angle between E and the direction of uw. If E is used, ¢ = 90°, and H = (E, sin 6,+E, sin 8;)/c. Hence eee ae oe = (i sin 6, +E, sin @,)(E, sin 6, +E, sin 0), and 2u os ; Qu Kte? (sin 6, sin 8 G=+-—+4 (zz, sin 0, sin 0, dr= + : Z FAO 2 e Ary) rr? BFR, Now 72 =P v7" sin*@)\ and 27 sin?e=y77 Let a=1/2 the distance apart of the electrons. Neglecting all terms in w?, and placing dr=2ydydz, we have us winte? a y? dy dz Ga 9 ‘i See tata"), \, wAlere! 49 lle ae which is obtained by making use of the symmetry of the at. involved in the Formation of Complex Atoms. 727 equation. The evaluation of this integral gives the value 1/2a, so | 2 ate? G= 4 Arrac? * The mass represented by this value of @ is ad e a pe Amrc?a” where Am is the change of mass due to the overlapping of the fields of the electrons. Now the longitudinal mass is | y 1 rR” where R=the radius of the electron. By division Am _ 3R Wipe Ga If Am is expressed in ordinary units, and a is replaced by d/2, where d is the total distance apart of the electrons, 2e? Am= -— od, and the total mass of a positive and a negative electron is 2 (3 + e* = 3e? M4.-=s3lu+-—- > ete 82\R ° 9 df» or in a general form 2 e,” €1€, 1 a: A eae ear {3 Va = s3E7e} : 3e ry 1,2 This equation in the form last given was derived by Nicholson in a paper presented to the London Physical Society on February 26, 1915*, almost the same time that the paper by the authors was sent to the National Academy. Nicholson derived the equation in a different manner. He points out the fundamental nature of this equation, and states that the effect of mutual mass must be considered in practically all of the work on the structure of atoms. It may be well to consider here one or two points of Nicholson’s paper which seem to be open to question. It is difficult to see how even the elements supposed to exist in * Proc. Phys. Soc. London, xxvii. pp. 217-229 (April 15, 1915). 728 Prof. Harkins and Mr. Wilson on Energy Relations the nebule, other than the system le, can have simple, or non-structural nuclei when they contain several charges of the same sign. It is contrary to our experience to find charges of the same sign packed close together unless there is some “ matter” to which they are attached. Moreover, in calculating the distances apart of the alpha particles in the radioactive elements, Nicholson uses an atom whose nucleus is made up of a simple positive charge of ne plus another positive charge of 2e, also simple, and two negative electrons. Around this rotate enough negative electrons to make the atom neutral. In the first place, the part of the nucleus represented by the term ne cannot be simple. As Nicholson himself states, it is recognized that there are negative electrons in the nuclei of the radioactive elements, and while in this case he assumes that there are two present, other considerations lead to the conclusion that these are not enough. It must be kept in mind that the mass of the elements has to be accounted for, and to do this it is necessary that the number of positive units (if they are hydrogen nuclei) in the nucleus must be approximately equal to the atomic weight of the element. Then in order that the resultant charge on the nucleus be equal to the atomic number, as Moseley’s work demands, a certain number of negative electrons must be included, and this number is not far from one-half the atomic weight. If the elements are built up according to Nicholson’s own system, they would have to contain a much larger number of both kinds of electrons. Also, the mass of the alpha particle, which is described as consisting of a simple charge of 2e, is entirely unaccounted for. If Nicholson is right, and the positive electron has a mass of about 0°081, the mass of the alpha particle could only be 0°324 at the most, using Nicholson’s Jaw that for the simple elements the atomic weight is proportional to the square of the atomic number. LHven if the hydrogen nucleus is considered to be the positive electron, a system consisting of two such charges would not (unless by Nicholson’s theory for simple elements) account for an atomic weight of four. It seems probable that the helium nucleus must be built up of four hydrogen nuclei and two negative electrons ; so it is difficult to see how any calculations of distances involved in complex atoms based upon the assumption that the helium nucleus is simple, can be correct. The consideration of the law of. radioactive change which was developed by Fajans, Soddy, and others, leads to a regular and simple system for the structure of the atoms. involved in the Formation of Complex Atoms. 729 When a radioactive element ejects an alpha particle, the new element lies two groups to the left in the periodic table, and therefore has an atomic number and a valency with values two less than before. Since this relation holds in regard to the elements of high atomic weight, it seems not unlikely that it may hold for the low atomic weight elements. If this is true, by beginning with helium and adding a weight of four—one helium atom—for each step of two atomic numbers, the atomic weights should be very closely deter- mined. This gives Atomic number even=4 8 12 16 20 24 28 32 mroammenamberodd = 7 Il 15 19 23 27 31 35 which are, on the whole, the correct atomic weights. Table I. gives the proposed structure of the first 26 elements of the periodic table. It will be noticed that there is a great regularity in the number of hydrogen atoms, H;, which must be included to give the atomic weights of the elements of odd atomic number. Taste I.—A Symbolical Representation of the Atomic Weights of the Elements in the first three Series of the Periodic Table. H=1:0078. | 0. 1. 2. 3. 4, 5. 6. 7. 8, at Ea Be a Se NO F |Ser. 2.|| He | He+H, | 2He+H|2He+H,| 3He |3He+H, | 4He | 4He +H, |Pheor.| 400] 700°| 90 | 110 |1200| 14:00 "|16-00! 19-00 |Det....| 4:00| 694 | 91 | 11-0 |12-00) 1401 |1600| 19:00 eo" | Ne Na Mg Al Si P S Cl Ser. 3.) 5He 5He+H,} 6He (|6He+H, 7He |7He+H,| 8He |8He+H, 20°0| 23:0 2400 | 27:0 |28:0)] 31:00 /|32:00) 35:00 20:0; 23:0 2432 | 271 |283) 31:02 | 32-07) 35°46 iq | Theor. i ; Det. eee | | ——— | A K Ca Se Ti V Or Mn Fe Co | Ser. 4.|10He 9He+H,;} 10He llHe |12He |12He+H,) 13He|l3He+H,/14He |14He+H, |Theor.|| 40°0| 39-00 40°00 440 /|480)/ 51:0 520] 55:00 |56:00) 59:00 | Det... 39°9; 39°10 40 07 44°1 481 51:0 52°70 | 54:93 | 55°84) 58°97 Increment from Series 2 to Series 3 = 4He. Increment from Series 3 to Series 4 = 5He (4He for K and Oa). Increment from Series 4 to Series 5 = 6He. An examination of this table shows that the elements in even-numbered groups, and therefore probably the radioactive elements giving off alpha particles on disintegration, are Phil. Mag. 8. 6. Vol. 30. No. 179. Nov. 1915. 3B 730 =Prof. Harkins and Mr. Wilson on Energy Relations built up entirely of helium nuclei, and hence would give only helium and no hydrogen on disintegration. Later it will be shown that there is another reason for helium only being given off, which does not depend in any way on any theory as to the exact structure of the higher atomic weight elements. The conception of mass as a function of energy leads to some very important and interesting conclusions. If the mass of a body is a function of the total energy, any body weighs more when hot than when cold, more w shen carr ying an electric char ge than when uncharged, and more when in motion than when at rest. In any chemical change where there is heat evolved, there is a decrease in mass. To cal- culate the magnitudes of these effects it is necessary to know the form of the function, and this will be considered later. The law of the conservation of mass, even on this newer conception, does not fail when applied to the entire universe. The energy liberated in any change of mass goes to increase the energy, and hence the mass, of some other body. It is only when a restricted system is considered that there are actual losses of mass. It will be seen later that for practically all of the ordinary chemical and physical pro- cesses with which we have to deal, this loss is negligible. From the theory of relativity, Binstein * in 1905 and Lewis + in 1908 derived the following equation expressing mass as a function of the total energy : iH M = @? where M is the mass, H the energy, and ¢ the velocity of light. Also Comer from electromagnetic considerations, has derived a similar equation which, how ever, differs from the one given here by a constant. Comstock ‘attempted to apply this equation ft to the problem of atomie weights, and recognized that if mass is a function of energy, the ‘elements may be built up of hydrogen even though the atomic weights are not exact multiples of the atomic weight of hydrogen. He considered, however, that the energy could change only through changes i in velocity. His calculations are rendered worthless, because in the application to actual cases he made * Ann. Phys. xvili. p. 639. + Phil. Mag. xvi. p. 705 (1908). t J. Amer. Chem. Soc, xxx. p. 683, p | j i ; 4 ; — ee CS ee involved in the Formation of Complex Atoms. 731 a serious error. He considered the deviations from whole numbers on the oxygen basis, all except two of which were positive, and attempted to explain them. It is very evident that if the elements are built up from hydrogen, the deviations which have to be explained are the differences between the atomic weights on the oxygen basis and the particular multiples of 1:0078 which are supposed to make up the elements in question. ‘Thatis, for carbon the deviation to be explained is the difference between 12 times 1:0078 and the atomic weight of carbon on the oxygen basis. It is found that all except three of these deviations are negative. Two systems which are at a distance apart may be con- sidered as being made to approach each other by some outside force. Let them approach until a new system is formed. If the total energy of this new system is less than the sums of the energies of the separate systems, it will be stable. In other words, the stability of a system depends upon whether or not its energy is less than for any neighbouring configuration. This evidently means that in the formation of the system energy was radiated, and there- fore, according to the definition of mass given above, there was a loss of mass. This energy which is radiated might be called the ‘‘free energy of formation” of the element in question. If the two systems described above were two hydrogen atoms, the system formed would be a hydrogen molecule. If, however, four hydrogen atoms should come together in such a way that the resulting system should be made up of four nuclei and two negative electrons, a helium nucleus would result. Just what conditions the configuration of the system which is formed is a question, at the present time. It seems very likely, however, that a process like the one described must take place. It might seem at first thought that if the free energy of formation of these elements is negutive, they should be formed in hydrogen under ordinary conditions, but there are several things to consider in this connexion. Whether or not a reaction will proceed of its own accord depends on two things: first, the sign of the free energy of the reaction, and second, the rate of the reaction. That it is impossible to predict whether or not a reaction will proceed is illustrated by considering the two reactions N,+ 0O,=2NO and N.+3H,=2NHsz, the first of which is exothermic and the second endothermic. In neither case does the reaction proceed if the substances are brought together under ordinary conditions. Just what the conditions are which are 3B 2 732 Prof. Harkins and Mr. Wilson on Energy Relations necessary to the formation of the elements out of hydrogen, ig not known. Using the equation given above, the energy involved in the formation of one mol. of helium from four of hydrogen is found to be AK=9 x 10” (4:00—4:0312) = — 2°808 x 10" ergs = — 6°708 x 10" calories This enormous amount of energy of formation of helium is well in accord with our knowledge of its stability. From Table I. it is seen that oxygen is considered as being made up of 4He. The energy change in its formation is four times that of He if it is considered that it is formed from hydrogen directly. It is evident that there would be some energy change when oxygen is formed, whether from hydrogen or from helium; but according to the above con- sideration there would be no final change in energy content if it were formed from 4He. However, an error of 0-001 in the atomic weight determination would cause an error of 9°0x 10" ergs in the energy calculation. As the atomie weights are known nowhere nearly so accurately, it may well be that oxygen is formed from helium with a con- siderable energy change. On the other hand, it may be possible that oxygen does not contain helium nuclei as such, although there are 16 hydrogen nuclei present. From these calculations it is clear why the radioactive elements should be expected to break down into helium instead of hydrogen. ‘The stability of the helium is so much greater than that of the system 4H; 2. e., the energy of the helium is so much less than that of 4H, that if the atom were about to break up, it would be much more unstable with respect to a helium disintegration than with respect to the hydrogen disintegration, In a recent paper on the “ Analysis of Gases after the Passage of Electric Discharge,’ Egerton* considers the energy change involved in the formation of 1He from 4H, using the same method as used above. However, when he substitutes the values for the weights involved, he inter- changes the two terms, and obtains AH as a positive quantity. That is, he considers that energy must be put into hydrogen to form helium, with a decrease of mass, which is obviously * Proc. Roy. Soc. A, xci. p. 180 (1915). | | | involved in the Formation of Complex Atoms. 733 wrong. He concludes that it is impossible that the helium found in vacuum-tubes is formed from hydrogen. However, it is evident that it is the slow reaction velocity, rather than the sign of the energy change, which prevents ordinary hydrogen from going over into helium, and when the evidence offered by stellar phenomena for the evolution of the elements is considered, it does not seem unlikely that part at least of the helium which appears in these tubes may be formed from the hydrogen. It might equally well, of course, be formed by the disintegration of the electrodes or the walls of the tube, by reason of the intense energy used. While, as stated above, there are changes in mass in chemical reactions, a simple calculation shows that they are far too small to be detected. In the formation of one mol. of water 68,000 calories, or 2°89 x 10" ergs of energy are liberated. Then AM — 259 x 1079 x 10" = —3'18x 107° gm. Thus the energy involved in our ordinary chemical reactions is extremely small in comparison with that involved in the formation of the elements. Since in the radioactive changes from radium to lead there is a certain amount of energy liberated, the atomic weight of lead should differ from that of radium by more than the weight of the helium atoms shot off. The heat effect for the complete series from radium to lead is not known, but the heat evolved in the decay of 1 gram of Ra in equilibrium with its products to RaO has been determined as 132 cal. per hour. Calculations using this value will not, of course, give the total change in mass which would result in the change of Ra to Pb, but will give an idea of the magnitudes involved. Since at any time the amount of heat evolved by any one of the radioactive elements which are in the equilibrium mixture has the same ratio to the total amount as at another time, the entire heat effect may be considered to decrease at the same rate as the activity of the radium. That is, Q' at any time ¢ is given by a Qe", where Qo is the heat evolved initially. If the time is expressed in hours, X=4°52x 107-8. The half-period for Ra is 1°5382x10' hrs. Q)=2:98x104 cal. The total heat Q evolved in the complete disintegration of 1 mol. of Ra equals 734 . Prof. W. M. Hicks on the twice the heat evolved in the half-period. Therefore 1532 x 107 =) , 98 x 10!)e— (452x107 *)t dt a 2 OS OS: lh ae ee: f_22810! 22 OLUT 0 Sao OX, 10 veal, 76x LO" enee. Then 9). 19 AM=~ 76 x 10 9-00 x 102° =0-0307 gm. The entire change of mass between radium and lead cannot be very much larger than this, since the only changes between these two Als aae which are not included in the thermal data are those between radium C€ and lead. May 29, 1915. LXXVII. Note on the Calculation of Serres in Spectra. By W. M. Hioxs, F.R.S.* N the October number of this Journal appears a joint paper by Mr. Savidge and Prof. Nicholson on the calculation of series in spectra, occasioned by a supposed difficulty in testing for series, and in calculating the formula constants. Asa fact, however, the usual method of testing for series by the aid of Rydberg’s table is quite simple. The determination of the formule constants requires natur- ally a certain amount of numerical work. Possibly an explanation of the method I have adopted in my own work may be of interest to others. I find it takes about twenty minutes’ work—including verification—to obtain all the constants involved. As an instance of test the complete set of OD,'’’ may be taken, the 2nd, 3rd, and 4th of which were used as examples by the authors. The wave numbers referred to vacuo are given in the first of the following columns :— * Communicated by the Author, = cis : Pn ae. Calculation of Series in Spectra. 735 10791°32 2°97 5462°20 1623352 397 2520°13 18753°65 4°97 1365°85 20119°50 597 821°81 20941°31 6°97 532°37 21473'88 7:97 36447 21838°35 8°97 261°356 22099°71 9°97 The second column contains their successive differences, the third, the denominators of the formule, given by Rydberg’s table, which cause these differences, e. g. N N Cee es re cae SE (E97 97 71S The equality of the decimal parts in the third column shows at a glance that the lines belong to a series which approximately follows Rydberg’s type of formula. If, as in the paper referred to, only the 2nd, 3rd, and 4th lines were tested, the Table would enable us at once to determine whereabouts the others should come. For instance, the line after 20119 should be about 822 ahead, the next 532 ahead of this, and so on. Really the example is not well chosen to exhibit the advantages of the method, since the series happen to satisfy approximately the Rydberg formula. Ina set which does not do this, viz. one in which, in the denominator m+p+a/m, « is comparatively large, the numbers corre- sponding to w in the Rydberg (‘97 in the example) would not be exactly the same for all the orders—only approxi- mutely so. With a numerator 4N all that is necessary is to divide the differences of the wave-numbers by 4 and proceed as before. In calculating the actual constants I have adopted the following method :—Taking any two successive lines the difference is found. Rydberg’s table gives the corre- sponding numbers with this difference. ‘Thus in the above the difference 2519-6 is caused by 6961—4442. Hence 16233 = A— 6961 or 18753 =A — 4442. The limit A is therefore close to 23194, say 23194+€, where & will only be a few units. 736 On the Calculation of Series in Spectra, The series of lines is then supposed given by N Grain (m+ w+ a/m)??’ where A is known and &, w, « are to be determined from three successive wave-numbers n. We have time OD) ERS en Ee (Ae | Ce = m+ dm—YmE (say) my + a = Mdy— My né. Differencing, pe = A(md,,) — EA(my») O= A?(md,,) — EAP), os A?(md,,) ~ A?(mym)? and mw and a follow at once. whence The method may best be exemplified by a numerical example—e.g., the first three lines of OD,’ above. The limit is 23194+ &, 10791°32 12402 68 4:0935156 4:37978 9465921 -0002397 "4732960 119°8 16233°52 6960'43 3°8426392 475610 1:1974685 -0005703 5987342 285°] 18758°65 ~4440°35 3°6474172 3°04893 1°3926907 °0011192 6965453 559°6 2'973693 3°969484 4969873 1947386 239°6 961066 615°7 2°908452 009974 855'3 767-4 971040 1383°1 3°879492 22384 9974 =°961066—18 x ‘0006157 008004 "953062 a= 1°947386—13 x ‘0002396 —2u “003114 1°909238 1906124 "033148 1:909238 n= 2320700 — /{ m-4-953062-4 095148 * Dispersion of Carbon Diowide in the Infra-red Region. 737 In the above columns, the first number is n, the second A—n, the third log (A—n), the fourth log N/(A —n) obtained by subtracting the third from log N=5-0401077, which, with practice, is easily done mentally as quickly as the figures can be written down. The last is N¥?