<2 = “= = —— =, a == ore aliens : ik that “tage Mdasiasey p f Ma | aus eS ip ‘ff tae f } 1] Mt iy Aba ) ‘lt th arias Hite) * ol ae * t Cais ih mi) aie a ie dette wel be Hien aA ey fh ays W ie ian seth ats ==* sé "4 ng eet nivel aay una) slant Lig pwn! Hey pagunie | Hy a a i) ) wide a) i Ki eee Raritan star pat 7 j s ty hae huh a: ai “i ving a Se Wears a itt ahtY vital Ser en He diate yh 8 (pins) ce nae HARE Beh eh Hi ae f yy vin Mitt iN a i sige \f sh iat fir -ol — - x ~' 2 a rs <= oan — = *~ = > < = 3 TF = ae sase Sete tin a tae 255 : = Sg a a bee SS ee = Oe ae Se Se ee Rowse ye S : ab = +52 aa . baa = : Sats 8p oe ae ' sine 3 LU asa, AY Be hed 4 A atid + ee j ; \ ; Hl : aa Pa it ; a eat i dys ny! , : i aati i a ; 2 i Sen ap ne bait APR “” ne ' i aney : a ot LS A ‘! Ries ‘ pia ‘ i f ales ; RHO ete ie ; | y i rue Ht Well Ly " i ‘ FA's ay ney i aie ra pS a AS aid le f vee : Seana stevens i i pl yh , Py as ha Bagi hy! hn P i , uN d fe i | ae) ba! oh be Fi ; , hy: at +i viet Aah iy ae. Hear A Mi a 4 ot) Hy welt 4 i} ; i ‘ Mary LOS eet h : ie eh ins Ah " 4 aie tt * ‘ 1 \ sit Res , ‘oq Ui ne ap ii i" j a A ih DU CARIN ba ane bt Pe bbs ran Py Fahy YF, ef beh oak co lhe Ra treat Mad i) ete is. yf tis a ai i 1 qh ‘ales oN Hea : Ly i] eon ‘ ‘ [cd ey j j - he iJ e! miteasake tiger ’ 3 ete Byte f ee n} l it, NYSE iy Me M1) ti heh ‘ $ Blass SH) ‘ 4 see rk Parte Rene ara picid: OG ae a age dat ietel ( Mitt byt ae "i oa a ath Giga Pee awe here Bann iki edd its Bredeate | sqvestt 4 Ht 4 an Ms Baheeean ten eye tonal Bini } fae a aCe sit deat at ie +} Wh Shee +h! RPhheac | Fa Ie RUAE rit! ey a it ; Tye a eage de phe at GL ' +i alt PA yay Set ha LI righ ea + ih ih i , if by oR ‘ails ait iio! i} Pt ee Vil ji rth dey iy (Ved iy at : cRY) que ave gant wip Mh vif 13% dali ; .) ‘ ne aa rte ey iy ) sini ‘ 7 ,ary dabdaleedl yee tle ey ta Sa @ pet ihe, \ Nh aa Aad RUA hs Why Hy A AEY ty Wwe W Rye ay THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. CONDUCTED BY SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.R.S. SIR JOSEPH JOHN THOMSON, O.M., M.A., Sc.D., LL.D., E.R.S. JOHN JOLY, M.A., D.S8c., F-.R.S., F.G.S. AND WILLIAM FRANCIS, F.1.S. “ Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster vilior quia ex alienis libamus ut apes.” Just. Lips. Polit. lib.i. cap. 1. Not. VOL. XL.—SIXTH SERIES. JULY—DECEMBER 1920. ¥ P$A0S = LONDON: TAYLOR AND FRANCIS, RED LION COURY, FLEET STREET. SOLD BY SMITH! AND SON, GLASGOW :— HODGES, FIGGIS, AND CO. DUBLIN;— AND VEUVE J. BOYVEAU, PARIS, eae , ee “Meditatiunis est perscrutari occulta; contemplationis est admirari perspicua.... Admiratio generat quaestionem, 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 ceelo, Quo micet igne Iris, superos quis conciat orbes Tam yario motu.” J.B. Pinelli ad Mazonium. CONTENTS OF VOL, XL. (SIXTH SERLES). NUMBER CCXXXYV.—JULY 1920. Prof. H. C. C. Baly on Light Absorption and Fluoreseence.— VI. Atomic and Molecular Energy Quanta ............ Prof. H.C. C. Baly on Light Absorption and Fluorescence.— VII. Molecular Phases ... ae Prof, Frederick Slate on a New Reading of Relativity Messrs. J. H. L. Johnstone and B. B. Boltwood on the Relative Activity of Radium and ce Uranium with which it is in Radioactive Equilibrium . ... Prof. R. A. Sampson on the Bearing of Rotation on 1 Kelativity. Pre ..b. Mohler, Dr. Paul D. Foote, and Dr. H. F. Stimson on Ionization and Resonance Potentials for Electrons in Vapours of Lead and Calcium ... Dr. Paul D. Foote and Dr. W. F. Meggers 0 on Atomic Theory and Low Voltage Arcs in Cesium Vapour. (Plate I.) Prof. A. R. Richardson on Stationary Waves in Water . Mr. A. C. Bartlett on Radiation from a Cylindrical Wall .. Miss Alice Everett on a Projective Theorem of Lippich’s in Geometrical Optics. (With a Note on the Equations of the Projection of a Straight Line on a Plane.).. Prof. P.O. Pedersen on the The ory of Ionization by Collision Mr. Bibhutibhusan Datta on the Stability of two Rectilinear Vortices of Compressible Fluid moving in an Incompressible targa <7)... Messrs. I. B. Young, rb cebyseie ad W. Devons. on Dee iteal Disturbanees due to Tides and Waves. (Plate II.)...... Dr. Megh Nad Saha on the Secondary Spectrum of Hydrogen. Dr. Norman R. Campbell on the Measurement of Time and S071 D2 GE AEs Pa ea een ee Mr. Balth. van der Pol, jun., on the Pr opagation of Hlectro- magnetic Waves formddhe Barth: eae lV CONTENTS OF VOL. XL.—SIXTH SERIES. Page Notices respecting New Books :— : L. Bairstow’s Applied Aerodynamics .............. 164 Sir T. Muir’s The Theory of Determinants in the historical-order of development’)... ..-.. 77 pees 165 F, W. Edridge-Green’s The Physiology of Vision with special reference to Colour-Blindness, and Card Test ter: Colowr- Blindness (9.0500). 5 aoe eee 168 W. Makower-et H. Geiger’s Mesures Pratiques en Radioactivite ee ee ee a 168 Dr. L. Silberstein’s Elements of Vector Algebra .. ... 168 NUMBER CCXXXVI.—AUGUST. Prof. W. Lawrence Bragg on the Bee of Atoms in Crystals. (Plate LI.).. . 169 Messrs. St. Landau and Ed. Stenz on the Dissociation of Jodine Mee and its Fluorescence .... 189 Messrs. M. P. Appl ee and D. L. Chapman o on the Equation Ot Sake: ee oS es SW ea eine vac oe er 197— Prof. W. B. eas on Statieshrietion,——-lla. o) 2. 201 Prof. A. W. Porter on the Specific Heat ot Saturated Vapours and Entropy-Temperature Diagrams of Fluids .. 211 Dr. Herbert Chatley on Cohesion. ;. 25 ....445 -..1.)) 22 eee 2ka Mr. R. Meldrum Stewart on the Adjustment of Observations. 217 Prof. A Anderson on a Method of finding the Scalar and Vector Potentials due to the Motion of Electric Charges.. 228 Messrs. R. W. James and Norman Tunstall on the Crystalline Structure of Antimony”... peux. 0255 9430 233 Mr. G. P. Thomson on the Spectrum of Hydrogen ‘a Rays. (Plate Lv.) 0-46 20 fe ee ee . 240 Proceedings of the Geological Society :— Prof. S. James Shand on a Hift-Valley in Western Persia: 2.2.30) a eS ata ne Bot ans er 247 NUMBER CCXXXVII.—SEPTEM BER. Prof. F. Y. Edgeworth on the Application of Probabilities to the Movement of Gas-Molecules .................. 249 Prof. J. ©. MeLennan and Mr. W. W. Shaver on the Permeability of Thin Fabrics and Films to Hydrogen EL @LUULINY 2 125 alo ee ee en gg he Rene ee 272 Sub-Lieut. A. L. Williams on the Blsctrical Conductivity of Copper fused with Mica. (With Introduction by Prof, J. C. McLennan.) (Plates NEVA) 2, ie 281 CONTENTS OF VOL. XL.—SIXTH SERIES. ¥ Dr. Rk. T. Beatty and Mr. A. Gilmour on the Measurement of Changes in Resistance by a Valve Method .......... 291 Mr. G. A. Hemsalech on the Character of the Light Radia- tions emitted by the Vapours of Magnesium, Copper, and Manganese under the Selective Actions of Thermo-chemical and ‘I’hermo-electrical Excitations. (Plates VII]. & IX.) 296 Mr. G. A. Hemsalech: Note on avery Convenient Method of Accurately Focussing and Adjusting the Image of a Laboratory Light Source upon the Shit of a Spectrograph. 316 Dr. W. T. David on Heat-loss by Conduction in aaa of Coal-gas and Air...... 318 Mr. G. B. Jeffery on the Path of a , Ray of Light in the eramiaiion Bipld of GheSun: ...) cai. i eR 327 Mr. L. C. Jackson on Variably Coupled Vibrations: Gravity- Elastic Combinations. — IJ. Both Masses and Periods ienalla, “Gerace Be. oo PP ee si a SL LG 329 Mr. E. 8. Pearson on the Advance of Perihelion of a Planet. 342 Prof. W. M. Thornton on the Ignition of Gases at Reduced Pressures by Impulsive Electric Sparks................ 345 Prot. G. W. Todd on the Variation of the Specific Heat of a er eae Pemmnerabure: oc ae lee eas ack es B07 Mr. Cyril H. Meyers on a Vapour Pressure Equation aK, 362 Mr. Alex. Frumkin on the Theory of Electrocapillarity.--I. 363 Mr. Alex. Frumkin on the Theory of Electrocapillarity.—II. 375 Mr. H. P. Waran on an Improved Design for the Friction Cones ot Searle’s Apparatus for the Mechanical Equivalent eS MAL eee Bia th ake nN ale a Laloe He 386 Notices respecting New Books :— Prof. J. H. Poynting’s Collected Scientific Papers .... 388 Prof. A. N. Whitehead’s The Concept of Nature...... 389 Prof. A. Gray’s A Treatise on Gyrostatics and Rotational Motion: thie, crue Ae Phe ee he CIE ae pees 392 NUMBER CCXXXVIIJ.—OCTOBER. Sir J. J. Thomson on the Scattering of Light by Unsym- metrical Atomaand: Molecules 5.25 65 oe je a air se 393 Messrs. G. Stead and B.S. Gossling on the Relative Ionization Potentials of Gases as observed in Thermionic Valves.... 413 Dr. H. Stanley Allen on Optical Rotation, ee Tsomerisn., ang the Hing-Hleelron ©. .....0.. 426 Prof. Frank Horton and Miss Doris Bailey on ie ‘Effect of a Trace of Impurity on the Measurement of the Jonization Wéelocisy tor Blectrons. ine Helium 1-4... tie es 440 Prof. W. M. Thornton on the Ignition of Gases at Reduced pressuies Owe Lrameient APCS oc esas ieieivleltane se eee 450 v1 CONTENTS OF VOL. XL.—SIXTH SERIES. Page Prof. Eric K. Rideal on the Velocity of Unimolecular Re- ¥ CHLOMS yi. sie a eee ee PR en et ee 461 Mr. J. H. J. Poole on the Radium Content of the Rocks of the: Woetschbero Mummy siete cee Sie he Dr. Megh Nad Saha on Ionization in the Solar Chromosphere. 472 Mr. Herbert Bell on the Helium-Hydrogen Series Constants. 489 Dr. 8. &. Milner on the Internal Energy of the Lorentz lGemrom ues se ae .... 494 Prof. A. Anderson on the Orbit of a Planet... 4). 499 Sir J. A. Ewing on the Specific Heat of Saturated Vapours and the Entropy-Temperature Diagrams of certain Fluids. 501 Mr. J. BR. Clarke on the Thermal Conductivity of some Solid . Insulators.) ee en 502 Prof. J. 8. Townsend on the Collisions of Electrons with Molecules of aiGase ee 505 Mr. 8S. Ratner on the Pressure on the Poles of an Hlectric HANG? ae ae oie Mr. T. C. Tobin on a Method of Finding « a Parabolie Equation ot the rth Degree for any Graphically Faired Curve .... 513 Prof. A. O. Rankine on the Dimensions of Atoms .. _. 516 Prof. 8. J. Barnett on a Double Solenoid for the Pr oduct of Uniform Maenetie Hields 9.00)... 4... ae 519 NUMBER CCXXXIX.—NOVEMBER. Dr. Jobn Prescott on the Torsion of Closed and Open Tubes. 521 Mr. Seibei Konno on the Variation of Thermal Conductivity during the Fusion of Metals ...... 542 Prof. K.T. Compton : Ionization and Production of Radiation by Electron Impacts in Heliuin nee by a New Method .... 203 Messrs. C. N. Tebagil ae orn #E. i Bowen on the. Bate of Chemical Action in the Crystalline State .............. 569 Mr. Humphrey D. Murray ona the Precipitation of Colloids . 578 The Research Staff of the General Electric Company on the Disappearance of Gas in the Electric Discharge. (Work conducted by N. R. Campbell and J. W. H. Ryde) ...... 585 Prof. Barton and Miss Browning on Triple Pendulums with Mutual Interaction and the Analogous Electrical Civemts.—-l. (Plates X1.-MLEE) ooo. eee 611 Mr. Satyendra Nath Basu on the Deduction of Rydberg’s Law from the Quantum Theory of Spectral Emission ..... 619 Dr. F. W. Aston on the Mass-Spectra of Chemical Elements. GBambioey ie Criate XV.) “0 2 Seay pseeaeee st acai sree 628: Prof. R. Whiddington on the Ultra-Micrometer : an appli- cation of the Thermionic Valve to the measurement ot very small my my k Mee the equation of motion P_R dv P—R_ dv ete ae Saat ae: (7) by simple recasting yields the forms mv dv mr? dv ea eer ie eet Stee ahh CS vy? —v? dt ve —wv dt ’ (8) noe dv Ae do PP dv le v av (9) MWe: p—vdt’? m, dt v2Z—vdt* ° But equations (7) one legitimately to net field-action com- bining a constant propelling field (a), and an automatically excited resisting field (kv?), the acceleration being then characteristically Py ube Be of the “ body coefficient ” (m,). And any actual physical linkage of (a, /), when (4) changes * See note {, p. 32; and cf. Silberstein, p. 194. + Among others this: it favours grafting forms due to variable inertia upon a root-idea of constant (m,). Relativity’s “ complex ” of force has the magnitude of (To) : e. f., for the tensor of its quarternion ; which the so-called “ rest-system” then renders coincident with a “ Newtonian ” force. Uf. Silberstein, pp. 193-4. The same thought applies at equa- tion (11) below. D 2 36 Prof. F. Slate on a with (a), that entails constancy of the ratio (a/k), would evidently fix also a common value of the terminal speed, whatever strength (a) has. Supposing this condition met,. and with (v,=c) given, equation (9) corresponds visibly to equation (5) save in one essential respect: the type of the original equation (7) makes (m,) necessarily constant. It is. true, the elimination of (m,) would permit assigning any series of epoch-values to that quantity, without influencing the kinematics through the consequent readjustments of force. They might differ arbitrarily or in conformity with a rule laiddown. They might follow, for example, the range of (m), as it varies under the rule of equation (4). Yet each such momentary value of (m,) must be rated constant as a. parameter, because equation (7) has required that. The proposition is defensible that liight-speed is also limiting speed for all electrons under any electromagnetic field-forces. It can borrow some confirmation from forms like equation (6),. which somehow standardize in terms of (c), and it is plausibly consistent with an idea impressed upon relativity regarding (c) as an unexceeded value. That this critical speed should coincide with the wave-speed does not look unreasonable. Only one of these réles for (c), then, would be common to light and gravitation, since the critical speed alone is known in the latter; it separates the ellipse as an orbit from the two other conics. The analogies of these two field-actions might go astray on such an ambiguity. As affording a provisional background for some electronic dynamics, we shall write the above assumption about (c) into our expressions: relative to (I). Note next how the first of equations (8) in effect suppresses (R), which is passive in the sense of not contributing to the total energy-flux, but merely diverting part of it. It is clearly immaterial whether the diverted energy is reversible or not. Read the second member as a product of the true acceleration by an effective inertia*. Since that inertia- factor is here variable, the equation in the form l he MOR Hh Pam,” = (my(0)) = 2 | ole eee must be incomplete, as judged by equation (1) which becomes. * Tffective inertia merits wider use perhaps; it is so variously adaptable to bridge gaps in knowledge or to secure compacter statement. The pure translation that equation (5) supposes may thus replace a. screw-motion of a rigid solid having constant mass and moment of inertia. The pitch of the screw would vary. an New Reading of Relativity. 37 its obligatory type. The artificial infinity of (m,.) for (v=c) is requisite to preserve the zero-acceleration at (P=R) with finite (m,). Within the freedom to utilize epoch-values for (m), choose a series that gives perpetual equality of (m, m,). That device amounts to completing equation (10), and establishing an exact step-by-step agreement in force-magni- tude with equation (5), Yet the two plans do not quite coalesce ; (dm) can never be primarily an increment of any constant (m,), also residual divergence from eqnations (3, 6) must continue. The Principle of vis viva is differentially valid; each constant (m,) absorbs a work-element wholly into change of kinetic energy. Remark finally that assign- ment of permanent value (mp) to (m;) brings in a significant instantaneous relation = Cok maps tie | ct os) cel ED) This line of possible connexion has been dwelt upon as preliminary at more length than its simple thought would justify, did it not strike centrally at the reasons—often left elusively subtle—which necessitate certain transitions peculiar to relativity. Its plain bearing will not be overlooked, it ean be hoped, upon the duplicated forces of ‘“ Minkowski type” and of ‘ Newton type” ; nor upon the introduction of “proper time” as well as “local time.” Both of which discriminations are given effect differentially, on the model of equation (11) *. Associated with the standard frame (F) are a group (U) of comparison frames (“legitimate frames”) having un- accelerated translations (2) relative to it, (u,v,) being colinear, and for standard sign codirectional. Transfers between (F) and (U) disturb fundamental expressions ; but we shall test there what degree of approach can be disclosed nevertheless to that permanence of type which is relativity’s victory. Velocity and tangential acceleration belonging to (F) will show in any (U) the magnitudes dv’, He pe as v= Uy—U; [ (w) is vector-constant. | * Relativity has invented manipulation of factors like (y(v)), where it needs to add or to suppress them, with admirable success. But no more will be expected here than to indicate clues at important turns of the ‘discussion, after which calculative detail is read with new meanings in mere routine. Moreover, the points made can appeal to those only who judge the argument with an interest due to knowledge of the subject. 38 ‘Prof. F. Slate on a Accordingly an “apparent” force (T,') estimated in that (U) is given by SS moe + (Up SS Bp =o (me ize ue, .) Ce) Add this explicit agreement: Effective inertia is inferred from physical relations in (I"), and passes unchanged through such purely kinematical revisions. Direct combination of equations (4, 12) under their condition of equal accelerations shows ee oe wae The “reduction factor”? (1—wv,/c?) connects the forces manifest in (U, F) through using (m) with the respective kinematical .data. Or equation (12) is evidently applicable to compare forces in (I), for the same inertia and acceleration, at differing speeds. In particular at (w=2,), Avs , a= me (w=, |.) =. Cone This introduces the familiar “rest-system” of relativity. And since we may write dis 5 & QU T,’/= [ (12) mort ) | aT = = m' — Tr under the suggestion of rigid dynamics, it would follow that AD a ee ro a 4 (h) y) a t= (S obs m!) oe = mS, ee 5.) c*>— Uv, c*— Uv, dt dt Identification of “ Fresnel’s coefficient’ («) in this second member fits the idea of its connexion with “ inertia-drag ” ot, CUE Dine Hinstein’s ‘‘ Addition theorem for velocities ” presents the same reduction factor as equation (13). But there it would reduce the velocity of (m) relative to (O’) the origin of (U) to dependence upon specifying frames. Newton makes that a constant difference of any pair of corresponding values. The intrinsic meaning of the theorem must go deeper, since its characteristic factor offers itself unconstrainedly in a dynamic relation of the two frames. This must quicken the surmise that what underlies the ‘‘ Lorentz transformation ” * Of, Silberstein, pp. 172-3. New Reading of Relativity. 3g is not exhausted when redefining simultaneousness [ Hinstein ] is added to the other avowed purpose of gaining in formal symmetry | Lorentz]. The same thought is continued, if we recall too how gravitation-foree (G,) is ‘“‘apparent”’ in weight (W,), with reduction factor that in the simplest case takes either form, Wi=G, (1-@); Gi=Wi (142), D165 This essential parallelism of equations (13, 16) throws new light on reading (w) in terms of imaginary rotation *. A second standard use of equations of motion adopts force as physically determinate (given) and predicts its kinematical effects by caleulation. This would impose (T,) invariantly upon (U,F): one stock instance is a convected compressed spring. Then we should pair with equation (12) dv’ ,dm l=, +v ae dy ern ee (17) where now the accelerations (dv'/dt, dv,/dt) fall away from equality, when the given values accompanying the same (m) are ise Woy a0 ole one wu: Hvidently also, the last relation would not be a permanent adjustment. What devices may look towards reconciling the divergent plans of equations (13, 17) is the next natural question. The answer will begin at considering the activity (v'T,) developed by the force (T,) of equation (17) at the working- speed (v'). In this aspect, both factors belong immediately to (U). Inview of equation (13) that activity is equivalently pe v ols == Dea En (18) (v,’) being a calculated (auxiliary) speed, but without de- parture from C.G.S. measure. Thus the activity within (U) of a force (T.,) implying unequal accelerations in (U, F) can be equalized with that of a force (I,’) implying equal accelerations, if at the same time the working-speed be * Minkowski; Sommerfeld. It is arithmetic to adjust a ‘“ length- contraction ” to producing numerical equality of the fields (G,g). The difference that these replace (c*) is provably superficial. 40 Prof. F. Slate on a modified from the observed value (v,'=v,— wu) as equation (18) denotes. Put into other words: Einstein’s rule yields a distorted speed in (U) which compensates exactly the dis- tortion of force into (T.’) from (T,). What may well be called a process internal to (U) is made in so far indifferent about the supposition of equal accelerations. Plainly the same rule holds between (U) and any frame (F"’) fixed relatively to the original(F). But observe how our derivation locates that compensation within the experimentally verified scope of equation (4). As a matter of algebra, we obtain the four forms : = 13 YY (2) CH == wu) = au, s dv! dt Y (6) = 3-7 (ve ys . (19) They all connect some Newtonian velocity of (m) relative to (O, O') the origins of (F, U) with Einstein’s specification of it. None of these alternative aspects is released, however, from its source in a strictly conventional expedient to the same end, which is here just nakedly announced, but which tle Lorentz transformation has masked under tle Hinstein variables. The completer activity (v,T,) in (F), expressed also by means of (v,/, Ty’) is Ao= volo = 97 (uw) Ta (we HU) os i. EO Therefore / ao @ayll 7 (1 + Se) 5 oye) Td =m y* (ve) 0 ae Equations (13, 21) being demonstrated corollaries of equa- tion (5), they are pivotal relations between the ‘‘ observing- frame” (IF), and the auxiliary quantities (appearances) in a comparison frame (U). Continuing, we may undertake to render equation (13) more fully reciprocal between the frames (F, U). Instead of depending upon observations in (F), let them be primarily taken in (U), the distinctive notation being (dv,'/dt, v, T.’). Correlate the force denoted by (T,’) on the one hand with its associated kinematics and on the other with a supposititious partner (T.) in (F). The plan of equation (18) gives New Reading of Relativity. 4] symmetrically, for equivalent activities now “internal” to (F), i Key Ferre as Wowk as lay ve wel sos + C22) where the factors in the second member are first any pair that satisfy the equality. Then equation (20) repeats in the form Ege te = te ey attig ey IE yi23) Since (T,) is a foree in (F) at the speed (v.), equation (5) holds ; so Ta = mgy*(t) de (24) a 0 c at . . ° . . . =~ Both equations (22, 23) are satisfied when (v,) is determined by the conversion rule that pursues the track of relativity : _ Uy +u , Ue é to! Ve= 53 Uy =—_ 7" (u) (ve — uu) = ——_; ‘a = yp we 14. we oa oo C c me de 3, \ 5) E(w! dt ¥y (v¢) ee (25) And provable corollaries of he assumed value are Uo. Ue 1+ x ) Ts are a ie . y I yf o- T,’ =moy(u)y? (2 ee U}.eerit) G26) The mathematics fixes upon equation (23) as committing us already to the consequences in equation (26), by requiring for consistency that (v.) which equation (25) calculates. The symmetry of (F, U) postulated by relativity is in fact seen to lurk in equation (23). The logical sequence may be reversedand begin by superposing the third of equations (26) upon equation (5). Whatever is hypothetical in equation (23) is reflected in equation (26), and vice versa. It would not be overbold analogy, as the outcome thus far shows, to anticipate through the second form of equation (16) the factor that ‘‘ reduces” from (U) to (F). In the view of local action, (T., T.’, T., T.’) of an electromagnetic statement would involve effective strengths of a ‘‘ motionless field”’ (F-space, medium), and of a ‘‘convected field” (U- space, medium). But whereas physics has long labelled (W,) without hesitation “‘ apparent.” the notation here need not be stressed hastily with similar decision. When relations are ascertained clearly between mathematical sequences whose * Connect equation (10) with this invariance, and one similar in equations (19). 42 Prof. F. Slate on a correspondence has been more obscure, and a way is opened to standardize in one frame by systematic reduction, the final issues in ‘‘ true physics versus correct mathematics” are not prejudiced by some postponement. Indeed, the closer scrutiny is revealing multiplied chances for illusion and encouraging suspended judgment. Certain critical values of the auxiliary velocities (v,', v) deserve mention. We find w=0, usu; v,,=0, w=Uus il Ww Oh Ue = 2 i=?) Up 6 | Ce amas OPEN. De ef a! 0, | Oe ue f parte ig lay (e2) reese erils aay Temp ee) (w) mt (27) These make plain first that the limiting speed (c), if affirmed for (v7), applies also to (v,’). The two series (%, v-) in F, U) respectively, and each within the range (0, ¢), are so telescoped relatively that the initial overlap (wv) is gradually annulled. Secondly, the relative translation (+w) of (I, U) links all aspects of the combinations as an unchanged magnitude. . There is just so much demand as can be read into these special ratios, for the “ practical infinity of (¢);” or for the “axiomatic necessity ” that light-speed relatively to all frames (U) stands invariant at (¢) *. Hquations (13, 21, 26) have connected force-values by four reduction factors ; with algebraically equal values shown by We Usa Uae ae Cues) “a Cc 1—- [= -— = SO c V¢ (ui) y(%o) Ae ol bait ce wt YO tu Y (Uo) | ( Vo y(w)y(v,") | (28) ae Uy) Uy te (wv) F | Ye Yuy(vo') UY, Ve—U y(v’) LRAT Tri to (uy (ve) J * The comment on equation (9) would lead to a finite activity of (P) expending work upon (2) during unlimited time of asymptotic approach to finite (c). Equation (6) also presents gross work (of (To)), with whose result relativity agrees. Iinding Wo=o; %=c; constant (mo) ; does not confirm the practical infinity of (c). Not even after redefining (Wo) as that “change of kinetic energy” whose disappearance from it we have traced. Attentive sifting of infinity-values is seldom misplaced in physics. 3 New Reading of Relativity. 43 The second members appear as “telescoping ratios” for velocities. The third members may reduce effective inertia conveniently. For instance, " GAGs 28 pave.) eva yi\) tare dm 7S Le Lk i SS ke oT . —_ aa oe as UV, uv, dt Uv, at ‘ees i l— =F tg é* adm ; ~ P | ae > = Moy (Wy (Ve ey (28a) The Fresnel coefficient («) of equation (15) is given also by mye) alr) Wie ee) (ve) (eo) (ely (to) y(w)y(ve') isd Nth g: Ct yo) Ped) ued? and by pct pa Seanad Fpl) wh. 1.28) Ce Equations (28a, 29) lead directly to the relation, significant like equation (15) of meaning for (x), [ moy(w)y(ve oe, ae = Tp (1- as )=xI). airy (29%) fe Because Uv ur, uv! 1 DOOD =p, ( =) cy y?(u) the product of each factor by y(w) modifies the reductions so that the original magnitude is restored by a combination of direct and reverse reduction between (F, U). Therefore our results reproduce formally the transformations for Minkowski force (K). If we identify T, = Ky; y(w)T. = K.; ae (i-* us 2) Ky; mo y(u) (14 we) K.. . (30a) [ yu)’ 1) Ky’; » “Ky =y(u) (1+" On the line of this parallelism, the ‘“‘ Newton force” of relativity and its ‘local time” do not enter directly. Per- haps they are secondary in a larger sense, aS more artificial alignment with Newtonian mechanics in part misread. Similarly, for Kp’=T.; K,/= UVe Mae K,/=7(u) eee \ Ky (30) 44 Prof. F. Slate on a It is a striking feature about the forms referring to the frame (U) that y({w) recurs as a sort of “ weighting factor ”’ which also designates the particular frame that is associated with (F). Comparing the last forms in equations (21, 26), we may say that y(w) disturbs the symmetry or completes it; we may seek symmetry in the first member or in the second ; and one additional step reaches either end. Consideration of the work-equations helps to analyse these connexions further. Direct integration of either extreme member in equation (18), referring to equations (5, 21), gives D Ve (To (oo—wdt= Ls 1. v. dt=¢ e/ 9 Mo san WI—-D- » BY) From equation (22) the work obtained is, remembering equations (24, 26), | “Tel(esl +ujat= | lad = erty (oye . (@2) ve 0 uU Complete the set ; adding in reliance upon equation (26) { Lap Sor Oran = 1). 2s) ea 0 Equations (6, 32) express work done in (I); equations (31, 33) work done in(U). The scheme of substituted velocities and new limits is symmetrical in all four. But, transition from (EF) to (U) divides by y(~) in one case and multiplies by it in the other. The deciding fact is patent: Hquation (6) presents the original (or physical) quantity as belonging to (F), and makes allowance for its ‘‘appearance”’ as judged in (U); equation (33) takes its start in the physics of (U) and calculates what appears in (Ff). Substituting for equa- tion (26) the form Ty (1572) Te Sg eee which would embody the reversed reduction with the same (T.), gives immediate verification. Finally it is plain that the various reduction factors, in combination with original totals and with fractions of them, constitute a mode of projection * The above group of relations brings out the idea that they approach a treatment of motion from rest in (U) as though * This furnishes a strong hint about their affiliations with imaginary angles, and the natural adaptation of hyperbolic trigonometry to relativity. New Reading of Relativity. 45 the initial inertia were (m'=y(v)m), which continuity of value with our assumptions for the standard (I°) would make it. The result that one vital difference hidden under identical kinematics in (EF, U) is due to altered inertia affecting work provoked early comment. In one dynamical analysis of relativity, it was shrewdly proposed as a chief postulate * The accompanying pairs of interchanges between (v,' +1, Yo, Vo) and (v_/, ve; Ve—u) must certainly not be ignored. Yet the commonsense reason for those is also one “key to Hinstein’s redefined simultaneousness. The same value of fluxion time (t) as an argument in time functions defines Newtonian simultaneous values of them. In this meaning, the auxiliary (v,’) and the observed (v,) are simultaneous : also (v., v). In addition, simultaneous observations in (F, U) would of necessity be recorded as (v,, u.—w), or as (v! +u, vo). It is clear that a pair of such observable data (vo +u, vo.) would not realize, for instance, the simplicity of equations (26), if (v.+u) were substituted for (v,) there. But if time-slip (lag or lead) between straightforward U.G.S. observations in (I’, U) be regulated under a known law of velocity-change, the paired values (v,, v,') requisite for sim- plicity are also attainable as observed. The same thought coordinates equation (13) [Newton] and equation (21) [Einstein]. Further, because this ‘ propagated simul- taneousness”’ and the Newtonian allowance for “‘ apparent force” establish identical reduction factors; for Newtonian velocity and for Hinstein’s force-law respectiv ely; it seems fairly proven that the two systems of procedure are effectively alternative—for the electronic case discussed. The reciprocal relations between the dynamics of the standard frame (F’) and that formulated for any one frame of the group (U) are to this extent conclusively settled. It remains to examine how much of such reciprocity persists, when neither of the compared frames is (F). This will confront the conditions where relativity can do best service through its simplifications. Choose then two comparison frames (U', U") with origins (O', O”’) and with translations tw’, w'’) sate to (IF), “still colinear . with (vo) of course. Denote observed velocities of (m) now by (v, v', v''), so that —e=vtewsvu' tu’; Yael =Hu'—w. . foo} A reasonable notation for other quantities will also disting euish them by accents. Thus simultaneously for im); UV Piss A | ESE fies =) Le 0 * Frank, Wied. Ann. (1912) vol. xxxix. p. 693. uly ger 43) 46 ; Prof. F. Slate on a Observations in (IF) would show the velocity of (m) relative to.(O") lav (ula) Sv uy. 2. . . (87) Equation (19) applied correctly to each part of the third member gives for velocities in (U’) pete to t(O- rae v—U u! =u Vel = ] 3 Up! — an ° e e (38) UU U ‘ou! te 2 i 2 C C Then without complications the identity is verified : ler. y(t") Te! y(n!) (I=) (yw) Te). (89) In verbal form: Direct transition (F’, U'’) and two successive transitions (I, U’), (U', U"), coincide upon the force fmally apparent in (U"). But in terms made consistent with equation (18) as regards all entries under (v'’); and only when each apparent force is properly ‘‘ weighted.” The previous developments furnish plain reasons for both con- ditions. Begin with (U’) as observation-frame, and under transferred guidance of equations (13, 26) construct the provable identity: yu) ( sate ihe ‘= | 1 (ue )-" we se (w)(1+ © oe yids | tee Again results agree exactly for direct transition Uy and for the two- -step process, (U’ F’) followed by (F, U"). In this use, corresponding to equation (38) we must have in their relation to (v) and (U'): v= — 43 Uu=—— .te:C«sSCSA*dL c PAS) C= Y es WU U Ug ee 1 2 C ¢ Without repetitive detail, therefore, the conclusion can be put generally: When these discriminations about terms are upheld, the calculated net distortion (into apparent force) depends upon the terminal frames alone. Hence it vanishes if the series of transitions closes at the initial frame ; the remark under equation (29) becomes a wider truth*. As bearing upon our immediate purpose, it places all transitions within the group (U) in a comprehensive setting of partial reduction to the same standard (F). * A consequence that one might borrow from the familiar theorem on superposition of colinear Lorentz transformations, by assigning present meanings to its symbols. New Reading of Relativity. AT Equation (14) records the special value of (T,') for that rest-frame to which relativity concedes unique prominence through a coincidence in magnitude for the two force-types (“ Minkow ski” and ‘Newton’ *) and a peculiar -<‘rest- aeceleration.””. Analysis of the transition trom (EF) to a rest-frame (R) in those aspects detects three stages : (a) Changing Minskowski-force into Newton-type. (b) Cancelltas a factor (y?(w)); or (7/°(v)) transiently undistinguished from it; with some freedom among assignable reasons. (c) ee eats the magnitude (v3(v)dv/dt); now as perclerauian in (R) under . the sliding-scale of units for time and length. A mathematically colourless regrouping of factors led to the important quantities (ve, Ue) of equations (18, 25); its con- venience has wide range. But such fully interchangeable forms are often unequally accommodated to physics. Hqua- tion (5) presents in effect this same rest-acceleration as incidental to a (probably fictitious) fusion of force-items. Only an inflexible view about constant inertia can force upon relativity the result of (c) as an acceleration. In comparison, is the latter not artificial? What is worse, it is superficial as well, if it obscures that inclusion of rest-frames in the ‘shift process’? common to many coordinate-systems and describable as a parameter-variation*. In that feature, it enforces the thought attached to equations (10, 11), and rompts a directer inter pretation of rest-frame calculations, though the formal results stand uncorrected. We build upon a more general relation for any frame (U) and specify a momentum ((,) determined from a velocity (¢;) and an inertia (m,) through Qi = my (wu) r= (rmgy(v) Jey = mj (y(u)v,).. . (42) This introduces the same weighting factor as equations (30, 31, 33) and it is written out to suggest a double possibility : associating (y(w)) with either (m,) or (v,). Regarding (Q;) as a function of (wu, v,), its exact differential and its total * For the elaboration of this idea see Slate, ‘ Fundamental Equations of Dynamics’ (1918); Index, under “ Shift,” Shift-rate.” Nothing but unformed habit bars embracing i in the same schenie continuous change in the unit-magnitudes. Other seed-thoughts of the present “ines are to be found at pp. 38-43; p. 211; where the need of equations (1, 2, 3 is emphasized, 48 Prof. F. Slate on a time-rate are, when (m,) is constant, SEN Chey mee dQ) = Ou du+ Ov, av, J d Ci) alu: cm, avy eal tau cue dt = 704,04 Clr dt St (c?—u’) 1j2 at e (43) Defining m’= my(w), the last equation is mathematically presentable as a measure of tangential force, d i's dm dv d dQy = Det ey +m ims = apse tu) > La Yet the first term in the third member is spurious for the physics, so long as (m) is constant ; it has itsesole origin in a shift-rate among the frames (U). In application to a finite interval, the name rest-system at equation (14) is a collective noun, comprising selected frames in the possible group (U); the earmark of a rest-frame is the transient equality of its permanent (w) and the velocity (v) of (m) at that one epoch. Identify (m,, mo) and (%, v); and let the shift-rate be so regulated that (v=w) perpetually. If the spurious force were real (2.e., physical), equation (44) would give (T,) correctly, on making the substitutions my = ny. . Ea Or it would repeat the essentials of equations (21, 24, 26), on exchanging (v' Uv.) for (v); while due use of the weighting factor harmonizes the forms completely. Thus an assumed constant inertia, plus illusory (imaginary, quaterni- onic) force due to a glossed-over shift, can replace with, mathematical precision our assumed variable inertia*, through continuous use of the rest-frames for the several epochs. Since the working-speed (v/, v,') becomes zero for a rest- frame, that cannot be self-sufficing for the expression of activity. The directest composite plan contains as well frames fixed relatively to (If) +; but more deviously, rela- tivity reduces the quasi-force (7,dm'/dt) by the factor (¢/2), coins an unrealized velocity-term (¢), attaches it loosely to (v,), and arrives correctly at the activity (A,) of equation (6) essen- tially thus: dm dm c\ ( dm = {Dye —_—p OP Rae SS = 0 — ae ° e ° Ay=o Ta cae (« Ae ¢ () (» di (46) * It is apposite to quote Sommerfeld’s remark that the element (dr) of proper time “ Ist kein vollstindiges Differential.” + Slate, ‘Fundamental Equations,’ pp. 147-54. That background of standard frame cannot be quite ignored. Lorentz remarks ‘ Bei einer absoluten Genauigkeit ware auch der Unterschied der ‘ Higenzeit’ der Erde von der Zeit des Sonnensystems zu beriicksichtigen.” New Reading of Relativity. 49 Ingenious device contrives a mathematical status for the last term in equation (2), whose recognition at first hand constancy of inertia excludes. The natural inferences that equation (13) has forced upon attention by restating the situation of equation (16), offer more than a glimpse of a self-selected frame, basic for physical phenomena in this sense: Departures from it, either by rotation or by translation impress distortions of common type upon dynamical magnitudes. To render that deeply-seated symmetry convincing might crown our instinctive search better than inventing for physical laws a formal indifference among base-frames. If such ideal outlook has restricted appeal, there is practical service in coordinating at the same time elements of potential. That a potential is available which gives coherence to all phases of our electron’s energy, the work equation indicates through its sole dependence upon terminals of the interval. Equations (39, 40) are another aspect of that thought. Moreover, the initial suppositicn of equation (5) and its probable connexions at equations (9, 10, 11) fit at sight the more standard form of the Lagrange function (lL) as measuring appropriately a difference of energy-states. Write eye, = — mc? ree — 1) = mc? —m(c?—v?). (47) c Fixing the zero-phase as rest in the frame (F), the last member sets down an expectation then bounded by bringing (mo) to the limiting speed (c). In the general phase, (my) has become (m) by gathered increment: so reading (mv?) as a fact of realization the last term exhibits what other energy of (m) at that epoch is outstanding for final conversion into kinetic form; at the limit-value (c), whatever (m) may be attained by accretion. The form (2K —L) for total energy agrees with equation (6) for work done from rest. Whatever cumulative pressure from these arguments is felt will carry us toward restoring to fuller proportions the shrunken claim that Lorentz urges for “ Wahre Zeit” and for frames ** Dieim Ather ruhen;” a passage remarkable for its scientific candour*. The complications due to curvilinear path, to radiation, and to other causes do not weaken this primary correlation of Einstein with Newton. Both lines of thought alike must cope with the superposed difficulties. University of California. * Sammlung, p. 75. Phil. Mag. Ser. 6. Vol. 40. No. 235. July 1920. EK IV. The Relative Actirnty of Radium and the Uranium with which it is in Radioactive Equilibrium*. By J. H. L. JonNsTONE and B. B. Botrwoopt. en ae the matter has been under consideration and discussion for a number of years, the genetic relationship between the earlier members of the uranium family of radio-elements is still a matter of considerable uncertainty. It is generally conceded that both ionium and actinium are products of the radioactive disintegration of uranium, but the exact point of origin of actinium and its immediate parentage have remained somewhat obscure and uncertain. The work of Boltwoodt on the relative a-ray activity of uranium minerals and the uranium which they contained demonstrated a constancy of relationship between the radio- active constituents of the older minerals and clearly indicated a close genetic relation between uranium and actinium. His determination of the activities of the more stable a-ray products relative to the activity of the associated uranium showed a simple and direct relation to exist between the products ionium, radium, and polonium, but showed an abnormally low value for the ratio in the case of the actinium series, which could only be explained on the assumption that actinium originates as a branch product and belongs to what may be termed a collateral branch of the ionium-radium- polonium family The values obtained by Boltwood in the course of his experiments showed that the activity of the uranium was about 2°2 times that of the radium with which it was in equilibrium, although at that time the range of the «-particle from uranium was supposed to be about 2-7 cm., which is less than that of the a-particle from radium. Since the ionizing power of an a-particle is nearly proportional to the range, and since, on the basis of the disintegration theory, an equal number of «-particles are emitted per second by each of two * The experimental results given in this paper and the general theo- retical conclusions are taken from a dissertation on the “ Relative Activity of Uranium and Radium” presented on April 27, 1916, by J. H. L. Johnstone in fulfilment of the requirements for the degree of Doctor of Philosophy in Yale University. The chief reason for the delay in publication was the entry of Dr. Johnstone into active military service with the Canadian forces in May 1916. The work was carried out in the Sloane Physical Laboratory of Yale University. + Communicated by the Authors. { Am. Journ. Sci. xxv. p. 269 (1908). Relative Activity of Radium and Uranium. oe -adioactive products in equilibrium with one another, it was necessary to assume that either the uranium atom emitted two «-particles simultaneously, which was improbable, or that two distinct a-ray changes existed in ordinary uranium. Neither of these assumptions, however, completely obviated the dithculty. The fact that uranium actually did emit twice the number of 2-particles to be expected on theoretical grounds was sub- sequently demonstrated by Geiger and Rutherford * and by Brownt who counted the number of a-particles emitted per second from a film of pure uranium oxide and a similar film of uraninite of known composition. Using the scintillation method, Marsden and Barratt{ made a careful examination of the a-radiation from uranium and concluded as a result. of their experiments that ordinary uranium consists of a mix- ture of two successive a-ray products in equilibrium with one another. Attempts to measure the separate ranges of tlie a~particles emitted by these two products were made by Foch§ and Friedm By the use of a better method of measurement, in which the Brage ionization curves for a uranium film were compared with the corresponding curves obtained with polonium and ionium, Geiger and Nuttall calculated the ranges of the a- particles from uranium to be 2-5cem. and 2°9 cm. (at 0°C.). Numerous unsuccessful attempts have been made to reduce the specifie «ray activity of uranium. In one experiment conducted by the authors about two kilograms of pure uranium nitrate were subjected to fractional . crystallization, and a least soluble “‘ head”? fraction weighing about twenty grams was obtained after about forty operations. The specific a-ray activity of the uranium in this material did not vary by as much as one per cent. from the specific activity of the uranium in the original nitrate. This shows that the two components are so closely allied chemically as to be inseparable, a conclusion which is supported by all the other known facts at our disposal. We may outline, therefore, the progressive disintegration of the uranium atom, considering for the present only tle products emitting a@-rays, as taking place in the following manner: the parent element uranium I[., produces the pro- duct uranium JJ. This in turn produces ionium, which * Phil. Mag. xx. p. 691 (1910). + Proc. Roy. Soc. A lxxxiv. p. 151 (1910). { Phys. Soc. Proc. A xxii. p. 367 (1911). § Le Radium, viii. p. 101 (1911). || Wien. Ber. cxx. p. 1361 (1911). HK 2 52 Messrs. J. H. L. Johnstone and B. B. Boltwood on the disintegrates to furm radium, followed successively by radium emanation, radium A, radium ©, and polonium. When these are all present in equilibrium proportions, as is the case in a non-emanating, old, radioactive mineral, then certain com- paratively simple relations will exist between the a-ray activities of the different constituents. It has been shown by Geiger* that the ionizing power of an e-particle is propor- tional to the two-thirds power of its range. The ionization produced by equal numbers of a-particles emitted by two different radio-elements will therefore be proportional to the . two-thirds power of the ranges of these particles. Ina series of successive products in equilibrium, each product emits the same number of a@-particles in unit time. The relative ionization (and therefore the relative activities) due to each of these products should therefure be proportional to the two- thirds power of the range of the respective a-particles. This relation has been shown+ to hold quite closely in the case of radio-thorium and its a-ray products, and also in the actinium and the radium series of products. The chief object vf the work described in the present paper was to apply the same methods to the case of the uranium-radium series with the expectation that the results would throw some light on the obscure relations of the earlier members of the series. The Radioactive Measurements. The determinations of the radioactivity of the different: solids examined were carried out in an electroscope which has already been described{. In the present experiments a telemicroscope was used for observing the position of the gold-leaf, The natural leak of the instrument was small and over a period of about six months varied from 0:4 to 0:7 scale division per minute. Before and after each series of measurements the sensibility of the electroscope was deter- mined by measurements of the activity of a standard refer-. ence film of pure uranoso-uranic oxide. This film was carefully preserved throughout the entire period of the measurements here recorded, «snd all the results given in this. paper are given in terms of this film as the standard. The method of preparing the radioactive materials for measurement was essentially the same, with certain modifi- cations, as that described by Boltwood. The material to be * Proc. Roy. Soc. A lxxxili. p. 505 (1910). + McCoy and Viol, Phil. Mag. xxv. p. 353 (1913); McCoy and Leman, Phys. Rey. iv. p. 409 (1914) ; ibid. vi. p. 185 (1915). { Boltwood, Am. Journ. Sci. xxv. p. 272 (1908). Relative Activicy of Radium and Uranium. D3 examined was ground as finely as possible in the form of a thin paste with pure ethyl alcohol in a small agate mortar. A sheet of aluminium 7°5x9em. and 0:01 em. thick was first carefully cleaned with liquid soap and distilled water, and was then placed in a drying oven at 65° C. for fifteen minutes. It was placed in a desiccator over sulphuric acid for half an hour, and then weighed on a sensitive chemical balance. The paste of material and alcohol was then thinly spread on the surface of the aluminium with a small camel’s- hair brush*. The coated plate was placed in the oven, cooled in the desiccator and weighed as before. The weight of the films could be determined in this manner with an accuracy of one per cent. The solid material adhered quite strongly to the plate and showed no tendency to fall off even when the plate was inverted. The measurements of radium emanation were made with a gold-leaf electroscope having an air-tight ionization cham- ber with a capacity of about three litres. The separation and collection of the radium emanation, its transfer to the electroscope, and the measurement of its radioactivity were carried out according to methods which have already been described fF. Fatio of the Activity of a Uranium Mineral to the Activity of the Contained Uranium. The relation of the activity of the parent element, uranium, associated with equilibrium amounts of all of its disintegra- tion products, to the activity of the parent element alone, is a fundamental quantity of great importance to any considera- tion of the relations existing between the individual products themselves. The actual progenitor of the series is uranium L., but this cannot be isolated from its invariable associate and isotopic product uranium IJ. The combined effect of these two elements when mixed in equilibrium proportions can be determined, however, and this can be compared with the activity of a similar mixture containing all the other disin- tegration products in equilibrium proportions. Such a mixture is furnished by a pure, primary, unaltered uranite. A mineral containing a low proportion of thorium is preferable since a correction must be made for the activity * The brushes used were carefully cleaned in advance by long immer- sion in alcohol and subsequent washing in fresh quantities of the same liquid. + Boltwood, Am. Journ. Sci. xviii. p. 378 (1904); Phil. Mag. ix. p. 599 (1905). Aerivi ty in Ovisions PER MINUTE 54 Messrs. J. H. L. Johnstone and B. B. Boltwood on the of the thorium products present. A specimen of uraninite from Spruce Pine in the possession of the authors was con- sidered to fulfil all the necessary requirements. It consisted of essentially unaltered material selected with much care from a considerably larger quantity. It contained less than 0-2 per cent of silica and residue insoluble in dilute nitric acid. A determination of the uranium content was made by one of the authors and by Ledoux & Co., of New York City *. The mineral contained 1°9 per cent. of thorium oxide. In determining the activity of uranium a very pure speci- men of uranoso-uranic oxide was used. This had been pre- pared from a specimen of especially pure uranium nitrate WEIGHT CF FitLr7 =104 obtained by fractional recrystallization of a much larger quantity (see p. 52). The oxide was made from the nitrate with all the precautions which have been mentioned in an earlier papert. It is very important to note that this oxide was used as a standard in the analytical determina- tion of uranium (both volumetric and gravimetric) in the * The authors wish to express here their obligation to Ledoux & Co. for this favour and to state their appreciation of the value of this carefully conducted analysis. 1 Boltwood, Am. Journ. Sci. xxv. p. 278 (1908). Relative Activity of Radium and Uranium. 5d mineral, so that the analytical determinations and _ radio- active measurements are in direct accord with one another, although it was assumed for purposes of calculation that the uranoso-uranic oxide contained 84°8 per cent. of uranium. It has been found by McCoy ™* that considerable absorption of the radiation takes place in the film itself when its thick- ness is appreciable. In earlier work by one of ust it was shown that by the use of thin films any necessity for an absorption correction could be avoided. To further demon- strate this fact a series of films of uranoso-uranic oxide weighing from 0:0019 g. to 0:04 g. were prepared and their activities were measured. The results are given in Table I. and are shown graphicaily in fig. 1. In films weighing not more than 10 milligrams the absorption of the @-radiation was negligible. In all cases where values of importance were to be derived the weight of the films used was less than this maximum, so that no correction has to be made in the results for absorption of the radiations. The average of the first eight values in the fourth column of Table 1. is 666, which denotes that the average activity of the eight lighter films was 666 divisions per minute per gram of uranium oxide. This corresponds to an activity of 785 (viz. 666/0°848) divisions per minute per gram of uranium. LABIA, cl. Film Weight of oxide Activity Activity Number. in grams. Div./Min. Weight. DA eats deta “00187 1:24 663 LAS es SSR Se 00432 2°91 673 NS oe eee ee "00566 3717 666 Doe ee Ne ea ‘00653 4°37 669 Be ee eh a3 ‘00740 4°95 668 es SLANT S| ‘00832 5°56 662 fi ese hs “00898 5°95 662 Zi ieee eee "00653 4°38 670 AGS: Aad 5 me “01405 8°89 633 oe ee “01380 8-94 648 rA5h oan tee ene "01440 9°30 645 | gah. babel 02150 13°45 624 oA bg eee & 04350 25°96 596 * Journ. Am. Chem. Soe. xxvii. p. 391 (1905); Phys. Rev. i. p. 393 (1913). + Boltwood, Am. Journ, Sci. xxv. p. 176 (1908). 56 Messrs. J. H. L. Johnstone and B. B. Boltwood on the The specific activity of the uraninite was determined by the measurement of four films weighing from approximately 2 to 6 milligrams. The results are given in Table II. Tasue II. Film Weight Activity in divisions per Nuinber. gram, © minute per gram. Dili, ec. nee segeedees 0:00387 2627 DS chee wage tyske 0:00240 2655 PLS Ra Area Ren Ae 0:00588 2627 SOP arc erceuyas 0:00170 2600 These results give the mean value for the specific activity of the uraninite as 2624 divisions per minute per gram. As already stated, the mineral contained 0:019 gram of thorium oxide (and other thorium products in equilibrium with this) per gram of mineral. The above value for the specific activity of the mineral includes the increment due to the thorium products which must be eliminated. To accom- plish this measurements were made of a series of films pre- pared from a specimen of thorite containing 52 per cent. of thorium oxide and 0°37 per cent. of uranium. ‘The results — are tabulated in Table III. Tasue III. Film Weight Activity per gram in Number. gram. divisions per minute, BO Wi imc'sane Bs eetE 0:0056 511 UE Sane A te 00076 515 Las eae AOE 0:00776 514 Av. 518 The activity of the uranium and its radioactive products contained in the thorite will closely equal 2624 a ae ‘ aoe x 0:0037 = 13°3 divisions per minute. Correcting the specific activity of the thorite by this number and dividing the remainder by 0°52 (the weight of thorium oxide per gram) we obtain 960 divisions per minute as the activity of one gram of thorium in equilibrium with its pro- ducts. We may now correct the value found for the activity of the uraninite by an amount equal to the activity of the thorium components (960 x 0:019), and obtain the value Relative Activity of Radium and Uranium. Xe 2606 divisions per minute per gram of mineral for the uranium series of products which it contains and a value of 3570 divisions per minute per gram of uranium present (2606/-73). The uraninite, however, contained less than the full equi- librium amounts of the uranium-radium products because of the fact that in the finely divided form in which it was used it spontaneously lost a small proportion of its radium emana- tion. ‘The loss of this and the absence of the proportionate amounts of radium A and radium © would cause a deficiency which must be corrected for. The relative proportion of radium emanation lost by the films of uraninite was determined by the method described by McCoy and Leman*. It was found to be 9:1 per cent. In applying the correction, the value for the ratio of the activity of the radium pr oducts (radium emanation, radium A and radium C) to the activity of the radium with which they are in equilibrium, as found by McCoy and Leman, namely 4°11, was made use of, as was also the ratio of thie activity of radium to the activity of the uranium with which it is in equilibrium (0°49) which was derived in the course oi the present investigation (see p. 60). The correction has the following form, 3570 + (785 x 4:11 x 0°49 x 0°091) =3715, which gives an activity of 3715 divisions per minute per gram of uranium as the activity of the mineral due to uranium and its products in a complete state of equilibrium. From this final result we are able to calculate the value sought, namely, the ratio of the activity of the uranium with its equilibrium amounts of disintegration products to the activity of the uranium (uranium I.+ uranium II.) alone. This is 8715/785 =4°73, which is in good agreement with the value 4°69 found earlier by Boltwood. Ratio of the Activity of Radium to the Activity of the Uranium with which it is associated. If the transformation of the atoms of uranium I. into atoms of uranium II. takes place directly without the pro- duction of any side products, and the transformations uranium I[.—ionium—radium proceed in the same direct manner, then the relative activities of the three members— uranium I., uranium II., radium—should be proportional to * Phys. Rev. vi. p. 185 (1915). 58 Messrs. J. H. L. Johnstone and B. B. Boltwood on the the two-thirds power of the ranges of the 2-particles emitted by the respective elements, namely, the activities should be proportional to CBO ATO) s, (Geta) 5, where the numbers in parentheses are the ranges at 0° C. of the «-particles from uranium [., uranium II., and radium, respectively. Any departure from this proportionality will indicate an irregularity in the mode of transformation and may serve to suggest the nature of the changes which are actually taking place. If the transformations are all simple the activity of the uranium (consisting of an equilibrium mixture of uranium [. and uranium II.) should be to the activity of the radium in the proportion (#8 2b 1-96) 224-100 ),0-ae An experimental determination of this ratio was carried out in the following manner :— A quantity of radium was separated from Colorado carno- tite and was carefully freed from other radioactive substances which can be separated by chemical operations. Since car- notite is free from thorium, the specimen of radium obtained did not contain any appreciable amounts of mesothorium or other products. A solution of this radium in dilute hydrochloric acid was then prepared and its approximate strength in radium was determined by the emanation method. Using this first solution as a basis, two other solutions (denoted hereafter as solutions B and C) were prepared, which contained about 0:025 g. of barium chloride and 2°4 x 107* g. of radium in 10 ¢.c. of solution. The quantities of radium were so chosen that the radium films ultimately obtained would have activities of the same order of magni- tude as the activities of the uranium films with which they were to be compared. An accurate determination was then made of the emana- tion in equilibrium with the radium in 10¢c.c. of the radium- barium solutions. The results were recorded in terms of the leak produced in the emanation electroscope in divisions per minute. The results were :— For 10 c.c. of Solution B For 10 c.e. of Solution C The radium emanation in equilibrium with the radium contained in one gram of the uraninite was also determined and was found to correspond to 853°5 divisions per minute in the same electroscope. Since the uraninite contained stone hare 81:7 div. per min. Seep ke 4A LO) disper maine Relative Activity of Radium and Uranium. 59 73°0 per cent. of uranium, the radium in equilibrium with 1 gram of uranium in the mineral was equivalent to a leak of 1170 divisions per minute. The quantity of radium in 10c.c. of solution B, therefore, was the same as that in equilibrium with 0°0698 @. of uranium, and in 10 ¢.c. of solu- tion C with 0:0607 g. of uranium. The a-ray activity of the radium itself was determined in the following manner :— Portions of solutions B and C 10 c.c. in volume were taken, and the barium and radium present were precipitated sulphate under standard analytical conditions. The precipi- tates were removed and ignited as promptly as possible and the time at which the ignition was carried out was noted. The weight of the precipitate was determined and the material was quickly ground to a fine powder with ethyl alcohol. Fig. 2 /Z00 pee aie Oe2 cea Bere eet 7100 CvRVE LZ cCvaRve Z- Ff). i, | Pie WOE 10 a ee Films of this material were then prepared in the manner already described and the activity of these films was measured in the a- ray electroscope. Measurements of the activity were continued over a period of six to seven hours, and the increase with the time (due to the growth of the emanation and other active products) was noted at definite intervals. Preparatory to the precipitation of the sulphates the radium-barium solutions were kept at a temperature slightly below the boiling-point for a period of four hours in order 60 Messrs. J. H. L. Johnstone and B. B. Boltwood on the to remove any emanation as formed and to permit the other products (radium A, radium C) to completely disintegrate. The activity of the material forming the films was therefore due to radium alone at the time of ignition, and this activity could be easily determined from the data available. The variation of the activity with the time for two typical films is shown in fig. 2, where the zero time is taken as the moment of ignition. By a simple extrapolation of the curve the aaa activity could be obtained with accuracy. The calculation of the ratio of the activity of the radium to the activity of the uranium was made by the use of the following equation, where w= the required ratio. Em, = the activity of the equilibrium amount of emanation from one gram of uraninite in div. per min. (in the emanation electroscope). Em, = the activity of the equilibrium amount of emanation from 10c¢.c. of radium-barium solution in div. per min. (in the emanation electroscope). U_— = the activity of one gram of uranium oxide in div. per min. (in the a-ray electroscope). R= =the initial activity of one gram of radium-barium sul- phate (in the a-ray electroscope). W =the weight of sulphate precipitated from 10c.c. of the radium-barium solution. The value of wx is given by W x Em, x R x 0°848 O37 3.5¢ Bim x UU: The advantage of this method of calculation lies in the fact that the question of either radium or uranium standards is not involved in the final value. The results obtained from the measurement of the radium films are given in Table IV. C= TaBe LV. Film Film. Solution. Number, Weight. U. R. W. > im, Bim, pe 123, ieee 101 0:00926 710* 1049 0-0278 853 81°8 0°493 Dr eestey 102 0-00711 710* 1050 0:°0278 853 817 0-497 By eelie. 103 0:00819 TLO* 1030) O:0277 eos" 87 tee Beet’: ae 104 0:00667 710* 1025 0:0274 853 81:8 0-480 Cheese? 110 0:00548 710* 1025 ©0237 853 71:0 0-478 Chavet 113 0:00300 710* 1050 0:0237. 853 70:0 0-495 Mean value of c=0°488. * The value here given is based on a different. sensibility of the electroscope from that which it possessed in the case of the values given in Table 1. Relative Activity of Radium and Uranium. 61. The value obtained by Boltwood* for this ratio (0°45) is somewhat lower than that given above. When the experi- mental conditions are taken into consideration, however, the agreement is as good as might be expected. A determina- tion has also been made by Mey er and PanethT, who com- pared the radiation from a known quantity of radium with the ionization produced by the a-particles from one gram of uranium. They obtained a value of 0°57 for the ratio. Aside from other objections to their method, the manner in which they obtained the uranium salt used as a standard is open to the most serious criticism. Comparatively crude uranyl nitrate was subjected to purely chemical methods of purification, valsehiclR which are generally recognized as un- suitable for obtaining a pure uranium product. Very little, if any, weight can therefore be attached to their deteaminanen of the value of the ratio. Discussion of Results. As already pointed out, if the entire series of transforma- tions from uranium I., through uranium II. and ionium, to radium is a simple and direct one, the value to be expected for the uranium-radium ratio is approximately 0°57. The value found in this investigation is 0:49, which is lower by an amount far in excess of the probable experimental error. The result suggests that the number of radium atoms which disintegrate with the emission of a-particles in the unit time is less than the number of atoms of uranium I. or uranium I]. which disintegrate in the same period. This indicates either (a) that a series of branch products is split off from the main series betore the radium atom is produced, or (6) that radium itself disintegrates in a complex manner, a larger proportion (but not all) of the atoms being trans- formed with the emission of a-particles. Unless the accepted values for the ranges of the a-particles from uranium are greatly in error (which appears to be rather improbable) the progress of transformation from uranium I. to (and including) radium is at some point irregular and is accompanied by the production of a collateral series. ‘This conclusion is sup- ported by the occurrence of actinium and its products in association with radium in uranium minerals, and by the impossibility of tracing the origin of actinium to any point in the series subsequent to radium. * Am, Journ. Sci. xxv. p. 269 (1908). + Wren. Ber. exxi. Abt. IIa (1912). 62 Messrs. J. H. L. Johnstone and B. B. Boltwood on the We will now proceed to a consideration of the two alterna- tives (a) and (4) mentioned in the preceding paragraph. As a preliminary to this the relative activities of the other members of the main-line series can be calculated from the ratio experimentally determined for the uranium and radium. Geiger’s equation is employed TIN oR and the results obtained are given in Table V. The calcu- lations are made on the assumption that the simplest (1 : 1) relation exists between all the products following and including ionium. TABLE V. Range of a-particle Relative Activity Element. in em. (at O° C.) Cale. Found. Wimannuiia oas s kee eee Or rama sees eee eee 2°75 1:00 1:00 Fomine 45, (ees ae 2°95 0°46 adidas 2% 3) eee ee eee mle 0:49 0:49 Bimanatiomlt.e cae eee 3°94 0:57 Hvarditiin (ACL fhe Seen eee 4°50 0:62 Radium: (© 2) eee rose 6:57 0°80 Radium. ieee pee eee 3°64 0°53 Sum=4°47 Relative activity of actinium and actinium products as determined experimentally by Boltwood* =0°28 Total=4°75 Total activity as determined (p. 57)............ 4:73 | The very close agreement between the total activities as calculated and as found is purely accidental and merely indicates that the value given for the activity of the actinium products by difference (4:°73—4:47=0°26) is practically the same as that suggested by Boltwood as a result of his experiments. The ranges of the «particles from uranium, radium, and each of Nie actinium products and also the valle of the aes thirds power of the range are given in Table VI. * Am. Journ. Sci. xxv. p. 297 (1908). Relative Activity of Radium and Uranium. 63 TaBLeE VI. Element. Range at 20°. R3. Peer MPR co. cn ass ste tens R, =2°54 1°87 Mere OE Soares ncaa ss aie dase Ri, =2'95 2°05 RRR cc terete Ghote denials R,=3'36 2°24 eC O ACHE oo dn as cecesccsensanse R,=4-29 2°64 BPMRTPOIUURIN 2, or Sie. Sh cas ct out esa apes 6 R,=4°34 2°66 Actinium emanation ........:...... R,=5'66 3°18 PRRTMINUUN: ce e.Seue canwebacssacces> R.=6°37 3°44 PROCITE. AL Aerob ce vagy ssccetues R,=o24 3°01 Applying the data available we may calculate what proportion of the total number of atoms of uranium II. would have to be assumed to disintegrate in a mode leading to the production of actinium in order that the ratio of the activity of the actinium products to the activity of the radium would have the value 0°28/0°49 indicated in Table V. There are five actinium products emitting a-rays as compared with a single e-ray change in the case of radium, and if equal numbers of atoms of each of the ele- ments were disintegrating in unit time the ratio of the activities would be* (2°64+4 2°66 +3°1843:4443°01) : 2°24=66: 1. The observed ratio is, however, 0°28:0°49. If 100 atoms of uranium Ii. are assumed to disintegrate in unit time of which 2 disintegrate to form actinium, we have the relation 1S Oo hie a 20 (100—.) 2°24 0-49’ which gives a value for « of approximately 8. So that, if, out of every one hundred atoms of uranium II. disin- tegrating, a total of eight atoms changed into actinium and co) : the remaining 92 changed into ionium (and ultimately foo) radium), the observed relations would exist between the activity of the radium and the activity of the actinium pro- ducts in a mineral. Based on considerations of this character a number of attempts have been made to devise a scheme of transforma- tion which will satisfactorily indicate the successive changes undergone by the uranium atoms. The most plausible of these have been proposed by Soddy and Cranstonf and are given on the following page. * Table VI. + Proc. Roy. Soe, A xciv. p, 384 (1918). 64 Messrs. J. H. L. Johnstone and B. B. Boltwood on the Ue =~ Wars cians War ee onVs . aie aps (Nes dl == (IT) UT*UX*-UXS UE x a RE UY+-U ZAG aE Ee It is not proposed to discuss these schemes in detail or to consider at any length certain minor points involved to which exception might be taken. It is, however, difficult to under- stand just what is implied by the dual transformation with the expulsion of a-particles in both cases which is suggested for UI. in the first scheme and for U II. in the second. It would appear that the loss of an «-particle in each case should lead to the production of one and the same kind of matter, namely, to a single product and not to two different products to be designated as U X, and U Y in one example and as Io and U Y inthe other. The main point under consideration is, however, whether either of these schemes gives us a clue to the explanation of the relative activities of uranium and radium as they have been found in our experi- ments, and it may be stated that they do not, since the first scheme would suggest a ratio of 1/55 for the relative activities of the uranium and the radium, while the second scheme would imply a ratio of 1/:53 for the same quantities. In order that Scheme I. might satisfactorily apply to the ratio as found by experiment it would be necessary to assume that about 26 out of every 100 atoms of U I. were trans- formed in the mode leading to the production of actinium. This in turn is contradicted by the relative activity of the actinium products in an equilibrium mixture. In order to fulfil the conditions involved in Scheme II. it would be necessary for 14 out of every 100 atoms of Relative Activity of Radium and Uranium. 65 uranium to be transformed in the mode leading to the production of actinium. Nor is there any apparent advan- tage gained by assuming that the transformation of either UI. or UII. into the first member of the side series is accompanied by the expulsion of a 8-particle instead of an a-particle. It might, however, be assumed that the branching of the series takes place ait some other point, as at radium, for example, and that 86 per cent. of the radium atoms disin- tegrate with the emission of a-rays to form the emanation, etc., while 14 per cent. disintegrate (emitting B-rays) to form actinium. Direct evidence of the emission of §-rays by a specimen of radium has been obtained by Hahn and Meitner*. Under these conditions the «-ray activity of ionium would be proportional to the uranium radiation and would equal 0°53. The activity of the actinium series would equal 0°56 and the activity of the radium + emanation + radium A, C, and F would be 3°01 (the sum of the values given in Table V.).. The sum of all of these together with uranium is 5°10 for the total activity of the uranium series (as in uraninite). There are, however, serious objections to the assumption that the side branch arises at radium, aside from the fact that the values mentioned are widely different from those found in Boltwood’s experiments and the value found in the present experiments for the total activity of the uranium products. The most significant objection is presented by the agree- ment of the value found for the disintegration constant er radium by Rutherford and Geiger+ and the value of this constant found by Miss Gleditsch{. The Rutherford and Geiger estimate was based on the number of e-particles emitted per second by the radium C in equilibrium with one gram of radium. If only eighty-six out of every hundred of the radium atoms disintegrate to form radium GC, then this estimate would. be 14 per cent. too low. The method employed by Miss Gleditsch depended on the production of radium from the ionium in equilibrium with a known amount of radium, and was measured in terms of the fraction of the total equilibrium amount which was produced in a known period of time. This would have given the true value for the disintegration constant irrespective of the mode of * Physik. Zeitschr. x. p. 741 (1909). + Rutherford, Phil. Mag. xxviii. p. 326 (1914). { Am. Journ. Sci. xli. p. 112 (1916). Phil. Mag. 8. 6. Vol. 40. No. 235. July 1920. Fr 66 Relative Activity of Radium and Uranium, disintegration. These two methods would therefore have given different and not similar values if the collateral series had originated at radium. Among other objections may be mentioned the experiment made by Soddy*, who examined a specimen of radium salt containing 13°2 mgs. of radium which had been sealed for a period of ten years. No evidence of the presence of actinium was obtained. Paneth and Fajansft examined a specimen containing 180 mgs. of radium which had been sealed for six years, but. were unable to detect the presence of any actinium products. We may therefore dismiss the possibility of the side chain splitting off at radium as highly improbable in the light of our present knowledge. The possibility that the collateral series originates at ionium may also be considered. The fact that the experi- mental evidence is all opposed to the emission of a §-radiation by ionium is in itself a decided objection to this view. Moreover, it would require (uranium taken as unity) an activity of 0°56 for the actinium products, an activity of 0°46 for ionium, and an activity of 3°01 for radium and its pro- ducts, with a total activity of 5°02. Paneth and Fajans{ have directly attacked this problem by seeking for the presence of actinium in a iBmong preparation of ionium- thorium which had been undisturbed for four years. ‘They were unable to discover the presence of any actinium pro- ducts. Lacking any support, therefore, the supposition that the collateral series arises at ionium is untenable at present. These circumstances compel a return to a consideration of the earlier members of the series, to U I. and U IL., in the hope of being able to find there an explanation of the con- ditions indicated by our experiments. At first sight it might seem that the conditions would be satisfied by assum- ing hes what we now call uranium consists of three radio- elements, a parent element and two isotopic products in equilibrium, all emitting arays. But if these are present | in relative amounts of the same approximate order of mag- nitude (2.e., 100, 92, 92, etc.), then the e-rays emitted by at least one of them would have to be of exceedingly short range and small ionizing power and the rate of change of this substance would be “excessively slow. It is not impos- sible, but it does not seem probable, that ordinary uranium may consist of what we know as U I. and U II., both radio- elements in the main line of descent, and a third isotope * ‘Nature,’ xci. p. 634 (1918). + Wien. Ber, exxiii. Ila, p. 1627 (1914). t. Wren, Ber. xxiii Ila, p. 1627 (1914). The Bearing of Rotation on Relativity. 