/(A—n)¥, Then the third is subtracted from the fifth and written down on the right hand. ‘This is log a n)??, Under this is N¥*/(A—n)??, which, divided | by 2 gives Ym. The next step is to multiply the mantissz at the bottom of the columns respectively by 2, 3, 4 and difference twice. Similarly with the y. These operations are done beneath as shown. The constants are then found at once. The method also enables the values of O0& Om, Oa caused by observation errors easily to be determined. Should it be desired to deal with another term §/m?, we use four lines, multiply d,.y, by m? and difference three times. Then A?(m?2din i i ek 4 2 p= A?(m? dn) — EA?(m*y,,), a=A(m?dm) —EA(m?ym) —(2m+ 1)p, B=Mdn—m'ynE—mMa— mw. The University, Sheffield. LXXVIII. On the Dispersion of Carbon Dioxide in the Infra-red Region of the Spectrum. By C. Svavescv, D.Se., of the University of Bucharest*. a have been few investigations on the dispersion of gases in the infra-red region of the spectrum. Jobn Koch f, using an inierisrerice method, determined the dispersion, for hydrogen, oxygen, alr, carbon dioxide, carbon monoxide, and methane, both in the visible spectrum and in the ubré-red. He found the latter by making use of the “ Reststrahlen” of cale-spar (A=6°7094 4) and of gypsum (A=8'6784 4). I myself used a direct method, that of minimum deviation, such as Biot and Aragot used in their determination of the refractive index of air. Mascart$ * Communicated by Sir J. J. Thomson, O.M., F.R.S. t J. Kock, “ Dispersionmessungen an Gasen im sichtboren und im ultraroten Spectrum,” Nova Acta Reg. Suc. Scient. Upsatiensis, ser. 4, vol. ii. No. 5. t Biot and Arago, Mémonres de la premicére classe de I’ Institut, t. viii. 1806. § Mascart, Annales de I’'Ecole Norm, Sup. (2) vi. p. 1 (1877). 738 Dr. C. Stateseu on Dispersion of Carbon Dioxide also used a similar method in his well-known work on the dispersion of gases. The following is the theory of my experiment. Let @ be the angle of the gas prism and D the angle of minimum deviation. Then the refractive index (n) of the gas is given by the formula :— a aeels = ples mae 4 din oae MOON RNG Eee aan So; Since D has only a small value for gases we may write a—1l= scot ©. Taking the results of Mascart*, Benoit, Chappuis and Riviéret, and of Sutherland § on the relation between tempe- rature, pressure, and refractive index, this value of n—1 may be reduced to that at 0° C. and 760 mm. pressure by the formula :— D 2 where mp is the refractive index at 0° C. and 760 mm., and pis the pressure of the gas in millimetres of mercury measured at ¢° C. and reduced, for greater accuracy, to the value it would have had at 0° C. P. Chappuis’ value of a (=0°003716) was taken. The angle of the prism, ¢, was 900 2a 29). In the actual experiment I measured not the absolute deviation and the absolute pressure, but the change in deviation produced by a definite change in pressure, and substituted these values in the above formula to obtain the refractive index. The carbon dioxide was taken from a cylinder. It had been used in a previous experiment, so that I was able to assume it had been freed from air impurities. The gas was dried over calcium chloride and phosphorus pentoxide before being admitted into the gas prism. The arrangements for the prism were as follows. A tunnel of rectangular section 4°5 x 5 sq.cm. was cut through a block of brass. The sides of the prism were formed by two plates of rock-salt 18°58 mm. thick. ‘These were placed inclined to one another in the tunnel. Thus the brass itself * Mascart, loc. cit. + Benoit, Journ. de Phys. (2) viii. (1889). {~ Chappuis et Riviere, Ann. de Chim. et de Phys. (6) xiv. (1888). § Sutherland, Phil. Mag. [5] ii. pp. 141-155 (1889). N—L= 760 cot ae nie in the Infra-red Region of the Spectrum. 739 formed the top and bottom of the prism. The latter was in communication with a U-tube manometer and a pump. The temperature was taken by means of a thermometer in contact with the prism reading to tenths of a degree. The height of the mercury column in the manometer was read by a cathe- tometer marked in tenths of millimetres. The arrangement of the apparatus is shown in the diagram below :— 8 The source of energy used was the Nernst lamp N. The concave mirror mand the plane mirror m’ were used so that the image of the filament fell on the slit I. Between m and m' was a metal diaphragm with double walls provided with a shutter. The rays from I were diverted by means of another concave mirror M to fall on the rock-salt prism a, set In minimum deviation for all radiations by Wadsworth’s method. This prism had been used previously by Professor I’. Paschen* in his experiments on dispersion by rock-salt. The gas prism itself was standardized for the infra-red radia- tions by means of it. The rays emerging from a wire were focussed by M’ on I’, which was the second slit of the spectro- scope IMM'IL’. The spectroscope could be read to 72, of a second. The gas prism A was placed one metre from I’ and was set in the position of minimum deviation. Immediately behind it was a metallic mirror M’’ with radius one metre. This helped to form an image of I’ at a point B on the same side of A as I, the radiations thus passing twice through the prism. At Bwasathermopile connected with a galvanometer of the Paschen type. The thermopile could be displaced by means of a Zeiss travelling microscope to which it was attached. This enabled the linear displacement (Ad) to be read to yz, mm. Since the distance M’’ B was known * ¥; Paschen, Ann, der Phys. iv. Bd. 26, p. 120 (1908). 740 =Dr. C. Statescu on Dispersion of Carbon Dioxide (991°3 mm.) the change in deviation (AD)is expressed by the formula AD=Ad/2M"' B, because the rays passed twice through the prism. Between 0°38 wand 13m rock-salt is quite transparent, and it was in this region that I measured the dispersion of carbon dioxide. Iwo galvanometers were used, one, less sensitive, for the region up to 5 yw, and, for the remainder, a more sensitive one. In order to make the linear displacement, which is pro- portional to the pressure, reasonably large, I worked with changes of pressure of the order of two atmospheres. Between the limits I took, the rock-salt plates underwent no deforma- tion whatever. Some details of the determination of the displacement Ad must be given. Ad for each wave-length is the distance between the positions of the thermopile for the maximum deflexion of the galvanometer for two different pressures. These maximum deflexions were found from the graphs plotted between galvanometer deflexions and the positions of the thermopile. For each curve six to eight positions of the thermopile were taken. Hach curve was plotted twice. The first time the thermopile was moved in one direction and the second time in the reverse way. The maximum for each curve was found as follows. Tangents were drawn to the curve at corresponding pairs of points. These intersected above the curve. The maximum was taken to be at the point at which the best straight line drawn through the intersecting points cut the curve. I found a certain lack of symmetry in all the curves. I was able to correct for this to some extent by attaching greater weight to the accuracy of the tangents drawn from points near the foot of the curve than from points near the top. After readings had been taken for a series of wave-lengtlis at a certain pressure, verified at the end of the observations, and at a temperature read from time to time, the same observations were repeated at a different gas pressure. ach series of observations was preceded by one made by the eye in sodium light, so that the refractive indices might be given in their absolute values. I was able to gauge the accuracy of my observations from the analyses nade by Kayser and Runge* of the errors in their dispersion experiments. They found that a ditlerence of temperature of one-tenth of a degree or a difference in pressure of one-tenth of a millimetre of mercury had very small bearing on the fourth significant figure of the result. * Kayser and Runge, “ Die Dispersion der Luft,” Abh. der K. Preuss. Akad. d. Wiss. zu Berlin, 1898. in the Infra-red Region of the Spectrum. 741 With the linear displacements known to 7,55 mm. the fourth figure might have been obtained. However, the above men- tioned lack of symmetry in the curves made it uncertain. The following table gives the results I obtained :— TABLE I. Ap’ in n,—1 at 0 din p. Adinmm.| mm. Hg oo N—1. and 760 mm. at 0° c, | ™ C. Absolute value. 0°5894 Na | 3:0334 1411°5 {18°27} 0-000439, 0:0004508 08 3-061 14331 |18°-25 436, 448, 1-0 3018 14331 |18°-15 430, 441, 2:0 2-985 1433'L [182-10 423° 434, 3-0 2/343 1433-1 [180-15 406, 47, 4-0 1-962 14331 |18%15 279, 291, 5:0 3-642 1433-1 |18°-20 519, 531, 6°7 3-366 - | 14331 [18°20 480, 491, 87 3-135 14331 |18°-20 447, 458, 11-0 3:012 14331 |t8°-20 429, 441, 05894 Na | 3:0551 | 14186 |180-35 440 . 05894 Na | 3:0690 | 1423-1 |182-20 441. \ 0-0004508 08 3-042 14275 |17°-33 434, 444, 1-0 3-018 1427°5 179-37 431, 441. 2:0 2-961 14275 |17°-35 423. 432, 3-0 2/865 1427-5 |17°-40 409, 419, 4-0 1-938 1427-5 |17°-45 276, 236, 50 3°655 1427-5 [179-45 522, 532, 67 3-260 14275 |17°52 466, 475, 8:7 3-131 1427-5 |17°-52 447, 457, 11-0 3-031 1427°5 [179-55 443, 453, 13°19 2°730 1427-5 [179-60 390, 400, Table II., below, contains the average of various values of M)—1 corrected by means of the known absolute value of m—1 for sodium. ‘This value is taken as the average of those given by Ketteler, Mascart, Chappuis and Riviére, Perreau, and Walker. TaBueE IT. A in ol tink Observer : - Average value, . 0:5894 Na -0-0004508 { Average from values . of Ketteler, Mascart, &c. 0°8 446, Statescu, 1-0 441. : | 2:0 433, - 30 418, e | 4:0 289, iS 5:0 531, = 6°7 483, 4 6-7094 48038 J. Koch. 87 458, Statescu. 86784 45792 J. Koch. 11:0 447, Statescu, 13-19 400, ' 742 Dispersion of Curbon Dioxide in the Infra-red Region. The diagram below is drawn using the numbers in Table II. a jsaateuesces ry a a2 8 Sue a3 See a 1 7 Hina ue i I repeated the observations for X=1, 2, and 3 w respec- tively in conditions of greater sensitiveness. As the heat- energy was great in this region I was able to make use of a metallic mirror M’”’ of radius 3 metres instead of M”. M'' B was 3148°7 mm., so that Ad was magnified more than three times. Table III. gives the new values obtained. Asie EE. Ap' in mm.) 4 og. | fers Ain p. Adin mm. n,— 1. reduced to of Hg. | : absolute value. —_—_—_————— | CU 0:5894 Na | 9:9499 | 14238 |14°20 0-000443, 92786 | 14066 |14°20| 445, \ 0:0004508 10 9-720 14090 |14°-65 | 438; |] Py i 9-810 1421-6 |14°-75 440. | j or 2-0 9-544 1409-0 140 430, |) age © 9-629 1421-6 [149-75 432, | f 4 3:0 9-274 14090 [149-65 418, \ aa R 9-283 1421-6 |14°°75 | 416, | 0 Analysis of the curve shows the existence of an absorption band at 4°270 w and a tendency towards another one at about 14°7 p. Now itis known that carbon dioxide shows anomalous dis- persion at 2°715 w*, 4270 w*, and 14-7 ut. I could not detect the presence of a band between 3and4y. It may be supposed that, owing to these bands being very near one another, the very marked band at 4°270 w completely obscured the other. # K. Angstrém, Phys. Rev. i. p. 606 (1892). + Bubens and Aschkinass, Wied. Ann. lxiv. p. 600 (1898). } Tonic Molbilities in Hydrogen. 743 If these observations are combined with those of J. Koch* and H. C. Reutschlerf given for the visible spectrum and carried as far as X=0°3342 «w, we must suppose the existence of another band of absorption in the ultra-violet, as the values of n increase with the diminution of 2. In fine, the study of the dispersion of carbon dioxide between 0°8 » and 13 gives a marked band of absorption at 4°270 » and the tendency towards another one at 14°7 yp. The values obtained for n may be used to a first approxt- mation to find the formula of dispersion for carbon dioxide, which formula I propose to try to find. Its discovery would be an interesting one as it would be an application of the theory of dispersion for gases. This work was done in the Physics Laboratory at Tiibingen. T am much obliged to Prof. F. Paschen for his kind advice, as well as for the excellent apparatus which he put at my disposal, and to Miss Buckley of Girton College for her assistance in preparing this paper for the press. LXXIX. Ionic Mobilities in Hydrogen. To the Editors of the Philosophical Magazine. GENTLEMEN,— Weim experiments on the mobility of ions in hydrogen, described by Haines in the October issue of the Philosophical Magazine, confirm and amplify some measure- ments published in 1910¢ by Dr. Chattock and myself. It was found that in very pure hydrogen the pressure of the electric wind from a discharging point was very minute. The mean mobility of the negative ions, deduced from this, on the assumption that the discharge was unilateral through a portion of its path, was about 230 cm./sec. volt/em. The influence of the addition of known percentages of oxygen in redueing the mobility of the negative ions was studied very completely, the percentages added ranging from 0°0002 per cent. to 3°7 per cent. The effect of these additions on the mobility of the positive ions was shown to be nil. At the time there was no evidence from the work of other observers of the existence of electronic negative ions in any gas at atmospheric pressure, and another explanation of the results was sought. It was suggested that the high values of mobility obtained were spurious, and the results due to the setting in of a back-discharge from the plate as the purity of the gas increased. * J. Koch, “ Dispersionmessungen an Gasen...” Nova Acta Reg. Soc. Scient. Upsaliensis, ser. 4, vol. ii. No. 5. + H. C. Reutschler, Astrophys. Journ. xxviii, p. 435 (1908). t Phil. Mag. [6] xix. p. 463 (1910). 744 Tonic Mobilities in Hydrogen. However, in the corresponding number of the Bericht. d.- D. Phys. Gesellschaft (xii. p. 291, 1910), Franck published © his work on the mobility of ions in pure nitrogen and argon. | The similarity between his results and ours was so striking that it justified the assumption that we had greatly over- estimated the part (if any) played by back-discharge, and . that, as in nitrogen and argon, we were in reality dealing F with true changes in mobility. This was pointed out by one of us in ‘Nature’ (vol. lxxxiv. p. 530, 1910), and it was later shown (Phil. Mag. xxi. p. 585, 1911) that other effects | in point-discharge in pure and impure hydrogen could be readily explained by the existence of electronic negative ions. Though Franck recorded a negative result in hydrogen, Haines has now confirmed the existence of electrons in the pure gas; but he also appears to have obtained, in addition, evidence of other types of negative ion. The wind-pressure method merely gives the average mobility of all ions present, but the proportion of normal or intermediate ions to electrons in the pure hydrogen we used must have been small. Possibly this may be attributed to the stronger fields in which the ions travel in point-discharge; or possibly to the efficacy with which the walls of the vessel and the surfaces of the elec- trodes may be “cleaned” by a preliminary point-discharge. In negative point-discharge the rate of combination per coulomb between hydrogen and oxygen is abnormally high when oxygen is present in minute quantities. [The rate of combination is probably a function of the number of electrons transferred, loc. cit.| As oxygen and air were the only impurities used, it was comparatively a simple matter to wash the vessel out with pure gas, carry out a preliminary heavy negative discharge, and then perform the mobility measure- ments before the diffusion of oxygen from the deeper layers in the surfaces had time to make itself felt. The efficacy of this procedure was shown by the fact that the effect on the mobility of the negative ion of adding 0°0002 per cent. of oxygen to pure hydrogen was readily observed, and its rapid disappearance with continued discharge traced. Haines observed that hydrogen was neutral when added as an impurity to nitrogen. We found that nitrogen was neutral when added as an impurity to nitrogen ; thus the effect on the mobility of adding oxygen was identical with that of adding air containing the same percentage of oxygen, This naturally follows from the view that nitrogen and hydrogen in the pure state at atmospheric pressure both yield electronic negative ions. sels Yours faithfull University of Bristol, Ys October 16, 1915. A. M. Tynpaut. — eH b<| x & Merce aaa Phil. Mag, Ser. 6, Vol. 80, Pl si ates eS y e4 oT TET ia, Ut, ia. 12. ae ~* P sical he - ‘ nee - =e fn i Ka, ees A “re one im ~ ah aR 22 Se ee eee: 5 we = Pee. ae Sat to > 5 ‘ RR % Pe bP Ee b ee ne => iia | cat AD =... oP - - . : 4 ; ; s i , i ~ 4 a a < 4 a t ci ' t y » . « me ‘ - ra cs, , = Z 4 . rs ’ P » ee - - he i > » rv eu , - i : ; ¥, ; f f é ’ - . ; a < y rc - e c - - Cis oe . # = -~ 2 ; ‘ : . a0 se i - bs 7 ; - ’ 7 a d ‘ bg : 4 * * oe C . . F; : * rig ie? - 1 s < ~ - Ee “ ¢ Wy bd > $ . eva & DEARLE. Phil. Mag. Ser. 6, Vol. 30, PI. XIV. COPPER | TERMINALS ° ra. 3: Phil. Mag. Ser. 6, Vol. 30, Pl. XV. ine Reaentat tai = acs HH eae oS enans seas cease nsase ry) 130 IMA. WAVE LENGTH 26 24 no mo ee OO) “SIN NI NOILOATAIAG 2 WAVE LENGTH ° fe) NAL 10. Fie Phil, Mag, Ser. 6, Vol. 30, Pl. XV, McLennan & Deanve x 2 = HHH — i= ; Hee 2 4 a = = : : ies: ae : HH z 5 = : Penta: q 2 ae - siestaitati ee 4 i a EE ae = = t rt i + = H per iee : & HH + i = Hs easseee : + ‘ai : : EEE sec EEESHEHsttaitae 2 : if: SHEE = HTH : we iz = ; & os t Ps : eats Hf a ~ Hy aH i ts | : i p HH + Q T + ow = H a EEEEE HE + Be Ht se — { x : i g : HH 2g 3 8 & a” 8 “CHU HI NOWLaUag : the ree ele = THERE ra FETTTEE EES Hs Pa F ie z aH Haar gnten na gdastez H z es Ef Ht ~ pastas Ete: : F igsuzezed tazezseras Hi iH i ft Hs HE ‘| oe . REE it ai tet fiee gy @ : Ht rau Es line iz a us = : : of ui bread tess eg i & if tH aa i Ht nu 5: 3 2 SHE n 2 + crs) HHH HEEE “2 bee 94 ti 7 a saad if +H $ HATE PSE Ss : as sa tassaseesacsesd HH Rye igeasueai faszais genet Be a : au see Ss aa aeateae 2 : = 9 est otezs feted tezeitece # x Ht E Ht HES Eee aE = + fH ee EEE Sees Rec if z 4 ait 2 fe i sper tseed 23 r Hae Et tise |is iff - zt - Hee : bette io he 2 tH |: Ht 2 Hi if 3 c et & ono w S Re 7H t- cana + A : sn +4 te H : 0 Ht ie : : Hf ¢ $ =m ot) HH Ha fey Ez : seeoses HHH op uw ae t + =a) = + i: : t o@ CTS + eH + mae H seers r : mueene + . my Us) : + K : i et i Ht if 3 - oO a 2) = Fa Hy g ° a it t ie Hf i i : : : : f ©: Ht i = = z : : me Soe essaatasevaes tetutesatosay i ol betea fesad estat se- 4p ik HH ° : = 2 ° o * ~ ° + + ° Sw Ml NOMJITaaIa saudadnvY o wo mn McLennan & Epwanps. Phil. Mag, den ONELEH si - {UL all 3 . il [a ry I i “6 FL We Sees el 5 ae Re oe eT esem. oO a) ® a a A 0 2a Nu 5 N i) ; 2) WI) “@ OS 0) a N fo) (>) i : * m ' g 8 3 3 wo oO 1S it) ] N N Te a dea i || EE I | | re ills le 3075.99 2536.72 2288.79 3 ot © © = 0 ® Se : S ) WM) Be u Oo rn Oe) eh SEs 3 “~ uw a oS A. B. Woop. Phil. Mag. Ser. 6, Vol. 30, Pl. XVII. . Hires 3: THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] DECHMBER 1945. LXXX. Note on the Velocity of Electrons expelled by X-rays. By ©. G. Barxnia, #.R.S., and G. SHEARER, M.A., University of Hdinburgh™. ic emp relation between the maximum velocity of ejection of electrons from substances exposed to ultra-violet light, and the frequency of the radiation, has been shown by » Richardson & Compton + and by A. L. Hughes t to be given by the equation 4mv?=kn—E,, where n is the frequency of the light and & and Ky are constants for a particular sub- stance, but vary from substance to substance. The smail irregularities in the values of k observed by Richardson & Compton were considered by them to be within the limits of experimental error, but in the case of the experiments of Hughes the variation seemed to be a real one ; indeed it was apparently a regular variation connected with the atomic volume and probably the valency of the particular substance. The extreme values of k were those found for calcium and zine, that for zinc being nearly 1:2 times that for calcium. When the frequency of radiation is very high, Ho is negligible and the maximum velocity of emission from zine is nearly 10 per cent. greater than that from calcium. If such a law held in the case of ejection by radiation of the still higher frequency of X-rays, this difference in velocity ought to be easily observable, for the maximum distance traversed in air by the electrons from zinc would be more than 1:4 times that of the electrons from calcium. (This * Communicated by the Authors. + Phil. Mag. Oct. 1912. { Phil. Trans. Roy. Soc. A. cexii. pp. 205-226 (1912). Phil. Mag. 8. 6. Vol. 30. No. 180. Dec. 1915. 3C 746 ~=Prof. C. G. Barkla and Mr. G. Shearer on the follows from Whiddington’s Law that the path varies as the fourth power of the velocity.) As the corpuscular radiation from substances which from their behaviour under exposure to ultra-violet light might be expected to give well marked variations had not previously been examined, the first ex- periments were made to test for such a variation. Cooksey & Innes *, experimenting with ordinary hetero- geneous beams of X-rays, had, by the magnetic deflexion method, found small differences between the velocities of emission from the various substances Zn, Ag, Pt, Au, Pb. The results, however, showed primarily the constancy of the value of the velocity for a given radiation ; the variations observed were slight and appear to have been due to the experimental inaccuracies. Sadler f, investigating the emission under homogeneous X-rays, had found the absorbabilities in atmospheric air of the electronic radiations from Al, Fe, Cu, Ag, when these were exposed to a definite homogeneous X-radiation, to be practically the same. The conclusion was that the velocities of emission were the same. Whiddington ¢ had later calculated the velocity of emis- sion by applying his Fourth Power Law to estimates of the maximum path of the electrons obtained from Beatty’s § experimental results. In these experiments the ionization in a narrow chamber due to the corpuscular radiation from the ends was found for a series of pressures of the contained air. In the experiments undertaken by the writers, the last method was adopted, pressure-ionization curves being obtained for air ionized by a homogeneous X-radiation in a short ionization-chamber backed in one case by calcium and in the other by zine. The ionization-chamber was in the form of a shallow square box, 10 em. edge and 3:0 cm. deep (fig. 1). An insulated electrode in the form of a square frame of aluminium crossed by very fine aluminium threads was situated about midway between the two faces of the ionization-chamber. ‘This was connected by insulated leads through ebonite plugs to an electroscope. Inlet and outlet tubes were connected so that the chamber could be exhausted or filled with any gas. The front window, which was supported inside by a strong perforated brass plate, was of thin aluminium lined inside with filter-paper. At the back of the chamber was placed a screen of any desired material with its face in a definite plane. The secondary X-radiation characteristic of a given metal * Roy. Soc. Proc. Aug. 1907. + Phil. Mag. March 1910. J Roy. Soc. Proc. Jan. 1912 § Phil. Mag. Aug. 1910. Velocity of Electrons expelled by X-rays. 747 was directed through the window and ionization vessel on to the screen from which the corpuscular radiation was to be studied. A series of observations of the ionization produced Fig. 1. To electroscope Fig. 2 lonization Pressure in the*chamber at various pressures of the contained air was taken, and plotting ionization and pressure of air, a curve of the usual form (fig. 2) was obtained. 3 C2 748 Prof. C. G. Barkla and Mr. G. Shearer on the Increasing the pressure, the curve became straight as at P when the corpuscular radiation from the ends was totally absorbed. Thus at a pressure p the greatest distance traversed by electrons from the ends was just the length of the chamber, 3°0 centimetres. As the front face was of paper, the gain or loss of electrons from this was negligible ; and the surface of the electrode exposed was so minute that its effect also was negligible ; so that the curvature of the graph was due entirely to the net gain of corpuscles from the screen. The curves obtained with screens of calcium, zine, and: tin when exposed to Ag and Sn X-radiations (series K) were carefully compared. Although the exact shape of the curve depended upon the substance of the screen, the pressure at which the curve became straight was identical for all screens exposed toa particular radiation. In some experiments it was impossible to determine this point with a possible error less than about 10 per cent., but in others (in which the observations were more consistent) it appeared that a difference of 5 per cent., if such existed, would be detected. The results of these experiments showed that the electrons which under exposure to Ag X-rays (series K) were emitted by Ca, Zn, and Sn in a direction straight across the chamber, were just absorbed at a distance of 3:0 cm., when the pressure was 17°5 cm. of mercury. From Whiddington’s law and the observed agreement, it was seen that the speeds of the fastest electrons from Ca, from Zn, and from Sn when exposed to Ag X-radiation (series K) did not differ by as much as 2 per cent. A similar agreement was observed under exposure to Sn X-rays (series K), the pressure p being 25 cm. of mercury. (There is very strong experimental evidence in support of the assumption generally made that when the X-radiation is homogeneous the maximum velocity of emergence is the actual velocity of every directly expelled electron as it just leaves the atom.) We conclude, therefore, that the variation in the maximum velocity of ejection observed in the case of emission under exposure to ultra-violet light is not characteristic of the * The absolute value of the maximum velocity obtained from Whid- dington’s formula vt} =2x10*°Xd, where v is the initial velocity of an electron of path d in atmospheric air (O.G.S. units) is 108x 10° em. per sec. for the electrons ejected by Ag X-rays (series K), and 118°5 x 10° em. per sec. for those ejected by Sn X-rays. These values agree well with Whiddington’s general conclusion that the maximum velocity is 10° times the atomic weight of the substance emitting the characteristic X-radiation. Velomty of Electrons expelled by X-rays. 749 emission caused by X-rays. The maximum velocity of emergence from a substance is dependent simply on the frequency of the exciting X-radiation. The experiments of Sadler * showed that the emission of a characteristic X-radiation by a substance under exposure to a primary X-radiation is accompanied by an emission of a corpuscular (electronic) radiation from that substance. One of us has indicated f that not only is this so, but that a well defined corpuscular radiation is definitely associated with each characteristic X-radiation—that is, with the cha- racteristic X-radiation of each series. The point needs further explanation. It was first shown by one of the writers{ that each element has its own characteristic line spectrum in X-rays. A characteristic radiation may be excited in any substance by the incidence on that substance of X-radiation of shorter wave-length only. These homogeneous X-radiations have been called the K, L, M, ete. radiations, each letter corresponding to a spectral line—or rather, as shown later by interference ex- periments §, a strong line with several fainter lines about it —in the X-ray spectrum. When a primary X-radiation is transmitted through a substance and excites this fluorescent radiation, the primary radiation absorbed may be definitely analysed as (a) that scattered (usually though by no means universally small), (/) that absorbed in association with the emission of K radiation, (¢c) that absorbed in association with the emission of L radiation, and so on. There is thus what we shall call “ K absorption” of the primary beam, meaning the absorption which is clearly con- nected with the emission of the K fluorescent characteristic X-radiation, though it is not entirely re-emitted as K radia- tion ; L absorption similarly associated with the emission of L radiation, und so on. In the same way the ionization produced in a given substance under the action of X-rays may be divided into several distinct portions, each portion associated with || the * Phil. Mag. March 1910. + Barkla, ‘Nature,’ Feb, 18 and March 4, 1915. { Barkla, “The Spectra of Fluorescent Roéntgen Radiations” (Phil. Mag. Sept. 1911). ‘This paper gaye a summary of the results of experi- ments made by the writer with Sadler, Nicol, and others. § W.H. & W. L. Brage, Proc. Roy. Soc. A. lxxxviii, 1913; Moseley & Darwin, Phil. Mag. Jan. 1913, and later papers. || Distinct from that produced by the characteristic X-radiation itself when this is absorbed. 750 ~=Prof, C. G. Barkla and Mr. G. Shearer on the emission of a particular characteristic X-radiation from the substance ionized. Further,—this indeed follows in part from the last ex- perimental fact—the electronic radiation emitted by a substance may be regarded as divided into several distinct groups, each group being definitely associated with a particular fluorescent characteristic X-radiation from that substance. When a fluorescent characteristic X-radiation 1s emitted, there is emitted also a particular group of electrons. Hach of these groups we shall denote by the letter assigned to the associated fluorescent (characteristic) X-radiation. Thus there is the K corpuscular radiation emitted in asso- ciation with the K fluorescent X-radiation, and producing the K ionization in a substance exposed to the primary radiation. All these phenomena are associated with and consequent upon the K absorption of the primary radiation. Similarly we have the associated LL absorption, L cor- puscular radiation, L fluorescent radiation, L ionization, &e. As K and L corpuscular radiations have distinct origins, it seemed possible that the velocities of their emission might differ appreciably. Experiments were therefore made in order to compare the velocities of the K and L electrons emitted by a given substance. To do this, the ionization pressure curves were obtained as before, when the radiating screen of silver placed at the back of the ionization-chamber was exposed to the X-radiations (series K) from Ag, Sn, Sb, and I. From these curves by subtraction of the ordinates corresponding to the line OR parallel to PQ we get curves of the form in fig. 3, representing the ionization-pressure relation for the ionization produced by the balance of cor- puscular radiation from the ends alone. Under the action of Ag X-rays, Ag emits only L and some M and possibly N electrons, but not K electrons; Sn X-radiation ejects a few K electrons in addition ; Sb and I X-radiations eject a large number of K electrons, so that there are in the compound corpuscular (electronic) radiation from Ag roughly as many K electrons as L and M electrons. Four ionization-pressure curves were obtained for the ionization by the electrons ejected from the silver sheet when exposed to Ag, Sn, Sband I radiations (series K). As these radiations are arranged in ascending order of frequency, the velocities of ejection of the electrons, and consequently the minimum pressures of air necessary to completely absorb the electronic radiations, are in ascending order of magnitude. By adjusting the scales of abscissee, we can, however, compare the shapes of the curves and thus the velocity distribution 2-0 a Radiatiom, Jonization by Corpuscular ce) Velocity of Electrons expelled by X-rays. 751 of the electrons inthe four cases. If the scales of abscissze (pressure) be adjusted on the assumption that the maximum distance traversed by an electron is proportional to the fourth power of the atomic weight of the source of the exciting 5: 10 15 20 Pressure., Showing that the Ionization-Pressure Curves for Ionization by K and by L corpuscular (electronic) radiations are identical. Discontinuous line indicates the form of the Sb or 1 curve if the velocities had differed by about 10 per cent. fluorescent X-radiation, we must reduce the pressures for the Sn, Sb, and I curves in the ratios 1 to °68, °66, and *52 respectively. This places the Sb and I curves slightly to the left of the Ag and Sn curves,—almost in the position of the discontinuous curve of fig. 3. We, however, know that this law gives only an approximation to the truth; the factors ‘71, °78, and °68 were found to give the closest agreement. Such an adjustment of scales has been made in fig. 3, from which we see that within a small possible error the curves are identical. The Ag and Sn curves are, however, curves 25 752 ~=Prof. OC. G. Barkla and Mr. G. Shearer on the for L, and the very much smaller number of M electrons together; while the Sb and I curves show the ionization- pressure relation for K electrons in addition. The similarity indicates that the K electrons have approximately the same velocity as the Lelectrons*. Interpreting the above results in the way most unfavourable to this conclusion,—that is assuming a real difference of shape between the Ag and Sn curves and the Sb and I curves,—there is indication that K electrons are emitted with a velocity about 8 per cent. less than that of the L electrons. This is possible, though we hesitate, especially after examination of the results obtained with Cu and As radiations, to say that sucha difference has been established in any case. If, for instance, the K electrons had a velocity 10 per cent. less than that of the L electrons, they would be completely absorbed at less than 2/3 of the final pressure necessary to absorb the L electrons, and a curve of the shape shown by the broken linein fig. 3 would be obtained for a compound radiation of the K and L electrons. The method is therefore a sensitive one, and the close agreement between the actual curves shows that the lonization-pressure curve for K electrons is almost idéntical with that of the L electrons. The only conclusion we can draw from this is that the velocities of ejection of the K and L electrons do not differ by more than about 8 per cent. if at all. Similar experiments were made with the electrons ejected from copper by copper and arsenic radiations of series K. Copper radiation (series K) when incident on copper, pro- duces an emission of L,a smaller number of M electrons, and possibly N electrons associated with the emission of fluorescent radiations of still lower frequency. The L elec- trons, however, predominate. Arsenic radiation when incident on copper causes the emission of K electrons as well, and in greater number than the L electrons. As the electrons ejected by copper K-radiation only penetrate about 1 mm. of air at atmospheric pressure, the ionization- chamber was reduced in depth to 1 em. and the contained air was replaced by hydrogen. ‘The direct ionization of the * It is evident that the curves might have differed in shape owing to a different distribution of the electrons in space when these were ejected trom the plate by X-radiations of different wave-lengths. This would have admitted the possibility of an approximate balance of two changes in the shape of the curve, in which case the different velocity of K electrons would have been hidden. Apart from the improbability of this we have the evidence that there was no appreciable change in the shape of the curve due simply to change of wave-length of the exciting X-radiation, in the perfect similarity of the Ag and Sn curves. ; Velocity of Electrons expelled by X-rays. 153 gas was thus enormously reduced and that due to the elec- trons from the copper became the principal part of the total ionization ; in addition, the range of pressure over which the ionization-pressure curve was obtained was greatly increased, and the sensitiveness was consequenily increased. These changes and the substitution of Wilson’s Tilted Hlec- troscope for that of ordinary pattern were necessary on account of the smallness of the total effect of the corpuscular radiation emerging from the surface of a plate exposed to copper K-radiation. In this case after applying the calculated factor, a com- parison of the curves showed them to be identical. There is thus no observable difference between the velocities of K and Lelectrons emitted by copper under exposure to a particular X-radiation. The maximum velocity and distri- bution appear identical. The general conclusion is that though the K and L corpuscular (electronic) radiations have many distinctive properties, or more correctly have distinctive associations, after emission from the parent substance they do not possess distinctive velocities, or the velocity differences are very small if not inappreciable. Summary. If the electrons in the corpuscular radiation from sub- stances exposed to X-radiation were ejected under the same conditions as those ejected by ultra-violet light, the experi- ments of A. L. Hughes would lead us to expect well marked differences in the velocities from various substances. Careful experiments have shown these differences do not exist. The maximum velocity of ejection does not depend upon the particular substance from which the electron is ejected. (Possible error 1 to 2 per cent.) The K and L electrons ejected from a particular substance under the action of X-rays, though those electrons are different in origin and in association, are emitted with approximately the same velocity. (In one case a variation which might be interpr eted asa difference of about 8 per cent. was observed, though in another there was no suggestion of any difference outside the possible error of 3 or 4 per cent.) | We should like to express our indebtedness to Mr. J. E. H. Hagger for his valuable help in these experiments. | tes LXXXI. Rotation of Elastic Bodies and the Principle of Relativity. By P. J. Dante, Assistant-Professor of Mathematics, Rice Institute, Houston, Tex.* T is well known that the rotation of a “rigid” body about a stationary axis is inconsistent with the Principle of Relativity. In fact, the circle along which any part moves is contracted while the radius is unaltered. It has been suggested that a rotating circular disk might buckle, but it could then no longer be regarded as rigid. Herglotz, ina paper on the mechanics of deformable bodies +, has shown a method by which problems relating to elastic bodies may be solved. The method depends on the variation of the integral of a potential function of the rest-deformations ; that is to say, if each elementary part of the body is referred to axes with respect to which it is at rest, the deformations in terms of these new axes are the rest-deformations. Here this method has been used to consider the question of the rotation of an isotropic elastic body. It is found that even when the elastic constants are infinite, rotation is possible, although the rest-deformations are not zero. The body is not “rigid ’ in the absolute sense of Born’s definition. It might perhaps be called semi-rigid. By some the Principle of Relativity has been extended so as to include the assump- tion that even elastic waves cannot be propagated with a velocity greater than that of light ; then the ratios of the elastic constants to the density cannot exceed the square of the velocity of light. ‘The elastic ratios of all material bodies, even of su-called ‘ incompressible” fluids, do not approach this value. But in electron theory the ‘ semi- rigid’ body whose rotation is not inconsistent with the Principle of Relativity may be even more useful a con- ception than Born’s rigid electron. Only small deformations will be considered, and the square of the ratio of the velocity of any part to the velocity of light will be taken as small. Squares and products of such quantities will be neglected throughout. Let + denote a position vector in the undeformed body referred to a system § ; +b denotes the corresponding deformed position vector referred to space-axes rotating with the body ; r’ denotes the corresponding deformed position vector referred to 8. * Communicated by the Author. t Herglotz, “Mechanik des Deformierbaren Korpers,” Annalen der Physik, xxxvi. p. 500. Rotation of Elastic Bodies. 755 If wu’ is the vector velocity of an element, dr'=dr+db+u'dt. Let dr = dr —_ da denote the differential of the corresponding position vector obtained by making a Lorentz transformation to space-time axes with regard to which the element is at rest. Then da is the rest-deformation differential vector. The Lorentz transtormation gives ! dry=dr' + = (u' .dr') —Buldt, where ee Jue * aoe and ¢ is the velocity of light, or = dryp=dr-+db+ = (u'.dr+db). Instead of w’ consider u= (wX7), where w is the rotation vector and (x7) is a vector product. Then, neglecting squares of small quantities, dry=dr+db+ u.dr), al uU or da=db+ pa (ear). Again, feces aC ' dr’) + Bat, =— B ay .dr+db)+ 8 (1-3 Jar. If we consider ¢) as changing while 7, and therefore 8, is fixed (i. e., uniform rotation) , dty=8(1— wat, =(1— ail dt, or Kdt let us say. 756 Prof. P. J. Daniell on Rotation of The potential function depends only on the rest-defor- mations a. It can be seen that oA wv (1.) div a=div b+ 32° (11.) curl a=curl b, (iil.) grad a,= grad b;+ i .U,; (iv.) at the surface if 6n denotes an element of normal to the surface, and if n, is a unit vector in the direction of the normal 0a 00 on On Only bodies symmetrical about their axes of rotation will be considered, and then w is perpendicular to 7. uU + (ny U) 5a: 0a _ Ob Or ae Let us put * ype sO egg i MUOGe ie) i Oe La aa Ou 5 yy —_ Oy b) ae Oz lL OG 20a, Lay oan * oa. * = Oaz eyz = 12 aR 02 > = 5(55 Ae ) Cry = 5) Al ah Oy ) The ordinary potential function is given by W’ where ZW! = (AF 2M) (Cre + Cy + Coz)? +AU (Crys + Cree + Cay — Cyylzz — Conbon — Cal yy) This can be reduced to the form 2W =) (div a)?