67 which is a product in the collateral actinium series. But the difficulties here are not inconsiderable aside from the fact that the existence of such an isotope is somewhat diffi- cult to imagine. If present in amounts proportional to the actinium this product would have to emit comparatively long range (7°2. cm.) a-particles and would therefore have a very Short life period. Such a conclusion does not seem at all probable i in the light of our present knowledge. It is not impossible that the values accepted for the ranges of the e#-particles from uranium are considerably in error and that this is the reason for the lack of agreement between theory and experiment. But until some more definite data have been obtained there seems to be little justification for abstruse speculation on the genetic relationship in the earlier stages of the uranium series. Summary. The relation of the activity of radium to the activity of the uranium with which it is in radioactive equilibrium has been redetermined. The results obtained indicate that if the activity of uranium is taken as unity the activity of the radium is equal to approximately 0°49. The total activity of uranium mixed with equilibrium quantities of its disintegration products has been compared with the activity of the uranium alone, and the former has been found to be 4°73 times the latter. A critical examination has been made of the various theories which have been proposed to explain the genesis of radium and actinium from uranium. None of these theories appears to satisfy the necessary requirements. V. The Bearing of Rotation on Relativity. By Prof. R. A. Sampson, #’.22.S.* ONSIDER two concentric spheres with a very small space between them so that we need not distinguish between their radii. An observer A is placed on the outer surface of the inner sphere and an observer B on the inner surface of the outer. All phenomena are supposed to pass in the space between the two spheres. Regard this system and its changes first from a purely geometrical point of view. A and B "will possess in common a natural unit of length, being the circumference of their * Communicated by the Author. F 2 68 Prof. R. A. Sampson on the sphere. Let the arc AB as it exists at any moment be determined as a fraction of this unit. Let it be determined again in the same way ata later moment. If the two do not agree, we can say that a relative rotation of the two spheres must have occurred, through a definite angle, about an axis. perpendicular to the plane of the great circle AB. Whether any relative rotation about an axis inthe plane AB has taken place, or whether both spheres have executed in common any other rotation about any axis whatever, the observers at A and B will be unable to say. We may express this position by saying that A and B are under circumstances of complete: geometrical relativity. The whole description is, however, an abstraction. It is the abstraction which hes at the basis of geometry; it eliminates time from consideration and supposes figures to exist with definite distances between the points. But in reality any distance assigned requires time for its deter- mination, and the standard case would be this: A and B. each have hold of a graduated measure, allowing it to slip through their hands, and as they watch its successive readings they signal them to one another; each will then only be aware of the other’s reading, that is to say, of the other’s distance at any moment, as complicated by the time of transmission of the signals. This aberrational allowance is. an inevitable attendant upon actual physical measures. It is inseparable from motion. It would not be surprising if the ideas of motion required a complete surrender of the scheme of abstract geometrical relativity defined above. How much the physical theory of motion affects our notions: of absolute and relative is very well known. Let the sphere: A be the Earth and the sphere B a complete opaque sheet of cloud rotating with it. A Foucault pendulum set up at the: north pole would of itself change its plane of oscillation with respect to the meridians, pointing out an absolute direction in space, and an absolute rate of rotation, of which the ob-. server would be unaware without this, or other similar, appeal to dynamics. Accelerating Potential Volts. Electron currents in calcium vapour. Curves 8, 18, and 22 “ total current”; curves 1, 3, 9, 14, and 15 ‘ partial current.” vice versa, and f is due to two collisions of type 1 and one of type 2 in any of the three possible orders. The positions of e and f are above the ionization point, and for this reason somewhat variable. The point > is too faint for accurate 78 Drs. Mohler, Foote, and Stimson on Jonization and measurement, so the resonance potentials have been com- puted from the points a,c, and d. The mean value of c—a gives for the first resonance potential 1°90 volts. The second resonance potential, d—a is 2°85 volts. The observed ionization potential is 6°01 volts. The occurrence of two resonance potentials has been suspected in the case of other metals, notably zine and magnesium, but this is the first instance in which the phenomena were unmistakable. It suggests the possibility that in all metals there may be many potentials of inelastic impact, of which the observed resonance potential is the most probable. For the metals of the second column of the periodic table previously studied, it has been found that the resonance potential is determined by the frequency of the combination series line 1°5 S—2 ps, and the ionization potential by the limiting frequency 1°58. The first line of the principal series of single lines 1°5 S—2P is, however, predominant in spectra of the alkali earths. The frequency 1°58 for calcium is computed by F. A. Saunders * to be v=49304'8, A=2027°56. This corresponds to V=6'081 volts in close agreement with the observed value V=6'01 volts. ‘The line 1°55 S—2n. has not been correctly identified in any published work, but is un- doubtedly X=6572°78, as this agrees with the best values of series constants, and the physical properties of this line are characteristic. Accordingly we obtain for the theoretical value V=1°877 volts, while the observed value is V=1-90 volts. The line 158S—2P, %=4226°73 corresponds to V=2-918 volts in good agreement with the observed value V=2°'85 volts. The spectral relations of the second resonance potential are not surprising. McLennan ft has found that the line 1:5 S—2P alone appears in the magnesium arc below the ionization potential, and that in zine and cadmium ares both 1:5 S—P and 1°5 S—2p, appear. Davis and Goucherft by their photoelectric method of detecting different types of radiation found evidence that 1°55 S—2P, as well as 1:5 S—2p., appears in mercury at the voltage corresponding to its frequency. Interesting light on these fundamental frequencies in the spectra of the alkali earths is given by * Data furnished by Dr. Saunders. + McLennan, Proc. Roy. Soc. xcii. p. 574 (1916). McLennan & Treton, Phil. Mag. xxxvi. p. 461 (1918). pi Phys, nev. xiip.) 101) (1917). Resonance Potentials for Llectrons in Lead and Calcium. 79 the work of A. S. King * with the tube furnace at different temperatures. In the calcium spectrum at high temperature the line X=4227 is predominant and X=6573 quite faint, while at low temperature X=4227, though still the brightest line, has lost considerably in relative iatensity and X=6573 has increased until it is second only to 7=4227. Mag- nesium shows the same phenomena, but in a more striking manner, for the line 1°5 S—2p.,%=4571 is the brightest line in the spectrum at low temperature though faint at high temperature, and in ordinary are spectra. That these two lines in calcium are related to the two observed resonance potentials is to be expected. However, the relative prominence of the 1st resonance potential due to 1:5 S—2p. in calcium and magnesium is difficult to reconcile with the researches of McLennan on low-voltage ares in magnesium where only the line 1:5 S—2P was observed. The problem is evidently complicated. The tube furnace spectra show at least that the two emission centres are affected differently by a change in physical condition. Summary. The resonance and ionization potentials of lead are 1°26 and 7°93 volts respectively. The line A=10291 gives the probable theoretical value for the former as 1°198 volts. Calcium has two resonance potentials, 1°90 volts and 2°85 volts, of which the first is more prominent. Tonization was observed at 6:01 volts. The following spectral frequencies determine these po- tentials :— First resonance.... 1°65 S—2p,, \=6572°78 ce V=15877 volts. Second resonance .. 1:5 S—2P, \=4226-73 A., V=2-918 volts. « ° a i fo} Tonia :...cc ov. o-/1°O BS; A = 2027°56 A., V=6:081L volts. Bureau of Standards, Washington, D.C., October 20, 1919. * Astrophys. Journ, xlviii. p. 13 (1918). ease VII. Atomic Theory and Low Voltaae Arcs in Cesium Vapour. By Pau D. Foorr, Ph.D.,and W. F. Mracrrs, Ph.D.* [Plate I.] I, THEORETICAL. HE spectrum of cesium is characterized by the principal doublet series 1°5s—mm:, where m=2, 3, 4, etc.; the Ist subordinate series 2p,—md and 2p.—md', where m=3, 4,5,etc.; the 2nd subordinate series 2p:—ms, where m=2°5, 3°90, 4°59, etc.; the Bergmann series 3d—mAp and 3d/—mAp, where m=4, 5, 6, etc.; satellites to the 1st subordinate series; and certain combination lines, those known lying in the far infra-redf. The spectroscopic data are sufficient to permit a schematic representation of the czesium atom, as illustrated in fig. 1. Fig. 1. — Convergence — Prin.Series 15 S$ —1S*and 2" Sub.Series 2p, and 2p,— Berqmann Series 30 — Ta 86 3.87 volts guste - : 2a 14S : 4 3 SP, Ps °s 2°, ae ASS 3.55 258 1.5/S 6d Sd 3d S4p SAp 4Ap 34 35 3.6 Si 3.8 3.9 4.0 41 4.2 43 44 45 log. wave number Schematic representation of cesium atom. This method was suggested to the writers by Dr. Raymond T. Birge; and, regardless of the theory of atomic structure, it affords a precise picture of the possible series lines in the spectrum of an element—a much clearer picture than may be * Communicated by the Authors, and published by permission of the Director Bureau of Standards. t The above notation is that employed by Dunz in his tables, except that for convenience we haye written p: for p, and p, when both lines are referred to. Atomic Theory and Low Voltage Arcsin Cesium Vapour. 81 gained by one unfamiliar with series notation from an exami- nation of the series formule. ‘The first doublet of the principal series may be represented by electrons falling from the 2p, and 2p, rings into the 1°5s ring, thus emitting the lines L'5s—2pr. The second doublet of this series is repre- sented by electrons falling from the 3p, and 3p, rings into the 1°5s ring, giving rise to 1: ds—3p2, and so on. The lines of the 1st subordinate series are “represented by electrons falling from the md rings into the 2p, and 2p, rings ; the lines of the 2nd subordinate series by electrons fall- ing from the ms rings into the 2p, and 2p. rings; the lines of the Bergmann series by electrons falling frem the mAp rings into the d rings. Combination lines are repre- sented in the same manner. Thus the line 2°5s— 3p, arises from an electron falling from the 3p, ring into the 2°5s ring. Few of the possible combination lines are probable enough to appear in spectroscopic measurements. Tor example, the line 2p,—4p,, while possible, represents an extremely impro- bable type of transition, and has never been observed in the cesium spectrum although it is known for sodium. There are other ways much more likely for an electron to leave the 4p, ring. On the basis of the Bohr theory, the lines shown in the diagram are really portions of the elliptical or ring orbits about the nucleus, which is located off the figure in the right. Surrounding the nucleus are the X-ray rings and many other orbits between the nucleus and the 1°5s ring in which electrons may be present. In the unexcited czsium atom no electrons exist outside of the 15s ring. This ring represents the outermost stable orbit and the innermost unstable orbit of the normal atom, and the diameter of this orbit should give the diameter of the normal cesium atom. The evidence for this is fairly conclusive. That there are electrons in this ring is shown by the existence of the principal series, which converges at 15s. That it represents the innermost unstable orbit for any ordinary method of excitation is shown by the fact that no spectroscopic series of csesium converges at a higher fre- quency than 1°5s, nor is any line of higher frequency known. That it represents the outermost stable orbit is evidenced by determination of the ionization potential for which Foote, Rognley, and Mohler* observed the value 3°9 volts. On the basis of the quantum relation, hy=eV, an electron falling through 3°877 volts possesses just sufficient energy to eject * Phys. Rev. xiii. p. 59 (1919). Phil. Mag. 8. 6. Vol. 40. No. 235. July 1920. G 82 Dr. P. D. Foote and Dr. W. F. Meggers on Atomic an electron from the 1:5s ring to infinity, in excelient agree- ment with the experimentally determined value of 3°9 volts. Thus the ionization of the normal czesium atom is produced by ejecting an electron from the ]‘5s ring. If the outermost stable orbit were the 2p rings the ionization potential would be 2°4 volts, unless we make the improbable (see later) assumption that electrons colliding with an atom could not displace an electron from the 2 ring. Thus it is safe to conclude that any displacement of an electron from its orbit in the stable condition of the atom takes place from the 1°5s ring. The question arises as to what type of displacements may occur in electronic-atomie impact. Inelastic conditions take place when the impacting electron has the energy eV’ where V is the ionization poten- tial, this kinetic energy being just sufficient to account for the increase in total energy of the ionized atom. Another type of inelastic impact has been observed in which the colliding electron has the energy eV’ where V is the re- sonance potential”, this kinetic energy being just sufficient to account for the increase in total energy of the atom arising from a displacement of an electron from the 1°5s ring to the 2p ring. The displaced electron being then in an unstable orbit falls to the 1-5s ring, giving up the quantum of energy eV received from the collision as a quantum of energy hy of radiation of frequency v, as is shown by the experiments described later. ‘Thus inelastic impacts are known when the colliding electron has sufficient energy to eject an electron from the 1:5s ring to infinity or to the 2p rings. In the former case the electron returning to the atom causes the emission of various lines, and with many electrons in different ionized atoms returning by various paths we obtain the com- plicated line-spectra in which the intensities of the lines represent to some degree the probabilities of each particular type of transition from orbit to orbit. Suppose the colliding electron possessed just sufficient energy to eject an electron to some ring intermediate to x and 2p. Could the normal atom absorb the entire kinetic energy of the impacting electron? The only known series converging at 1°5s is the principal series 1-5s—mp;. Hence it is extremely probable that, if a displaced electron falls into the 1°5s orbit, 1t comes from one of the p rings. Accordingly it is reasonable to assume that in the original displacement from the 1:5s ring the electron is ejected to some p ring. Thus inelastic impacts might occur when the colliding elec- tron had fallen through any of the following potential * Foote, Rognley, and Mohler, loc. cit. Theory and Low Voltage Arcs in Cesium Vapour. 83 differences, the kinetic energy being in each case equal to the increase in total energy of the atom corresponding to displacements to the 2p, 3p, ot to oa 7 Tings: Displacement from 15s to mp). Me. Volts, eG |) ee eee Pe 1448 ch g Odd Lee 2709 Apa INS 3°184 ee GA ATRPPO G, TES a a7 ait ptieuk: ahs. tt 3°049 COPIA LY AUUL EDL) 3°877 The first and last of these values are readily observable, as above mentioned. ‘hat the values corresponding to m=3, 4, etc., have not been observed may be due to the fact that such displacements are very much less probable than the displacement corresponding to m=2 and the method of detection accordingly insufficiently sensitive. This is sup- ported by the fact ‘that the probability of a transition from the mp ring to the L-5s ring decreases as m increases—evidence for which lies in the relative intensities of the lines of any series. The displacement to the © ring may take place in an infinite number of ways, and hence is readily produced. The value 2°709 is nearly double the value 1°448, and accord- ingly might be mistaken for two successive collisions with diferent atoms, each resulting in a displacement to the 2p, ring, the probability of which is guite high under ordinary experimental conditions. Suppose that the inelastic impact resulted in a displace- ment to the 8p ring. The electron may return to the 15s ring directly, resulting in an emission of the line 1-5s—3p, or it might fall to the 3d ring giving the line 3d—3p, then to the 2p ring giving 2p—3d, and finally to the ls ring giving 1:5s—2p, the total value of all the quanta being equal to the increase in total energy arising from the original displacement to the 3, orbit. Other modes of transition from ring to ring are possible. Hence, when a large number of atoms are considered, a cer- tain group of spectral lines may result from a displacement of the type considered. The important point on the basis of Bohyr’s theory is that for the collision mentioned, lines resulting in displacement from rings beyond 3p could not exist although the size of the quantum involved might be G 2 84 Dr. P. D. Foote and Dr. W. F. Meggers on Atomic many times less. Thus an electronic impact of 2°7 volts could not excite the visually intense line X 6972 (2p,—5d) although the quantum involved requires but 1°8 volts. The energy (43x 10°" erg) is sufficient to produce this line (29x 10- erg), but is not sufficient to displace the electron to the 5d ring (5'1 x 10-” erg)—a condition necessary for its excitation. The above conception is in agreement with fluorescence phenomena observed in sodium vapour. Thus D-light is absorbed by sodium vapour, and for each quantum absorbed an electron is displaced from the 15s to the 2p ring in an atom. ‘The electron in falling back to the 15s ring emits the D-line (1°5s—2p), which may be observed at right angles to the beam of incident radiation. Recently, Strutt* has found that the sodium line 23303 (1°5s—3yp:) similarly stimulates both 1:5s—3p: and 1:5s—2p:. Thus, after absorb- ing a quantum of frequency 1°5s—3y, an electron is displaced to the 3p: orbit. In returning to equilibrium it may fall directly to the 1°5s ring, in which case the line 43303 is emitted, or it may fall to the 2p: ring and then to the 1'5s ring, the second step involving an emission of the D-lines. This leaves a quantum of frequency 2p:—3p: (A 7519) looked for but not observed by Strutt. However, this line represents. a very improbable type of orbital transition, as is evidenced by the fact that the frequency is never observed in any sodium spectrum. A more probable method of falling from 3p: to 2p, is first to the 3d ring and then to the 2p ring, giving rise to the lines 18196, 8184, 9048, and 9085, or from the 3p; ring to the 2°5s ring and then to the 2p: ring. In both cases the quantum relations are correct and the lines. are well known, but, lying in the infra-red, they would not have been observed by Strutt. The fact that an atom can absorb radiation of frequency 1:5s—mp (absorption has been observedt for m=2 to 60 in the case of sodium), thus resulting in displacement of an electron to the mp ring, for each absorbing atom suggests that similar displacements may be produced by electronic collision. On the other hand, the faiiure to detect such dis- placements is not an argument against the Bohr theory. It is possible that the impacting electron must have the correct velocity as well as energy to produce any displace- ment at all. Thus, while the mass of an electron happens to. be such that when its velocity is that obtained by falling through 1°45 or 3°88 volts, a collision with a cesium atoin is. = Proc. Noyes Soc. xevi. p, 252 (IONS), + Wood and Fortrat, Astroph. Journ. xlii. p. 75 (1916). Theory and Low Voltage Arcs in Cesium Vapour. — 85 inelastic, resulting in a displacement of a bound electron to the 2p, and cop rings respectively: at other voltages, such as 3°18, the velocity condition is not fulfilled. This assump- tion, as far as displacement from the 1'5s ring is concerned, is, however, scarcely justifiable, since the range between 1-45 and 3°88 volts is small ; and since it appears that when the electron possesses 3°18 volts velocity, 1°45 volts velocity may be absorbed at the impact, this going into increased total energy of the atom and the remainder into kinetic energy of the colliding electron. Also ionization is believed to occur from the 2p ring, since ares may be operated below the ionization potential. At some stage, however, the mass of the colliding particle is effective, since the quantum relations do not appear to hold for low-speed positive or negative ions. Indeed, it is questionable whether a definite ionization potential for ions exists. An electron of 1°45 volts velocity colliding with a czesium atom ejects an electron from the 1°5s ring to the 27, ring. In returning to equilibrium the atom emits the frequency 1-5s—2p,. This radiation, however, is capable of being absorbed by a neighbouring atom (resonance radiation effect) resulting in a displacement of an electron to the 27, ring. Hence, in an electron-tube there may be a building-up of the number of electrons in the 2p, ring, and if all radiation emitted were absorbed, every atom might have electrons in this ring. Thus further collision with 1:45-volt electrons would result in an ejection of an electron to an extreme outer ring. A collision with a 2°4-volt electron would ionize co} the atom. Van der Bijl* has suggested as an explanation of low-voltage ares operated bélow the ionization potential, a theory of successive impact. The atom collides with an electron, causing a displacement of a bound electron to the 2p, ring. Before it is able to return to the 1*5s ring it collides again with a second electron, and the atom is ionized. The probability of a second collision under such conditions is, however, very small. ‘lhe above suggestion of the absorption of radiation which was first proposed by Comptont isa much more plausible explanation of ares below the ionization potential. In fact, one must consider how ionization could take place otherwise than from the 2p, ring. It must be accordingly necessary that only a small proportion of the radiation emitted when electrons fall from the 2/, to 1-5s ring is absorbed by the vapour. This is a reasonable * Physical Review, x. p. 546 (1917). + KX. Compton, Chicago Meeting Am. Phys. Soc. 1919, Phys. Rev. 86 Dr. P. D. Foote and Dr. W. I. Meggers on Atomic assumption, for a large amount of radiation escapes from the tube. Thus with sodium at 22 volts the D-lines are visibly intense, and with cesium at 1°5 volts the infra-red lines A8943 and 8521 may be readily photographed with properly sensitized plates. ‘The ionization of a vapour below its ionization potential is observed when the vapour and electron densities are high. ‘This fact substantiates the present hypo- thesis, for more collisions occur, more radiation is emitted, and more is. absorbed, when. dhe electron, earecutiamnineme vapour pressure are increased. Thus, as pointed out by Jompton, in the operation of the mercury are at 5 volts and high-current density, electrons are maintained in the 27, ring, partly by electronic impact, but mainly by absorption of radiation of the frequency 1° or ep and a d-volt impact is sufficient to ionize from the 2p, ring. An extension of this hy pothesis leads to interesting con- clusions in the case of absorption of radiation. The lines of the principal series of the alkali metals are all absorption- lines, the energy absorbed increasing the total energy of the atom by a displacement of the electron from the 1°5s ring to some pring. If, however, the vapour is excited by electronic impact (or radiation) corresponding to Ve=hy, where y=1:5s—2p, electrons normally in the 1°5s ring are driven to the 2p ring, and are no longer capable of absorbing 1'5s—mp, but “rather series lines which converge at Qn. Hence, if the excitation is sufficient (7. e., electron curreut very dense), the principal series should show less absorption, and the 1st and 2nd subordinate series should tend to become the prominent absorption-lines. It appears from the above that the general impression of the existence of a single-line spectrum at a certain low voltage, and an abrupt transition to the many-line spectrum as the voltage is increased to the ionization potential is scarcely justified a priort on any theoretical basis. If a mechanical theory is accepted, it is quite conceivable, in spite of Stokes’ law, that the lines of higher frequency may be excited by resonance when a fundamental line is produced. We have analogies to this effect in the relation between transverse and longitudinal vibrations of a rod, or in the “tripling” of the frequency of alternating current, etc. Thus, instead of a single line, a single series might appear-— the fundamental frequency and its harmonics: a view held by certain spectroscopists. On the basis of the Bohr theory various groups of lines may appear as the excitation is increased. There is no experimental evidence on this question. In all previous Theory and Low Voltage Arcs in Cesium Vapour. 87 work on single-line spectra the observations would not have shown the presence of other lines had they existed. Thus the observed single-line spectrum of mercury is 12537, but the second line of the series 11436 might be present, and, lying in the extreme ultra-violet, it could not have been detected by the methods employed *. Il. EXPERIMENTAL. The present work was undertaken with the object of obtaining information as to the existence of single-line ye 3) Fig. 2. | | Cathode Tonization tube. spectra, single-series spectra, and group spectra. Czesium appeared to be a de- sirable material, since all of its spectral lines, except certain combination lines in the far infra-red, may be photographed by properly sensitized plates. ‘The appa- ratus is shown in fig. 2. The ionization chamber consisted of a pyrex glass tube having an optically ground pyrex plate fused to one end, through which the are was viewed. The anode was a _ nickel cylinder, and the cathode a lime-coated platinum cylinder about 5 mm. in dia- meter, heated by a smali heating-coil inside. The cathode was thus made an equipotential surface, and because of its large area the current was high (150 milli- amperes at six volts), even though the temperature was maintained at 500° C. It is quite desirable to operate the cathode considerably below a red heat, as otherwise, on long exposure, stray reflexion of the emitted light and heat may obscure the ceesium lines in the range 1.6500 to X 9000. All lead-wire seals were tungsten in Corning G 702 P glass. The ionization tube was evacuated and maintained at about 0:0001 mm. of mercury gas pres- sure, and heated to 200° C., thus giving a considerable vapour-pressure. The radiation emitted near the anode where the electrons had gained their maximum velocity was photographed. * Cf. McLennan, Proc. Roy. Soc. Lond. xcii. p. 305 (1916); xci. p- 485 (1915). 88 Dr. P, D. Foote and Dr. W. F. Meggers on Atomic The spectrum photographs were made on Seed 23 plates stained with dicyanin. A few of tlhe plates were sensitized with Hoechst dicyanin, but most of them were prepared with dicyanin made by the Bureau of Chemistry, Department of Agriculture, since these American dyes appear to be equal or superior to the German ones in photo-sensitizing action. The process of staining has been described several times, and it has been used almost continuously in spectroscopic inves- tigations at the Bureau of Standards for the past four years. This‘experience in staining has made it possible to reproduce sensitized plates which are remarkably similar, so that very nearly the same spectral sensitivity is shown by different plates. This uniformity is of importance in the comparison of spectroscopic data obtained from different plates. The spectrograph was made by Carl Zeiss, and has a large flint prism of 10cm. base length and 60° angle. The colli- mator and camera lenses have 5cm. aperture and 26 cm. focal length. On account of the very low intensity of the low-voltage are a spectograph with the highest attainable light efficiency is advantageous. About thirty lines with wave-lengths from 4555A to 8943.A were recorded with an exposure of one minute to the 6-volt cesium are. An exposure of 5 minutes showed additional lines and extended the spectrum from 3878A to 9208A. The exposures were much longer when the are was operated at 3 volts or less. Most of these averaged about 15 hours, although in some cases exposures of 40 to 60 hours were employed. The length of the spectrum on the photographic plate was about 40 mm. between 3878 A and 9208.A, and the dispersion in this range was sufficient to separate all of the important doublets in the various spectral series, even when a relatively large slit width (0-2 mm.) was used. Plate I. shows a photograph of several typical exposures at different voltages. Lower voltage exposures were made, but were not suitaole for reproduction although the origina! negatives were measurable. It appears from visual observa- tion that the first doublet of the principal series alone appears below the applied potential of 2°4 volts; but this fact, as has been heretofore, with other elements, assumed con- clusive, should not be accepted as proof of the existence of a single-line spectrum. If the plates possessed a constant sensitivity to light of various wave-lengths, the line » 8521 could be made to appear alone on very short exposare even in « high-voltage arc, because in absolute energy it is the most intense line of the spectrum. The sensitivity of the plate, however, varies with the wave-length, as shown by Theory and Low Voltage Arcs in Castum Vapour, 89 fig. 3, so that on short exposure the line 1 6973 may appear alone. With ordinary blue sensitive plates the line 14555 Fig. 3. DENSITY > 2S > o Si ay S oS So S S Ss So S oO o a Se a ae a ee R WAVE -1_ENGTH Sensitivity of dicyanin plates to equal energy spectrum. is the most strongly reproduced. It is, however, possible to draw some conclusions by visually comparing various nega- tives. Thus, if the density of X 8521 witha high-voltage are is the same as that with a low-voltage arc, and in the latter case no other lines appear while in the former case the entire spectrum is shown, an argument is obtained for the single- line spectrum. How ever, it is desirable to eliminate the plate sensitivity from the data and reduce all measurements to the same scale of intensity of the lines in the arc. To do this the densities of all the lines photographed were carefully measured by a disappearing filament micro-photometer*. This instrument is essentially the micro-pyrometer described by Burgessf, except that a microscope of higher power is employed. The photographic plate was mounted imme- diately below the objective of the microscope on a horizontal bed movable with a graduated screw, and was illuminated * This simple and effective instrument is described by the writers in a separate paper. } Burgess, Bureau Standards Scientific Paper 198. 90 Dr. P. D. Foote and Dr. W. F. Meggers on Atomic beneath by an intense beam of light from a tungsten-ribbon lamp. Light transmitted by the portion of the plate, the image of which was adjacent to that of the photometer-lamp filament, was matched against the filament brightness by adjusting the current through the lamp. The current read- ings are readily translated into measurements of photo- graphic density, correcting, of course, for the “ fog ” of the velatine films. Several series of exposures from 2 to 200 seconds were made at different times on the 6-volt ceesium are, and density measurements of the spectral lines showed that the dicyanin- stained plates have approximately the same development factor (gamma=1°45) for all wave-lengths from 4555A to 8521A. To calibrate the plates in terms of absolute intensity of the source, series of exposures of different durations were made by sighting on a black body, the temperature of which was measured by an optical pyrometer. The D-lines of sodium were superposed on each spectrum as fiducial marks. The densities of the plates for various wave-lengths were measured with the micro-photometer, and from the “ charac- teristic curves’ of the plates and the computed distribution of energy in the spectrum of the black body, the photo- vraphic density as a function of wave-length was corrected for an equal energy distribution in the spectrum, a typical curve thus obtained being illustrated by fiy. 3. With the general characteristics of dicyanin-stained plates thus deter- mined, the relative absolute intensities of lines in the ceesium- are spectra were obtained from their measured photographic densities and the durations of exposure. Since relatively long exposures were required to photograph the svectra of the low-intensity ares, the reciprocity law of the photo- graphic plate (density=function of intensity x time) was corrected* by giving the exposure time an exponent of 0°8. Ill. ReEsuurs. Fig. 4 shows the logarithms of the actual intensities of various lines emitted in the cesium arc plotted against applied accelerating potential. At slightly below 3 volts the entire line spectrum appears, only a few of the lines of which are shown in the figure. It is noted that the intensity of the lines 18521 and 278943 persists at voltages below which the intensity of all other lines vanishes. This is more clearly illustrated by the following table of ratio of intensity * Schwarzschild, Photographische Correspondenz 1899, p. 109. Theory and Low Voltage Arcs in Cesium Vapour, 91 of X8521 to that of the prominent lines 16973 and 24555 for various voltages :— J— J J— VJ Applied Voltave S521 6973" 8521" 4558 ap, 105 3d50 AF SA PC (130) (400) A (200) (600) Sad ee ceca: 560 2100 Bree Rs 1100 8300 Pe er) Skee 620 3400 pt ee eee 3600 10500 ce Pe eee > 10000 > 10000 pA a ieee Oe bis} 2, is excited at a sacrifice in intensity of 48521 and 28943, since each line of this series requires that electrons fall from the mp ring into the 1:5s ring. The subordinate * Loc. cit. + Phil. Mag. xxxvi. p. 64 (1918). 94 Dr. P. D. Foote and Dr. W. F. Meggers on Atomic series lines, however, converge at 2p, and hence do not affect the intensity of 1°5s—2p , which may still be excited at the same collision. In fig. 6 we have plotted the logarithm of the intensity for 18521, A6973, and 74555 per unit number of electrons reaching the outer cylinder as the accelerating voltage is Volts Applied accelerating potential Logarithm of intensity per unit number of electrons reaching anode as a function of the applied accelerating voltage. increased. At higher voltages this ratio becomes a constant for each line in agreement with the work of Jolly*, who observed that in discharge through hydrogen “the radiation is proportional to the current density both for the whole spectrum and for any portion of it.” This follows (approxi- mately) directly from the quantum theory. Above a certain minimum voltage the number of electronic atomic collisions, and hence the number of quanta of any particular frequency produced, is proportional to the number of electrons present. The existence of 1°5s—2p: when the intensity of all other lines has dropped to zero is well illustrated in this plot. The above law is not rigorously applicable in the present case, since some of the electrons reaching the anode have been produced by ionization, and, in general, would not be able to accumulate sufficient velocity to cause radiation upon further collision. An interesting phenomenon was observed in the operation of the caesium are at 120 volts. It was found that, after once * Phil. Mag. xxvi. p. 801 (1918). Theory and Low Voltage Arcs in Cesium Vapour. 95 Y e i} being started, the are operated with either electrode as cathode, and thi at no meetiie ation of alternating current was cheer able. This phenomenon is well aca in certain types of mercury are. With the outer cylinder as a cathode a quiet dazzling- olow discharge is obtained, while with the small hot eathode. the are forms streams of Pent discharge which flicker back and forth along the tube. A much discussed question as to the ratio of intensities of the components of a doublet when the exciting voltage is varied may be answered by referring to fig. 4. In absolute measure the ratio of intensities A8521/X8943 is constant and equal to 1°5 within the errors of observation. This value was further confirmed by observations on a 120-volt are. Although the czesium employed was made from cesium chloride of supposedly high purity, it contained traces of both sodium and potassium. ‘All exposures above 2°2 volts applied potential show the presence of the first doublet of the prin- cipal series 1*5s— 2p. of both these metals, the ionization potentials of which are 5:1 volts and 4:3 vo Its respectively. Accordingly, in the light st the more extensive work deseribed with czesium, we may eon chetec that the doublets 1:5s—2p: are the single- the spectra of sodium and potassium. LV. SuMMARY. Theoretical. On the basis of several possible theories of atomic structure, it is shown that the normal operation of an are below ioniza- tion might result in the excitation of a single-line spectrum, a single- -series spectrum, or a group spectrum consisting ot certain lines of different series. ‘his latter conclusion fol- lows from an extension of Bolhr’s theory. Thus, if inelastic electronic-atomic impact occurred resulting in the ejection of an electron to the pth ring, the electron in returning to the nth ring or equilibrium may produce any combination of lines represented by inter-orbit transitions within this range, the single-line spectrum being a special case where pants. A simple explanation is offered of fluorescence phenomena in vapours of the alkali metals. A mechanism of absorption of radiation is described, and the theory proposed by K. Compton, that the ionization of an atom below the ionization potential may be explained by absorption of radiation arising in other atoms from electronic- atomic impact of insufficient energy to ionize, is further 96 Atomic Theory and Low Voltage Ares in Cesium Vapour. discussed. This hypothesis suggests that vapours of the alkali metals may be so stimulated that the lst and 2nd subordinate series lines, instead of the principal series, tend . to become absorption-lines. Heperimental. The cesium spectrum was photographed for various accelerating voltages, from A3878 to 79208, by use of dicey anin-stained plates. The sensitivity of the plates was investigated by density measurements of the spectrum of a black body having a known energy distribution. The general characteristics of the plates were determined and all lines of the cesium spectrum were reduced to an absolute scale of intensity by means of density measurements made with a micro-photometer and consideration of the plate sensitivity. No evidence of group or single-series spectra could be obtained. Thus the ratio of intensities of the first and second lines of the principal series, both of which should appear in a single-series spectrum or above 2-7 volts in a group spectrum, rapidly approaches infinity as the accelerating potential in the are is decreased. This ratio is 350 in a 7-volt are, 2100 at 4 volts, 10,500 at 3°4 volts, and as near infinity as can be measured at 2°8 volts. Similarly, the intensity ratio of either 1°5s—2p, or 1°5s—2p. to any other line approaches infinity at low voltage, proving for the first time the existence of a single-line spectrum rather than a singie-series or group spectrum—in the case of ceesium the doublet 18521 and V8943. The doublet 1°5s—2p: is alone produced under excitation of 1:5 to 3:9 volts accelerating field. The intensity of both of these Te gradually increases approximately proportional to the total number of electrons reaching the anode until the ionization potential is reached. At ‘this point a pro- nounced decrease in intensity of these two lines occurs, amounting to the factor one-third. ‘This decrease takes place at the voltage at which the com- plete line spectrum is produced, and is readily explainable on the basis of Bohr’s theory—in fact, it affords a strong argu- ment for this theory. Thus the lines 1°5s— 2p; are the result of inelastic collision with electrons having velocities between 1:45 and 3°9 volts, but as the latter voltage is exceeded, elec- trons, which at a slightly lower velocity would have given rise to 1°5s—2p:, now produce the compiete-series spectrum ; On Stationary Waves in Water. 97 and any line of the series 1*5s—mp:, where m>2, is neces- sarily excited at the sacrifice of 1*5s—2p,. Above a certain voltage the intensity of any line per unit number of electrons reaching the anode attains a saturation value, in agreement with the quantum hypothesis, which requires that the number of quanta radiated be proportional to the number of collisions, and hence (approximately) to the number of electrons present. Curves are given showing the relative intensities of the prominent czesium lines at various voltages. The ratio of intensities of the components of the first doublet of the principal series X8521/2 8943 is constant and equal to 1°5 from 1°5 volts to 120 volts. The czesium are of the type employed does not rectify alternating current of 120 volts. Sodium and potassium occurring as an impurity of the cxsium similarly exhibited the single-line or doublet spectrum 1-5s—2p2 below their respective ionization potentials. Only two types of inelastic impact between electrons and atoms of the alkali-metal vapours oecur, at potentials known as the resonance and ionization potentials and given by the quantum relation hv=eV, where v=1: 5s—2p, and y— 1-Ds. Bureau of Standards, Washington, D.C., January 14, 1920. VIII. Stationary Wavesin Water. By A. R. Ricwarpson, Imperial College of Science and Technology * LTHOUGH the subject is of very great practical im- portance, and has received much attention at the hands of engineers and experimentalists, very few exact solutions have been obtained of problems involving the flow of a liquid under gravity. In this paper some exact solutions are obtained, and existing results are linked together through a differential equation which forms the subject of the first part. Amongst the problems discussed is that of the flow over a weir, and an approximate calculation is made of the constants which appear in the Francis formula. | * Communicated by the Author. Phil. Mag. S. 6. Vol. 40. No, 235. July 1920: H 98 Prof. A. R. Richardson on Part I. I. Stream-line flow, under gravity, with a free surface. Adopt the usual notation : c=a+iy, y being measured vertically upwards, w=o+i, q, O refer to the fluid velocity. Consider the equation dn iL nea27 1 Dan aval i@ | Fe ital G(w)}++iG (w) |= al If along part of any stream-line, say w=, G(w)s, G'(w), {1—G(w)}* are real, and finite, over that part 3 2 y=- gs fo+{em@yl, 2... @ P= e{G(w)}? Hy 2 py Sle oak 1G G+ 29y= =O a ee oe Oe if pi 3g.. 0d) (loi Hence there will be a free surface, and (1) will give the ° solution of the problem provided G(w) is chosen so as to satisfy the conditions over the rigid boundary. ‘The presence of the term {G(w)}s makes this choice difficult, and in esas Aes re dz : : ae addition to singularities of a which arise where a rigid boundary bends, or meets a free surface, others may occur at places such as the crest of a wave of maximum elevation. If w=0 is such a place, near w=0 G(w) = 3 (1taw+ Ri ) be eM ill ‘eo V3 ‘a ad (S+ ye dealt = () 0=7/6 <0 @=—7/6, and at the crest g=0. .) approximately, Stationary Waves in Water. 99 (a) It is not without interest to notice that G(w) = — gives a complete free surface with straight boundaries inclined at +30° to the horizontal. (b) G(w)=w? gives flow in a deep tank having an inlet and outlet pipe in the bottom as in fig. 1. Rie. J; 4 —> Outlet Inlet (ce) Michell’s examples in his paper on the Highest Waye (Phil. Mag. 1893, xxxvi. p. 437) are given by G(w) = A(—isin we”) {1+ ce? + cet? +...3° There is, however, no simplification in the calculation of the constants by the use of (1). It appears that the method used by Michell is the most direct when the form of the stream-bed is given. Equation (1) is useful when the shape of the free surface is known approximately, and it is desired to obtain the general characteristics of the motion. {d) Flow in a stream with a finite drop. Consider G(w)=B—tanhaw B>1, «<1. dz 1 me ee | dF — 2? sech* Att — 2a sech’ zoo | , dw w{B—tanh aw}s [it ; H 2 100 Prof. A. R. Richardson on The singularities of Tp ate given by W B=tanhaw, and cosh?aw=-ta. These limit the depth of the stream, and different cases arise according to the relative magnitude of the roots. The stream-line y=0 is a free surface over which the ome g T= Ou B tanh aay : Fig. 2 has been drawn for the special case B=2, «=H, with w=1, i.e g=}. The extreme cases are given by a=1, when the free surface will be vertical at ¢=0 and will tend to curl over | a Fig. 2. Depth 1°57 = 1:73 Veloc.=! 2-08 (max.) ‘ 4 YZ, tsi Numbers on stream-bed ‘8 (min.) refer to Velocity. Vv -85 ey I-! Depth f =1/2 : 1-44] and break ; and B=1 when the fluid is at rest at b=a oa the free surface. If B<1 a rigid boundary is necessary over part of the stream-line =O if this particular form of free surface is possible. Referring to fig. 2, it is interesting to note how the velocity changes over the stream-bed, and how quickly a nearly uniform regime is established down streaam*. The tendency for a more or less quiet pool to form at the bottom of the drop is apparent, as is also the tendency to erosion at the upper edge of the fall, and a short distance up-stream. Evidently a form of G(w) containing several factors of this type would give rise to cases of flow over more irregular shaped stream-beds. (e) Flow over a corrugated stream-bed. This problem has been solved approximately by Lord Kelvin (‘Mathematival and Physical Papers,’ vol. iv.) on the as- sumption that the irregularities are small compared with the depth of the stream. * Searle’s assumption is justified. Phil. Mag. May 1912. Stationary Waves in Water. 101 Hquation (1) with G(w)=B-cosaw gives the exact solution of a case of flow illustrated in fig. 3, drawn for the yalues: B=? a="9. fies eas refer fii” to Velocity. jj re woe as dz The singularities of Tuo tte given by Ww : 1 B=cosaw, sinaw='+ —. a dz if 10 ae Sige ae: alee: — 22 Se eee Mame = ere | (1 a” sin? aw}s +1e sin 7 at : if «<1 and B>1 Ww=0 gives a free surface. Now on the. surface 0 depends only on a, and not on B, so that an altera- tion in B will not alter the concavity or convexity of the free surface. is am abn Qn |i] an. esi) pa Gl (ho ost emo > Ow The shape of the stream-bed, however, alters as indicated in fig. 4, an elevation of the surface taking place over a depression of the stream-bed. 102 Prof. A. R. Richardson on In such cases the velocity of the stream from surface to bed is not monotonic, but rises to a maximum at some point below the surface. The following table has been calculated for the values a='9, B=1:11, @=0, and represents the distribution of velocity across AB in fig. 4. ea 25: es | 85 ‘4 | 45 | 10) | Oona q | -479 |-499 516 15 529 536 536-527 |-499 |-440 | -288 | | II. Flow over a Weir. Take the origin at O, the sharp edge of the weir (fig. 5). Let y=0 be the stream-line from the edge, ar=(Q be the upper stream-line ; Q being the quantity of fluid flowing over the notch in unit time. | Let v be the speed of the liquid at a place on the upper stream surface where y=H. Then over ~p=Q q+ 2gy=0" + 29H, and using equation (1) g = M1 G(P+7Q) fs, G(o+iQ) =| — Cae y] ' ht ley Stationary Waves in Water. 103 Similarly on the lower free surface close to the edge O Be BL 5 WOR epee a See a bh aC) where G is the value of G which will give the lower stream surface. Hence, if y be measured at the point on the upper surface where ¢=0, (iQ) -G(0) =3 vza[(5, +H-y)— -(5, +8) |, (7) ° bo SIGS TO 1 aaa ee Raa where K is the right-hand side of (7). Evidently Q depends on the form of the function G(w), 2.e. on the complete rigid boundary conditions both up and down stream, and not merely on the state of affairs at the edge and surface. As a first approximation, however, it may be assumed that the function G(w) is dominated by the terms giving its development near the point w=0 corresponding to the edge of the weir. Jonsider now Tao [(1-@2w) BP +i1G@'w) |. - - ) Under the conditions postulated as to G'(w), this gives a flow with a free surface along which g=1 and over which y=G(w) +C. Moreover, G(iQ)—G(O) is the y-component distance between the points ¢= -() on the stream-lines a=) in the associated flow (9). For example, take G(w) =e"7®, 2 G(iQ)—G(0) = "= Q= 6110, or if G(Q) be assumed=G(Q), (8) gives Q= 327 | (H+, x) —(H+55—m) | rel: for flow over such a weir. 104 Prof. A. R. Richardson on The Francis empirical formula for such cases shows 3 35 as the coefficient. (a) Extension to the case of depressed nappes. lf the pressures over the upper and lower surfaces are different, Gro(bo+iQ) — Gao) = [2(5, +H-y)) ei [= pt lee ec In most cases Gy(w) is not the same as G, fer the nearest singularity to do on the down-stream side occurs where the nappe ceases to be a free stream-line, and this will be different in the two cases. However, so long as this place is not close to the edge O ¢ will not be altered very much, and the effect of a partial vacuum behind the nappe will be to increase the flow. Experimental results show that the flow is approximately given by Q=cK. Hence, if c’ refers to a different shaped weir face and the nappe springs clear in both cases, ¢’/c should be nearly inde- pendent of the head. This agrees with Bazin’s experiments*. 3 (b) Case G(w)=B—e” (approximation to flow over a weir). dz it 2w 2 __ °|. i dw ~ p{B—e® \3 | {1 we 1€ | Take B>1. The singularities of 2 in the finite part of the w-plane are ; wW=inT, w=log B+i2n7. Confine attention to the strip in the w-plane for which OSWv=7. The singelarities are then on Y=0, d=log B and d=0, ony Wea 0): * Bovey, ‘ Hydraulics,’ p. 101. Stationary Waves in Water. 105 Over w=0. Start with those determinations which at w= —o give (B—e”)s, (1—e?”)2 real and positive. —«22g9n. * I have not succeeded in constructing a case of this kind.—A. R. R. Stationary Waves in Water. 107 Suppose this is the case. Put p= kat the roots of (4) are given by ru? (X?—1) (2gAX— 6") —p?=0. ae There will always be one real root, say a, >s~. If J F 2gr there are three real roots the other two will lie between +1. Now cospw=ay gives ww=2nm7+icosh! a, 2 COS MW= J = ay gives ww=2n7 +i cosh} a, Y and ay < a, 24. ‘Cosh? ay < cosh™* ag. Case (a). Two imaginary roots E+in. Let cos (ho + eo) =F + i. Hence the singularities all lie on the lines ap= + cosh™*a,, +cosh~1a,, +o, and different problems will be solved according to the range of values taken for Ww and the relative magnitudes of wo and 4). Let w, sinh paw 7 g cosh® paw’ cP and =? + 2gh— 2g F(w) =c?(1—2a?sech? paw). . . . (8) P@w)=—- Stationary Waves in Water. 109 This will give a wave very closely resembling that of Lord Rayleigh (Lamb, ‘ Hydrodynamics,’ art. 248), : : oe dz ; The finite singularities of ~~ are given by = dw = Gosh paw=— 0, te. yaw =i(2n+1)7/2, SoS he Mea skh VI. OG a Maat, des!) CO} gaa Dee G D2 * Rea So) Oa" 1 GR. (10) where XN =cosh? paw, eee Pp o? Two main eases arise according as 2a°721, i,e. as ?Z2gh. Gase\(2) 2a? <1, 1.e.c7 <2gh. The roots of (9) are paw =i(nr+4,) where cosaj=av/2, 01 occurring. Hence the singularities are paw =1(2n+1)7/2+a, where sinha,=6,, paw=i(nm+e,) where cosa=6,, 0XSa,<7/2, paw=a+iP. No motion of this type is therefore possible in an un- obstructed liquid of infinite depth. There are no singularities on the real axis in the w-plane and the smallest zeros on the ¢-axis are +tia,, -+ia,, and on the stream-line p= at d=-+a. Different cases arise according as @Za, and to the range of values taken for wp. The most interesting case is ahicae B1, i.e. c?>2gh, the motion with a free surface of this form is impossible: the discussion follows the same lines as in the previous example. IV. Progressive waves in deep water. [t is not without interest to obtain the expressions given by Stokes (Lamb, ‘ Hydrodynamics,’ art. 230). Interchange the variables w, < by writing F(w)=H(z), (1) becomes t= 6) OAH (ey jie? gh — Bas) and will include cases of flow with a free surface. There will be no singularities in the finite part of the w-plane if 1—27H'(z) =r {e+ 2gh— 29H (2)}, Wee ik N= C4 Be, ae lw iL jen ih We (LO Re dz Te ne” wre. pany) vt Bev sin ha k te . eee yan/! y+ Be’ cos kx V. Conclusion. where gr=k. The above analysis shows that although the problem of the flow of liquid under gravity with a free surface, and given form of rigid boundary, is very difficult of solution, yet the main characteristics can be determined by noting the form of the free surface in such cases and using the methods of this paper. My thanks are due to Prof. A. R. Forsyth and Prof. A. N. Whitehead for their assistance. (pd ele 8 IX. On Radiation from a Cylindrical Wall. By A. C. Barrier, B.A.* O*: page 359 and succeeding pages of the Phil. Mag. of March 19 20, in a paper ander the above heading, an expression 1s abieaned for the amount of heat inte from the inner walls of a vertical cy linder on to a horizontal coaxial disk. The result is obtained from first principles by quadruple integration over the surfaces of the disk and cylinder, and the process is long and laborious. It is, however, possible to deduce the result, by a method which is not only Hite simpler but much more pow erful. It has been shown by Sumpner (Phys. Soc. 1892) that if an element dS of the surface of a sphere i is radiating according to a cosine law, and ds is any other element of surface of the sphere, then the radiated energy received by ds from dX is independent of the position of ds on the sphere. It can be shown that the amount of energy received by ds is Sds(T,* —T.4) A ; y, where N is the normal radiation from dS; T, is the tempe- _ rature of dS and T, of ds; A is the area of the sphere. This result will obviously hold for finite portions of the spherical surface; therefore, if S and s are two non-intersecting curves lying on a sphere enclosing spherical areas 8! and s' respectively, and maintained at ‘constant temperatures i and Ty respectively, the energy received by s' from 8’ is nee (T,*—T,*). This result will still be true it the surfaces s’ and §8’ are replaced by any two surfaces s'’ and 8" provided that s” and S" satisfy the following three conditions :— (1) Their boundary curves s and § lie on a sphere. (2) If V is any point in 8”, then no portion of the surface s'’ visible from V lies outside the cone vertex V passing thr ough the curve s. (3) If condition (2) is true when § and sand 8" and s“ are interchanged. * Communicated by the Author. 112 On Radiation from a Cylindrical Wall. Provided these conditions are satisfied, the energy received by s” from 8” will be aN8's' A From this result the problem can be readily solved. With the same figure as on page 360 and using the same notation, consider any point V on the disk. It is receiving radiation from a surface of temperature T, occupying the solid angle between the two cones having V as vertex and the circles M and K as bases. The disk therefore is receiving from the inner walls an amount of energy equal to that which it would receive from a circular disk K, less what it would receive from a circular disk M, both disks having the same temperature and emissivity as the walls of the cylinder. Construct a sphere of which the circle K and the receiving disk O are small circles. The radius of this sphere is 2, (a =b'— a7)", \/? ahts We By substitution of the spherical areas of the two disks in the formula, the result obtained after reduction is that the energy received by the disk O from a disk K is: (T,*—T,”). = é (w+ a? + 1°)? — 4076? — (a? + a? +b?) | x (1 4— Te). Similarly, the energy received by the disk O from a disk at M would be Ti Onae = = bys bgteng. he al NAC =F a* —+ b?)? — Aa76? == (Gaon + == b?) | (T,4—T,"). Therefore the energy received by the disk O from the walls of the cyiinder is TO a (T;4- T 4) V (a? +a? + b?)? — 4.07)? — VV (al + e+ P—40rb? — 2 + 2,7, This is the result obtained on page 364. In the same manner an expression can be obtained for the energy received by a circular disk from a surface bounded by any number of circles, provided that each of these circles is co-spherical with the disk and conditions (2) and (3) stated previously are satisfied. as This result can be generalized for the case where the radiating Lippich’s Projective Theorem in Geometrical Optics. 118 surface is bounded by any curves and the receiving surface is bounded by a circle provided the same seeditians are satisfied. Any further attempt to generalize by allowing the bounding curves of the receiving qittane to be non- _planar reduces ap once to the original theorem, since a non-planar curve cannot lie on more than one sphere. Research Laboratories of the General Electric Co., Ltd., London. X. On a Projective Theorem of Lippich’s in Geometrical Optics. (With a Note on the Equations of the Sener of a Straight Line on a Plane.) By Autce Everert * HE theorem referred to is the following :—If the corre- sponding incident and refracted portions of a ray, which is infinitely near toa chief ray lying in a principal section of a refracting surface, be projected upon either the tangential or sagittal sections of the chief ray, then the projections also correspond. By near rays are meant rays which are nearly parallel and are incident at near points where the normals are nearly, parallel. The tangential section is defined as that principal section ot the surface, at the point of incidence of the chief ray, which contains the chief ray and normal ; it coincides with the plane of incidence. The sagittal plane is defined as a plane through the chief ray perpendicular to the plane of incidence. ‘The sagittal sections for the incident and re- fracted chief rays are in general different, and not principal sections. The theorem was proved by Lippich (1877) for the sphere only, in an essay on Refraction and Reflexion of Infinitely Thin Ray Systems by Spherical Surfaces (Vienna Academy, Denkschriften, Band 38, p- 176 (1878)). Culmann extends it to non-spherical surfaces in his chapter in von Rohr’s ‘Theorie der optischen Instrumente,’ Band 1, pp. 183-185. Its mterest lies in the fact that it enables the path, after refraction at a non-spherical surface, of any oblique ray infinitely near to a ray in a principal section to be found by applying to its projections the ordinary method adopted for rays in an axial plane of a spherical refracting surface. We * Communicated by the Author. Phil. Mag. 8. 6. Vol. 40. No. 235. July 1920. i 114 Miss A. Everett on a Projective Theorem have only to replace the radius of the sphere by the proper radius of curvature, find the refracted paths of the projec- tions, and then the required refracted path of the oblique ray will be the ray of which these paths are the projections. The refracted ray is usually found in terms of its inclination Fig. 1. to the optic axis, and the “schnittweite” or distance from the last surface of the intersection of ray and axis. If these quantities be S, s for the sagittal projection, T, ¢ for the tangential, and the planes of projection be taken as the xy and «wz planes, then the required ray is given as the line of intersection of the planes x—s=ycots, a—t=zcotT, ae | Qarsians yeaeu ian > EHO. ee ann ee In general the theorem cannot be applied to tracing a ray through a series of surfaces, because the principal sections will vary from surface to surface. Culmann, however, applies it to a series of surfaces in two special cases :— ([.) A series of noa-spberical surfaces which are all normal to the chief ray, and have their principal sections coincident. (II.) Two infinitely thin non-spherical systems with principal sections not coincident. As an instance of (JI.), Culmann mentions crossed cylinders. As another instance may be mentioned a coaxial series of tores having the centres of their generating circles all lying on a right circular cone with its axis on the axis of revolution (a tore being defined as the surface generated by a circle revolving about an axis in its plane). ‘hus in an axial section (fig. 2) the centres of the generating cireles would lie on a pair of straight lines meeting on the axis, each line being a common normal to all of Lippich’s in Geometrical Optics. 115 the tores. The cone may include a eylinder or plane as a particular case. Suppose a telescope to sweep the horizon, Fig. 2. rotating about a vertical axis, then a vertical axial section of its lens surfaces would trace a series of tores with the centres of the generating circles in a plane, and a ray along the optic axis of the telescope would be a common normal to these tores. Lippich’s essay is based on the principle of collinear correspondence. This simple geometric relationship is, of course, far from being satisfied for rays making finite angles with the axis, but he remarks :—‘‘ The following properties (developed in the first place for a single refracting surface) of an infinitely thin bundle of rays become very simple, and approximate very closely to the properties of paraxial bundles, when the axis of the incident, and consequently the axis of the refracted, bundle lies in a plane through the optic axis.” He then proceeds to establish a series of theorems, of which the first few are identical with Young’s well-known properties of the aplanatic spheres. Homocentric pencils are first dealt with, and then theorem No. 12, the one here discussed, paves the way for the treatment of non-homocentric pencils. The proofs given by Lippich and Culmann involve the assumption of certain properties of small pencils and Sturm’s focal lines. Owing to the elementary nature of the theorem, the idea naturally suggests itself that it should be capable of proof directly from first principles. Hence the following attempt, in which no optical assumption is made except the E2 116 Miss A. Everett on a Projective Theorem natural law of refraction The question being really one of differential geometry, this method has been chosen as the most suitable to apply, though another very elementary proof by spherical projection is appended. The analysis brings to light the following facts overlooked by Lippich. In the case of the tangential projection, the theorem has a wider application than he assigned to it, for here the variable ray need not be nearly parallel to the chiet ray itself, but only to its plane of incidence. On the other hand, in the case of the sagittal projection, small quantities of the first order have to be neglected unless the angle of incidence is small, whereas in the tangential projection only small quantities of the second order need be neglected. Thus the theorem holds less accurately for the sagittal than the tan- gential projection. When the chief ray is incident soatalls the sagittal plane is a principal plane, and the two cases are interchangeable. It seems improbable that a theorem of this nature should have been left undiscovered till Lippich’s time. The writer, however, has so far been unable to find any mention of it by earlier investigators, and would be obliged for references. PROOF. 1. Projection on the Tangential Section. Hour, uf Zz The condition that the plane of projection shall be a principal section is clearly necessary to ensure that the of Lippich’s in Geometrical Optics. 117 normai to the surface shall remain in the plane of the section as the point of incidence moves along the curve of section. As will be seen later, the principal plane selected must be the one in which the chief ray lies. Suppose it to coincide with the plane of the paper, and take the principal sections as two co-ordinate planes. Let O be the point of incidence of the chief ray, taken as origin, P be the point of incidence of an infinitely near ray. Take the axis of w along the normal at O, the axis of y in the plane of the paper, the axis of < perpendicular to the paper, the co-ordinates of P as a’, y', 2’. Then the equation of the refracting surface may be written 2u = by? +cz?+higher terms in y and z, b, ¢ being the curvatures at O. In the neighbourhood of the origin O we have in Oe O=—wt ol oe 2°= E(x, y, 2), say; showing that w is a small quantity of the second order. In what follows.small quantities of the second order are considered negligible, hence 7=0. The direction cosines of the normal at a point a’,y’,2 near O are proportional to dik di dE dy dz Let the direction cosines of the chief ray at incidence be [cosy, siny, 0], aud the direction cosines of the incident ray at P be [L, M, N], and use dashed letters to denote the same quantities after refraction. The direction cosines of the normal at O are [—1, 0, 0] ! I, and ” ” 9 >) 9 i ” | —1, by 9 CF i; The general equations of refraction give, denoting the refractive indices by pw, p', pL’ —pl wv M'’—pM — pw N'—pN Se a a if 118 Miss A. Everett on a Projective Theorem The equations of the ray incident at P(0, y’, 2’) are ) OL onl logae eae es ele es NG the equations of the ray refracted at P(0, y’, 2') are 1 ea M! nnaa N’ 3 the equations of the normal at P are ® yy’ 2-2 Ee ' . (2) : The projection of the incident ray on z=0 is | oie | ihe PAN . the projection of the refracted ray on z=0 is | (3) A bey | I ve ae J the projection of the normal on z=0 is oli No death —1l by’ The quantities in the denominators may be taken as actual direction cosines, since N is small by hypothesis, and therefore L?4+ M?=L?4 M?+N?=1. And by (1) wN'—pN= —cz'(y'L' — wh), hence N’ is of the first order of small quantities like N and 2’. Also y’ is small, therefore 1 Oy? S11; The projections (3) of the incident and refracted rays meet at the point (0, y/, 0), or @ say, which satisfies the equation of the surface, neglecting y”. The equations of the normal at Q are Uf v may, g —1l7 ba R0” of Lippich’s in Geometrical Optics. 119 which agree with the expression found for the projection of the normal at P. Hence the normal at P projects into the normal at the projection of P. The condition that the projections (3) should be conjugate, or form part of the same ray of light, is pli Ni == fall pe —pM, — =l by' which is true by (1). Thus the theorem is proved for the tangential section, whatever the values of L aud M may be. If the projection had been upon the plane y=0, the condition obtained for conjugacy of the projected paths would have been a aN 5 pL! 4 wh * a/ LL? 4 N” 124 N? MENT LAr Ne =e , or neglecting N? and N”, ING a teiN, nice enh as, + (mw +p) “ which does not agree with (1) generally. The expressions agree if L'=L=+1, 1.e. if the ray is parallel to the normal at O. If the chief ray is normal, then evidently it is immaterial which of the two principal sections is regarded as the tan- gential section. ‘The ray lies in both. II. Projection on the Sagittal Sections. The general equations (see Note at end of discussion) of the projection of a straight line e—#- -y—y' 2-2 0 ee a a on a plane lr +my+nz=p (p being the perpendicular on the plane from the origin) are aw—a'+1(le'+my'+nz'—p) _ Lo ‘+ m(la'’ + my'+nz'—p) L—lcos8@ M—mcos@ c—2'+n(la' + my! + nz yl) = N—n cos 0 3 6 being the angle between the straight line and the normal 120 Miss A. Everett on a Projective Theoreni to the plane, so that cos@=LIi+ Mm+Nn, and the sum of the squares of the denominators=sin? @. In the present case the equation of the first sagittal plane is xsinw—ycosw=0, thus t=siny, m=—cosy, n=0,)\p=0, 2 = Wee cos0=Lsinysy—Mcosy. Substituting these values, we find for the projection of the ray (L, M, N) on the first sagittal plane “2—y'.sinw.cosy. _ y—y' sin? pr ge! cos u(Licosyt+ Msiny) — sin W(Leosy + M sin ar) ney or OE 28 a Een al N ) CLeeos y+ M siny) ty sina... (4) The projection cuts the axis of z at the point where Ny’ sin Ny’ sinw Mee eee Sig! ee Ea, Se ee ~ (Leosy+ M sin yw) COS a am a being the angle between the two incident rays, assumed small of first order. Similarly, if « be the angle between the two refracted rays, the projection of the refracted ray on the second sagittal plane cuts the axis of z where p. Ny sina cosa! Shr Hence, to the first order, the rays will meet in a point, N, N’, y' being all small. If, in addition, W is small this will be true to the second order. The point (0, 0, 2’) satisfies the equation of the refracting surface O= —2a+ by?+ cz’, and the equations of the normal at this point are a = —— pome & / a Y “—z Site (sae 2 thus it lies in the plane of za, i.e. the plane through the normal at O perpendicular to the chief incidence plane. The equat‘ons of refraction for the projections are, from (4) p' cosy’—poosp — p’sinw'—psin wp — |] es 0 ce eG pN L'cosy’+M'siny’ Leoswy+Msiny OZ = p’' cos f’— “cos d, of Lippich’s in Geometrical Optics. a. od, 6 being the angles made by the projections with the normal at their point of intersection. ‘These are equivalent to two equations. ‘The second expression is indeterminate SI ‘sind =ps!I ;o the fourth is used instead. ‘I'w since mw’ siny =p sin yy, so the fourth 1s used instead. wo conditions are necessary in this case, because projection did not take place on a single plane as in the tangential case, so we have to show that the rays and normal are coplanar, besides satisfying the angular relation. Now INT cz’ N cos 6 = —cos —— = -— C03 ? wih COS a sv since N, z',and «are small. Similarly, cos ¢' = —cos y’. Hence one condition holds, namely bh’ cos Ww! —wcos = —p' cos db’ + cos d. The remaining condition is po N' wN cosa¢’ cosa pe’ cosy’ — uw cos w= a aA, The equations of refraction of the original ray give Eliminate 2’, since the result is to hold for all small values of z’, and the required condition becomes (fae ae eS B' cosy’ — woos HN’ BN cose cosa This evidently holds, since a, a’ are small arigles so that a = approximately, and differentiation of the identity w'siny’=psiny gives pL’ —pL=p' cos >! —p cos yp. More exactly, differentiating p’ sin’ =psiny we get pw’ cosy’ .dw'=pcosp.dyp ; also L=cos(v+a), L'=cos (f'+2’'), LD Miss A. Everett on a Projective Theorem hence i Ibt sei le p' (cosy! —a' . siny’) — pw (cosy — a.sinyp) fe cos wr’ — woos pe’ cos vr — w cos W foe / res I/ a eu a) 2 Va tapa = pw! coswW’ (a— a’) Now if w>p, tany may become infinite, but tany’ cannot numerically exceed the tangent of the critical angle, a finite limit. The equality etany’=a'tany, which may at first sight appear paradoxical when w=90°, is explained by a’ then becoming of the second order, as follows from p' cosy’ .dyy'=pcoosy.dy, so that «’ tany is not of order 0x «, but 0?x «, 7. e. of first order of smallness. When tan ap! is infinitely small, a tan yy is of the second order ; but for finite values of tan ap! , atan wy’ is of the same order as a, that is the first order by hypothesis. If p'<,, the same reasoning will apply by transferring the accents. Thus Lippich’s theorem applied to the sagittal projection requires neglect of small quantities of the first order, unless incidence is “nearly normal ; whereas in the tangential pro- jection only small quantities of the second order have to be neglected. In the following numerical examples a=dyp is taken as 1", or ‘00000484814 in circular measure, so that millionths are regarded as first order quantities. Also w’=1°5, w=1. p. Wy’. a —ay'. atany='a' tan wp. fe) ! " 6) ! 20 13 10 48°51 00000312 00000114 20 0 1 13 10 49°16 30 19 28 16:39 00000297 ‘00000172 30 0 1 19 28 17-01 80 41 2 11:08 00000074 00000421 80 0 1 41 2 11:28 88 41 46 44-776 00000015 ‘00000432 ae) dl 41 46 44:807 89 59 59 41 48 37:135 ‘00000000 00000434 90 0 O Taking a as 1’ or (000291, and W=45°, would give a tan yp)’ =e’ tan P='0001555. of Lippich’s in Geometrical Optics. 123 ELEMENTARY PROOF. The following is an attempt at an Elementary Geometrical Proof of Lippich’s Theorem. In order that two rays may be conjugate :— (1) They must meet in’a point on the refracting surface. (2) Their plane must contain the normal to the re- fracting surface at that point. (3) The angles $, ¢’ which they make with the normal must satisfy the relation «’ sin d’=psin d, where uu, uw’ are the refractive indices. Projection on the Tangential Section. Let O be the point of incidence of the chief ray. If P’ be the projection of P, the point of incidence of a neigh- bouring oblique ray, upon the plane of incidence of the chief ray, itis well known that the distance of P’ from the tangent plane at O (the point of incidence of the chief ray) is of the second order of small quantities. Hence we may regard P’ as lying on the surface, and the projections of rays and normal through P pass through P!. This disposes of con- dition (1). Condition (2) is satisfied if the plane of projection is in a principal section at O. . Fig. 4. K N' M D' Suppose a sphere described about O as centre (figs. 4 and 6). Let the (chief) ray incident at O cut the sphere at A, 93 99 99 ray refracted at O pe) 99 399 yj Bp >» 93 normal at O Page We rg: a radius parallel to the ray incident at P ,, _,, Pai Pied 3 ” ” a3 95) (Dy refracted at P bid te) ” ” D, 2 ” Re normal at P 59 ”? ” N. 124 Miss A. Everett on a Projective Theorem Let K be the pole of the great circle ABM, L the point of intersection of the great circles ABM and CDN, and @ the angle between them. From K through N, ©, D, draw great circles meeting AB at N’, C’, D' respectively. Since the incident ray, refracted ray, and normal at a point are coplanar, the points ABM lie on a great circle, and so do ODN. 7 In the case of the tangential projection, the arcs AC and BD need not be assumed small, but the angles MN between the normals, and CC’ between the second ray and the chief plane of incidence, are assumed small of first order. NN’ will also be small, since cos MN=cosNN’.cosMN’, ... NN'’) 99 OOF 9? 