—(curl a)? + 2u[ (grad a,)?+ (grad ay)? + (grad a,)*]. The “action ” would be given by T— W’, where T is the kinetic energy, : 2 We can bring this kinetic energy into the expression of the potential function by assuming it to be W=—M-—W’ per unit 4-volume in (py oto) space. W=-—3sArCdiv a)? + 3(curl a)*—p[ (grad az)?+ (grad ay)’ + (grad az)?] —M. rX, @ are the ordinary elastic constants, M = c? x the density p. pl ea te ee 9P pe eae * Love, ‘ Elasticity,’ 2nd edition. Elastie Bodies and the Principle of Relativity. 757 The required differential equations are found by making the variation 8 (\\\ Wdzdydzdt, =0, or 8 ({\\ Wade dy dz(Kdt)=0, or 6 (iV W Kda dy dz ==N) if the rotation is uniform. In the elastic terms of W, K can be put equal to1 ; K is retained only when it is multiplied by M, and then the kinetic energy term appears. This variation is 0 for all possible variations 66 if 2 d grad div a+pcurl curl a+ 2u (div grad) a+ M grad a Je throughout the volume, and if at the surface AN, div a+ u(y X curl a) + 2y Se —(, nr These are exactly the equations which occur in: the Newtonian mechanics (cf. Love, ‘ Elasticity’) ; but they refer to a, the rest-deformation, not b, the echiel defor- mation. We must use the transformations (i. to iv.) above. Then dr grad div b+ wcurl curl na a grad) b +(A+M) g grad 55 5+ (grad .u) — =0, 6 (grad.u) u=(u. oh ut+u div u, div u=div (wx r)=0, (wu. grad) w= (w xu), % grad w=(u. grad) u+(ux curl uw). But curl u= 2 or (wx curl vw) = —2 (@ xu) =—2 (u. grad) u, so (grad .u)u= —4 grad w?, Hence the volume equation becomes (v.) (A+2p) grad div b—yp curl curl # +(A—“n+M) g orad — 52 a and the surface equation becomes (vi.) An div b+m(mx curl bd) 1, oe +2 an +rAn —A) 20? 758 Prof. P. J. Daniell on Rotation of These are the same as the ordinary equations obtained for a body with the same elastic constants d, «, with density A— M M a instead of p or 2? and acted on by a surface force 2 U nage? : normal to the surface=r 5,2 Pressing inwards. The solutions in any particular Bee can be found by the ordinary methods, if these two changes are made. For the general case Betti? S agree Theory * can be used. We shall denote by @,, the average value, not of €,, as defined above in terms of a, on of an e,, = Ob: defined in terms of the actual deformation. There will be apparent 2 volume forces (M+%—y) grad a and apparent normal 2 (imamate surface forces X 5-5 inwards. ZC Let E denote Young’s modulus, o Poisson’s ratio, V the volume. 2 ae c d 3%) pace Al a hae FACE Vig ~ ay i pare —aym—azn)dS, where dS denotes an element of surface, 1, m,n are the direction cosines of the outward drawn normal, (e 2 2 2 But {{| a 5 (52) de dy dz = |\jeete- (\\ sesde ay ac Then 3 d a) Cr, = aaa) Lz da ~ (53) Sei = oe 5 (50 2] dx dy dz oe: 53 (L—2¢) de dy dz. d But qa (5) =3 ae [ (oy? +0,)e—w,0,2—0,0,y]. * Love, ‘ Elasticity’ (2nd ed.), p. 170. r | Elastic Bodies and the Principle of Relativity. Let x=0, y=0, z=0 be principal planes of the body and let pe (\{ 22 dx dy dz, ete. Further, let Oz be the axis of rotation and suppose the body to be svmmetrical about this axis. 759 Then o,=0, o, =0, o,=o, kesh P=gky’. Then i = EE wk,” dee by = OFF [(M—p)(1—0)—2(1—20)] oh? pee ie mira: ehh: 7 2c) ~ Newtonian mechanics would give the first term only, and the average @,, @,, are less according to the Principle of Relativity. é.. = OF [—20(M—p)—A(1—20)) = —w*h;? — This is the same as in Newtonian mechanics. Similarly, using Betti’s theorem it can be shown that é,. = €., = €, = 0, as in Newtonian mechanics. We can define a “ semi-rigid ” body as one in which E is infinite, not only as compared with p, but with pc? or M. Then é,.=0, @,=@,,=é,=0, = ke wk? oe eee FA *, Thus a body with infinite elastic constants, or infinitesimal Wii, can rotate, but it does not remain “rigid” in the absolute sense. >, » M(l—oc) Mo iii ‘eee — GE M(ii-— —20) ae 2h 760 Rotation of Elastic Bodies. This is 0 when 2 is infinite, but a itself is not necessarily 0 throughout. For all known material bodies, although H/p is large, H/M=pc? is very small. Hence no experiments on the rotation of bodies could decide between Newtonian and Relative mechanics. Two interesting special cases can be worked out by methods practically the same as in Newtonian mechanics (of. Love, ‘ Elasticity’ (2nd ed.), p. 144). Case 1.—Long cylinder, assuming a uniform longitudinal extension e so that w=ez. When r=a the relative radial extension . is given by uw oe Mo) ue Ge Ae [ E | CE ortad NP: a ae, For a “‘semirigid ”” body u wa e=0, -=-— i a Case 2.—Thin disk, with the same assumptions as in Love’s ‘ Elasticity.’ When r=a, z=0, u wa? ao 1 i Mo(l+c) 2? s iD 2 K 3a? a When 7=0, z=1, w_ wf MA+Ey) My | [Th 82) LOZ DCN ae Oa i one For a “ semirigid” body According to some an electron is supposed to possess no non-electromagnetic mass. In this case M would be infini- tesimal and the electron would appear, at first sight, to be “semirigid” rather than absolutely “rigid.” Such an On Certain Problems of Two-Dimensional Physics. 761 electron or even a positive ion (of smaller radius so as to possess greater mass) could rotate and something like a magneton would be the result, even if the elastic constants were not supposed to be infinite. Actually, however, the electromagnetic ‘ potential”? energy will produce effects analogous to those due to a mass density varying from the centre to the circumference. By supposing > and wp to be infinite, the “ semirigid ” rotating electron (electron- magneton) could still be used as an hypothesis consistent with the Principle of Relativity. LXXXITI. On the Solution of Certain Problems of Two- Dimensional Physics. By J. R. Witton, M.A., D.Sc., Assistant. Lecturer in Mathematics at the University of Sheffield *. ily A GENERAL method of solution of certain types of physical problem, in which the boundary considered consists of a single analytical curve, may be founded on the obvious remark f that the transformation +t =X(r)+cY(r), in which r=y—v&, and X and Y are real when 7 is real, makes the real axis in the 7 plane correspond to the curve Ee PVs eo EE in the «+vy plane. We may therefore take the equation of any analytical boundary in the form a Dean Site ie: tale) ankle 0=r. For the sake of brevity, we shall denote X(n) by X, X(r) by Xy, and X(@) by X,, with a similar notation in the case aL XY. or, if 02=7+ &, we have In the simplest type of problem we are required to deter- mine a function yr from V*r-=0, together with the conditions ~=/ (vy), oF =F (m) on the boundary, where dn is an element of the outward drawn normal. The solution is Y=HFO+FO}+ F( (K+Y)FOD)ao, * Communicated by the Author. + Cf. Forsyth, ‘Theory of Functions,’ § 265, p. 624 (2nd edition) ; also Jeans, ‘ Electricity,’ p. 264. Phil. Mag. 8.6. Vol. 80. No. 180. Dec. 1915. 3D 762 Dr. J. R. Wilton on the Solution of But, in general, the boundary conditions are not alone sufficient to determine Ww, and we have to resort to other means in order to obtain the final form of the solution. In the examples that follow the endeavour has been to give a consistent exposition of the mode of attack on the problems of most frequent occurrence; hence the inclusion of a number of well-known results. Hydrodynamical Problems. 2. If in hydrodynamical steady motion under forces whose potential is O(a, y) the curve (1) is a free surface, we obtain the stream function (Harnshaw’s) w by means of the boundary conditions y~=0, G2) +) oven where ( is constant, together with V*,=0. The result is 0 yas, | P+ Vy (C+20(%, Yay, ee The theory of plane progressive waves may be based on this result, but the work is practically identical with that of the well-known method due to Stokes. 3. The motion of a cylinder of any form in perfect fluid at rest at infinity may be obtained with equal ease. Let the cylinder be moving with a velocity whose com- ponents parallel to the axes are U and V, and let it be rotating with angular velocity . Then we have V*»p=0, and on the cylinder Oy ee Oe ee ee Be ae Bs o(#Si +5"), 2. @. r= Uy—Vae—to(2?+y’) when 6=7. And thus Le a = 3, {F(@)—F Cr) $+$U(Y14 Yo) —$ V(X, + X,) —Jo(XP4- Xs + Ve+ Vy), where the function F is to be determined from the fact thai a is nowhere infinite and vanishes at infinity. As an illustration we take the familiar case of the elliptic cylinder, for which X=ccoshrAcosy, Y=csinhAsin 7». Certain Problems of Two-Dimensional Physics. 763 So that v= 5 (FO) —F(7)}+40U (sin 0 +sin tr) —3aV(cos 6+ cos 7) — ac’ (2 cosh 2X.+ cos 28+ cos 27), where a=ccosha, b=csinhnr. Hence, finally, omitting a constant, w=e4(bU sin yn —aV cos n) — fac? e~* cos 27, 4. In the case of a vortex filament bounded by the curve (1), we have, within the vortex Vi;=2¢, and without the vortex V/*,=0, where &, supposed constant, is the vorticity. And on the boundary we have oe ~ Yor piano ty"), where it is assumed that the vortex rotates with constant angular velocity w. At infinity yy) must take the same form as the gravitational potential of a cylinder of the same form and of density —27¢. Further, in order to avoid infinite ae and “Sn must vanish at the singular points of the transformation z=x+u=X(tT)+cY(r), velocities, namely, the points where ps =, 2. @. X'()=—0¥"(n), and therefore X'(0)=cY'(8). We easily find Ate ty?)— 59 (K+ Re4- V+ Ve)+ 5 (F@—FO}, r9 oa to (KP XP4+ V4 Va)+ 5) KY XY) d+ FF) -FO}, where F is to be determined by the conditions given above. The case of variable vorticity may be treated in the same way. Itis evident, however, that the problem is precisely the same as that of determining figures of equilibrium of rotating fluid, where w is put for —w? and € for —2mp. (See §§ 10 and 11 infra.) 3 D 2 764 Dr. J. R. Wilton on the Solution of We take, by way of illustration, the hypotrochoid of n oscillations, z=acosy+6cos one ; (2) y=a sin 9—b sin (n—1)y. This includes the ellipse when n=2, and when 0 is not too large—the greatest possible value for 4 is 1/(n—1)—it represents a circle disturbed by an harmonic inequality with m maxima. We have, in fact, r= a?+6?+ 2ab cos nn, and if 6 is so small that squares of b/a may be neglected, r=a+6 cosnn, and » differs from @ by a multiple of b/a, where » and @ are polar coordinates. In general, we have atoy=a et 4b e~@-YE+m), a+ yaa? c% +b? e—2"—DE 4 2ab e— — 7) cos nn, 4(X? + X.? + YP4+ Y.”) =a? + B? + 2ab cosh né cos nn, 5 | (XY —X'Y)dn={a—(n—1)b?}E— (1—2/n)ad sinh n€ cos ny. And therefore, provided that the boundary does not eross itself, 2. e. provided that b } 1/(n—1), Wi=40(2? + y?— a? —b? —2ab cosh n€ cos nn) + wab e~"é cos nn + (1—2/n)£ab sinh n¥ cos nn vy =aab e~ cos nn + {a? —(n—1)b?} E82. The singular points of the transformation are given by n&E= log {(n—1)b/a}, nn=2kr, where & is an integer or zero. Thus @ is determined by the condition that dy;/O&=0 at these points. We have a) 1 (n—1) & DP Me as ot n (een: and i= 3t(at + y—a? — #9) —Lab)n){o% + (n—1)*(B[a2) e-"} cos ne, y= Cf a? — (n—L)B*}{E+ [| (n—1)/na] e~ cos ny}. Certain Problems of Two-Dimensional Physics. 765 In the case of the ellipse, n=2, we have the well-known result o/S=3(a + 6%) /a’, where a+6, a—b are the semi-axes. And in the case of the n-cusped hypocycloid, for which b=a/(n—1), o/f=1—2/n. As a corollary to the general case we notice that a= 304 2? +9? —2[a?— (n—1)0? | E—2ab| e-* + (4/n) sinh n£] cos ny} satisfies the equation V/**"=2¢, which is a particular form of V*h=f(y), and is such that the velocity vanishes on the cylinder (2). It therefore represents a possible motion of viscous fluid within this cylinder. 5. In the case of viscous fluid motions so slow that the squares of the velocities may be neglected, we have mee hers. on | kop ; ae aaa weer aay +vV/*v. And in the case of steady motions this leads at once to Vay — 0, ° : ° : e . e (3) with the conditions w=v=0 on the boundary. The general solution of (3) subject to these conditions is easily seen to be cbs ) dn l J together with a similar term, with y, Y in place of a, X, which may be written down by symmetry. In the case of the ellipse, for which xv +uy==c cosh (X+&+4en), | 1 ty r= 9; RO) —F(7r)—2 | b we have 1 | Y= 5,4 Pent) -F-e) a —sech r cos 7 cosh (A+ p| FE’ (m) sec ndn \ + &e. n—wé In accordance with a result first given by Stokes, it appears to be impossible to determine a solution corresponding to the 766 Dr. J. R. Wilton on the Solution of case of the elliptic cylinder in a steady stream. Consider, however, the stream function nue B w=(A+ B)E— al Ge sec 7+ — cosec n) dy n—& = (A+ B)e— ral tan7! (22) ee an Gee a) a COs 7 b sin We obtain the velocity at any point from the formule DUDE, OF DE BYP DE | DYPDE OE dy t On Oa” OF Oe | On Oy’ And we have immediately By =A+B— ae § sinh (A+ &) cos tan1(22 ®) cosh £ connie 0& CA cos” / © 1+ (sinh? €/cos? ) Be : ee *) cosh & sinh (A + &) Ne at coh ae veniy ian sin pate et (sinh? &/-in? 7) J” oy = cosh (A+ €) sin n} tan wks | AWE t cosy / sinh? + cos? Be... een Asante 3 sinh & sin n v a sinh (1+ &) cos 4 1 tan ( Ani) | eink ae At infinity we have ie = — Teu(> cos 9 + ; sin n), wy Alin B oe = 1 ate(= sin 7 — 5- cos n)- And therefore v= 9 Further, on the ellipse £=0 we have w=v=0, provided that neither cos y nor sin 7 vanishes. In the neighbourhood of the extremities of the minor axis put ee 5b’ Qi) i 1 (= — — 2 sinh &= tan 4u.cos ». Then, at these points, A : A SEY 5 u= TAC sin a) Oe TACHI COS 1) + a? where —7 <= w <7, but p is otherwise arbitrary. Certain Problems of Two-Dimensional Physics. 767 Similarly, in the neighbourhood of the extremities of the major axis, put sinh &= tan 4y sin n, Then at these points, u= + S — 94 b+ cosv), v=— Se sin v), where, again, —r p=xlu,y)—4 1x, Yy) +x( Xe, Yo) ; = al (¥ a — X'S ain a is 9 + Uy—Va—Jo(Xi+Xar+-Yit4 Yat) 2 | "(RY XY ae, Remembering that Ps) : fo) fo) ! ! 1 Tae? OF a3 (SY SE) (a! 4 1V') = 5, (uo) (40, we may write down the velocity in the form | ye OX Ox ONX1 Ox1 Hoy ta Ya OX ee where = V(X, Y;) . If itis possible to choose y so that this expression vanishes at infinity, we obtain a solution for the case in which the cylinder moves through the fluid at rest at infinity. (Cf. §§ 5 and 6.) If in addition to making yf vanish at infinity y satisfies the equation V*y=0, we have V/*f~=0, and the analytical form of the solution is precisely the same as in the case of slow motion of viscous fluid with the same boundary conditions. In the problem under consideration in the present paragraph we are not limited to the case of slow motions, but it must not therefore be supposed that the solution may be applied to problems of viscous fluid motion in which the velocities contemplated are not small. It is only in exceptional cases that this is true. j hy fi : Certain Problems of Two-Dimensional Physics. 769 Electrical Problems. 7. The potential of a freely electrified cylinder is evidently g=V+ 5, {FOF }, where F is so chosen aie at infinity o=— 2H log, E being the charge per unit length. Take, for instance, the case of the cylinder io cosy, | Y —Olemiege YS (4) in which n must be a positive integer. We have and at €=+, w& +ly=a cos” (n —t&) + ub sin” (n—2); pe’ ya re e4 aaah, ‘so that log r=né, approximately. Thus the potential of the cylinder bounded by the curve (4) is o= V—2nHé, provided that the singular points of the transformation do not fall within the field of ee OF.» Tb n> I there are singular points at E=0, n= 6, re =, 2. €. the electric density is infinite at these points. 8. The potential of a cylinder magnetized transversely may be determined in the same way. For simplicity we shall assume that the components of the intensity of magnetization are derivatives of a single function J, so that i een ale ae V70,=0, VW?2(0;+47J) =0, and on the boundary We then have where Q) is the external, 0; the internal magnetic potential. 770 Dr. J. R. Wilton on the Solution of In addition we have 0)=0 at infinity, and 30,/o£=d0,/a7=0 at the singular points of the transformation. We have immediately Q=% | (7) 40K, Vs) +f @)—Ad (Ky Ya) | a oat F(@)—F() }, O=84 f)+/@ 4rd) +51 FO-FO |, where F and / are to be determined by the conditions stated above. As a very simple illustration take the case of an elliptic cylinder uniformly magnetized. We have, in this case, J=—Axa—By. And it immediately follows that Oo=e = {(H+47aA) cos n+ (F+475B) sin 7}, 0;=0,—47 sinh &(bA cos 7 +aB sin 7). In this result HE and F are constants to be determined from the values of 90 /d0& and 00./dn at the singular points. We have 00: 0& sae Bee { (i +4aA) cosn+ (E+ 47bB) sin n} — 4 cosh &(bA cosy +aB sin n), Oeae tf — (H+ 47aA) sin y+ (Ff +47dB) cos n} +4 sinh £(bA sin 7—aB cosy). And these must both vanish when E=—), sinn=0. Thus we have 7 H=—4rA(at+becoshr) = —47aA(a+ 26)/(a +5), F=—47B(b+ae™sinhr) = —47dB(2a+6)/(a+0). And, finally, OQ = —47{ab/(a+b)}e-*(A cos n+ Bsin 7), O;= —4m sinh £(bA cos n+ aB sin 7) + O,. Certain Problems of Two-Dimensional Physics. 771 9. In the case of a cylinder of dielectric material, we have directly V=t{rO+/@) f+ 5° [FO-F@ }, vi=i{so@tsin! ae (@)-F@) }, where Vo, K, are the ve V., Ki the internal potential and specific inductive capacity, and fand F are to be deter- mined from the usual conditions of finiteness, &c. For the boundary conditions are a Re aI Lb 9 OE t OE when 0=7. If, however, the boundary be charged to surface-density o, we must add to V; the term tr] | o(X'?+¥)¥dn. 2a WK; Let us consider the simple case of an uncharged elliptic cylinder of radius a and specific inductive capacity Kk, surrounded by air, and in the presence of a line charge, H per unit length, cutting the ay plane at the point whose coordinates are 2%, yo. Let i ae } (w7—ay)’ + (y— yo)" }3, be the distance of the point x, y from the line charge. Then in the neighbourhood of 7;=0 we have Vj =—2E log 7. But, putting Ly + UYy= Ce cosh (A+ Ey+ ey), we have ry = (¥@—2o)" + (y— Yo)” =c?{cosh (V+ &+0)—cosh (A+ &,+ ep) }? =c”{cosh (E—£,) —cos (y—m)) } {cosh (24+ & + £) —cos (y+) }- And therefore, in the neighbourhood of the point &, mo, log 7; = 4 log {cosh (E—&o) —cos (n—)} =4 log {2 sin 3(0--7 ,—v&,) sin (7 — +48) } =} log { [cosh (£ — &)) —cos (7 —m9) ][ cosh (E + &) — cos (9-0) J} cosh (E—&) —cos (9 — 7) +4 °8 cosh hey cee eorae 172 Dr. J. R. Wilton on the Solution of and to get the corresponding term of V; we have to divide the coefficient of the second logarithm in the expression last printed by K. Hence a term of Vg is —H log {cosh (€—&) —cos (n—70)}, and the corresponding term of V; is —}H(1+1/K) log {cosh (££) —cos (7 —19)} — $Hi(1—1/K) log {cosh (E + &) —cos (n—19)}. But we must remove the second logarithm, since (if & < 2) it becomes infinite at —&, 7. By precisely the same analysis as that just used we thus find a term of V; in the form —3H1+1/K) log {cosh (€—&) —cos (9 — 7) $, and the corresponding term of V, is —{(K+1)?/4K} log {cosh (§ —&) —cos (n—)} + {(K2—D/4K}H log {cosh (+ £,) cos (7 0). in order that this last term should take the correct form at &, 7 we must divide through by (K+1)?/4K ; and we thus obtain a term of V, equal to —E log {cosh (££) —cos (7 —)} +{(K—L)/(K +1) }B log {cosh (E+ £6) cos (na) } and the corresponding term of V; — {2H /(K + 1)} log {cosh (££) —cos (q—) }. At the singular points £=—)dA, 7=0 or zw, the part of 0V:/0é arising from the logarithm in this term is —sinh (A+ &) /{cosh (X+ &) = cos yo}, and the part of 0V;/07 is + sin 9,/{cosh (A+ &) + cos7)}. Hxactly the same terms arise from the expression —log {cosh (2X +&+&)—cos (n+0)}. . - (6) Hence a possible term for the internal potential is —{48/(K+1)} log ; and we have to determine the effect on V, of the addition of Certain Problems of Two-Dimensional Physics. 773 the proper multiple of the term (6) to V; This term is equal to — {2H/(K +1)} log {2 sin $[0 +) +4(2A, 4+ &)] sind[7-+,—1(2X4 &) ]} — {E/(K+1)} log {4 sin (¢ + ea) sin (6 — za) sin (W— tx) sin (Wr + 1a) } aD 1 2sin (+a) sin (r— we) K+1 2 2sin (d—wa) sin (~r+sa)’ where $=2(0+%), Y=H(t+m), a=A+4£) The term to be added to V, is found by multiplying the coefficient of the second logarithm by K. Hence Vo>=—E log {cosh (E—£&) —cos (9 — 99) } + {(K—1)/(K + 1) }E log {cosh (€ + &)—cos (n—1) ¢ — H log {cosh (24+ £,+ &)—cos (n+) } + {(K—1)/(K+1)}E log {cosh (244+ & —£&) —cos (7 + 9) }- And we have Vi=— {4H/(K+1)} log 7. This, however, is not the solution of the problem we set out to solve, for there is a Jine charge of strength eee) KR + 1) at E=2r+ €,, n= 27 —. In the case of the circle X is infinite (¢>0 in such a way that tce\=a, the radius of the circle) and the terms given are sufficient for the complete solution. In the more general case we superimpose the solution for a charge {((K-1)/(K+1)}H at 20468, 27-7, thus obtaining the solution of a problem involving a charge —{(K-1)/(K+1) PE at 4048, no. And proceeding in this way we have, finally, Vi=—{4E(K+1)} 3 {(K-D/(K+)}" x Slog [cosh (2nX+ £,—£)—cos {n—(—)" 7] + log [cosh {(2n+ 2)X4-& + &}—cos{n+{—)*}] t, Vo= —Elog {cosh (E.