99 99 x >) 99 D +9 9) oy) Dp" 9 99 ) KB. In this case MN and AC are assumed small of first order. Hence MN”, AU”, are also small, being less than MN, AC. MK is a principal section at M, and therefore at N” near M, hence NN” at right angles to MN"'K is the other principal section at N'’, and the line represented by N”, being the line of intersection of two principal sections, is a normal to the refracting surface. Tf GN is small of first order, so is DN, and as before all the small ares in the region NCC’AM may be taken as straight lines, and equal to their sines. Hence, since NC:ND::MA:MB, it is easily seen that the points N", 6", D” lie in a straight line, and therefore the lines they represent are parallel toa plane, and having been shown to be concurrent, must be coplanar. Also N’C": ND": : wy’: uw. Hence, if the angle of incidence is of the first order, the sagittal projections are conjugate, neglecting second order quantities, and obviously the smaller the angle of incidence the closer the approximation. If the angle of incidence is not small, then by neglecting small quantities of the first order, N may be taken as co- inciding with M, C with A, and therefore D with B. Hence the theorem holds with that proviso. Angle w between the planes of incidence. Because two near incident rays are nearly parallel, and the normals also nearly parallel, it does not necessarily follow of Lippich’s in Geometrical Optics. 127 that the planes of incidence are nearly parallel. Suppose one ray to be incident normally, and the other nearly so ; then the plane of incidence of the nearly normal ray is fixed, while that of the normal ray may have any inclination to a given normal plane. If, however, both rays are incident at 90°, and the normals coincide, then =the angle between the rays, which is small by hypothesis. An idea of the connexion between @ and angle of incidence may be gained by supposing the great circles CN, AM, capable of rotating about fixed diameters through N, M, and CA to be a sliding bar with its ends moving on the circles. The remarks above on @ at the close of the discussion of the tangential pro- jection apply in this case also, NN’, CO’ (fig. 4), being less than MN, AC, and therefore small. It is seen that for finite angle of incidence, w must be small ; but for small angle of incidence, w may be either small or not. Note on the equations of the Projection of the Straight Line. w—u' yy _ e—! A M won the Plane le+my+nz+p=0. The equations employed were originally obtained as follows :-— Let (2”, y', 2") be « second point on the line, at distance r fepan (2, y' , 2’ ). Let p’, p’’ be the perpendiculars from these points on the plane; (&, 7’, &), (E", 7", &”) the co-ordinates of their feet, then’ g''—g'=Lr, &e. p =p —le’—my'—ne, Ez + lp’: &e. es ee lp", &e. The equations of the required projection are Pere YT wea. Be ee aay Ae and by substitution EU — B= oe" — 2! —Ir(Ll+ Mm+ Nn)=r(L—leos 0). (If, as suggested below, w’’, y'’, z'’ had been taken as the point where the line meets the plane, we should have had a i =e a cos and E" — 8 = 2" — 2! — lp'=Lr—Ir cos 6.) 128 Lippich’s Projective Theorem in Geometrical Optics. The denominators were also derived by means of the relations aL+6M+cN=0, al+bm+en=0, adr+bu+ev=0, IN+mp+nv=0, where A, mw, v are the direction cosines of the projection,and a, b, c of the normal to the projecting plane. The result might also be obtained by changing the co- ordinate planes so as to make the plane of projection one of them, but it seems simpler to proceed directly. For the following much neater method the writer is indebted to Mr. T’. Smith. Make use of the point in which the line meets the plane. (In the particular case where there is no intersection the projection result is obvious without investigation.) If the point of intersection is e+Lp, y'+Mp, 2+Nop, then la’ + my! +n + pcos 0 =p. The projection lies in the same plane as the line and the normal, and therefore its direction cosines are of the form a¢utBl, aM+ Am, a«aN+Bn. Since it is perpendicular to the normal to the plane, 2cos 0+ 8=0, the direction cosines are a(li—leost), .2(M—m cos?) a( N—nicosd) from which it at once follows that a=1/sin 0. The projection is then ! L all ! patel : e-a« t+ cos OMe ie We eae =) ee - = uC. — oer . L—teosé i : ; lx’ + my! + nz’! — 9 or, adding to each fraction — ——— cake ae | cos 0 v—u' + l(la! + my! + nz'—p) ak . —— ~~“ = two similar expressions. L—lcos@ The writer desires sincerely to thank Mr. T. Smith for reading and eriticizing the manuscript before publication, and suggesting (besides the matter just mentioned in the note) the clearer expression for the final analytical condition in the case of the sagittal projection, which led to the dis- covery that the theorem is less accurate for this case than for the tangential projection. XI. On the Theory of Lonization by Collision. By P. O. PEDERSEN *. f. A an article in this journal Norman R. Campbell f treats of the mathematical theory of ionization by collision, making the simplest possible assumptions with regard to the nature of this ionization. Only ionization due to collisions between electrons and molecules is con- sidered, and his assumptions may briefly be stated as follows :— (1) An electron will produce a fresh pair of ions (electron + positive ion) when it collides with a neutral molecule, if previously to that collision its velocity v satisfies the following condition : feet duicw ctw where m is the mass of the electron, —e its charge, and V, a certain constant voltage depending on the nature of the molecule. (2) Collisions between electrons and molecules take place unelastically, so that after collision the electron is free again but has the velocity zero. (3) The electrons set free by ionization have also the velocity zero. (4) The velocities of the electrons due to the electric field are so great that the thermal velocities of the electrons and the molecules may be disregarded tf. 2. Norman R. Campbell does not give the exact solution. He says§: ‘‘ Now it is possible to calculate accurately according to the theory what should be the current between electrodes a distance a apart (fig. 1) in a gas at pressure p, * Communicated by the Author. + Phil. Mag. (6) vol. xxiii. p. 400 (1912). { With regard to the question of the validity of these assumptions, see J. S. Townsend, ‘Electricity in Gases’ (Oxford, 1915), and the discussion between Townsend and Norman R. Cam pbell, Phil. Mage. (6) vol. xxlil. pp. 856 & 986 (1912). § Loc, cit. p. 404. Phil. Mag. 8. 6. Vol. 40. No. 235. July 1929. K 130 Pps VOL eellerecal ae ale when (the electric field) X has a given value, but the calculations are extremely complex, and the resulting ice le formula for the relation between 7, a, p, X is not of a form to which Townsend’s measurements can be applied.” He therefore assumes that the number of electrons n, varies continuously with the distance w from the cathode and thus obtains an approximate solution. The exact solution is, however, quite simple and may be given a form to which Townsend’s. measurements may easily be applied. The deduction of this solution may therefore, perhaps, be of sufficient interest to be given here. 3. Let L be the mean free path of the electron; then the probability s that a new collision results in ionization is determined by lo ve i (| where ne V, omg 1h eS ee (4) The electrons coming from A (see fig. 1) fall into two groups 1 and 2. In the first we shall reckon the electrons whose initial free path is greater than /) and which will thus ionize at the first collision. The second group includes the electrons whose initial free path is less than J): of these the first collisions which occur between z=0 and «=1, will not result in ionization. The number of ionizations which take place in the space element between the planes # and w+dz, and which is Theory of Tontzation by Collision. lol due to the first group of electrons, we shall call No(w) . de, where the sufhx 0 denotes that the electrons come directly from A without intermediate collisions. Fig, LS : " 2, We have then fore iy Noe) =. 0, ee eee. 2 Ih), (5 and for «= ly: N,(2) =7-¢ L, t ae) ) We shall then proceed to the determination of the first ionizations which are due to group 2. In the element of space (y, y+dy), where y in the same way as a signifies the distance from A and where 02l): Ne-,)(a) = f N,(#).dy==(e— te =), y=0 L J The total number of ionizations N(#).dz in the element of space (a, v+da), which are due to electrons which either come directly from A or the direct collisions of which took place within the distance /) from A, is thus determined by : LOL.) lye Ni@) 0; > | th -b 2l): NG = ae By) It is obvious that the number N,(x)dz in the element of space (vw, +d), which are due to collisions with electrons originally coming from A, is determined by : fore 0-2 0:4 06 0:8 LO Lil ena: a= Sy Ise bg 2°86 83 24:2 81:0 373 2250 Wo Ss =) netics 1:05 2°12 319 4°39 0°92 772 0 Theory of Ionization hy Collision. Lot : Mi . : . The values of log— are drawn with a as abscisse (fig. 6). No : One puts after a preliminary trial J,=0°05 cm., and draws through the point c on the abscisse axis determined by Oc=ly) a straight line ch which for small values of a Fig. 6 _5 ‘meen 4 “6 ag 7): LE CHE Wo-- ++ +--+ +--+ + H-- --- - - - -- -- ----- - ---- 2 - -- = LY L0 Ci agrees as closely as possible with the found values of log — (For greater values of a, log“ lies above the line dg a 9 account of the ionization caused by the positive ions of which no account is taken.) From fig. 6 is obtained ie ya=9'°8. To 1,=0:05 cm. corresponds y= 10°28, to which again according to the preceding table corresponds 78> ‘44, From this is found by using equation (16') J,=0°046 cm., and as this value is sufficiently near to the one previously assumed, namely 0°05 cm., there is no reason to make a new determination. For the ionization voltage we now obtain, Voi 0046 x 350 = 16°1 volt. A further discussion of this problem is to be found in a Danish paper * Copenhagen, November 1919. * Det Kgl. Danske Videnskabernes Selskab. Mathematisk-fysiske Meddelelser, i. part 7 (1918). y=0°78. Consequently «= falas. XII. On the Stability of two Rectilinear Vortices of Com- pressible Fluid moving in an Incompressible Liqud. By Brsuuripausan Darra, M.Sc.,. Lecturer in Applied Mathe- matics, University of Calcutta, India*. () ieee stability of the circular form in two rectilinear vortices has been discussed by Sir J. J. Thomson f for incompressible fluid. Dr. Chreet attempted to extend his treatment to a compressible fluid but did not succeed except in some special cases. The object of the present paper is to complete the work begun by Dr. Chree. | When the vortex-lines are straight lines parallel to the axis of z, the velocity components u, v at any point (w, y) in the fluid are given by § ges OD. OME lleles sO Ohare Ones 1 iy: ae i o= a + yar eng ( ) hence V1o= —9@, Wie Ret Orieeet sy. Nt (2) Way & ao A O° Ox where = ae ap Vi + aa Then by the theory of attraction, o=— Fal 6' log rdu' dy' + do, j (3) = male Pas ay. ng 6’, & being respectively the value of 6 and € at the point (zal) and p= f(e@—a't (y—y')?, Vrd=0, Vivo=0. If S denote partial differentiation, and a differentiation * Communicated by the Author. + ‘Motion of Vortex Rings,’ p. 71. t Messenger of Mathematics, vol. xvii. p. 115 (1888). § Lamb, ‘ Hydrodynamics,’ 4th ed. p. 213 (1916). Stability of Rectilinear Vortices of Compressible Flwd. 139 following the fluid, the equation of continuity is dp d put dpv ae ae A aK or p De b= a(n it logrda'dy'+do. . . + (4) If po be the mean value of the density of the fluid at the cross section o of a circular vortex, defined by the relation \\e' da’ dy'=poo, aliay ald obuetn(s) we get =| — Slog r+ dp. Again, the equation of continuity is D. poo Dt or 1 Dp , 1 De Po *pe a: DE =(, Hence o=— sy ae ema) eee. (6) Since o always refers to a definite column of fluid, there 4 o do is no difference between —— and —. sity Di dt Let us consider a truly circular cross section of radius a ; if the vorticity be the same at any instant at all points of the section, we have, when r>a, (7) where « is the strength of the vortex, C and D being 140 Mr. B. Datta on Stability of two Rectilinear Vortices of constants. Also when r(a,cosnO+Brsinnb), . - - (9) where a, and £, are functions of t, independent of 8, and very small compared with a. Outside the vortex, r>a, let o=C— = a plogr+ | E,, cos nO + F, sin 8) (2), em! alo 1) ve log r+ 3(A, cos nO + B, sin n8) (C): inside the vortex, (a, cos né + 8, sin 6) Lowy, 0d: 1 _lody' , o¢' rod or r oo or OW OG al Wed ma Br re” Ore! r8e Compressible Fluid moving in Incompressible Liquid. 141 Since products of small terms are neglected, these con- ditions become Bye LOW Wosmr oa) er ON” Om, a en Substituting the values of w, W’, ¢, ¢’ from (10), (11) and equating coefficients of cos né and sin nd, we get K K Be ana BO Seng Om l wide 4 ies de: i - sz, Sn kk a Cyaan pie lew ° ° (13) 2rrna at 2arna dt Hence outside the vortex, we have do 1 da dt . a n | o=C— =e logr +35 (a,cosn0 + B, sin n@) (<) ; | : eae K K . a\” w=D+ 97 108? — 25, (tn cos nO + 8B, sin nO) (*) hs) Suppose there are two vortices of strengths «, «’ ; let their cross sections at time ¢t be o and a’ respectively, the distance between their centres being c. Jet the radii of their cross sections be given by R=a+ (za), cosnO+ £8, sin n8), l R'=b +3 (a,' cosnb’+B,'sinng’).f °° Let ¢, y be the functions at an external point for the first and ¢', y’ for the second. If (r, 8) and (7, 0’) be the coordinates of a point referred to the centres of the two vortices, @ and #& will be given by (14), and doa! \ hee. Soest ' dt ee rece DN d =~ oa 7H 1°83 1 be $y (an! COS nd' + B,, sin n@ ) (=) ) hae / / n y= = log oe (en cos nd’ + B,,' sin n8')(-) ; | We shall fix our attention on the second vortex. If 3 be the radial velocity of a point on it and © the velocity per- pendicular to the radius vector, both relative to the centre 142. Mr. B. Datta on Stability of two Rectilinear Vortices of of the vortex, we have from (15) : db det! ! v dB, a = - + = ( a COS nO’ + “ae sin nO! } —2nO(a,’ sin nO’ — BG, cosnd’).- | eee Since products and powers of «,’, 8,’ are to be neglected, in (17) we may put ! K OF amt e ° . ° ° e (18) Again su ONe oe or dad aT a ak eo S777 Sao5in (8 e) laa; Bs ee Al ee Of a eos (6) =e). 20.) a where v’ must be put equal to b+ >(a,' cos nO’ + B,' sin n8’) after differentiation, and ¢€ is the angle made by the line joining the centres of the two vortices with its initial position. If b/c be small, @ and wean be expressed in terms of 2”, 6, and e, such as TG; e = (==ly |" e Ss aay [log Sieh ye ; liber € ih ] n—1 Sa = | (a cos ne+ 8, sin ne) = = (- i. ar lhe ue + Creel) cose 2Qan dt ¢ fo a) + (8, COS NE— Ay, sin née) om —1)*- -] 2a ye 2 at) je AED) (TY sin (6 (6-6) |, ‘5 ¢ Wee D+ p. [loge+ S eC Joos s(@ =e | Sal Ss J il = K C— S A ree Ge | (an COs ne + Gh, sin ne) > (= Lye } s=0 +1)...(n+s— r'\s ye Wet ) = S )(")'c0s s(6' 6) c + (By, cos ne— a, Sin ne) Y (— 1)s-1 s=i é HH ears) (”)sin (6'-6) |. KG s! Compressible Fluid moving in Incompressible Liquid. 143 Substituting in (19) we get ! lo’ ee i (a, sin nd’ — 8,’ cos n8') + Bh) ae 2b 2Qarb dt has py Ra SEY TES oe x [1+ 7 aie (2, cos nO’ + B,' sin nd) | K an a +38 [Cencosnet usin ne) &(—1)™ n(ntl)...(a+ 3 ae , ge MR Tn SP) au = sin s(0' —e) + (8B, Cos ne — a, Sin ne) S (—1)s-} s=1 n(n a ye (n+ See) Ue I | (s—1)! iat el Phuc = a pee ae eS fist ae s(0’—e) —— oe x ¢ 1)s-! cos s(@’—e) FE: y, [ ' cos n0’ + 8,’ sin nd) | 1. Vadoxae= s => Tae | (2, cose +B, sin ne) & (— 1) n(n+1). eet) val -1 Seppe wor beber dd a ieete cried (—1! +(B,, COs HE 2, sin ne) 7 (—1)s~} esi cos s(0' —e) ; Elke sl OD lato IM 5g 9] ~sin (0'—e)— ss © cos (0’—e). (20) The two values of 38 as given by (17) and (20) must be the same ; so that we can equate the terms independent of g' in either expression as well as the coefficients of the sines and cosines of multiples of @’. The terms independent Dar m do’ db of @' simply verify the known result that Frm re Equating the coefficients of cos 6’, we have day! K / Ls de 7 = 92 (B, cos 2e— a, sin 2e) + a SP (a, cos 2e+ (, sin 2e); 144 Mr. B. Datta on Stability of two Rectilinear Vortices of ! f e e da, ° e so that to our degree of approximation —~- =0; similarly dt d s =(). Hence, so far as these terms are con- ( cerned, compressibility of fluid makes no change in the motion or shape of the vortices. Equating the coefficients of cos 20' and sin 20’, we get to the same degree ee approximation con _ b deo = ~——>sIn 2e— =; —— d Bit gi Sag 28 Te gy 8 Cl Kb ben dou = — ea ae f D) Wy TDR ao mre COS Zé are aE sin 2e (2 ) About these equations Dr. Chree remarks : “It is scarcely likely that these equations admit of a complete solution.” * But the solution of them is of essential importance, for on it depends the determination of the exact change in the form of the cross section. I shall presently show that they 3 can be solved. The terms in as, 83 will involve “> and thus will be relatively unimportant, so we shall not determine them here. Let = +105 a e U hee where j= v=1l . ay V = ay — 1 Be’, Multiply the equation (22) by 2: the two equations (21), (22) become, by addition and subtraction, Gig ene. EO ars b da 5. Smet —— GC Saye ate’ Cus at 2arb 277 2nc? dt (24) / / 4 im dv +4 BA ppl? dee a) mice de —2ie dt Peale 2c? 2arc? dt These are linear differential equations. On integration we get ae ee be ae a) j bs t WH yh é ( ania) fines Pee u 1) ae, | 0 i 4 bone oo 5 5 KL l l Aen eile ( ale) &. ay i at Qmrc* = Dac? dt where wo and v) are initial values of wand v. If pay’, 9B’ be the values of a,', 8,’ respectively when ¢=0, we must have Uj = hes + tose hn to—02 lose. |. re eu) IE NGS {dey WS IIS Compressible Fluid moving in Incompressible Liquid. 145 Now, substituting the values of wand v from (23) in (25), we get by addition and subtraction, ao C= |) + B,' sin (5 V5) Eb 9 Kk! (dt =o + Ee 2sin ( é— om a b6 do Kk’ (dt Din) See ~Qre dt cos (2¢ =e )] an ; lt oy' sin Gis) —B,! 008 (5 id \z) Pa aN Ya LO | rete CL oe: {" lee vos (2¢ ore B) bi do K! (dt a Ine dpe (2e— ell dt. Solving these aaa for a’ and 8,’, we get ee =n ‘eos (5 Bi A — (Bo' sin (¢- AF) + gos fo =) —sS [isin (2e— i cae 2 )\ x ad | ( \z) = Tp 005( 26 eA) dt F ng Gore ” S) a 2 + 0 ° Zz Zs mn ao o Neon 2 : 3 2 ow oi fi ‘c = oO sx Argon 02 e9 ‘ re) al wy a > a 2) (eo) 2 fe) Log] Ww i)) aq ro) S im) Se 4g Boer Gow, Sia? S7 os ss es ss og — iy) "Atomic Radius’ * VO cr. Arrangement of Atoms in Crystals. L709 10. In fig. 3 the elements are arranged in the order of their Atomic Numbers. The ordinates represent the diameters of the “Atomic Domain” measured in Angstrém Units. The figure summarizes the empirical relation which has been found to hold, namely, that the distance between neighbour- ing atomic centres in a crystal is the sum of two constants characteristic of the atoms. ‘The crystal may be imagined as an assemblage of spheres packed together, the constants then representing the radii of the spheres. The atomic diameters lie on a curve resembling Lothar Meyer’s curve of atomic voiumes. The alkali metals head each period with the greatest diameter, followed by the alkaline earths. The diameter diminishes steadily as the atomic weight is increased, reaching a minimum for the electronegative elements at the end of the period. In other words, w hen the atomic arrangement of compounds is taken into account, the periodic relation between the atomic volumes shown by Lothar Mever’s curve can be extended to the compounds of the atoms. A list is given below of the “ atomic diameters” assigned to the elements, and for convenience the ‘atomic radii”’ are added. The second table is a comparison of the observed distance between atoms in crystals with those obtained by adding together the radii of the two atoms concerned. It will be seen that the difference between observed and calcu- lated ee is never large, the average difference being 0:06 x 10-8 It is ays “jatendea to assign any physical significance to these “diameters” other than that discussed below. Sodium, for instance, has been given a diameter much larger than that of chlorine, yet it will be seen that there is every reason for supposing that the group of electrons surrounding the sodium nucleus in sodium chloride has smaller dimensions than that surrounding the chlorine nucleus in the same crystal. The way of regarding the atoms as spheres packed tightly together is useful in * constructing models of crystalline structures. Such models are aetraren in Plate III., the crystalline structures being those of sodium chloride, calcium carbonate, and zinc-blende. 180 Prof. W. L. Bragg on the Atomic Atomie 5 Atomic Number. Element. Diameter, in A. Radius, in A. 3 Tnthvumal baccarat 3°00 1-50 4, Beryliswin’ 26) Leer. 2°30 115 6. Carboni) e)- tak... Vere 1°54 0:77 7 Nittogen cue eaten meee 1:30 0°65 8 ORV ge) Occ. snore aeaes ee 1-30 0-65 9. Eluorines®. .A0%. wake. © 1°35 0:67 hele Sodiivany ‘}yceeep eee soe 3°55 Lh 12. Miaparesiuiiay peeeenae. sco: 2°85 1:42 13% PANT MUNCIE fs sconsan cass deeseee- 2°70 1°35 14. SiltCOns jcc oe eee ae 2°35 1:17 16. Slip lows wees see A ete 2:05 1:02 Wie Chilorinererec eee a 2°10 1:05 19. Wotassiunntes peeeee ee es. <- 4:15 2:07 20. Caleiuaa ti ME heo0 7, (9 3°40 1:70 22) Di tarviuay pe eee ia oes eee os 2°80 1:40 24, Chiro mimi 2°80 1:40 (electronevative hance... 2°35 1:17 25. Mempgatiese Poh asec. sesse.enee 2°95 1-47 (“electronegative”) ......... 2°35 17 26. POM, gi.) 6. oee eeIe eee s 2°89 1:40 FBT 3 Cobalthacd-cceeen cee eee 2-75 1:37 28. Nickel, 2 Fee Wee ee. 2°70 1°35 20... Copperi.d ceca eerreee eee 2°75 1:37 30. PANY Oyo Biker 2 AR IO Oe 88 2°65 1°32 33. ASYSCING. 2cc toes eames 2°52 1:26 34. Seleniuniee cee en) 2°35 LEW; 31). Brpaiinee Wiser aie eek oe 2°38 1:19 37. Rubidium fo5-24oe ee 4°50 2°25 38. PSL ROMNATTLIN Wig tdadackeha sdottaade 3°90 195 47. Silver (i Seer ee Sale. BORE Soe ae 48. Cacdlinminainieens Gay nas-cea ses 3°20 1:60 50. ARTE TWO ors oN OE. aa (a 2°80 1:49 D1. SAUTNGTUNOTNY/ acerctee feae ease 2°80 1:40 52. Tel liuanethun rn eee ce oeresncaiss os a 2°65 ise 53. ToOdiiGyton Ss peek ee tenes. 5 2°80 1:40 a Cres iia eee ae 4°75 Don 56. BATU Gacsscen nue eoee: 4:20 2°10 81. Tues tua Ae ee LYE Pee 4°50 Die 82, Meads) Sheek ea: Ae ee 3°80 1:90 83. Bismilithi 3. fics skh eeichacn 2°96 1°48 Atomic Sum of radiz, Observed Difference, oO (@) Compound. centres, in A. Distance, in A. im A. Hlement Li. 3°00 3°03 +0:03 LiF. Li,F. 1:50+067=2°17 2°05 —0312 LiCl. Li,Cl. 150+ 1:05=2°55 Days +0°02 LiBr. Li, Br. 1°50+1:19=2°69 2:79 +0:10 endl. Li I. 1:50+-1:'40=2-90 Sd +0°25 Element Be. 2°30 2°52 +0°22 * Assuming close packing. BeO. Be,O. 1°15+-0°65=1°80 1-78 —0'02t t+ Assuming ZnO structure. Atomic Compound. centres, Element Na. NaF. Na,F, NaCl. Na,Cl. NaBr. Na,Br. Nal. Na, I. NaNQ,. Na,O. Element Mg. MgO. Mg,O MgCo,. Mg,O Mes. Mg.8. Element Al. Al,O,. Al,O KF. K,F KCL K,Cl KBr. K, Br KI, Ga CaO. Ca,O CaCO,. Ca,O Ca8. Ca,S. CaF,. Ca, F MnCO,,. Mn,0, MnS,,. Mn,S Element Fe. FeCO,. Fe,O Fe,O,. Fe,O Fe. Oi: Fe,O FeS,. Fe,S Element Cu. CuO, Cu,O ZnO. Zn.O ZnCO,. Zn,O ZnS. Zn,8 RbCl, Rb,Cl RbBr, Rb, Br RblI. Rb, I. SrO. Sr,O. Sr8. Sr,S. CdCoO,. Cd,O Cds, Cd,$ CsCl. Cs,Cl CsBr. Cs,Br OsI. Cs, I. BaO. Ba,O. BaS, Ba,S. Arrangement of Atoms in Crystals. Sums of radii, 2) m2 A. 3" 35 2°85 1°42+0°65=2:°07 J°42+4-0°65=2°07 1°42 +1°02=2°44 2-70 1°35 +0°65= 2-00 2°07 -+0:67 =2°74 2:07 +1:05=3'12 2°07 +1:19=3:26 2°07 +1:40=3°47 1:70+0°65=2°35 1:70+0°65= 2°35 1-70+1:02=2-72 1:70+0:65=2°35 1:474065=2'11 1-47-+1:02=2-49 2°80 1:40 +0°65 =2°05 1:40+0°65=2:05 1:40+0°65=2°05 1:40+1:°02=2°42 2°75 1°37 +0°65=2°02 1:32+0°65=1°97 1°32+0°65=1-97 1°3241-°02=2°32 1°95 +0°65=2°60 195+ 1°02=2°97 1:60+0°65=2°25 160+ 1:02=2°62 2°37 +1:05=3'42 2°37 + 1°19=3°56 2°37+1:40=3°77 2°15+0'65=2°80 2°15+1:02=3:17 Observed Z . 0 Distance, in A, boty ty wotresty pety in A, +0°17 —0:05 +0°02 +0°01 —0-14 —0'09 +0°37 +0:03 —0:07 +0:lu +0°17 +002 +0:04 +0:01 +0:02 +0°05 +005 — 0:05 +0°05 —0°01 —0-01 +0:10 —0°33 —0°01 +0:05 —0:05 —0:14 —0:20 —0°15 0°00 +002 +0:03 —0-02 0-00 +001 + 0:03 +0:02 — 0-04 +0:05 —0:14 —0°16 —0°'16 +001 +0:03 181 Difference, ° 182 Prof. W. L. Bragg on the 11. These empirical relations will now be considered with reference to the theory of atomic structure proposed by Lewis*, which has been greatly extended by Langmuir. Briefly stated, some of the principal features of the theory in the form in which Langmuir presents it are the following :— (a) The electrons surrounding the positively charged nucleus of an atom are either stationary, or oscillate about certain fixed positions. (6) The electrons are distributed in a series of approxi- mately spherical shells surrounding the nucleus. (c) Certain arrangements of electrons around a nucleus, those of the atoms of the inert gases, are very stable. ‘These arrangements are:—Helium, where a nucleus with two unit positive charges is sur- rounded by a shell containing two electrons. Neon, nuclear charge 10, surrounded by an inner shell containing two electrons and an outer shell con- taining eight electrons. Argon, nuclear charge 18, surrounded by three shells containing two, eight, and eight electrons. Krypton, nuclear charge "36, shells containing two, eight, eight, and eighteen electrons. Xenon, nuclear charge 54, shells con- taining two, eight, eight, eighteen, and eighteen electrons. Niton, nuclear charge 86, shells con- taining two, eight, eight, eighteen, eighteen, and thirty-two electrons. (d) The chemical properties of the elements depend pri- marily on the tendency of the atom to surround itself with a more stable arrangement of electrons. The most simple chemical properties are exhibited by those atoms which revert most easily to the inert gas form, z.e., the atoms nearest the inert gases in the periodic table. An electropositive element is one which tends to give up electrons in doing this, an electronegative element one which tends to take up electrons. 12. Broadly speaking, the theory distinguishes between two different types of combination in chemical compounds. The first type is represented by a compound such as KCl. The potassium atom has a nuclear charge of 19 units; and is surrounded by 18 electrons arranged in the same w ay as those of the argon atom, with in addition an electron which finds * G, N. Lewis, éoc. cit. + I. Langmuir, doe. cit. Arrangement of Atoms in Crystals. 183 no place in the stable argon. arrangement. Chlorine has a nuclear charge of 17, and is surrounded by 17 electrons, one less than the number required to form the stable argon arrangement. When an atom of potassium combines with one of chlorine, the chlorine atom absorbs into its system the additional electron from the potassium atom. Both atoms are uow surrounded by the argon shells, but as the nuclear charges are 19 and 17, and each atom is surrounded by 18 electrons, there will be a resultant positive charge of one unit on the potassium atom and a negative charge of one unit on the chlorine atom. The electrostatic attraction of these charges holds the molecule together. The nuclei, surrounded by the stable argon shells, compose the mono- valent kations and anions of potassium and chlorine. The other type of combination is represented by a com- pound of two electronegative elements. In this case both atoms have fewer electrons than correspond to a stable system. They compiete the required number of electrons in their outer shells by holding one or more pairs of electrons in common. In such compounds as SO,, CO., the atoms complete their outer shells of eight electrons by sharing them with their neighbours. The crystalline structure of a compound such as KCl is very simply explained by this theory. As has been pointed out by Langmuir, the crystal is to be regarded as an assem- blage of potassium and chlorine ions arranged on a cubic lattice*, The ions consist of the stable argon shells, but as the nuclear charges are 19 and 17 the ions have resultant unit positive and negative charges. Each ion tends to sur- round itself. with as many ions of the opposite sign as possible. This is realised in the KCI strneture, where each ion is surrounded by six ions of the opposite sign. There are no individual molecules in the crystal structure, the potassium ion has exactly the same relation to the six chlorine ions surrounding it, and wee versa. Some repulsive force must exist emer the outer shells, which holds the atoms apart against the electrostatic attraction. In the structure of calcium sulphide,: the calcium atom Joses two electrons, and the sulphur atom gains two, in reverting to the argon form. The resultant charges on the ions are twice as great as those in the KCl structure, and as a result of the increased attraction between the ions the structure, while being similar to that of KCl, has all its dimensions reduced. * Cp. Debye and Scherrer, Phys. Zeit. xix. (1918), where evidence is given that an electron has passed from the [i atom to the IF atom in LiF. 184 Prof. W. L. Bragg on the In the diamond, each carbon has a nuclear charge of six units. The atom has two electrons in its inner shell and four in the outer shell. In order to complete the number of electrons, eight, which would make its outer shell correspond to the stable neon form, it shares its electrons with the tour carbon atoms surrounding it in the diamond structure. Hach pair of atoms holds two electrons in common. The forces binding the atoms together are of a different type, the atoms are united because they share electrons, not as a result of opposite charges on ions as in KCl. A crystal of an electropositive element, such as sodium, consists of an assemblage of the stable “inert gas” shells with an additional! electron associated with each in order to neutralize completely the nuclear charge. ‘These electrons have no fixed positions in the structure, they move under the action of an electromotive force and convey a current of electricity through the metal. On the other hand, a crystal in which the atoms are bound together by sharing electrons, so that there are no free electrons, is a non-conduetor. This is the case for the typical electronegative element. 13. The empirical relations of fig. 3 are readily explained by this theory. In each period, the alkali metal which follows one of the inert gases has been assigned a large ‘“‘diameter.” This expresses the fact that it appears to occupy a large space in any crystal structure; the centre of the atom is separated by a considerable distance from the centres of the neighbouring atoms. Successive elements are assigned smaller diameters, and at the end of the period the electronegative elements immediately preceding the next inert gas have diameters which approximate closely to a limiting minimum value for that period. The small diameters have been assigned to the electre- negative elements on account of their proximity in a crystal structure. In sodium nitrate, for example, the distance between the oxygen and nitrogen centres is 1:30 A., that between the oxygen and sodium centres 2°38 A. In all these atoms the nuclei are surrounded by the stable neon arrange- ment of electrons, and presumably these electrons are ap- proximately the same distance from the nucleus in sodium, nitrogen, and oxygen. The oxygen and nitrogen atoms have realized the stable arrangement, however, by sharing elec- trons, and their centres are correspondingly close together. The sodium atom is already surrounded by a stable shell and is isolated in the structure. In sodium nitrate there is the same arrangement of positive and negative ions us in sodium chloride, except that the negative ion in this case is the Arrangement of Atoms iu Crystals. 185 complex NO; group. In order to complete a stable arrange. ment around the four nuclei, the NO; group has borrowed an electron from the sodium atom, leaving it a positively charged ion. ‘These ions are arranged in the same way in NaNO, and NaCl, each ion being surrounded by six of the opposite sign. The form of the NO; group has, however, distorted the structure so that the crystal is rhombohedral instead of cubic. In MgCQ; the arrangement of the atoms is the same as in NaNO;. The magnesium ion has a double positive charge, the CO; ion a double negative charge. As a result of the greater electrostatic forces, the dimensions of the structure are reduced, the distance between magnesium and oxygen centres being 2°00 A., as compared with 2°38 A. in the case of sodinm nitrate. This will make it clear why the divalent element appears to occupy a smaller space in a erystalline structure than the monovalent element preceding it in the periodic series. The large diameters assigned to the electropositive elements as compared with the electronegative elements do not imply a corresponding difference in the dimensions of the atomic structure. They are an expression of the fact that the electropositive element does not share electrons with neigh- bouring atoms, it is always surrounded by a complete stable shell. The repulsion between this outer shell and the shells of neighbouring atoms keeps the atom at a distance from its neighbours, so that it appears to occupy a large space in the crystal structure. It is interesting to compare the structure of graphite with that of diamond from this point of view. The graphite erystal has been analysed by Debye and Scherrer*. It corresponds to a diamond structure in which, firstly, the dimensions of the whole structures parallel to a trigonal axis have been lengthened in the ratio 0°598: 1, and, secondly. the carbon atoms in the pairs of (111) planes of the diamond have been so displaced that they lie very nearly in the same plane. The atoms in a (111) plane are therefore very much closer to each other than they are to the atoms in the next planes. This may be explained by supposing that they are sharing electrons with their neighbours in the (111) planes but not with the other atoms, the very ready cleavage parallel to (111) lending support to this view (cp. Debye’s paper). In such a case as this, the analogy of the crystal structure to a set of spheres packed together obviously * Debye and Scherrer, Phys. Zeit. xviii. (June 1917). Phil. Mag. Ser. 6. Vol. 40. No. 236. Aug. 1920. O 186 Prof. W. L. Bragg on the breaks down. The distance between neighbouring carbon atoms in graphite is 1°45 A. 14. It has been seen that in each period the diameters of the electronegative atoms appear to approach a lower limit. If it is true that these atoms share electrons when combined together in the crystal, the diameters which have been assigned to them should give an estimate of the diameters of the outer shells in which the electrons are situated. In the first short period the diameters assigned to the atoms of carbon, nitrogen, oxygen, and fluorine are 1°54 A., 1:30 A., 1:30 A., 1°35 A. The first three of these have been calculated from compounds in which the atoms share elec- trons, the nitrates, carbonates, and diamond. No compound in which fluorine shares electrons has been analysed, but evidence has been given that it occupies the same volume as oxygen. The outer electron shell which these atoms tend to complete is that of Neon. We may therefore estimate the diameter of the outer neon shell as being 1°30 A. Since two electrons at least are held in common by the elements this estimate may be somewhat too large. In the second short period the diameters of silicon, sulphur, and chlorine are 2°35 A., 2:05 A., 2:10 A. The structure of phosphorus has not yet been analysed. The diameter of the outer Argon shell appears to be 2°05 A. . In the first long period, the lower limit to which the diameters tend is 2°35 A. ‘The structure of arsenic has not been analysed, but it crystallizes in a form isomorphous with antimony, the structure of which has been recently deter- ° mined by James and Tunstall. If its structure is that of antimony, the distance between the nearest atoms is 2°52 ie Selenium has been assigned a diameter of 2°35 A., bromine a diameter of 2:38A. Other elements in the same period tend to approach this limit. When manganese and chromium act as acid-forming elements and so share electrons with other atoms, they enter into compounds isomorphous with the sulphates and selenates, and the molecular volumes of the compounds are very nearly those of the selenates, so that the atoms appear to have dimensions identical with those of selenium. The distances between atomic centres in iron, nickel, and copper are 2°47 A., 2°39 A., 2°55 A. These figures confirm the estimate of 2°35 A. as the lower limit to which the diameter tends. In the second long period, the distance between atomic centres in gray tinis 2°80 A., in antimony 2°80 A. Tellurium Arrangement of Atoms in Crystals. 187 and iodine have been assigned diameters of 2°66 A. and 2°80 A. The evidence for the lower limit is imperfect, but it may be estimated as 2°70 A. The diameters of the outer electron shells of as inert gases therefore appear to be— Reames Pie Ae eee ks. oe 1:30 A. agen |y sas eee 2s 205 A. irintan aut ict: 2:35 A. Dreegre) fy test. eeetee sl eS 2°70 A. On Langmuir’s theory, the crystal of an electr opositive element consists of an assemblage of positively charged ions held together by electrons which are free to move in the structure. The empirical relation between inter-atomic dis- tances In compounds is less accurate when applied to the metals, perhaps as a result of the different nature of the forces in this latter case. For instance, in a number of isomorphous series the substitution of magnesium for iron decreases the molecular volume, yet the distances between atomic centres in metallic magnesium and iron are Balas and 2-47 A, respectively. Silver and sodium form many isomorphous salts of nearly identical molecular volume. The distances between, atomic centres in crystalline silver and sodium are 2°87 A. and 3:72.A. Isomorphous salts of the same molecular volume are formed by rubidium and thallium, by strontium and lead, substances whose atomic volumes differ widely. The relations shown in fig. 3 hold most accurately for compounds and for the electronegative elements. The electropositive elements crystallize in the cubic or hexagonal systems. This was pointed out by Barlow and Pope, and used as a basis for the theory of close-packing in crystalline structures, since an assemblage of equal spheres packed together in the closest manner has either cubic or hexagonal symmetry. It is now known that the atoms of some metals are not arranged in a close-packed manner. Nevertheless, the idea of a metal as an assemblage of positive ions held together by electrons indicates a reason for the simple crystalline structure. Hach atom has the same relation to its neighbours, it is not bound in any way to one rather than another of them, and the assemblages will take a form like the arrangement of a set of equal spheres. The crystal of an electronegative element, on the other hand, where atoms are linked: by holding electrons in common, will have a more complicated structure, as is the * A.W. Hull, Phys. Rev., July 1917. 2 188 On the Arrangement of Atoms in Crystals. case for salphur, selenium, tellurium, iodine, arsenic, anti- mony, bismuth. : 15. In order to obtain a more complete knowledge of the distances between atoms which hold electrons in common, the examination of salts such as the nitrates, chlorates, bromates, sulphates, and selenates would be desirable. The investigation of these salts presents some difficulty, since their crystalline forms are complex. The symmetry of the crystal is of less assistance in determining the arrangement of the atoms than it is for the simple crystals, as it is of a much lower type. It is hoped that the empirical relations formu- lated in this paper will help in this investigation. The conception of the atoms as a set of spheres of appropriate diameters packed tightly together limits the number of possible arrangements and aids in deciding the correct disposition of the atoms. The scheme may be of assistance in analysing the structure of crystals such as quartz*, sulphur *, and the alkaline sulphates+, crystals for which the dimensions of the lattice are known, but which have so far proved too complicated for complete analysis. Summary. 1. An examination of the distances between neighbouring atoms in a crystal leads to an empirical relation determining these distances. The distance between the centres of two atoms may be expressed as the sum of two constants charac- teristic of the atoms. ‘The arrangement of the atoms in a crystalline structure may therefore be pictured as that of an assemblage of spheres of appropriate diameters, each sphere being held in place by contact with its neighbonrs. “This empirical law is summarized by the curve of we a where the constants for a number of elements, arranged in the order of their atomic numbers, are plotted. The curve 1s periodic and resembles Lothar Meyer’s curve of Atomic Volumes. Each atom occupies a constant space in any crystalline structure of which it forms part. The space occupied by the alkaline metals and alkaline earths is greatest, that occupied by the electronegative elements least. 3. The accuracy of the relation is discussed. Variations of the order of 10 per cent. between the calculated and observed distances occur, so that the law is only approx1- mately true. Nevertheless, it is of considerable assistance * W.H. Bragg, Proc, Roy. Soe. A. vol. lxxxix. (Jan. 1914). + Ogg and Hopwood, Phil. Mag. [6] xxxii. p. 518 (1916). Dissociation of Iodine Vapour and tts Fluorescence. 189 in the analysis of the more complex crystal structures, since the conception of the atoms as an assemblage of spheres of known diameters packed tightly together limits the number of possible arrangements which have to be tried in inter- preting the diffraction of X rays by the crystal. 4. The physical significance of the relation is examined with reference to Lanemuir’ s theory of atomic structure. From this point of view, it follows that two electronegative atoms are situated close together in a crystalline strueture because they share electrons, and the spheres representing them are therefore assigned small diameters. On the other hand, an electropositive element does not share the electrons in its outer shell with the neighbouring atoms, and is therefore situated at a distance from other atoms so that it oT to occupy a greater space in the structure. It is shown that the relation is less accurate when Mapiied to the crystals of metals, which, on Langmuir’s theory, consist of an assemblage of positive ions held together by electrons which have no fixed positions in the structure. 6. From the distance between electronegative atoms holding electrons in common, an estimate 1s made of the diameter of the outer electron shell in the inert gases. Manchester University, April 1920. XIX. The Dissociation of Iodine Vapour and its Fluor- escence. By Sr. Lanpav, B.Sce., Lecturer in Physics at the Governmental Technical Scllook Warsaw, and Ep. SteEnz* I. The aim of this work. | ee researches of R. W. Wood on the fluorescence of the vapours of sodium, mercury, and iodine are generally known; he discovered the remarkable phenomenon of optical resonance in these vapours. The most complicated relations were found by Wood + in the case of iodine vapour ; the number of lines in the absorption spectrum of iodine is estimated by Wood to be 40,000-50,000. Different “ reso- nance spectra”? may be obtained, when the exciting line covers different absorption lines. We put the following question: Is the complicated vi- brating system, which corresponds to these various resonance * Communicated by the Authors. Presented by Prof. L. Natanson to the Polish Academy of Sciences (Cracow) 18th Noy. 1919. + Phil. Mag. March 1918, p. 236. 190 Messrs. St: Landau and Ed. Stenz on the spectra, inherent to the atom or tothe molecule? In this last case we should suppose that the vibrating mechanism lies in the bond of the atoms. The experiments described below show that the atom of iodine is incapable of fluorescence, at least in the visible part of the spectrum. It seems interesting to notice that the result obtained is in accordance with the work of Dunoyer* on the fluorescence of sodium vapour. This author showed that the fluorescence of pure sodium vapour is reduced to the D-lines; he adinits that the complicated fluorescence spectrum hitherto observed with sodium vapour is due to the aggregates of molecules of sodium with the molecules of the impurities, perhaps hydrocarbons. Thus in the case of iodine and sodium, as well as in the case of mercury, one would think that the structure of the atom is not of an excessive complication. It was impossible to foresee the results obtained, as the absorption spectrum of the monatomic iodine is yet unknown. Konen f{, who did the most complete work on the spectra of iodine, states that he only observed that the absorption spectrum of iodine becomes less marked with increasing temperature, but he never remarked its complete vanishing ; he also never observed a separate absorption spectrum belonging to the dissociated iodine vapour (J. ¢. p. 259). The plan of our work was very simple. We raised the temperature of iodine vapour at a known low pressure and observed its fluorescence, the degree of dissociation being calculated. Three factors come then into account: tempe- rature, pressure, and dissociation. ‘The pressure remained at a constant value in the majority of our experiments. It was important to study separately the effect of the temperature and the effect of the dissociation. The separation of the two factors can be only partially done, as will be seen in the following. Il. The effect of the temperature; observations below 400° C. R. W. Wood and W. P. Speast have made a photometrical study on the fluorescence of saturated iodine vapour between —30° and +75° C. In that research it was found that the maximum of the fluorescence corresponds to 20°=25° C. With higher temperatures the fluorescence decays rapidly ; at 75° C. the fluorescence is quite invisible. It is clear that in these experiments it was vapour density that played * D. Dunoyer, Le Radium, 1912, p. 177. + Annalen der Physik, vol. lxv. p. 257 (1898). { Phil. Mag. vol. xxvii. p. 531 (1914). Dissociation of Iodine Vapour and its Fluorescence. 191 the most prominent role. The light cannot penetrate dense layers of vapour ; the light of the fluorescence is also absorbed on its way to the eye of the observer. When our experiments were terminated, we learned that W. H. Westphal* had also observed below 400° C. the effect of the temperature on the fluorescence of iodine vapour, kept at constant pressure. Butas that author has accomplished only few and rough measurements (he attributes a certain sense only to the mean value), we give herea short description ef our experiments, which seem to us more complete and more conclusive. In all our experiments we used iodine carefully purified by distillation in vacuo. The apparatus joined to the pump is shown at fig. 1. Purified iodine crystals were put in part (; Hig. 1. V LT. G the Gaede mercury pump working, the whole apparatus was strongly heated during 1-2 hours, excepting the lower part of C, containing iodine crystals; the U-tube immersed in solid CQO, served to exclude mercury vapour from the apparatus and iodine vapour from the pump; the apparatus was then sealed at M. During 24 hours some of the iodine sublimated from A to B; then the apparatus was closed at N. The bulb A, which served tor observations, was from Jena ylass (called ‘ durobax’’—with red stripe), as ordinary glass was found to be attacked by iodine vapour at high temperatures. I'he photometer we used was the same as is described by Wood and Speas (J. c.) with little improvements. Through * Verhandlungen d. Deutschen Physik. Ges. 1914, p. 829. During the war (and even now) it was impossible to receive regularly scientific journals at Warsaw; the authors excuse themselves for having perhaps omitted some papers concerning the question treated. 192 Messrs. St. Landau and Ed. Stenz on the the nicols the eye looked at a surface of paper, which was painted with a colour resembling closely the hue of fluores- cence (p, fig. 2). This screen was illuminated by light, Fig. 2. iS 4 oN reflected by a glass mirror, coming from the source, which was a 900 watts incandescent lamp. A little fragment of Dissociation of Iodine Vapour and its Fluorescence. 193 galvanometer-mirror m occupied the centre of the field of vision. Thus we realized something like the Lummer- Brodhun photometer. The experiments were also performed using a mercury- quartz lamp (Westinghouse, Cooper-Hewitt); in this case the screen, painted in orange-red, was illuminated separately by a little incandescent lamp. The oven for the heating was an electrical one; in the case of the mercury lamp the observations were made “‘end”’ on, and a gas-furnace supplied with suitable cuttings for the passage of light was adapted. he pressure of the iodine vapour was kept constant by placing the tube B (fig. 1) in a water-bath of the temperature GE 20°C. The results obtained were generally the same, whether we used the incandescent or the mercury-lamp, and whether the vapour was saturated or unsaturated. A typical curve is shown in fig. 3: in this case, which corresponds to the mercury light excitation, saturated iodine vapour at a pres- sure of 0°25 mm. was used. In this figure the observations taken with rising temperature are marked by points; the observations obtained when lowering the temperature by crosses. Fig. 3. ‘SI - 100° 200° g00° C It will be seen that the intensity of the fluorescence dimin- ishes with rising temperature, but the diminution is not as great as might be supposed. At 360° C. the fluorescence is still very intense*. | * We findin the Fortschritte d, Physik, ii. p. 385 (1915) an abstract from a paper of McLennan (Proc. Royal Society of London, A. xci. pp. 23-26 (1919)). Itis said there that the author has found a dis- appearance of the resonance radiation at the temperature of 826°C. It will be seen that our observations are opposite to that, as we did observe the resonance radiation very clearly even above 700° C. It was im- possible for us to obtain the original paper of McLennan (see footnote p. 191). ? 194 Messrs. St. Landau and Ed. Stenz on the III. Observations above 400° C.; influence of dissociation. The dissociation of iodine vapour was the subject of a very | thorough research of Bodenstein and Starck*. Denoting by K a constant, by pr and pr, the pressures of the mono- and diatomic iodine, by T the absolute temperature, they obtained 2 4 =K and log K==— = + 1:75 log T— 0:000416 T+ 0°422. Ty That formula, based on the thermodynamical theorem of Nernst, is in good accord with the experiments. We applied that formula in our calculations of the degree of dissociation of the iodine vapour. The vapour we had to do with was kept at a pressure of 0:25 mm. mercury. Theformula above permits us to calculate K for different temperatures, and then to find the ratio # of the number of dissociated molecules to the total number of molecules which would be present were there not any dissociation. Temperature .\.t =900° Ci. 600%; 70023, S007; aso ee aa ee 036; 0-76: 0:95; "Oem dissoc. mol. The main difficulty of such observations is due to the fact that every furnace heated above 500° C. emits so much light that the weak light of fluorescence vanishes. We remarked that a gas flame, when sufficiently supplied with air, is pale enough to permit observations to be made, the bulb with the vapour being in the flame. A great Méker furnace was fed with the gas exhausted from the gas pipes by a Gaede rotatory oil-pump ; the gas passed through a vessel of 20-30 litres capacity to reduce oscillations of pressure. The lower part of the burner was supplied with a ring to regulate the quantity of air drawn into the burner. The flame had a length of 25 cm. and a diameter of 4 cm., although the gas pressure in the pipes was very low. It may be mentioned that such a flame was found to be useful in many cases for the glass-work ; it is longer and hotter than the blowpipe- flame. The quartz apparatus we employed was similar to that shown in fig. 1; the part corresponding to A was a sphere of 34 cm. in diameter, the part B was likewise of quartz, the * Zeitschrift f. Llektrochemie, vol. xvi. p. 961 (1910). Dissociation of Iodine Vapour and its Fluorescence. 195 other parts were of glass. Glass and quartz were joined together by chalk sealing-wax; a compact layer of clean asbestos interposed between the quartz and glass tubes prevented the iodine vapour from coming in contact with the sealing-wax. The iodine was introduced in the apparatus in the way described above. In an experiment the tube with iodine crystals was held in a water- or air-bath of suitable temperature. An intense light- -beam was directed onthe bulb, and then the bulb heated in the flame of the Méker burner. The fluorescence disappears yradually ; the bulb seems optically sau After removing che flame, the fluorescence reappears. The phenomenon was observed as well with carbon-are as with mercury-are light, and employing iodine vapour of different density. The temperature of the flame exceeds 1000° C.: using iodine vapour of the pressure of + mm., we had complete dis- sociation. It seemed to us to be of interest to determine the tempe- rature of extinction of the fluorescence. For that purpose we employed a large iron pipe of 70 cm. length and 8 cm. diameter. That pipe was vertically set; the large flame of the Méker burner was introduced into the lower part of the pipe. The pipe was covered by a piece of asbestos provided with an opening, the size of which could be changed. In that manner the draught and also the temperature in the different parts of the iron pipe were regulated. The walls of the pipe scarcely emitted any light, although the tempe- ‘ature near the axis in the middle part of the pipe reached 800-900° C. The middle part of the pipe was protected externally by a thick layer of asbestos. Four openings were made, two of a diameter of 9 mm. for the passage of the hight; a large window of a diameter of 36 mm. served for the observation of the fluorescence, it was placed somewhat above the openings mentioned; a fourth little opening per- mitted the introduction of a thermocouple platinum—platino- rhodium. The three larger windows were covered with thin mica sheets. The joint of the thermocouple was placed very near to that place on the quartz bulb where the light came out. The lateral tube containing iodine crystals was kept at a constant temperature of 20° C, by a stream of water. Using the light emitted by a little carbon arc-lamp we were able to state, that the fluorescence is very distinct still at a temperature of 700° (.; at 780° C., when the fraction of dissociated molecules was equal to about 0°9, we were able to distinguish a faint glimmer of fluorescence. 196 Dissociation of Iodine Vapour and its Fluorescence. Thus dissociation and disappearance of the fluorescence are parailel to each other. One more experiment may be wa altionted, which seems to Eonfrm our ides.) The amentnratedl vedine. vapour, heated while the volume remains constant, is less dissociated at a given temperature than the saturated vapour kept at a constant pressure. This is obvious without any calculation, as the raising of the pressure causes a diminution of the dissociation. We isolated the quartz bulb containing iodine vapour from the tube with the crystals. A quartz rod was sealed to the bulb and the last placed in the iron pipe. In that case we observed the fluorescence at 825° C. To be quite conclusive the experiment should be arranged asa differential one. We were impeded by material diffi- culties from accomplishing this, but we intend to do it next time. The calculation shows that the experiment is a hopeful one, as is shown in the table below containing the fractions of dissociated molecules in the case of a vapour kept ata constant pressure of + mm. mercury, and in the ease of vapour kept at constant volume (quartz vessel), having originally (20°) a pressure of + mm. mercury. Temperature..... 500°C. 600° 700° 800° 900° Fraction z of diss. mol. at const. press. O-1 O36 076 095 0:99 OO” » fe volume 0:06 0:20 047 O76 0:90 LV. Conclusions. A. The raising of temperature does not produce as much effect on the fluorescence of the iodine vapour as has hitherto been thought. We have observed the fluorescence even above 800° C. B. Dissociation destroys the fluorescence and the resonance spectra. Thus the complicated vibrating system, corresponding to the thousands of absorption lines in the visible part of the spectrum, is not inherent in the atom, but in the molecule. The structure of the atom should be relatively simple. That idea based on our observations is perfectly in accord with other facts from the domain of the fluorescence of vapours. The monatomic mercury vapour gives a simple resonance ; the complicated fluorescence spectrum of the sodium vapour, as shown by Dunoyer, is due to impurities; pure sodium gives D-lines only. It seems also nearly certain that the absorption lines, which are so characteristic for the diatomic iodine and so sensitive On the Equation of State. 197 to the action of monochromatic light, do not belong to the absorption spectrum of monatomic iodine. Prof. Joseph Wierusz Kowalski has lent us some apparatus and assisted us ina most generous manner; the same did our colleagues from the different physical laboratories of Warsaw. It may be permitted to express here to those gentlemen our best thanks. A great part of the expense was covered by a subvention of the Mianowski foundation. Warsaw, State Technical School, founded by H. Wawelberg and 8. Rotwand. January 1920. XX. On the Equation of State. By M. P. Apriupry, IA., Fellow of St. John’s College, Oujurd, and D. L. CHapman, MA., PRS. Fellow of Jesus College, Oxford *. T has long been known that Clerk Maxweil’s law of the equipartition of energy between the degrees of freedom cannot hold for the molecules of gases. The specific heats of solids at low temperatures prove that the translational kinetic energy of the atoms ig not subject to the law, and even the energy of free translation of the atoms of helium has been shown to fall below the theoretical value 3/2R at high pressures and low temperatures Tt: We are therefore no longer justified in assuming that the mean kinetic energy of translation of a molecule is proportional to the temperature. Apparently the dynamical definition of temperature must be abandoned. The best known equation of state is that of Van der Waals. It does not accurately represent the behaviour of gases ; but this was scarcely to be expected, since in its deduction it was assumed that the volume occupied by the molecules is small in comparison with the volume of the gas,and the quantities “qa” and “)” were taken as constant, whereas it is probable that these magnitudes vary with the temperature and density of the gas. On theoretical grounds, however, it seems to us that fewer objections can be urged against Van der Waals’ equation than might be raised against other equations of state which are in much closer accord with the facts. Van der Waals in deriving his equation assumed that the * Communicated by the Authors. + Eucken, Ber. Deutsch phys. Ges. xviii. p, 4 (1916), 198 Messrs. M. P. Applebey and D, L. Chapman on mean kinetic energy of translation of a molecule was propor- tional to the temperature. Strictly this is not true. We are only justified in concluding that (p+ 5) (eb) = gnmv? ae where the letters have their usual significance. In other words, the equation relates the pressure and volume of the gas with the total kinetic energy of translation of the mole- cules, but not with the temperature. If any relation is to be established between the pressure, volume, and thermodynamic temperature of a gas, some assumption will have to be made in order to replace that which is no longer strictly valid. We assume that in a clased space which contains a large number of like molecules the ratio of the number of molecules per unit volume whose potential energy is A to the number of molecules per unit volume whose potential energy ts zero is given A by the expression e ™, k being the gas constant for a single molecule, and t the thermodynamic temperature. It will be observed that the distinction between this assumption and the corresponding proposition deduced from Maxwell’s laws of the partition and distribution of energy, is that ¢ is substituted in the assumption for 3 of the mean kinetic energy of translation in the corresponding proposition. Starting from this assumption, a modified form of Van der Waals’ equation can be deduced in the following way. Consider a column of gas in a field of force—let us say gravity. Take the axis of « in the direction of the force. The density of the gas will increase in the direction of increasing w. The increasing density of the gas will set up te) a field or cohesive force acting in the same direction as c : : dp ‘ gravity and equal in magnitude to a where p is the density of the gas and a a constant. Let m be the mass of a molecule of the gas. The sum of the forces due to gravity and cohesion, acting in the direction of #, on dx of gravity. From this must be subtracted the upthrust due to the fluid displaced by the molecule. Let the volume of fluid displaced be vepresented by (604, Then the total downward force acting on the molecule be- this molecule will be at gee ay g being the acceleration comes : (m—b'p) (gra?) If the molecule is displaced the Equation of State. 199 through a distance dw the increase of potential energy is d —(m —0'p)(9+ aif) du or—(m—b'p)(gde+adp). Whence, introducing the assumption which relates the temperature, the potential energy, and concentration of the molecules, we obtain ptdp bi a p 3 ean (m—b’p)(gdx+adp) pS kt Substituting dp for pgdw# and solving the equation, we find 4 ie ie re ie m—v'ip a 2 17 2 Ca If in this equation we put p=~, v being the volume occupied by a molecule, the equation reduces to kt v—b’ a®*® 1 Sag arm ’ = 2 : C (2) am? i ae Tf in the above equation (2} P is substituted for Pt where a= the total internal pressure, the relation expressed by the equation Pb! ht ——— = P(v—8’) ° . : ° ° (3) e kt —1 can easily be shown to hold. The similarity between the left-hand side of equation (3) and the expression deduced by Planck for the mean energy of a resonator at temperature ¢ will be immediately re- cognized, and the similarity suggests that there may be a close relation between the product Pd! and the quantum hyp. * If aand dare assumed to be constant the above equation has the defects of Van der Waals’ equation. For example, it leads to the relation Ree 1.0.6 Pee 200 On the Hquation of State. In this connexion it may be pointed out that when v is large in comparison with 0’, b/ is the volume of the sphere whose radius is the mean minimum distance between the centres of two molecules during a collision. Bearing this in mind and comparing equations (3) and (1), it will be seen pI that P(o=5) is two-thirds of the mean kinetic energy of a molecule, and therefore PU! kt PB’ 2mvV? kt Ppa + coal — > 2 0 A 5 e 5 (A) et —1 The left-hand side of (4) with Ay substituted for Pd’ is Planck’s modified expression for the mean energy of a resonator of period v. ) | The distinction which is drawn above between thermo- dynamic temperature and the kinetic energy of a degree of freedom we regard as a provisional hypothesis which needs to be more carefully examined. Norr.—The above communication is a portion of a paper written in January 1919. It has been kept with a view to testing its practical applicability. However, in the corre- spondence of the April number of this Journal, M. N. Shaha and S. N. Basu refer to a paper on the Equation of State published by them in 1918 which had escaped our notice. In this communication the authors deduce from the well- known Boltzmann’s entropy theory the equation ni Rey v—2b a ob 0g ec = which (since 2b=6') is identical with the characteristic equation deduced by us from apparently entirely different considerations. This agreement in the final results of two different modes of attacking the problem makes it desirable that the publication of our method of obtaining the above equation should not be further delayed. Jesus College and St. John’s College, Oxford. 0? aed XXI. Static Friction. —Il. By W. B. Harpy *. A. CHEMICAL CONSTITUTION AND THE LUBRICATION oF BISMUTH. el aig experiments are a continuation of earlier work on the static friction of glass faces (Phil. Mag. ser. 6, vol. xxxviil. p. 32 (1919)). A slider having a curved surface was applied to a plane surface, both slider and plate being of bismuth +. This metal was chosen because it is highly crystalline and at the same time takes a high degree of polish. It therefore offered unusual facilities for com- _ paring the friction of the amorphous state of the metal found on a burnished face { with that of the crystalline state. One of the faces being curved, contact was over an area defined by the weight and the elasticity of the material. The extent of this area is unknown, and the normal pres- sure over it not uniform. There are therefore unknown quantities which make it impossible to express the static friction in terms of a normal pressure. The observations of Burgess § show that nothing would be gained in this respect by employing two plane surfaces. The best ob- tainable surfaces touch only at points, and if of glass, show Newtonian colours. The tractive force was applied slowly, and in such a way as to make the slider rock forwards. The static friction therefore is that between freshly applied faces. The method is apt to give undue importance to viscosity, and care is needed to distinguish between transitory effects due to viscosity and true lubrication (see the earlier paper). Passing from glass to bismuth faces, one enters another world. Meticulous care perfected by practice is needed to secure clean glass faces, and, once secured, they readily contaminate by the spreading of a film of matter from solids with which they are in contact or by condensation from the air. The edges of a plate oppose the spreading of most if not all fluids, and it is this interesting property of edges which renders exact work with glass possible. Bismuth, on the other hand, can be cleaned by simply rubbing the face with wash-leather, and the clean plate * Communicated by the Author. + Iam indebted to my friend Mr. Heycock for a specimen. t Beilby, Proc. Roy. Soc. Ixxii., p. 218 (1993). § Proc. Roy. Soc. A. lxxxvi., p. 25 (1911). Phil. Mag. 8. 6. Vol. 40. No. 236. Aug. 1920. gE 202 Mr. W. B. Hardy on may be handled freely if actual contact with the burnished face is avoided. To clean the faces the plate and slider were washed in ethyl alcohol 98 per cent., drained, and burnished with wash-leather. Wash-leather clings to a clean face of bismuth. When the surface is rubbed with it, there is a characteristic harsh feel and characteristic notes of high pitch are given out if the surface be clean. Harely a lubricant is found which will not be displaced from the surfaces in this simple way: it is then necessary to clean them with rouge. What is the test of cleanliness? There is no criterion other than static friction itself. When static friction is maximal it is assumed that the surfaces are clean, or, to put it more exactly, ‘‘ clean” faces are defined as those whose static friction reaches a certain level such that a force of 34,300 dynes just fails to produce movement in the slider used throughout, whose weight was 70°5 grammes and radius of curvature 25°5 millimetres. This assumption is justified by the fact that every substance tried was found to reduce this datum value when applied to the surface. In this bismuth differs from glass. Many substances are neutral to glass in that they do not alter the static friction. No substance neutral to bismuth was found. This section, with the limitations noticed in paragraph 2, is confined to a study of one variable—namely, the chemical constitution of the lubricant. One point should, however, be mentioned and reserved for future discussion. A few lubricants appear to abolish the static friction of bismuth altogether (e.g. ricinolic acid). In these cases the value given is that at which the tractive force produced sliding so slow as to be just detectable by unaided vision. In testing a fluid the surfaces were flooded so that the slider moved in a pool. The thickness of the film of lubricant then was that determined naturally by capillarity acting in opposition to the normal pressure. Solid lubricants were deposited ina layer, thick enough to dull the burnished surfaces, from dilute solutions in ethyl alcohol, benzene, or ether. There isa danger lest the friction of bismuth against the solid in mass be mistaken for what is sought for, namely, the friction of bismuth against bismuth lubricated by the solid. The former obviously is an important limiting value when the lubricant is a solid. Whether it is the only value and identical with the latter will be considered on some future occasion. In the meantime it may be noted here that the value of the friction of bismuth lubricated with Static Friction. 203 cholesterol was the same for an obvious smear of the solid and for the invisible film left when the smear had been wiped away by cotton-wool moistened with alcohol. On the other hand, the value of the coefficient for a visible layer of stearic acid deposited from benzene or ether was ‘20 and for an invisible film +15. I incline to the view that the difference is due to the difficulty of getting rid of the last traces of the solvent from some substances. The force which just fails to move the slider ammediately it ts applied is taken as the measure of the friction. Precision in this matter is needed, because static friction is sometimes a function of the time during which the external force has acted. The significance of this time factor is not yet clear. Sold Bismuth.—Surface burnished: some of the fluids used, however, etched the surface so as to expose the crystals: such are the acids formic, acetic, propionic, and valeric, and the sulphur compounds thiophenol and benzy]- hydrosulphide. Temperature 11-14°C. Measurements made in a current of dry air. Weight of slider 70°5 grammes ; radius of curvature 25°5 millimetres. The results given in the column headed Static Friction are the values of the ratio applied force in grammes divided by the weight in grammes of the slider. Static friction is a function of the molecular weight of the lubricant; and in a simple chemical series of chain compounds such as fatty acids and alcohols or paraffins a good lubricant will be found if one goes high enough in the series. But it is not a simple function, as “inspection of the charts and curves shows. ‘The friction, for instance, rises sharply in moving from CHCl, to CCl, and from phenol to catechol and quinol. The influence of molecular weight is overshadowed by the influence of chemical constitution. In some simple chemical series the relation appears to be a linear one. Hxamples are paraffins ; the series benzene, naphthalene, anthracene ; and, making allowance for the fact that the ammonia was a solution in water, the series ending with propylamine. In the aliphatic alcohols and acids the chain is weighted at one end with the ‘OH or :COOH group, and the simple linear relation to molecular weight is disturbed thereby. The relation of lubricating qualities to viscosity broadly resembles that to molecular weight. In a simple chemical series lubrication and viscosity * change in much the same | = 204 Mr. W. B. Hardy on way with molecular weight, but that there is no funda- mental relation between viscosity and Jubrication is shown by the following figures :— Viscosity at 20°. Static friction. Carbon tetrachloride ...... "0096 43 Chloroform: s.co.ces eee "0056 "30 Acetic acidic. ecunccaccucee’ "0122 “40 Octyllic acid assests °0575 19 Berizene {2.328 kero eee "0065 at) Toluene. sic. Sere eee ne "0058 "28 Benzyl alcohol............... 0558 “Bil o!3 C;4,(0H), CoO EOne Ts C,,H,COOH 12 C.H,(0h)_,° 4 ° C,H, (Coon) . ae TPS C,H,COOH C,oH,0H © 20 CH O@ ™ @ @ 10 C.HCH.O ne C_H.C He o@ p Cymene Cehace ee Nes bers ncutch c.cIoToA a 6's°2"'5° eC H_CH.OH © C,H, CH,6 jerelaanel>n ie F 4 a SAS “CH°CH: H.-CH,CH,-COOC H CeH_CH,e ICH (CH)> e!2C.H,.CH,OH = PC OH, GANS) CoH CHACH CCOCSHEie @ @ C.H_OH USGS) el sree a i THYymoLe® MENTHOL Cato 6S ©c_H_CH c.HsH C,H.CH,SH. ry i CARVACROL ! OH Cen, RING COMPOUNDS "70 80 30 100 110 120 130 140 150 160 170 180 e ACETONE di CARBOXYLIC ALCOHOL Ne — dj ETHYL ESTER GLYCEROL ce LACTIC ACID ORDINATES STATIC FRICTION ABSCISSA MOLECULAR WT, "30 40 50 60 70 80 90 100 NO 120 130 140 150 160 170 180 190.200 z2lo 220 230 240 Fluidity of the lubricant has no constant significance, as indeed might be expected on the surface-energy theory of lubrication. The curves for acids, aleohols, and paraffins show no break where with Increasing molecular weight the lubricant becomes a solid at the temperature of observation. Compare also benzene, naphthalene, and anthracene ; menthone and menthol ; thymol and carvacrol, Static Friction. 205 The upward trend of the first part of the curve for the aliphatic alcohols is in agreement with the fact that methyl alcohol is abnormal in some of its physical properties such as specific gravity. In their qualities as lubricants of bismuth, ring compounds are the converse of chain compounds: thus the effect of a double-bonded atom is to decrease the lubricating value of the former and to increase that of the latter. As examples, compare naphthoic acid with double-bonded oxygen, with naphthalene, menthone with menthol, cyclohexanone with cyclohexane, benzoic acid with benzene. As examples of double-bonded carbon compare cinnamic ester with hydro- cinnamic ester, dipentene, having two unsaturated carbon atoms, with menthol and cyclohexane. On the other hand, the presence of unsaturated atoms increases the lubricating qualities of chain compounds, whether it be the double- bonded oxygen of ketones or acids, or carbon of olefines and alcohols ; but this rule is departed from (in the case of acids) when the chain becomes much elongated. Whatever view be taken of the structure of the benzene ring, it must be admitted to be less saturated than cyclo- hexane, and we find consistently that the more saturated cyclohexane and its derivative are the better lubricants. When ring and chain are joined as in butylxylene, the result is a better lubricant than cither, When the atoms are disposed with complete symmetry about a carbon atom the result is a very bad lubricant, as we see in carbon tetrachloride and the alcohol pent- erythritol C(CH,OH),. In the ring compounds the replacement of hydrogen decreases lubricating power in the case of N, : O, or -COOH, an] increases it in the case of other groups in the order O,H;< CH; < OH. The effect of a second group of the same or of a different kind is to decrease the effect of the first. Compare for instance toluene with xylene ; catechol, quinol, and cresol with phenol; and methyl cyclohexanol with cyclohexanol. The simpler the group the more effective it is. Compare cymene with toluene or xylene; and benzyl alcohol with phenol. The esters occupy a quite unexpected position. The simple aliphatic esters are much worse lubricants than their related acids or aleohols. On the contrary, the ring esters are better lubricants than are their related acids (e. g., ethyl benzoate and benzoic acid). Perhaps the most interesting substances are the hydroxy 206 Mr. W. B. Hardy on acids with OH and COOH groups. This conjunction pro- duces a remarkable increase in the lubricating power of a chain compound (a. lactic acid and ricinolic acid), and almost destroys lubricating action in the case of the ring compounds (salicylic and benzylic acids). It will be noticed that no ring compound is a good lubricant. Liven cholesterol with the molecular weight 366 is no exception. The group SH acts much as OH, thiophenol C,H;0H and benzylhydrosulphide OgH;. CH,SH resembling phenol and benzy! alcohol respectively. It need hardly be stated that these conclusions are pre- sented not as generalizations, but as a summary of the relations which actually obtain amongst the limited number of compounds studied. The fact that the influence of chemical constitution differs widely in degree if not in kind when glass is lubricated would alone enforce caution. Mixtures of. lubricants were not specially studied, but it is certain that they will reveal complex relations of great practical importance. In the experiments solids were deposited in thin Jayers on the plate from very weak solutions in ether, ethyl aleohol, or benzene. In most cases the change in friction was followed whilst the solutions were drying and nothing abnormal noted, the friction changing rapidly to the value for the solid in the last stage of drying, and in such a way as to suggest that the combined eftect of the two substances was merely additive. In some cases, however, e.g. phenol and cetyl alcohol, abnormal values were noted. Just before drying was complete the triction fell to a comparatively low level (‘07 cetyl] alcohol, :15 phenol). The values for paraffins can be considered only as approximate. Octane was the purest specimen employed. The solid paraffins were identified by their melting-points and were not wholly free from unsaturated substances, but the displacement of the observed frictions due to this may be considered as being small. The lower members of the fatty acid series etched the surfaces so as to expose the crystals, the action being slow and slight in the case of valeric acid. Capryllie acid did not etch the surface. This action no doubt contributed to the bending downwards of the first part of the curve. I am indebted to Professor Lapworth for examples of cyclic compounds, to Sir George Beilby, Mr. Dootson, and Dr. Ida Maclean for various specimens, and especially to Sir William Pope for permission to raid his large collection of substances. Statice Friction. 207 Faces clean. Static friction °5. CHAIN CoMPOUNDS. Alcohols. Statice friction. Static friction. 1100 AEA Se "29 Tsopropylt csc dsc saresk eee "32 2b ee ene "32 Tsobatyls Hteeee.e. <.: “30 J Ee Cee "34 oA Tg ch caninuentetow ae shs< “20, LS ie aes 30 GV CON eeiecs sscuasmataenssn 30 IR ee a Cas ene ctins a2 “27 Gilycordl 0.5 5.5 af bani Ae Pate. Sas LG dathdae “25 Penterythritol” ..; anes “40 sith. ci ccieainn Li Acids. DORI ais cin 5 wn a nen vale 45 SECRIIG seinw cane osebice tee on "15 CE Oh ae eee “40 O12 (Re te Gone Se SS 10 ISPOPEGING: vc ccncnenndsses "Ol IRICHMONIG acento at anes soe "02 (EU ci (eed aes es "28 ce: Toate hic eta Se el Bee ‘20 Capryllic, fluid ......... 19 Gly. ceriel Vy 2. da. "22 frozen on plate ‘05 PEOHOME 5 ce. see ohn one 32 [oe RGM OLRER: ec euecs son oe. 33 Methyl ethyl ketone ... ‘29 | Ethyl acetate ............ “36 BURY Sparatiin: eax! “20 = valerianate——... “OD. Solid paraffin, m.p. 30°5 ‘09 BERROAMITE, 2... vcness 22: 24 a mae, 46 ‘07 Le CGT) eee ae “14 Carbon tetrachloride... *43 Acetone di carboxylic di Chloroform) 25 .2.- eect “30 BiMyL ester «....... «02. "29 AGU IOTIO. c.cieenes a6 so" ‘26 Mee RMON oe. waa os “37 I = *@Octylene i; ss .72:0e: tere ce “28 ge Ttoptane. ..02..0.. 2220.) 346 ee Wchanehyatacs2s: cake. “32 '.. Batyl xylene | .2.:csai 27 Rivne ComPounps. Static friction. Static friction. Pomme tLe ees "34 Ethyl benzene ......... "32 Todo benzene ............ "30 Rintophienol .2/d25.5440254% "22 ERGEUC: «es ws sun Boo asnigee "28 Benzylhydrosulphide... — °28 5 EL ee TS eel “30 3 oa “Ol PYGIGUIC eadade ctiboncesane 33 PEL IGM’ caer. scenes "32 ee era “25 OU 1 i soe gee Naphthalene ............ “29 2 LOIEG A RS StS ee a “40 Anthracene ..... ........- 26 ue Ue (0) ee ee ee 26 GU Nam tolls? dello deei=t « "33 Benzyl] alcohol. .......... “31 Naphthoic acid ......... ‘39 Benzoic acid ............ ‘B38 CARY ACTO ys. c wan ub oes a0 "23 Phthalic acid ............ 37 ity mal) i ddses esa. "24 Cinnamic acid ......... Hf TOMO cs daaes gdmecnee «x "26 Benzilic acid. ...........- ‘45 di PenteMfe ............... “31 Salicylic acid ..:......... “41 Cannio 2186 550.226i508 24 Ethyl benzoate ......... 3 3] Active ethyl ester of o-Phthalic ester ...... “27 Camphor oxime ...... 33 Ethyl hydrocinnamate. 28 Iso-Cholesterol ......... “a7 Ethyl cinnamate......... 32 208 Mr. W. B. Hardy on Cycric CoMPpouNDS. Cyclohexane ............ “31 Methyl cyclohexane ... 30 1.3. di Methyl cyclohexane °29 Cyclohexanone ............ 79) | 1.2. Methyl cyclohexanone ‘32 Cyclowexanol |. <.. 2. :.20.- 20 Leos aes: a 519) 1.2. Methyl cyclohexanol ‘28 lee ee a "33 1 5 #5 "25 Ammonia fortiss. ...... "O4+ 1) Castor: Onl ssc sscreeeneams 03 Triethylamine............ °30 Tripropylamine ......... "26 Water 2.2 .scsetisumesuanan "33 B. INFLUENCE OF THE BEILBY FILM. In 1903 Beilby described how, in the process of burnishing, or polishing, the substance of the solid actually flows so as to cover the surface with a film of amorphous material. The formation of the Beilby film can be readily followed on bismuth, and in order to test its influence some measurements of friction were made after it had been etched off by dilute acid so as thoroughly to expose the coarse crystalline structure of the metal. The following values were obtained :-— Burnished. Etched. Ratio. Benzeite: nae awe ere eee "34 39 87 Pyrideneé. 4-2 e Se eee 33 4 83 Hithylvalcolole ater eeeereeet "32 39 "82 Butylxyleney essere Tf o7 “i Octyltalcololteeseeeeee etter "25 "36 at Cyclohexanol .......... EAL Sane ‘20 33 6 The substances are arranged in the order of the value for static friction of burnished faces. The values for etched faces do not follow this order, whilst the ratios do. This may be merely coincidence, but it raises questions which must be reserved for discussion until more facts are available. C. ADSORPTION OF LUBRICANTS. The theory outlined in the first paper (this journal, ser. 6. vol. xxxvili. p. 32 (1919)) embodies two propositions: the first, that resistance to slipping is due to cohesion even when a lubricant is present, and that a Inbricant decreases friction by partially or wholly masking the cohesive forces of the solids; the second, that a lubricant maintains its position against the normal pressure because its surface energy is a function of the thickness of the layer. The capacity for decreasing friction, then, is a function of the potential of the attractive forces between lubricant and Statice Friction. 209 solid integrated through the depth of the layer, whilst the stability of a layer of a given thickness is a function of the diflerential coefficients of the interfacial energy taken with reference to the thickness. The integral is the work done by the cohesive forces acting between lubricant and solid when the layer of the former is interposed between the faces of the latter. This may be expressed in terms of a tension, and thereby become mea- surable, if one solid face alone is considered. Let ¢ be the work in ergs done per unit area in removing a layer from the surface of a mass of fluid, the layer being so thin that the fluid composing it is not in mass; ¢ will then also be the tension of the free layer. Now apply the layer to the solid face. The forces of attraction between the two will do work. Let this be ¢’ per unit area. The tension T of the composite surface so formed will then be T=i+4+T,-t, where J’, is the tension of the solid. For the difference between the tension of two composite surfaces formed on the same solid but with different fluids a and } we have T—l) = (ta— te) + (ty)'—t,'). If we assume that ¢, bears the same ratio to ¢; as do the tensions of the fluids in mass and choose for the purpose of experiment two fluids whose surface tensions are equal, the term (t,—¢,) will vanish, and the left-hand side of the equation be positive or negative according as the term (t,’—t,') is positive or negative. In an earlier paper (this journal, ser. 6, vol. xxxviil. p. 49 (1919) ) I described how films of insensible thickness form on a plate of glass about drops of certain fluids, and how the drops ace moved over the surface of the plate by the con- tractility of the films. This property may be utilized to measure the sign of the term (t,/—t'). Let one drop of each of two fluids a and 6 be placed on the plate: there will form about each a composite surface of tensions T. and T respectively. If T, is greater than T, the drop a will move away from drop 6 and the latter will pursue it and, if the surface tensions of the fluids are equal, such movement will show that the adhesion of the film of 6 to the surface is greater than that of a. Benzene and propionic acid are a pair of fluids whose surface tensions are practically equal, while the former 210 On Statice Friction. does not and the latter does lubricate the surface of. glass. A. drop of each was placed on a clean glass plate a few millimetres apart with striking results. The drop of acid chased the drop of benzene to the edge of the plate where, owing to the characteristic edge repulsion, the latter was split into two. We have, I think, in this observation direct evidence that the forces of attraction operate more strongly between a solid face and a good lubricant than between it and a bad lubricant. The better lubricant is more strongly adsorbed by the solid face. Olefines are better lubricants than paraffins, and one of the methods in use for freeing the latter from the former is by taking advantage of the fact that olefines are more strongly adsorbed by a solid. The impure paraffin, in commercial practice, is filtered through a dry powder to clear it of unsaturated subtances. Of the pair of fluids acetic acid—water, a drop of the latter pursues strongly a drop of the former on a plate. The surface tension of water is much higher than that of acetic acid, and, since water does not and acetic acid does lubricate glass, the decrement due to interaction with the solid face is, according to theory, less for water. Therefore, in respect of both terms of the right-hand member of the equation the advantage lies with the insensible film of water. In both of these pairs the result was the same whichever one of a pair was placed first on the plate. Another pair, benzene and acetic acid, gave uncertain results. According to theory benzene should move away and the acid follow, and this usually happened. Sometimes, however, when a drop of acid was placed on a plate on which a drop of benzene already stood, the latter darted away, as though the vapour of acetic acid had lowered the tension of the insensible film of benzene between the drops ; sometimes the drops simply moved away from each other. These complications are to be expected, for we have the tensions of three films to consider—those of the relatively pure films outside each drop, and that of the film of mixed origin between them. If the vapours condense in the last in such proportions as to produce a film of tension less than that of either pure film, the drops will be pulled apart. Reasons are given in the earlier paper for believing that an insensible film about a drop is formed always by condensation of vapour and not by direct spreading. Brand «| XXII. On the Specific *Heat of Saturated Vapours and Entropy- Temperature Diagrams of Fliuads, By ALFRED W. Porter, D.Sce., F.R.S* ‘iy a paper with the above title in the June number of the Philosophical Magazine (p. 633) Sir J. A. Ewing extends our detailed knowledge of the specific heats of saturated vapours, and at the same time corrects certain loose statements which have appeared at various times. I desire to point out that the question (as might be expected) is very thoroughly discussed by Duhem in his Traité Elémentaire de Méchanique Chimique, t. ii. p. 211, &e. Duhem shows that if the specific heat of the saturated vapour is plotted against temperature, its form is that of an inverted, unsymmetrical U ; that sometimes this lies wholly in the negative region ; but that it may lie higher up so as to cross into the positive region, and that if it does this it crosses twice. These are the two possibilities of which Sir J. A. Ewing gives detailed examples. He asserts, however, that positive values do not occur for sulphurous acid. ‘This statement itself appears to be in error. The case of sulphurous acid (SO,) has been directly studied by Mathias (Comptes Rendus, t. exix. p. 849 ; Ann. de la Faculté des Sciences de Toulouse, t. x. (1896), quoted by Duhem, loc. cit.). The importance of this substance in con- nexion with mechanical refrigeration warrants one in quoting his results in the following table :— Pore: K,. t° C., Ki (ae ee —0°410 LOOM, 2: +0:027 Hier & fore 390 ia ee +0062 PA, esccene ‘357 120° eee —0:078 Bee -330 TD Bee ‘176 Atay Fite 5. -300 130. PSs 506 FF | enti 270 ieee 452 WO, xt Arias "335 A ee, 620 7 BO). ageee( 8 ‘205 PAD Ab SG “848 BO nten: 165 1500 ee. 1:253 LR opaaes —0-095 Tin Seater —3:850 The temperatures of inversion are 97°5 C. and 114° C. In order to free the subject from all possible misunder- standing, I wish to add that the entropy of the liquid is not é C,dT ry y] yo I where C, is the specific heat at constant pressure, even with * Communicated by the Author. 212 Specific Heat of Saturated Vapours. the proviso that the value of C, taken must be that at saturation. The value C, must be replaced by K,, the specific heat of saturated liquid. The difference between C and Kis certainly negligible at temperatures sufficiently remote from the critical temperature, but it tends towards + infinity as the critical temperature is approached. The relation between the two quantities is 1 Ko —1($5,) dp w= U, Oly, al or K,, may be given conveniently in terms of C,, the specific heat of the liquid at constant volume ; Op T($h Riya st), ds w v e al’ Ou) +r where s is the specific volume of the liquid at saturation. The values of K,, obtained by Mathias for SO, are given herewith :— t° C. i AO” “°C. K,. Ax 10% Oy +40°315 20 1 110 4.0°442 EO. 316 32 1 1a... 470 0 317 40 2-5 130 510 10 3195 110 4-5 140 620 20 394 252 6 150. 872 30 330 480) 8 ae 920 40 338 600 9 152. 980 Bh 5G 347 900 12 TES 1-070 60 359 2850 13 | gaa 1-355 70 372 | Maren 500 15 eae 1-800 80 387 10500 : 16 Lelie. 2 85 ape 403 100/ee 499 «19 I have added the increase per degree, A, in order to show that K,, for SO, is not a linear function of the temperature even in a region remote from the critical region. P2834] XXIII. On Cohesion. By Hersert Cuattey, D.Sc. (Lond.), Harbour Investigation Office, Huangpu Conservancy Board, Shanghai, China*. is en objects of this note are as follow :— (1) To disprove the Kelvin theory of Newtonian cohesion. (2) To indicate dimensional relations of cohesion to gravitation and electric chemical affinity. (3) Tosuggest an empirical gravitation-cohesion formula. (4) To indicate cohesion values in certain hydrogels. (1) Kelvin’s theory of Newtonian cohesion t+ states that “Cohesion will be a necessary consequence of gravitational attraction provided only that the space occupied by the atoms of a material body is sufficiently small in comparison with its bulk” (de Tunzelman).- If two cubes in “ perfect contact” be considered, each can be imagined to consist of three sets of n straight bars, each set mutually perpendicular. The bars perpendicular to the interface each have a mass = of the original mass of either cube, and “however small may be the masses of two such bars, the attraction between them, per unit of sectional area, may be increased without limit by diminishing the sectional area of the two bars while keeping their masses constant. Now, the total attraction between the two groups (of bars) is greater than the sum of the attractions between the pairs, that is to say, greater than n times the attraction between any pair of conterminous bars. ‘The whole attraction between the two cubes may therefore be made to attain any value, however great, by sufficiently diminishing the sectional area of the bars while keeping their number and the mass of each constant”’ (author’s italics). As far as the writer is aware, this reasoning has not been definitely challenged, and has certainly been accepted by many physicists. Nevertheless, it appears not only uncon- vineing but incorrect. The area for which the attraction between two of the postulated conterminous bar elements is * Communicated by the Author. ; + Trans. Roy. Soc. Edin. Apr. 21, vol. iv. (1862); Pop. Lect. vol. 1.5 Dewar, Encyclopedia Britannica, 11th ed. Article, “ Liquid Gases” ; de Tunzelman, ‘Electrical Theory and the Problem of the Universe.’ 214 Dr. H. Chatley on Cohesion. effective is the whole section of the bar, and the actual size of the atoms, independent of the space they occupy, has nothing whatever to do with it. One can scarcely conceive how Kelvin overlooked the fact that the mass of the atom is a function of the spacing, so that in the molecular realm the gravitational stresses still remain minute. If we consider two 1l-centimetre cubes of a substance of density three and an atomic interval of one Angstrém unit, the mass of a bar of atoms of the type postulated is 107!® gram. The mutual attraction of two collinear bars of unit length end to end is ‘ , 1+6)? l= M, ° M, . log. STS | > where M,, M, are the masses per unit length and 6 the distance between the ends (Harnshaw, ‘ Dynamics,’ p. 327). In this case, when M,=M, and 1/6 is very large, we can write f= 23026 ME. logwo(55) Feo? andi Mi 10s, f=about 2 x 107?! dyne. If the number of bars, “n,” is (10°)’, then ¥',=nf=2 x 10~¥ dyne on a square centimetre. The additive effect of the various bars on those other than their own partners must also be considered. ven if we assume that each bar of the cube acts on all those of the second cube other than its own mate with the same force as on its own mate, then the total attraction, due to the longitudinal bars only, is but Fo 1) of mean atomic interval to that at absolute zero ; G, m, and d as in the Newtonian formula. This formula has three possible advantages :— (a) It is continuous with the Newtonian formula, 2. e. ICD p= 69) ea Cr |e | (b) It is dimensionally fairly correct for many cases of cohesion. (c) At absolute zero (i. e. atomic “ contact”) it practically agrees with chemical affinity. (d) Its space rate of decrease is not inconsistent with known strengths of materials. Near absolute zero the index of the distance is the inverse sixth. (e) It is polarized (possibly not correctly), since for all powers other than the inverse square the force is not centralized. [| Note.—The inter-atomic repulsion appears to be of the order LA coat al 17 Nay Gee where d)=atomic spacing at absolute zero; R, T, and N as in kinetic theory of gases. ] ; (4) The writer is especially interested in this subject in connexion with problems of accretion, erosion, and stability of river-mud, and has obtained certain indications of interest from conditions in the Huangpu River :— (a) Fine mud resists erosion until the hydrodynamic tangential drag on a surface particle equals about 10~* dyne, 2. e. each particle, containing some millions of molecules, is retained by at least one inter-molecular “cohesion ” bond. t The Adjustment of Observations. 217 As a result of this, fine mud beds are not easily eroded, whereas coarser sand is easily moved. (b) Colloidal suspension and plasticity in gels becomes of importance when the weight of the particles is about equal to one inter-molecular cohesion bond (e. g. with cubic particles, s. g. 2°5, each has a diameter 1 y/ pete | og d= (25 —1)981 * er L Onze) This agrees with the dimension usually assigned as a maximum to colloidal particles. ———— XXIV. The Adjustment of Observations. By R. MELDRUM STEWART. [: a recent article f on this subject, Dr. Norman Campbell describes a method of reduction of observations which is proposed to replace the “‘ method of least squares.” Among other statements of « similar tenor, it is claimed that the latter is ‘‘an intolerably cumbrous method tor obtaining quite misleading results,” and that the method proposed “ is incomparably simpler and gives results which are not mis- leading” ; and the intention is expressed of attempting to entirely subvert the method of least syuares. In the interests of mathematical accuracy, and to prevent the acceptance without due examination of misleading theory, it does not seem right that such statements as these, and such reasoning as occurs in parts of the above article, should be allowed to pass unchallenged. ‘The ‘‘ method of least squares” is based on two funda- mental postulates, neither of which appears to be questioned by Dr. Campbell: (1) that the probability of the occurrence of an error of any given magnitude is a function of that magnitude ; (2) that in the case of several observations to determine a single quantity, made under the same con- ditions, the arithmetic mean is the most probable value of the quantity sought : these postulates are the mathematical equivalent of the Gaussian law of errors. Whether or not this Jaw of errors, and therefore these two assumptions, be strictly true or not, is a question which it is not proposed to argue at present; the result of experience, however, shows the law to be at least a close approximation to the * Communicated by the Author. + Phil. Mag. February 1920, p. 177. Phil. Mag. 8. 6. Vol. 40. No. 236. Aug. 1920. Q 218 Mr. R. Meldrum Stewart on the truth. The point which it is desired to emphasize is that, if these two postulates be granted, the whole method of least squares, as properly stated and applied, follows inevitably by rigorous mathematical reasoning. It is quite true that in most, if not all, of the present-day text-books on the subject, the reasoning which purports to establish the principles of the method is anything but rigorous; and the method itself is at least in some in- stances grossly misapphed. There is perhaps no subject in the realm of mathematics where clear and careful thinking and rigorous reasoning are more necessary than in the theory of probability and its application to the theory of errors ; and yet it would be difficult to find a subject where con- fusion of thought and loose reasoning are more prevalent : among the most obvious examples of this might perhaps be mentioned the common confusion of the terms “‘ true value ” and “‘ most probable value,” “error” and “residual,” with the loose reasoning that inevitably arises from such confusion. But from the tact that loose reasoning is frequently used in attempts to deduce the principles of the method, it by no means follows that these principles have not been, or cannot be, established by rigorous reasoning. Most, if not all, of the fundamental principles, and many of the details, of the theory have been correctly developed by the classic authors on the subject, as may be readily verified by reference to the original memoirs. It seems worth while to emphasize this fact because, by most of those who have occasion to adjust observations, these classic authors are seldom read, and the prevalent ideas on the method of least squares are those garnered from the more or less faulty text-books of the present day. It remains to consider briefly the three types of problems mentioned by Dr. Campbell. The first type is the ordinary case of the arithmetic mean. It may be remarked in passing that the problem before us is not the determination of the “ true value,” even with the limitation™ imposed by Dr. Campbell. The true value must for ever remain unknown ; all that we can hope to attain is the ‘‘ most probable value,” and this is all that is, or can be, claimed for the arithmetic mean. Further—and here we arrive at the crux of the whole theory of errors—the adoption of the arithmetic mean as the most probable value is a pure assumption ; in fact, if the law of errors is any other than the Gaussian haw, the arithmetic mean is not the most * Loc, cit. p. 178, footnote. Adjustment of Observations, 219 probable value. It is therefore (as also on practical grounds) absurd to speak of the possibility of establishing it directly by experiment. The second type of problem is the familiar one of direct conditioned observations. Dr. Campbell makes the indefinite statement * that in one type of this problem the method of least squares is not applicable: the statement is supported by no reasoning, and his objections are not clear ; the point will not be discussed here, but it will appear from the discussion of the general case below that the allegation is incorrect. It is, however, to the problem of what he calls the third type that he pays more particular attention. As enunciated f, this is the perfectly general case of indirect observations ; as such, it includes not only the case of the arithmetic mean, but also those of both direct and indirect conditioned obser- vations: in fact, the first and second types are merely particular cases of the third ; the solution given {, however, and stated to be that of fiemethod of lense squares, 1s entirely incorrect. It may be worth while to consider this general problem in some detail, and to indicate the correct method of solution, more especially since the treatment in the text-books is less general, and does not appear to cover all cases Q. It is assumed that measurements lave been made on quantities whose true values are 2), yj, 21 ~~~. &o) Yo, Zo ++ 235 Y3, 23 --- 2N, YN, 2N---, and that these quantities are known to be related by the equations Ai (@y Yio 1 +o + A, De Cs 2) ==) : to (22, Y25 <9 ~-. A, b, Ges ) = 0, i F alee eee 2 1ECE) : * f | ; ty(an, YUN) -N--- G, b, C29 Ay = 0, 4) the forms of the functions fA, fo... fy being known, but not the values of the constants a,b,c.... ‘It is required to determine the most probable values of GOs Gg bey BIE number of which we shall suppose to be qg(q. = do, be | EyAty + ynAyng..- taxyAat+ ByAb+...= dn. | Substituting the same values of 7, y,, 2, etc. in equa- tions (2) they become Az, = 2y' —X, Ay; = Yi. —Y, etc., } At, = # —X, Ay = Yo —YXo » | small iii 431 xinaldon, TaLayeatle, othe (4) Avy = ty'—Xn Ayy= yn’—Yn airit The equations (3) and (4), Nn+N in number, and involving Nn+y unknown quantities, are all linear, and the ecoeficients and absolute terms are all known numerical 2D} Mr. R. Meldrum Stewart on the quantities. From the N exact condition equations (3) we now proceed to determine the values of any N of the unknown quantities in terms of the Nn-+ qg—N remaining ones; and substituting these values in the equations (4) there result Nn observation equations for the determination of the most probable values of Nn+q—N unknown quantities. Having formed the normal equations in the usual way and derived the solutions, we then find the most probable values of the remaining N unknowns from the exact equations (3). If the corrections so obtained to the assumed approximate values A, B,C... X,, Yq, Z, ... ete. are sufficiently small, this may be accepted as the definitive solution; if not, we assume as new approximate values the quantities A+Aa, B+Ab...X,+Aa,, Y;+ Ay, ... ete. and proceed exactly as before to a further approximation. The solution of this general problem is thus perfectly straightforward and unambiguous, but is entirely different from the erroneous solution given by Dr. Campbell. It may of course be claimed that when either g or n 1s large (and indeed even when they are small) the rigorous solution of the general case as described above is cumbersome and tedious, and that in many cases more approximate results are quite sufficiently accurate for all practical purposes. This may readily be granted; and Dr. Campbell’s paper may serve a very useful purpose in drawing attention to a fact which is undoubtedly true, that when the number of obser- vations is large compared with that of the unknown quantities required it is comparatively easy, by the application of common sense, to devise approximate methods which will lead to very nearly the same results as the more rigorous method of least squares, and with much less labour. It is necessary, however, to exercise considerable caution in the choice and application of such methods; and the fact remains that (provided we accept the principle of the arithmetic mean) itis more than an even chance that the results obtained by least squares are nearer the truth than any other results from the same data, however obtained. And it may be worth emphasizing that the statement (see pp. 191, 192, 193) that there are certain classes of problems in which the method of least squares 1s not applicable, or gives ambiguous results, is quite untrue. It will also do no harm to insist once again on the fact that the method of least squares makes no claim to deduce the true values of the quantities sought, but only their most probable values. Though all cases are covered by the solution of the general Adjustment of Observations, 223 problem as given above, it may be interesting in conclusion to apply it to one or two specific cases. The first is an example taken from Merriman%, and has been chosen partly as illustrating the occasional misapplication of the method of least squares mentioned earlier in this article. In introducing the example Merriman says :—“ In the last article the quantities y and w were both observed; but the latter was regarded free from error, because .... The following example gives an outline of a method that may be used when both observed quantities are affected by accidental errors.— In order to determine the most probable equation of a certain straight line, the abscissee and ordinates of four of its points were measured with equal precision. The observed co- ordinates are pos OrGe 10signd 12; z= 04% 026; -048; ‘and. 0-9, And the most probable straight line for the four points is expressed by the equation y =Saet+T in which § and T are constants whose most probable values are to be found.” Then, notwithstanding his preamble, he proceeds to a solution which is the exact equivalent of the assumption that the measured values of « are free from error. That is, he forms normal equations which are the equivalent of ay = 247 Sw, Sy =S22,-+47, obtaining the solution See wleooe. T = —:029. Had the equation of the line been written in the equivalent form if di OE to and a similar solution made, the results would have been Sj. 354, T= +039: * ‘Theory of Least Squares’ (original edition), Art. 107. 294. Mr: R. Meldrum Stewart on the He would thus have been confronted with exactly the same difficulty as experienced by Dr. Campbell in the problem of the determination of density*. The first solution corre- sponds to the suppositious case in which the abscissze have been measured with absolute precision, while the measures of the ordinates are subject to accidental error ; the second to the converse case. When both the co-ordinates are subject to errors of measurement neither solution is correct. Denoting, for the moment, the numerical measured values of the co-ordinates by 2’, #.'...y,', y)'... we have the elght observation equations Y= Bay Yi = One y= @o Yo = Yo» ay = X53. Ys = Ys 5 hy hy Yn = Ye and the four condition equations yy = Sa,+T, Uo t= Sav, +T, ¥Y3 = Sa,+T, YA aa Sa,+T, from which to determine the most probable values of ten quantities, the eight observed co-ordinates and the two constants S and T. i To reduce the condition equations to the linear form we put Ss = Sot AS, a= X,+Ax,, V9 = Xo+ Axy, v3 = X3+ Aas, E ae X4 ar Ax,, where So, X;, X_... are approximate values and AS, Aa, Az,...the most probable corrections to these. Neglecting products of the latter, the condition equations become YN — Sov +X, AS + Ay Yn = So¥e + X,AS+ 7, ) etc. ; and substituting in the observation equations these take * Phil. Mag. doc. cit. pp. 192, 193. Adjustment of Observations. 225 the form 5 Nee ms ae Res 1? > as, ck: aaa 4" Sot: + X,AS+T = y/, Sote + X,AS+T = yy’, Sot3 + X3,484+T = yx;/, Sotit X,AS+T = yy’, from which the following normal equations are obtained : (SP +L)a,4+8.X,A8 +8,T = Syy,/+ ey, (So? + 1)agt+8pX,AS +8)T = Soys' +22’, (So? +1L)a3+8oX3A8S +8 .0 = Soy,’ +23’, (Sor? +1) ty+SoM4AS +8oT == Soya’ +24’, Spe 2 SAS -pS XT > XY’, Sapo ) ue oe 2D Fe ys or eliminating 2, @, #3, &4, > X?.AS43X.T = > Xy'—S,) > Xa’, SAS AT Sy! = Sy a’ As a first approximation we assume X,;=2,', X,=2,', ete, and obtain the values Sp) Ma oNs T- == 029, i «yd f= he ea Ta) oegiean Odd Using these in turn as approximate values and re-solving, we obtain : AS O18, Sood, Ti = = -035, which are correct to within one unit in the last place *. * The same result would of course have been arrived at by the methods of analytical geometry, assuming y=Sz+T as the equation of the line required, and imposing the condition that the sums of the squares of the perpendicular distances from it of the points (1' y,'), (X2' Y2'), (xs ys'), (v4! y,') should be a minimum; the solution by this method would have been direct. 226 Mr. R. Meldrum Stewart on the We now proceed to the solution of the problem of de- termination of density from measurements of a mass and a volume, in which case Dr. Campbell states* that the method of least squares is clearly inapplicable. The mass (2) and the volume (vy) of a number of different samples of a substance are supposed to have been measured. Then if 6 be the density, the general condition equation con- necting the measured quantities is w=by. It is required to find the most probable value of 0. Hliminating 2%, a, #;... by substituting their values from the condition equations, and putting 6 = B+Ad, n= Yyt+An, Yo = Yet Aye, etc., the observation equations become y= yr By, + Y,Ab = ay’, Yo = Yo By2+ YoAb = 2", Assume that #,', 7... are measured with weight unity, and y;', ¥... with weight p; then the normal equations are (B?-+ p)y + BY,Adb = Bz,’ + py)! (B? + Pye + BY.Ab — Ba,! -++ pYya! BSYy +S Y?.Ab= Sev. We may now make use of a simple device to obviate the necessity of successive approximations. Assume that the approximate values B, Y,, Y.... have been chosen exactly correctly : then necessarily Ab=0, B=b, Y;=y1, Yo=¥o, etc., and the normal equations become (O° +p) = bay! + pyy' (0° +p) yo = bay! + py,’ DivSyty ta WS lalay: * Loe. cit. p. 192. Adjustment of Observations. 227 Eliminating y;, ys, y3 --- we obtain easily (—p)Sa'y! = b(2a?—pdy”), or putting Sa?—pSy?=2myza'y', b?—2mb—p = 0; whence b= mt /m +p," the negative value being obviously inadmissible. Had the relation between 2 and y been written ax=y, a being the specific volume, we should have obtained by an ae 1 Le alle exactly similar process a=-(—m-+ Vm? +p), which is equal 2 1 sin, a to i Thus the two methods of attack give identical results, ) as should be the case. It follows from the form of the result (as is indeed evident a priort) that to obtain an intelligent solution we must know the value of p, that is, the relative accuracies of the measure- ments of mass and volume. If p is infinite (measurements Le | ie" ke ; = Z while if p is zero, or rather y 9 ae Pa, — infinite (measurements of mass exact) ir These Dp u of volume exact) b= are the two results given by Dr. Campbell, and ure of course in general, as is to be expected, different; when p has a finite value, that is, when both sets of Peat reienis are fallible, the value ak b lies between these two extremes. Summary. Dr. Campbell’s article serves a useiul purpose in drawing attention to the fact that a rigorous least squares solution is sometimes laborious, and that more approximate methods will often yield results sufficiently accurate for most pur- poses. On the other hand, many of his statements in regard to the principles of the method of least squares are erroneous, and his application of them to several classes of problems which he considers is incorrect. The method of solution for the general case, including as particular cases all the problems treated by him, is indicated, and is applied to two specific examples. Dominion Observatory, Ottawa, March 4th, 1920, Liv 228 rel XXV. A Method of finding the Scalar and Vector Potentials due tothe Motion of Electric ‘Charges. By Prof. A. ANDERSON™. KT the coordinates of a moving electron be =¢(t), y= Wt), z=x@). let AG be its paths amdayiies position at time ¢: also let Q’ be its position at time i“, iew"!, where r=Q'P, and ¢ is the velocity i light; then the co- ordinates of ()’ are w= $(t—" ‘, y= (e -*), z=x(—"), In what follows, we shall use the symbols ¢, Ww, y to denote these iunchtene of t—- Thus the component velocities of Q’ i are @, 1, x Lot ¢2($'e—@)4 G4 Gol! ee show that if A= te A satisfies the equation of wave- propagation, A ‘ eM. V A—4 aplai 0. Since 7? =(a—¢)?+ (y—wW)?+ («— x)”, we have (eae c& 0€ M AC aa Ee as iy where X denotes the quantity [pie h) + Wy —) +x" (&—x) Me. * Communicated by the Author. Method of finding Scalar and Vector Potentials. 229 Thus oA a iL \ ae Et and pra 3(cE + Ar)? ty Ov oF (r—€) yee pa(errse N(BA +¢)7? + 4c - A c(8c+rA)E ib pr 2) C255 where w= [d(x —b) +'"(y—W) +x" (2-x) —3(¢'¢" tab ap! +yly Pave Also r _a—¢ or y-¥ Olan 257% A 88 = 8D (e +N) (e-9)], and aA Ooo = GEC Sepa AD si Oe Nal @ 2 a tcp ae? (w = eer —e(e+a)]; with similar expressions for ¢? ws and or Adding these three expressions, we easily find that po Oe Oe ms (So+8 a OY” a a _ _2cnr _ bc&(e+A) _ 3e(e+Ar) 3(e+r)?7? ae O=8) (8 BP 8 08 ab ae oh ONT! rag hgh AONGES pr? as, rey Cr — £74 ee A(3A-+ ¢)7? + 4nrcEr Can pr? (eae, re —€&)* , and, therefore, 230 Prof. A. Anderson: Wethod of finding Scalar and Thus a satisfies the equation of wave-propagation. g If A be any function of «, y, z, é that satisfies this equation and F be any other function of w, y, 2, t, then AF will also satisfy the equation if wok, ~omw Ol aA Se +? 3¢- 30) = AV/7Hh 2( OA OF OA or. Oe >F Oz Ov OY’ OY OZ” 5: ): Let A be and F any function of i". We have oA OF, 0A. OF om OF) oF a Oz dx Oy OY Bebe rey Ot) ® eee and 1 ea Oe ae 3K" Bi ee ; Tey ee)". 97) an ves Ce (Ee) cee ( dere f| c(r—&) and thus On OM OE or oe ot 2H AVE +2 (57 545, oy ae phy? a ss P@—Bt pag OM 2h a Also OA == os Ae —— ae (cE+Ar), and oF re ae oF GHEE GEO Fe) Therefore Opler oe ol Eine EK’ . ee ED Lae = Perey eee en Neer a 162) A et AA OE Gage eee a + 2cEr + c&) Bee) Hence —— satisfies the equation of wave-propagation. ele Veetor Potentials due to Motion of Electric Charges. 231 The co-ordinates, velocities, and accelerations of (' are all functions of i", and, therefore, we have a number of expressions satisfying the equation. ‘Thus, if the co- ordinates of Q’ are 2’, y’, 2’, its velocities uz, uy, uz, and its accelerations f;, fy, fe, Uy ; Ts: Gi") ‘toe r( =) C ¢ all satisfy the equation of wave-propagation at any point P at a distance 7 from the position Q' of the moving electron, u, being the velocity of Q' resolved along Q’P, ‘and ¢ the velocity of light. CO ereisshe Suppose, now, we have a distribution of moy ing eleciricity, and we wish to find solutions of the equations of wave- propagation that will hold at any point P. A | {\ ee ia I el Tua oo Dalaman ca at any point P where there is no electricity. If there is electricity at P moving with velocity u, we draw a small sphere round P as centre. As this sphere diminishes it A is clear that A and Lee contributed by the charges inside the sphere tend to zero, but V/A tends to the value 2 , al ee eee, UU C= U Hence A = a eee dy! de! is the solution of the equation Qarr Bad 1 ban ¢ c+u NS eee, coke nae St ee V 7A mae = p> lo c—uU : : ; er ke But, since w is a function of t—-, it follows that C 232 Method of finding Scalar and Vector Potentials. is the solution of the equation ; 1 o?A eae = =P) Ol LIA = —/?p, and the solution of a A= =)5 is “ipa \\y pu uz dx’ dy’ dz’ pit ve Inc foe [r(1—"*) ] C—U C with similar expressions for the solutions of the equations U CYA ce —p2 and OA =—p a These are the scalar potential and the components of the vector potential at P. It would seem from the above, if correct, that for a single charge e at Q moving in any manner the scalar potential at any point P is eu | 2c log aoe E (1 -")| c—U C where r=Q’'P, and wu is the velocity of Q’, and the com- ponents of vector potential at P are EU Ur CU Uy +5) 7 ) ctu ( Up c+u Up 27c* log E iL = “| 2c" log rat (et 6= 0 x ¢ c—U Cc euu, 27c* log anes E (1 +") GS {Uh U. ° If the square of — is neglected, we get the expressions C given by Liénard and Wiechert. ieee ia] XXVI. The Crystalline Structure of Antimony. By R. W. JAMES, W.A., B.Sc., and Norman TunstTauy, B.Sc., Lec- turers in Physics in the University of Manchester*. eee erystallises in the dihexagonal-alternating class of the hexagonal system. As in the case of calcite, the crystalline symmetry is that of a rhombohedron, the three edges of the rhombohedron which meet in the trigonal axis being taken as the axes of the crystal. The angle between any two of these edges is 86° 58’, and the angle between the rhombohedral faces 100:010 is 92° 53’, so that the antimony rhomb is a slightly distorted cube. The crystals have a very perfect cleavage at right angles to the trigonal axis and parallel to the (111) planes, and a somewhat less perfect one parallel to {110}. From some observations made in 1914, Sir W. H. Bragg and Prof. W. L. Brage+ concluded that the arrangement of the atoms in Antimony was similar to that in diamond, except that the whole structure was distorted along the trigonal axis. As the observations on which this conclusion was based were incomplete, it was suggested by Prof. W. L. Bragg that it would be worth while to repeat them more carefully. This has now been done, and it has been found that a struc- ture simiJar to that of diamond will not explain the spectra observed from the various faces. The observations were made with the X-ray spectrometer, a bulb with a palladium anticathode being used, and five faces in all were examined. The (111) and (110) faces were obtained by cleavage from a large mass of crystals, and surfaces parallel to the (100), (110), and (111) planes were ground on slices of the crystal, the angles which they made with the cleavage planes having been calculated. Antimony is very brittle, and no distorted surface layer appears to be formed by grinding, for the ground faces were found to reflect X-rays nearly as well as the natural ones. The positions and relative intensities of the spectra observed from the various faces are shown diagrammatically in fig. 1. The glancing angles for the first-order spectra were as follows :— GOO YAAO). xo GAO) of ALIjuie ¢(12T) a” 27 7° 24! (aes AZ 2G 19214, The sines of these angles are in the ratio Beh 36 2 1°44.2-0°31di: 1:70. * Communicated by Prof. W. L. Bragg. + ‘X-rays and Crystal Structure,’ p. 227. Phil. Mag. 8. 6. Vol. 40. No. 236. Aug. 1920. R 234 Messrs. R. W. James and N. Tunstall on the Now, for a face-centred rhombohedral lattice, as calculated from the geometry of the rhombohedron, the corresponding ratios should be 1: 1°377 : 1°450 : 0°820 : 0°880, while for a simple rhombohedral lattice they would be Li: OF: PASOe U-650 2 17 6a Glancing angles and intensities for Spacings of planes. wave-length 0-584 A.U. Distances in Angstrom Units. The small glancing-angle for the first-order spectrum from the (111) face indicates that the underlying structure is the face-centred lattice. Assuming this to be the case, we find from the glancing angle for the (100) face, that the edge of each unit rhomb of the face-centred lattice has a length of 6°20 A.U. Taking the density of antimony as 6°70, and calculating the number of atoms in such a unit rhomb, we find the number to be eight. (The number actually found was 7:96.) Now, if one atom is associated with each corner and face centre of the unit lattice, the number would be four only. Thus it appears that the structure, as in the case of diamond, consists of two interpenetrating face-centred lattices. The * W. H. Bragg and W. L. Bragg, Proc. Roy. Soc. A, vol. lxxxix. perce Crystalline Structure of Antimony. 235 arrangement of these lattices is, however, different, as may be seen from a consideration of the intensities of the spectra from the different faces. The following structure will be found to account com- pletely for the observed angles and intensities of the spectra. The antimony atoms lie in two interpenetrating face-centred lattices. Suppose for one of the lattices, diagonals are drawn parallel to the trigonal axis for each of the eight equal rhom- bohedral cells into which the unit lattice may be divided. Fig. 2. From considerations of symmetry, it will be seen that the atoms of the second lattice must lie in these diagonals. If they lay at the unoccupied corners of the first lattice, the structure would become a simple rhombohedral one. This will not fit in with the observed facts, but if the atoms of the R 2 236 Messrs. R. W. James and N. Tunstall on the second lattice are all displaced from these corners along the diagonals in the same direction by a distance equal to -074 of the length of a diagonal of one of the small cells, all the observed facts can be explained. This structure is shown in fig. 2, in which, for the sake of clearness, one only of the small cells 1s shown. The Arrangement and Spacing of the Planes. Gril) planes. (3 Wr awiG EE “ioe2.)) _ Parallel to the (111) face, the planes containing the atoms occur in pairs. The planes belonging to one of the lattices. divide the distance between those belonging to the other in the ratio 0°389:0°611. The planes belonging to one lattice form a first-order spectrum at a glancing-angle of 4° 26’, but. those belonging to the second lattice add a contribution to: this spectrum, differing in phase by 27 x 0°389 or 140° from that from the first set, thus reducing the intensity of the first-order spectrum. On this assumption, taking the inten- sities of a normal set of spectra to be in the ratio IOO 3 Bas 6 1453 We de the intensities of the first four orders should be date: ZO"b.: LOI O22 2 328: If the intensity of the second-order spectrum is put equal to 100, the calculated ratios are Die mOOR oe leas sks, The observed intensities have the ratios CO OO AS er0e aloe which is quite as close an agreement as can be expected. (b) (100) planes. The planes occur in pairs containing equal numbers of atoms. ‘The spacing of the pairs is 0°074:0°926. Thus the contributions to the first-order spectrum from the two sets of planes differ in phase only by about 36°. This will corre- spond to a nearly normal series of spectra, the intensities faliing off rather more rapidly, which is, in fact, what was. observed. The intensities of the spectra from this plane show clearly that the structure cannot be similar to that of the diamond. Crystalline Structure of Antimony. 237 (c) (110) planes. The planes are again in pairs, each plane containing an equal number vf atoms. The spacing is 0°141:0°859. The intensities of the first three orders should have the ratio LOO psi. 0 sl. The observed ratio was about 100 : 20:0. (d) (110) planes. These planes are parallel to the trigonal axis. They are evenly spaced, and should show a normal sequence of spectra. No certain third-order spectrum was found, but the glancing angle is rather large, and even the normal intensity will be very small. (e) (111) planes. (AEG, BDF, fig. 2.) The planes are in pairs, very nearly evenly spaced, the spacing being in the ratio 0°463:0°537. For the first- order spectrum, the phase-difference between the contribu- tions from the two set of planes is 167°. The first-order spectrum should, therefore, be very small. No first order could be observed at all. The structure assumed approxi- mates to that of a simple rhombohedral lattice, for which the glancing angle for the first-order spectrum for the (111) planes should be double the angle for the face-centred lattice. Thus the spectra from the (111) face confirm the conclusion that the atoms of the second lattice must lie very close to the unoccupied corners of the first lattice, and not somewhere near the middle of the diagonals of the small cells, as would be the case for a structure approximating to that of diamond. Accuracy of determination of Position of the Atoms. The planes parallel to the (111) face occur in pairs, and it is from the spacing of these planes that the relative positions of the atoms in the crystal are determined. The accuracy with which the spacing ratio of these pairs of planes can be determined depends on the accuracy with which the relative intensities of the spectra of different orders can be measured, on our knowledge of the intensities of the orders of the spectra from a series of equally-spaced planes, and also on the actual spacing of the planes under consideration, since a 238 Messrs. R. W. James and N. Tunstall on the small change in the spacing has a far greater effect on the relative intensities in some positions than in others. In the present experiments, the intensities of the orders were measured roughly by the method of sweeping the crystal round at a constant speed *, and for the first three orders are probably known to within about 10 per cent. The fourth order was too faint to be observed at all, and, although an undoubted fifth order was found, its intensity cannot be con- sidered as accurately known. The intensities of a series of normal spectra were taken as TO Mee eae ae These figures are based on comparisons of some measure- ments on rock salt and on galena, and probably only represent very roughly the normal intensities from antimony. ‘The only planes in this crystal which are evenly spaced are so close together that the higher-order spectra occur at large angles, and are therefore faint, so that no reliable intensity measure- ments could be made for them. Assuming the normal intensities stated above, and calcu- lating the intensities of three series of pairs of planes spaced so that the phase-differences, 6, between the two sets for the first-order spectra are 135°, 140°, and 145° respectively, we get, taking the intensity of the second-order spectrum in each case as 100, Vee 86 200 Tae Ow 0 OSIM? og onoe 57:100:51:1:°4:18 $= 145°. 29 nO ae eae The observed intensities were 60: 100:48:0:15 Supposing the intensities to have been measured to within 10 per cent., it will be seen that the value of 6 can hardly vary by more than 2° on either side of 140° without causing intensity changes greater than the errors of observation. Assuming now a different normal intensity law, supposing the intensities to be inversely proportional to the square of the order of the spectrum, which gives a ratio 100) 25 ees 6:2 53.4 * W. H. Bragg, Phil. Mag., May 1914, p. 881. Crystalline Structure of Antimony. 239 the corresponding calculated intensities are BO Vasa es LPG’: TOO%2 76.2.0; 2 27 6=140° ...... 79 :100 :56:1°3: 26 e462 arn . 542 100: 42 242 : 24. The most likely value of 5 is now 143°, but the general “fit” is not so good. Making: this large alteration in the assumed normal intensity series only makes 3° difference in the probable value of 6. But it is practically certain that the true normal intensity series is much more nearly the first series than the second, and we can take as the probable value of 6 o= 140° £22 which gives a range of only 0-062 A, within which the Anti- mony atoms of the second lattice may be displaced along the diagonals of the first lattice. Distances between the Atoms. Any particular atom lies in one of the (111) planes, and is equidistant from three atoms in each of the (111) planes on either side of it. The three atoms in one plane are, however, nearer the given atom than those in the other plane. The distance between the centres of two atoms, such as A and B in fig. 2, in the close pair of planes, which is the closest distance of approach of two antimony atoms, is 2°87 A with a probable error of +0:01 A., assuming the limits of error for 5 stated above; and this can, perhaps, be considered as the “atomic diameter” of antimony. For the wider pairs of planes the closest distance of approach of two atomic centres, such as B and fi, is 3°37 A. The planes thus occur in pairs in which the atoms approach one another closely, the pairs being separated by a wider spacing. It should be noted that the very good cleavage of antimony is parallel to these pairs of planes. In conclusion, we wish to express our thanks to Prof. C. A Edwards, of Manchester University, who kindly supplied us with some excellent specimens of crystalline antimony, and to Prof. W. L. Bragg, who suggested the work, and to whom we are indebted tor much helpful advice during its progress. [24004 XXVIII. The Spectrum of Hydrogen Positive Rays. By G. P. TuHomson, M.A., Fellow..and Lecturer Corpus Christi College, Cambridge *. [Plate IV.] 1. INTRODUCTORY. A* is well known, hydrogen gives rise to two spectra. Bu One, the Balmer series, is a group of lines of which only a small number are easily obtainable under ordinary conditions and whose frequencies follow a simple law; the other, the so-called second or many-lined spectrum, consists of a large number of lines whose frequencies have not yet been reduced to any law. The theoretical researches of Bohr make it seem very probable that the Balmer series is produced by the neutrali- zation of a positively charged hydrogen atom by an electron moving through successive orbits. If this is true, it would seem natural to ascribe the more complicated second spectrum to the neutralization of the more complicated molecule. Stark}+ has obtained results from the spectra of different parts of the strie of the positive column in a hydrogen-filled discharge tube, which lead him to the same conclusion, but the evidence is not so direct as could be desired. On the other hand, Fabry and Buisson { from measure- ments of the width of spectral lines came to the conclusion that the second spectrum was atomic in origin; while Wolfke § showed that a formula closely resembling that of Bohr could be obtained for the Balmer series on the assumption that it was due to the molecule. In addition, there is the difficulty, which will be dealt with more fully later, that, according to both Stark || and Wilsar{], the second spectrum of hydrogen shows no Doppler effect when hydrogen positive rays are examined in the direction of their motion ; while it is well known that when the positive rays of hydregen are analysed electro-magnetically by Sir J. J. Thomson’s method, a considerable proportion generally consists of rapidly moving hydrogen molecules with a positive charge. * Communicated by Sir J. J. Thomson, O.M., P.R.S. + Annalen der Physik, vol. lil. p. 221 (1917). t Journal de Physique, vol. ii. p. 442 (1912). § Physikalisher Zeitschrift, vol. xvii. p. 71 (1916). | Annalen der Physik, vol. xxi. p. 425 (1906). q Loc, cit. vol. xxxvii. p. 1251 (1912). The Spectrum of Hydrogen Positive Rays. 241 2. MerHop or EXPERIMENT. In order to obtain direct evidence on these questions, it occurred to the author that it might be possible to combine the electro-magnetic analysis of “the positive rays with an investigation of their spectrum. ‘The method first tried was to analyse a narrow pencil of positive rays by magnetic and electric fields in the ordinary way (see ‘ Rays of Positive Hlectricity,’ Sir J. J. Thomson, p. 20), and then examine the spectra of the various constituents. It was found, however, that the light obtainable from a pencil sufficiently small to give sharp resolution was totally inadequate for spectroscopic investigation. It was therefore decided to make use of the fact that by changing the conditions in the discharge tube it is possible to vary the proportion of atoms to molecules in the positive rays over a wide range. ‘The apparatus used is illustrated in fig: 1. or < u, any assigned velocity. Say U>u: and put Av=At(U—u), where Aé is an interval of time so short that the number of molecules making more than one collision during that instant may be neglected. Then the expected number of collisions produced by a specimen of class U overtaking one of class uv, during the instant Aé, is NAiWU —w) AU AuF(U) 7G). 2 12 ee alien That is the number of members lost to class U by collisions of that type during At. Let the respective velocities resulting from such a collision be U' and w' (u'>U’). When a molecule of class u’ overtakes one of class U’, the resulting velocities are the original w and U. Accord- ingly {, the loss to class U through collisions of the first * The reasoning is readily extended to the more general case in which the mean square increases with the number of the elements ; the elements have not all the same mean square, and are not measured from their mean value. + Cp. Watson, ‘ Kinetic Theory of Gases,’ § 14 et seq. t See below, p. 258 (16). Probabilities to the Movement of Gas-Molecules, 257 type, in the time ¢, will be compensated by the gain to class U through collisions of the second type, if At(U—u) AU Au F(U) fu) = ane —U VA Ag Deu rw)... "U13) Of the factors on each side (U—u)= (u’ —U’), the colliding bodies being perfectly elastic; and AU!Aw’ is equatable to AU Aw independently of the variables U and u*. Accord- ingly, the equation will be satisfied if F(U) fw) = FU) f(), ULV SD G i CL) consistently with the conservation of energy. ‘This func- tional equation may be solved by putting ® = log F, p=log f: 0(U) + ¢(u) = ®(U')+4(w’), . . . (15) subject to the condition MU? 4+ mu? = MU? 4+ mw”. The condition supplies the solution J, viz. the left-hand side of the equation multiplied by a constant, evidently negative, say —h. That is, not taking account of the conservation of momentum ; otherwise it is proper to add a term of the form k(MU+mu). The constants may be determined by well-known considerations. This reasoning has uo claim to novelty. It is re-stated here only in order to draw attention to two points. One is the relation between this third argument and our first. Whereas the third argument does not prove that the normal dis- tribution will be set up in a molecular medley, but only that if it is set up it will be maintained—this method of approach has hitherto been completed by the “ H” theorem of Boitzmann. The error theory of Laplace is now suggested as an alternative complement. Another point to which attention is called is the implied use of that sort of Probability which has been called in the ‘ Philosophical Magazine’ “a prion” ¢ and elsewhere * As may be shown by differentiating with respect to U' and wu’ the expression for U (1) and that for uw; and more generally from the consideration adduced below (34). + Cp. Jeans, ‘Dynamical Theory of Gases,’ p. 25, ed. 2. If (18) is obtained by the reasoning employed in the first argument to determine the constants for the normal function ((4) et seg.) the solution of (18) avails to prove that the function is necessary. t See Phil. Mag. 1884, article by the present writer. The term “a priort” is infelicitous so far as it is sometimes used with a different , connotation in the calculus of probabilities. As to the alternative designations, see article on “‘ Probability,” Encyc. Brit. (ed. 2) p. 377 258 Prof. F. Y. Edgeworth on the Application of “unverified ” or, after Boole, “intellectual.” An assump- tion of this kind was tacitly made in the preceding argument when it was taken for granted that the appearance (in co sequence of collisions) of additional couples with velocities. U'andw’, u’>U', within a distance of (u' — U')At from each other, the slower M molecule behind the quicker one, was. tantamount to, or attended with, the appearance of additional couples corresponding in all respects to the above description except that the slower body is now before the quicker one. The underlying assumption might perhaps be formulated as follows :—(16) A molecular chaos must present the same appearance, both as to the relative position of molecules and as to their velocities, whether contemplated from above or below. Thus corresponding to couples of the first type just now defined with M behind m, there are (on an average) at any time an equal number of couples in other respects similar but with M before m. The preceding arguments are readily extended to the case in which there are several varieties of mass, each pertaining to a large set of cylindrical molecules. B. Coming nearer to reality, let us next consider the random movements of disks on a plane. I. The simplest case under this head—that in which the molecules are equal not only in volume but also in mass—has. already served to introduce our first argument to readers of the ‘ Philosophical Magazine’ *. It has been shown that if a disk with velocities Uand V overtake one with velocities w and v, the line joining the centres at the moment of impact making an angle @ with the axis of U and u, the new velocities of the former are as follows : U’= Usin? @—V sin 6 cos 8 +.u cos? 8+ cos sin 6, | i V’ =—U sin 8 cos 84+ V cos?6+u cos 6sin 8+ vsin? 0. | (17) If now with these acquired velocities the said disk collide with a fresh disk of which the velocities are wu, and vj, the angle made by the centre-line with the axis of U being now 6;, we shall have for the new velocity = sin? 2 Ne si, Cosi, +u, cos* 6,+ 1, cos 6, sin 6,,. . (18) ‘with a corresponding expression for V’’; where for U’ * See Phil. Mag., January 1913, p. 106. The “spheres” there con= sidered are here replaced by disks ,and the notation is altered. Probabilities to the Movement of Gas-Molecules. 259 and V' their values in U, V, wv, » above given are to be substituted. It was thus shown that after many, say n, collisions the velocity of the disk under consideration, say U™ and likewise V™, is a linear function of numerous elements. It is now to be added that the plurality of constituent elements is further secured in the more concrete case of disks differing: in type: say, one large class of mass M, another of mass m. For then, after the first collision, U' = (M—m)(U cos? 6+ V cos @ sin 8) + U sin? 0 —V sin 0 cos 0+ 2m(u cos? P+rcos@sin@), (19) with corresponding expression for V’. Accordingly, except in the limiting cases when M and m are very unequal or nearly equal, the ultimate velocity of any disk is largely made up of fragments of orders reduced, not only through repeated multiplication by powers and products of sines and cosines, but also by factors of the type (M—m)’, where M—m is a proper fraction. The penultimate constituents— the less remote progenitors—of the ultimate velocity do in- deed bulk more largely. But in general, and except initially, they are themselves the product of numerous antecedent col- lisions, and are therefore on the way to normal distribution *. The system may be conceived as approaching that ultimate state by successive steps as in the simpler case f, or rather more rapidly. As before, we may argue that the normal law in a general form will beset up; and may determine the con- stants from the conservation of energy, and the condition that the velocity-distribution of the molecules which come into collision during Aé should be the same before and after collision. The frequency-function thus found may be written Cone Oe ie Lire in vt sfc ® tx 20) Here U, V and w, v are velocities relative to the centre of eravity—which is at rest or moving uniformly. The trans- formation required when the velocities relate to a fixed point has already been indicated f. II. This conclusion is reached more quickly by the second -argument. Putting /(U, V, u,v) for the sought function, we have now to determine f so that the integral (over all possible yalues of the variables) of jflogf should be a minimum—subject to the condition that the ¢o-extensive * t bere, note to p. 251. t Above, p. 250. ft Above, p. 253. 260 Prof. F. Y. Edgeworth on the Application of integral of f and of fA(M(U?+ V*)+m(w? + v?)) should be constant, and the two additional conditions that the integrals of 7 MU+mu) and that of ((MV+mv) should each be constant. Varying jf, as before, we find for the required function, : f = Const. exp.—h(M(U? + V2) +m(v?+v?)) +k(MU+mu)+UMV+mv); . (21) where the constants are to be determined from the data by familiar considerations. ITI. The third argument purports to show that the loss to the contents of any class (U, V) through collisions with molecules of class (uw, v) is repaired by collisions between molecules of certain other classes—provided that the dis- tributions of velocities, say F(U, V) and f(u, v), are normal. Following the analogy of the simpler case, consider the species defined by an m molecule of class w occurring in such neighbourhood to an M molecule of class U that a collision will result within the time At. And let the species now be divided into varieties according to the relative orientation of the molecules at or on the eve of a collision, defined, say, by the angle @ which a line joining the point of contact makes with a horizontal axis. ‘ Buigeuis By parity of reasoning it may shown that the distribution will be stable if F(U, V)f(u, vr) =F(U',V')f(w’, v'); or putting as before ® for log F, ¢ for log f, ®(U, V)+¢(u, v) =M(U?+ V")4+m(u?tv?), . (22) subject to the condition M(U?+4 V?) + m(u? +07) = M(U? + V"?) + m(u? +07). Probabilities to the Movement of Gas-Molecules. 261 As before, one side of the equation multiplied by a (negative) constant supplies the solution *. That is, not taking account of the moments. When they are similarly treated, there is obtained for the required function one identical with (21). The character of the reproductive collision may be seen by looking at figure 1 from the top of the page as if it were the bottom. If the velocities just after collision are reversed and the figure is then looked at from above, the circle with centre ¢ which was at first overtaken now overtakes the other. The dotted circle with point of impact at I’ is meant to represent such a collision viewed from below. The argu- ment involves an assumption of the kind exhibited in the simpler case (16), namely that (in a molecular medley) of the couples with respective velocities U, V and w, v there are (on average) as many with one molecule before as behind the other. The arguments which have been employed with reference to collisions between disks of different masses may of course be narrowed to the case of perfectly similar disks, and may be extended to the case of more than two types of molecules, and to three dimensions. C. We go on from Cartesian to Lagrangian co-ordinates, beginning with the case of two dimensions and two sets of molecules. I. Let the generalized co-ordinate of a molecule for one set be Q), Qo... Qn; for another q1, q2...9¢n3; and the corre- sponding components of momentum P,, Ps... py, po.... Let R be the impulsive force at the moment of collision between two molecules of different type—its direction being normal to the contour of each figure at the point of impact, say I. Let X be the abscissa of I with respect to fixed rectangular axes =@®(Q), Qo... Q,,), and likewise Y=WV(Q,, Q.... Qn). Let ®, and likewise VY; denote the respective differential coefficients of ® and WV with respect to Q,. If @ is the angle made with the horizontal by the normal at the moment of collision, R(®:cos 0+: sin 9), say (23) RL;=(P:—P’,) ; where, R being positive, L is negative when P’>P. JSike equations are obtainable for the other P’s. Likewise for the p’s R(d:cos A+; sin 0), * The argument may be more closely assimilated to that under head A by transforming to new axes, one making an angle @ with the horizontal and the other with the vertical. Only the velocities in the first of these directions, say W and w, will be changed by the collision, say to W’ and w’. W—w=w'—W’, as the argument requires. Cp. C.II. below. ’ 262 Prof. F. Y. Edgeworth on the Application of | say (24) Rl=p:—p ; where | is negative when p< pi. One more equation connecting R with the P’s and p’s is obtained from the principle that the relative velocity of the two points of impact is the same (in absolute magnitude) before and after collision: say (25) W—w=w’—W’, both sides positive. Now W=1,Q,+ 1,Q.4+ ... + LnzQn = L,(By P+ ByP.+...) + L,(ByP) + BoPo+...)4+...35 when for each of the Q’s is substituted its value in terms of the p’s*. And W'is an analogous linear function of the dashed P’s. Rearranging and adding, we have W+ W!'= (P, 4+ P’))(L, By, + LeBy +...) + (P2+ P’9)(L1 Bo, + L,Boy+...). Likewise (w+w') is expressible in terms of p’s and 0’s. But by (25) (W+ W’)—(w+w’')=0. And by (23) and (24) P’,= P.—RL, pe=pit+ RL. Whence R is found asa linear function of the P’sand p’s. Accordingly P’, what P; becomes through the collision, is a linear function of the P’s and p’s. Likewise P", (the result of a fresh collision) is a linear func- tion of those P’s and p’s, and of fresh p’s themselves t the result of numerous previous collisions. There is evidenced that plurality of components which, the composition being linear, tends to generate the law of error. But a scruple may be raised with respect to the condition of independence. That condition is threatened by the cir- cumstance that the coefficients of the P’s in the expressions for P,, P,... (and likewise the coefficients of the p’s in P15 P2--. ) are not, as usual in the statement of the law of error, constants, but functions of the co-ordinates which vary between successive collisions. It is not necessary, how- ever, for the genesis of the normal law that the “ constants ” in the linear function should be rigorously constant. It suffices that they should be random specimens of a medleyt , as the coefficients here may be regarded. The distribution resulting from this jumble will be in a form which mav be written Z=J exp 4,T/[T]; where J is a constant securing that the integral of Z between extreme limits is unity ; ,T is a quadratic function of the P’s and p’s with numerical coefficients § ; [T] is the mean energy of all the molecules (in a large unit of area). To show the relation of I to T, the expression in terms of P’s and p’s for the mean energy of molecules having the same P’s and p’s but * Cp. Watson and Burbury, ‘ Generalised Co-ordinates,’ § 7. + Cp. note to p. 251 above. + “ Law of Error,” Camb. Phil. Trans. 1906, p. 128. § As in the simpler cases, Cartesian (« and y) co-ordinates do not appear in the expression for P,, Ps, etc., nor in the expression for the energy. Probabilities to the Movement of Gas-Molecules. 263 different Q’s and q’s, transform T so as to form a linear function of squares, say C,117 + Cork + eee Clare + Gime + o- Onin 3 ° (26) where the coefficients are numerical. Then, as in the simpler case, ;T must be of the form feel? tn ach Om lbs E1611 sae atin 1C,=Cy(m+n)/2[T], ...q4=10.(m+n)/2[T], ... the condition analogous to (4) will be satisfied. Also as ,T=,1" (the mean energy of the set which has come into collision during At) the other equations for the constants must be satisfied (see (13) and (34)). ‘Thus the distribution may be written a xa TG, Seance eo) (27) 2 and if The complex molecules with different co-ordinates to which this distribution refers form a ‘‘ universe,” which may be divided into genera delimited by values of the co- ordinates. Consider first extensive genera, e. g. all the eases which have co-ordinates between Q, and Q,+a, Q. and Q.+ 8, where a, 8, etc. are considerable. If the normal law is very well fultilled for the “ universe ” it must be jairly well fulfilled by those genera*. As the law becomes still better fulfilled for the universe it must become very well fulfilled for those genera. So by parity of reason it must be fairly well fulfilled for subordinate less extensive genera ; and so on. Ultimately the law must be fulfilled for every genus detined by neighbouring values of the co-ordinates Q, and Q,+AQ,, Q, and Q,+AQ,...... The mean energy for each such genus is presumably the same f. II. The second argument gives at once the distribution in terms of the co-ordinates, with reference to each particular configuration which may be assumed by a pair of colliding molecules. Let T be the energy of a pair expressed in terms of the co-ordinates, and either the generalized velo- cities or the components of momentum ; and let P, be the momentum in the direction of a fixed axis w, and likewise * The constituents of the universe being taken at random from those genera, supposed few. + The energy would be the same for any one molecule moving without collision through different phases defined by the varying values of the co-ordinates; and there is no reason why it should not be the same for the average molecule in different phases. 264 = Prof. F. Y. Edgeworth on the Application of P, in the direction y. Then, by reasoning of parity with that employed in the simpler cases, the required function is of the form Const. exp —A(T+4P,+/P,). . .-~ (28) Hach genus being thus distributed normally, the universe which comprises all the genera must be so distributed, whence (27) can be deduced. IT]. The assumptions involved by the third argument may be illustrated by a simple example. Let the molecules of one set be shaped each like a flat ruler with one side AIB in fig. 2 A rectilinear, but the other side not*. And let another -_ Sw Se ae es ow oe ¢ D set of molecules consist of disks, both sets moving in the same plane. lig. 2 A represents a collision between two such bodies. Say U.V are the velocity components for the centre of gravity of the ruler; Q© is the velocity of rotation about the centre of yravity; @ is the angle made by the straight line AB with the vertical; u and v are the (translational) velucities of the disk. If I is the point on the ruler which comes into contact with the disk, let the velocity of I just before impact resolved in the direction normal to the colliding surfaces be W, and that of. * To secure that the velocities of the bar after a collision of type A might not be replenished by an impact from behind, as in the case of a symmetrical disk (above, p. 261). Impact from behind, say at iI, is to be distinguished from the “ back-hand” collision at I’, Probabilities to the Movement of Gas-Molecules. 265 the disk w. Let the velocities after collision be W’ and w'— say, W>w,w'>W’. Analogy with the simpler cases suggests that we should equate F(U, V, Q) flu, v) = F(U4, V', O')f(u',v’). . (29) But there will not now be secured thereby the replenish- ment of the class (U, V,Q) by means of collisions between molecules of the dotted classes. The sort of collision which might be expected to act thus is shown by reversing the velocities of the points of contact, and then looking at the figure from above (16). The view thus obtained is shown in fig. 2 B, mee the disk has the post-collision velocities ul, a; but the velocity of I’ in the direction perpendicular to A' B' is U'cos $+ V'sing minus cO, where ¢ is the perpendicular distance of the normal a I from G, whereas in W the sign of cQ is plus. The relative velocity of the points of contact at (2. e. Just before) this back-hand collision is thus not the same as it was for the collision shownin fig. 2 A. The reasoning before employed * thus breaks down. ‘The break is obviated by the assumption that the class (U, V, —Q) occurs as frequently in the medley as the class (U, V,+0). _ We have then (50) F(U, V, ©) =F(U,V,—©). What is lost to class (U, V,Q) through the collision shown by fig. 2A is gained by the class (U', V', +Q’). And by the postulate the content of that class keeps equal to that of (U’, V’, —Q"’). And collision between molecules of the latter class and those of class (u’, v’)—the collisions shown in fig. 2 B—results in the class aag Vv, —Q). If then equation (29) is satisfied, what is lost to class (U, V, ) through a back-hand collision will be gained by Ue CW Va —0) +. But the contents of these two classes are equal by (30). Therefore if equation (29) is satisfied, the class (U, V, 2)—and likewise any other class— will not be increased or dimmniahed by collisions with m mole- cules of the class (uw, v) and likewise not by collisions with those of any other class. The example may be generalized by supposing the ruler AB to be connected by a hinge at B with another flat bar BC, and that again with another link CD ; while the disk is likewise connected with a complicated mechanism. By parity of reason, the velocities lost to any class (Q,, Q, . ..) through a direct collision with any class (4), qo .. beach class of an * Above, (22). + By analogy with the particular cases above or by the general reasoning below (p. 267). Phil. Mag. 8. 