—£) —cos(y).—7) $ +4(K—1)/(K+1)$E log jeosh (& + £) —cos (n,—7)} — {4KE (K+1)?} 2 {(K-D/(K+1) og [cosh (20+ f+ §) = —cos {7—(—)" } ], 774 Dr. J. R. Wilton on the Solution of provided that these series converge,—as they evidently do, for all finite values of & except that at E=&, n=, Vo becomes infinite like —2E log7,;. And when € is infinite we have Vo=—(2E/(K +1) }é— (4K B/(K +1} & {((K-)/(K +) pe == — 2H log r;. a So that all the necessary conditions are satisfied. There is little interest in the case of the circular cylinder, as the solution can immediately be derived by the method of images. But there is some interest in the case of the thin plate of dielectric, X=0, for the series for Vo and V; can then be summed. [For all points outside the plate (which cuts the wy plane in the line y=0, —c < # < c) we have V=— Blog {cosh (&)—&)—cos (m-—7)} — FE log {cosh (£,+ £)—cos (qo +7)}, where a+wy=c cosh (§ +47). Gravitational Potential. 10. If Vo and V; be respectively the external and internal potentials of a gravitating cylinder bounded by the curve (1), we have V?Vi=—4nmp, V?Vo=0, and when 0@=7, Vi= Vo, 0V./0E=0 V/0&. Also Vo>—Alogr asr>o, where A is a constant, which in the case of constant density is 20 x the area of cross-section of the cylinder. Let yy be determined so that —4rp=V7. Then Vi=vo, +Hy (0) +/(0)} + 5 FO-FO)}, Vo=dig(®) +9(0)} + 5 (G0) —G(n)}. And we have immediately . 9=f + W(X, ¥), [fen BN} Lae at » Oo. . G'=k —(X' te Certain Problems of Two-Dimensional Physics. 775 Thus with the previous value of V; we have Vo= Hf (8) +f (0)} + 5 (FO) —Fin)} +H(Sp Vi) +¥Rs, Y)} 1 (°(v OF _ yO where f and F are functions to be determined by the form of V, at infinity, and by the conditions that 0V:/0E=0 V:/O0n=0 at the singular points of the transformation, where ie een GAS a= —X,'. But 3 a ae ke OW yx OY es Y,) +W(Xs, Va)—e | (w' SY -X sy) t Oy Oy OF yr, OV yx) ao + SY, ©? me > Y: alae daa =(), at the singular points. In the same way the differential coefficient of the same function with regard to 7 vanishes at these points. And it is obvious that the differential co- efficients of W(z,y) with regard to & and 7 also vanish at the singular points. Hence OV)/d& and QV>/d7 vanish at these points. This result greatly simplifies the work of obtaining the potential of a cylinder of any given form. For example, let p be constant and let ati =a t+ be Et) 4,02 te 4 4b em Eten)s Then MXY+XP+Ve4+¥e) and > i) "(KY'—X'V) dn contain 7 only through multiples of cosy, cos 2n, ...,cos(n+1)7. And therefore V, is of the form — 2mp(a?— bi? — 2b. ... —nbn?) {E+ Aye—= cosnt ... + Aniie FUE eos (xn+1)n}, and both 0 V,/dé and OV)/d7 must vanish when ae th, e~ Et — 25, 0-28 tM | — nb, e~ 2E+) =O, : iL Vo= —2arp(a?—b2— 2.2 ... —nb,2) | é+ 5 = On —(n+l)E cos (n+1)y r and V; may be written down by means of the general formula. The form taken by Vp) at infinity assumes that the curve does not cross itself. This, of course, restricts the variability of the coefficients 61, 5,... bn The value of Vy) may be similarly written down in the rather more general case of the transformation ie COS On tech mapas e+ wy =a, ert) +a, se =I) Ga i +04 étng - +be7 Et) 4 pe 2E te 4. 4b em Eten), But in this case we have, at infinity, log r+ n€, and therefore the coefficient of in V,) is —27pn x the area of cross-section, so that a term in & alone occurs in V;._ For example, for the evolute of the ellipse, we have “+ ty =a cos® (n—st&) + 0b sin® (n—2é) =c{3 cosh (X+&+ 1m) + cosh (A—3E—3xm) }, on putting a=4ecosha, b=4csinhn. At the singular points | 1 4 g&h—2E—2iy_,—-4E— Aen) _,2A—-GE—6e9 0), Hence we obtain, by integration, | Vo= —IrpablE—ke*—*) cos 2n + te ** cos 4n + fe” \—**) cos 6m}, and therefore Vi — 7p(a? +4?) + Srrpabé + Vo— 7%s7rpab sinh 4€ cos 4n + 4rpc?(15 cosh 2& cos 2n + cosh 6€ cos 6) + 3apc* cosh 2X cosh 4€ cos 47. It is not possible to pass to the case of the four-cusped hypocloid by making > 0 , 2ce*->a, for the transformations take different forms at i™finity. j ; : i f ] Certain Problems of Two-Dimensional Physics. 777 Figures of Equilibrium of Rotating Fluid. 11. If the gravitational potential of a cylinder take the form* C—tw?{(x—k)?+y"} on the surface €=0, 2. e. if it is possible (in the case of constant density) to have Vi=—m7p{(a—k)? +y°} +4(27p — @”) (Xi? + Xo? + Yi? + Y,?) 1 +5. {F(8) -F(}, Vo=—}o%(X 2+ Xi 4 V+ Yo") +40%(Ki +X) 9 +mpe( (XY'—X'Y)dn+ : {F(@)—F(7)}, A then the cylinder is a possible form of equilibrium of liquid rotating, under the influence of its own attraction, with angular velocity w. In particular, the hypotrochoids of equation (2), § 4, are possible figures of equilibrium if 4=0 and ee. fare { 1—-(n—1) 4} ‘ 27 p n y a Thus for any given positive integral value of n, and for - values of e varying from 0 to 1, the hypotrochoids ar/ n(n —e) =c(n cos 7 +€ cos nN), yr/ n(n—e2)=c(n sin n—e sin nn), [with ?/27p=(n—e’)/(n+1) | form a linear series of figures of equilibrium (unstable if n >1) passing out of the circle of radius ce. The case of n=1 is that of the elliptic cylinders, which are stable if e< 4, the bifurcating ellipse being that for which e=3. The case of n > 1 is that of the hypotrochoid which passes, as € increases to unity, into the n+1-cusped hypocycloid, after which fluid escapes at the cusps, as is easily verified in any particular case. For example, take x + Ly =a ee ren +6 oe (Erm) a C e—XE+en) and therefore 3(X1?+ X,?+4+ V+ Y,?)=a?+b?+¢? + 2be cosh Ecos + 2ab cosh 2€ cos 27 + 2ca cosh 3€ cos 3, Ve : at (XY'—X'Y)dn = (a? —b? — 2c?) E— 3d¢ sinh & cos n—Acasinh 3£cos3n- z * By taking the surface-condition in this form we are really making use of Poincaré’s theorem that there is a plane of symmetry (y=0) through the axis. Phil. Mag. 8. 6. Vol. 30. No. 180. Dec. 1915. 3 778 Dr. J. R. Wilton on the Solution of Thus 3 Vo= —21p(a?— b? — 2c?) E—w?(bee-* cos n + ab e~* cos 2 + ca e—*é cos 3n) + kw? {(a + b)e—* cos n+ ¢ e~7§ cos 2n}. Hence aE ae — Qarp(a?—b?— 2c?) —w*{ [be—k(a +b) Je E+) + 2(ab—ke) e~ 27E+'N) 4 2cq eo 8EtM) and this must vanish at the singular points, where a—b e— 2+) _ 9¢ oe SEF) — (9. Consequently we have k=be/(at+b), 4c°=a(a+b), w"/2ap= (a-++b)(a— 2b) /3a7.. a Thus the curve w= (a+b) cosn+ta ¥(1+6/a) cos 2y, y= (a—b) sinn—4a (1+ 0/a) sin 2n, which is the three-cusped hypocycloid when 6=0, is a possible form of rotating figure of equilibrium, provided that @ is given by (7). But if 6 >0 the curve possesses loops, and it is therefore not a proper solution, but must be regarded as indicating that as the angular velocity diminishes the fluid escapes at the cusps of the hypocycloid. More general cases of figures of equilibrium of this type may be found without difficulty, but there is no great interest in carrying on the investigation as all the figures so obtain- able are, with the exception of the ellipse, unstable. Problems of Elastic Equilibrium. 12. It isalso possible to obtain solutions of certain problems of elastic equilibrium, namely, the torsion problem*, the flexure problem, the problem of plane strain for a cylinder bent by its own weight t, and the approximate theory of the equilibrium’ of a plane plate clamped or supported at the edge§. But, except in those cases in which the solution is well known, the analysis is tedious, and the results do not appear to be of sufficient interest to repay the labour of investigation. * Love, ‘ Elasticity ’ (second edition) p. 301, §§ 217-8. This is merely the hydrodynamical problem of fluid in a rotating cylinder (with w=—1). + Loe. cit. p. 317, § 229. t Loe. ert. p. 347, § 244, § Loe. cit. p. 465, § 513. oy q : Certain Problems of Two-Dimensional Physics. 779 Using the notation of the section last quoted, consider as an example the problem of the bending of a plate by its own weight, the edge being clamped in a horizontal plane. We have, in this case, Viw=Z'/D=W/AD=640, say, where W is the weight and A is the area of the plate, with the conditions w=0odw/d&=0 at the edge €=0. The general solution is WHO} (8 +y?)—2(Ki+ Vy HR Ve)? "@ +20) ° (X2+¥2)(KY—X'V)dy } i ; ay [XF"() + YG (») ]dn + [ F (0) —F(r)] +y[G()—G@)] \, where F and G are functions to be determined from the conditions of finiteness, &c. A sufficient illustration will be furnished by the considera- tion of the case of the elliptic boundary, for which we may evidently take F(0) =Q(A sin 8+ Bsin 30), G(@) = —Q(Ccos 6+ HKeos 38), where A, B, C, & H are constants which may be determined by equating to zero the coefficient of §—since & becomes infinite at the centre when the ellipse is a circle—and the value of OW/O& when €=—2. The equations thus obtained lead to A=2b(a?+8), C=2a(a?+l?), _ 2b(8a? + 6?) (a*—D*), Bee 2a(a? + 3b?) (at$— ~ Bat+2a%b? + 304 mae eee On substituting these values in the expression for w, we obtain after reduction the—otherwise obvious—result, = (W/8rD) a0" (1-5 E -4) | | (Bat + 20°? + 359), The solution of the problem here given is, of course, to be regarded merely as an illustrative example of the general process. Rageor! | LXXXIII. A Method of Finding the Coefficients of Absorption of the Different Constituents of a Beam of Heterogeneous Rénigen Rays, or the Periods and Coefficients of Damping of a Vibrating Dynamical System. By Sir J. J. THoMson, OM oF BIS | HAVE found the following method useful in finding the absorption coefficients of the various constituents of a beam of heterogeneous Réntgen rays. In this case the intensity of the rays after passing through a thick- ness @ of aluminium can be represented by the expression Pe cA Ne | cg Reagan the problem is to find from the 2n observations which are necessary for this purpose the values of Aj, Ag,...An; Ai, Ae,--+An Though I arrived at the method from the consideration of the absorption problem, it is of much more general application and applies to any quantity which can be represented as the sum of a series of exponentials of some variable, whether the coefficients of the exponent are real, imaginary, or complex. Since the amplitude of the vibrations, damped or undamped, of a dynamical system about a position of equilibrium can be represented by a series of the form SA Y%, the method can be employed to find the periods and damping coefficients of a dynamical system, and is especially useful when we have a graph representing the variation of the displacement of the point in the system with the time. In the case of undamped vibrations it gives a simple method of finding the simple pericdic terms into which the motion can be resolved. The method depends upon the following theorem. Suppose Yoo Yt> Yar +++ Yan—1 are 2n observed values, the observations being made at equal intervals 0, of the variable: in the case of the X-ray problem this variable is the thickness of the aluminium leaf through which the radiation has passed ; in the dynamical problem the variable is the time. We shall denote this variable by ¢t. Then yo is the observation when t=0, y,; when t=0, 2 when ¢=20, and so on. Since y = Aye” + Age’? + Aged” + Ane, Yo = Ay + 7 ob As + Aa yy = AE + Asn +As3¢ alae e898 yo = Ag+ Ag+ Ae t+..., 62 + + + @ ye = Agk* + Agni Ashi +. eo J * Communicated by the Author. 4 if Absorption of a Beam of Heterogeneous Réntgen Rays. 781 pare \,0 \,9 \30 i ea eee NG eA 5). 4) Kliminating A,...A, from the first (n+1) equations, we have Yoo ie ns 1 Yny She n", ate oye Expanding the determinant, we get yoDo +%7D,+y4,D.+ occ — 0, = . A - (1) where D, is the minor corresponding to y;, and is equal to BSD i a the row &, 7°, f°... being left out. We easily see that D, = P . Ss, where P is the product of all the differences of &, n, &, 2. e. (Pn) (EC) Ee) «+ (n—€)(n—e) .-. (¢—<€) Re, and § the sum of the products &, n, € taken n—s at a time. Thus < So = Ene, 8, = Eno+Ene+... Since P is a common factor of the D’s, we may write equation (1) as Yoo Y1 + Yo Seo sO. By eliminating A,, A,,...An from the n equations beginning with y; instead of yo, we get Yio Ya + y3S2 ess eC) ; e ° >] and similarly ysSo—ysSitysSs.-. =O, down ” Yn—1S0— Yn1 + Yntide vent ee Ok 782 Sir J. J. Thomson on the Absorption Coefficients But if # is equal to any of the quantities &, 7, €, &c., So—28 +278, 504 = 0. Hence, eliminating S, S,,...8, from the n+1 equations we get TO! G1. | Yaa wie Ga Ys Y2 Y30 0+ + Ynt1 var T8s es Tee ey Yn-19 Yno +e 22 Y2n-1 il L Hau) HN be Thus the roots of this equation for # are the values of E, 7, €—t.e., of &+,e?. Hence this equation determines XO 6 : 3 the values of €', Pe ; and since 0 is known, the values of the ’s can be immediately determined. We can find the values of the coefficients A,, Az, A3, &e. as follows :— From the first n equations of the system a we get Ay =a Yos i, 1 Roa et OG E, ; G =, n, ec? Pt, ge YoSn—-1— Y1Sn—2 + YoSn—3 eee (3) E-NE ROC Scrat | ae where §, is the sum of the products of the (n—1) quantities n, ©, ¢,...taken ratatime. Thus Sn-1 = ne, Sy =ynt+S+et+... The value of a coefficient can be expressed in terms of the root corresponding to the coefficient, and does not require a knowledge of the other roots. For 7, ¢,¢ are the roots of the equation a1 8 2"-27 + 8,a7-8—... = 0; so that the polynomial is equal to (w—n)(a@—€)(@—e).... of a Beam of Heterogeneous Réntgen Rays. 783 Writing equation (1) in the form L— 90") + ppv" -27—... = 0, since the roots are &, 7, ¢,..., the polynomial is equal to ni (w—£)(u—n)(#—6).... (2— &)(a"-1—S,a"-? + S.a"-3.. .) 2 =i Bas) bate <= Fy fa — py” + pox” — P3k n = and equating coefficients we get Suté = Pi; Sot &8, = Pr» SstES. = Dz, En = Pn Thus all the S’s can be found in terms of &. Differentiating the identity a" — pia") + pox”? = (w—£)(a@—)(w—E)..., and putting v=€ after differentiation we find ne! —(n—1) prB-? + (n—2) rok, = (€—n) (E—6)(E—e) «+5 Thus both the numerator and denominator of (2) can be expressed in terms of & without knowing 7, &,.... If two roots are equal, say & and 7, the expressions for both A, and A, become indefinite. A,+A,, the quantity we require, has, however, the definite value ae YoOn—2— Y19n-3 t+ Y29n—4 ee ) le) sen ea mt+&é =, gat Eq, = 8x, 93+ &q2 = Bs, where the S’s are the same as in equation (3). The de- nominator 1s (n — 1) &"-?— (n—2)S,E"-F + (n—3)S.E"... Mr. R. A. Herman of Trinity College, Cambridge, has pointed out to me that the determinant in equation (2) is of the type discussed by Sylvester in a paper in the Philosophical Magazine, Nov. 1851, and called by him a canonizant. where EBA. ol LXXXIV. Radiation from an Electrie Source, and Line Spectra. (Third Paper.) By lL. Sivperstein, Ph.D., Lecturer in Natural Philosophy at the University of Rome*. [Plate XVIII.] CONTENTS. Large Permittivity and Atomic Dimensions of Source. Generalities on Distribution of Spectral Lines. Typical Form of Atomic Dispersion. The Simplest Typical Dispersion, and the Balmer Law. The Diffuse Series of Hydrogen and the Principal Series of Sodium and Lithium. Large Permittivity and Atomic Dimensions of Source. T° see the Justification of contemplating large permit- tivities at all, and enormous ones especially, start from the simplest case of unit permittivity. The radiation curve for this case is drawn in the upper part of fig.1 (Pl. XVIIL.), with A and J as abscisse and ordinates ; A,, A», Ke. are the positions of maxima in the first and subsequent branches, and 14, v2, &c. are the successive zeros of J corresponding to the critical frequencies of the source, as in the Second Paper}. The dotted ordinates, within the first branch, have the length of one half of the corresponding maximum ordinate, so that their distance apart givesa certain kind of measure of “breadth,” as recently proposed by some authors for proper line spectra. Thus, the “breadth” in the case of the first branch would be, roughly, 1°8 a, and for the second branch, alittle more than ja, &c., a being the radius of the spherical source. But it will be enough to fix our attention upon the first branch. The above “ breadth” being (for small K) a rather coarse concept, it may be well to quote here also the numbers for A; and 4. To four decimal figures I find dy /a=2°2907, v;/a=1°3983, so that the distance from zero to maximum-radiation is dy—, = 8924 a. (K=1) Jumping over intermediate values, pass at once to the case K=100, z.e. w=10w, to which the lower part of fig. 1 corresponds. In order to obtain a just comparison the radius a@ is now taken equal to one-tenth of the previous * Communicated by the Author. + Phil. Mag. vol, xxx. pp. 163-178 (1915). v7 ‘| <1 Radiation from an Electric Source, and Line Spectra, 785 one, so that, for instance, vy; is rigorously the same wave- length as before. The scale for J is chosen in an arbitrary manner, the essential thing being the distribution of maxima and zeros of intensity. ‘The breadth is so small in this case that it is impossible to more than indicate it in the figure, but this is sufficient for the reader to be able to form a rough judgment of the ‘“‘ breadth” in comparison with the previous case. The distance from maximum- to zero- radiation is My —,='01527.a VK, (K=100) 2.e., remembering that 10a is the previous a, nearly one- siatieth as large as in the previous case. The two radiation curves represented in fig. 1 may suffice. To quote some further numerical results, I find, for K=500, y/a YW K=1-4013, and v,/a / K=1:3983, so that A—n='0030a VK, (K=500) and, what may be more interesting, the wave-length for which J falls to less than 7, of J,,. (more exactly, 400 iS J max? 469) is found to be X=1'4008.a/K, so that the corresponding distance is only dA='0005 .a VK. (K =500) The physically observable breadth of the first band or branch would therefore certainly be much smaller than (26 or) one-thousandth of aWK. Hxamples concerning much larger permittivities will be given later on. Hrom the above examples the reader will see that in order to obtain bands sufficiently sharp to represent actual line-spectra, the permittivity of the source must exceed values of the order of 107. This being granted, it is easy to show that we are compelled at once to assume much greater permittivities, even apart from the ultimate physical requirement of sharpness of the lines. In fact, imagine a number of similar spherical sources, all of radius a (or nearly so), homogeneously distributed in empty space. ‘To fix the ideas, let this assemblage represent a given volume of hydrogen gas. Call Ky the value of K* for infinitely long waves, for each of the spheres, and let Ky be the observable, macroscopic, permittivity of hydrogen, under normal conditions. Then Ko (being at least 10°) will certainly be large enough to justify the application of * In the preceding examples K has, of course, been assumed constant only for the sake of simplicity. 786 Dr. L. Silberstein on Radiation from Poisson’s formula * in its approwimate form, which is known as Mossotti’s formula, 7. e. ey y) mo 14+ 2 ton where w is the volume of the spheres per unit volume of gas. Consequently, remembering that Ky is in our case but slightly greater than unity, w=(Ky—1)/(Ky+ 2) =4(Ky—1). Here K)>—1=2°64.10~4, and therefore w=0°88.1074. Let us suppose for the moment that each spherical source contains more than one molecule of hydrogen, say, 2* mole- cules. Then, the number of molecules (or atoms, if the gas be dissociated) per cubic centimetre being 2°76.10°, we should have 3 wo= 2-76 10°" — 0-88. 10-4 & whence, in round figures, aje—9 told; em. Now the molecules or atoms being distributed homogeneously, it is obvious that if each sphere a contained (on the average) more than a single atom, the spheres would practically occupy all the volume of the gas, or would even overlap with one another. But if such were the case, its macro- scopic permittivity K, would, instead of being but slightly greater than unity, be practically as great as Ky itself. We must conclude, therefore, that e=1, and consequently a=913.10-°cm. . . 5. ee Thus the source shrinks to molecular or atomic dimensions T. Using this result, it will be seen at once that the corre- sponding value of permittivity of the spherical source in the case of line spectra is enormous. ‘Take, for example, the value of k,=477a?Ky as required by Table III. for the = ee , of. Maxwell’s ‘ Treatise.’ + The values of the radius of a hydrogen molecule calculated from the deviations from Boyle’s law, and from the coefficients of viscosity, of thermic conduction, and of diffusion are 1:025, 1:024, 0°995, and 1:01.107% cm., respectively. Cf, for instance, J. H. Jeans’s ‘ Dyna- mical Theory of Gases,’ Cambridge, 1904, p. 340. * kK, an Electric Source, and Line Spectra. 787 hydrogen series (Phil. Mag. vol. xxix. p. 714), i.e. ky=5°05 microns’. This gives, by (30), _ 505.1078 Ky = Aqr?q? in round figures, valid for \ =~ according to the dispersion formula k=2+8/(1—y*A~?). For the first hydrogen line, H., we should have, according to the same formula, K=2°59.10', and so on. Similar figures would follow from the constants of Tables I. and II. In short, the permittivity of the atomic sources responsible for the line spectra turns out to be of the order of 10’, as was mentioned previously. Under these conditions the “lines” emitted are eaceed- ingly sharp, as may be easily judged from the above results obtained in the case of the comparatively small permittivity K=500, and still better from the following Table for K= 10", in which the factor cé)?/a has been taken as unit intensity of radiation. is 10% 8 ee TABLE V. we ) A/a. | J 2 a | 19.1077 a ae 61 .1077 [cl eek le ea Ve. 107 * LE ae eed | ot) 10° Pree a ae 59 .107° Berets Gio Asee 2s 1:53.107° SOB ll ich lath hy) eee) 107° (139°8450 oo... 741.1078 | 139°84493 ...ececeeee. | 2877 units. | 4 189-8449] o.c.ececces | 495°36 (max.) | (1998447 soesecs.. | 399.1072 J 139°8438 oo... Pi $8! 10% os | 49 .10~° 139-829 (2) see | 0 The second and following branches (from v, to v:, &c.) have very much the same character. The above, first, branch consists practically of a very sharp “line,” or narrow band, ranging over (or less than) the bracketed figures. The corresponding wave-length interval is 64='0003 a, so that 788 Dr. L. Silberstein on Radiation from the detection of its breadth would require a “ resolving power ” exceeding nr on Such being the state of things for K=10*, we should have for K=10° to 10’, as required for reducing the source to atomic dimensions, even considerably sharper lines, possibly much sharper than the finest line yet observed. Should the latter be the case, then the actually observed breadths of spectral lines would have* to be accounted for by special assumptions, such, for instance, as the attribution to the various atoms or atomic sources of an emitting gas, of slightly different radii a}, or the assumption of sources slightly flattened. And since the required differences be- tween the individual radii a would be exceedingly small (even compared with 107° cm.), there would certainly be no serious difficulty in accepting such or any equivalent hypothesis. The reader will remember that in most, or all, attempts to construct spectral theories, the difficulty—as pointed out by Lord Rayleigh—has been in obtaining (not sufficiently broad but) sufficiently sharp lines. And it is certainly an advantage of the proposed theory that it yields, without particular artifices, such exceedingly sharp lines. Remembering that, in the case of line spectra in general, K is, as in (31), of the order of 10’, let us estimate the prodigality or the relative emissivity e (i.e. the fraction of the average stored electromagnetic energy emitted per period) of an atomic source, which in the case of such spectra seems to acquire a particular physical interest. The general expression of the relative emissivity is given by (26), Second Paper. Since, in the present case, w= w™/K is moderate and w?=u?.10~’, we have, by (26), Li Aen g? (a) WK Gu) | The value of K being enormous, we know that the values of the variable uw, or wave-length X=A,, for which the radia- tion J attains a maximum, are but slightly different from the values of u, or wave-length X¥=y,, for which J=0; in short, that the maxima are very near the corresponding = 500,000. (32) * Apart from the cooperation of the Doppler effect due to molecular agitation. + Say, distributed round a given value a@ according to the “law of error.” an Llectric Source, and Line Spectra. 789 zeros of radiation, the wave-lengths X, A», &c. being almost imperceptibly greater than v1, v2, &c. Thus, the maxima of J will be attained for u=u;— du, where 6,u are very small fractions. By (23), Second Paper, J attains its maxima when wu becomes, very nearly, a root of the equation cosu = ia BLM ae 1. @. when 2a peels u ane Here we can, with sufficient accuracy, put on the right hand u equal to the successive critical values w,, u., &e. Thus, the relative emissivity of the source when emitting waves (A=A,, i=1, 2, 3, &c.), for which J attains a mazi- mum, will be given by An uj® cos? u; a7 Ka? aig 2 ra i=l, 2, 3,&c., - + + (33) z where, by (9), and because sin u;/u;=cos ui, e+ PSI ei s,s) (38) ly G; = G (u,) =uU;— Introducing here the known values of w, us, &c., and using the tables of Si already quoted, we find, for the (common) logarithms of G, to Ge, 0°83000, 1°01337, 1°13163, 1°22583, 1°30359, 1°36721. (35) The values of 5;u, determining the distance of the maxima - from the corresponding zeros of radiation, will be very nearly given by dq U;” COS U; u.—6.u)= —{—= )d.u=— — ——__—_ g(4s—o31) ey : K Now, g=sin u/u—cos u, dg i dg \ ; — =sinu——g(u); (—%) =sinw,. du ud )3 ey) Ath Therefore, Bye. U;" COT U; 4 ou= aa oa PST BS ey!) ee) These approximate formule, (33) and (36), are more than 790 Dr. L. Silberstein on Radiation from sufficiently accurate in the case of line spectra, when K is of the order 10’. Thus the relative emissivity e; is inversely proportional to the fifth power, and 6,u to the square of the intrinsic refractive index (K*) of the source. To have a numerical example, take the first line of the diffuse hydrogen series, H,, as treated in Table IIL. for instance (First Note). Then k=47?a?K =8°527 and, by (20), Ke = 2°50 ats Roe 2 ae aay Agri,” COS’ Ug = 4882 10? = 9G = bol): and the relative emissivity for H, * becomes, by (33), é,=2°113 1008) Again, w cot u;=4°494 ; therefore, by (36), us 1734.07. enn i. e. the wave-length difference between maximum- and zero- intensity, 6A=y\-Yy = PrN" 50 = 2508 . 10-8 micron, (88a) 1 so that the detection of this difference would require an instrument of resolving power ),/6,A=26,000,000. This extraordinary sharpness of the theoretical lines has already been hinted at. (It is needless to remark that the experi- mental H. is a very complex line.) Of equal interest is the extreme smallness of the relative emissivity as exemplified by (37). From (37) we see that this energy emitted per period zs only the 5. 10¥-th part of the mean energy contained within the sphere a in electromagnetic form. |The latter energy (U, belonging to the internal field EZ, M) will not be confounded with the source’s store of generally non-electro- magnetic energy, say W, which makes good the (almost evanescent) losses due to emission. If the amplitude e of the “impressed force” is, as hitherto supposed, to be rigorously or at least practically constant, we have to imagine an enormous store W. But, in general, if e) and hence also U, and the corresponding external energy U!, are slowly varying with time, we cannot assert about W any- thing more than do Ara? ah a tO) tes where W is the average of W taken over a period of oscil- lation. Here it is assumed, of course, that all energy J 0) * More exactly, for the place of maximum radiation-intensity in the narrow region of spectrum thus denominated. an Electric Source, and Line Spectra. 791 supplies are derived from the store W. But the last energy equation will hardly be needed in the sequel. This matter has been mentioned only in order to prevent any misunder- standing about the meaning of the store U and of the relative emissivity. ' Suggestions as to the possible process of excitation of line spectra. For large K the maxima of intensity of radiation J Fay Giese tht ce.” f general J;= aK vt [formula (256), Second Paper], occur at U=U,—6,U, U,—d,u, K., where 6,u are given by (36), and, as in (38), are very small fractions. ‘The zeros of radiation intensity are in the immediate neighbourhood of the corresponding maxima, since they are attained rigorously for U=U,,U,, &e. Now, a gas—whose atoms are here identified with similar electric sources—is not luminous in ordinary circumstances. It becomes so, and emits its characteristic spectrum, when it is excited in a peculiar manner, say, by sparks. One might think that the impressed forces ¢, and therefore the emitted oscillations are produced by the sparks or whatever the external agent might be; that oscillations of all possible frequencies being thus called into existence within the atoms, those and only those whose frequencies are nearly corresponding to w =u;—odu are radiated out with sufficient intensities to be observable as spectral lines. But, in connexion with the explained properties of the electric sources, the following simple and peculiarly fascinating view of the process of exciting spectrum emission naturally suggests itself. Let us suppose that in the normal, z.¢. non-luminous, state of a gas there are, in the atomic sources, oscillations of rigorously x : a fs : ct ; critical frequencies only *, i.e, e==e, sin (WE ui), i=1, 2, 3, &e. a , Either all of these frequencies may be present in each atom, or some in some atoms, others in other atoms, whose radii are, in the normal state, such that _navK _2raVK c r U are roots of tanw=wu. Then, by what has been shown previously, there will be no trace of emission, whatever the amplitude e, of * As if all other frequencies, just because they have not been critical or strictly internal, were exhausted in the course of ages by continuous radiation, so that out of all possible frequencies only the critical ones could survive. To2 Dr. L. Silberstein on Radiation from the internal oscillations. Now, let through the agency of the ex- ternal stimulus, such as sparking, the radu a of all, or of some, atomic sources undergo slight variations, da. Then, the frequencies n remaining as in the normal state, the values of u will be slightly varied, and each source will begin to emit its waves, the emission of each of them being more or less intense according to the magni- tude dua of the individual variation *. The maximum emission of a given atom will be attained when the change of its radius happens to be such that ou = Qn VK IN a= — Ou, 2. €., by (86), when the radius of the atomic source shrinks by the amount Ni & iy” COL Us. 5 a ae eee D7 K3/2 “% t (39) 6 had 64 = 0,4 = — For any smaller or larger shrinkage or expansion, the emission will be weaker, ranging from 0 toJ; The observable spectrum would be an average result of all such individual contributions, which—ceteris paribus—would be unequally distributed among the various lines (7=1, 2, 3, &.), since 0,a, 6,4, &c. have slightly different values. The shrinkage may in one atom be most favour- able for the emission of the waves ,, say, and in another of X,, and so on. Details of this kind, however, need not detain us here. But what seems worthy of notice is the smallness of shrink- age of an atomic source, (3%), required for even the most efficient release of imprisoned energy, 7z.¢. for the passage from zero to maximum emission. Thus, for instance, returning to the numerical example (38), we have, for the first member of the diffuse hydrogen series, 0°6563 .1°734.10-7 Oo Ve aor iis: and taking @=9°13.10-® micron, as in (30), ff 5 3.90.10-*. . er = — 3°56 .10-} micron, Thus the shrinkage would amount to four hundred-millionths of the ‘‘ normal” atomic radius only. It will be understood that the above is here offered merely as a vague suggestion whose acceptance or rejection would be a matter of indifference to the essential parts of the proposed theory. * The same result would follow on the assumption that K undergoes a change. On any electronic theory the permittivity of our sphere would depend on the configuration, and chiefly on the density of spacing, of the subatomic entities, so that both K and the radius a would be changed simultaneously. To fix the ideas, we have supposed above that the latter only is being varied. | eo! F " ; an Electric Source, and Line Spectra. 793 Generalities on Distribution of Spectral Lines. As we have repeatedly remarked, the position of the lines of the spectrum emitted from a source of high permittivity K is given, with more than sufficient accuracy, by the roots themselves of the transcendental equation tan u=w, where De aie u= 12 Fs Y) X being the wave-length reduced to vacuum, and a the radius of the source oratom. To abbreviate let us write, as before, An Opinio ist Geeta let) ws (AL) then De, galeria ak, (42) Now, let the permittivity be any function of A, k=k(d), representing what we have previously called “the intrinsic” or “the atomic dispersion.’ Jf the function k(d) be given, we can draw, with as abscissa and w? as ordinate, the curve uw = au SPN, Saye! a a: oy (aa and a series of parallels to the axis of abscisse, u?=w,?, uw’=uy", &c., in general, w?=u,?._ Notice that the equation tanu=wu has certainly no purely imaginary roots *, so that all these parallels will be contained in the upper part of the plane A’, w?. The position of the spectral lines will be deter- mined by the points of intersection of these parallels with the curve (43), the corresponding wave-lengths being given by the square roots of the abscissze of these points. Accord- ing to the nature of the curve (and therefore of the dispersion law) there may be one or more spectral lines, or none, corresponding to the 7th root, vu; Should any branch of the curve lie in the lower part of the plane X?, uw’, there will nevertheless be no points of intersection and, therefore, no spectral lines corresponding to this branch f. * In fact, g(w) =sin u/u—cos u can be written 2 4 6 Serre! Par Now, if u? were negative, all the terms would be negative. Thus, there is no negative w” satisfying the equation 7g=0. + It will be remembered that the physical meaning of the real part 2 Pee c ' : of K is, in general, not c?/v* but 7 (1—o*), where o is the ‘coefficient of extinction.” Thus a negative real part of K would correspond to o>l, i.e. to very strong extinction. Phil. Mag. 8. 6. Vol. 30. No. 180. Dec. 1915. 3F 794 Dr. L. Silberstein on Radiation from If the dispersion-curve, as (43) may conveniently be called, has (anywhere but at the origin) an asymptote A?=y’, as in fig. 2, then there will be an infinity of lines which will crowd towards the asymptote from the red to the violet end of the spectrum (in the case of fig. 2,a) or in the opposite sense (in the case of fig. 2,b). The experimentally known line-series, properly so-called, correspond to the former case, but in band spectra both cases occur. In either case we shall have what is called a “head” or a convergence-point, of wave- length A=y. If the asymptocy is of such a nature that k is, in the neighbourhood of y, proportional to any positive power of = then—especially if behind y there is another similar convergence-point—the dispersion curve may well emerge again from the lower into the upper part of the plane giving rise to fresh lines beyond the convergence- point y, the first of these lines being, in the case represented by fig. 2,a, of wave-length L. But what seems interesting to remark is that a certain region behind the convergence- point (from A to B) will be entirely dark, 7. e. free from lines. If there is a certain number of such asymptotes, then we shall have a succession of such gaps, following upon crowded lines. Multiple lines could be accounted for by narrow protuberances of the dispersion curve, and so on. But let us leave these generalities. It will be understood that it is only in some cases that we might expect to be able to guess the precise form of the dispersion curve belonging to the atom of a substance, and deduce from it, by means of equation (43), the lines of its characteristic spectrum. But as a rule, what is given by experience are the lines themselves, and the dispersion curve has to be constructed from these data. The former is certainly the more fascinating task of the mathematical physicist, but even the latter does not seem useless. The knowledge of atomic dispersion curves thus defined and constructed by means of spectroscopic material may well throw some new light on the intrinsic properties of atoms. It is not my purpose, however, to enter into investigations of the latter kind just now. I shall limit myself, therefore, to give here only one such “ experimental” curve (fig. 3), which corresponds to hydrogen, as far only as its diffuse or Balmer series is taken into account. The abscissee of the centres of the small circles are the squares of the observed wave-lengths of H,=H,, H,=Hg, &c., and their ordinates tbe squares of the successive roots of tanu=u, beginning with wu,”, that is 20°1906. The vertical bar represents an Electric Source, and Line Spectra. 795 Balmer’s theoretical limit or the convergence-point of the series. A smooth curve on which the centres of the circles may be placed, is not drawn in the figure. The meaning of the drawn curve (hyperbola) will be explained later on (p. 801). Typical form of Atomic Dispersion. Guided partly by the aspect of curves such as that given in fig. 3 (and the corresponding curve k =k(X*)) and partly by analogy with the refractive properties of molar bodies, I propose to investigate the chief consequences of assuming the particular form of dispersion, t=kK b, eon al eas . ° . . ° (44) which, for the sake of short reference only, I shall call the typical form of atomic dispersion*. Here ky is the statical value of k, as before, and b,, y, are constants. Originally I left the values of these latter constants free in relation to kp. Certain general reasons, however, along with some numerical exainples, have suggested the introduction of a simple relation between &, and the remaining constants. It is well known that in some at least of the molar dispersion formule, which (apart from absorption bands) are all of the type K=K,+> vom ey “ the constants approximately satisfy the equation Rey, e and that this equation is also a consequence of every theory of dispersion based on electronic assumptions. In our case, Ko=k)/47r7a? being of the order 10° or 10’, unity can be omitted. Let us postulate, therefore, the relation h=>-— See BG: > e ° . e ° (45) to be introduced in (44). This supplementary relation, besides being familiar from * Some numerical applications of (44) are given in the Preliminary Note, Phil. Mag. vol. xxix. 1915, p. 709. The symbols there employed are thus related to the present ones: b e+B=k; y=y; B=5. oF 2 796 Dr. L. Silberstein on Radiation from molar optics, has in the present connexion a remarkable analytical property. In fact, by (44) and (42), the equation determining 2, that is the position of the lines of the corresponding spectrum, is 2 . by, iat cise that is an equation of the «+1 order in )’, say 7 NG aE We) Vout Pha eee Nec. Now, by multiplying out, & y b, Ge — ( ae oe ye Ay2 ag {3% —Iy } . Thus, when (45) is satisfied, Ay vanishes, and the equation for ” is, with the obvious rejection of the root A=0, reduced to the x-th degree. Now, since each y, gives rise to ‘a convergence-point with its series of lines, it is obviously desirable to have, in the case of dispersion with « terms, « and not «+1 lines to correspond to each root u=u;. And this is precisely what is secured by the relation (45). If we desire to obtain the more general dispersion law (44), with any ko, we have only to introduce a («+1)st convergence- point y~>9%o with such a coefficient 6 that B=b/y? should have the required finite value; thus, the relation (45) is ultimately no real restriction. There is no need of expanding here the expressions of the remaining coefficients A, in general. It will be enough to write down, for the sake of illustration, the formule for k=2. Thus, when the dispersion is ba by OS enG als Pate halo + ao + ea heh it sie we have for the determination of the eee the quadratic equation By conven, APH—yP Amy”? or v—ae+a=0, «=r, ) 1 | DEL Nets 2, + | Ay —— 7 + Y2 + 7281 + B2)> ‘ ern. (46) 1 ay = 1°72? + ua Bie + Bay’) giving for every root u; (‘=1,2,3,...) two different roots z‘ and, in as far as both of these are real and positive, two an Electric Source, and Line Spectra. 