6. Vol. 40. No. 237. Sept. 1920. bh 266 ~=Prof. F. Y. Edgeworth on the Application of assigned genus (defined by the values of the Q’s and q’s)— will be restored by a back-hand collision of type B: granted postulates analogous to (16) and (30). The new axiom may be stated thus (31) :—For any assigned values of the co-ordinates pertaining to a set of molecules, the class defined by particular values * of the velocities of the centre of gravity (or the corresponding momenta), and of the other components of velocity (or momentum), occurs with the same frequency as the class defined by the reverse of the components for the centre of gravity, together with the same values as before for the other components. Which frequency is the same as that of the class defined by retaining the components of velocity or momentum for the centre of gravity and reversing all the other components ; as follows from the received axiom that (32) the frequency of a class is not altered by changing the signs of all the components of velocity or momentum fF. To combine this datum with the leading equations analogous to (13) and (29), transform the co-ordinates so that one of them may be the abscissa of the centre of gravity of the molecules of mass M measured on an axis parallel to the normal with corresponding velocity U at the point of impact and another the ordinate thereto with corresponding velocity V, and likewise for the molecules of mass m. Considering any species defined by the proximity of two molecules of assigned classest, say (U, V, Q,, Q....) and (u, v, 1, d2...), suppose the points of impact on the contour of the two molecules to be moving just before the collision with respective normal velocities W and w (say, W>w). Let these be changed by the collision to W' and w', the velocities of the points in a tangential direction remaining unchanged. By (32) the frequency of the species with the dashed components which results from the collision will be the same as the frequency of the species formed by reversing all those components. To secure that the loss to the original class (U, V,+ Q) through * TI. e., values between limits separated by small finite differences, e.g. Q and Q+AQ. + Cp. Burbury, ‘Kinetic Theory of Gases,’ ed. 2. p. 21; Bryan, “Report on Thermodynamics,” Brit. Assoc. Report, 1891, p. 92. The generally admitted (82) is to be distinguished from the now proposed (31). 4 + Classes defined by velocities (or momenta) and species constituted by members of two classes within a certain proximity. (Cp. above, © pp. 256, 260.) Probabilities to the Movement of Gas-Molecules. 267 oe with (uw, v,+91...) is the same as the loss to class (U', V',—Qy...) from collision between members of that olde ‘a class (w’, v', —q'...) (w’ reversed overtaking W’ reversed), that is, the gain to the class of (U, V, —Q, ..), we have—granted the postulate (31)—the condition AQ, AQ, i408 Aju Ags ater AUAVAQ, AQ, hake Au Av Ag, Age ee ay W—wl F(Q,, (Qe Yo U NG Q:; Qe...) (91 a. +.Uy v, qs Go... )s (33) equal to the expression which is formed by retaining all the symbols relating to co-ordinates and affecting all the velocities with dashes. ‘he equation may likewise be expressed in terms of the components of momentum, say WOW, NEV) ORG, Eom; miu; p13 Do cals This equation relates to the genus (or pair of genera) which is defined by values of the co-ordinates between assigned values of Q, and Q,+AQ,, Q, and Q.+ AQ»... g, and gi t+Aqi...3 which are the same on both sides of the equation, unaltered by the collision. F and f, as before, denote fre- quencies of classes, for the definition of which it is necessary now to specify the coordinates as well as the velocities ; AU AV AQ... AuAvd g;...=AU' AV' AQ’... Au'Av' AQ!... (34) ; since each P (and so each Q) is a linear function of P’s * and accordingly the right side of (34) = the left side multi- plied by a constant —which constant cannot be other than unity. Also(W—w)=w'—W’. Thus (33) will be satisfied if F(QiQ:... UVQ «..) Ags G2 - ii lll Grucci) eee. UV, "Qs. yA Gat eal arma EN 710 (35) subject to the condition that T’ the energy of the two molecules after the collision equals I’ what it was before. Whence, as before t, it is deduced that the required distri- bution is of the form J exp —AT, where T is a quadratic function either of the components of velocity or those of momentum. For the co-ordinates it is convenient to take, besides the velocity components of the centre of gravity, * As pointed out by Jeans, op. ert. p. 18. + Cp. (15) and (22). T 2 268 Prof. F. Y. Edgeworth on the Applicateon of a third co-ordinate which does not figure in the expression for the energy, viz. the angle made with a fixed axis by an arbitrary line through the “centre. of gravity ; and for the remaining co- -ordinates such as do not relate to points or lines fixed in space (but only to parts of the complex molecule) *. The characteristic defect of the third argument, that it does not prove the necessity of the normal distribution, may, as before, be remedied by the first argument. The analogues in three dimensions+ of the preceding problems, and other generalizations not involving other principles of Probabilities, are here passed over. D. So far we have supposed two molecules colliding with each other and separating with velocities determined by the laws of impact for perfectly elastic bodies. We are now to entertain the more general conception of an encounter t between two molecules supposed to come within each other’s sphere of influence and to be deflected from their previous path according to some unknown law of repulsion (perhaps preceded by attraction). Considering first the simple case of equal disks moving in a plane, we suppose the circles of influence to be so small and rare that the path of a molecule after leaving one of these circles, and likewise before entering it, is reetilinear, and that throughout a considerable space (in a large unit of area) there occur only molecules moving in free straight paths. The remainder of the unit area-~a constant proportion——consists of regions within which the movements of the couples are correlated as Dr. Burbury * The energy will then be of the form MU2+ mu2-+ Mk2O?+T,, where © is the velocity of the moJecule about its centre of gravity, k2 involves co-ordinates other than that of which Q is the differential coefficient. Ty, the remainder of the energy, is a quadratic function of the remaining velocities (or momenta) with coefficients involving the corresponding co-ordinates. ‘The segns of the P’s for the back-hand collision are obtained by observing the signs of the L’s and /’s in equations (25) and (24). + Analogous to the view of the molecular system from above (16) is now the view in any direction. The configuration at a “back-hand”’ collision will now depend not only on the configuration, but also on the velocities of rotation about each centre of gravity, at the direct collision between two molecules. | Aa t+ Cp. Watson, p. 36, Jeans, etc. A good representation of encounter is given by Clerk Maxwell in his ‘Dynamical Theory of Gases: Collected Papers,’ vol. ii. p. 42. Probabilities to the Movement of Gas-Molecules. 269 conceives *. We are now concerned only with the free molecules. ‘Their number per unit area and their mean energy are to be considered constant. I. The velocity (in either direction) of any one in particular may be considered as depending on an immense number of preceding velocities, as in the simpler examples. But the dependence is not now known to be linear. Let Ff be the function whereby wu,’, the present velocity in the horizontal direction of any particular molecule, is connected with the velocities of n molecules at a prior epoch. Say / ‘ ty Yt een ena ers Or onli, where the variables involved in the functions are velocities at the prior epoch. Expanding fin ascending powers of the variable, write a 7M = Oy DU + by Dre,? — Cine +b ie Ao dp + bo tw,2 + Date +terms involving combinations of the w’s and of the v’s inter se, and of the w’s with the v’s. Here the symbol Sis for the moment used to denote weighted sums; since with respect to any particular value of u,’ a particular coefficient attaches to each wu. Similarly interpreted, Me = Ou, + 20," > uu, + OWA. ba d0,2 + Yas Do; + Cee, ae Seis + 20 Go Ups + a But in forming the mean value of u’, or any of its powers, it is proper to treat one value of uw as on a par with another. Accordingly we may write fu’? ] =a,?[ Su? | + 2a,;?[Su,as| +... +a?| S07] 4... + 2a;dg[ 2u,vs], where © is interpreted as usual, and square brackets denote mean values. Put v=u/n, y=v/n, A=na, [ w’? |= A,?[Sv?| + 2A,?[Su.ms|+...Ae?[ S77] +... | + 2A,A,/ Surns |, and we haye (neglecting some quantities of the order 1/n) uw’? |= A,?[ Sv? ] + 2A,°[ Supvs) Fe. Ay? [>n? | +... * According to his‘ Assumption B,” ‘ Kinetic Theory of Gases,’ p. 11 et passim. That within the spheres of influence the distribution of velocities must be normal may be concluded by the use of an argument like our second. Presumably the whole class of molecules within such regions may be broken up into species characterized by the stage of approach—some couples for instance at the initial stage, others as close together as possible, at the very acme of the encounter. To each stage there would pertain a different coefficient of correlation. = 270 Prof. F. Y. Edgeworth on the Application of Of the terms on the right some disappear because affected with first powers of uv and y, the means whereof are respec- tively zero, e.g., Yu,vs; others for that reason combined with the presumption that the v’s and 7’s are independent ™, é. g., 4un; others because the number of the combinations is of an inferior order compared with that of some which are retained, e. g., }v,* as compared with Xv,7v,2 t. There are retained when n is large, only combinations equatable to powers of [=v?] and [2n?], say of k, and ky. Thus [w'?] = Arh + 2B hy? +... + Ae?ke + 2Bok,?. Also [w'?] =k, (by the conservation of energy). Therefore ky = Ah) + 2BY hy? +... + Ag?ha + 2Be7ho? +... with a like equation for kp. Differentiating with respect to k, and ky separately, we find A,=A,=1, and every other coefficient =0. Thus w’, and likewise v’, is shown to be in effect a sum of numerous independently fluctuating, randomly selected constituents ; and accordingly each of these velocities is distributed according to the normal law. To generalize this reasoning: consider first a simplified example of two correlated co-ordinates. Imagine corpuscles each consisting of two links AB and BC, as in fig. 3; each A x rod is of negligible mass, but there is a nucleus of mass m at the extremity of each, at Band at C. There are two degrees of freedom, the link BC turning about a joint at B, * Cp. Tait, cited above, », 252. Clerk Maxwell’s similar judgment (Phil. Mag. 1860) will surely not be questioned with respect to the free molecules here considered. 1 For the proof of these propositions, see the proof of the daw of error given by the present writer in Camb. Phil. Trans. doc. cit. Part I. § 1. Probabilities to the Movement of Gas-Molecules. 271 and the link AB turning about a fixed vertical pin at A ; the same pin for all the corpuscles, which move in horizontal planes that are indefinitely close to each other. Hach nucleus repels any other within a minute circle of influence with a force depending on the horizontal distance only, oa according to some unknown law. Let the angles made by AB and BC with the horizontal be @ and w ‘respectively. The present velocities of any molecule $! and wW’' are con- sidered as each an undetermined function of previous q¢’s and ws; the two velocities of any molecule among the progenitors being not now independent of each other. Identifying the mean energy of a molecule as deduced (by expansion of the functions) from that consideration with what it is as given by dynamical theory, we find by parity of reasoning that d’, the present velocity of a molecule, is in effect the joreta pied) sum of innumerable prior } sand w’s, and likewise w’, of y’s sand¢’s. Accordingly, the distribution of $! and yy’ is normal ; but a8 among fines constituents, each pair, e.g., o, and w, are now correlated, the exponent of the resulting error-function may be expected to involve the product, as well as the squares of velocities * As before, we may pass from the distribution of the ““ universe ’ ati that of the genus f. Parity of reasoning is applicable to molecular motion of the most general character, admitting movements of trans- lation and other degrees of freedom. II. The result is reached more readily by the second argument with regard to the free molecules moving (with ny number of degrees of freedom) in the space outside the spheres of influence. We have now to determine the la-w of distribution 7, so that Sflog 7 should be a minimum, subject to the conditions that }/=const., £f/T= [T], where > 3 is used to denote integration with respect to all the velocities (or components of momentum), but not the co-ordinates. T is the quadratic expression in terms of the velocities (or com- ponents of momentum) for any assigned values of the co-ordinates, say = AQ? + 2AjQi Qe + AvoQs? +... + ang + 2aroqgo. “49 “ As to the formation of correlated compounds from correlated elements, and other propositions implied in this paragraph, see Camb. Phil. Trans. loc. cit. p. 116 et seq. Cp. above, p. 263. 272 Prof. McLennan and Mr. Shaver on Permealility of where the A’s are, in general, functions of the co-ordinates ();, Qo...3; and the a’s are likewise functions of the gq’s. Whence at once there is obtained for the i form J exp—AT. As before, we may pass from this expression for the law of distribution of the genus to that of the universe * III. The third argument in the simple form so far adopted is not applicable to cases in which the co-ordinates are changed by an encounter. Recourse must be had to that theorem of Liouville which leading writers have called in at an earlier stage. E. It would be possible to advance further in other directions—in particular, where a field of force oceurs—on the lines of the first and second arguments, without the aid of Hamiltonian Dynamics, by mere Probabilities. XXX. On the Permealility of Thin Fabries and Films to Hydrogen and Helium. By Prof. J. C. McLennan, PRS. and W. W. SH aver, B.A., University of Toronto F. Ll. Introduction. N a recent paper by R. T. Elworthy{ and V. I. Murray the diffusion of hydrogen and helium through thin rubber fabrics was discussed, and the results of ‘east renee made by them on several samples of balloon fabrics were given. In these experiments the amount of gas diffusing through the fabrics was measured by a Shakespear Katharo- meter and by a Jamin Interferometer. As the method was one capable of wide application it was decided to use it in determining the permeability of liquid films to various gases, and the following paper describes some experiments made upon the passage of hydrogen and helium through soap films. The study of gas transfusion throagh membranous tissues is an important physiological problem, and it was thought on this account that it would be useful and might prove interesting to measure the rate of gas diffusion through films of various materials, with a view to formulating a more exact theory of the process of gas transfusion than exists at present. * Above, p. 264. + Communicated by the Authors. t Proc. Roy. Soc. Can., May 1919. Thin Fabrics and Films to Hydrogen and Helium. 273 Il. Preliminary Experiments. In order to test the apparatus and to acquire a working familiarity with the instruments, a preliminary study of tlie diffusion of hydrogen through the fabrics used by Elworthy and Murray was made. The apparatus used and the method of assembling it was the same as described in their paper. The fabrics used by them were inserted as a separating diaphragm in an air-tight drum-like vessel. Two gases were brought into this drum, one on either side of the fabric, and their transfusion was determined by tests on the gases by means of the instruments mentioned above. For a full description of the Shakespear apparatus the reader is referred to the paper by Elworthy and Murray. It will suffice here to say that this apparatus was made by the Cambridge Scientific iiekeninient Co., and that its principle is based on the variation in resistance of a heated platinum coil, consti- tuting one branch of a Wheatstone Bridge circuit, when the gas mixture surrounding the cell has its thermal conductivity varied by changes in its component parts. The two methods adopted were (1) to pass a continuous stream of pure air and one of pure hydrogen on opposite sides of the fabric as a dividing diaphragm, and (2) to enclose a known quantity of pure air on one side and to pass a continuous stream of pure hydrogen past the other side of the fabric. In the present experiments both methods were followed, but gas tests. were made with the katharometer only. It was found that 20°C. was a more suitable temperature for working at than 15°75 ©. as previously used by Elworthy and Murray. The measurements obtained were made by keeping the permeameter and connexions in a thermostat at 20°0 C., the variation in temperature being not more taan 07:2 C. TL, Calibration. The katharometer used to detect small percentages of hydrogen or of helium in air had alre aidy been calibrated for both gases ; but this calibration was checked by noting the galvanometer deflexions for a given sainple of gas, deducing the percentage of helium or hydrogen present from the calibration curve and then checking the result by actually weighing a known volume of the sample studied. It was found that the values obtainel by the latter method fitted in very closely with the calibration curve of Elworthy and Murray. It may be stated here that in their work it had . 274 Prof. McLennan and Mr. Shaver on Permeability of been well established that the curve obtained by plotting galvanometer deflexions against percentages of hydrogen or helium present in air was a straight line through the origin. The calibration showed that (1) 259 mm. deflexion on the scale 1 metre from the galvanometer represented 1 per cent. hydrogen in air, and (2) 163 mm. deflexion on the scale 1 metre from the galvanometer represented 1 per cent. of helium in air. The following table gives a comparison of the results obtained in the present experiments with those obtained by Elworthy and Murray when using the same fabrics. In each case the permeability is given as being the number of litres of gas permeating 1 square metre of a fabric in 24 hours :— TABLE I. Results obtained in this Results obtained by Elworthy investigation. Temp.20°C. and Murray. Temp. 15°°5 C. Method I. Method ILI. Method I. Method IT. Fabric No. Using Using Using Using Katharometer. Katharometer. Interferometer. Katharometer. HEB 25-0 96 9°8 84 a5 WGA 2. 8-4 80 | 0 8-6 iT Vge Bacore 5:0 at. 55 AT We By.o.. 6:3 ie 6-7 6-4 Wit. A, e.. 8:0 soe 6.2 8:1 Wav Crss' 2 78 75 8:1 EXGINE Peat o'4 IV. Permeability of Films. After the preliminary experiments had been made, an attempt was made to employ the same method in making a determination of the transfusion of hydrogen and of helium through a soap film. Sir James Dewar + ina paper presented at a meeting of the Royal Institution of Great Britain in Jan. 1917, described many interesting experiments with long-lived soap bubbles and films, among them being a determination of what he calls “ gas transference ”’ through * The fabric numbers refer to samples of balloon febrics described in the papers by Elworthy and Murray. t Dewar, Paper, ‘‘Soap Bubbles of Long Duration,” presented at weekly meeting of the Royal Institution of Great Britain, Jan. 19, 1917. Thin Fabrics and Films to Hydrogen and Helium. 275 a soap bubble, by blowing a hydrogen bubble in hydrogen and noting the decrease in diameter as time went on, due to the slight excess pressure inside the bubble. What he measured was the excess of the rate of gas diffusion outward over the rate inward through the film, and he found that as the soap bubble became thinner the gas transference became greater. In the present experiment the endeavour was to determine the actual rate of gas flow per square centimetre through the film, keeping the film as nearly constant in composition and thickness as possible. V. Description of Apparatus. A small cylindrical brass chamber (see fig. 1) was made for the film in two sections with a ground nese joint, which, when covered with soft wax and pressed together, made ne vessel air-tight. Hach section was 4:1 cm. in diameter and 7-0 em. in height, having inlet and outlet tubes as shown in the diagram. The top section A, fig. 1, was closed by a window of plate glass, G, put on with hard wax, so that when a source of light was held directly over the chamber, its image in the film could be distinctly seen and in this way the character of the surface of the film—whether concave, convex, or plane—was known at once by the character of the image produced. Knowing the curvature of the film one could adjust the pressures of hydrogen and air on either side very accurately and so‘as to keep the film plane and there- fore eliminate the diffusion due to excess pressure on either side. The brass ring (, fig. 1, supporting the film was 4°95 em. in diameter, and ground down to a sharp edge. An annular channel, D, was made in the outer part of the supporting ring, and the whole soldered in the lower section of the film chamber, leaving about 0°6 em. of the brass ring projecting above the wax surface. In this way the soft wax used in making the joint air-tight was prevented from contaminating the film and destroying its surface tension. To overcome the difficulty of evaporation and drainage from the film, that is to keep its composition and thickness constant, the air and hydrogen used were both saturated with water vapour before entering the chamber, and, in addition, a means of adding solution to the film was provided in the following way. A bent tube, T, was inserted in the upper chamber as indicated in the diagr am, having a thistle tube connected to the outer end by rubber tubing. A small amount of the same soap solution used in making the film was poured into the thistle tube and a drop ei this was - 276 Prof. McLennan and Mr. Shaver on Permeability of allowed to fall on the film at short intervals (say, every two or three minutes), the flow being regulated by a clamp on the rubber tubing. The excess solution drained off the edges Bigs de —n | OY fk Cs y WN i NE N N N N Q NY Q NN NA NZ N ® N 4 N A amy yA NA Y A G yf fy SY = . So &y | ' y ST Y ' 7 oe A : ates SS g SP SS Z od p SS Z lai dei ETE Z) XSSSASSSS ZZ | ‘ VZZ2 CLLLLIA WZ, of the film into the lower section of the chamber. In this way the film was kept at a practically constant maximum thickness. and the variations in diffusion due to changes in the film were eliminated as far as possible. Thin Fabrics and Films to Hydrogen and Helium. 277 VI. Air Circuit. The air was let in through a T’-tube A (see fig. 2), which permitted the rate of flow to be varied by raising or lowering the water level. With any given level the pressure was adjusted so that the air always gently bubbled through the base of the T-tube. In its course through the system the air passed through a copper warming coil B and two tubes filled with cotton wool moistened with glycerine so as to eliminate all dust particles, as it was found that the intro- duction of dust particles soon caused the soap films to rupture. It was then bubbled through water and led through a gas- meter which measured the rate of flow. rom the gas-meter the air, saturated with water vapour, was led through the lower section of the film chamber, D, sweeping out with it the hydrogen gas which diffused through the film. From there the air was dried in the phosphorus pentoxide tubes E and H',and tested by the katharometer K, finally bubbling out through the water in H at a pressure of about 1 cm. of water. 3 VII. Gas Circuit. The gas cireuit shown in fig. 2 was somewhat similar, except that the gas was tested before entering the film chamber by a purity meter P, of the katharometer type, Fig. 2. Wires fe cell and ladicaliag mcler i/ arr ey pate Shei A OOD —S} i/) : A ee JV and then passed through the cotton wool tubes R and R’, bubbled through water in the bottle S, and thence to the film chamber, and, finally, bubbled out through water in U, at a pressure of about 1 cm. of water. The pressure of the gas in the circuit was altered very slightly by adjusting the tube U, so that the film surface remained plane. ‘The rate of flow used was about 2 litres per hour. 278 Prof. McLennan and Mr. Shaver on Permeability of VIII. General Procedure. When readings were taken the air and the gas under test were allowed to flow past the film until the katharometer reading giving the percentage of gas in the air was steady. This required one half hour, and then readings were taken at two minute intervals for forty minutes, and note was taken of the rate of flow ofair by means of the gas meter. This ranged from 2to 10 litres per hour. After one set of readings was taken, if the film still remained intact, the rate of air flow was changed, and after conditions became steady again another set of results was obtained. In this way as many as four sets of readings were taken without renewing the film. IX. Purity of Gases. The purity of the gases under test was in both cases comparatively high. The hydrogen was obtained from a commercial supply which was guaranteed to be of 99 per cent. purity. The helium used was first purified by passing it through a set of four charcoal tubes at the temperature of liquid air. Its purity was tested by means of a quartz density balance, and found to be 99:2 per cent. X. Soap Solution. The soap solution used was one made up according to Boys’ formula, and contained 2 per cent. sodium oleate, 24 per cent. glycerine, and 74 per cent. water, with a few drops of strong ammonia. XI. Results. The following are the results obtained. The last column gives the number of cubic centimetres of gas transfusing through one square centimetre of film per hour. The readings were taken at room temperature which varied slightly as shown in the table. However, taking the average values obtained for the two gases, we find the ratio of the transfusion of helium to that of hydrogen to be 0°70. In the case of balloon fabrics, Elworthy and Murray found this ratio to be 0°67. Hxpressing the average results tor hydrogen and helium in the case of a soap film in the same terms as the permeability of balloon fabrics, we find the permeability of films to these gases given by Hlworthy and Murray for Thin Fabries and Films to Hydrogen and Helium. 279 TaBLeE II. (a) Hydrogen. Transfusion of gas in c.c. Duration of film. Temp. per sq. cm. of film per hour. H. min. OF NE | ptaeneiliae patio. seater ce 18°6 48 Lh, CSG geet arate atd 20°0 5:2 LO) EES AR a A aed bedi fi 20°1 42 Whe Sole ee ae eo ae 19°6 4°3 nae. FL SI 20°8 39 PEO? 22 i3. ohh eek 19°97 4°1 BeOe SRA a & 19-1 3°8 28s G1 eed eR ASE REE 19°2 ou 19°1 31 19:2 38 19h 36 Average ...... 40 2 Ut ere 18°4 30 18°3 2-4 18:0 31 UD ior fas wb see leercevieiee 19°5 22 19'3 2'8 187 2°8 187 31 Average ...... 2°8 hydrogen to be 960 and for helium 670 litres per square metre per day. For the most highly porous balloon fabrics tested by Elworthy and Murray the transfusion of hydrogen was only about 10:0 litres per square metre per day, and of helium 7°1 litres per square metre perday. It is interesting to note that while soap films were very much more permeable to hydrogen and helium than were the balloon fabrics tested, the ratios of the permeabilities of both fabrics and films to the two gases were practically the same. This is the more interesting when it is considered that while in the case of the films the membrane was of the continuous type, in the ease of the fabric there was a possibility of the diaphragm being discontinuous. It may ba, however, that on account of the fabrics being ‘‘ doped” the discontinuity referred to was negligible. In this case the process of transfusion of the gases through the substance of the fabric would probably be of the same nature as that of transfusion through the films. 280 = Permeability of Films to Hydrogen and Helium. XIT. Diffusion of Hydrogen through Wet and Dry Cotton Fabries. Some experiments were made on the transfusion of hydrogen through a closely woven cotton fabric when wet and also when drv. When this fabric was dry the gas diffused through it so rapidly that it was impossible to obtain a measure of the rate of transfusion with the katharometer. | On the other hand, when the fabric was thoroughly wetted with distilled water it was found that the transfusion of hydrogen through it was so slow thatit could not be detected with the katharometer, even when the rate of flow of the air past the fabric was reduced to as low a value as 2°4 litres per hour. It was noted in the experiments on the transfusion of hydrogen through soap films that as soon as the film became thinner than the red-green stage the rate of diffusion rapidly increased. It is evident, therefore, that the rate of diffusion depends very largely on the thickness of the films used. In the case of the wet cotton fabries the thickness of the water films fillmg up the interstices was very much greater than that of the soap films investigated. XLT. Summary of Results. 1. The rate of diffusion of hydrogen through a series of balloon fabrics has been determined. 2. The permeability of soap films whose thickness corre- sponds to the red-green stage has been found for helium to be 670 litres per square metre per day and for hydrogen 960 litres per square metre per day at 20° C. 3. The rate of transfusion of helium through soap films has been shown to be 0°70 of that of hydrogen through similar films. 4. The diffusion of hydrogen through water films filling the interstices of a wet cotton fabric has been shown to be very low; with soap films showing interference colours the rate of diffusion of both hydrogen and helium was found to be considerable. The Physical Laboratory, University of Toronto. May 15th, 1920. page | XXXI. On the Electrical Conductivity of Copper fused with Mica. By Sub-Lieut. A. L. Witiiams, R.., with Intro- duction by Prof. J. C. McLennan, F.R.S.* [Plates V.-VII.] INTRODUCTION. 7 HILE acting as Scientific Adviser to the Admiralty, I had my attention drawn by Sub-Lieut. A. L. Williams, R.N., to some experiments made by him in the early part of 1919 at Cambridge, in which he found that samples of copper when fused with mica exhibited a remarkably large fall in resistance when gradually subjected to rising temperatures. During a short furlough he was given an opportunity at the Admiralty Physical Laboratory, South Kensington, to develop this discovery and, on going back to duty, he left with me some notes embodying the results of his work. I have not had an opportunity of communicating with him again, but as the results are interesting it is thought they should be duly recorded. His experiments are described below, and accompanying them are some additional notes of results obtained at the University of Toronto by Miss Isabel Mackey and Miss I. Giles, who have followed up the subject still further. J.C. MeL. A. EXPERIMENTS BY SuB-Lievt. A. L. Wiutiams, R.N. I. Preparation. The samples for test were all prepared in the open ona piece of iron or copper plate—used as an anode—and a carbon rod as the cathode, the are being struck at first between the plate and carbon, and then, when hot, to the mixture. The mica was first melted, then the copper added. In making up the samples studied, about equal proportions of copper and mica were used. Il. Effect of Temperature. Resistance temperature measurements for two samples were made for a range of temperatures from 27° to 850° C. For sample A, the curves of which are attached, Graphs 1 and 2 (PI. V.), the resistance fell from 16,000 ohms at 27° C., to 0°5 ohm at 850° C. * Communicated by Prof. J. C. McLennan. Phil. Mag. Ser. 6. Vol. 40. No. 237. Sept. 1920. 13; 282 Sub-Lieut. A. L. Williams on the Electrical III. Notes. ; (1) It was nected that the material was malleable at about 2000° C. (2) A specimen piece was rolled at this temperature into a small rod 2°5 mm. in diameter for the purpose of ascer- taining the specific resistance of the mixture. This was found to be as follows :— 25° C., Specific Resistance, 10,400 ohms. 30. Ce ie a 8,000 ohms. (3) An attempt was made to obtain a sample of the mix- ture in the form of a very thin film for delicate temperature measurements, etc., and it was found possible to squeeze it out to about 1/1000 of an inch between platinum foil. It was not possible, however, to separate the film from the foil; but two pieces of foil cemented together by this fine film were found to be extremely sensitive to heat. They quickly responded to the action of infra-red rays from an are about one yard away, notwithstahding the comparatively large volume of platinum to be heated first. It is thought that with suitable films of the copper-mica mixtures enclosed in hydrogen it may be possible to use them for signalling purposes. It is also suggested that these films may be used instead of wires in microphones for sound- ranging, as the changes of resistance, due to changes of temperature, are quite considerable, being some thousands of ohms per degree centigrade with some samples. (4) Attempts were made to make thin sheets by mixing the copper-mica material, finely powdered, with fine carbon, in the form of cane-sugar, and driving off the water by heating. ‘The resistance of the resulting material was extremely high, but very regular thin sheets could be obtained in this way. It is possible, when the density of this mixture is increased by compression in an hydraulic press, that it may be obtained in sheets, rods, or other forms having a moderate resistance and yet possessing a high resistance-temperature coefficient. 5. Attempts to cast the material in various forms were not successful, partly owing to the difficulty in working with the requisite high temperatures. The material, when molten, is absorbed by such porous substances as porcelain, and if glazed porcelain is used the glazing melts and mixes with the material. It is possible that castings could be obtained by using fused quartz as a moulding material. Conductivity of Copper fused with Mica. 283 6. Attempts were made to make up similar compounds with the following metals and mica :— Tin. The metal vaporized at too low a temperature. Silver. Did not combine. Platinum. Did not combine. Iron. Combined, but no resistance temperature measure- ments were made. Bb. EXPERIMENTS BY Miss MAcKEY. I. Experimental Arrangements. (a) The samples to be tested were all made in the open on a piece of iron plate used as an anode and a carbon rod as a cathode. The current was controlled by a large rheostat giving up to 30 amperes on the 110 D.c. circuit. An are was struck between the plate and carbon and, when hot, the mica was melted and the other material added. (6) A quartz tube closed at one end and. covered with nichrome wire was used as a receptacle in which to melt the material and form it into a regular cylindrical shape for experimental work. (c) A small electrical furnace was used to heat the material. It consisted of a circular porcelain foundation covered with wire and all was covered with asbestos except the two binding posts. Il. Results. (a) Mica and Copper.—Mica and copper were fused on the iron plate into small lumps, and some of these were then finely ground into powder. No traces of’mica or copper could be detected, only a uniform dull black powder. The powder was put into a quartz tube and heated, but this did not prove a satisfactory method of obtaining the mixture in the form of solid rods, as part of the mixture fused with the quartz, and it was found impossible to separate the two substances. When the quartz was broken, the copper-mica was found to be very brittle and not at all suitable for resistance measurements. Platinum wires were then fused into the ends of the copper-mica lumps which had not been powdered, and the variations in the resistances of these lumps were observed when they were raised to various temperatures. = U2 284 Sub-Lieut. A. L. Williams on the Electrical Two different samples of copper and mica were tested in the furnace for variation in resistance with temperature (but only up to about 400°C.). In Case No. 1 (Graph No. 3), the resistance was found to vary from 4400 ohms to 300 ohms, while the temperature varied from 25° ©. to 400°C. The specific gravity of the specimen was found to be 5:1, and as copper is given by 8:9, it will be seen that the specimen contained considerable mica. GRAPH No. 3. Temperature, Resistance Temperature, Resistance 2) (Ohms). e (Ohms). 22 4,400 148 1,250 46 3,600 154 1,150 54 3,200 164 1,100 58°5 2,940 206 780 61 2,900 260 611 65°95 2,800 275 450 70 2,460 315 400 72 2,400 344 340 93 1,950 364 281 127 1,550 In Case No. 2 (Graph No. 4) the variation in resistance was from 95,000 ohms to 8000 ohms, while the temperature changed from 100°C. to 400°C. The specific gravity was. found to be 4:3. GRAPH No. 4. Temperature, Resistance Temperature, Resistance 2 (Ohms). ss (Ohms). 139 45,550 228 11,370 178 23,330 211 13,585 236 11,645 208 14,570 262 F120 200 16,738 314 6,000 192 19,090 345 4,590 172 23,445 364 3,790 165 26,230 078 3,605 155 29,670 395 3,280 147 34,050 333 4,200 139 40,505 323 4,740 13 44,350 313 5,490 122 51,340 280 6,330 118 57,110 278 7,030 113 64,070 267 7,790 108 74,033 256 8,610 103 81,740 245 9,410 99 91,010 232 10,790 From these results it would appear that an increase in the mica-content of the mixture raises the resistance at ordinary Conductivity of Copper fused with Mica. 285 temperature and causes the fall in resistance with temperature to be much more rapid. While more brittle than copper, the copper-mica is not as brittle as iron-mica compounds described below. The hardness is almost the same as that of glass. X-ray photo- graphs showed the composition to be quite homogeneous. The mixture was black with a dull metallic lustre. (b) Zron and Mica.—Two mixtures were made as in the ease of the copper and mica, and the temperatures and resistances were measured as before. In Case No. 1 (Graph No. 5) the resistance fell from 1300 ohms to 100 ohms on being heated from 25°C. to nearly 300° C. GRAPH No. 5. Temperature, Resistance Temperature, Resistance (Ohms). OG: (Ohms), 26 1,350 303 91 42 980 308 90 475 890 165 235 52°5 825 138 310 54 790 116 380 104 410 97 490 134 350 75 650 138 340 83 590 178 219 64 765 195 180 55 860 209 160 51 905 218 150 45 1,010 165 280 40 1,070 214 170 38 1,110 263 115 33°5 1,203 282 105 31 1,240 294 95 In Case No. 2 (Graph No. 6) the resistance fell from 32,000 ohms on being heated from 160° C. to 380°C. GRAPH No. 6. Temperature, Resistance Temperature, Resistance > (Ohms). = (Obms), 250 6,100 294 3,380 228 9,050 282 3.900 280 3,870 255 5,970 318 2,410 242 7,390 335 1,980 231 8,440 344 1,780 222 10,380 360 1,550 214 11,720 377 1,280 204. 15,750 360 1.550 197 15,900 336 1,980 187 19,550 322 2,360 175 25,210 309 2,730 164 32,100 286 Sub-Lieut. A. L. Williams on the Electrical The hardness was above that of glass, and the material was much more brittle than copper-mica and had more metallic lustre. X-ray examinations showed the mixture to be homogeneous. The specific gravity in Case No. 1 was 3°7, and in Case No. 2 was 4. The specimens studied were quite irregular in shape, but from a rough examination of the sizes of the samples, it appeared that the sample which had the higher mica-content was the one which had the higher specific resistance. | (c) Aluminium and Mica.— No fusion was obtained between aluminium and mica. The two seemed to remain entirely separate. (d) Antimony and Mica.—The antimony when heated gave off dense clouds of vapour, leaving nothing to fuse with the mica. (e) Bismuth and Mica.—The same results were obtained as with antimony. (£) Cobalt and Mica.—Cobalt and mica were fused on the iron plate in the same manner as the copper and mica. The cobalt-mica had a very dull black colour and was very brittle, but hard enough to scratch glass. Platinum wires were fused in the ends with difficulty, and the resistance at ordinary temperatures was very great. When heated red hot with a bunsen flame, a current of about ‘020 ampere was obtained, using the 110 circuit. (2) Mickel and Mica.—When nickel and mica were fused, the substance produced was very similar to cobalt-mica. When it was heated red hot, a current of about ‘001 ampere was obtained, using the 110 circuit. (h) Manganese and Mica.—Mica and manganese did not seem to mix at all. In one test, the manganese was found to form a complete shell around the mica, and in other cases an X-ray photograph showed the two to be quite separate. (i) Stlicon and Copper.—lt did not seem at all easy, if indeed possible, to fuse copper and silicon. The two sub- stances appeared to be quite separate after fusion. (j) Seleniwm and Copper.—These fused quite readily and formed a dull black substance with very little or no lustre. The resistance was found at various temperatures and a graph, No. 8, drawn. The specific gravity was 6°6, and the hardness less than that of glass. With this mixture it will be seen that a discontinuity occurred in the resistance tem- perature measurements at about 150°C. The explanation of this result does not appear evident at present. Conductivity of Copper fused with Mica. 287 (sRAPH No. 8. Temperature, Resistance Temperature, Resistance 9 O c: (Ohms). | (Ohins). 370 “219 124 ‘261 335 ‘207 115 *200 315 200 108 ‘176 298 188 104 "172 260 173 101 169 247 "164 98 169 240 "164 95 168 230 "162 92 163 221 157 90 164 204 152 87 164 190 "tay). ~ 84 163 180 139 | 80 165 173 "136 78 167 167 "136 65 ‘176 160 "135 61 "176 154 "130 59 a hey 136 "333 58 "176 130 ‘317 44 185 127 “300 (k) Ferro-Silicon.—A sample of commercial ferro-silicon was also investigated. It was found to be very brittle and difficult to grind up into regular form for examination. In studying a sample, leading wires of iron were used, as platinum fused readily at the junction when the ferro-silicon was raised to a high temperature. When a graph was drawn between temperatures as abscissze and resistance as ordinates, the result was a straight line showing that the resistance varied directly as the temperature, just as in the case of ordinary pure metals. (See Graph No. 7.) GRAPH No. 7. Temperature, Kesistance | Temperature, Resistance ma 6 (Ohms relative). | a Oe (Ohmis relative). 280 “092 148 ‘O78 259 ‘089 139 OE - 246 ‘088 126 ‘O76 226 ‘O85 118 ‘O75 214 “084 109 ‘O74 202 083 96 073 185 082 84 ‘O71 176 ‘O81 19 ‘070 165 ‘O79 a ‘O70 CG EXPERIMENTS BY Miss GILEs. In these experiments a micrographic study was made of the plane polished surfaces of the fused copper-mica mixtures referred to above. These were made both when the mixtures 288 Sub-Lieut. A. L. Williams on the Electrical were at room temperatures and when their temperature was gradually raised by means of an electric furnace. The object in view was to see whether the fused mixtures possessed any crystalline structure, and if they did whether the increased conductivity observed with them on raising their tempera- ture could be connected in any way with observable modifi- cations in their crystal structure. I. Preparation of Specimens. In preparing these specimens they were first of all filed off to an approximately flat surface. The surfaces were then ground on a carborundum wheel, and after that on several successive grades of aloxite of increasing fineness. The grades used were those commercially known as Nos. 90, 150, 220, and 3F respectively. The polishing was then started with optical alundum and finished with jewellers’ rouge. The two coarsest grades of aloxite were used on a flat metal plate, while the finer grades and the optical alundum were used on fine even linen fabric stretched over a smooth glass plate. The rouge was used on a piece of soft, smooth broad- cloth stretched over a glass plate. The plates used were fastened on a horizontal revolving table rotated by a small electric motor. In some cases the surfaces were etched with nitric acid of various concentrations ranging from strengths of 10 per cent. to 25 per cent. and even to 50 per cent. Better results, however, were obtained by the use of ammonia in solution, with a specific gravity of about 0°93. With this solution the specimens were found to be uniformly etched by an attack of about one hour. Il. Optical Equipment. The microscope used was one of the instruments especially designed by Bausch and Lomb for micrographic work. For normal illumination the type of illuminator used was the - usual reflecting disk of thin cover glass. In this method the light was projected at right angles to the optical axis of the microscope, reflected from the cover glass alony the optical axis of the system to the specimen, and then back through the microscope. Yor visual examination the source of light was a frosted electric light bulb, while for the photo- graphic work a small carbon arc wasused. The photographic plates used were rapid panchromatic, and the shorter wave- lengths in the illuminating beam were cut out witha Wratten and Wainwright filter. Oblique, in place of normal illumina- tion, was used in some cases. 3 Conductivity of Copper fused with Mica. 289 ITI. Lesults. When examined under the microscope different specimens were found to exhibit different appearances. Most samples appeared to be quite uniform in structure, while in some many little globules could be seen, which from their lustre appeared to be pure copper. Specimens which possessed a high temperature coefficient were found both under high and low power magnification to show no change in structure, either by normal or oblique illumination, when heated to temperatures as high as 400° C. PVE, fig. 1 shows the appearance of a specimen at room temperatures with a magnification of 46. The resistance of this sample, which was 3200 ohms at 21°C., fell to 1600 when at 95°C. The structure of the specimen appeared very uniform, and no copper could be discerned in it judging by metallic lustre. Pl. VII. fig. 2 shows the appearance of this specimen when etched with ammonia solution for an hour. As pure copper was found to require approximately about seven hours’ exposure to ammonia to bring out its crystalline structure, the markings on the plate may be taken to indicate the boundaries between copper and mica or the constituents of the latter. The regularity of the markings would indicate that the copper and mica fused into an intimate and homo- geneous mass. A specimen, whose resistance at 100°C. was found to be 95,000 ohms and only 3000 ohms at 400° C., was polished and examined previous to etching it with ammonia, both with high-power and low-power magnification, and with oblique and direct illumination. Pl. VII. fig. 3 shows its appearance when illuminated obliquely under a magnification of 46. Pl. VII. figs. 4 & 5 show the same region when illuminated by normally reflected light under magnifications 46 and 205 respectively. The structure in this case, as will be seen, is quite different from that shown in Pl. VII. fig. 1. With the sample illustrated by figs. 3, 4, and 5 there appeared to be a great many streaks of light and dark, bounded by straight lines running in all directions, while in other specimens there appeared to be nothing uniform in the shapes of the patches. The portions of the surface which are dark in Pl. VII. fig. 3 it will be seen are light in PI.VII. fig.4. In this specimen much detail was brought out with the low- -power objective. It was therefore used among others with low magnification to study the effect of any increase in temperatures. A water-cell provided with 290 Electrical Conductivity of Copper fused with Mica. running water was placed between the specimen and the microscope objective, in order to cut off the heat from the objective, and the specimen was heated up to 400°C. No change could be discerned in the appearance of the etching. Pl. VII. fig. 6 shows the appearance of a portion of the surface at a temperature of 350° C. The specimen was then etched with the ammonia solution. Here, again, the surface was found to be marked by fine lines after an attack of about an hour, but no copper could be detected. It was heated again to 400°C. after etching, but no change in structure could be observed due to the rise in temperature. IV. Resistance-temperature coefficient of Glass. In studying these specimens one gained the impression that they possessed a number of the characteristics of glass. In most cases the specimens were very hard, and one could easily produce scratches on a glass plate with many of them. [t is known, too, that many glasses when strongly heated become electrically conducting, and with a view of making a comparison between the behaviour of these specimens and that of a sample of glass, some measurements were made on the resistance of a rod of glass when its temperature was vradually raised. In these experiments a rod of “Schmeltzglas ” about 80cm. long and 5:0 mm. in diameter was used. Short platinum wires were attached. These were then joined in circuit with the mains of the 110 volt p.c. circuit, and the glass portion was placed within an electric furnace. As the temperature rose observations were made on the current which passed and on the fall of potential between the ends of the glass rod, contact being made with the circuit at the platinum junctions. In these observations practically no current was found to pass through the glass until a temperature of about 300° C. was reached. Hven then the current was only of the order of 10-7 ampere, which showed that the resistance of the glass rod at this temperature was very high, practically about 109 ohms. From this result it would appear that the high resistance temperature coefficient possessed by the fused copper-mica mixtures is something specific, and it does not appear that the remarkable property they exhibit finds a direct parallel in the behaviour of glass. The Physical Laboratory, University of Toronto, May 15th, 1920. XXXII. On the Measurement of Changes in Resistance by a Valve Method. By R. T. Bearry, M.d., D.Sce., Lecturer in Physics, and A. Gitmowr, ALSe., 1851 Lehilition Student, Queen’s University, Belfast ™. HEN a valve circuit is arranged, as in fig. 1, to generate oscillations, it is well known that on the insertion of a high resistance shunted by a condenser, in series with the grid, the oscillations will be broken up into equally spaced groups, and the groups thus forined will give rise to a note whose frequency will be equal to the number of interruptions per second. If the frequency of the note be plotted against the grid resistance R (fig. 1) a curve is obtained showing decreasing frequency with increasing R, and when R is very large the sound in the telephone may only occur, perhaps, once per minute. Bre: Tl. 80 V, =a a RN, ar! OS | TEIO 077 25:0 04 | 1460 | 0°95 30:2 0°5 1380 | 1:05 340 324 Dr. W. T. David on Heat-loss by Conduction TasLe 1V.—Heat-loss by Conduction in a 9°7 per cent. mixture of Coal-gas and Air. Heat of Combustion of Coal-gas in Vessel= 10,600 calories. Heat lost by Con- Time Mean Heat lost by Con- | duction egpressed as a after Gas duction per sq. cm. percentage of the Ignition Temp. of Wall-surface. Heat of Combustion (sees.). | (°C. abs.). (Calories. ) of the Coal-gas in | Vessel. 0°18 1660 0:26 11:0 (max.temp.) | 0:2 0°31 12:8 O25) aos 0-4 16:5 0:3 1520 0:47 19°4 0-4 1390 0-59 24°4 0:5 1310 0°67 27-7 The conduction-loss up to the moment of maximum tem- perature, it will be noticed, amounts to 51 per cent. of the heat of combustion in the 15 per cent. mixture, 5°5 per cent. in the 12°4 per cent. mixture, and 11°0 per cent. in the 9-7 per cent. mixture. The much greater proportion of the heat of combustion lost by conduction to the vessel-walls up to the moment of maximum temperature in the weakest mixture is due to the much greater “time of explosion ” which occurs in this mixture; it amounts to 0°18 sec., whereas in the 15 per cent. mixture it was only 0°05 see. At 0°5 sec. after ignition the 15 per cent. mixture has lost by conduction about 38 per cent. of its heat of combustion. The 12:4 per cent. mixture at the same instant has lost about 34 per cent. and the 9°7 per cent. mixture about 28 per cent. The curves in fig. 4 and data in Table V. show the rate at which heat-loss by conduction is proceeding during the cooling of the various mixtures after explosion. It will be noticed that the weaker mixtures in the initial stages of cooling lose heat by conduction rather more rapidly than the stronger mixtures when they have cooled to the same mean temperatures as the weaker mixtures have in this epoch *. This is probably mainly due to convection currents, which, having been set up during the explosion period, are more vigorous in the initial stages of cooling than in the later stages. * A similar result was obtained in an investigation into the radiation loss (Phil. Trans. A. vol. ccxi.). in Hexplosions of Coal-gas and Air. 325 TasLe V.—Rate of Heat-loss by Conduction in 15 per cent., 12-4 per cent., and 9°7 per cent. mixtures of Coal-gas and Air. Rate of Heat-loss by Conduction per ay sq.cm. of Wall-surface.—Calories per sec. Temp. .|- a sick laa ° = Ria)! 35 °/, Mixture. | 12:4 °/, Mixture. | 9-7 °/, Mixture. 2440 87 2400 79 2200 5:0 | 2050 38 | 4°] 2000 3-4 | 3:7 | | 1800 | 2°4 2°5 1660 | 1-9 i) 21 16U0 ry | Liner 1:8 1400 1-2 | 1:2 Li! -| Fig. 4 10 @ or) b Rate of Loss of Heat & Conduction - Calories per s9.cm. per Sec. oO 2600 2400 2200 2000 1800 1600 1400 Mean Gas Temperature — °C. abso/vte. 2300 2100 1900 1700 1500 1300 100 Approx. Difference of Temperature between Gas & Vesse/ Walls (@ -300), “C. Curves showing Rate of Loss of Heat by Conduction per sq. cm. of Wall-surface per sec. in 15:0 per cent., 12-4 per cent., and 9:7 per cent. mixtures of Coal-gas and Air. 326 Heat-loss by Conduction in Explosions of Coal-gas. At temperatures above 2000° C. abs. the rate at which ‘conduction-loss proceeds in the 15 per cent. mixture is roughly proportional to the fourth power of the difference in temperature between the gaseous mixture and the vessel- walls, while at temperatures below 2000° C. abs. the rate of loss is more nearly proportional to the third power of the temperature difference. That this is so will be seen from the dotted curve which, above 2000° C. abs., is proportional to the fourth power of the temperature difference and, below ~ -2000° C. abs., is proportional to the third power of ‘this difference. The equation to the dotted curve is C=4 x 10°-4(6—8@,,)4 for temperatures above 2000° C. abs., and C=7 x 10-(@—8@.)° for temperatures below 2000° C. abs. Where C=rate of loss by conduction in calories per sq. cm. of wall-surface per sec., @=mean absolute temperature of the gaseous mixture, and @.=absolute temperature of the walls of the explosion vessel which in these experiments has been taken to be 300°. The equations given above are sufficiently accurate for rough calculation, but on closer examination the law of cooling by conduction would appear to be more complicated. In the neighbourhood of 2400° C. abs. (when convection currents are probably somewhat vigorous) the rate of loss is proportional to the fifth power of the temperature difference, whereas in the neighbourhood of 1600° C. abs. it approaches proportionality to the square of the temperature difference. . An inspection of the curve indicates that C is proportional to (0—6w)® in the neighbourhood of 2400° ©. abs., (8—Ow)* a fe OO aloes (0—Ow)? y » yy kU Gs, alos. and (0—@.) ae a e002 .calose At still lower temperatures it is probable that C would become proportional to the temperature difference simply. The experiments described in this paper were made in the Engineering Laboratory at Cambridge in the years before the war. The Laboratory was-then under the control of the late Prof. Bertram Hopkinson, and I desire to place on record an expression of my indebtedness to him for the valuable advice and assistance he so readily gave me. Pesan! XXXVI. On the Path of a Ray of Light in the Gravitation Field of the Sun. By G. B. Jerrery, M.A., B.Sc., Fellow of University College, London *. i a paper published in the May number of the ‘ Philo- sophical Magazine,’ Prof. A. Anderson gives a modi- fication of the usual theory of the motion of a planet under Hinstein’s theory from which he concludes that there should be no advance of the perihelion. The particular integral of Hinstein’s contracted tensor equations for a particle of mass mat rest at the origin of polar co-ordinates gives for the line element in a plane through the origin . df= — oy eae FOO" di, 2 (1) where y=1—2m/r. For the propagation of light we have ds=0, which gives aif pa : Ce ee a) ee so that for a ray of light making an angle y with the radius vector the velocity of propagation is v where v= ycos Ves VY) Nas abe s (2) In order to avoid this dependence of the velocity of propagation upon the direction y, it is usual to write r=?r,+m and to interpret 7, as the actual measured radius vector t. Then, neglecting the square of m/r, the velocity of propagation becomes 1—2m/7;, and is independent of the direction. The problem thus becomes identical with that of the determination of the path of a ray of light in a medium of variable refractive index w=1+2m/r,. This leads quite simply to the conclusion that a ray of light passing at a distance R from the Sun would be deflected through an angle 4m/R. Einstein’s equations of motion for a planet give ) m aoe +u=5 ere Se Se 1g 93) where w=1/r and h is a constant. * Communicated by the Author. + The introduction of 7, is also defended on the ground that to the first order in m/r it reduces (1) to the form ds? = — y—1(dr,?+7,2d6") +-ydt*, so that the scales of measurement along and perpendicular to the radius vector are the same. 328 Path of Ray of Light in Gravitation Field of Sun. If we write w,=1/r,, so that neglecting squares of m, u=u,(1—mu,) and substitute in (3), it is readily seen that to the same degree of approximation the lasi term disappears and we are left with the ordinary Newtonian equation giving no motion of the perihelion. This is Prof. Anderson’s result. It would seem, however, that criticism might be directed against the “aia ale adn of the co-ordinate r" into the first problem rather than against its omission in the second. It is, therefore, of interest to inquire what the effect of the gravitational field would be upon the path of a ray of light if we regard ras the actual measured radius vector. If we write 9’ =d0/dr, then tany=r6" and (2) becomes p 14776" i? 4 U1l+y770" § and the time between two points is = (or 1+ y7r?d'? 2 dr, where 7, 6 are given at both limits. By Huyghens’ principle ¢ is stationary for small variations of the path. We have “lenin m8 dy 7°6'60 a Ra 726! c. $1 +7"! Teese dr \ (1+ 776")? \ Cin The first term taken between limits vanishes, and the vanishing of the integral for 60 arbitrary gives 740"? = 0?(1 +776”), where ¢ is a constant. Writing uw=1/r, we have after differentiating au 792 1 wa Ome’. Se — It is interesting to compare this with the differential equation of planetary orbits. The first term on the right- hand side of (3) represents the ordinary Newtonian attrac- tion, and the second the small Hinstein correction. Equation (4) might then be interpreted by saying that a ray of light is not subject to the Newtonian attraction, but that the Einstein effect is the same as fora planet. The advance of On Variably Coupled Vibrations. 329 the perihelion of a planet and the deflexion of a ray of light thus appear, according to this view, as different manifesta- tions of the same effect. Solving (4) by successive approximations, we have, neglecting (mz)?, 1 2n m™ u=—cos (@—a) + =, — —, cos? (0—a R ( ) Lei ed ors ( ); where R, « are constants. Putting w=0, we obtain for the directions of the asymptotes, to the same degree of approximation, 7. 2m Hence the ray is deflected through a total angle 4m/R which agrees with the usual theory. It appears that the deflexion of a ray of light is the same whether we regard r or 7, as the actual measured distances, but that the difference is important in the problem of the orbit of a planet. XXXVI. Variably Coupled Vibrations : Gravity-Elastic Com- binations.—II. Both Masses and Periods Unequal. By L. C. Jackson, F.P.S.L., University College, Nottingham *. [Plate X.] I. INTRODUCTION. yr a previous paper the author described a model of a coupled system consisting of a gravity pendulum and an elastic pendulum, the separate periods and the masses of the two being equal. The present paper deals with the same system for the case in which both the masses and periods of the two pendulums are different. The model may thus be considered as analogous to the case of coupled electrical circuits, in which the inductances and periods are both different. The apparatus, though still the same in principle, has been entirely re-made and im- provements effected in certain details, as experience showed them to be desirable. The chief points in which the present apparatus differs from the one previously described, are such * Communicated by Prof. E. H. Barton, F.R.S. + Phil. Mag. vol. xxxix. p. 294. Phil. Mag. Ser. 6. Vol. 40. No. 237. Sept. 1920. Z 330 Mr. L. C. Jackson on that we can now obtain simultaneous traces of the motion of each bob on the same photographic plate. As the traces showed that the damping for the present arrangement was not negligible, it was thought desirable to take this into account in the theory. The paper includes twenty-four photographic repro- ductions of the double traces obtained for the motions of the bobs under various conditions of starting and coupling. Il, THEory. Equations of Motion and Coupling. Using the notation of the previous paper, the equations of motion may be written : Mod 4 Minty = Miiaz, - (. 3). es + (Nn?+ Mm?a?)e=Mmay. . . . (2) 2 Putting x =P. “5 =n and inserting the frictional term 2k af in the first, we have € d’y dt? d?z m* ee, 2 2 ae a + pma” jz=m*apy. dy as 2k — remy y =miaz If m?a is put equal to a and — i apes equal to h, we have : 4 d?y dy we 9 s 25,—)> qa Pek a, bi y 42) velo eee aie dae aur t a= AAW ail enliven dake ene From (3) and (4) we may write for the coefficient of coupling y, ee ee ee To solve (3) and (4), try in Ss C=C, Ls Variably Coupled Vibrations. 331 This gives = Bae nH (6) ap Put (6) in (3); then we obtain the auxiliary equation in a w+ 2ha? + (b+ m?)ae* + 2kbe+mb—a’p=0. . (7) This is a complete equation of the fourth degree, but since & is small compared with the coefficients of x? and a, we may assume a solution in the form r= ee hor aa-tegy (8) where r and s are small quantities whose squares are negligible. Thus we have the equivalent equation (x+r—ip)(e+r+ip)(e+s—ig)(@+s+ig)=0, or w+2(rts)aret(p?+ +7? 4+5?+4rs)x? +2(p?staer+r’s t+ rs*)at+ (p?+r’)(q? +s?) =0. This, on omitting the negligible quantities, becomes the approximate equation sufficiently accurate for the present purpose at + 2(r+ sae + (p?+@)a® + 2(p's + gr)a+ peg? =0. (10) The comparison of coefficients in (7) and (10) yields Pel See aus pape aw. id a4 UES Di BA ee nth ret gain ML) Pisces ehh 2 ee ar oe CB) PGi O87 p:) «oleiatiny rie (14) From (12) and (14) we may eliminate g? and obtain a biquadratic for p?, whose roots may be called p? and q?. We thus find 2p? =b+ m?+,4/{ (b—m’)? + 4a’pt 2g? =b+m?—,/{(b—m?)*+4a’p} \, ' (11) and (13) give gic p—b (19) eke Wen =->5 aoe gaan Fe k, pe pq LZ 2 332 Mr. L. C. Jackson on and by the use of (15) these become Dy tar AO ii) ae, 1 2/ {Sty $ 4a%} cb b—m?+/{(b—m?) +4a7pt lk hi, 2,/{(b—m?)? + 4a} ee Using (6) and (8), and introducing the usual constants, the general solution may be written in the form gaze" (Aer + Be?) 4+ e-8*(Cet + Dea), . (18) i, ae (17) and Aayep ab | y= Pp FO ri A epit 4. Ber Pi) ms gq +b (Cet + De-*4) ap ap a Cee Aer + Be-Pit) + SON yews Der) ay ap ap ] If small quantities are further neglected, these will simplify to e=He-™ sin (pt+e)+Fe-"sin (gé+¢),. . (20) and y=Ge "sin (pi +e) +He“sin(gi+¢). 2 (Qi (21) and (22) are equations representing two superposed simple harmonic vibrations, of which the frequency ratio is Pp ae ee (22) ae b+ m7 — V/ {\(b—m’)? + 4a} Initial Conditions. Let us consider now the form of the general solutions (20) and (21) for various conditions of starting the vibra- tions of the pendulums. (1.) Upper bob struck. We may here write y—0, 20. ia for, t=O: faaneen Introducing these conditions in (20) and (21) and into the Variably Coupled Vibrations. 333 differentiations of these with respect to the time, we find O0O=Esine+F sing Dr. 2» ; (24) 0=Gsine+Hsing v=Epcose+I*q cos oe 0=Gpcose+ Hq cos @) Equations (24) and (25) are satisfied by ines, ae. C26) Then from (18), (19), and (21), we have ed ie PN Ls A Ce M2 | ap 2ap if pra PHP potomet Viana datph. (EP 4 Ce i 1 Qap ae 3 (26) and Qa} in (25) give mines ; , (28) =Gp+ Hq whence — =-g+b if 4 i PAG ee —p' +b ia (29) 1 9ee tr) eg Thus we see that the ratios of the amplitudes of the quick and slow motions for the y and z vibrations respectively are given by > a b+m?— V {(b—m’*)?+4a?o} 7? 9 P b+m? + / {(b—m’*)? + 4a’p} | Mi 4 _ (b—m? + V {(b—m?)? + 4a’p} )(b + m? — V{(b—m’)? +407} )? | f he (b~m?— 4 (b—m’)? +40? pt)(b+ m+ / {(b—m?)?+4a’p})2 | or writing 6 for / {(b—m?)?+4a’p} G AY a = ae b+m?+8 _ (6—m?+6)(b+m?—8)? ~ (b=m?—6)(b+m?+8)2 Bint eat | | f 334 Mr. L. C. Jackson on Gi.) Lower bob struck. We may here write dy AP dz = = for t=0- UO 2): 6 ie Ree mE Then, proceeding as before, we find O=E sine+F sin $I eee (32) 0O=Gsine+Hsing) DNB eG hes Basta , (33) v=Gp cose+Hacosdh ) Equations (32) and (33) are satisfied by e= 0, o=0e k= a (34) From (27) and (34) in (33) we obtain 0O=Hp+Fq, v=Gpt+ Hq, whence = Clete Meo wee “ap ae” | | fa a ah ee = ae Ciena § Thus we see that the ratios of the amplitudes of the quick and slow motions for the y and z vibrations respectively are given by =G( =p e) be Abo (b+ m? + 0)(b+m?—6)? } 12 Gen’ Wan Wii Oca) em ites at: ess = eae ag Pa Pp a5 b+m?+ 6 (:11.) Upper bob displaced ; lower bob free. cr -. (36) ies] fes| \a=)| So We may here write yao a = 10); = =) forj—0), Then, proceeding as before, we find f= sine+F sin $ | af=Gsine+Hsing ts O=Hp cose+Fq cos $ | O= Gp cos acre a (37) (38) Variably Coupled Vibrations. 935 (37) and (38) are satisfied by | 1 T ‘ (a » 3 C= 2 5 . ° ° ° ° (39) From (27) and (39) in (37) we obtain q =H+ Rt, whence G (b-—m?+6)— 2am | | ; ~ Qaap—(b—m?—8) | r (40) _ (St + 0) —2aco 7 2aap—(b—m?—6S) } Note a iol (iv.) Lower bob displaced ; upper bob free. We may here write "0 dy dz Pa) a ==) ae =() for ¢=0. Then, wey as before, we obtain f=Esine+F sing | (E+) on ad: beet) f=Gsine+Hsing ; 0=Hp cose+F'q cos | 0=Gpcose+ Hq cos ¢. Equations (41) and (42) are satisfied by e= 5, p= 5- Ti sip, tT. - (43 ) From (27) and (43) in (41) we obtain seh hid f=G+H, 336 Mr. L. C. Jackson on @) | Qa (- ak a’p )—ar(b—mi +8) | {o—me—s t } | - Se * (b= m*—8) — 2a (~ +atp)} {bam +5 | . E_ 2a (- +a%p ) —a’(b—m'?+6) rE ‘ L b— m2 — Sh See a?(b—m*— 0) a(-+ p) | III. RELATIONS AMONG VARIABLES. ee relations between the coupling y and either the values a (the ratio of the distance from the base of the elastic aes to the point of suspension of the gravity pendulum to the whole length of the elastic pendulum) or the ratio of the frequencies of the superposed simple harmonic vibrations, i, e. p/g, may be more conveniently visualised by means of a raph. . In fig. 1 the graph showing the relation between y and « is plotted for the three cases given in the Tables I. to IIL. a separate graph being needed for each combination of p and 7. TaBLE I.—Masses 3:2, Periods 12: 7. Coupling Length Ratio Frequency Ratio Y: a. p/9- Per cent. 0 1:339 29:01 0°324 1:563 36°72 0-422 1-682 43°74 0-520 1-839 50°25 0-618 2014 Tape IT.—Masses 1:1, Periods 12:13. Coupling Length Ratio | Frequency Ratio y: a. P/q Per cent. 0 1°041 22°81 0:225 1°266 31°95 0°324 1°393 40-22 0°422 1°539 54:10 0°618 1°862 Variably Coupled Vibrations. 337 Fig. 1. 50 40 30 20 O-l 0-2 6-3 O-4 0°5 Q- ido Oo 6 Q°7 a Tasie III.—Masses 2:3, Periods 4: 7. Coupling Length Ratio Frequency Ratio Y: a. p/q- Per cent. 0 0 1311 12°69 0°127 1°338 18°68 0176 1-353 23°62 0°225 1°372 28°09 0274 1:393 338 Mr. L. C. Jackson on Vig. 2 is a graph showing the relation between y and p/q. Mig 2: 10) 5 fe) 1S 20 25 30 35 40 45 50 55 , per cent LV. EXPERIMENTAL ARRANGEMENTS AND RESULTS. The apparatus used by the author consists of a gravity pendulum coupled to an elastic pendulum, the motions of the two being recorded photographically. Fig. 3 is a general view of the arrangement. For the purpose of recording the vibrations, each bob carries a small electric lamp and means Variably Coupled Vibrations. 339 for focussing a spot of light therefrom on a photographic plate, B, placed sensitive side uppermost in a frame which can be moved uniformly in a direction perpendicular to the motion of the pendulums. The experiment is performed in a dark room and the plate developed immediately. Fig. 8. \ The elastic pendulum consists of an oak lath clamped in a vice at A and carrying a bob of special construction at its upper end. The details of the latter can be seen from fig. 4, which is shown partly in section. The end of the lath 8 is fitted with a brass cap carrying a screw T. This screw serves for the purpose both of clamping the lead weights ht between washers, as shown, and of forming the central 340 Mr. L. C. Jackson on contact of the small lamp-holder G. The light from the lamp U passes through the pinhole V and the lens W, thereby allowing a spot of light to be focussed on the photo plate by means of the mirrors D and the lens L (fig. 3). The mirrors D can be adjusted to any angle with one another, and can be clamped in such a position on a hori- zontal slide that they occupy the correct position relative to N, the latter movement being necessary, as the equilibrium position of N varies with the distance of the point of sus- pension of the gravity pendulum M from the base of the lath. The gravity pendulum consists of a steel rod which can slide in a steel tube carrying a cylindrical lead bob which supports the optical arrangements. ‘These latter are exactly similar to those of the bob N. The gravity pendulum swings on a knife edge attached to the rod P, which screws into the lath at various distances along its axis. The electrical connexions are as follows:—A wire is fastened to each side of the lath throughout its whole length, while a third wire (GH in fig. 4) runs down the back of the Variably Coupled Vibrations. 341 lath. One side wire is connected to the brass cap of the lath, and hence to the central contact of the lamp-holder, while the upper end of the other wire is free. The current enters by the wire GH (which is bent into a rectangular form at N to avoid the lead weights—see fig. 3) and passes through the lamp U and down the side wire Q to the point P, where it passes to the lamp in M via a clip on the lath, and special leads to the base of M. ‘The spiral leads to and from the lamp to the insulator IF are not shown in fig. 3 for clearness. ‘To avoid interfering with the swinging of M, the leads from the clips to F are bent into a circular form near the knife edge. The current returns through a second clip and down the side wire K to the battery at E. The masses of the lead weights which can be attached to N are such that each is approximately equal to half that of M. Thus the ratio of the masses, and with it the ratio of the periods of M and N, can be altered within certain limits. Figs. 1-24 (Pl. X.) are reproductions of the traces obtained with the apparatus under various conditions. It should be noted that while the trace of the lower bob gives the actual amplitude of the vibrations of this pen- dulum, the trace of the upper bob is proportional but not equal to the actual amplitude of its vibrations, owing to the occurrence of a proportionality factor due to the reflexion of the light at the mirrors D and its passage through the lens L. The proportionality factor is the same in any pair of photos, but is not in general the same for any two photos selected at random. The reproductions are arranged in pairs, one for the upper bob displaced, and the other for the lower bob displaced as initial conditions, and each with the same coupling. The dependence of the details of the traces on the initial con- ditions is thus strikingly illustrated in certain cases. Figs. 1-8 (Pl. X.) are traces obtained for the case in which the ratio of the masses p was 3:2, and that of the frequencies n 7:12. Except in figs. 4 and 8, the traces of the lower bob are seen to be very nearly simple sine curves, the amplitude of the second component vibration being so small as not to affect the curve appreciably, while the traces of the motion of the upper bob are throughout complex, showing distinctly the existence of the two superposed vibrations. Figs. 4 and 8 show component vibrations, the frequencies of which are very nearly as 2:1, the characteristic kink of the 2:1 curve showing plainly. Comparison with Table I. . 342 Mr. B.S. Pearson on the Advance of shows that the agreement between theory and experiment is satisfactory. Figs 9-16 (Pl. X.) are traces obtained for the case p=1 and 7=1:083. The figures show clearly the pheno- menon of beats with a progressive increase in the difference between the frequencies of the component vibrations. Figs. 17-24 (Pl. X.) are traces obtained for the case p=0'67 and »=1°45. They show well the dependence of the traces on the initial conditions, the trace of the bob which was started being in this case almost a simple sine curve throughout, while the other trace is complex. The figures show that the bob which is first started can have a very considerable amplitude of vibration without pro- ducing any great amplitude in the coupled vibration of the other bob. V. SUMMARY. 1. In the present paper, the mathematical theory of a coupled system consisting of a gravity and an elastic pen- dulum is developed for the general case, in which the masses and periods of the two pendulums were both unequal. 2. The paper is illustrated by 24 photographic repro- ductions of the double traces obtained for the motions of the pendulums under various conditions of starting and coupling. _ 8. This mechanical case possesses features analogous to the electrical. case of coupled circuits of which both the inductances and periods are unequal and can be used to illustrate the latter. Physical Department, University College, Nottingham, June 10th, 1920. XXXVI. Advance of Perihelion of a Planet. To the Editors of the Philosophical Magazine. GENTLEMEN,— N the May number of the ‘Philosophical Magazine,’ Prof. Anderson, applying the transformation, 2 r=n(1+39.), D Cieiieatec wins lyst. JeeiaeaiiteD) the Perihelion of a Planet. 343 to the equations for the motion of a planet "\ 2 dé 42m 2mh? a (5 A *(5-) =2 —t ich car eae (iT,) Pe ee ee OOM oe). Gin) reaches equations from which he argues that no motion of perihelion is to be expected, but only non-constancy in the rate of description of areas, although he does not appear to show where the supposed fallacy i in Hinstein’s argument lies. 2 2 . ee e . . Now the small additional term x in (ii.) which is intro- i eS duced by Hinstein’s theory is of order -) (for h will be of the same order as in Newtonian mechanics, where h= Vma(1—e?)); and therefore, as Prof. Anderson is ad- mittedly neglecting squares of m—1i.e. terms of order ee “it is to be expected that (i.) will transform (11.) into the ordinary Newtonian equation (7) +n “) eb lausn qlee siubleini rug) ds Ty v —_ n where c?—1 is identified with ep In fact, we are not making use of Hinstein’s theory at all, and the equation 2m dé (eyed rd Pee which Prof. Anderson obtains by applying transformation (i.) to (ii1.) has no more significance than it would have if we were treating the problem on purely Newtonian lines. {t is perhaps of interest to note that if we substitute for x from (i.) in (i.) and (iil.) and do not neglect terms in 2 nd . . . . b=) we obtain, as is to be expected, an equation of motion i giving the advance of perihelion. Equations (ii.) and (iii.) 344 Advance of Perihelion of a Planet. when combined give Behine mh? j Ges pee e 8 we Li ees r? dé r pe ? and then using (i.) and neglecting terms of order higher than CG my we obtain after reduction h? (dr, he 24m 2m "ome ie at) hipaa —l+e ry TO Plane 3 (vil. ) or writing w= ; , and differentiating with respectito 0, 1 au ca yee? a SP ua dé? MPU (vill. ) A n? If we identify c? with 1— “and note acne —, and ie u a a Ca are second order terms, we obtain as an approximate solution: u= ja(1+e cos (O—a@ )), 2 ice eee (ix.) as in Newtonian mechanics, and proceeding to a second approximation on the usual lines, integrating and neglecting terms which will have no appreciable effect in the solution, we have: = m? i j2( 1+2 cos (6—a) + 55 -e.@.sin(@— =)) me = Ta(t+e eos (@— yen ? 2 where oe and gives the advance in perihelion per revolution. Yours faithfully, Trinity College, EB. S. Pearson. Cambridge. [ 345 J XXXIX. The Ignition of Gases at Reduced Pressures by Impulsive Electric Sparks. By W. M.Tuornton, Pro- fessor of Electrical Engineering in Armstrong College, Newcastle-on- Tyne*. 1. Introduction. GNITION by discharge across a fixed spark-gap is complex function of the nature of the gas, the mode of production of the spark, and the gas pressure. ‘The discharge is intermittent ft and in part oscillatory, the first spark of the train being a bright active capacity discharge, the remainder tailing off in brightness and igniting power. Paterson and ‘ampbell’s researches deal fully with most of the variables. Their result that the first spark of the train alune can deter- mine ignition is of the first importance. The object of the present work was to examine whether the singular variations of intflammability previously observed in coal gas{ were regular phenomena of ignition of gases at reduced pressure, and if so to determine their cause. A large induction-coil was used, with the primary con- denser short-circuited, and the current broken by a switch of very rapid action. Since the speed and nature of the primary break is copied in the manner of rise of the secondary voltage, a clean rapid separation by a large quick- break switch gives a sharper and higher voltage than a break made by a slow scraping contact, and ignition phenomena are known to depend upon the rapidity of break. Platinum poles 2 millimetres diameter with rounded ends were used throughout and kept bright. The gases examined were confined to hydrogen, methane, ethane and propane, carbon monoxide, and coal gas as a mixture of some of these. 2. Nature of Disruptive Sparks. Ionization in gases has three well-marked stages which occur in succession as the electric field is increased; an approach to saturation, the electronic state, and ionization by collision. The last begins with values of X/p, volts per centimetre per millimetre of mercury pressure, greater than 50. Higher values than this were used here, at least twice as great. Before the electronic state is reached the current * Communicated by the Author. + C. C. Paterson and N. Campbell, Proc. Phys. Soc. vol. xxxi. Part IV. June 15, 1919, p. 177. See also J. D. Morgan, ‘ Electric Spark Ignition,’ Chap. IT. t “The Reaction between Gas and Pole in the E Poe ical Ignition of Gaseous Mixture,” Proc. Roy. Soc. A, vol. xcii. p. 16 (1915). Phil. Mag. S. 6. Vol. 40. No. 237. Sept.1920. 2A 346 Prof. W. M. Thornton on Jqnition of Gases at is known to be carried by ions, with a group of molecules surrounding each, which groups become smaller as the field and velocity of ions increase until there is a state of free electrons in equilibrium with inert molecules. Above this ionization is caused by collision of electrons, moving with very high velocity in the electric field, with molecules. This then is the state in which, on the eventual passage of a spark, ignition occurs. The molecules of combustible gas and air in and near an electric are are both, according to the work of Lowry* and Haber and Koenig T, ionized by oondaee with it. They then form groups in which electric recom- bination occurs. When in the present case of disruptive discharge there is a complete bridge of ions between the poles, a spark passes and ignition may or may not follow. The dividing line is extremely sharp. A change of one per cent. of the primary current frequently converts a spark, apparently brilliant but still inert, into one that gives certain ignition. Something more than ionization is needed. There is a critical intensity of spark, that is a certain number of ions produced per second to ensure ignition. Time enters as i principal factor as it has been shown to do with the slower transient ares which form circuit break sparks. The spectrum of disruptive sparks contains lines both of the gas and material of the poles, but the latter lines are secondary and arise from finely divided metal carried over by the first current rush from the poles. There is, however, as seen by the luminosity, intense energy of vibration as well as translational energy of electrons in the spark, and to start an explosion of gas certainly requires a finite liberation of energy at the source, first in the spark and then by com- bustion of the gas, sufficient to continue the latter by self- ignition after the spark ceases. 3. Ignition by Disruptive Sparks. There is then in ignition by jump sparks the following sequence of events. Tirst, ionization leading up to ionization by collision probably of all gases present, increasing until a spark passes which releases the electrostatic energy at the terminals, and by introducing a conducting path across the gap allows a current to pass equal to the momentary resultant voltage divided by the resistance of the high- tension windings of the coil. Around this current there is a magnetic field, which as the induced voltage impulse dies down decays with oscillations and prolongs the duration of the spark. * T, M. Lowry, J. C. 8S. 101. p. 1152 (1912) ; Faraday Soc. ix. p. 189 (1913). + Haber and Koeniz, Zeitsch. Evectrochem. xiii. p. 725 (1907); xiv. p. 689 (1908). Reduced Pressures by Impulsive Llectric Sparks. 347 4. Pressure and Temperature in the Spark. The temperature of a disruptive spark, like that of an are, is probably not the same throughont. ‘Towards the positive pole the spark is brighter. This brightness falls off as the pressure is reduced and eventually forms a glow corre- sponding to the positive column in a vacuum- tube. Ina steady are the heat is greatest at the positive pole, and the temperature is at most that of vaporization of the electrode. The observation* that the first spark of an oscillatory series passes through air and the succeeding sparks through the vapour of the metal of the electrodes, indicates that this initial spark has a higher voltage gradient and tempe- rature than those following. A higher temperature would to some extent explain the fact, quoted ah Le bhatt Hints first spark has greater power of ignition than the others ; it certainly has a higher voltage gradient. When the circuit is inductive the metal lines appear in the spectrum of the first spark also. Apart from energy, all records of ignition show that the discharge of a condenser with relatively low inductance in series has greater activity as a source of ignition than a single jump spark from an induction-coil. From photographs of spark discharge the actual break- down which constitutes a disruptive spark takes place along a localized narrow chain of ions. The volume of the spark as such does not appear to be inversely proportional to the pressure but is more nearly constant. ‘Thus the lower the pressure the fewer the number of molecules affected. Haschek and Mache’s conclusion} that pressures of 60 to 120 atmospheres occur in induction-coil spark discharge can be taken to indicate transient temperatures higher than those of the steady carbon arc. The pressures in condenser discharge sparks have been estimated{ to approach 1000 atmospheres, and these sparks pit platinum freely, single induction-coil discharge does not. The temperatures in dis- ruptive spark discharge are in any case much higher than are required for ignition by a steady source of heat. At pressures lower than atmospheric the rise of pressure in the spark and the temperature of discharge fall steadily. If ignition depended upon temperature alone it should increase in difficulty as the pressure is reduced until temperature is reached where a gas could not transmit flame. When ignition is by hot wires, inflammation is obtained a pressures as low as 10 centimetres of mereury, or 0°13 of < * Schuster and Hemsalech, Phil. Trans. vol. exciii, p. 189 (1899). + Wied. Ann. lxviii. p. 740 (1899). + Sir J. J. Thomson, ‘ Conduction through Gases,’ p. 517. 2A 2 j 348 Prof. W. M. Thornton on Jgnition 07 Gases at atmosphere. In the present case it was never obtained below 0°25 atmosphere. In vacuum-tubes the temperature of the bright bands of the striated positive column is about. 100° C., too low in any case to ignite gas. At some point about an eighth of an atmosphere ignition appears to fail, but the fact that with different kinds of ignition the lower limiting pressure may change from 0°13 to 0-4 of an atmo- sphere shows that the argument of a simple thermal limit to ignition will not hold. That there is thermal effect cannot be doubted, and the hyperbolic form of some of the curves. suggests that in certain cases it may be more marked than others, but the singular variations, from which no mixtures are free, show clearly that there are reactions other than thermal between the spark and gas which profoundly modify the process of ignition, especially of the simpler gases such as hydrogen and methane. 5. General nature of results. The curves of figs. 1 to 6 have as ordinates the magnitude of the primary current which when broken gives a secondary spark just causing ignition. Considering figs. 1 to 6 as a whole, and remembering that the secondary voltage of a transformer having an open @ AMPERES | hg 2 Gamage” ec ee. gwen GU” 2 4 6) 8 10 42. 14 0) 92) 4. 6 6 ee ee ATMOSPHERES. magnetie circuit and carrying a small current is proportional to this primary current, the first observation is that as the gas pressure is lowered difficulty of ignition rises, on account Reduced Pressures by Impulsive Electric Sparks. 349 partly of the lower thermal energy per unit volume of the mixture. The curves were all obtained with the same break, and the causes of any singularities are to be looked for in the gas and not in the spark alone. The second significant cr Cha eek SS ie Be COAL-GAS ce) ae ‘4 ‘Ss 8 fo 2 &4- feature is that all the curves have sudden changes or points of inflexion at about 0°4 of an atmosphere. Now it has been shown elsewhere thatthis pressure is a critical one in all ignition by transient sources, as distinct from hot wires, and since there is ionization in all these cases and no other electrical action . 350 Prof. W. M. Thornton on /gnition of Gases at that might give rise to singularities of the kind observed, the inference is that there is a change of type of ion, from clusters of molecules to the electronic state for example, or of numbers of ions, which retards ignition, finally in the case of methane, partially in hy ‘drogen ‘and the other paraffins, and gives rise to the point of inflexion in coal- -gas.. The evidence of the next two sections supports the view that any such critical points depend more on group formation than upon duration of spark, though in ignition by transient ares the latter undoubtedly has great influence. But the first and most decisive condition is that the temperature of the point from which the spark starts should be sufficient to. permit discharge of ions from the heated surface under the momentary high voltage gradient, and this discharge appears. to have a critical value at *4 of an atmosphere. 6. Hydrogen. The points on the curve of fig. 1 are the results of many observations of explosions. There are in the curve five distinct stages. Here as in steady arc phenomena atmospheric pressure is a point of transition. In passing through it the eurrent is doubled. There is from A to B no change in the current. It then begins to rise, slowly at first, suddenly at C, approaches another steady state and then goes through a violent fluctuation in the neichbourhood of 0-4 atm., reaching the limit. of inflammability at a little over 0°3 of an atmosphere. This well-marked oscillation of inflammability of hydrogen at low pressures has been pre- viously observed in the ignition of coal gus by condenser discharge sparks*, but the approach to it was then smooth. It does not occur in hot-wire ignitiont. The two steps from A to B and C to D do not occur in break-spark ignition of hydrogen. There is no sudden change of the chemical or thermal properties of the mixture as a whole to account for them. They appear to be a consequence of critical relations between time of molecular contact in collision and time of action of the spark, of the same nature as the steps in methane. From the values of the ratio e/m clusters of molecules are known to be formed around an electric charge and can to some extent be identified. The groups change in size by the successive loss of molecules, and it is in the integral relations * “The Reaction between Gas and Pole,” oe, eit. fig. 5 t “The Ignition of Gases by Hot Wires,” Phil. Mag. vol. xxxviii. p. 628 (Nov. 1919). Reduced Pressures by Impulsive Electric Sparks. 351 between the numbers of molecules in such groups that the observed sudden stages in ignition are here considered to arise. There are in hydrogen-air mixtures two such possibilities, the clusters may be of hydrogen molecules around an activated oxygen molecule, or the reverse. The molecules through which the spark immediately passes may even be dissociated by its extremely high temperature so that, as in ares, transient groups in which there are atoms may be formed. The ratios of collision frequency between hydrogen molecules in a mixture can be varied in two ways, by changing the proportion of gas to air, or by changing the pressure “of a mixture of fixed composition. A variation of action which takes place as a consequence of change of pressure can be expressed in terms of the corresponding variation of composition, ‘There was no a priori reason to expect that such a mode of interpretation would explain the salient points of the ignition curves. but it appears on making the comparison that there is a suggestive coincidence of this nature. The composition of the mixture was 29:2 per cent. of hydrogen by volume in air, giving the proportion H,+0. By raising the proportion of hydrogen a series of mixtures is obtained in which the ratio of combustible gas to oxygen has the values of the natural numbers in succession. Taking the collision frequency in the H,+0 mixture at atmospheric pressure as unity that in the other mixtures is given below. TABLE I. Ratio of Per cent. collision Observed Mixture. in Air. frequencies. steps at ts lO ae eae 29°2 120 1:0 3 GIRS 2 are 45:2 0°64 6 Bat EO winnie: oot 552 0°53 — 205 Oe eee eee 62:2 0-47 "48 BRO snake 67°4 0:43 = ce eS @ eee 70-2 0-41 “4 os Fe od 9 a 768 0°38 ‘as SO EO ©0058. bes 80°7 0°36 — The third column is the pressure in atmospheres which has the same ratio of collision frequency or partial pressure as the mixtures in the second column at atmospheric pressures. 302 Prof. W. M. Thornton on Ignition of Gases at On the other hand, the ratio may be reduced by lowering the proportion of hydrogen to oxygen as follows :— TABLE IT. Per cent. Collision Mixture. in Air. frequency. O4CG ao. tes. See A 10 CO re PRM de coro. geil 0-58 OF He, Be eee eee Pay 0-41 O 7 FH! kk. cee 9°35 0-32 On sb, es (2 0°26 Ob Ei, Sei nedes neon 6°44 0:22 Certain of the numbers in the third column of Table I. are close to those in Table II., so that in the event of these coin- ciding with the critical stages in ignition it would be difficult to discriminate between them. The curve rises suddenly at 1:0, 0°6, 0°48, falls through a mean value at about 0°4 and has a minimum at 0°38. The evidence is if anything in favour of Table I., that is, of ignition rising suddenly in difficulty with each added molecule of hydrogen. This is in general agreement with the facts of ignition by con- denser discharge sparks, which passes through critical rising steps when the proportions of hydrogen in air are 29°2, 45:2, 55°2, and 70 per cent.* And in all kinds of ignition by direct sparks previously examined excess of oxygen is favourable to electrical ignition, though it is not to final combustion, while excess of hydrogen retards it. It is therefore improbable in the above argument that the sudden jumps of difficulty of ignition are due to excess of oxygen, though they may be to excess of hydrogen. 7. Methane. The curve of fig. 2 has similar points of interest. There are steps at 1:0, 0°8, 0°6, and 0°45 atmosphere. The mixture for perfect combustion taking air as 4°85 times the con- tained oxygen is 9°35 per cent. Proceeding as in hydrogen we have :— TasBuLE III. Per cent, Observed Mixture. in Air. Ratios. steps at, CHF Oz! esses 764 1:22 1425 CAO) eiiinc sacha 9°35 “1:0 1:0 CUT AOS. sateregcas «inate 12:0 0:78 0:8 CTH AO) cele ociestaae ile} 0°54 0°6 2XO)18 Vici OS BR Be MERE ee ee 21°6 0°43 0:45 * “The Ignition of Gases by Condenser Discharge Sparks,” Proc. Roy. Soc. A, vol. xci. p. 17 (1914). Reduced Pressures by Impulsive Electric Sparks. 353 There is here an agreement between the possible ratios and the pressures at which steps were observed which is on the whole too close to be accidental. It is possible that the slower rate of initial combination of methane and oxygen is favourable to the production of these well-marked stages. Similar steps have been shown to coincide when ignition is by condenser discharge. An action of this kind is not to be expected, though it cannot be ruled out as impossible, when ignition proceeds by general bombardment over the wave-front as in an estab- lished travelling flame. It is most likely to occur when in addition to the normal time of combination of the two gases another element is introduced, in this case the time of duration of the spark and of the ionization that leads up to it. The explanation now offered why the electrical spark impulse necessary for igniting gas changes suddenly when the proportion of gas to ¢ oxygen goes through certain values, is that in the passage of a spar k ions necessary for chemical combination are formed, and that before these have recom- bined electrically and become inert, combustion has begun. The proportions of gas and oxygen in the upper and lower limit mixtures show* that in these mixtures there is a sudden change of inflammability determined solely by the ratios of the numbers. of oxygen atoms to one of combustible gas. What is now found is that when the source of ignition has a very short duration these steps occur whenever the proportion of oxygen atoms to combustible gas passes through integral values such as the first few natural numbers. Change of gas pressure has the same effect on the rate of group formation as change of proportion of the mixture. From a purely electric point of view steps might occur when the conductivity of the sparking path changed suddenly. It is only possible to imagine such a change. taking place after chemical combination has begun, but if the rate of development of the latter is so slow that a definite time of duration of spark is necessary to allow it to work up to a state of self-ignition, it might appear to have an electrical origin and to explain Paterson and Campbell’s stepped ignition curves as depending on an instantaneous change of capacity, but the origin of their steps and those now observed would be the same. * “The Limits of Inflammability of Gaseous Mixtures,” Phil. Mag. vol. xxxili. February 1917, p. 190. 354 Prof. W. M. Thornton on Lynition of Gases at 8. Ethane and Propane, and the cause of their oscillation. When there are no singularities such as those first dis-* cussed there is the oscillation of inflammability found in the neighbourhood of 0-4 atmosphere pressure. Such a variation has been observed in the case of ignition by transient ares and it is there much more pronounced than in fies. 3 and 4, for ethane and propane. As the pressure is lowered the are goes through a remarkable change of condition and is for a time unstable. Its length and dura- tion suddenly diminish almost with discontinuity. As the pressure is still further reduced the are again lengthens and changes type from a sharply defined discharge to a glow. This eventually extends so that when the mechanical break is about one and a half centimetres the are persists as a brightly luminous though cool discharge, showing striations or pulsations of brightness as it is drawn out. From examination of some eight hundred photographs of transient arcs at low pressure it became evident that as the pressure is reduced the temperature of the point of the cathode trom which the arc starts is lowered until a point is reached where it is insufficient to start a discharge of metallic vapour or of Jarge ions through which the are can be maintained. The extreme suddenness with which this state is reached resembles that observed by Sir J. J. Thomson at lower pressures*, where a small change of cathode temperature suffices to lower the current to a hundredth of its previous value. ‘his point corresponds to an increase of difficulty of ignition because of the smaller size and less duration of the are. Following this period of inaction there appears with almost. equal suddenness a discharge similar to an are but without the distinctive features of one started by a hot cathode. It is rather the commencement of the positive glow and is a relatively low-temperature discharge in which ionization is active. Disruptive spark discharge at low pressure does not show any such changes of length, the product of spark length and pressure is constant. When, however, the spark-gap is fixed and small the magnitude of the primary current is a measure of the current in the spark, and the temperature of the points as affected by the magnitude of the current changes with it. So far as this is the case the action is not unlike that of a transient are. The ordinates of the curves of ignition of ethane and propane show sudden fluctuations at nearly the same point. * “Conduction of Electricity through Gases,’ p- 480. Reduced Pressures by Impulsive [electric Sparks. 355 That of ethane is a definite increase of inflammuability, that of propane is not more than an indication of its presence. ‘The propane curve is otherwise an approach to the hyper- bolic form associated with ignition by condenser discharge at low pressures. Comparing the hydrogen and methane curves in view of the above, the commencement of the glow stage, which is probably that of ionization by collision rising suddenly to saturation, is seen to be unfavourable to the ignition of methane, favourable to that of hydrogen. Now ‘according to Bone’s accepted theory*, the combustion of methane pro- ceeds by the successive addition of atoms of oxygen to the hydrogen of the molecule. In the normal high- -temperature are or spark oxygen is dissociated and atomic oxygen is free to combine and start self-ignition. As the pressure is reduced and the sparks become ‘cooler there is less of this action and in the glow discharge high-temperature dissocia- tion will almost cease. There are then present molecules and free electrons, a state which is perhaps less favourable to combination than when oxygen is atomic. On the other hand, Bone has shown that “in explosion flames hydrogen is directly burnt to steam as the result of trimolecular impacts,” and a phase in which there is ionization of molecules of gas by collision with electrons without disso- ciation is favourable to such an action. It has been shown elsewhere that as the gases rise in the paratiin series lenition, which became suddenly more difficult in the step from hydrogen to methane, becomes easier, and that pentane approaches hydrogen in sensitiveness and type. Ethane and propane are intermediate, and it is not difficult to follow the transition from the methane curve of fig. 2, by that of ethane to that of propane, falling in towards the origin and showing, though to a much smaller extent, the fluctuation so marked in hydrogen. 9, Carbon Monowide. The curious cusp in fig. 5 has been found in carbon monoxide with more than one kind of ignition. It is a sudden but not permanent increase of difficulty of ignition occurring at much the same pressure as the change first discussed. It would appear to depend more upon the varia- tion of spark intensity, or temperature of the pole at the * W.A. Bone and others, “Gaseous Combustion at High Pressures,” Phil. Trans. ser. A, vol. 215. p, 304. 356 Ignition of Gases at Reduced Pressures. point of sparking, than upon anything else; in this resembling the oscillation of hydrogen at the same pressure. ‘he cause of this has been discussed in § & 10. Coal Gas. This is a mixture containing about 50 per cent. hydrogen, 35 per cent. methane, 10 to 15 per cent. carbon monoxide. Its curve of ignition, fig. 6, shows clearly the influence of hydrogen and methane but not of carbon monoxide. The magnitude of the currents and the shape of the curve at pressures not much below atmospheric resemble hydrogen. The straight part of the curve is a mean line through the methane steps. The point of inflexion occurs at the same pressure as the oscillations in hydrogen and carbon monoxide, and gives to the curve a form similar to that of the pressure-volume curve of a gas at the critical temperature. In the present case the analogue of temperature is the reci- procal of time of duration of the igniting impulse. The curve is of interest in showing the modification of the hydrogen type by methane, though the gas in all essentials behaves as hydrogen. 11. Conelusions. Ignition by impulsive sparks is on the whole more difficult at low pressures, in this differing from ignition by hot wires. In hydrogen and methane the increase of difficulty takes place by well-marked steps or stages. In ethane and propane these do not occur, but there is an oscillation of the curve, found also in hydrogen and carbon monoxide. ‘This appears to be associated with the temperature of the sparking point during the passage of the spark. It has been shown previously that at pressures well above atmospheric sudden changes of inflammability occur when ignition is by impulsive sparks or condenser discharge. It is now found that there are such steps at reduced pressures and that there is a close agreement between the partial pressures corresponding to the | points at which oxygen has ratios to the mass of combustible gas expressed by successive natural numbers, and the pressures at which sudden changes of inflammability occur. fii BBM. 1] XL. The Variation 07 the Specific Heat of a Gas with Temperature. By Grorce W. Topp, M.A.(Cantabd.), D.Se.(Birm.), Professor of Experimental Physics, Arm- strong College, Newcastle-on-Tyne*. VENUE criticisms which are frequently brought up against the principle of the equipartition of energy among the degrees of freedom of the molecules of a gas are the experi- mental observations (a) that the molecular heat varies with the temperature when it should be C= &(5R8) =;k, and (b) that the ratio of the specific heats of a gas also varies with the temperature when it should agree with the 2 equation y=1+—, n being the number of degrees of n freedom. It is often argued that since one cannot have a fractional degree of freedom, then if C and vy do change with temperature, the changes should take place in steps. It will be shown that the criticisms are unjust and that the principle will account for the available experimental facts, as well as does the quantum theory which discards altogether the equipartition principle. The Specific Heat at Constant Volume. Adopting a method of treatment similar to that used in previous papers t, we will assume that the gas molecules have three degrees of freedom so long as their velocities are below a critical value. Above this critical value we assume that the collisions result in added degrees of freedom (e. q. rotation produced). Let the number of degrees of freedom added be g. The number of molecules per c.c. with velocities between e and c+dc is me fai el Bry, 9 ee Sad (sho 496 & where the symbols have their usual significance. . AC, * Communicated by the Author. + Todd & Owen, Phil. Mag. vol. xxxvii. p. 224, and vol. xxxviii. p- 655, 358 Prof. G. W. Todd on the Variation of the The translational energy of these molecules is cheretnne mec? m \22 ——~ Nol ong RO ot | de, So that the kinetic energy of molecules in 1 c.c. having velocities greater than a critical value c is y) mez ae -Nm (ona a a @.dc, where 6=e — BRO ot The total energy of these molecules is 3+q 2 m 2 ( ages mloea) |, ¢-a The translational energy of molecules in 1 ec.e. with velocities less than ¢ is and since by re hen these have only three degrees of freedom, this is also their total energy. Hence the total energy of 1 c.c. of the gas 1s 2 m \3/? (a . if \ =~ Nm (— ee sal OC ee (To) yee ona) bap! @ . de re ( es: It follows that the Specific Heat at Constant Volume will be given by Bf LO sae! me Ldn “CFL, # det (oa) f- When g=0, the equation becomes 2a seaike aa B18 ye Oe | m 2m which is the usual expression for a monatomic gas. Now the integral 2ROV? (2 = / ae \¢ .do= ( a4 { ee wide, where v=c TL i= Specific Heat of a Gas with Temperature. 359 -so that The integral | b.de= | b de | dh .de a9 v 9 Ngee Shee wow ma eee 6 | ey lia ela st \ eee reat) Hquation (2) now becomes 2 eee a Ky= /t 56 | m | 8 E ee sob e 8 | {Elgt+ SBie-? 2m er 6 6 3 p-tg-34 2 49! t | +p tty Se-igtl |. (8) MC where K= ~—. 2. Eucken + has determined experimentally the molecular heat of hydrogen at low temperatures, and he obtains the very important result that at very low temperatures the gas behaves as if it were monatomic. Some of his values are given below :— Cane” <. din 401545, 50), 1605 65, 70 80), 90) 100 1110 196-5 273 mol, ht.... 2°98 2°98 3°00 3:01 2:99 3:04 3:10 3:14 3:26 3:42 3:62 4:39 4-84 If we put E=625 and g=2 in equation (3), we get the following results :— ee ee 9 36 64 81 100 121 144 196 256 7S (5 =): 1-000 1-000 1-009 1:043 1113 1-215 1:385 1542 1-677 Zm * See Todd & Owen, Phil. Mag. vol. xxxvii. p. 227. + Eucken, Sitzungsber. kon. preuss. Akad. Wissenschaft. 1912. 360 Prof. G. W. Todd on the Variation of the These figures give the continuous line in fig. 1, and Kucken’s experimental values are shown as small. circles. i ae) ae evas eo [beh Dae LL ee edie 0 100 120 140 160 180 200 220 Kiowa 45 40 35 Mol. Heat. 3:0 The agreement is good. Of course, if further degrees of freedom develop at higher critical molecular velocities, the expression for KX, will be modified, and the theoretical curve, instead of running asymptotically to the value representing five degrees of freedom for every molecule, would go on rising. The development of further degrees of freedom would explain Bjerrum’s* values for hydrogen at very high temperatures. The Ratio of the Specific Heats. Whatever we assume about the number of degrees of freedom of the gas molecules, the ratio of the increase in translational energy to the increase in total energy of the gas molecules is NOE 3) (Ko a2 where y is the ratio of the specific heats of the gas. If every molecule has the same number (8+ 9) of degrees of freedom, then Gal), 3 é Me mkle 2 po tia gt = Ss ee o1 (eae te * Bjerrum, Zeit. f. Electroch, xvii. (1911), and XVII aol Speciic Heat of a Gas with Temperature. 361 Take our case. The increase in translational energy for a rise of 1° C. is 3/2 Fe = Nm “sre o) J, Ke . de 2 m \22 Ce” Be ae Nm (sre) { gp, - de Ar = SNR. The increase in total energy for a rise of 1° (, is a (open a dallvtecd “dour pe Nm (sR¢ Meals hid i+ |" $y-de =2nr4 SOR [ety (Saderv + ome ™ ree: =—7,2 1 —72 =; ark —x? ce *) — 8 (5.0% side Pee iit 1 ae 6 Therefore A(Tr) _ wiC emitter! i Fy. aA A (To) ws ue [ (@+1) (a2, terms) —0@(a terms) | 3/1 2 ana = «y= f+ 89 - (4) at aa [ (@ +1) (a, terms) — O(aterms) | We are now in a position to calculate the values of y for hydrogen at low temperatures. Putting H=625, we find 625 (8+1)(x,terms) =e °+147800(6+1) +18°75(0+1)?+:015(6+1)3}, and a similar expression for @(xterms), the only change being @ for (@+1). After substituting these values in equation (4), we obtain the following table :— Calculated values of y for hydrogen at low temperatures. @ able, .:.... 25 81 100 121 144 169 225 Ee etidasian ess 1:666 1-645 1-608 1-545 1-498 1°458 1-408 The change in y with the absolute temperature @ is shown Phil. Mag. Ser. 6. Vol. 40. No. 237. Sept. 1920. 2B 362 A Vapour Pressure Equation. in fig. 2. Unfortunately there are no experimental figures to put beside these calculated values. , Se 2. er 60 — 60 —— i Apparently hydrogen is the only gas for which there are experimentally determined values ‘of the molecular heat at low temperatures. It will be interesting to see, when the data become available, whether the kinetic theory will suffice for other gases and to see whether any relations exist between their critical energies and their physical or chemical properties. Armstrong College, Newcastle-on-Tyne, April, 1920. XLI. A Vapour Pressure Equation * To the Editors of the Philosophical Magazine. GENTLEMEN,— le je the paper by George W. Todd and 8S. P. Owen on ‘A Vapour Pressure Equation,” which appeared in the ‘Novena issue of the Philosophical Magazine, the authors developed the equation a2 B N,=Nie *(14 5), elie, ie. oe boeken (1) and from the fact that N, «

p, the potential difference between mercury and solution and the charge of the mercury surface are both negative ; when the mercury surface is increased ions enter into the solution and the potential of mercury therefore increases. If P 0 (2) we obtain OY : Ow =e-+ 1 Hel, ° ° . . ° (4) an equation which can be regarded as a generalized equation of Krueger. It is easy to show that the second term of the right-hand side of equation (4) may sometimes be of importance. Let us consider a drop of zinc amalgam immersed in a solution of zine sulphate. In the solution and in the surface-layer * The position of the dividing surface is chosen as to make I'y 6 zero, 372 Mr. A. Frumkin on the there are ions of two kinds: Zn‘: and SO,", so that we can ae e=(—Ty,. + Tso.) F 5 further T= Dee ae and if we neglect the influence of undissociated molecules rt ? whence ay ap =P go .F ee With the help of this equation we can calculate the maximum possible value which oY ean have when zine amalgam is immersed in »/1000 ZnSO,. The amalgam is negatively charged, in the surface-layer there is therefore a depiciency of SO," ions. We may assume the thickness of the double layer to be approximately equal to 10-7 -em. The absolute value of Ty, is at any rate less than 1077 em. x 10~° gr. eq./c.cm.=10-™ gr. eq./em.?; this corresponds toa value of OY equal to 10~* coulomb/em.?= 1071 dyne/volt em.» which can be neglected. The surface-tension of a zinc amalgam immersed in a dilute solution of a zinc salt does not depend therefore on its potential, nor, in consequence, on the concentration of the solution. In this particular case OY . aN Si is zero because the quantities « and I'y,F have equal absolute values and opposite signs. Let us suppose now that in the solution there is a great excess of a salt with another cation, for instance, Na,SOx,, then €= — Cie —= pec + Deon )F, or, if the concentration of ZnSO, is low as compared with that of Na,SQ,, c= ( Th gt THE As there is now an eacess of Na: in the surface-layer, the value of € is no longer limited and the surface-tension will vary with Ww and in consequence with the concentration of ZnSO, To verify this inference I measured with the * The reasoning above is not quite correct, the concentration of Zn being, contrary to the assumption we have made previously, of the same order of magnitude as the concentration of SO,". Nevertheless, as it is easy to show, equation (5) holds if we denote by w the potential difference between an imaginary SO,” electrode and the zinc electrode. Besides that does not affect the following considerations. Theory of Eiectrocapillarity. 373 capillary electrometer the surface-tension of a zinc amalgam in solutions of ZnSO, and Na,SO,. The results are given in Table IJ. (the maximum surface-iension between mercury and water is assumed to be 100). Tasie IT. AE EN 2) GAS a ae A i EOCO wetesO iL 91°2 mn, ZaSO =. Na, SO Ke ‘91:0 1/1000x. ZnSO,+ 7. Na,SO,......... 89°8 The experimental values are in agreement with the theory; unfortunately, the lack of mobility of the meniscus does not allow of rendering these measurements more accurate or of extending them to still more dilute solutions. Let us now consider the mechanism of working of a dropping electrode in these solutions. ‘To simplify the problem we may neglect the presence of SO,’ ions. If a drop of amalgam is immersed in a solution of zine sulphate, zinc ions enter the solution, but, as no other ions are present, they must remain in the double Jayer to counterbalance the negative charge of the amalgam. ‘The formation of a new surface is there- fore not accompanied by any change of concentration, and the potential of a dropping electrode is equal to the potential of a still one. The conditions are quite different when there is in the solution a great excess of sodium ions; these take the place of the zinc ions in the double layer, and the zine ions enter the bulk of the solution. Thus, in presence of an excess of sodium ions the formation of a new surface lowers the concentration of sodium ions and increases the concentration of zinc ions. The result isan alteration of the potential. Table III. contains the potential difference between dropping and still amalgam in different solutions of ZnSO, and Na,SO,, the amalgam jet being surrounded bya hydrogen atmosphere. TABLE III. G00 Ls ZESOW 3: k eis pee.» 0:003 volt 0-001n. ZnSO,+2. Na,SOy......... 0:037 ,, WOOO, AaSOy. cee. cs. hos etd GOT; 0:0001n. ZnSO,+7. Na,SO, ...... 0:055 _—, We see that according to the theory, the addition of Na,SQ, greatly increases the potential difference; the small potential difference observed without Na,SO,is due probably to the presence of SO," ions, which we neglected in our . 374 Mr. A. Frumkin on the reasoning. Thus the term I'F'is of importance if the double layer is built up by the same ions whose concentration determines the value of the potential difference, but so far as we are dealing with pure mercury its influence is pro- bably very small. In fact, along nearly the whole of the electrocapillary curve the mercury concentration in the solution is very low as compared with the concentration of other ions. If we suppose with Krueger that the anomalies ot the electrocapillary curves of anorganic electrolytes are due to the adsorption of mercury, we must assume that salts of mercury are much more adsorbed at the mercury surface than any other known substances. Thus, the addition of n/100 KI to n/1 Na,SO, lowers the surface-tension whicb corresponds to yr=0'6 volt by 5 per cent. ‘To produce such a lowering effect, a very active substance like amyl alcohol ought to be present at a concentration as high as M/10, whereas the corresponding coneentration of mercury in the solution, as we may easily calculate with the help of Nernst’s formula from the experimental data of Abegg and Sherill*, is 10-16, Therefore it seems to me reasonable to admit that the lowering effect is not caused by the mercury salts, but by the KI (viz. by I’), whose concentration is high enough. Krueger quotes in favour of his theory that the anomaly of the electrocapillary curves increases with increasing stability of the corresponding complex salts, 2. e. with the concentration of mercury in the solution at constant potential; but it is easy to show that this relation does not hold. In fact, the stability of complex salts increases in the following order: nitrates, sulphates, iodides, cyanides, whereas the maximum surface-tension of normal solutions are: KNO;, 98°95; K,SO,, 100:17; KI, 94:0; KCN, 96°6. In the following we will make the probable assumption that the adsorbability of mercury salts isa quantity of the same order of magnitude as that of other substances and neglect the term [y,F. Equation (4) becomes then identical with the classical equatiom (1) oY =, |. eae sales heen OT ae (1) eS (CP a ironed In equation (1) ¢ is a function of wf, whose form is deter- mined by the composition ef the solution. In particular, the value of ww which makes é zero may vary between very wide limits (0°2 v.-l v:). * Abegg’s Handbuch der anorganischen Chemie, Bd. ii. Abt. 2, p. 648. where Theory of Electrocapillanity. 3795 When mercury comes in contact with the solution, ions of mereury will enter the solution if e<0, although their osmotic pressure may be greater than the ionic solution pressure of mercury; if e>0, they will be removed from the solution, their osmotic pressure may thereby be less than the ionic solution pressure. ‘Thus we see that, whilst the value of the potential difference between the solution and the mercury is determined only by the osmotic pressure of ions of mercury, contrary to Nernst’s theory the direction of the reaction Ho _, Hg9" + 28, which takes place when a new surface is formed, depends on the sign of e, and therefore on all active components of the Seiion. Also, contrary to Nernst’s theory, the existence of potential difference between solution and mercury is not at all connected with the exchange of ions of mercury. In fact, if the concentration of the ions of mercury corresponds to the zero value of €, no ions at all are exchanged when a new surface is formed ; the potential difference which we must assume between solution and mercury to account for the displacement of the maximum can be caused only by adsorbed layers of anions and cations: we are justified in calling it adsorption potential difference. In conclusion, I gladly take this opportunity of expressing my gratitude to Professor A. Sakhanow for the interest he has taken in the progress of this work. Laboratory of Physical Chemistry, The University, Odessa. XLII. On the Theory of OL ie Il. By ALEXANDER FRUMKIN * S Gibbs has stated, there exists a simple relation between the adsorbed quantity of a substance and the lowering effect on the surface-tension, expressed by the equation re ta et Oy where I’ is the excess of the solute in gram-mols. per cm.” of the dividing surface, c the concentration of the solute, and y the surface-tension. * Communicated by the Author. 376 Mr. A. Frumkin on the Lewis’s* investigations have shown that lowering of surface-tension at a liquid/liquid interface is really accom- panied by adsorption, but the observed value of T’ was nearly always much greater than the calculated one. Only non-electrolytes of a small molecular weight like caffeine gave satisfactory results. Lewis supposed that the observed discrepancy is due to gelatinization and to electrical effects. Unfortunately, the accuracy of Lewis’s method was very limited ; further experimental investigations would be of great interest. The reasoning of Gibbs neglects the electric charge of the dividing surface. Gouy f showed that an analogous equation may be deduced for charged surfaces, if the potential difference between the two phases remains constant when the concentration of the active substance is varied. Gouy deduced therefrom that the electrocapillary curve of a solution becomes larger with increasing dilution by a constant quantity, independent of the value of y. We will show that, using Gibbs’s equation, it is possible to calculate the horizontal distance between the ascending and descending branches of electrocapillary curves at different concentrations, and that the calculated values are in fair agreement with the observed ones. We must first consider what form Gibbs’s ie. equation will have if we take separate account of the absorp- tion of the anion and the cation. For this purpose let us use anarrangement already employed by Chapman } (fig. 1), A mercury drop and two electrodes from Na and Cl] are * Phil. Mag. (6) xv. p. 499 (1908); xvii. p. 466 (1909); Zeit. phys. Chem. 1xxiii. p. 129 (1910). + Journ. de Phys. (3) x. p. 245 (1901). t Phil, Mag. (6) xxv. p. 475 (1913). Theory of Electrocapillanity. 377 immersed in a solution of NaCl. The electrodes are con- nected with the mercury by means of two condensers. The potential differences between the solution and the Hg, Na, and Cl electrodes are respectively w, Wi, Wo. The quantity of electricity which has passed through the solution since a certain moment from 1 to 2 is K,, and from 1’ to 2' E,. The concentration of the NaCl, which we shall assume to be completely dissociated, is ¢ and the surface area of the mercury s. The state of the solution is wholly determined by the values of yw, c, and s. If we increase s by ds, keeping ar and c¢ constant, and ¢ by de keeping wy and s constant, the work performed will be dA= [yt (pow) 4 +4) Se | as + [rw + pw SP] ae whence, as yr, and Wy, are functions of c and yf, but not of s, By_ Bhd WW» ay Oc noes 0c Os where OWT) Oa OE ‘ Aone Resim 14 corde “dis ¥)> (3) E When a quantity of electricity equal to clef passes 1 dE es from 1 to 2, F “a, a8 g oram-equivalents Na enter the solution and as many gr. eq. Hg are removed from it ; likewise, when a aoe of electricity equal to On ds passes from Oe to 9! : 1 OH ’ F Os removed from the solution. On the other hand, when the surface is increased by ds, E, and E, being constant, T'y,ds, ds gr.eq. Hg and as many gr.eq. Cl are Todds, and = ds gr.eq. Na, Cl, and Hg respectively, are 7 removed from the solution ; in consequence, as and ¢ must remain constant during the increase of s, St ee a a S Liang) (Ol 4 BBs) pie Hchielt ats x. 28 a) Phil. Mag. 8. 6. Vol. 40. No. 237. Sept. 1920. 20 y= eae Da ap P 378 Mr. A. Frumkin on the Oo, Ovni gov E f fe) 1 On substituting the values o a) ae an Me from (3) and (4) in (2) we obtain res G RP =— Cmts - es 2. €. Gibbs’s equation. Let us now suppose that the valency of the anion is n, and the valency of the cation n, ; instead of (3) we must i Ovi RT oy, __ RT Oc y, Hamel Fy Oc ~ i me If we express the quantities F in gr. eq. per cm.” equation (4) will keep its form, hence a oY -(-) -(). ae Let us compare (5a) with the equation of the electro- capiulary curve I, Descending branch. The mercury is charged negatively. In the surface-layer there is an excess of cations and a deficiency of anions, so that [, has a negative value ; the absolute value of I’, is at any rate less than cé, where 6 is the thickness of the surface- layer and may be neglected if the solution is a dilute one * This agrees very well with experiment, as the form of the descending branch, at a certain distance from the maximum, does not depend on the nature of the anion. In consequence we may put I’, =0, whence Oy _ Foy Oo RT 93 9 loge’ c The integral of this equation is Rll v=f(e+ pes e). 9a) + tesa K If y1=%05 RT RT y+ (ee C= Wet pare’ C95 whence RT 206, any ee ae roy naif OE, eee ey In consequence, the horizontal distance between the descending * In fact, if a cé will be of the order 10-1? gr. eq./cm.?, whereas the quantity ran oe is of the order 10-?° gr. eq./cm.?. o¢ Theory of Electrocapillarity. 379 branches of two solutions, whose concentrations are ¢, and ¢s, . c . is 0°057 logiy — (¢=15°) for a monovalent cation and C9 0°029 logy) — for a bivalent one. If we take account of the Co ~ incomplete dissociation, we must replace c by ac, where a is the degree of dissociation. In Table I. are given the values of w.—h, which correspond to different values of y (the maximum surface-tension between mercury and water is assumed to be 100). The measurements were carried out with a capillary electrometer as described by Gouy™*, the large mercury electrode being always immersed in a 7/10 solution of KCI. Let us now denote by the potential difference between the mercury in the decinormal calomel electrode and the mercury in the capillary tube, by 2 the value of the current which passes through the capillary electrometer, and by w its internal resistance. Then, obviously ~y=applied E.M.F. —iuw. The value of w was calculated, that of 2 determined ier 2. 100 ‘90 80 70 05 ’ rs with an Edelmann string galvanometer. The term ww could be neglected at higher concentrations, but with /1000 solutions it amounted to 0:01 volt and more. The electro- capillary curves of K NO; are plotted on fig. 2. * Ann. chim. phys. (7) xxix. p. 178 (1908). aC 2 J ad 380 Mr. A. Frumkin on the TABLE I. | KNOg & 215 9iChiae2 )) y- m—A/10n. ~ 1/102—1/100n. 1/100 2 — 1/1000 ». 99:0 ae 0-045 0:049 98:0 0:037 0:051 95:0 0:021 0:052 90:0 0:022 0-048 85:0 0:025 0051 80:0 . 0:035 0-052 75:0 0:035 ie i a 0051 0-054 | 0:056 1 2C4.G Ba(NOQ3)>. #=15°. Y: 1/10 n—1/100 ». 1/100 2 —1/1000 n. 99-0 0:020 0:032 98:0 0:020 0:027 97:0 0°023 0-029 95:0 0:026 90-0 0:022 , 86:0 0-025 ik a ee ut 0-028 We see thus that the observed values of w.—, are in a satisfactory agreement with the calculated ones, except for normal solutions. At higher concentrations the term Ll may be of importance, or perhaps in this case Gibbs’s equation is no longer valid. Let us now consider a zine amalgam, immersed ina solution of zine sulphate. Here ele wr =const. — Ti log ¢, whence mn -— = loo ¢— const: v mie Se 4 and y=const., i. e. the surface-tension of the amalgam does not vary with the concentration of tne solution, a result which we have already obtained in a different way. Theory of Electrocapillarity. 381 II. Ascending branch. Applying to the ascending branch a reasoning similar to the above, 7. e.assuming T, to be zero, we obtain the equation ; RY y=/ (¥- 7 FOR :) ? whence RI It appears that in reality, especially with active * electro- lytes, the assumption T,,=0 does not hold for the ascending branch and that there is an excess of both anions and cations in the surface-layer, when a is positive. We are induced to admit this if we consider :— (1) Lhe position of the maximum.—lIn solutions of electro- lytes with an active anion like Br’, I’, ON’, SCN’, the maximum is displaced to the right, as compared with solutions of electrolytes with an inactive anion, like NO,', SO,”, OH’. Thus the maximum surface-tension in n/10 KNO; corresponds to w=0°57 volt, in n/1 KNO, to Ww=0'61, and in n/1 KI to ~=0°87. Let us consider the portion of the electrocapillary curve of n/1 KI between 0°57 and 0°87 volt, supposing that the first value of w really corre- sponds to the zero potential difference between mercury and solution. If 0°570, and, in consequence, the potential of the mercury must be higher than the potential of the nearest layer of the solution; as the whole potential difference between solution and mercury is positive, the potential in the surface-layer must vary with the distance from the mercury surface in a way shown by fig. 3. The rise of potential can be caused only by free positive charges, and in consequence we must assume an excess of anions immediately at the mercury surface and at some distance from it an excess of cations, a circumstance which has already been pointed out by Gouy t. * We call an electrolyte active if it gives an electrocapillary curve with a depressed maximum. ‘The activity of anorganic electrolytes depends on the anion. Tt C. &, exxxi. p. $39 (1900). 382 Mr. A. Frumkin on the Fig. 3. Potential we (2) Salts with active cations—Gouy* has shown that similar to the active anions Br’, 1’, CN’, there exist active cations like N(C.H;)4, S(CH3)3. The electrocapillary curves of salts of these anions have a normal ascending branch and an altered descending one. For instance, the ascending branch of a solution of | N(C,H;)4].5O, coincides with the ascending branch of a solution of Na,SO,, whilst the de- scending branch of [N(C,H;), |,5O,is considerably depressed. But if the anion is active, the activity of the cation manifests. itself in the ascending branch as well as in the descending one. Thus, comparing the electrocapillary curve of two. salts with an active anion like N(C,H;),Br and KBr, we see that their ascending branches are different. We must, there- fore, admit that in presence of an active anion, the cation is adsorbed even in the ascending branch. (3) The value of the horizontal displacement of the ascending branch.—-lt T,, >0, Zar OY een RTO loge evr, and the horizontal distance between the ascending branches of two solutions whose concentrations are c, and c, must be greater than elon it ni * Ann. chim. phys. (8) ix. p. 87 (1906). Theory of Electrocapillanty. 383 The experimental determination of y,—f, presents some difficulties caused by the lack of mobility of the capillary meniscus, especially at low concentrations. The results ob- tained do not pretend therefore to a high degree of accuracy. Nevertheless, we may see from Table II. that the value of the displacement with SO," and Cl’ is approximately equal to RT log at low concentrations, whereas with an active anion 8 ihe? like I’ much greater displacements are observed. The same phenomenon is shown to a less degree with the Cl’ ion in a concentrated solution. The data for H,SO, were obtained in the same way as the data for KNO,; and Ba(NQ3). in Table I., a correction for the potential difference between H,SO, and the n/10 KCl of the calomel electrode being made; the approximate data for NaCl and KI were calculated from the measurements of Gouy *. Tasce II. best 220". 7: n—1/10 n. 1/10 n-1/100 x, 95:0 0:042 0:051 90:0 0-038 0:028 85°0 0-032 0°022 80°0 0-027 0-023 75°0 0-027 0°023 70:0 0-027 be. Nak 418°. 90°0 0:09 0-05 Rat pe be 90:0 0°15 O11 75°0 0:09 ‘ III. The maximum. In the neighbourhood of the maximum T, and T, are quantities of the same order of magnitude, and no definite inference can be drawn from (ja). . aes (8) where K is the specific inductive capacity of water, p the osmotic pressure ot the ions in the bulk of the solution, and V=W—W,,,,. AS Chapman’s reasoning involves the assump- tion that the ions in the double layer behave like perfect gases, equation (8) especially at higher concentrations can be used only for small absolute values of V. It seemed to me therefore of interest to test eq. (8) in the neighbourhood of the maximum. ‘Table III. contains the experimental values of Yun, —Y in C.G.8. units for KCl, KNO;, and Na,SO,, and the values calculated by means of equation (8), assuming K=81, t=18°, and p=acRT, where « is the degree of dis— sociation of KNO,. Moreover, Table III. contains values of ¥yr,..— calculated by means of the classical formula Yer OH YSOV", . Oe a being determined from the value of y,,,.—y which corre- spond to V=1 volt. Table III. shows a very great discrepancy between the calculated and the observed values: the influence of concen- tration is much less pronounced, as it ought to be according to (8); moreover, yy,,.—Yy is approximately proportional to V7. Thus, the results of experiment are unfavourable to Chapman’s assumption concerning the conditions of the equilibrium in the double layer. As Chapman’s reasoning is thermodynamically correct, equation (8) must be in agreement with'(6). In fact, for great values of V, we have Y Max. With the help of equation (5a) we may calculate the value of absorption of any ions, except those of mercury, as we cannot vary their concembration without varying y. It is easy to give to the equation of the electrocapillary curve oY Ov =e+T,,,F FV ya ly / em 2RT — const. Vc e28? = const. oon pee). Theory of Hiectrocapillarity. TaBLeE III. Caleul. by means of (8). 385d Caleul. by means of (9). KNO. KNO,. ec ec s. || a er ba Sin Rieter Vv 0-:05.v.| Oly. /O15v, = V 0.05 ¥.| Oly. 0-15 y. nw, | 267 | 138 ria eal eR n, | O28 114 2/10. O96 | 4:96 | yiat n/10. | 0°26 | 1-08 n/100. 0°32 | 167 | 5°57 | | n/ioo0, o-10 | 0-54 | 1-79 | yD VORP ORD 1 Be | | | | Observed KNO,° Observed KCl. (6 c ry 0:05 v:| O-1 v. | 0-15 v. V 0:05 v.| Ol v.) 0°15 v.! n. |0:34"| 1°36 n/10. | 0-43 | 1:38 n/10. | 0-29 | 1:20 n/100. 0-85 | 22 n/100.| 0°25 | 0:98 | 2:3 ‘n/1000.| 0°37 Observed Na,SQ,. Cc | | | V 0-05 vy. Uly. O15 v. | | n/10. | 0:34 | 1°35 | | | | n/100.) O34 10 235 In the form of an equation of adsorption of mercury ions. fact, let us put RT de ee en a RM Qe Hon (e He }? a formula quite analogous to (5a). Laboratory of Physical Chemistry, The University, Odessa. P.S.—After this paper was already forwarded to the editors we received here Grouy’s article (Ann. Phys. (9) viii. p. 129, 1917), where similar considerations are developed. reese. XLIV. An Improved Design for the Friction Cones of Seurle’s Apparatus for the Mechanical Equivalent of Heat. By H. P. Waran, M.A., Government Scholar of the Uni- versity of Madras * N Dr. Searle’s alle known apparatus we have two cones of gun-metal, one inside the other, ground to a good fit so that by rotating the outer cone about the inner the fric- tion between them converts the work done into heat, and it is estimated by noting the temperature rise of water con- tained in the inner cone. Though the apparatus is extra- ordinarily efficient for its size and simplicity, yet it is not — without a few drawbacks. The present modification is an attempt to get over them. The thickness of the walls of each of the metal cones is about 3 mm., and the heat generated by friction at the surface of contact has to be conducted through this thickness of metal before it can raise the temperature of the water. And in an ordinary laboratory experiment to minimise the effects of heat loss due to radiation the temperature rise is kept very low, and consequently the temperature gradient in the metal is small and the rate of conduction of heat to the water is slow. Further, the employment of water with its low conductivity of heat as the liquid to absorb the heat, makes the situation only worse. In the presence of these drawbacks very vigorous stirring of the water in the inner cone is essential to ensure a rapid equalization of temperature throughout the system, and in that stirring probably we are adding an amount of unmeasurable work and consequently heat to the system, which is probably very small. Further, this is an extra operation the experimenter has to do very efficiently, in addition to his duties of turning the wheel at a steady rate with one hand and noting down the thermo- meter reading with the other. The following i is an improved design for the cones calcu- lated to overcome the above disadvantages. As will be evident from the diagram, the inner cone I is made open at the bottom, and has, in addition, a few helical grooves G cut on the contact surface of the inner cone, these grooves terminating in a ring-channel S cut near the top of the cone. From this channel a few holes are drilled sloping inwards into the cone. The grooves G are cut sloping upwards in the direction of rotation of the outer cone. further, for the liquid in the cones any essential oil of known specific heat and large conductivity for heat is * Communicated by the Author. Improved Design for Friction Cones. 387 used in preference to water. These comprise in brief the essential points of the improved design. If we look into the action of these improved cones we shall see that, as an effect of the rotation of the outer cone, the oil in the cup is Pie? I, forced up through the grooves G on to the channel S and through the holes 2 back into the inner chamber again. Thus the liquid is stirred efficiently and automatically and a liquid of high conductivity brought into intimate contact with the surface where the heat is generated by friction, thus ensuring a uniform rise of temperature of the whole system in the minimum of time. The liquid also gives a steady and uniform lubrication for the friction surfaces, and in a class-room the experiment can be done by an average student solely by himself, without any extra aid to stir the liquid or to observe and take down the readings of the thermometer. This modification has also the great advantage that it can be directly introduced on all the existing type of millions of the apparatus already in use. The Cavendish Laboratory, Cambridge. ARG XLV. Notices respecting New Books. | Collected Scientific Papers. By Professor Joan Henry Poyntine, F.R.S. Pp. xxxu+768. Cambridge University Press. 1920. 37s. 6d. net. JROBABLY few are aware of the wide range of studies covered by Poynting’s researches. His faine was made in 1884 by papers on the transfer of energy in the Electromagnetic Field (Phil. Trans. Roy. Soc. A) and on electric currents and the electric and magnetic induction in the surrounding medium (Phil. Trans. 1888). The Poynting vector specifies the direction and magnitude of flow of the energy in an electromagnetic field; it is at right angles to both the electric aud magnetic forces, and is proportional to the product of these forces and the sine of the angle between them. This vector has taken an important position in any modern theory ; and it may be said also from the practical point of view that no clear idea of the propagation of energy in wireless telegraphy could have been obtained without it. But in three other directions at least Poynting made investi- gations of fundamental importance. In 1887 his attention was concentrated on tue phenomena of change of state, and at other times he came back to this and the allied problems of osmotic pressure. He showed thermodynamically that vapour pressure must increase with isothermal increase of pressure; that the lowering of freezing-point with pressure must depend upon whether both phases are subject to the pressure or one phase alone. Experiments on this latter point did not bear out very well his theoretic conclusions, but there is no doubt that he was on the right lines; the difficulty was in trying to reproduce the theoretical conditions. With regard to the former point it is now known to be one example of a very general theory of pressure- influence in connexion with which much theoretical and experi- mental work has been done. Much more importance attaches to his investigations concerning the pressure of radiation. Along with Dr. Guy Barlow he estab- lished the existence of a tangential force when light is incident upon an absorbing surface, and also the existence of a torque when light passes through a prism. In 1910 he showed that a radiating body recoils from the radiation it emits. His fourth main subject was that of gravitation. First (1878- 1891) he undertook a difficult investigation into the mean density of the earth using a balance method. Although his method was in the end overshadowed by the exceedingly neat method adopted by C. V. Boys, yet his extreme skill and perseverance overcame many of the difficulties which were minimised in the later method. He also sought whether the attraction between two quartz crystals depends upon their orientation ; no difference was Notices respecting New Books. 389 detected. And lastly, in conjunction with Dr. Phillips, he showed that gravitation is independent of temperature ; at least to within 1 part in a thousand-million between 15° and 100° C. This brief summary has referred only to investigations of physical interest. The papers here reprinted include one on the Drunkenness statistics of the large towns in England and Wales ; and one on a comparison of the Huctuations in the price of wheat and in the cotton and silk imports into Great Britain ; and also twenty-five addresses and general articles. In all, seventy com- munications are reproduced. Poynting was above allan experimental philosopher. Although he had a theoretical equipment of a high order he was somewhat atraid of theoretical results unless well tested experimentally. The present volume is a memorial one edited by Drs. Shakespear and Guy Barlow. It contains biographical notices by Sir J. Thomson, Sir O. Lodge, and Sir Joseph Larmor. It is a worthy memorial; it will make better known to all the work of a man who was one of the least assertive of men; but who by his quiet and pertinacious labours (with a body enfeebled by disease) has enriched the world. The Concept of Nature. Tarner Lectures delivered in Trinity College, November 1919. By A. N. Wutrznxnap. Cambridge University Press. Prorsssor WHITEHEAD asks us to regard this book as a com- panion to his recently published Enquiry concerning the Principles of Natural Knowledge (see Phil. Mag. June 1920). The two books are independent, but they supplement one anotber. It is quite clear why he wishes us todo so. In the Enquiry he accepted, not uncritically, the principle of relativity, and particularly the expression given to it by Minkowski in his concept of a four- dimensicnal universe constituted of events. The purpose of the Enquiry was to demonstrate the fundamental character of the event, fo show how events are related and measurable, how objects are derived from them and to settle in the form of a definition what an event and what an object is. His book, however, was hardly published before we were all discussing the new general relativity of Einstein. It is not surprising therefore that Pro- fessor Whitehead has taken the opportunity of the Tarner Lectures to make his own position in regard to the generalized principle clear. Without being at all unreceptive to the new theory, and while accepting its particular applications, the new formula for gravi- tation for example, he is very anxious to dissociate himseli from the extremist mterpretation. His keen philosophic vision warns him lest a position of absolute negativity towards independent objective reality should bring upon physical science a similar impasse to that which Hume’s scepticism brought upon philosophy. . 390 Notices respecting New Books. I refer to the insistence by Einstein on the impossibility of pre- senting the reality of nature in any purely objective form, that is, in any form which does not take account of the observer and his system of reference. Professor Whitehead will give up absolute space and absolute time, he has no need of material or stuff, the hypothetical ether of the physicists he dismisses with scant respect, but he must have an ether, an ether of events, if, as he holds, the universe consists of events. His scientific instinet will not let him entertain the possibility of a purely subjective reality. To ‘proclaim monads as the real atoms of nature would signify for him the death of physical science. ‘There is now reigning in philo- sophy and in science an apathetic acquiescence in the conclusion that no coherent account can be given of nature as it is disclosed to us in sense-awareness, without dragging in its relations to mind.” ‘lo counteract this is the inspiring motive of this book. On one definite point only it joins issue directly with Einstein. . Professor Whitehead will not have the bending of space. The gravitational field is not in his view equivalent to the curvature of space in the field. Otherwise he accepts the new formulation and does not challenge the experimental tests by which it is confirmed. The first two chapters, entitled “Nature and Thought” and “The Bifurcation of Nature,” are introductory and meant to lead us to the central problem. In the five chapters which follow,— “Time,” “ The Method of Extensive Abstraction,” “Space and Motion,” ‘Congruence,” and ‘‘ Objects,’—the author is com- pletely at home, using his own peculiar method, working at what he has happily named the organization of thought. The two last chapters,—‘ Summary ” and “ The Ultimate Physical Concepts,”— are additions which, we are told, formed no part of the original course. They emphasize the conclusion in regard to Einstein’s eeneral relativity. By the bifurcation of nature, Professor Whitehead means the division of the science of nature into two classes of entities, those disclosed to sense-awareness, and those disclosed to conceptual thought. Materialism he rejects outright. It 1s so absurdly in- adequate that it is becoming matter of amazement that the scientific opinion of the last century should have taken it, almost universally, as axiomatic. On the other hand, he is vigorous in denouncing metaphysics. A metaphysics of reality is, in his view, completely out of place in the philosophy of science. ‘It is like throwing a match into the powder magazine. It blows up the whole arena.” Philosophy of science is the philosophy of the thing perceived, while metaphysics confuses everything by embracing within one reality both perceiver and perceived. The whole book is valuable therefore in the indication it gives us of the line along which the issue is likely to be joined by physicists who agree in accepting the principle of relativity and yet disagree profoundly in its interpretation, The line of demarcation will be Notices respecting New Books. 391 metaphysics in Professor Whitehead’s meaning of the term. For according to Einstein, not only is it impossible to denote any reality save in the terms of some observer’s observation, but also when you abstract from the observer and his system of reference there is nothing to denote. If there be, what is it? Shall we say, for example, that colour is sense-awareness and relative to the observer, while the conceptual entity, the light waves of a certain amplitude and frequency, is not relative? We are at once brought to book by the fact that the essential character of this entity is its dimensions, and dimensions depend on the relative movement of systems of reference and vary with their acceleration. If it is not the dimensions what is it that is absolute in this entity 2? You find you must answer and you can give no definite answer. You tall back on the indefinite something or other I know not what. Einstein says there is nothing. I find in Professor Whitehead’s discussion of the theory he accepts, namely, that space-time is four-dimensional, a great deal that puzzles me and even seems at times incongruous. It suggests to ne that he is putting new wine into old bottles. A four- dimensional space-time is the groundwork of his concept of nature. Yet he is continually talking about ‘‘ instantaneous space ” ‘‘ time- less spaces” “time systems,” these last being apparently inter- changeable without affecting the space. It leaves me wondering whether his four dimensions mean anything more than that time and space are never dissociated. I will give an actual illustration. On p. 97 he says “the meaning of saying that Cambridge in the appropriate instantaneous space at 10 o’clock this morning for that instant is 652 miles from London at 11 o’clock this morning in the appropriate instantaneous space for that instant beats me entirely.” A little further on, after discussing the boundaries of . events, he says(p. 100) ‘Thus the boundary of a duration consists f two momentary three-dimensional spaces.” Now have we not here precisely the difference between Professor Whitehead and Einstein? Itcould hardly be more complete. For Einstein there is no appropriate instantaneous space for any moment, and no momentary three-dimensional space boundary of any duration. Time and space are solidary. The only meaning you can give to the statement that 52 miles in space separate Cambridge from London is that the train journey occupies an hour, the light journey (if you are in a position to see London from Cambridge) an infinitesimal fraction of a second. In fact I find in this exampie the essential meaning of Einstein. Reduce the time interval to zero and there is nothing, not only no time but also no space. I will conclude with one word concerning the relation of meta- physics to physical science. Probably I have already given myself away and shown that I come under Professor Whitehead’s censure as one who contuses the problem of the thing perceived with the problem of the perceiver. The philosophy of science 392 Notices respecting New Books. stands to gain and. not to lose by insisting that every scientific fact: is what it is, always for some observer in some system of reference, and that every description of scientific fact must take the observer and his system into account as factors. If this be metaphysics, science has nothing to fear from it, but everything to fear from disregarding it. There is, however, a metaphysics which brings sterility alike to philosophy and to science. This is the affirmation of an existence which is not what it is to any observer. It is named in philosophy the thing-in-itself and physical science has no need of it. I have only to add that the central portion of the book is certainly difficult but the whole is in the author’s inimitable style and bright throughout with his wonderful humour. H. Winpoy CaRR. A Treatise on Gyrostatics and Rotational Motion. By Anprew Gray, F.R.S. (Macmillan & Co. £2 2s. net.) Pror, Gray has produced in this volume avery complete account of gyrostatics, which will be welcomed by students of applied mathematics and technical students generally. All who are engaged in the study of this subject will be grateful to the author for bringing together in such an attractive form the many inter- esting matters set forth in the pages of this book. Following the discussion of elementary principles in the earlier chapters, a number of illustrations of the practical applications of eyrostatic action are given. The reader is also provided with a concise treatment of elliptic integrals, with the numerical solution of problems relating to rotating bodies. The numerical examples occurring in various parts of the book will be especially valuable, and should do much to encourage further investigation. In suc- ceeding chapters the author gives an exhaustive treatment of the general dynamics of rotating systems, and a comprehensive dis- cussion of more advanced gyrostatic problems, including the whirling of shafts and chains. Weare glad to commend this excellent and authoritative treatise, the work of one who has done much towards the development of this branch of science. It isto be hoped that Prof. Gray will find it possible to issue in the near future the supplementary volume dealing with gyrostatic devices of use in engineering and naval and military affairs. - The excellence of the printing and diagrams should not be overlooked. The book is well produced, has a useful index, and as a work of reference will be found indispensable to all interested in this fascinating subject. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. /,, [SIXTH SERIEB.] / a Tae PBCTOBER 1920. \ = ~»>* XLVI. On the Scattering of Light by Unsymmetrical Atoms and Molecules. By Sir J. J. Toomson, 0.11, P.RS* HEN the arrangement of the electrons in an atom is quite sy mmetrical, the displacement of an electron due to an external force will always be in the direction of that force and the ratio of the displacement to the force will be independent of the direction of the force. These are the conditions postulated in the investigations hitherto given of the scattering of light by small particles, and we know that when they are fulfilled and the incident light is plane polarized the intensity of the scattered light vanishes along the direction of the electric force in the incident light. When the incident ight is not polarized, the scattered light when observed in a direction .at right angles to the incident light is completely polarized. These results, however, will not hold when the atom is unsymmetrical and the displacement of an electron is not necessarily in the direction of the disturbing force. We shall see that under these conditions the intensity of the scattered light does not vanish in any direction : it, however, varies with the direction, and experiments on the law of variation such as those made by Lord Rayleigh give valuable information about the structure of the atom. Suppose that OA, OB, OC are three directions in an atom, at right angles to each other such that the electrons * Communicated by the Author. Phil. Mag. 8. 