198 spectral lines ;, say, A;) and A,’’. To these latter we shall refer shortly as the i-th lines. Similarly for any number of convergence-points. The relevant combination of dispersion constants, 8, = es might conveniently be called the intensity of the s-th convergence-point. The simplest typical Dispersion, and the Balmer Law. Postponing the fuller discussion of the cases of two and more y’s, let us consider the simplest case in which there is but one convergence-point. Then the dispersion formula (44), with (45), becomes b Br and therefore the equation of the spectrum, b oP (A2—y?) =r?, oe TNS Bp 0 (47) Up’ We shall call this infinity of lines, converging towards A=y, a stmple series of lines. Its relationship to those series known from spectroscopic experience will be shown presently. It will be remembered that u,=4'4934, w.=7°7253, Ke., so that the second term in (47) decreases rapidly with the increasing ordinal number of the lines. The most characteristic feature of a series of lines is doubtless its convergence-point itself and its immediate neighbourhood. For, apart from other reasons, this feature remains unaffected by the simultaneous existence of other convergence-points. It is therefore natural to inquire first of all into the properties of the simple series \47), especially when i; approaches y. We have, rigorously, b 17? u=Y [1+ aM hi Now, for high values of 2, that is for u,;* large as compared with b/y*, we have, neglecting terms with 1/u‘, etc., 1 pee! Oy. Ae mt Loar ae i By (27), Second Paper, 94 )7 2 2 2 : fetes arene ~3( ana) 7 798 Dr. L. Silberstein on Radiation from When the first and second terms only are retained (which even for i=6 or 7 is sufficiently correct), then (47 a) becomes of a similar type to the Moggendorft-Hicks formule for sodium, lithium, &. But assuming that 7 is as large as, say, 15 or 20, we can reject even the second term, writing simply w;= (22 + 1) > Then we have, from (47 a), 1 ik Lea) where 2b R= =) ry" 9 e . r . e e (48) that is a frequency formula of the Balmer type. Thus, the simple typical dispersion (44a), or the corre- sponding simple series (47), approaches Balmer’s law asympto- tically, with increasing i. In view of this property it seems legitimate to assert that this dispersion formula expresses something which is essential about spectra, and as such deserves some particular attention. Again, the structure of the coefficient k in (47 b) has a suggestive meaning. As has already been stated, it is not the purpose of these investigations to enter upon any intra-atomic mechanisms. But, incidentaliy, the reader may find some interest in interpreting our formule electronically. Now, according to either Drude’s or Planck’s electronic theory of ordinary (molar) dispersion (cf. Drude’s. Lehrbuch der Optik, 2nd ed. 1906, pp. 376-377), the coefficients B in K=K,+23B/(’—y’) have the physical meaning where q is the charge, in ordinary (irrational) electromagnetic units, and m the mass of the ‘electrons,’ or more generally of the electrified particles responsible for the particular term B/(\’*—y’), and N the number of these particles per unit volume. In our case, remembering that the part of B is taken over by 6/47°a’, the electronic interpretation would give a =4ra°N aL fs Y m Now, our coefficient R contains, besides a purely numerical factor, precisely the combination 6/y* which has such an intrinsic and simple meaning on any electronic theory. To wit, by (48), where a is the radius of the spherical (atomic) source, or, if Jt be ‘S 4 an Electric Source, and Line Spectra. T92. the number of electrical particles of the given kind contained in the source, ne 9 eee er it ia), oat an me Such would be the significance of R in terms of subatomic entities. If we wish, we can goa step further by assuming that the mass m of the particles in question is of purely electromagnetic origin. ‘Then, supposing, for example, that the particles are spheres of radius 7 and of homogeneous surface-charge, m=43q’/r, and Ot r ro Be ay RN aS ee Aa (49a) i. €. proportional to the number of particles of the y-kind contained in the source (atom) and to the ratio of their dimensions to those of the source. But it must be expressly stated that the above electronic inter- pretation would hamper our progress; for cases are known, in which the lines crowd towards the infra-red (instead of towards the more refrangible) end of the spectrum, so that some of the coefficients 6 and therefore R may be negative, while all the factors in (49) are essentially positive. We shall, therefore, as a rule not assume this, nor any other mechanistic interpretation. Using the coefficient R, defined by (48), apart from any subatomic interpretation, the rigorous formula (47) can be written wh rt=y(1+ 55), af 2, By 02 .De Sut Pita (47) Formula (47 4) is what (47) becomes for large 7’s. From this fact we may expect, without numerical calculation, that the higher lines of the simple series represented by (47) will coincide with the corresponding members of observed series of the hydrogen type. How far down in the scale of 7’s this agreement holds, can be seen only from a detailed numerical comparison of (47) with spectroscopic experience. It will be well, therefore, to insert here a few examples. 1. The Diffuse Series of Hydrogen.—Let us take for y=rw the wave-length of the convergence-point which, used in the Balmer formula, gives the best representation of the experi- mental series, 2. ¢. 1n microns, Let us determine the value of R=2(\?—y’)u;? : (my)? from the wave-length of the highest observed line Hs;, measured recently by Mitchell (Astrophys. Journ. 1913, 800 Dr. L. Silberstein on Radiation from vol. xxxviii. p. 407), coordinating this line with 72=35, 7. e. writing A3;= H;,='365680. By Huler’s formula, Us, 111517574 * == 35497146 or. These figures give QD mp2 R=13-36722 ; logio(™ # ~ ) =0°9432206. . (50) With these values of y and R formula (47) gives the results collected in Table VI., which requires no further explanations. Tasie VI. 1e) 0 A, cale. by (47). Aops, (Mitchell). Ad in ALU, fo) SOL MISS a ey ee een ie lla 39 "365680 °365680 0:00 34 5737 5740 — 0:03 33 5799 5819 —0:20 32 5866 5880 —0'14 31 5941 5988 —0-47 30 6022 6047 —0:25 29 6112 6142 — 0°30 28 6212 6237 —0°25 27 6323 6356 —0°33 26 6446 6480 —0°34 25 6584 6623 —0°39 24 6740 6791 —0°51 23 6915 6960 —0°45 22 7114 7145 —0°31 21 7342 7396 — 0:54 20 7604 7648 — 0-44 19 7907 7948 —0-41 18 8260 8296 —0°36 17 8675 8697 —0-22 16 9167 9178 —O-11 15 369757 369735 +0:22 14 370471 "370403 +0°68 13 1349 1220 +1:29 12 2443 2220 +2:23 The differences Yeaic.(47) — Aops. for the next members of the series are (11) 3°67; (10) 5°85 ; (9) 9°17; (8) 14°41; (7) 22°79 A.U., &e. From the last column of Table VI. we see that our * It may be worth noticing that here the third term of Euler's series, 2 s(aaes) ° amounts only to 5.107%. é an Electric Source, and Line Spectra. 801 ‘“‘simple series” (47) represents fairly well the higher members of the observed diffused series of hydrogen from H;; down to Hy=He, with a difference not much exceeding 4A.U., and generally much smaller. For the remaining lines, H,; to H,= Ha, the ditferences AX mount up with increasing rapidity. Thus the said approximation holds at least for twenty or twenty-two members of the series. This is shown also in fig. 3 (Pl. XVIII.) where the continuous curve is drawn according to (47), 2. e. represents the equilateral hyperbola uw? (A? — 9") = Ty R/2, the value a R being as in (50). 2. The Principal Series of Sodium.—This is, owing to the admirable work of R. W. Wood*, the most extended and most complete series hitherto observed. It is well-known that this series, similarly to those of lithium, potassium, &c., is more closely represented by formule of the Moggendorff- Hicks type f (involving four constants) than by the original Balmer formula. In view of this fact and by what has been remarked above, one could expect that the asymptotic agree- ment with experience of our “simple series” (47) will extend in the present case further down in the scale of 2’s than in the preceding case of hydrogen. A detailed calecu- lation has fully corroborated this vague expectation. Prof. Wood estimates the convergence-point of his beautiful series to be at about y=0°241ly, which would give y°='05812921. After some preliminary trials I have found most suitable for the majority of lines the slightly greater value, y?="0581814. By doing justice to several members of the observed series at the same time, the corresponding value of 7?y?R/2 comes out as 2°705044. With these figures, formula (47) becomes 2°7 05044 = Af=-05818144 — pe) Cent Uj; The second column of Table VII. contains the wave-lengths calculated according to this formula, and the third column the wave-lengths (means of doublets, whose components down to 1=4 are exceedingly close to one another) measured by Wood in the absorption spectrum of sodium vapour. The fourth column contains the differences, Neale, —Aobs, In A.U. * Phil. Mag. vol. xvi. 1908; Physik. Zeitschrift, vol. x. 1909, pp. 88— 90, and p. 913, where some errata in the figures are amended. t Cf. Dr. W. Marshall Watts’s paper in Phil. Mag., vol. xxix. 1915, p. 775. —_—_ 802 Dr. L. Silberstein on Radiation from TABLE VII. te Neale: Aops, Wood. AX. CO SA QBS ah ye US a ita dae 48 241449. 241450 00) AT 1460 1464 — *04 46 1471 1478 — ‘07 45 1483 1494 = fd 44 1495 1506 — ‘ll 43 15U8 1521 SS 42 1523 1537 ee 4) 1588 1552 ee 40 1554 1570 = he 39 1572 1589 — ‘17 38 1591 1611 nt) OL 1612 1633 — 21 36 1634 1656 ee), 35 1659 1680 Maa) | ae 1685 1710 mee. 33 1714 1738 eee Be) 1746 Arar eo Sill 1780 1809 — 29 30 1818 1844 ==) 25 29 1860 1900 — 40 28 1907 1950 — 43 27 1959 2002 — 43 26 2016 2060 — 44 25 208} 2129 es 24 215 2204 = ol 23 9235 22590 = "boa 22 329 2388 SES) 21 9435 2500 Silo: 20 2557 2628 er 19 2699 Qi he — 73 “18 2864 2942 STS iy 3058 | 3143 = ES 16 3288 3385 == 97 1h 3564 3670 —1:06 14 3898 4006 —1:08 13 4309 4424 SS 12 4822 4946 —1'24 11 Oy etre 5602 — 1:29 10 6317 6453 Sa 9 7437 7560 yes 8 248969 *2AG070 =O 7 *251140 251215 ee) 6 "254361 "254382 Seiya! Bi) Ole Ne sanosety 259405 4. -25 4 | "268054 *268046 08 1 The difference for the next member, i=3, is AXN=—8°05, and the remaining two members are, in this connexion, -wholly out of question. Thus, the simple series (47) agrees in a very satisfactory degree with Wood’s observations from the head, or at least an Lilectric Source, and Line Spectra. 803 from 72=48, as far down as 2=4. The deviation swells up to — 1°36 A.U. for i=10 (which difference can be reduced a little by slightly retouching the two constants), but then decreases down to ‘08 A.U., forz=4. One can hardly fail to perceive from this Table that the simplest typical dispersion br? and the corresponding series (47) do express some very essential property of the observed series, and therefore of the sodium atom. It may be worth mentioning that there is still a certain kind of qualitative agreement between the proposed theory and observation. In fact, the atomic source has, according to the form of the present dispersion law, properly but one free or “‘natural” period, e. g. that corresponding to the wave-length >, =y itself (none of the remaining 2,’s having anything essential about it). Thus one would be led to expect that at or just behind the convergence-point y of the series there should be strong absorption. Now, in the second of his papers, just quoted, Prot. Wood points out with particular emphasis that there is ‘a general absorption beginning at the head of the series and stretching to the end of the spectrum” *. It would be interesting to verify by experiments whether anything of the kind happens with the series of other elements. k= 3. The Principal Series of Lithtwm.—In this case the agreement of (47) with observation is equally if not more satisfactory than in the preceding example. ‘The second column of Table VIII. contains the wave-lengthis calculated by the formula N2=052855 te ee BU OBS. ae a and the third column those measured by Bevan (:=41 to 8) and by Kayser and Runge (:=7 to 4) +. Bevan uses, for the Moggendorff-Hicks formula, a convergence frequency corresponding to the wave-length A, =0°229975 pw, ‘ but points out on his latest measurements indicate a higher frequency,” 7. e. a smaller X,. ‘The value we have adopted in the above formula is d, = =y=0°229902, and would, therefore, not be far from the truth. * Translating literally from Prof. Wood’s paper in Phys. Zettschr., and italicizing the few words of special interest. + I take the tigures for X,),. as quoted by Dr. Watts, doc. ct. p. 782. obs. 804 Radiation from an Electric Source, and Line Spectra. TABLE Vii . Neale. \obs. AX. 00 S229 902 «Mil ee Fa Wes eee 41 230220 -230220 0-00 40 0236 0238 — 02 39 0253 0259 — ‘06 38 0272 0283 — ‘ll or 0292 0303 — ll 36 0313 0324 =e 35 0337 0346 — 09 o 0362 0373 — ll 33 0384 0400 — ‘16 32 0420 0429 — ‘09 3l 0454 0463 — ‘09 30 0491 0499 — 08 29 0531 0541 — 10 28 0576 0587 — ‘ll 27 0626 0648 — +22 26 0681 0690 — ‘09 25 230744 °23075 =. 24 23081, *23083 — 1. 25 089, ‘23090 =: 22 098, *23100 ele 21 108, 23111 See 20 120, *23122 — i'l Ls 134, *23136 — 2, 18 149, 23152 es Ly 168, 23171 = 16 190, 23193 — 2, 15 | PAG "23219 ee 14 249, "23252 — 2, 13 289, 23290 =— 507 12 338, 23343 — ‘4, es 401, "23405 — 3; 10 482, *23485 — 2, 9 590, *23594 =e 8 738, "23739 — 0, 7 *239480 239454 ae AS 6 *242585 °242555 + -30 9) "247470 2475138 — “45 4 ‘255780 °256260 — 4-80 | The agreement extends here from the highest observed member, 2=41, down to 1=5, and the differences AX are, in general, much smaller than in the preceding example. It may be interesting to remark, in passing, that the values of the coefficients R= ca 5 for sodium and lithium, 7 used in (47, Na), (47, Li), and in the Tables VU. and VIII. On the Spectra of Helium and Hydrogen. 805 respectively, turn out to differ but little from one another. In fact, Rya=9°42140 Ry= Sado, so that, in round figures, Ry: Ry,=1:01. Any attempt to find the physico-chemical meaning of this approximate equality, either with or without the help of such formule as (49) or (49a), would not answer the purposes of the present paper. We have treated with some detail the properties of what has been provisionally called the simple series, taken by itself. The coexistence of two or more such series, which offers some interesting points, will occupy our attention in the next paper, in which also the few hereabove ignored lower members of spectral series will be taken into consider- ation. ‘Meanwhile I should like to point out only that, as far as terms involving 1/u;? are concerned, there is no mutual disturbance of the lines of two coexistent simple series ; their mutual “action,” to speak figuratively, is chiefly determined by a term of the fourth order in 1/u;, and the corresponding law is of a very simple nature. October, 1915. Corrigenda.—In the Second Paper on this subject, Phil. Mag., vol. xxx. July 1915, p. 178, line 5, instead of tann=+o, read tanyn=0; in equation (256) replace the exponent —1 by —2 and we? by w*, and consequently, in line 22, (a/AX)? by (a/r)*. LXXXV. A Comparison of the Positive Rays with the Spec- trum of the Positive Column in a Mixture of Heliwm and Flydrogen. By Harouip Smira, 6.A., M.Se., late Royal (1851) Exhibition Scholar*. Introduction. HEN an electric discharge is made to pass through a mixture of gases ata low pressure in a Geissler tube, the relative intensities of the spectra are by no means pro- portional to the relative quantities of the gases present. For instance, when the uncondensed spark-discharge is made to pass through air ata low pressure, nitrogen yields its spectrum very brightly, while no trace of any of the oxygen spectra is * Communicated by Sir J. J, Thomson, O.M., F.R.S, 806 Mr. H. Smith : Comparison of Positive Rays present; this in spite of the fact that 20 per cent of the mixture 1s oxygen. It is not always a simple matter to say which of two spectra is the stronger. In the case of two spectra yielding lines evenly dispersed throughout the same regions of wave- lengths in each case, it is of course easy to decide which spectrum is the stronger. In the case, however, where one epee contains a few bright lines, and the other a large number of faint lines, any decision must be arbitrary. It is possible, for instance, by varying the conditions of excitation, to vary the relative intensities of the primary and secondary spectra of hydrogen. But at what stage can it be said that the twa spectra are equal in intensity? Hxperimenters confine themselves generally to making observations on the more prominent lines of the spectra dealt with; and in most cases where the masking effect of one spectrum upon the other is large this method is satisfactory, and has led to some important generalizations. Previous Work. It has been known for some time that when metallic vapours, such as those of cadmium and mercury, are present in the Geissler tube as impurities, the spectra of these metals may be produced in strength, while the spectrum of the gas under examination may be considerably diminished. Indeed, Prof. Lewis has shown that the presence of one molecule of mercury for every three thousand of hydrogen will cut down the spectrum of the latter by one half. It has been shown by Nutting that non-metallic elements such as sulphur, bromine, jodie, &c., had similar effects. Even the spectrum of a metallic vapour, like that of sodium, was reduced by the presence of a halogen. Nutting examined altogether fifteen different elements in elghty different combinations; and the conclusion at which he arrived was that if there were an appreciable difference between the atomic weights, then the spectrum of the heavier element masked the spectrum of the lighter. Moreover, the masking effect was roughly pro- portional to the atomic weight. Nutting also observed that on increasing the current density, the spectrum of the lighter element underwent a slightly greater relative increase. It would appear therefore that ‘the masking effect is more noticeable at the smaller current densities. It will be noticed that Nutting’s result isalso confir nee by the case of the electric discharge in air. Under the ordinary induction-coil discharge at low gaseous pressure, the only { ir if with Spectra of Helium and Hydrogen. 807 spectra produced are the band spectra of the molecule* of nitrogen; if there were no nitrogen present, the series-line spectrum of the oxyen atom™ would be produced. Conse- quently we have here the masking effect of the heavier constituent. Withthe condensed discharge, however, or in the case of the spark in air at atmospheric pressure, the line spectrum of the nitrogen atom™ occurs. In this case there should be no masking effect; and we find that the elementary line spectrum of oxygen appears as brightly as the spectrum of nitrogen. The case of a mixture of hydrogenand helium is somewhat of an exception of the rule given above. Ramsay and Collie have shown that at a pressure of about 3 mm. as much as 30 per cent. of helium is necessary before the spectrum of helium appears in the mixture. At the lowest pressures the amount necessary falls as low as 10 per cent. For the spectrum of hydrogen to be detected in the mixture as low a percentage as ‘001 per cent. is sufhcient for all pressures. Nature of the Problem. In the case of a gas whose spectrum is masked by that of another, it is interesting to inquire to what extent it takes part in the discharge. In the case of air cited above, is the current so far as it is due to the motion of the positive ions, earried by the nitrogen molecules alone? It seems unlikely that this should be so, but it would leave one without an explanation for the absence of the oxygen spectrum. The present research was undertaken to compare the spectra of various mixtures of helium and hydrogen with the relative intensity of the positive rays of the same mixture. As a large proportion of helium is necessary before its spectrum appears, it would be possible to observe whether the positive rays of helium appeared just when the same proportion was present. If there were a direct con- nexion between the spectrum of helium and its positive rays in this manner, it would be strong argument for supposing that a gas whose spectrum is completely masked takes no part in the discharge. Apparatus. The type of apparatus was the usual type used for producing positive rays (fig. 1). The bulb and camera were connected by a side tube so that both could be exhausted to a low * Itis generally supposed that band spectra are produced by molecules, line spectra by atoms. 808 Mr. H. Smith: Comparison of Positive Rays pressure by means of a mercury pump, and also by means of charcoal cooled in liquid air. This tube could then be closed by means of the tap T. Another charcoal tube, CT, attached to the camera, was then immersed in liquid air in order to keep the pressure in the camera as low as possible during the whole exposure. By this means sharp positive-ray photo- graphs were obtained. The discharge was made to pass through a capillary tube (about 1 mm. in diameter) C, and a photograph of the spectrum was obtained by attaching a small direct-vision spectroscope to a camera (S in fig. 1). Bie.) d. TO HYDROGEN ¢ GENERATOR~ ))| Qa TO HELIUM GENERATOR ) < ENC: Y 770 PUMP. . 8 “eon daw J SASK ~ ix i of = RC. K Pp P = GT, A. Anode. K,. Cathode, Water-cooled. P. Wires from electrostatic plates to storage-cells. Horseshoe magnet for magnetic field not shown. The helium was prepared from thorianite by heating with sulphuric acid. The gas evolved was passed through a charcoal tube cooled in liquid air, and its spectrum observed in a small Geissler tube. As soon as the spectrum of helium became strong and the spectrum of hydrogen difficult to detect, the helium was considered pure, and the generating apparatus together with the charcoal tube were shut off by means of a tap. The helium was now contained in a tube connected to the apparatus, and by turning a tap could be let in at will. The method of filling the positive-ray bulb was as follows :—The whole apparatus was washed out several with Spectra of Helium and Hydrogen. 809 times with pure dry hydrogen. The bulb was then pumped out and the pressure—amounting to about -01 or ‘(02 mm.—was then measured with a McLeod gauge and alittle helium then admitted. After waiting a fair interval of time to allow the pressure to become constant all over the apparatus, the pressure was measured again, and the partial pressure of the helium and its proportion calculated. As charcoal does not absorb helium, the apparatus had to be pumped down to a low pressure by hand. ‘The new pressure was noted and the pressure of helium calculated. Photographs of the positive rays and of the spectrum of the positive column were taken simultaneously. Some pure dry hydrogen was then admitted, the pressure measured, and the proportion of helium present calculated. The apparatus was then pumped down to a low pressure and photographs taken as before. This process was repeated several times, so that a series of photographs was obtained; each member of the series being taken for a definite proportion of helium present in the mixture. Altogether three sets of positive-ray photographs were taken on the same plate, when the apparatus had to be opened up to remove the plate and prepare for a new series. Photographs of spectra were given from three to five hours exposure, and this was by no means too much. Longer time could not be given for fear of air leak. Photographs of the positive rays were generally given about one and three- quarter hours. Results. The photographs of the spectrum in residual air when the pressure is sufficiently low to give good photographs of the positive rays (‘005 mm. say), show the mercury lines strongly and the negative band of nitrogen at X=3914. I have not been able to detect the positive bands or the line spectrum of nitrogen, nor could I observe any of the spectra of oxygen. The positive rays, besides showing the lines due to the hydrogen atom and molecule, also show the line due to air very strongly. The apparatus used was not powerful enough to separate the lines due to nitrogen and oxygen, but other photographs taken efSewhere in the laboratory show the oxygen line frequently stronger than the nitrogen line. The lines due to the molecules of oxygen and uitrogen also occur. Thus while we get the positive rays of atomic oxygen and nitrogen in residual air, as well as the lines due to their molecules, we get only one of the spectra of molecular nitrogen present in the light from the positive column. In Phil. Mag. 8. 6. Vol, 30. No. 180. Dec. 1915. 3G 810 On the Spectra of Helium and Hydrogen. this case no connexion can be traced between the positive rays and the spectrum of the gaseous mixture. Helium and Hydrogen. lt was not found possible to get rid entirely of the air line from the positive rays, or of the nitrogen band from the spectrum, though both could be considerably diminished. In the first experiment there was about 60 per cent. of helium present, but the pressure of the gas was rather high, falling from °048 mm. to'025 mm. during the exposure. As a consequence, while the line due to the hydrogen atom was strong, all other lines were weak; the lines due to the hydrogen molecule and to helium could just be detected. The spectrum of helium, however, was just as strong as the spectrum of hydrogen. In the next photograph, with a smaller percentage of helium at a lower pressure, a strong positive ray photograph was obtained in which the line due to helium was as strong as the line due to the hydrogen molecule. The spectrum, unfortunately, gave no trace of either helium or hydrogen. A series of photographs was taken in which the amount of helium was varied from 30 per cent. to 2 percent. of the mixture. In the positive ray photographs I could not detect any helium at a smaller proportion than 10 per cent., though here the line due to it was as strong as the line due to the hydrogen molecule. In the spectra obtained not a trace of the helium spectrum appeared in any, though the spectrum of hydrogen was clearly visible. In a final case, where the proportion of helium was so small that I could not estimate the amount, T obtained a positive ray photograph of four hours’ exposure. The line due to helium, though faint, was visible. The spectrum, as was to be expected, showed only the hydrogen and mercury lines with a faint negative band of nitrogen. As far as these experiments go, it would appear that the helium spectrum disappears at a much earlier stage than the helium positive rays, which are remarkably persistent. It is difficult to obtain satisfactory results, as the pressure must be kept so low to produce good positive ray photographs that the light from the positive column becomes too faint to render good spectrographic work possible. Conclusion. No simple connexion between the spectra of mixed gases and their positive rays has been observed. It would seem that a gas, whose spectrum in the positive column is more or The Recoil of Radium D from Radium C. 811 less completely masked by the spectrum of the gas with which it is mixed, may still be taking a very important part in carrying the current in the dark space. It is doubtful, however, whether this conclusion can be carried further so as to include the positive column. Until more is known definitely of the manner in which an atom emits its spectrum, it will be very difficult to suggest a cause for the masking effect of oue spectrum upon another. But it is interesting to note in this connexion the discovery of multiply-charged atoms. Sir J. J. Thomson has ob- served that in the positive-ray tube the heavier atoms can lose more corpuscles than the lighter atoms. As the masking efect only occurs when there is an appreciable difference between the atomic weights, it may be that it is connected with the property of the heavier atoms to become more completely ionized. If these multiply-charged atoms occur in the positive column, they would form the more strongly attractive centres, and recombination with subsequent spectral emission would be more frequent in their case. In conclusion, I desire to thank Sir J. J. Thomson for suggesting this research, and for the kind interest taken in its progress, Cavendish Laboratory, Cambridge. ——_$—____ = - LXXXVI. The Recoil of Radium D from Radium C. By A. B. Woon, M.Sc., and W. Maxowrr, M.A., D.Se.* ANY attempts have been made to detect its photo- graphic action of the recoil stream emitted by radioactive surfaces, but hitherto they have been unsuc- cessful in detecting any dwrect action. The position at which the recoil stream from radium A i impinges on a photo- graphic plate has, however, been fixed by allowing the radium B reaching the plate to decay in situ, and obtaining a photographic record by means of the @ radiation sub- sequently emitted by the radium C producedf. It seemed possible that the failure to procure any direct photographic evidence of a recoil stream might be due to the extreme facility with which this type of radiation is absorbed by matter ; for an ordin: ary bo ahd el film contains so much gelatin ‘that most of the recoil particles striking the plate are absorbed by the an, before colliding with a grain of * Communicated by the Authors. t+ Walmsley and Makower, Phil. Mag, Feb. 1916, 3G 2 ad 812 Mr. A. B. Wood and Dr. W. Makower on the silver halide, so that no photographic action can be expected In order to overcome this difficulty experiments were made by Wood and Steven * to detect photographically the recoil stream from polonium by means of Schumann plates, in which the amount of gelatin is reduced toa minimum. Thisattempt was unsuccessful; but the failure to obtain any photographic record may well have been due to one or both of the following causes. ‘The experiments were performed by allowing a con- fined beam of the recoil stream and «radiation from polonium to pass through a strong magnetic field, and it was hoped thus to separate the recoil stream from the e rays; but unless the recoil particles and « particles carry different charges, it is of course impossible to do this. There is, however, no evidence to indicate what charge is carried by a particle of radium G recoiling from polonium when it disintegrates, unless it be justifiable to argue from the analogy of the recoil of radium B from radium A, in which case the recoil particle carries a single charge in contact with the double charge carried by the « particle. But evenif it were possible to separate the recoil stream from the # rays by means of a magnetic field, it seems doubtful whether any photographic action could be expected with the sources of polonium available except with an exposure of prohibitive duration. Further experiments therefore seemed desirable. In order to obtain as strong a source of radiation as possible, a length of about one centimetre of platinum wire "14 mm. in diameter was exposed in a fine capillary tube to about 300 millicuries of radium emanation which had been reduced to a sufficiently small volume by condensation in liquid air. After an exposure of three hours the emanation was pumped out of the capillary, and the wire removed and subsequently mounted in the apparatus used by Walmsley Fig. 1. SP ene P | | —— jf, eee d, fe and Makower to measure the magnetic deflexion of the recoil stream from radium Af. The arrangement is shown in fig. 1, which is reproduced from the paper just quoted. * Wood and Steven, Proc. Phys. Soc. 27th April 1916. + Loe, cit. Recoil of Radium D from Radium C. 813 The wire mounted at O at right angles to the plane of the paper acted asa line source 3 cm. distant (/,) from a slit § of width 0°14 mm., and the radiation which passed through was allowed to impinge on a Schumann plate placed at P. The distance SP (l.) was 2°8cm. The whole apparatus was enclosed in a brass box which could be completely evacuated, and was situated between the pole-pieces of a strong electro- magnet, which when excited produced a uniform field of about 13,000 gauss at right angles to the plane of the paper. To carry out an experiment the active wire, upon which was deposited some 30 milligrams of active deposit, was placed in a tube which could be evacuated and heated to drive off any adhering emanation. The wire was then left for twenty minutes, during which time the radium A on it had decayed to a small fraction of its original value. It was then mounted at QO, and the box containing it was exhausted as rapidly as possible. An interval of eight minutes was then allowed to elapse, during which the radium A still remaining on the wire decayed to an insig- nificant amount. During this time the radiation from O struck the centre of the Schumann plate P and produced a photographic impression. The magnetic field was then excited and kept at a constant value for 75 minutes. The photographic record obtained in this way is shown in fig. 2. Fig. 2. On the left of the photograph is a strong line giving the undeflected position of the beam of « rays coming through the slit before exciting the electromagnet. On the right is another somewhat stronger line fixing the position of the beam of # rays when deflected by the magnetic tield. Between these two lines is a much fainter one due to the deflected recoil stream. It is evident that this line must have been produced by the direct action of the recoil stream, since radium D emits no « rays and its period of transformation is so long that any photographic action due to this or any subsequent disintegration product is precluded. 814 The Recoil oy Radium D from Radium C. It is interesting to note that the line due to the recoil stream does not appear to be accurately midway between the two a ray lines, which is remarkable if radium D carries a single positive charge when it recoils from radium C; but careful consideration of the experimental conditions under which the photograph was taken leads to the conclusion that a slight dissymmetry such as is observed is to be expected. For when the wire is exposed to the emanation, there is deposited on its surface radium A, which subsequently disintegrates into radium B and then into radium C. During the process of disintegration of the radium A, which is accompanied by the expulsion of particles, the radium B is shot into the wire by recoil, with ihe result that the radium C subsequently formed is situated, not on the surface of the wire, but at varying depths below its surface. The recoil stream of radium D subsequently emerging from the wire is therefore heterogeneous; for the particles of radium D must have traversed different thicknesses of platinum and so escape with all possible velocities. The densest part of the recoil line in the photograph is therefore produced in the region corre- sponding with the most probable velocity of emission of the particles*. In support of this view it will be noticed that the left-hand edge of the recoil line is much sharper than the right-hand edge, indicating that there are no particles travelling with velocities greater than that corresponding with the left-hand edge of the line. On the right-hand side the line gradually fades away, indicating the presence of recoil particles of all possible velocities. If instead of fixing attention on the regions of maximum density in the lines, the positions of the left-hand edges of the three lines are measured, it is found that the recoil line is much more nearly midway between the two a ray lines. But since even the left-hand edge of the recoil line is not perfectly sharp, and since the displacement of the line from the centre is small, it is not possible to make this measurement with very great accuracy. From these considerations there seems to be no room for doubt that the particles of radium D which leave the surface of the wire traverse the magnetic field in a path the radius of curvature of which is double that of the « rays from radium C. Thus the radium D is projected with a single positive charge, just as is the case when radium B recoils from radium A. It should, however, be noticed that no * If this view of the displacement of the recoil line is correct, the efficiency of recoil of radium D from radium C should be low, and experiments are in progress to test this point. Variation of Emanation Content of Certain Springs. 815 general conclusions as to recoil phenomena can be deduced from this fact since radium B and radium D are isotopic elements, and must therefore be expected to behave alike. Possibly in other cases of radioactive disintegration the recoil streams may consist of particles with multiple charges; but unfortunately there are no other cases which lend themselves to easy experimental investigation. We are indebted to Sir Ernest Rutherford for putting at our disposal the large quantities of radium emanation without which successful experiments would have been impossible. We have also to thank Mr. C. W. Gamble, of the Manchester School of Technology, for helping us to make the Schumann plates used in these experiments. Physical Laboratories, The University, Manchester. LXXXVII. The Variation of the Emanation Content of Certain Springs. By R. R. Ramsey, Ph.D., Associate Professor of Physics, Indiana University, Bloomington, Ind., U.S.A.* HILE measuring the radioactivity of springs in the neighbourhood of Bloomington, Indiana, I found that measurements made at one time did not agree with those made at another time. The discrepancy was so great, that I set out to determine if the difference was due to errors or due to an actual change of the value. Iselected two springs‘ and have tested them once a week during the last nine months. The method of measurement was the Schmidt shaking method (Phys. Zeit. vol. vi. p. 561, 1905). This method has been found to have an accuracy of about 5 per cent. when measurements are made at the spring, and an accuracy of about 3 per cent. when measurements are made in the laboratory (Am. Journ. Sci. vol. xl. p. 311, 1915). An emanation electroscope was used. This electroscope was at first calibrated by means of Duane and Laborde’st empirical formula, which holds for an electroscope of one litre capacity. Later, the electroscope was calibrated by means of emanation standard E 54, which is certified by the Bureau of Standards to be accurate to within 3 per cent. The two methods of calibration agreed within 5 per cent. * Communicated by the Author. + Le Radium, vol. xi. p. 5 (1914); Ann. der Phys. vol. xxxviil. p. 959 (1912); Compt. Rendus, vol. cl. p. 1421 (1910) ; Journ. de Phys. vol. iv. p. 605 (1905) ; Indiana Acad. Proc. 1914. 816 Prof. R. R. Ramsey on the Variation of the Variation of Emanation Content of Certain Springs near Bloomington, Indiana, Hxpressed in curies per litre. Hottle Spring. Ill. Cent. Spring. Date. Temp. | Flow. Curies. Temp| Flow. - Curies, Sept, 24 ...... 13° ©. 65010717 | 18°06. 445 10-12 Ockes1G) ie. 8 13 695 12:8 166 Oct. 23 ...... 133 700 13 120 Océ. 3050. .c% 13 10000 665 12-7 13000 20 NONE}. 13 650 126 40 Nov. 13..... 13 705 13 20 Nov. 20%.....1 18 520 13 20 INOW. 20 %.5.. 13 550 13 33 ECE A: ccs .eu|), LS 535 13 60 Decwbte s. 13 510 13 20 dD iciciel earaae 13 450 13 00 WDes.s26 13 10000 445 13 5000 00 aMe cece. 13 20000 560 12:8 32000 40 Janey OT vewan 12°6 1000 12 135000 | ~ 340 Tia) CNS 13 770 13 40000 270 Sams 2h cis. 13 680 12°8 40000 100 Jam. 28 00. 12 610 | 12 32000 20 Heb wee cle 12 62000 850 12 250000 750 Bob I eis 12 40000 875 12-6 | 125000 166 Hobs Secu 11:8 915 12 100000 285 Reb 25.2.2). 11:3 890 12 75000 170 Mian} 13: S.Sca 11:5 1010 11-9 | 160000 140 EV en 11-7 900 12 90000 220 VR Re eee 11:3 920 12 62500 160 Mar 20 (ihos, 11:3 800 12 40000 90 ANI TE Ass 11 670 12 30000 45 - April 8 s...3: 11:3 690 12 28000 60 OPO neni!) aS 830 12 30000 60 Aprilad cy. 12 890 12 30000 256 April 28 \:..5.. 12 750 12:2 25000 410 AMEE Wu cate 11:4 1140 12 400000 365 May 14.0.5. 11:9 825 12:4 60000 365 May 21 ...... 11 1050 12: 42000 25 May 28 ...... 12 1340 121 | 500000 750 1c eee 11-6 1420 12 400000 820 June 1Q...... 11°8 1120 12 76000 355 June 25 ...... 12 1280 12-5 30000 | 715 ee - — ——- ————— — ee pn tae rternenafeemeenenma One, the Hottle spring, is due north of the square, near the corporation line of the city of Bloomington, latitude 39° 10'-6, longitude 8€° 323. Hlevation 770 feet. Bloom- ington Quadrangle U.S. Geological Topographic Map. The other, the Lllinois Central spring, is due west of the square a little beyond the corporation line, near the Ill. Cent. R.R. pumping station. Latitude 39° 10'1, longitude 86° 33':2. Emanation Content of Certain Springs. 817 Hlevation 820 feet. The two springs are about 1°3 miles apart. Hach has an average flow of from 10,000 to 40,000 gallons per day. ‘The flow of both springs is affected by the rainfall. The Hottle spring issues from a crevice in the solid rock. The Ill. Cent. spring issues through coarse gravel or stones. The measurements are given in table form. The date, the temperature of the spring, the flow in gallons per day, and the emanation content per litre of water is given for the two springs. The variation of the radioactivity with the variation of the flow is better shown in graphic form. The diagram shows the flow of the Ill. Cent. spring and the radioactivity of the two springs. The flow of the Hottle spring varies in the same manner as the flow of the Ill. Cent. spring, but not through so great arange. The radioactivity of the Hottle spring is also more constant than the Ill. Cent. spring, although its variation is about 300 per cent. a a or ws at Han a a all ek Md OD TR | " fadiqact ivity, Cc Radidact vity, , Flow, [1}.Cent.Sprin x | :