6. Vol. 40. No. 238. Oct. 1920. 2D 394 Sir J. J. Thomson on the Scattering of can be displaced along any one of them without causing displacements along the other two. Suppose that when the electrons are displaced — by distances &, 7, € parallel to OA, OB, OC respectively, the restoring forces are respectively A&, By, C&; then, it F,, F., F; are the forces acting on an electron in these directions, the equations for &, 7, € are respectively 2 d m Tat AE =H, 2 Ae 7+ Bn = Fy, alee GAG yl aa CE = ies lf the applied forces vary as Pt the solutions are F, Fy FE P= Kemp? 7 Bomp? § = a oO C—mp~ When A, B, C are small compared with mp’, =F, _ =f, Me elle are mp? ilitr Ue @ oo mp?’ and 1S ifn el ealagbe dE opt Eig ple The electron is thus displaced along the direction of the force and the usual theory will apply. Thus, as long as the frequency of the incident vibrations is large compared with the free frequencies of the electrons, the system of electrons will behave as if it were quite symmetrical and the light will be scattered in accordance with the usual theory. When however A, B, © cannot be neglected in comparison with mp’, we see that unless A= B=C the direction of the displacement will not be that of the force and the theory requires modification. Let A, B, C be the points where the axes OA, OB, OC of an atom cut a sphere of unit radius; X, Y, Z the points where this sphere is cut by three fixed axes OX, OY, OZ: let us calculate the displacements along these axes cf an electron acted on by a force Z parallel to OZ. Light by Unsymmetrical Atoms and Molecules. 395 With the notation indicated by the figure, the direction cosines of OC with respect to the fixed axes are sn@cosy, sin@sinyw, cosd; Bigs L. the direction cosines of OA are | cos @ cosy cos d—sin ¢ sin yf, cos @ sin cos 6+sin d cosy, — sin 0dcos¢; those of OB are —cos.é cos sin d—cos ¢ sin ff, —cos @sin sin @+cos ¢ cos f, sin 6 sin @. Thus the components of Z along OA, OB, OC are F,=—Zsin@écosd, F,=Zsin@sind, F;=Zcos 0. &, 7, €, the displacements of the electron along OA, OB, OC due to these forces, are given by Zsin @ cos _ Zsin @ sin } Z cos 6 ane A—mp*? ° ied B—mp? ’ $= C= mp* 2) 2 396 Sir J. J. Thomson on the Scattering of If w, y, z are the displacements of the electron parallel to OX IO VNOZ, x=E (cos Ocos yy cos d—sin ¢ sin yr) —n (cos 0 cos sin 6+ cos¢ sin W) + sin 6 cosh =Z [sin 6 cos 6 cos W(c—a cos* 6—b sin’ d) +(a—b) sin 6sin sin ¢ cos ¢ ], where ia2oee 7. pm. ; A—mp” b—mp” U—mp?’ y=Z [sin 6 cos @ sin W(c—a cos? 6—b sin? ¢) +(a—b) sin @ cos ¥ sin ¢ cos $ | ; z=Z (c cos? 0+bsin? @ sin? ¢+a sin? 0 cos’ d). If Z varies as ¢’?', the components of the acceleration of the electron will be —p?2, —p*y, —p’z respectively. If a, 8, y are the components at a time ¢ of the magnetic force at a point P due to the motion of an electron whose distance from P is 7, then when r is large compared with the wave- length of the light, Clee € e ; a= — {jn—km}, B= {hl—hn}, y= fhm, where J, m,n are the direction cosines of the line joining the electron to P, and h, 7, & are the components of the r ; acceleration of the electron at the time t—-, where ¢ is. G the velocity of light and e the charge on the electron. The energy per unit volume of the scattered light at P is equal to ; it Q mete ty) 2 = Sl + f+ (ht mj + nk}. oot ee ae Now h, j, k are respectively —p?x, —p’y, —p*e when «, y, 2 have the values given above. Substituting these values in (1) we get the energy in the scattered light from a single atom. We cannot observe the effect of a single atom or molecule; we must take the sum of the effects. due to a large number of molecules with all possible values for 0, ¢, %. When the atoms are uniformly orientated Light by Unsymmetrical Atoms and Molecules. But in all directions the average values are as follows :— 2 = ay Lipa—0' + 0c +o—a )=j=ptfZ?, say; ee = Z? p'(3(a? + 0? +c?) + 2ab + 2ac+ 2be)=p*gZ? ; > bsg The energy of the scattered light.per unit’ volume at P is thus equal to 7 sy iene Fe PZ gO +m) + F(R Ft 2n2)} 5 _ a since f and g are positive quantities, this expression cannot vanish whatever may be the values of J, m, x unless f=0, 2.e. unless a=b=c: this requires the system of electrons to be quite symmetrical. We may write the expression for the energy in the scattered light in the form hes 4 Ie 2 P*Z{2f+ (g—f) sin? 3}, where 3 is the angle between the direction in which the scattered light is observed and the direction of the electric force in the incident light. Since g—/f is positive, the energy is a minimum when 6=0 and a maximum when 0=7/2. The ratio of the minimum to the maximum energy is 27/(7+ 9) ; this is equal to 2 a?+/?+0c?—ab—ac—bc) (2) 4(a?+l?+0c?)+ab+actbe ~ °° | 3 If a, b, c are positive the greatest value for this ratio is 1/2 when ab+ac+bc=0; so that two out of the three quantities a, 6, ¢ must vanish. When the incident light has the same period as one of the free vibrations of the atom, mp” will equal one of the three quantities A, B, ©, so that either a, b, ¢ will be infinite: in this case the minimum value of the intensity is one-half the maximum value. For stili greater frequencies the values of one or more of the quantities a, b, ¢ might be negative, and in such a case the minimum intensity might be more than half the maximum. Hitherto we have been considering polarized light. If the incident light is not polarized the intensity of the light scattered in any direction can be found as follows :—Let a~ OI be the direction of the incident light, OP that of the . 398 Sir J. J. Thomson on the Scattering of scattered light, OZ the direction of the electric force in the incident light ; IZ is always equal to 7/2. The intensity of the scattered light is, as we have just seen, proportional to 2/+(g—f) sin? @. But cos 8 = cos PZ = sin @ cos. Fig. 2. Hence the intensity of the scattered light is pro- portional to f+g—(g—f) sin’ $ cos? p. If the light is not polarized and all values of wW are equally probable, the mean value of cos’ is 1/2. So that the intensity of the scattered light is f+g—3(9—f) sin’ ¢. Since g is greater than 7 the intensity is greatest when d=0, when it is equal to f+g, and least when 6=7/2, when it is equal to $(3f+g) ; since / is positive the minimum intensity is always greater than (f+ )/2, and thus always greater than half the maximum energy. For the sym- metrical atom f=0 and the minimum intensity of the scattered light is one-half the maximum. Another point of difference between the scattering of light by symmetrical and unsymmetrical atoms occurs when the distances between the scattering atoms are small compared with the wave-length of the light. With sym- metrical atoms or isolated electrons the scattered light reaching a point P will be in the same phase from which- ever atom it may proceed; so that if there are N atoms the intensity of the magnetic force will be proportional to N and the intensity of the scattered light will be proportional to N’. In the case of unsymmetrical atoms, though they may be packed close together, the phase of Light by Unsymmetrical Atoms and Molecules. 399 the scattered light reaching P from one atom need not be the same as that reaching it from another atom. [or this phase depends to some extent on the orientation of the atom. Hence if the orientation of these atoms is quite irregular, the phase of part of the light scattered by different atoms will also be quite irregular, and in calculating the intensity of this part of the light we have to add the intensities of the light scattered by “the different atoms, and not, as in the case of the symmetrical atom, add the magnetic forces and take the square of this sum as proportional to the energy. ‘Thus the energy in this part of the light will be proportional to the number of atoms and not to the square of the number. It must be remembered, however, that even with unsymmetrical atoms the phase of every part of the scattered light cannot be made to change sign by altering the orientation of the atom. If we refer to the equations on p. 396, we see that the part of the light arising from the accelerations along x and y can be made to do so, but that depending on the acceleration along < cannot: hence part of the light scattered by N unsymmetrical atoms will be proportional to N? and another part to N. ‘The light due to the acceleration parallel to z vanishes along the axis of Z—that is, along the direction of the electric force in the incident light ; so that in this direction the scattered light would be proportional to N. In other directions there would be a part proportional to the square of the number of atoms, provided the linear dimensions of the space occupied by these atoms were not large compared with the wave-length of the light. For large volumes we see, by Huyghens’ principle, that the intensity of this part of the scattered light would be comparable with that scattered by the atoms contained in a layer whose thickness was of the order of the wave-length. To see the conditions necessary for this statement to be true, we notice that the magnetic force in the scattered light at a point whose distance r from any of the atoms is a large multiple of the wave-length is given by an expression of the form 4 Qa Mais @tiasomsnk) 2a +a cos —— (vt —(lag+ myy +nze)) +... I. where «,, y,, 2, are the co-ordinates of one of the atoms, and /, m, n are constant. 400 Sir J. J. Thomson on the Scattering of The energy being proportional to the square of the magnetic force is of the form 2 2 = | 5 cos (vt — (le,+ my, + nzr)) Qar x (vt — (lits+ mys+ ne) ) } ; where there are n terms under the single summation and n.(n—1)/2 under the double, n being the number of atoms. The average value over a considerable time is 20 + 22> cos ST (vt — (lw, + my, +nz,)} Cos a’ ¢ 27 /,. 92 a 12> Sicos ct (U6 2,— ts) + M(Yp— Ys) +(e — zs)) ; : If &— Xs, Yr—Ys, 2r—2Zs are all small compared with 2, each of the cosines will be unity and the energy will be proportional to g - = 1) Z Det ay OTRO Ile If the atoms are arranged in a lattice, so that v,—a,, Yr—~—Ysy 2r—2% ; are always integral multiples of constants a, 6, c, then for certain values of 1, m, n the angles will be all multiples of 27, and the energy in the directions corre- sponding to these values will be proportional to n®. If, however, there is not this regular crystalline arrangement of the atoms, but, as in a gas, the atoms are distributed at random, we can easily see that, when the space containing the atoms is bounded by planes, the magnitude of the term involving the cosines is comparable with the square of the number of atoms in a layer whose thickness is proportional to the wave-length, and thus, when the atoms occupy a considerable volume, becomes small compared with n; so that we may take the scattering due to a gas or a liquid as proportional to the number of atoms. From equation (2) we see that if we know the values of a, b, c we can calculate the ratio of the minimum to the maximum intensity of scattered polarized light ; and since we can calculate these values when we know the arrange- ment of the electrons in an atom we have a method, and a very powerful one, of testing any theories of the constitution of the atom ; for Lord Rayleigh has shown that the variations of the intensity of the scattered light in different directions can be determined with considerable accuracy, and that the magnitudes of these variations vary considerably with the nature of the atom. We shall therefore proceed to find the values of a, 0, ¢ for Light by Unsymmetrical Atoms and Molecules. 401 some specified type of atoms. The method of calculation will depend upon whether we suppose that the electrons in the undisturbed atom are describing circular orbits, or whether they are in equilibrium under a complex type of force. We shall begin with the first supposition, and consider the effect produced by a small disturbing force X cos gé acting parallel to w in the plane of the orbit, the orbit being described under the action of a central force equal to pr/(r?+d?)= ; where y is the distance from a point in the plane of the orbit, and d a constant. If d vanishes, we have the law of the inverse square; it is convenient, however, to take the more general law, for we shall require it when we consider molecules where there is more than one centre of positive electricity. The orbits we shall consider are those which are approxi- mately circular. Consider first the undisturbed orbits. The circular orbit is represented by “2 =acos(nt+e), y =asin (nt+e), oe p= n*(a? + d?)3. If the orbit is not circular, but only approximately so, and © if r=a+&, where & is a small quantity, we can easily show that gear a en qt (4p: n E = 0, D?=a’?+d?; if p> =4—3a?/D’, we see that we may write where r = a—ae cos { p(nt+e)—o}, where ¢ is small. The values of « and y for an orbit of this type are repre- sented if pant+e, p=(p—1)(nt+e)—o, y=(pt+1)(nit+e)—o by the expressions 2=acosp—kyae cos W + kgae cos y, y = asin $+, ae sin W + Kae sin x, psa ta | i 1 _——ae — pve 2] eS 1 ay 2 p. 3 where Following the methods of the Planetary Theory, we shall 402 Sir J. J. Thomson on the Scattering 07 suppose that when the force X cos gé acts on the electron « and y are still represented by equations of this type, but that the parameters are variable and are functions fe) of the time which it is our object to determine. If ar denotes differentiation with regard to the time only so far as the time enters into these parameters, and if we take the more general case where the electron is acted on by disturbing forces X’, Y’ parallel to the axes of # and y respectively, we have by the principle of Instantaneous Orbits dry BR ey ad fol G2 am P ovnae a ar Obcarr where 2’ = the parameters being considered CP otal dy Rp ae constant for these differentiations ; the mass of the electron has been taken as unity. From the equation Qa/dt = 0 we get VE et coap— = asin p6— © (ae). {hy cos y—k, cos y} — ae Ee (k, sin w—k,sin yx). In the coefficient of de/dt terms which are small compared with asing have been omitted. Ole : From Sia! we y on" sing tacos. _ (ae)(k; sin Ww+h sin y) ae = (ky cos +k, cos x). From sae? da ¢ dn . de =F, 1” +aT 7 £08 o—ansin 2 d + 5, (ae)((p—1)nki cos h+(pt+1)nk, cos x) dw ae, ((p—1)nk sin w+ (p +1) nk, sin x). Light by Unsymmetrical Atoms and Molecules. 403 02 vis From —-=X’ fols ; da xX=-— ak has ia bin $—an cos $. 5 Ee d + ©, (ae)((p—L)nk sin y—(p + 1) nky sin x) di -—ae = ((p—1) nk, cos ~—(p+1) nky cos y). Gam these and from the equation PE eile (a 24 2)¥ we get da a a ca? Ey cos@—X'sing), de ee a] I = ar sin @+ X' cos d), oe = sap t\ (he cos y+ hy cos yp) + X/(hy sin p—hy sin y)}, dw ' 0 TE gy LY (he sin x + hy sin Y) + X'(h 008 y— k, cosr)}. The third and fourth of these equations are equivalent to d a (2 C08 @) = a8 (I 008-y Ait C08 4h" ) + X'(k; sin yp’ —k,sin x’), ae ( 7 Ce Sine) = =o (42 sin x' + ky sin w’) + X!(k, cos y' —k, cos wv’) }, where = (p—l)(nt+e), yx’ = (pt1)(nt+e). 404 Sir J. J. Thomson on the Scattering of “Take the case when Y'=0, X’=X cos qt. Then, integrating these equations, we get xX +n)t ecos(gq—nit a= ayy Cos ae es - n) }. np* q qd Nr x {= (g+n)t sin —— — (ear gee ahaa Sn Ae nap g+n q—n ee 2S cos(ptlnt+q)t cos Cae Siccttesl? aT ( (pt tn+q. (+ Insg a CAs In+q)é cos (p—1 ln—q)t )} Salat ¢ «= 2) ae ésin@ aes , Cates euibla pai m 2nap |? (p+1)n+¢ (p-F il ny se sin (p—Intg)t , sin ( p— p—In- ~q): yt. : (p—l)nt+q (p—l)n—G _ Substituting these values in the expression for z, we find that ask 2 1°. (p—1) b2(p+1) i.e fsa Ad nalad 008 984 aa 7) or cs 1) *— 9} plri(p +1) a kyk a 2 12 cos (2n+q)t pre—(n+q)? kik = a) a 12 X cos (2n—q)t “pn (n— 9)? + dy COS nt. To calculate the intensity of the scattered light of the same colour as the incident we need only consider the term containing cospt. As this term is proportional to the applied force, we see that an electron rotating in an orbit acts, so far as its contribution to specific inductive capacity is concerned, like a statical system, where the displacement of the electron is proportional to the force. Hence for forces in the plane of the electron we have, using the notation of p. 396, 2 ky?(p—1) ko pan SoS p(n? — 9?) p{r?(p—1P—_} * p{n?(pt+-1)P?—¢_})? where bate a at Light by Unsymmetrical Atoms and Molecules. 405 If a force Zcos gt acts at right angles to the plane of the orbit, which we may suppose described under the influence Fig, 3. of two positive charges at A and B, the plane of the orbit bisecting AB at right angles, we have Oz, be 2 wy eae pt 3 cos? @)z = Z cos dé, where 6= Z PAO in the steady orbit ; : Came." 1, €., qe te a3 cos? @)z = Z cos gt, ee Z cos pt aK n*(1—3 cos? 0)—q? hence 1 os n*(1 — 3 cos? @) —q? Since a/D=sin @, p? = 4—3 sin? @ = 1+3 cos? @. Knowing the value of a, 6, c, we can by equation (2) on p. 397 calculate the ratio of the minimum to the maximum intensity of the scattered polarized light. Let us now take some special cases. The first is that of an electron describing an orbit under the attraction of a positive charge: here the charge will be in the plane of the orbit, so that 9=7/2, p=1, Sepals eel or when q is small compared with n, a=b = (2+), 406 Sir J. J. Thomson on the Scattering of Hence cis thal OTS 9) wh 4(a?+b?+0?)+abt+ac+be 98° So that by equation (2), p. 397, the minimum intensity would in this case be about 5 per cent. of the maximum. Let us take next the case of an electron describing a circular orbit under the attraction of two equal positive charges, the plane of the orbit bisecting at right angles the line joining the charges; we have found the values of a, b, c as functions of @. We see from the expression for ¢ that there cannot be a state of stable steady motion unless 1—3 cos? @ is positive ; so that @ must be between 90° and 54° 42’. We have already considered the case when @=90°. The values of o the ratio of the minimum to the maximum intensity, fon some intermediate values of @ are given in the following table :-— 0. o in per cent. ot alin es coc 22 (LOR ede Ne cok 16 Lene Sag repre ty: “46 DIONE Nate de aes 2 Se eee Die 50 We have supposed hitherto that only one electron was describing a circular orbit, and with a law of force of the type we have assumed it is easy to show that if we have more than one electron travelling round the same circle steady motion is unstable. For take the case of two electrons travelling round a circle of radius a: the steady motion, if possible, will be one where the two electrons are at opposite ends of a diameter. If 7, 72, 6;, @. are the co- ordinates of the two electrons when disturbed, we may put 1 =A+ 2X4, Tg=A+ Xo, hg aa lG O,=nt+7+ Yo, where 2), @, Y1, Yo are small quantities. The equations of motion are dt? dt Keg ke, ey eo) aa ee r, at (n dt Si, where P and © are the radial and tangential forces exerted Light by Unsymmetrical Atoms and Molecules. 407 by the second electron on the first. We find, by a simple calculation, that 1 Pi = ga(1—J(aite)) e2 0, = aa (%— 43). Hence for the steady motion we have —ar = Eto or fie Me = aa and for « and y we have aie — xn? —2an ‘= = Ha 1) = (71+ 2), aan +21 ek = ves (Yo—91)> with similar equations for x9, yp. From these we get d? (y+ &) dy, _ dy” es (a+ .x)n? a e =-(a+0){ fs (1- ae ti eae ee el =. UO: The solution of these equations is, 23,24 F (72%) & tig? 3r + Dp (1 D) ae 543? a+, = Acos(gt+e), 2H OP. Wty = B- ™ A sin (gt +e). If E= &y—Xy, N= Yi-Yas we have 408 Sir J. J. Thomson on the Scattering of or supposing & and 7 to vary as e’?’, we have f oa” e” ‘ (ro B(-8)} fendap ma a quadratic equation in p*. We have seen, p. 405, that for the equilibrium to be stable for displacements at right angles to the plane of the orbit 1—3cos?@ must be positive, where @ is the angle between the line joining the electron to a positive charge and the normal to the plane of the orbit. Since a/D=sin @, 1—3a?/D?=1—3 sin?@; and this is negative if 1—3 cos? @ is positive. Hence ee dsin?@) must be positive, so that the two values of p? given by this equation must have opposite signs ; thus one value of p? must be negative, so that the steady motion will be unstable. Thus if the two electrons repel each other, the system sometimes supposed to represent an atom of helium with two electrons revolving at the same distance from a central positive charge cannot be stable, nor can one when there are two electrons revolving in a circle midway between two positive charges, which is sometimes supposed to represent the constitution of the hydrogen molecule. If we make the extravagant assumption that the two electrons do not repel each other, the problem is the same as that of the single electron already discussed: the steady motion would be stable, and since the relative values of a, b,c would be the same as for a single electron the ratio of the minimum to the maximum intensity of the scattered polarized light would be that given in the table on p. 406. If the hydrogen molecule is a system like that represented in fig. 3, with two positive charges and two electrons at the corners of a rhombus, we see by considering the equilibrium of one of the positive charges that @ must be 60°; so that the ratio of the minimum to the maximum intensity of scattered pularized light would only be °4 per cent. Lord Rayleigh’s experiments show, however, that for hydrogen the ratio is at least ten times greater. We shall now go on to consider the case when the electrons, instead of revolving in circular orbits, are in equilibrium under forces between the positive charges and the electrons which do not, at the distances which separate the positive charges from the outer ring of electrons, vary inversely as the square of the distance, but at these distances may vanish and change from attractions to repulsions. If there is only one electron in the atom, the atom would Light by Unsymmetrical Atoms and Molecules. 409 have a finite electrical moment measured by the product of the charge on the electron and its distance from the positive charge. The atoms would “set” under the action of an external force, and under such a force the collection of atoms would have a finite electrical moment and therefore a specific inductive capacity greater than unity. If it were not for the collisions between the atoms their axes would all point in the direction of the electric force and the moment would be independent of the strength of the electrie field. These axes are, however, knocked out of line by the collisions between the atoms, so that the resultant moment diminishes as these collisions increase ; in fact, we can show that the resultant moment and therefore the excess of the specific inductive capacity over unity will, on account of this effect, vary inversely as the absolute ‘temperature of the gas. There are gases such as water vapour and the vapours of various alcohols which vary in this way; but the specific inductive capacity of gases such’ as hydrogen, helium, nitrogen, or oxygen varies much less rapidly with the temperature, showing that in the normal state the atoms or molecules of these gases have no finite electrical moment. One form of such an atom is that of a double charge at the centre A with an electron on either side of it. If such an electron is acted on by a force X at right angles to the line joining the central charge to the electrons, then, if du is the displacement of either electron in the direction of this force and d the distance of an electron from the central charge, we see that faba = ea, or Deon = Sa we. Now 2e6x is the electrical moment of the atom when it is exposed to the electric force, and thus 8d? is proportional to the quantity we have denoted on page 397 by a+ if the axis of y isalso at right angles to the ine joining the central charge to the electron, fea. We must now consider the effect of a force along the line of electrons: if dz is the displacement caused by a force Z acting in this direction, 2e?F the force exerted by the central charge on an electron, we see that yee! on a Phil. Mag. 8. 6. Vol. 40. No. 238. Oct. 1920. 2 i A10 Sir J. J. Thomson on the Scattering of Let } a oases ee ae EF when z=d, then Ld 26 06 .= BE? and from the equilibrium of the system, we have e2 2e? i = a) 79° Le 1317 Hence oasis oe Hence 6 = (8d) Garand a —=(0 78d The ratio of the minimum to the maximum intensity of the scattered light is therefore, by equation (2), equal to 2(@—1)? 98742844 Another system of which the hydrogen molecule may be taken as a type is that of two positive charges A and B: and two electrons C and D, arranged as in fig. 3. Tf the plane of the system be taken as the plane of a, y, then, if dz is the displacement of an electron due to a mee Z at right angles to the plane of xy, we have, by taking moments about AB, — dz = ZetCD, or 2e.62 =Z.CD?; thus ¢ is proportional to CD®. If the system is acted on by a force X parallel to AB, and if ¢e?F is the force between a positive charge and an electron, pps (F =) 8u = Xe, CONG where 7 is the distance between the positive charge and an alt electron, from this we get, if we put a =P, where F and » have the values corresponding to the undisturbed position, a 2e za —(1—£) cos? 6) d2 = X Xr or 2e6u = 4 F(1—(1—£) cos? 6) Light by Unsymmetrical Atoms and Molecules. 411 For the equilibrium of the system, 2F sin? = — F 1 or cb Tig OEW Ma’ iitihese ABD Hence 3 CD? Zz pa EN a ae iets 1—(1—8) cos? @ Hence CD? a meee a) cota Similarly we may show that if dy is the displacement parallel to CD caused by a force acting in this direction, CD met ets "1—(1—8) sin? 0° Hence hi CD* ~ 1—(1—8) sin? 0 From these values of a, 6, ¢ we find that o the ratio, expressed as a percentage, of the minimum to the maximum intensity of the scattered polarized light, is for a series of values of 8 and @ given by the following table :— 8@=90°. 6=60° or 30°. 9=4650. Oo. oO. oO, ar ie per cent. per cent. Ba eee 4-5 (oi Meee ae 7 2°5 1 se aes 0 0 0 B —— hE 1°8 if av, ice) iene Set 4:5 3°3 24 B == ay BS &°8 C2 7 = eat £15 iat 11 We can apply these results to check the validity of any assumed law of force between the positive charge and an electron in the following way :—Suppose that when the distance of the electron from the positive charge is 7, the attraction between them is equal to 2 p(2), where w=c/r, c being a constant. (a) must possess the 2H 2 412 Scattering of Light by Atoms and Molecules. following properties: (i.) when #=0, 2. e. when r is infinite, (a) =1; and (ii.) (wv) must be of the form which will give rise to the Balmer series in the hydrogen spectrum. These conditions are not sufficient to determine ¢(w). We shall show, however, that it is possible to write down an expression for @ which will satisfy these conditions, and in addition lead to the scattering of polarized light in a way consistent with Lord Rayleigh’s experiments. Taking the case of a hydrogen molecule, we see from equation (3) that if 2, be the value of # in the hydrogen molecule 1 Pa) = oe? and if the electrons and positive charges are at the corners of a square 9=77/4, so that in this case 1 (#2) a 9 / Since when «=, d¥ _& F ry where F is the force, we see, putting e2 Fa P(x), that ¢ dh’ (x) = —2x at pte yen c= Tye B p(2) d in? O(2) = aa \ cos? w+ — Hh This satisfies the condition that ¢(v)=1 when #=0, and it is of the form which would correspond to a Balmer’s series (see J. J. Thomson, Phil. Mag. April 1919). From the condition il f(2;) = an I find by the method of trial and error that 2,=°58, and substituting this value of # in the expression for 8, I find §8='3 approximately. This gives a scattering of a little less than 3 per cent.; according to Lord Rayleigh’s ex- periments the scattering is a little less than 4 per cent. Other expressions for @ could be assumed which would also be in agreement with experiment as far as the scattering goes. Assume Relative Ionization Potentials of Gases. 413 It has been urged against the idea of a value of force not varying inversely as the square of the distance that the scat- tering of the @ particles indicates a force of this type. The scattering of the « particles, however, practically all takes place at distances from the centre very small compared with the distance of the electrons which scatter the light or affect the emission of visible light ; and it is quite possible that the law should be the inverse square both at distances which are very large compared with ¢ and again at distances which are very small compared with it, and yet be quite of a different character when 7 is comparable with c—in fact, the law expressed by the value of ¢ given above is at both very great and very small distances that of the inverse square. XLVII. On the Relative Ionization Potentials of Gases as observed in Thermionic Valves. By G. Stead, M.A., and B.S. Gossuine, 17.A.* Pretiutviry Norr.—The observations descrioed in this paper were made during the summer of 1917, in connexion with the improvement of thermionic valves of a type in which the action was in considerable measure dependent on ionization. ‘They are now published with the permission of the Admiralty. (1) Method employed.—Some of the earlier methods f of determining the ionization potentials of gases have been the subject of criticism as indicating, not the appearance of positive ions, but some other effect such as photo-electric emission of electrons from the electrodes whcse intended function was to collect the positive ions when formed. In the experiments here described the method used was similar to that employed by Bazzonif, and consists in observing the effects of the presence of gas on the form of curves repre- senting the variation of the current leaving an incandescent cathode with change in the potential applied to the collecting electrodes. When no gas is present, the chief factor deter- mining the value of the space-current is the modification of the collecting field due to the applied potential by the addition * Communicated by Professor Sir J. J. Thomson. + £. g. Franck and Hertz, Deutsch. Phys. Ges. vol. xv. (1918) ; Pawlow, Proc. Roy. Soc. vol. xc. p. 398 (1914). { Phil. Mag. vol. xxxii. p. 566 (1916). 414 Messrs. Stead and Gossling on Relative Ionization of the field due to the electrons at the time in transit across the space. (Space-charge effect.) If the electrons are pro- jected with negligible velocity from an equipotential cathode, it can be shown* that the current will vary as the 3/2 power of the applied voltage up to the point when the current reaches a saturation value dependent on the temperature of the cathode, as shown in fig. 1. The current-voltage cha- racteristics of tubes in which the voltage is great enough for the corrections for initial velocity and for variation in the Fig. 1. ; AMeBAAND voltage potential of the starting point on the cathode surface to be negligible, afford ample and accurate confirmation of this theory, which has further been applied with uniform success in the design of more complieated instruments, such as high-voltage thermionic valves. When, however, the applied voltage is limited to low values, as in the case of the investigation of ionization effects, the effects of initial velocity and variation along the cathode are no longer to be neglected. In these experiments the same cathode heated to approximately the same temperature was used throughout. The results therefore indicate values of the 1onization potential relative to any one of the gases observed taken as standard. It is, however, necessary to assume that replacement of one gas by another does not give rise to serious changes in the contact potential between the emitting and collecting electrodes. The tube used was a three-electrode valve of the well- known form, the cold electrodes being connected together for the purposes of the experiments. Next the tungsten * Physical Review, vol. ii. p. 455 (1913). Potentials of Gases observed in Thermionic Valves. 415 filament there was a spiral molybdenum wire of diameter O-4 mm. coiled into a helix of diameter 4°5 mm., having four or five convolutions per em. length. Outside this was a nickel cylinder 10 mm. in diameter. In such a tubeif the vacuum is high enough for the effects of ionization to be inappreciable, it is necessary to apply a potentiat of some 60 volts to the combined collecting electrodes in order to reach the saturation value of the space-current when the cathode is, as in these experiments, hot enough to give a saturation current of 30 milliamperes. The lower parts of the current-voltage curve, well away from this saturation value, are observed to have a form sensibly independent of the cathode temperature, but showing a slight bodily move- ment, not accompanied by change of form, in the negative direction parallel to the potential axis when the temperature is Increased. This movement is ascribable in some measure to increase in the average velocity of projection of the electrons consti- tuting the current, but in large part also to increase of the length of the active part of the filament owing to a closer approach to the negative end of the filament by the point where the filament surface, in spite of the cooling effect of the end supports, first becomes hot enough to emit. If, however, gas is present in considerable quantity, a point can be found, by giving to the applied potential a woltage. suitable value dependent on the gas used, where there is a marked departure from the form of the high-vacuum characteristic, an upward turn of the curve making its appearance, as shown at A in fig. 2. The high-vacuum - 416 Messrs. Stead and Gossling on Relative Ionization characteristic is reproduced in the lower parts of the curve; but the subsequent upward turn indicates, of course, that the higher appiied potentials are competent to produce a larger current than in the high-vacuum case. In view of Langmuir’s theory of the effect of the space-charye it seems difficult to account for this larger current except by the assumption that positive ions have appeared in the neigh- bourhood of the cathode. Positive ions would be expected to produce just such an effect as is observed, for they would tend to neutralize the opposing field due to he electrons in transit, and so permit the passage of alarger electron current for any given valne of the applied voltage. As Professor Sir J. J. Thomson * has pointed out in a slightly different connexion, a very small proportion of positive ions produces a marked effect on the field by reason of their low velocity and consequent large contribution to the space-charge in their neighbourhood. (2) Discussion of Method.—As mentioned above, some of the methods which have been used in attempts to measure ionization potentials are open to the serious objection that they fail to distinguish between the picking up of positive ions by the collecting electrodes and the photo-electric emission of negative electricity from these electrodes. The method described in the present paper is free from this objection, as the collecting electrodes are always at a positive potential with respect to the filament, and their function is to collect electrons, so that photo-electric emission of electrons would diminish rather than increase the current. If photo- electric emission from the filament itself is supposed to occur, this will merely slightly increase the saturation current from the filament, and will have no influence whatever on the current through the tube. There remains the possibility that the Haden increase in the slope of the current-voltage curve is due to the production of positive ions, not by the thermions from the filament, but by more rapidly moving photo-electrons emitted by either the filament or the collectin v electrodes under the influence of ultra-violet radiations from the gas molecules. Such a view, however, appears to be inconsistent with the quantum relation Woes for this relation not only determines the fre- quency of the ultra-violet light necessary to cause an emission * Roy. Inst. Lecture, “ Engineering,” 1038. p. 568 (1917). Proc. R. Inst. xxii. p. 175 (1917). Potentials of Gases observed in Thermionic Valves. 417 of electrons having energy equal to Ve, but also determines the energy which the bombarding electrons must have in order to stimulate the gas molecules to radiate ultra-violet light of frequency v. In other words, if the quantum relation holds, the photo-electrons could never possess more energy than the thermally emitted electrons which are responsible for the radiation of the ultra-violet light, so that photo-electric emission could not cause positive ions to appear until the velocity of the thermions was sufficient to cause ionization of the gas. Finally it is to be noticed that the value of the current through the tube at the critical point is of the order of ) milliamperes, and this in itself appears to render any photo-electrie explanation very improbable, unless the marked increase in currents of this magnitude is taken as repre- senting a photo-electric emission far larger than any hitherto described. (3) Experimental Results —The method here described for the determination of ionization potentials has been applied to six different gases, with the following results :— ° Mereury vapout.«s : £ s 3 rs d < se fe 2 Spl é Bion, 45 Curvent-veltoqe curves for valves containtng mercury vapour. carton monoxide and ~nityogen. SeeanEbIIEEEEEE COO 8} 8 E calcul es s A } E ‘ an : : eh & ji lng A : me 419 Potentials of Gases observed in Thermionic Valves. a e. (quesanr) Spy ‘ ° 2 6 ray ° na or carbon monoxia ercu Ly Jo) oy) oa ni =|9 ae & 3 yy ale # vu o-/ [oy ° ro] yY op) d a. ° dl KS i ) J § =| J uv te pal an > d o ° ) 4 ~~ Qo - icf log (current Fd - 7 ressures. tlament in nitroqen at various or Current-voltagqe curves Na o 420 Messrs. Stead and Gossling on Relative Ionization Experiments have been made with most of the six gases over a considerable range of pressure, and it has been shown that the critical point in the curve, for any given gas, occurs at a voltage which is sensibly independent of the pressure. The change of slope at the critical point is, of course, dependent on the gas pressure, and becomes less qenised| as ene pressure is diminished, but otherwise there is no alteration. This is brought out clearly i in fig. 6. In all the experiments ‘the valve used was highly exhausted by means of a Gaede rotary pump, followed by charcoal cooled by liquid air. During the evacuation the valve was baked at a temperature of about 400° ©. for at least an hour, and the cold electrodes were subjected to sufficient electron bom- bardment to raise them toa red heat, in order to remove occluded gas. The gas to be experimented upon was then introduced in as pure a state as possible. The argon used was prepared from liquid air residues by sparking with oxygen over caustic potash, the excess of oxygen being finally removed by means of phosphorus. The argon when examined spectroscopically appeared to be free from nitrogen. Hydrogen was prepared by the electrolysis of a solution of barium hydroxide in water. Nitrogen from air was tried, but for most of the experiments it was prepared by mixing strong solutions of ammonium chloride and sodium nitrite and warming the mixture. Carbon monoxide was obtained by the action of strong sulphuric acid on formic acid, and also by heating nickel car bonyl. The helium used was obtained from Thomas Tyrer & Co., Ltd., and was purified by being left in contact with charcoal cooled by liquid air fora considerable time. In all cases mercury vapour was excluded by placing between the valve and the pump a U-tube immersed in liquid air. When mercury vapour itself was being studied this liquid air was removed after the valve had been thoro ughly evacuated. It may be remarked that, from the point of view of expe- riments on valves, and other apparatus in which a hot cathode is used, a knowledge of ionization potentials is often of great service, as the presence of a given gas in a sealed valve may frequently be detected by measuring the ionization potential in this way. Moreover, the test is quite sensitive. Thus the presence of mercury vapour is readily shown, even when it cannot be detected spectroscopically. Of course the con- ditions in a valve are not good for spectroscopic observation, as the discharge obtained through the bulb by means of an induction-coil is not very bright, but, nevertheless, this gives some idea of the sensitiveness of the ionization test. Again, Potentials of Gases observed in Thermionic Valves. 421 many valves which nominally contain helium, or nitrogen, give ionization voltages of about 15 after they have been in use fora short time. This indicates the presence of electrode gas, consisting probably of a mixture of carbon compounds and hydrogen, and shows that the bombardment of the anode and grid during the evacuation has not been carried far enough. (4) Corrections.—The values of the ionization potentials obtained by the method here described require some cor- rection, owing to the fact that the filament of a valve is not all at the same potential, nor at the same temperature, whilst correction is also necessary for the initial velocities of the emitted electrons, and for any contact potential difference which may exist between the filament and the collecting electrodes. As already remarked, the potential differences recorded are between the anode and the negative end of the filament, 2. ¢., they are the muzimum potential difference in the valve. Asamatter of fact the number of electrons which fall through this maximum difference of potential is entirely negligible, owing to the cooling of the ends of the filament bythe leads. The magnitude of the cooling effect due to the leads has been fully discussed by one of the writers in a previous paper*; and it appears that if a point is taken on the negative limb of the filament such that the electron emission from it (per cm.) is one-tenth of that from the hottest part of the filament, then the potential difference between this point and the negative end is about 0°3 volt for filaments of the diameter used. The fraction 1/10 is of course chosen arbitrarily, but it probably gives the order of the quantity to be subtracted from the measured voitage to allow for the cooling due to the leads. As regards the correction for the initial velocity of the electrons, it has been shownf that the average kinetic energy of the emitted electrons is equal to 2kT, where T is the absolute temperature of the filament and & is Boltzmann’s constant. Now the average kinetic energy of the electrons within the filament is the same as that of a gas molecule at the same temperature, viz. $41. In the experiments here considered the value of T for the central part of the filament was about 2500° K., and at this temperature the value of 3 kT, expressed in equivalent volts, isabout 0°33. Hence 2kT is equivalent to about 0°44 volt. At the point on the filament, * Stead, Journal of Inst. Elect. Hngineers, vol. lviii. Jan. 1920, p. 107. Tt O. W. Richardson, ‘Emission of Electricity from Hot Bodies,’ pp. 140, 141. 422 Messrs. Stead and Gossling on Relative Ionization considered in the previous paragraph, where the electron emission is one-tenth of that corresponding to the central part of the filament, the temperature would be considerably lower than 2500° K., and the correction would be reduced from 0:44 volt to about 0°40 volt. This correction must be added to the measured voltage, and it is seen to be almost compensated for by the amount (0:3 volt) to be subtracted on account of the non-uniformity of potential along the filament. It is to be observed that the velocities of the electrons emitted from a hot filament are distributed according to Maxwell’s law, and the correction of 0-4 volt applies only to those electrons which are emitted with the average velocity corresponding to the given filament tem- perature. According to Maxwell’s law of distribution an appreciable number of electrons, some 10 per cent. in fact, must come off with a velocity corresponding to at least one volt. Itis clear that both this effect and also the want of uniformity of the potential of the filament must tend to make the position of the critical point in the current-voltage curves less sharply detined, and diminish the accuracy with which the ionization potential can be determined. Goucher™ has overcome one of these difficulties by using as a source of electrons a platinum thimble heated to the required temperature by the radiation from a spiral of tungsten wire inside it, the tungsten wire being itself heated by an electric current. Such a source would not be satisfactory for the experiments here described because only a comparatively low temperature of the platinum is possible, and hence the electron emission is very small compared with that here employed. The whole question of the contact difference of potential between a hot filament and a cold anode in a moderate or high vacuum appears to be obscure at the present time, and much work remains to be done before the correction to be applied to ionization potentials on this account can be com- puted with any degree of confidence. It seems quite likely, however, that a correction of the order of 0°5 volt may be necessary T. (5) Discussion of Results —All these experiments on ionization potentials were carried out in May, June, and July, 1917, with the exception of those on helium, which were not undertaken until September 1917. Some of the * Physical Review, viii. p. 561 (1916). + Richardson, ‘ Emission of Electricity from Hot Bodies’; Stoekle, Phys. Rey., vil. p. 534 (1916). Potentials of Gases observed in Thermionic Valves. 423 results obtained were not in agreement with the values generally accepted at that time. Thus the ionization potentials of hydrogen and nitrogen, as measured by the Franck and Hertz method, came out at about 11 volts and 8 volts respectively, and for mercury vapour opinion was divided between 4:9 volts and absut 10°5 volts. By a modification of the Franck and Hertz method, designed to eliminate photo-electric effects, Davis and Goucher * have established the fact that no appreciable ionization occurs in mereury vapeur under 10 volts. In the same paper Davis and Goucher came tothe conclusion that ionization occurs in hydrogen at about 11 volts, but that a second type of ioni- zation sets in at about 15°8 volts. Shortly after the appear- ance of Davis and Goucher’s paper Bishop} published an account of some experiments on hydrogen in which he confirmed Davis and Goucher’s result that a second (and stronger) ionization appears in hydrogen at between 15 and 16 volts. The higher value agrees with the value of the ionization potential of hydrogen as determined by the method described in the present paper. No trace of ioni-- zation at 11 volts, however, appears in a valve containing hydrogen. The only measurements of the ionization potential of nitrogen which are in agreement with the value here given (17-2 volts) are those made by Davis and Gouchert, and more receutly by H. D. Smyth §. These observers have come to the conclusion that the true ionization potential of nitrogen is between 17 and 18 volts, and not about 8 volts, as had been previously supposed. The value 12°5 volts for argon is in agreement with the result given by McLennan in the Physical Review for July 1917, but very recently Horton and Davies|; have come to the conclusion that ionization does not occur in argon until about 15 volts, and Rentschler {] gives the value 17 volts. It is very difficult to reconcile this with the behaviour of argon in valves, in which it is hardly possible to avoid the conclusion that strong ionization occurs at about 12°5 volts. The result (20°8 volts) obtained for the ionization potential * Physical Review, August 1917, p. 101. t Phys. Rev. , September 1917, p. 244, t Phys. Rev., , January 1919. § Phys. Rev. a p- 409 (1919). l| Proc. Roy. Soc. vol. xevii. A. p. 1 (1920). | Phys. Rey. xiv. p. 503 (1919). 424 Messrs. Stead and Gossling on Relative Ionization of helium is in agreement with that of most other observers * except Horton and Daviest, who have recently found a higher value, as in the case of argon. Richardson and Bazzoni tf deduced a value of 29 volts from a consideration of the ultra-violet spectrum of helium. As regards carbon monoxide, the value, 15 volts, here given at present lacks confirmation by other observers. Hughes and Dixon§, employing a method of the Franck and Hertz type, obtained the value 7-2 volts. It seems probable that, as in the case of nitrogen, this low value represents a photo-electric effect, rather than the true ionization potential. At any rate there is no measurable ionization, in a valve containing carbon monoxide, below about 15 volts. It may be of interest to note that the increase in slope of the current-voltage curve which oceurs at the critical point is not accompanied by any visible radiation. The conditions for observing a faint glow between the electrodes are, however, not good, on account of the glare from the filament. All workers with soft valves are familiar with the fact that when the anode voltage exceeds a certain value a general glow suddeuly fills the bulb, but the appearance ‘of. this general glow must on no account be confused with the critical point which has been taken by the writers to denote the formation of positive ions by collision. The general glow sets in at a much higher voltage than the critical point— usually not less than twice the voltage corresponding to the critical point. Moreover, whilst the position of the critical point is independent of the form of the electrodes and of the gas pressure, and nearly independent of the filament tempe- rature, the potential at which the general glow appears depends very much on all these factors. Thus, in an audion type valve, with plane anodes and grids, the general olow occurs much more readily than in a cylindrical type valve, and the appearance of the general glow depends so much on the gas pressure that a rough “estimate of the softness of a valve ay be made by observing the voltage at which the glow appears. Again, if a current- voltage curve 1s plotted up to a potential at which the general clow appears, there is at this point a complete discontinuity, the current suddenly Jumping to many times its previous * McLennan, Phys. Rev., July 1917, p. $4. Bazzoni, loc. cit., and others. + Proc. Roy. Soc. vol. xevi. A, p.408. t+ Phil. Mag. xxxiv. p. 285 (1917). § Phys. Rev., November 1917. Potentials of Gases observed in Thermionie Valves. 425 value without, apparently, passing through any of the intermediate values. If the potential is now oradually lowered there is a marked hysteresis effect, and the general glow suddenly disappears at a potential considerably lower than that at which it appeared. The disappearance of the general glow is accompanied by a sudden and dis- continuous fall in the current. This behaviour is in marked contrast to the behaviour at the critical point. At this latter point there may be a discontinuity in the slope of the curve, but there is certainly no sudden jump in current without passing through intermediate values. Also there is little or no genuine hysteresis, the only observable effect being that the portion of the curve ‘dependent on the 3/2 power of the voltage may be slightly displaced, but not the position of the critical point. This displacement is probably due to some change in the surface of the filament resulting from its bombardment with positive ions. The point at which the general glow appears also shows up in ordinary three- electrode valve characteristics as a sudden discontinuity i in the value of the current, the occur- rence of which makes the adjustment of some “‘ soft” valves troublesome. (6) Summary.—A simple method is described of deter- mining the differences of potential which are necessary to cause positive ions to be produced in soft thermionic valves. This method has been applied to six different gases, viz.: mercury vapour, argon, hydrogen, carbon monoxide, nitrogen, and helium. The work was carried out in the Cavendish Laboratory, and the writers wish to express their best thanks to Professor Sir J. J. Thomson. Note added September 3, 1920. Since the above was written another valve method of determining ionization potentials has been described by Hodgson and Palmer*. For mercury vapour, nitrogen, and helium these observers obtained results substantially in agreement with those given in the present paper, but for argon a higher value (16°6 volts) was obtained. * Radio Review, vol. i. Aug. 1920, p. 525. Phil. Mag. 8. 6. Vol. 40. No. 238. Oct. 1920. 2F fc abu XLVIII. Optical Rotation, Optical Isomerism, and the Ring- Filectron. By H.Sranuey ALLEN, W.A., D.Sc., University of HKdinburgh *. Introduction. HEN a beam of plane-polarized hght passes through certain pure liquids and even certain vapours, there is produced a rotation of the plane of polarization, which implies that there must be some asymmetry in the molecule of the substance concerned. ‘This asymmetry has been studied mainly from the chemical side, the difference between isomeric molecules of the same structure being explained by a different arrangement of atoms in space. On the physical side progress has been less satisfactory, and although a formal connexion between optical rotation and the electro- magnetic theory of light was established by Drude f, it is only recently that successful attempts have been made to explain why asymmetry in molecular structure should in- volve difference in the velocity of propagation of circularly polarized light. In this paper the subject is approached from a somewhat different standpoint from that usually adopted, for it is assumed that the electron, in addition to its electrostatic action, behaves like a current circulating in a closed ring, and consequently acts as a small magnet. This conception involves a modification of prevailing views, both as regards the action of light on a system of electrons, and also as regards the constitution of the molecule. The paper therefore divides naturally into two parts. The first is con- cerned with the propagation of an electromagnetic wave through an assemblage of ring-electrons, whilst the second (which may prove of more interest to the chemist) deals with the question of the arrangement of the electrons in the molecule of an optically active substance, assuming the theory of the ‘‘ cubical atom” developed by G. N. Lewis f and Irving Langmuir §. * Communicated by the Author. + Drude, ‘The Theory of Optics,’ part ii. Chapters vi. & viii. + G. N. Lewis, Journ. Amer, Chem. Soe. vol. xxxviii. p. 762 (1916). § I. Langmuir, Journ. Amer. Chem. Soc. vol. xli. pp. 868, 1543 1919). Dr. H. Stanley Allen on Optical Rotation. 427 Part [. Optical Rotation. Rotation of the plane of polarization of light by a pure liquid or a vapour has presented serious difficulties to the theoretical physicist. Drude and Voigt have shown what type of electromagnetic equations are required to account for the rotation, but ‘“‘there is no satisfactory representation of the mechanism by means of which an asymmetrical molecular structure turns the plane of polarization” *. In the ordinary theory of dispersion the equation of motion of an electron {mass m, charge e) is of the form gS Oa ae an ae 6 s where &is the w-component of the displacement from the equilibrium position, X is the x-component of the exterior electric force, and the last term represents the restoring force called into play by the displacement of the electron. Drude ineludes also a frictional term which represents a force retarding the vibrations. In an isotropic medium the only possible extension of the equation is by the introduction mor’ OZ O2 ,0Y equations thus modified with Maxwell’s equations for the electromagnetic field, it can be shown that when plane- polarized light falls on the medium, two waves are propa- gated through it with different velocities, the first representing right- handed circular polarization, the second left-handed circular polarization. The superposition of the two waves yields a plane-polarized wave in which the plane of polariza- tion rotates through a definite angle for each unit length of optical path. The terms which have been added to the equations may be taken to represent a torsional electric force. Drude gives a graphical representation by conceiving that because of the molecular structure the paths of the electrons are not short straight lines, but short helices twisted in the same direction, with their axes directed at random in space. A rifle bullet lying in its rifle barrel would be displaced in a similar manner along the barrel both by a pulling and twisting force. But if we take the dimensions of a single electron to be very small, we exclude the possibility of a constraint which would enable a couple to cause a motion in one direction. We must in that case draw the conclusion * Schuster, ‘The Theory of Optics,’ § 162, 1904 edition. 2H 2 of aterm of the form ef ) By combining the 428 Dr. H. Stanley Allen on Optical Rotation, that the vibrations of the electron which give rise to the rotatory effect are motions of systems of electrons united together by certain forces which are such that a couple of electric forces produces a displacement of the positive electrons in one direction or of the negative electrons 1 in the opposite direction along the axis of the couple” (Schuster). In the last few years peter mathematical physicists, notably Born * and Gray f, have developed the latter hypo- thesis, and have shown that the optical activity of liquids and gases can he explained by regarding the molecules as coupled systems. Born views a molecule as a system of coupled electrons, the coupling and the restoring forces being identical. In the paper by Gray the atom is looked upon as a particle of dielectric. The atoms are coupled according to the ordinary laws for a doublet, and the restoring force on an electron is not identical with the coupling, but may be influenced by it. To the present writer it appears that a simpler and more realistic mental picture of optical activity may be obtained by abandoning the limitation that the dimensions of the electr on must be very small, and employing the ring-electron or “magneton” of A. Ll. Parsont. Such an jeleemen vibrating backwards and forwards along a straight line seems admirably adapted to replace the electron moving in a spiral path as imagined by Drude. The Ring-Hlectron. The ring-electron may be looked upon as a charge of negative electricity distributed continuously around a ring which rotates on its axis with high speed, and therefore bebaves like a small magnet. In an important paper on the electromagnetic mass of the Parson magneton, Webster § has shown that the ratio of the radius of the cross-section of the ring to that of the ring itself is extremely minute, and that most of the energy and momentum of the field are concentrated very closely around the ring. Thus, to a first approximation, the ring-electron may be regarded as a current in a circular wire of negligible thickness. Ina later paper || it is shown that the gyroscopic effect of the magneton * Born, Phys. Zeitschr. vol. xvi. p. 251 (1915); Ann. der Phystk, vo lv: pedi7/ (1918). + Gray, Phys. Rev. vol vii. p. 472 (1916). A. L. Parson, “ A Magneton Theory of the Structure of the Atom,” Smithsonian Misc. Coll. No. 2371, Nov. 1915. § Webster, Phys. Rev. vol. ix. p. 484 (1917). | Webster, Phys. Rev. vol. ix. p. 561 (1917). Optical Isomerism, and the Ring-Llectron. 429 is exactly the same as for an electron of the classical type moving in an orbit equal in size to the ring with a speed equal to that of the electricity of the ring. These results simplify very greatly the consideration of problems con- nected with the ring-electron. In the case of an ordinary electron exposed to light, the ineident vibrations of the light bring about forced vibrations through the action of the electric vector in the wave-front. With the modification here proposed, the effects are more complicated, for the ring-electron will be acted upon both by the electric and magnetic jet Any rotation of the plane of the ring, which may be produced by the magnetic force of the light-wave, will be neglected. There is a more important effect due to the alternating electromotive force acting round the ring and producing changes in the magnetic moment of the equivalent magnet. Consider a fixed ring- electron, the axis of the ring being parallel to the axis of w. When this is exposed to a light-wave, there will be an electro- motive force * in the ring given by —— o(\- age Ee an as SI ARON 3 OF: : = — Ac ( a ger approximately, where A is the area of the ring. If this alternating E.M.F. be represented by E,) cos pt, there will a round the ring an induced current differing 1 in phase by 4 from the electro- i EK, sin pt ; motive force, and represented by oe The magnetic dp ; moment of the electron will be increased by an amount AK, sin pt f : : ae , and in consequence there will be a mechanical p force acting upon the electron pr oportional to this increase and to the space variation of the controlling magnetic field. Thus the equation of motion of the electron will contain a term of the form of (Se oe s) required by Drude’s theory, the coefficient ef’ being proportional to A?/L, and depending also on the character nt the magnetic field due to the re- mainder of the molecule or to any external magnetic system. It then follows, as shown by Drude, that there will be two * Following Drude, E, a, 8, y are in electromagnetic units, whilst e, X, Y, Z are in electrostatic units. 430 Dr. H. Stanley Allen on Optical Rotation, circularly polarized waves travelling through the medium with different velocities, according as the rotation is right- handed or left-handed, and in consequence the plane of polarization will rotate uniformly about the direction of propagation of the light, the amount of the rotation per unit length being O=2iG_f ). where A is the wave-length of the light (in vacuum), and pis aes Here N denotes the number of electrons of the type con- sidered, in unit volume, 7 is 1/2a times the period of vibration of the light, and 7 is 1/2m7 times the natural free period of the electron. It is not claimed that the foregoing discussion gives a complete account of the behaviour of ring-electrons under the influeuce of ight-waves. There are other actions which may be briefiy referred to. In connexion with the magnetic rotation of the plane of polarization, Drude has given a theory based on the hypothesis of molecular currents, as conceived by Ampére and Weber. It is assumed that these molecular currents are made parallel to one another by the action of the external field. Drude points out that the displacement of the molecular current when «a light-wave falls upon it, produces a displacement of the magnetic lines of force which arise from it, so that a peculiar induction effect takes place. It is to be observed that this theory calls for rotations of opposite sign on opposite sides of an absorption band. It would seem that a complete theory of the action of light- waves on an assemblage of ring-electrons would have to take into consideration also the ‘‘ Hall effect,’ which has been found to yield a satisfactory explanation of rotatory disper- sion ina magnetic field. Thus the hypothesis of the Hall effect explains the result that the rotation is in the same direction on opposite sides of the absorption band in the case of sodium vapour, and also predicts an effect when the rays of light are perpendicular to the lines of magnetic force. I have discussed elsewhere * the possibility of accounting for the Zeeman effect by means of the ring-electron. * H. 5S. Allen, Proc. Phys. Soc. Lond. vol. xxxi. p. 49 (1919). Optical Isomerism, and the Ring-Electron. 431 Part IT. Optical Isomerism. In the theory of atomic structure first put forward by G. N. Lewis and afterwards developed by Langmuir, the electrons, instead of rotating in rings as in Bohr’s theory ,are supposed to occupy, or to oscillate ‘about, positions which are fixed in space with reference to the atomic nucleus. This fixity of the electrons is a characteristic feature of the magneton theory of the atom advanced by Parson—a theory which has not met with the recognition it merited, partly because it is based on the notion of a positive sphere, partly because Parson did not accept the atomic numbers of Moseley, which are now regarded as being determined not merely in a relative but also in an absolute sense. The magneton, or ring-electron, makes it possible to have stationary electrons. The most stable groupings of electrons, according to Lewis and Langmuir, are (1) the pair, as illustrated in the helium atom, (2) the octet, or group of eight electrons arranged LAER eee at the corners of a cube. Parson showed that sucha group of eight magnetons formed a system possessing very low magnetic energy and producing a very weak external field. This assumption is In agreement with the “rule of eight,” to which I have drawn attention in connexion with atomic and molecular numbers *. The number of unit electric charges in the atomic nuclei of related atoms or molecules frequently differs by 8 or a multiple of 8. In the formation of chemical compounds only an even number of electrons can be held in common. ‘‘ Two octets may hold 1,2 or sometimes even 3 pairs of electrons in common. A stable pair and an octet may hold a pair of electrons in common. An octet may share an even number of its electrons with 1, 2, 3 or 4 other octets. No electrons Fig. 1. can form parts of more than two octets” (Langmuir, Postu- late 11). Thus the single bond commonly used in oraphical formule involves two electrons held in common by two atoms (fig. 1); the double bond implies that four electrons * H.S. Allen, Trans. Chem. Soc. vol. exiii. p. 389 (1918). 432 Dr. H. Stanley Allen on Optical Rotation, are held conjointly by two atoms (fig. 2). According to a suggestion made by Lewis and adopted by Langmuir, the electrons, which are held in common between two octets or an octet and a stable pair, are drawn together to form pairs, perhaps by the action of magnetic forces. Thus in methane (CH,) the 8 electrons are located in pairs at the 4 corners of a tetrahedron, each hydrogen nucleus being held by one air. ‘ On this view, dextro- and levo-rotatory forms of a com- pound may be represented, as | have pointed out previously *, by mirror images as in fig. 3. In this diagram the letters N and § may be supposed to indicate the polarity of the ‘ exposed face of the ring-electron. Fig. 3. A suggestion has been made by W. EH. Garner + that a large number of optical isomerides may possibly exist amongst organic compounds in consequence of the right-handed or * ‘Nature,’ vol. cv. p. 71 (1920). + W. E. Garner, ‘Nature,’ vol. civ. p. 661 (1920); vol. ev. p. 171 (1920). See also letters from A. EH. Oxley, zd. vol. ev. pp. 105, 231 (1920). Optical Isomerism, and the Ring-Electron, 433 left-handed rotation of a valency electron around the direc- tion of a chemical bund. This suggestion does not receive support from the arrangement shown in the figure which seems to vield exactly the same number of isomerides.as the ordinary structural formule. I[t is true that it is possible to reverse in the diagram the magnetic polarity of one or more pairs of electrons, but even if the arrangements so obtained were stable, it is doubtful whether they would represent different isomerides. If such a reversal of the magnetic polarity were accompanied by a change in the nature of the compound, it does not seem possible to explain the pheno- menon of free mobility about a single bond which is assumed in stereochemistry. It should, however, be nentioned that although the prevailing view is that the single bond between carbon atoms does not fix the positions of (he atoms connected by it as regards rotation about the common axis, the contrary opinion has been supported by Aberson (C ohen, ‘Organic Chemistry for Advanced Students,’ pp. 116, 133). This raises a question in the theory of the cubical atom which requires further elucidation. It is clear from fig. 3 that the electrons associated with group a are related to ‘the electrons of group } in a manner different from that in which they are related to the electrons of ¢ or d. Lewis assumes that each pair of electrons is drawn together so as to represent a single corner of ‘‘the model of the tetrahedral carbon atom which has been of such signal utility throughout the whole of organic chemistry.” But even if the electrons in a pair are drawn closer together than in the diagram, the lines joining the centres of the pairs are oriented differently when a and 6 are compared with c and d. The difficulty might be got over by supposing the electrons in a pair to rotate about a point midway between them, or in the case of the ring-electron one electron might be supposed to move over the other, giving the arrangement ee “> —_ : gs Me a seragat eae instead of a . r e we 4 a ——— > ee Dr. Langmuir has been kind enough to express his views on this question in a letter to the author. ‘¢Whena pair of electrons acts as a bond between two adjacent atoms, the relationship between the two electrons has certainly becom: changed. The very fact that we never have one or three 434 Dr. H. Stanley Allen on Optical Rotation, electrons held in common between two atoms is proof that the electrons in the pair are bound into a kind of unit, in which the relationship between the two electrons is very different from that between two adjacent electrons in the neon atom.... In practically all the compounds of carbon, there are eight electrons arranged around the kernel of the carbon atom, but these electrons are gathered into four pairs, each pair constituting a unit or bond between the carbon atom and atoms surrounding it.. The nature of this unit is evidently closely related to that of the pair of electrons in the helium atom and hydrogen molecule, and my guess is that such a pair consists of two electrons revolving about a line oennelnae the kernels of the adjacent atoms, whereas the electrons in the neon atom are revolving about eight Eolen located at the corners of a cube. We are thus led to a conception of the carbon atom which is practically identical with that of the organic chemist, namely : that the carbon atom exhibits four valence bonds arranged i in ce in a symmetrical way, por Resjpomerne to the corners of ¢ tetrahedron.’ According to the present theory, optical activity arises from a difference effect, and can be manifested only when there is lack of compensation amongst the electrons associated with the various parts of tlhe molecule. If the chemical bond is to be attributed to a pair of electrons arranged side by side, it is easy to understand how such compensation can be brought about in the great majority of chemical compounds. In the case of a single asymmetric carbon atom, the sym- metrical arrangement of each of the four electron pairs 1s disturbed by the presence of the adjacent groups, resulting in only partial compensation. Thus, in the compound Cabed, the pair of electrons associated with group a@ is under the influence of the unlike groups ¢ and d, and the condition of symmetry is absent. But if ¢ and d are made alike, the whole molecule will have a plane of symmetry indicated by the broken line in the left half of fig. 3. Thus the molecule will be inactive through “internal compensation ~ with respect to the electrons which form the outer shell of the carbon atom. This arrangement of the electron pair side by side gives readily an explanation of the fact that most chemical com- pounds show diamagnetic properties, for such a pair would produce a very weak external magnetic field, and this configuration has in fact been utilized by Oxley * to account for the diamagnetism of the hydrogen molecule. * “Oxley, ‘Nature,’ vol. cy. p. 827 (1920). Optical Isomerism, and the Ring- Electron. 435 The evidence at present availabie does not justify a decisive verdict as between the two possible arrangements Yn ie ae Kehet and for the electron pair’ It may be that doth are possible, with corresponding differences in the optical and magnetic properties of the compound. On the basis of the latter con- figuration, a large number of isomerides may exist in accord- ance with the suggestion of Garner, referred to above. “‘ The electrons rotating in pairs around the four carbon valencies may possess either clockwise or anti-clockwise rotation with respect to the central carbon atom. On the assumption that two of these pairs of electrons rotate in a clockwise and two in an anti-clockwise direction, it is possible to deduce that eight isomerides of cinnamic acid may exist.” If the clock- wise rotation of the electron gives a north-seeking character to the valency and the anti-clockwise rotation a south- seeking character, isomerides may exist which may be represented graphically as below R, ki R, Ry * / DS / S N : N Noes NG SS ind SF BOS 7 C C ss aes SN BL PN N S N Ss fe \ we N The existence of the isomeric compounds suggested by Garner depends upon the somewhat arbitrary assumption that two of the pairs of electrons are rotating in a clockwise sense, and the other two in an anti-clockwise sense as viewed from the carbon atom. In a three-dimensional model the four valencies may be supposed directed towards the corners of a regular tetrahedron. It will then be observed that Garner’s assumption implies that the valency electrons in such a compound confer upon it paramagnetic properties, since the suggested arrangement of electrons would have 436 Dr. H. Stanley Allen on Optical Rotation, a resultant magnetic moment. It would, of course, be possible to overcome this difficulty by a further assumption that magnetic compensation arises from other parts of the molecule. Optical Rotatory Power. The theory of optical rotation for liquids and gases here proposed appears to be in good agreement with the experi- mental facts as summarized by Gray. 1. Any substance of which the molecules are truly asym- metrical causes rotation. 2, Antipodes, a molecule and its mirror image, rotate in opposite directions. 3. Symmetrical molecules are not active. Absorption systems in an active substance cause anomalies in the rotary dispersion, although the origin of the band may lie in a part of the molecule distant from the central carbon atom. 5. The magnitude of the activity is profoundly influenced by the molecular structure, and 1s roughly dependent on the magnitude of the asymmetry of the molecule. 6. Activity varies with temperature and pressure, and with the concentration of active molecules. From what has already been said it is obvious that the first three results are in accord with the theory. The fourth result follows alike from the electrostatic theories of Born and of Gray and from the electromagnetic theory here given. In the first case the rotation depends on a space derivative of the electric force, in the second on a space derivative of the magnetic force. The fifth result requires somewhat closer examination. Guye * and Crum Brown jf have discussed independently the relation of optical activity to the character of the radicles united to the asymmetric carbon atom. “It is obvious that the amount of optical activity of a given com- pound containing an asymmetric atom of carbon depends upon the amount of difference in character among the four radicles united to the asymmetric carbon atom, so that if two of them are very nearly equal we come very near to a com- pound of a symmetric carbon atom, in which the optical activity is zero. The question suggests itself, How are we to measure this difference of character?” Guye regarded * Guye, Compt. Rend. vol. cx. p. 714 (1890); vol. exvi. pp. 13878, 1451 (1893). + Crum Brown, Proc. Roy. Soc. Edin. vol. xvii. p. 181 (1890). Optical lsomerism, and the Ring- Electron. 43 } ) l the mass of each radicle and the distance of its centre of gravity from the centre of figure of the tetrahedron as all that need be considered—a view which could not be held at the present time ; but Crum Brown assumed merely “ that there is a function, capable of numerical representation, derivable from the composition and constitution of the radicle and the temperature of the substance, and that it is the ditference between the values of this function in the case of two radicles which gives us the difference of character referred to.” According to the present theory, it would be the values of the magnetic and electric fields produced by the radicles which would determine the degree of asym- metry. Amongst the papers read at the discussion on “ Optical Rotatory Power,” held before the Faraday Society in 1914, one of the most interesting from a theoretical standpoint was that by Leo Tschugaeft. He came to the conclusion that the electrons which are most active in producing rotation are attached to the asymmetric carbon atom itself, or are situated in the immediate neighbourhood of the centre of activity. As regards the other electrons which come into play in rotatory dispersion, we can assume that their activity is diminished with increasing distance from the asymmetric complex. Again, if it be admitted that the degree of asymmetry of the molecule depends upon the differences of four constants K,, K,, K;, and K,, corresponding with the four groups attached to the carbon atom, the value of the unknown function K must depend on the degree of satura- tion. “Thus, it would be expected that strongly unsaturated radicies containing free-movable electrons would exert a considerable influence on the electric field produced by the molecule, the differences K,—K,... being much larger if K, corresponds with a saturated group and K, to an un- saturated one than if both the groups are nearly equally saturated.” ; The present theory is in general agreement with the views just described, but the equation representing the views of Guye and Crum Brown would require some modification. Since the rotatory power as well as the dispersion is assumed to be equal to the sum of the effects produced by the several active electrons, as is expressed by the formula k O/'N ie a. 1-(7 it is necessary to consider the contributions from the four pairs of electrons associated with the four attached groups. , 438 Dr. H. Stanley Allen on Optical Rotation, lf the arrangement represented in fig. 3 be adopted, the asymmetry of pair a is due to groups ¢ and d, and the resulting contribution to the rotation will be equal to the algebraic sum of two terms of the type given above. In order to explain the absence of rotation in a symmetrical molecule, it is necessary to assume that the sign is positive for one electron and negative for the second electron of the pair. Let us further assume, at least as a first approxima- tion, that the natural free period is the same for each electron, \2 so that the denominator 1-(=) is the same. The con- tribution to the rotation made by the pair of electrons may then be written ou ON Py Ne = ae. =A(f'a—f'a,)- Hence the total result may be written P=A(K,—K,)+ B(Ka—K,) + CCK, — Ke) + D(K,— K,), since the difference between /’,, and /",, is due to the presence of the unlike groups ¢ and d. It follows that P=(A—B)(K,—K,)+ (C—D)(K,—K,). In the case in which the groups ¢ and d become identieal, K.=K, and C=D, so that there is no resultant activity. If, on the other hand, the alternative arrangement of coaxial rings be adopted, the resultant effect would have to be calculated by methods similar to those employed by Gray * in the ease of purely electrostatic forces. According to the principle of optical superposition formu- lated by van’t Hoff, the total rotatory power in a compound containing several asymmetric carbon atoms is the algebraic sum of the various radicles taken separately. This rule seems to be valid at least to a first approximation Tf. The work of ©. S. Hudson ¢ and his collaborators shows that the principle holds fairly closelv in the case of certain amides. The approximate validity of the rule is to be expected on such a theory of optical rotation as that here put forward. The variation of optical activity with temperature, 9 Ce ary ey) Reloc, cv. + Tschugaeff, Joc. cet. { C. S. Hudson, Journ. Amer. Chem. Soe. vol. xli, pp. 1140, 1141 (1919). Optical Isomerism, and the Ring- Electron. 439 pressure, and concentration presents difficulties whatever view may be adopted as to the origin of the rotation. The subject has been discussed by Livens *, who has examined the ettect of the presence of inactive substances, as in the case where a simple active substance is dissolved in an inactive liquid, and also by Gray t. Assuming that the atoms are vibrating about points of equilibrium inside the molecule, the average value of both the electrostatic and the magnetic field produced will change with the amplitude of the oscillations, and conse- quently the degree of asymmetry will be a function of the temperature. Finally, it may be noted that the optical activity of compounds containing asymmetric atoms other than carbon atoms follows at once from the theory. Such active compounds are known in the ease of nitrogen, phosphorus, sulphur, selenium, tin, silicon, cobalt, chromium, rhodium, and ironf{. It is | not without significance that the three elements cobalt, rhodium, and iridium occupy a peculiar position in Langmuir’ s theor y, as in each case there is an odd electron in the outer shell which is assumed to occupy a position at one end of the polar axis. Summary and Conclusion. In the present paper a theory of optical rotation has been advanced in which the electron, instead of being regarded asa point charge, is looked upon (as sugoested by A. L. Par son) as an anchor ring of negative electricity rotating rapidly toa itsaxis. Sucha ring-electron vibrating in a kivade path tak the place of an ordinary electron moving in a spiral cath as postulated by Drude. ‘It is shown that rotation of the plane of polarization of light will result, and an expression is found for the amount of rotation per unit length. Employ- ing the theory of atomic structure due to Lewis and Langmuir, a eraphieal representation may be obtained for dextro- and levo- -rotatory forms of a compound. The ex- perimental facts with regard to optical activity are in good agreement with the theory put forward, which may be applied not merely to carbon compounds, but to any com- pound containing an asymmetric atom. * Livens, Phil. Mag. vol. xxv. p. 817 (1913). T Loe. cit. { A. W. Stewart, ‘ Stereochemistry,’ chapter x. (1919). [Os XLIX. The Lifect of a Trace of Impurity on the Measurement of the lonization Velocity for Electrons in Helium. By Frank Horton, Se.D., Professor of Physics in the Uni- versity of London, and Doris Batiey, M.Sc., Assistant Lecturer in Physics in the Royal Holloway College, Engle- field Green”. T has been found by Horton and DaviesT that radiation is produced when electrons having a velocity of 20°4 volts collide with helium atoms, and that this effect is not acecom- panied by any ionization of the gas. Ionization of helium was found to occur when the electron velocity was raised to 25:6 volts. An account of the experiments from which these results were obtained was given to Section A of the British Association at the meeting in Bournemouth in 1919, and in the discussion which followed, the view was expressed by some of the speakers, that ionization of helium occurs at the lower critical velocity mentioned above. Dr. Goucher reported that experiments made by him with an improved form of his original apparatus for distin- guishing between the photoelectric effect of radiation on the electrodes and the ionization of the gas by electron impacts, had resulted in the detection of some ionization as well as radiation at about 20 volts, and copious ionization at about 25°5 volts. He stated that the helium used by him was possibly not as pure as that used in the experiments of Horton and Davies. The evidence of other speakers was mainly to the effect that when helium was used in a therm- ionic valve the current-H.M.F. curves showed a rise at about 20 volts which could only be due to the production of ionization at that point. The ionizing velocity for electrons in helium was first investigated by Franck and Hertz, who gave 20°5 volts as the critical value at which ionization occurs{. It was pointed out by Bohr that the method used by these experi- menters was not capable of distinguishing the effects of a production of radiation in the gas from those of ionization of the gas by electron collisions. From theoretical considerations Bohr had deduced the value 29 volts as the minimum ioni- zation velocity of helium, and he suggested that the effect. detected at 20°5 volts was not a genuine ionization of the helium, but a secondary effect due to the production of radiation in the gas. He pointed out that such a radiation * Communicated by the Authors. + Proc. Roy. Soc. A. vol. xev. p. 408 (1919). ile Franck and G. Hertz, Deutsch. Phys. Ges. Verh. vol. xv. p. 34 (1918). Ionization Velocity for Electrons in Helium. 44] would not only act photoelectricaliy on the metal parts ot the apparatus, but would also be able to ionize any Impurities present in the gas*. It is clear that the presence of easily ionizable impurities would thus result in the detection of ionization at the radiation velocity, even when the experiment is performed in an apparatus such as that of Dr. Goucher. In the early experiments with helium-filled valves, which were made at the Royal Holloway College in 1915, evidence of ionization at about 20 volts was always obtained, and it was only during the course of the investigation referred to at the beginning of this paper, that the view that helium can be ionized by electron collisions with this velocity was finally abandoned. Franck and Knipping have also arrived at the conclusion that the earlier eg tos of Franck and Hertz measured the minimum radiation velocity and not the minimum ionization Waracity of the gas; and in a recent paper + they have given values of these two critical velocities in good agreement t with those obtained by Horton and Davies. Several forms of apparatus were used in the present expe- rimentst, some of these being thermionic valves of the eylindrical pattern, in which ie filament was completely enclosed by the grid. In another form two parallel grids Pie~ | oe Vs i eeses Setaeauces Vp Cc Ses eeaateaaneas & V, D + = A, Anode; B,C, Grids; D, Vilament. were used, the filament being enclosed ' below the first, and the anode, a small platinum plate, being situated about 1°5 cm. beyond the second grid and parallel to it (see fig. 1). N. Bohr, Phil. Mag. vol. xxx. p. 410 (1915). ' ‘ Franck and P. Knipping, Phys. Zeits, xx. p. 481 (1919). t 1am indebted to the Government Grant Committee of the Royal Society for the means of purchasing some of tle apparatus and materials used in this research.—F’. H. Phil. Mag. Ser. 6. Vol. 40. No, 238. Oct. 1920. 2G - 442 Prof. F. Horton and Miss Doris Bailey on Measurement With this form of apparatus, a magnetic field parallel to the axis of the tube was used to prevent the electron stream from spreading laterally. All the metal (except the tungsten filament) was platinum and had been boiled for several days in strong nitric acid before being used. In every case the complete apparatus was baked for many hours, and at the same time kept evacuated to a low pressure by means of a mercury-vapour pump; the filament was also heated to a high temperature to rid it of occluded gases. During this process mercury vapour from the pump was prevented from passing over by an intervening U-tube containing carbon cooled in solid carbon dioxide. In filling the experimental apparatus with helium, the purified gas was allowed to pass very slowly along a fine capillary tube and then through the charcoal tube which was cooled in liquid air. The electric currents between the various electrodes were measured by sensitive moving-coil galvanometers, while the potential difference accelerating the primary electron stream was gradually increased. It was found that the current due to photoelectric action of the radiation on the electrodes was too small in comparison with the original electron current to be detected by the anode galvanometer, for when the apparatus contained perfectly pure helium there was no sudden rise in the current-.M.F. curve until ionization of the gas occurred. On account of the velocity of emission of the electrons from the filament and possibly of other causes, the measured potential difference does not give the velocity of the electrons passing through the gauze. In order to obtain the minimum ionization velocity it is necessary to determine the correction which must be added to the applied potential difference at which ionization is first detected. For this purpose the type of apparatus with two parallel grids has an advantage over the type with one grid only; for with the former type it is possible to obtain the correction to be added to the applied potential difference so as to give the velocity of the electrons which actually produce the effect measured by the anode galvanometer, whereas with the single grid the correction which is obtained gives the velocity of the swiftest electrons present under a given applied accelerating potential difference, irrespective of whether these swiftest electrons are sufficiently numerous to produce a measurable amount of ionization when their velocity reaches the critical value. A special investigation made with the apparatus repre- sented in fig. 1 showed that the minimum nuinber of electrons which could be detected by the galvanometer could also produce a detectable ionization current when their velocity of the Ionization Velocity for Electrons in Helium. 443 was above the critical value, so that, using a galvanometer as the current-measuring instrument, the two “corrections” re- ferred to above are identical. ‘The experiments were made us follows:—A potential ditference V, was applied between the filament and the first grid in such a direction as to oppose the emission of electrons from the filament. By the variation of this field the number of electrons passing through the o oauze could be controlled. In order to determine for what value of the potential difference V, the minimum number of electrons capable of giving a current measurable with the anode galvanometer was obtained, the electrons after passing through the first gauze were aceelerated by the fields V, and V;, towards the collecting electrode, the sum of the potential differences V, and V; being less than the ionization potential difference. While wa and V3 remained constant. V, was gradually reduced from a value too great to allow any electrons to pass through the first gauze, until the anode galvanometer gave the first indication of a current passing through it. Thavalue of V, at which this first indication was obtained is the correction to be applied to give the velocity of the swiftest electrons present. In the second part of the experiment, the field V, con- trolled the number of electrons employed, but the potential difference V, was made a little above the ionization value, so that the electrons which passed through the first gauze attained sufficient velocity between the first and second gauzes to be able to ionize helium atoms on collision with them in the s space above the second gauze. The potential difference V3 in this space oppesed the electrons from the filament and was maintained sufficiently great to prevent any of them from reaching the collecting electrode. As the photoelectric effect of the: radiation on the electrodes does not produce a measurable current, the only current measured by the anode galvanometer is that due to the positive ions produced in the space between the second gauze and the collecting electrode, by the impact of the electrons with helium atoms. Keeping the potential differences V, and V; constant, V,; was adjusted to the limiting value at ae a detectable ionization current was measured by the galvano- meter. It was found that this limiting value of ve was the same as the value of V, obtained in the first part of the experiment. Thus itis safe to assume that when a single- grid valve was used, the ionization velocity deduced from the swiftest electrons detectable, was actually the velocity of those which first produce a measurable ionization. In investigating the ionization velocity for electrons in 9 G ) 444 Prof. F. Horton and Miss Doris Bailey on Measurement the gas, the currents to the anode were measured as the accelerating potential difference was gradually increased, and the currents were plotted against the corresponding values of the electron velocities, which were deduced by applying to the accelerating potential differences the corrections found in the manner just described. With perfectly pure helium in the apparatus, curves similar to I. of fig. 2 were obtained, indicating that the minimum ionization velocity of helium is 25-0 volts. 1S 20 25 30 Electron velocity (volts) The figure also contains curves showing the effect of a. small but gradually increasing quantity of impurity upon the electron velocity at which ionization was first detected. The observations from which these particular curves were plotted were taken with a valve in which the grid and anode were connected together and formed the positive electrode, the glowing filament being the negative electrode. The pressure of the helium was about 0°85 mm., and the distance of the filament from the grid—about 5 mm.—was several times the mean free path of an electron in the gas, so that when the ionization potential difference was reached, ioni- zation would take place mainly in this space. When the observations plotted in curve I. had been made, a wide-bored tap, which separated the discharge chamber from the puri- fying charcoal tube, was ciosed, and after an interval of 5 minutes, during which the heating of the filament was continued, the characteristic curve was again taken. Curve II. of the figure represents the result obtained. It is seen that of the Ionization Velocity for Electrons in Helium. 445 this curve breaks away from the straight line at a lower value of the electron velocity than curve I. An hour later tne observations were repeated, and, as will be observed from curve III., ionization was then detected at an electron velocity of about 21 volts. After these observations the gas was withdrawn from the apparatus and made to circulate repeatedly through carbon cooled in liquid air, the gas being thus re-purified ; on again testing, ionization did not occur until the electron velocity had been increased to 25 volts. These experiments show that the detection of ionization at electron velocities less than 25 volts is due to impurity, which can be removed from the helium by repeated circu- lation through carbon cooled in liquid air. It will be noticed that in curves II. and III. of fig. 2 in which the indication of ionization occurs before 25 volts, there is no indication of additional ionization when 25 volts is reached. This shows that the direct ionization of the helium by 25 volt impacts does not result in the production of more ionization than the indirect ionization of the impurity by the helium radiation. The amount of impurity collecting in the apparatus under the conditions indicated must have been very small in view of the long treatment to which the apparatus had been sub- jected before these observations were begun, and it might seem unlikely that so small an amount of impurity should cause so marked an effect in the current-H.M.F. curve. It must be observed, however, that when the helium is pure, ionization is caused only by electron collisions, and is therefore limited to that part of a column of gas in the path of the electron stream, in which the electrons collide with atoms with enough velocity to ionize them, while with the impure gas the impurity is ionized by the helium radiation, so that in the presence of sufficient radiation, ionization can take place throughout the whole volume of the gas in the apparatus, thus giving a considerable current through the yalvanometer when only a very small percentage of impurity is present. The same considerations also explain why the ionization of the impurity by the radiation from the helium atoms is marked when there is no indication of the direct ionization of the impurity by electron impacts, although the impurity must be ionizable by collisions with electrons having less than 20 volts velocity, since it is ionized by the helium radiation, If there were enough impurity present in the helium for an indication of its ionization by electron impacts to be given in the curves, it is probable that the extra - 446 Prof. . Horton and Miss Doris Bailey on Measurement ionization produced by the helium radiation would not show in the marked way it does is fig. 2. In curves which were taken at various time intervals after the manner of those of fig. 2, the electron velocity at which ionization was first detected decreased vradually from 25 volts to about 21 volts, as the amount of impurity present increased, but the rise in the curve never occurred before about 21 volts, indicating that this is the minimum velocity at which radiation is pro- duced from the helium atom. With a very small amount of impurity (such as that collecting in a few minutes heating with the tap closed) the current-EH.M.F. curve rises at some value of the electron velocity intermediate between the ionization potential difference and the radiition potential difference. This is explained by the facts that the radiation is produced throughout a gradually increasing layer of gas as the accelerating potential difference is raised, and that with a very small “amount of impurity a large amount of radiation is necessary to produce sufficient ionization to give an indication on the galvanometer. Another instance of the marked effect of a small amount of impurity was obtained in the course of some experiments in which a speck of lime ona platinum strip was used as the source of electrons. To make the lime adhere to the platinum it had been mixed with a small proportion of powdered red sealing-wax. The filament had been raised to incandescence by an electric current, and the organic matter burnt up before the filament was sealed into the apparatus, and the heating of the filament had been continued for some days. It was, however, found that with this filament, ionization was always detected atan electron velocity of about 21 volts, even when the apparatus was in connexion with a carbon tube cooled in liquid air. By examining the spectrum of the luminosity which appeared in the gas at an electron velocity slightly higher than that necessary for ionization, it was found that the mercury green line 15461 was faintly visible. The small trace of mercury present in the otherwise spectro- scopically pure helium must have come from the vermilion colouring matter in the sealing-wax used, for it was im- possible for mercury vapour to have passed into the apparatus from the pump, as the connecting stopcock had never been open without the intervening U-tube being cooled in liquid air or in solid carbon dioxide. Moreover, with a tungsten filament, the merenry lines were never seen in the spectrum of the luminous discharge when the same precautions were taken to keep the helium pure. Fig. 3 is an example of the curves obtained with a of the Ionization Velocity for Electrons in Helium. 447 single-grid valve having a lime-coated filament and containing helium contaminated with a trace of mercury vapour from the sealing-wax used in coating the platinum strip. In taking the observations recorded in this figure a constant difference Fig. 8 12 Current cae Electron velocity (volts) of potential of 8 volts was maintained between the anode and the grid and the difference of potential between the grid and the filament was gradually increased. The currents measured by the galvanometer connected directly to the filament are given (Curve |.), as well as those measured by the anode galvamometer (Curve II.), the latter being also plotted on a larger scale (Curve III.), so that the rise in the current at about 21 volts may be seen. The curves show that both the measured currents increased abruptly at an electron velocity a few volts above that at which ionization of the helium can occur. The current measured by the filament galvanometer is mainly that carried by the stream of electrons from the glowing lime, and the sudden rise in the curve must be due to an increased emission, which no doubt results from the neutralization of the space-charge near the filament by the positive ions travelling to it. It was found that the point at which the sudden large increase in the current occurs depends very much upon the electric fields used between the electrodes. It could be prevented by adjusting the fields so as to prevent the positive ions formed in the - 448 Prof. F. Horton and Miss Doris Bailey on Measurement anode space from travelling to the filament, and at the same time arranging so that no ionization could take place between the first gauze and the filament. In order to secure the sharp rise in the curve it is necessary for the electric fields to be arranged so that the neutralization of the space-charge occurs suddenly. If the neutralization occurs gradually, the increase of electron emission from the filament is also gradual and a sudden rise in the current does not occur. This con- dition is attained when the grids and anode are connected together to form one electrode, as in the experiments the results of which are shown in fig. 2. The rise in curve ILI. of fig. 3 at about 21 volts is due to the ionization of the small trace of mercury vapour which the otherwise pure helium contained, although the ionization chamber opened into a purifying tube containing carbon which was cooled in liquid air. The impurity present in the apparatus during the experiments described earlier was pro- bably gas evolved from the glass in spite of the long treatment to which it had been subjected. The spectrum of the luminous discharge in this case showed no trace of the mercury lines but a very taint indication of a band spectrum, probably due to oxides of carbon. The view that the ionization which has been detected in helium when electron velocities of less than 25 volts are used, is due to the indirect ionization of traces of impurities by the helinm radiation, is also supported by the results of some experiments in which the spectrum of the luminosity produced in helium by electron bombardment was investi- gated for different velocities of the electron stream. These experiments were performed with a single-grid apparatus, the metal parts of which were not all of platinum and had been contaminated by contact with mercury vapour while they were being used in another research. It was con- sequently found to be impossible to eliminate mercur vapour completely during tle investigation, although the amount of this impurity present in the helium must have been very small, for the apparatus was in direct connexion with a charcoal tube cooled in liquid air throughout the experiments. A series of photographs of the spectrum of the luminosity produced in the gas between the anode and the grid was taken by means of a Hilger direct wave-length reading spectroscope, for various values of the applied potential difference. The lines seen on the photographic plate in a typical case are given in the following table. With the three lower electron velocities, namely 21°4 volts, 23°7 volts, of the Ionization Velocity for Electrons in Helium, 449 and 25°8 volts, the time of exposure of the plate was from 4 to 5 hours. With an electron velocity of 50 volts a bright luminosity was obtained and the exposure was only about 1 hour. Velocity of the electron stream. 2 ‘Seen aa a) 21°4 volts. 23°7 volts. 25°8 volts. 50 volts. 5048 (Ud) 5048 (5) 5016 (2) 5016 (8) 4922 (1) 4922 (6) 4713 (5) 4713 (6) 4472 (5) 4472 (10) 4438 (0°5) 4438 (6) 4388 (0:5) 4388 (6) 4559 very faint 43859 (2) 43859 (1) 4399 (5) 4848 (0°) 4348 (1) HIST (O"5) 4169 (1) 4144 (2 4121 (8) 4026 (3) 3868 (0°5) The numbers in brackets indicate the relative intensities of the lines on the photographic plate. The wave-lengths printed in italics are due to mercury. The lines 4359 and 4348 are due to mercury, and these were the only lines obtained so long as the maximum velocity of the electron stream was less than 25 volts. The line 4337, which was faintly visible on the plate when an electron velocity of 50 volts was used, is also due to mercury, but all the other lines given in the table are due to helium. On the view, now generally accepted, that the many-lined spectrum of a gas can be produced when ionization has occurred, these results indicate that ionization of helium does not occur below 25 volts, but that ionization of mercury was taking place between the resonance and ionization velo- cities of helium. From the first column of the table it will be observed that the only line seen on the photographie plate when the electron velocity was 21:4 volts was the mercury line 1. 4359, and as this was extremely faint, itis evident that there was very little ionization of the mercury vapour by direct impact, even although the electron velocity was well above the ionization value for mercury (10°4 volts). The greatly increased intensity of this line when the voltage was raised to 23°7 volts must therefore be due to the largely increased amount of helium radiation then produced. These two spectra thus indicate that the amount of ionization of mercury vapour by direct electron impact is extremely small 450 Pref. W. M. Thornton on the Ignition of compared with that produced by the helium radiation at the higher electron velocity, a result which is in agreement with that found by means of the eurrent-E.M.F. curves. The experiments described in this paper thus emphasize the importance of maintaining helium perfectly pure when attempting to investigate the ionization of the gas by electron collisions. The only satisfactory method when a glass apparatus is used is to have a slow circulation of freshly purified helium through the ionization chamber during the experiments. It is possible that if fused silica were sub- stituted for glass, contamination of the gas would be less likely to occur. . he curves which have been given indicate that the minimum ionization velocity for electrons in helium is 25°0 volts, a rather lower value than that obtained by Horton and Davies. In their experiments the correction applied to the accelerating potential difference to give the critical velocity was that found from the swiftest electrons present, and, although a sensitive electrometer was used to detect ionization, it is possible that the swiftest electrons were not sufficiently numerous to produce a measurable lonization current. The results given in this paper also indicate that the minimum radiation velocity for electrons in helium is about 21 volts; but this value is no doubt too high, for the point at which ionization of the impurities by the helium radiation was detected depended on the amount of impurity present, and was probably always higher than the point at which radiation was first produced “from the helium atoms. L. The Ignition of Gane at Reduced Pressures by Transient Ares. By W. M. ‘THornton, Professor of Llectrical Engineering in Armstrong College, Newcastle-on- Tyne”. 1. Introduction. HE momentary are formed at the point of break of a current-carrying circuit is an active source of ignition. It differs essentially in character from disruptive discharge, and consists mostly of a stream of charged particles passing from the negative to the positive pole. At the latter there is always a bright spot formed corresponding to the crater of * Communicated by the Author. Gases at Reduced Pressures by Transient Arcs, 451 maintained ares. This is more noticeable when the inductance of the circuit is low ; when it is high there is a flaming are which, though it has a resistance low compared with that of the film of gas at the cathode, obscures the effect. The ignition of gases by such noninductive transient ares between platinum’ poles is remarkably uniform in type, differing in this from the more varied phenomena of impulsive spark ignition. It is in one sense simpler than the latter in that there is no preliminary ionization of the gas before the spark passes. The action is therefore confined to contact between the explosive mixture and hot metallic vapour in which there is intense electrification under the combined influence of high temperature and strong field. The mode of ignition is then a chemical combination of the gases started partly by high temperature collision, as in fiame, aided by the direct electrification of the incandescent vapour with which the gases collide. At normal pressures heat alone may be more important ; at low pressures the tempera- ture of the are falls, but ionization by collision rises in value—the electric gradient being kept the same. It was therefore to be expected that the limiting con- ditions of ignition would vary and show critical points or phases, characteristic respectively of heat and ionization, though not to the same extent as with impulsive discharge, and the results obtained show this to be the case. 2. The nature of Transient Arcs. The voltage across the ass of break of a circuit is the sum of the inductive voltage i and of part of the circuit volts, the remainder being that absorbed in the resistance. In the present work the inductance was made as low as possible by the use of flat woven grid resistances. Oscillo- grams of break reveal no voltage overshoot, the circuit break sparks were therefore miniature arcs extinguished by air cooling. The influence of gas pressure on the properties of such ares is now being investigated. Jt has been found that as the pressure is lowered the length of the arc falls to a minimum, almost to zero, after w hich the discharge changes character, spreads into a luminous glow, lengthens and increases in duration. This change is always exceedingly sudden, and is possibly in the first place to be associated with the sudden drop in the number of ions emitted from a 452 Prof. W. M. Thornton on the Jgnition of hot surface when the temperature reaches a critical value* The electronic state is set up at values X/p from °2 to °d, where X is in volts per centimetre and p millimetres of mercury. In order to have this at pressures of say 300 millimetres in air on ares 5 mm. long X must be from 60 to 150, or the voltage on the are 30 to 75. It is clear that this will be passed through sometime during the break, for the gap voltage starts at zero and ends at that of the cireuit, in this case 100. 3. Results with Direct Current. ‘The break was made electromagnetically between platinum rods by drawing an iron plunger, to which they were attacbed, into the core of a coilt. The curves obtained, figs. 1 to 4, have for ordinates the currents broken, and for abscissee the pressures in atmospheres. They all approach a lower limit at a little above 0-2 atmosphere, and pass through an extraordinary fluctuation of inflammability from 0-4 to 0°5 atmosphere, after which they fall smoothly to a minimum reached in most cases at about 1°5 atmospheres. The most interesting feature of this set of curves is the progressive change of the paraffins as the order rises. Hydrogen is in a class by itself; the currents are much smaller and the oscillation greater. Regarding the latter as an indication of instability about a mean ordinate similar to that of the curves of change of state, the following values of the ordinates are taken to give equal areas above and below the mean line: elaydhyo oem He Se 0-30 ampere. Methane isan tate iQ) Bithane es seat ee ileal < Propanet2) ae 0-87 is Pentanes any ere 0-66 7 For ethane, propane and pentane, these currents are 93°5 mol weight’ the higher paraffins approach hydrogen in ease of ignition. expressed by 2=0°334+ that is, in the limit * J. J. Thomson: ‘The Conduction . of Electricity in Gases,’ p. 479-480. + “The Influence of Pressure on the Ignition of Methane,” Brit. Assoc. Newcastle; ‘The Electrician,’ Sept. 8th, 1916. Gases at Reduced Pressures by Transient Arcs. 453 ‘8 ANIPERES. ni sth PROPANE Pah _ ith . fhe Lt PEEP EERE a RS IN Fe laa ae ee OO teal othe i297, 4 OD 2 "4 g@ 10 “7 go to ta 14 4. Relative Influence of Carbon and Hydrogen Atoms on the Ignition of Paraffins. A curve similar to fig. 5 has been obtained for the same gases ignited by incandescent wires*. ‘The presence of a carbon atom increases affinity for oxygen, methane having a stronger attnity than hydrogen for oxygen as shown by analys sis of the products of combustion at high pressures. * “The Ienition of Gases by Hot Wires,” Phil. Mag. vol. xxxviii. November 1919, fig. 3. - 454 Prof W. M. Thornton on the Ignition of It retards ignition either by ares or hot wires. But if the readier ignition of the higher paraffins is to be regarded as caused by the hydrogen atoms their influence must follow a higher power than the first, or that of the carbon must remain constant. For the curve, fig. 5, may be taken as a straight line through the origin combined with an hyperbola. Writing (@—%)=ano/(nx)*, where mg is the number of carbon atoms and ny that of the hydrogen atoms in a molecule, and noting that ng=2nc+ 2, then ~ ae: 1 (t—%) =a/4 ( no+ 2+ th When the value of the coefficient a, derived from the currents in fig. 5, is constant, it is evidence that the igniting current 1s proportional to the ratio no/(mm). From the figure 7="06, or is negligibly small. For ethane a=20-08, propane 18°55, pentane 19:0. It is therefore probable that the relative influence of carbon and hydrogen atoms on the ignition of the higher parafiins is defined by (@—2%) =ane/(mx)?, where a is constant. 80. o io9 20 30 40 ~50 60 d. Lhe Form of the Curve of Inflammability. There is a close resemblance between fios 1 andl hie curve of change of state. The shape of the ionition curve can be varied by the rate of break or by fie Gee of alternating current. Change of frequency has an extra- ordinary influence on the current required for ignition*® . * “The Electrical Ignition of Gaseous Mixtures,” Proc. Roy. Soe. A, vol, xe. 1914, fig. 9. Gases at Reduced Pressures bu Transient Ares. 455 The reciprocal of the time of duration T of an are is the analogue of temperature, and the equation | (i+) (p—b) =C/T can be used to express fig. 1 in general form, where a is not the constant of § 4, but a function of T. The simplest ignition by a break spark is that in which the energy of the spark necessary for ignition is inversely proportional to the number of molecules which come into contact with it in unit time. In that case pqV is constant, where p is the gas pressure, V the circuit voltage, and g=7T the quantity passing while the spark lasts. V is constant and pil=constant is the first equation of this kind of ignition, or ip=C/T. There is a lower limit of pressure 6 at which the flame cannot travel by conduction and 2 (p—b)=C/T takes account of this. So far there is no action considered but the heat energy of the spark. The influence of calorific value per unit volume is included in the term p. Any action other than thermal, such as that caused by a change in the number of ions emitted from the hot pole or produced in a second froin any cause, must be expressed by a modification of the current. If such an effect occurs which retards ignition the current required will rise, if it accelerates ignition it will fall, more or less suddenly according as the action develops gradually or with critical sharpness. The temperature of the are without doubt falls with the pressure, its brightness is less. The volume of the arc, as shown by its photographic image, at first falls and then at low pressures rises. The one action that is definitely electrical which might increase the igniting power of a transientarcis a greater production or diffusion of ions in the gas in contact with it, that is, an increase in their velocity. Judging by the colour of a spark in gases with distinctive coloration, such as cyanogen, there is free penetration of the are by the gas. The suggestion now made is that in ignition the “activation”’ agreed by chemists to be necessary for combination in gases and thought by physicists to be a blend of ionization and high temperature collision, becomes suddenly more intense. The oscillation of fig. 1 is found in ignition by disruptive discharge, where there is ionization by collision before a spark can pass. It is not found in the same mixtures when ignition is by hot wires*, where ionization by collision does not occur. * Loe. cit. fig. 8. 456 Prof. W. M. Thornton on the Ignition of In an arc the gap is saturated with electrons from the circuit, and these have high temperature velocity apart from that given by the field. A lower voltage fall than 87 would then be sufficient to ionize a molecule by collision with an electron*, and in flame the velocity of combination is such that ions are produced with no electric field. In the case of normal ionization by collision change of gas pressure does not affect the total current passing between plates. ‘The velocity and rate of diffusion are both increased. As shown by the influence of traces of moisture, ignition is started by internal combination of a few molecules rather than by general action. The variation of ignition, if in- fluenced in any case by ionization, is not a function of the ionization current taken as a whole; but it is known that when the velocitv of an electron reaches a certain value ionization by collision begins suddenly. The velocity u== ee where m is the mass of the ion carrying the current, and D a coefficient expressing the dimensions of this mass which varies with the field and pressurey. As = increases m diminishes and — rises in value.) \Hor small ML changes of pressure D/m may be taken as a first approxima- : >. x AN tion proportional to a and writing it wee Uske(7 ) ) 2 P and here &, e and X are constant. Thus as p: falls U increases until, as a critical value is approached, ionization begins. When ignition is made easier by such a strong ionization by an electric field in addition to that eaused by the high temperature of the arc, a term of the form a/p? added to the circuit current expresses the influence of ionization by collision on the igniting current. This applies to ions moving in gases at relatively high pressures, such as half an atmo- sphere. When, however, X has values of the order of 1000 = wal xX volts per cm. and p=1 mm., U varies as ae t, and“‘the P whole character of the motion then changes.” Over the range of pressure within which ignition 1s possible D/m is in all probability a more complex function of * Ti. Rutherford and R. K. McClung, Phil. Trans. A, 196. p- 28 (1901). See Townsend, ‘ Electricity in Gases,’ p. 263. + Townsend, Joc. ext. p. 290. Zt Townsend, p. 312. Gases at Reduced Pressures by Transient Arcs. 457 r ~~ and of thetime of duration of theare. The velocity which an ion or a number of ions can acquire while the arc lasts depends on its duration, so that T enters directly and the term a'l'/p? expresses the tact that when the break is extremely rapid ignition becomes more an energy effect, dependent upon 2 alone, the ionization then being negligibly small. It is again shown by the relation between e/p and X/p for air* that ionization approaches an upper limit as X/p is raised, so that when p is very small the term does not become great. There is then a constant to be added to p, so that the final ; (p—6)=C|T expresses rm i wanda term is al/(p+ po) } ieee)? all the essential facts of break spark ignition. It may be remarked that this closely resembles Clausius’ equation for the curve of change of state, though the present work is clearly in too early a stage for any equation to have more than general interest as collecting the facts for a typical case. 6. L'wo phases of Ignition. Such an expression as that given above is only valid when the accelerating forces, molecular attraction in the change of state, activation in the case of ignition, are developed gradually. The action of spark ignition is too local for this to be generally applicable except in the case of a gas with high velocity of translation such as hydrogen. If in any other case the critical stage is reached suddenly, as might be expected, the term added to the current would come as suddenly into effect. The result would be that for a small change of pressure the igniting current should fall sharply, with almost discontinuity, as it does in methane, ethane, and propane. At pressures below this ignition proceeds with the ionization term fully active. There are asit were two distinct phases of ignition, and this is specially marked in ethane and propane, It is not in the facts of ionization alone or of the thermal changes in combustion that a sudden change in the conditions of ignition arises, but in their combination, having regard to the influence of the mean free time in allowing an ion to acquire sufficient velocity for collision to become decisive. * Loc. cit. fig. 50. Phil. Mag. S. 6. Vol. 40. No. 238. Oct. 1920. 2H 458 Prof. W. M. Thornton on the Ignition of 7. Ignition by Alternating Current Break Sparks. ‘The curves of figs. 6 to 9 contain the results obtained by the use of alternating currents at a frequency of 36 anda voltage of 200. There is a general modification of the shapes of the curves of considerable interest in its bearing on the views advanced in the last two sections. +4 AMPERES A.C. ‘6. poner 1G Hydrogen no longer has the extreme oscillation of fig. 1, and the extent to which it has been wiped outisan indication of the change in the relative values of the two phases, in which heat and ionization are respectively dominant. The Gases at Reduced Pressures by Transient Arcs. 459 latter is losing influence, because presumably of the ‘cup and ball’ action of an alternating field on ions from the are. Though the frequency is low it appears to be sufficiently high to check the activ ity of the arc in promoting ignition. When in the case of methane the current is continuous 0°5 of an ampere at 200 volts will cause ignition, while a current alternating at 100 periods a second requires 20 amperes to be broken (with bright coruscations) before the gas explodes. In the present case the change is not so great. Fig. 1 bears to fig. 2 much the same resemblance as fie, 6 to fig. tie bhe' position of the maximum at *65 atmosphere i in the first is at ‘4 in fig. 6, and the minima have similarly moved. That is, ionization now becomes critical at pressures lower than with direct current. Methane has changed only in magnitude of the oscillations. There is little sign of the ionization term here, the curve is almost hyperbolic. The interesting point is that so small a change of ordinate is significant. If fig. 7 had been observed first the kink in it would have been possibly regarded as an experimental error, instead of being as it is an indication of a process reaching its maximum in fig. 1. The resemblance between the curve of change of state and those of figs. 1 to 4 has been mentioned previously. Figs. 8 and 9 are illustrations of the flat stage corresponding to liquefaction at constant pressure. All that is wanted to complete the series is a curve having a point of inflexion corresponding to a critical isothermal. This has been observed with disruptive spark ignition*, and with break sparks at a frequency of 60 when the gas is methanef. The meaning of a flat stage between two phases is that as the influence of one falls it is exactly compensated by the increase of the other. The cause of the fall of the thermal term is the smaller number of molecules in unit volume; the rise of the ionization term is the increase in velocity due to the longer free path. It isclear that when these are the two chief factors such a compensation is not improbable. Coal Gas and Carbon Monowide. The ignition of these by continuous currents is given in figs. 10 and 11] and by alternating currents in figs. 12 and 13. Fig. 10 is a fair average of figs. 1 and 2. The magnitudes of the ordinates are those of hy drogen and this is again found by a comparison of figs. 6 and 12. As might have been expected hydrogen is the cause of the more sensitive ignition * Proc. Roy. Soc. A, vol. xcii. 1915, figs. 6 and 7. { Proc. Roy. Soc. A, vol. xc. 1914, fiz. 9. ZH 2 ; 460 Ignition of Gases by Transient Arcs. of coal gas. Carbon monoxide is not so readily ignited as hydrogen by continuous current break sparks, but it is little behind it at atmospheric pressure. On the other hand, it is much more difficult to ignite by impulsive sparks. The 7.0 LENSE FIG.} 1. Fiel3. practice of using low tension break sparks for the ignition of large slow speed blast-furnace gas-engines is here justified by the sensitiveness of this gas, largely carbon monoxide, to ignition by such sparks. There is a marked difference when alternating currents are used, coal gas is then ten times easier to ignite than carbon monoxide, on account, probably, of the greater mobility of the hydrogen ions. CARBON: MONOXIDE CARBON -MONOXIDE ea LI. On the Velocity of Unimolecular Reactions. By Eric K. Ripeat, Professor of Physical Chemistry at the Uni- versity of Illinois, U.S.A.* [ a recent communication (Phil. Mag. vol. xxxix. p. 26, 1920) W. C. M. Lewis has drawn attention to the anomalous results obtained in the calculation of the velocity constant of a unimolecular chemical reaction, the decom- position of phosphine, from the standpoint of the radiation hypothesis. This lack of agreement in a monomolecular reaction is all the more serious, since if the correct solution could be found the radiation hypothesis could be extended to the vaporization of metals, thus permitting us to calculate the so-called ‘‘ chemical constants” of substances with the aid of the Clausius-Clapeyron and Knudsen relationships, in terms of the molecular diameters and the natural radiation frequencies. Drs. Dushman and Langmuir have recently indicated that Trautz’s values for 2 in the following general equation, ae ~RT dt = ve - are proportional to Q. This purely empyric relationship has, however, a theo- retical basis, and in the light of the radiation theory leads to remarkable conclusions. In the general equation for a monomolecular reaction, Q is replaced by the quantum relationship Q=Nhiv. Hence the rate of change per molecule per unit of time is dn hy — = Ke a apie Teas p log t+ v0, 5 log K= — where the reaction proceeds according to the scheme, vyAtwHyB+...... =v,M+v,N +... and K is the ‘* Reaction-isobar,”’ ue p.m... being the partial pressures of the reacting sub- stances—M, N, etc. . In the present cases, viz., for a reaction of the type, Cag Cal ap ees esa ee we have vCp= (Cp )oat + (Cr)e— (Cp) ca- We can take (Cp) ca= (Cp) cat, and (C,)-=3R, the electron being supposed to behave like a monatomic gas. in the Solar Chromosphere. 479 Eggert calculates the chemical constant from the Sackur- Tetrode-Stern relation, (2M) *k4 GC =log he N2 =o slog My. (4) where M=molecular weight, the pressure being expressed in atmospherés. Now (© has the same value for Ca and Ca,. For the electron M=5°5 x 10-5, and C= —6°5. _ We have thus SLU 2 as olan a 5) To calculate the ‘“* Reaction-isobar ” K, let us assume that P is the total pressure, and a fraction w of the Ca-atoms is ionized. Then we have U i ee oe De " jae e i] eae eo. 9 log K =log7—, This is the equation of the “reaction-isobar ” which is throughout employed for calculating the “ electron-affinity ”’ of the ionized atom. lonization of Calcium, Barium, and Strontium. With the aid of formula (1), the degree of ionization for any element, under any temperature and pressure, can be calculated when the ionization potential is known. Asa concrete example, we may begin with Calcium, Strontium, and Barium. A glance at equation (1') shows that pressure has a very great influence on the degree of ionization, which does not seem to have been anticipated. This is due to the occur- rence of P in the first power in the expression for the ‘“‘Reaction-isobar.”’ A reduction in the value of P is attended with greatly enhanced ionization. This will become apparent from an inspection of the following tables, which show the ionization of Calcium, Strontium, and Barium under varying conditions of pressure and tem- perature. 480 Dr. Megh Nad Saha on Tonization TABLE LV. Ionization of Calcium (in per cents.). U=6:12 volts=1-40 . 10° calories approximately. Pressure in atmospheres—Temperature on the Absolute Seale. Bressuren.. 0) 02s gol siealOm cael alent Ones: 1074.) “one hitias Temp. 2000°...... bi Or” 46 Ome D500 2.1042 Teme 3000 0s... SO 1 9 AQ OO) Mean 2-8 9 26 93 BOWO! a.ae 2 6 20 55 90 cn 6000 ...... 2 Sf 96 HEA 93 99 7000 ...... 7 93 68 91 99 ie -——_— 7500 eee 11 SA 5 Gy a OCT yas 16.9 846 OST oR KOC aan 99. 70, + 195, aaa Complete OOOO)... oe 46. 85 985 acenl TOO: a 68 93 ——— Tonization. 12000 ...... 76 965 13000 ...... 84 =: 98H 14000 ...... 90 TABLE VY. Ionization of Strontium (in per cents.). = toe 107 calories. Presstare 2.3) 10.4 di | pODHo (107Fiuy | 2OTP80y]) 105A We tO eee 2000°...... I? 10m. 4ed Ome AS OO sae Ge lOm sae Ome B000 ce PAU D Gate R02 2:5 AOD aes 16 5 15 45 = 985 ROO ghee a 1° 13:2 ig 32 73 O68 ae —— 6000 ...... A 37 78 iis aa Hes FOOO? s..5-: 10 32 73 Qo —=——= 7500. 00... 15 45 84 98:5 8000 ...... 92 58 91 99 9000 ...... 38 79 97°5 Complete 10000 ...... 56 90 98:5 ai 1A COO Me: "1 95 - lonization. 12000 ...... 82 97°5 13000) 2. 89 985 14000N eee 93 wn the Solar Chromosphere. 481 Tasie VI. Tonization of Barium (in per cents.). U=1:2.10° calories approximately. Pressure ... 10. 12, 105%, 10-2: 10-2, 10-4. LOS, TORS: Temp. 2000°...... Got0m .2 ats) 2eto B00... LSsi0e> tin tO oe 6 SH... 16 5 4G 4000 .. ... 1 3 9 28 68 99 5000 ...... 17 Pe a Oe 48 86 OOM esi oe 6000 ...... G2) t19" Puneet es 99 eae Ut 1h aR ORM eee = 7500 ...... SOT hgs PON se 5), eee 30 70 94 | Complete 9000 ...... 47 85 hi Peete 10000 ...... 65 93 : 41000 2... at ican 13000... 99 We are not aware how the temperature and the pressure (partial pressure for a particular element) vary with height in the solar atmosphere. According to F. Biscoe*, the temperature of the photosphere is about 7500° K., while the pressure in the reversing layer varies, according to different investigators, from 10 to 1 atmospheres. If we suppose that the variation in temperature is entirely caused by radiation, the temperature of the upper layers should tend to the limit oe +, or a little more than 6000° K. The partial pressure may be supposed to vary from 10 atmo- spheres in the reversing layer to 10~” atmosphere in the outermost layers. An examination of Tables IV., V., VI. shows that, under the above-mentioned assumptions, about 34 per cent. of the Ca-atoms are ionized on the photosphere. When the pres- sure falls to 10~* atmosphere, almost all the atoms get ionized, so that up to this point in the solar atmosphere, we shall get combined emission of the H, K, and the g-line, but above this point, we shall have only the H, K lines. This is in very good agreement with observed facts. * F, Biscoe, ‘The Astrophysical Journal,’ vol. xlvi. p. 355. + Schwarzschild, Gott. Nachrichten, p. 41 (1906). 482 Dr. Megh Nad Saha on Jonization In the case of strontium and barium, owing to their com- paratively low ionization potential, ionization at 6000° is practically complete at 107% atmosphere, and the heights shown by the lines of the unionized atoms of these elements are still lower. Compare the Tables IV., V., VI. The results of the flash-spectrum observations are thus seen to be very satisfactorily accounted for on the basis of our theory. Laboratory experiments also, as far as they go, are in qualitative agreement with our theory. It is well known that in the flame, the flames due to the ionized atom either do not occur at all, or even if they do occur they are ex- tremely faint compared with the lines of the unionized atom. As the temperature is increased, the “enhanced lines” begin to strengthen, until at the temperature of the are they are comparable in intensity to the lines of the normal atom. We give below the results of King™* on the relative intensity of the “enhanced” and ordinary lines of the Tasue VIL. Temp. Photo- Chromo- Actual ... 1923 2273. 2623 Arc. sphere. sphere. Element. Approx.... 2000 2500 3000 4000 7500 6000 Line. Intensity. Cae aZZaT| (GI) abe 300 500 1000 500 20 25 Cart ts. sO GEE. 18 25 50 = 350 700 80 3933 (KX)... 20 30 60 400 1000 100 Proportion ofionized 1:4.107° 7.107? 1 23 75 93 atoms in per cents. (P=107}) (P=107°) Sree. QU saan. 300 400 600 600 1 2 Seeeiees | Fa Z1G ees 6 15 30 = 400 D 40 A073) 12 25 40 400 8 40 Proportion of (Sr 9) 4elOmP @ 2a 0mm 2:5) eet 84 97 in per cents. (P=10-1) (P=1073) Bafta: ae DaBGy Cake 400 500 1000 1000 2 1 4934 2 || 50 60 109 700, 7 12 4594 1. | 70 80 100 1000 8 20 Proportion jof Bay 9 2.10=" 97 1054 ip E68 91 98 in per cents. (P=1072) (P=1u7%) N.B.--The intensity scale under the headings photosphere and chromosphere is different from the scale in King’s furnace spectra. * King, ‘The Astrophysical Journal,’ vol. xlviii. p. 18. in the Solar Chromosphere. 483 alkaline earths in vacuum-tube furnaces at varying tem- perature. Unfortunately, the pressure, which is a vital point, is not mentioned. The last line shows the percentage of the ionized atoms under a pressure of 107* atmosphere, > *] mm. of mercury. The tables show that an increase of temperature causes an increase of ionization and tke proportion of emission centres of the enhanced lines. The increasing intensities of the double lines are mainly to be ascribed to this fact. These become comparable in intensity to the principal lines of the normal atom only when the degree of ionization is rather large (comp. the figures at 4000°). Comparing the relative intensities of the corresponding lines of the calcium and barium group, we find that for the same temperature the enhanced lines of barium are relatively stronger than the calcium lines ; and this, according to our theory, is’ due to the comparatively lower ionization potential of barium. The objection may be raised whether the proportion of ionized atoms at low temperatures, as given by the theory, is not rather too low. ‘The tables show that at 2000° K., only 1 in 10° calcium atoms is ionized. Is this small number of ionized atoms capable of affecting the photo- graphic plate by the emission of the H and the K lines? No definite answer can be given to this point. We may, however, point out that, according to Ladenburg and Loria* when a hydrogen vacuum tube at a pressure of a few mms. of mercury is excited by a spark, only 1 atom in 50,000 is found to be radiant, 2. ¢., capable of emitting H, and Hg. A very low proportion of radiant centres may therefore affect the photographic plate. It should also be remembered that at low temperatures the principal line of the normal atom is not only relatively more intense, but very broad and diffuse, when the enhanced lines are aT narrow in addition to being faint. A reduction in pressure will cause ave relative intensity of the H—K line to increase, but not the absolute intensity, because the total available number of radiant particles will now decrease. JI am not aware whether there is any Jaboratory experiment for testing this point. § 4. Hydrogen in the Sun. It has been mentioned in the introduction that hydrogen is not appreciably ionized at even the highest levels of ‘the * Ladenburg and Loria, Ber. d. D. Phys. Gesellschaft, 1908. 484 Dr. Megh Nad Saha on Jonization solar chromosphere. We should add to this the fact that hydrogen exists in the Sun only in the atomic state, for, if there were molecular hydrogen in the Sun, we could have detected some at least of the lines of the secondary spec- trum. But this is not the case; hydrogen enters into chemical combination with calcium and magnesium in the sun-spot, but does not probably form molecules of its own. We shall consider in this section whether these facts are reconcilable with our theory. ‘This requires a knowledge of the heat of molecular combination and the ionization potential of hydrogen. These data already lie available in a recent paper by Franck * and others. ‘They find evidences of the following chemical and electronic reactions :— H,=H + H+3°5 volts (=84,000 calories), . ( Hy=( EH.) +e sllOstiwolliss:. josh i. eee H =H, +e+13°6 volts (=3°2.10° calories),. (©) H,=H. +H, + 2e--30°7 volts (7:2. 10° )5 5 By The first is a purely chemical reaction, and the heat of molecular combination has been directly measured by Langmuir f and found to be 82,000 calories. The ionization Gia. 13°6 in (C), can be calculated from the relation yee : °, taking v)>=convergence frequency of the Lyman L series v=N [+ — aE 2.¢.,v=N. The actual occurrence ef this process in the ionization of Hy, is indicated by a sudden increase of ionization at 17:1 volts (13°6+3°5 volts). The ionization voltage 30°7 in process (D)=2.13°7+3°5, corresponding tv the complete breaking up of the H,- molecule into 2 atoms, and of these again into the core and the electron. Let us first consider reaction (A)—dissociation of the molecule into atoms. ‘Taking the equation of the reaction- isobar, U LvCy log K=— pa tp £571 T ae he * Franck and others, Ber. d. D. Phys. Gesellschaft, vol. xxi. + Langmuir, Zetts. f. Elektrochemie, vol. xxiii. p. 217 (1917) no. 20. in the Solar Chromosphere. 485 3K a? Ca,=—3°40*, Co=—16, 2wC=-2, U=8:2 .10* calories. we have XvC,=2(Cp)a — (Cy )a, = » Fy iat i ine where P=total pressure, and a fraction & —v has been dissociated. We have thus ) 2 a S32, ho? log ——_, P=— +l dlog T+:2. ©1—2 A571 ? Table VIII. shows the dissociation of hydrogen under different pressures and temperatures (in per cents.) :— TanuEe VII. Dissociation of the H,-molecule. Pressure...... 1@ 1o—! 10, 10-2. 10> Temp = 0. genes 1 3 9 29 70 POR ete csns tu Bo 75 97 eee BUUO ff. 22... 46 85 OSD i hiant sigii ® <5) 0) Ae oe 85 98°5 a HOO! FL. ols: 96°5 7H Complete Lonization. The table shows that under the conditions prevailing in zs . Wipeets |e, ; the Sun the dissociation is complete. Even in the umbra of sun-spots, assuming that the temperature is 4000° K. and the pressure is of the order of 1 atmosphere, the dissociation is almost complete (96'5 per cent.). Lonization of Hydrogen. For a rigorous treatment of the case, we should start with the process (D). But since in the Sun the hydrogen is entirely in the atomic state, we may use the process (C). The results will be but approximate, for the equation which follows dees not hold over the whole range of temperature. The case is quite analogous to the ionization of calcium. We have only to put U=3:2x 10’ calories (approximately ) * Reiche, Ann. d. Physik, vol. lviii. p. 657, and Leon Schames, Phys. Zeits. vol. xxi. p, 41. 486 Dr. Megh Nad Saha on Jonization corresponding to 13°6 volts. We have then ; ae bh Ba 2 10° i) ‘ OF -< bee > eer T en —6:020 for T=7500 = 9.279 for Teun | These figures show that at a point where T= 6000°, hydrogen can be completely ionized if P=10-'* atmosphere. Thus only at the highest points of the chromosphere, where the partial pressure falls to 10-4 atmosphere, can the ionization be complete, and the vanishing of the H-lines be expected. Helium. The previous work on the ionization of H atoms will have made it clear that the higher the ionization potential of an element is, the less will be its degree of ionization under a given thermal stimulus. This is best exemplified in the ease of helium, which has got the highest ionization potential of the elements so far investigated. The experimental results, however, are rather discordant. According to Bohr, the ionization potential should be 29 volts, while most investigators have detected the com- mencement of a distinct ionization at 20°5 volts. Some investigators have detected two distinct stages of ionization, one at 20°5 and another at 25 volts. In addition to this last, Rau detected a rather strong ionization when the potential is raised by 54°6 volts, 2. e., to about 80 volts. These processes probably take place according to the following schemes :— He =He.+Vy >. :. \ ere Parhe = He.siVeu ciel fy sore He =Parhe+Vj—Ve . » . ue Hel = Meg ae Ve ee (D) | The distinction between He and the so-called parhelium is taken as one of relative configuration of the steady orbits of the two electrons *. V,) may be identified with 20:5 volts, V, with 25 volts, and V3; with 4.13°6=54'6 volts f. * See a paper by Lande, Ber. d. D. Phys. Gesellschaft, 1919. + Itis not possible to deduce V from the quantum relation eV=A(1, s), for the fundamental term (1, s) is unknown both for helium and par- helium. What are generally called the principal series of helium are really the series y=(2, s)—(m, p) (the leading lines being 10880, for He and 20587 for parhelium). in the Solar Chromosphere. 487 Taking V,=20°5 volts, U=48x 10° calories approxi- mately, we have the following table for the first step ionization of parhelium. If the ionization voltage be taken =25 volts, the degree of dissociation will become still less :— TaBe IX. Ionization of Helium (in per cents.). U=4'8 . 10° calories (approximately). Pressure ... Lo 105". 10-2, 107-3. 10-4. 10-6, Temp. 6000 ...... FlOn Leeco bolus” I7ln! fel0 << BY10-° "O00 ..:... Lita Sate t10- Soe 81.10 “t.: 7AOO. *...... ANOSe Pog. 4.10 | T2°I0- BOO... £9N0-9 “Se. LAO 37.10 “1-2 40"? 3-7. 107! 9000 ...... TGs CO We eee 2S tOS | Too 7 FOOOO 2... 210% “Ste “Bar 1 3 31 11000 ...... L1G SG if 3-4 11 7 T2000 ...... $10 * 1 3 10 28 93 13000°...... tO 2 ri 22 58 re 14000" S.... 15 4 15 43 83 15009 ...... 3 10 28 68 94 16000 ...... 6 17 47 CU LEAE eimai The table shows clearly the ionization of helium is too slight under the conditions in the solar atmosphere, both in the reversing layer (T7500, pressure=1 atm.), as well as in the high-level chromosphere (T=6000°, p=10~® atmos.) But somewhere between the two (T 7000°, p=1073 atm.), there may be some slight ionization (1 in 10,000) which may account for the occurrence of the line of ionized helium X=4686, which has been detected by Mitchell. The calculations are, of course, of the roughest nature. The investigation also incidentally shows that the Pickering 1 1 lines v=N E eae occur as absorption-lines only in stars having the highest temperature, exceeding 16,000° K. This seems to be in- dependently borne out by the investigations of Eddington and Russell. The application of the method and the results obtained in the present paper to the problems of temperature radiation of elements and of the different spectral types of stars naturally suggest themselves, and will be taken up in a future communication. | and the Rydberg line 4686 can 488 Tonization in the Solar Chromosphere. Summary. 1. In the present paper it has been shown from a dis- cussion of the high-level chromospheric spectrum that this region is chiefly composed of ionized atoms of Calcium, Barium, Strontium, Scandium, Titanium, and Iron. In the lower layers both ionized and neutral atoms occur. 2. An attempt has been made to account for these tacts from the standpoint of Nernst’s theorem of the ‘ Reaction- isobar,” by assuming that the ionization is a sort of reversible chemical process taking place according to the equation ken) y r at SERN Veet! 2 Caz*Ca, +e—U. The energy of ionization U can be calculated from the ionization-potential of elements as determined by Franck and Hertz, and Machennan. For determining Nernst’s chemical constant and the specific heat, the electron has been assumed to be a monatomic gas 1 1836° 3. The equation shows the great influence of pressure on the relative degree of ionization attained. The almost com- plete ionization of Ca, Sr, and Ba atoms in the high-level chromosphere is due to the low pressure in these regions. The calculated values are in very good aecord with observa- tional data and the laboratory experiments of King. 4. Hydrogen has been shown to be completely dissociated into atoms at all points in the solar atmosphere. 5. ]t has also been shown that the greater the ionization potential of an element, the more difficult ionization will be tor that element under a given thermal stimulus. Calculations have been made in the case of hydrogen (V=13°6 volts) and helium (V=20°5 volts), which show that these elements cannot get ionized anywhere in the Sun to an appreciable extent. Helium can have appreciable ionization only in stars having the highest temperature (>16,000° K.), which only are therefore capable of showing the Rydberg line 4686 and the Pickering lines y= N Ee a | : 2 (m+ 4) In conclusion, I beg to record my best thanks to my students for their alla help in the calculations, and to my friend Dr. J. C. Ghosh for revising the proofs. having the atomic we el@ht of Calcutta, India, March 4, 1920. fy 489 ./] LIV. Vhe Helium-Hydrogen Series Constants. By HERBERT BELL * S is well known, Bohr proposed to modify the Balmer formula for these series by a term depending on the relativity-effect on the moying electron +. The formula for Helium thus becomes 1 =) ee oa I y = 4Nne (7 ae {ite ae, 5 . (1) where p and m are integers, m > Pp, Nue= RMa-/(Mire+ «), R being Rydberg’s constant 277e*u/ch*, Ma. the mass of the helium atom and uw that of the electron, while a= 27e Ihe. The Helium lines, hewever, display a fine-structure under high resolving power, and Sommerfeld { in explanation of this developed a theory of adjacent (elliptical) orbits with eccentricities depending on quantuin considerations. He gives « a physical meaning as the ratio of angular momenta, und proposes to measure it from fine-structure observations. We know from other considerations that a has approxi- mately the valuesd°3 x 107°. Paschen § tested this theory by an exhaustive set of measurements in the helium spectrum and_ substantial verification was apparently obtained. His value for « from the line 4686 is 5°315 x 107 According to Sommerfeld’s theory the series (1) when p=3, m=4, 5,..., which is the one most easily re- solved, consists of a set of triplets having components I, II, IL, in diminishing order of wave-length, their wave- number differences being constant. III— nig=alr 78 enisa II1—I="58 em.~1. Further, each component is bordered towards the red by a set of fainter lines rapidly closing in on them as higher terms of the series are reached. Formula (1) is to apply only to I, and can therefore he accurately tested. Paschen of course did this, implicitly, bat the new table from the Bureau of Standards || for the refractive index of air slightly alters his derived constants. His measurements are given in Table [., along with the new additive correction to reduce to vacuum, the figures in italies referring to the series p=4, m=—, 6,7, ..., the others to p=3, m=4, 5, * Communicated.by the Author. t Phil. Mag. Feb. 1915. { Ann. d. Phystk, li. No. 17 (1916). § Ann. d. Physik, 1. No. 16 (1916). || Bureau of Standards, Washington, ‘Scientific Papers, No. 327 (1918). Phil. Mag. 8. 6. Vol. 40, No. 238. Oct. 1920 2K 490) Mr. Ey Bell on the Re-writing (1) in the form 2m? ale Wel Nina fear oo : a Ae we see that if the formula is to hold the second or relativity term must be just large enough at each value to reduce the corresponding first-one to a constant. ‘Both cermeame given in the table, and their difference is under Ny. For the series p=3 a constant value is reached, average 109722°31 ; but such is not the case for the unresolved series p=a. iow this series is according to the theory one of quadruplets I, II, 111, IV, each bordered towards the red by fainter lines as before ; but I, II, III were not resolved, and it is most likely that Paschen’s readings apply to some kind of a mean among them, and consequently the wave- numbers are all too high for I to which the formula applies. Theoretically, IV —T1l="730° em? “Wl — 24 ene iJ) Cris If we assume the wave-numbers to be all too high by ov, then by (2) we need to apply a correction : 2 ee Le ie onan A(m* —p*) ar é ’ feneaern ad It is rather surprising * that this should so exactly, as far as present accuracy in measurements goes, be the relativity 2 correction. To equal it we require 6v=Na?4/p*=:091 cm.7?. However, it is readily seen that we cannot correct these terms by subtracting a suitable multiple of the relativity correction, for although a multiple 4 or 5 would do away with the diminishing trend the resultant Ny, would be much lower than that from the other series. It will be seen from the following that this oe. is due to systematic error among tmecolved lines The components ILI of the tri iplet series and IV of the quadruplets scum farthest out, differing from I by 2°31 em.+# and 1:09 cm.~? respectively. Their corresponding N values. have therefore to be corrected by these differences multiplied * If the differences in waye-number among the standard iron lines are all counted correctly (by interference methods} but the reference Cadmium line is in error by év, we should have a spurious relativity effect OA =—A*odv throughout the spectrum. This error ‘09 cm.~+ would, however, represent a miscount by about one part in 200,000 for: ared line. An accuracy of one part im ten million is claimed. Flelium-Hydrogen Series Constants. 49] by ae m ; 4(m? — — Pp’) these lines in some cases, as shown in Table II. It will be seen that the tendency to lower vaiues in the higher terms has disappeared in the series p=4. The results are not very consistent among themselves, but this is readily explained if we assume that in some cases only the bordering satellites were observed. The errors are all on the same side of the determined Nye. Puaschen’s estimates of possible error of observation are also given, reduced to our scale. Turning now to the Hydrogen (Balmer) series, we have the formula in each case. Paschen was able to record i 4m?p iL Ne= 3 70 27 (G+ = ai is (3) Nu being RM/(M+ yz), M mass of hydrogen nucleus. The careful measurements of Curtis * and Pasehen (loc. cit.) are given in Table III. Unfortunately they exhibit a small systematic drift—Curtis’s values increasing relatively to Paschen’s roughly linearly in terms of wave-lengths. The first line H, is easily resolved into a doublet, e. g. by Paschen in the third-order spectrum 6A='124 A.U. ov='289 cm.~!. The second line has also been resolved. In Sommerfeld’s theory the series is a set of doublets I, II, of constant wave-number difference [IL—I=°365 cin.“1, the stronger component I being towards the red, but both components being bordered on the long-wave side by fainter lines rapidly converging in the higher terms. Paschen’s measurements are for ‘‘centres of gravity,” while Curtis endeavoured to record the centres of the diffuse lines (they were not resolved). Now for the fainter lines component II would probably vanish first, and the diffuse mean of the photographic plate would tend towards I—that is, longer wave-lengths. This may partly explain the minute drift. Again, Curtis used Burns’s secondary iron lines, while Paschen used a variety—Fabry and Buisson, Neon lines by Meissner, Kc. The values of 4m? v/(m?—4) for the two observers are given under ©. and P., and in each case we find that a minimum seems to have been reached for Paschen at Hy, Curtis at He. Now the stock corrections to the Balmer formula all have the property of steadily decreasing, 1. @. no maximum or minimum, as for example the relativity correction given in the table. Such a correction could * Proc, Roy. Soe. vol. xc. (1914) and vol. xevi. (1919). 2K 2 Mr. H. Bell on the 492 GLG SOT 76-TG460T Hy FE-GeLo0I = "= Ni SI- SI. GI: 90- eG. 11 18-G LO *“L10Z) Pa 00-F€ AT kG 1786 OEE Ry ‘(uoyoseg ) soury wumnijey—']] Alavy, Tg. 49-6C4607 NGI =) F lig Were 10-G€ E686 68-SGL607 X(gd— pup gig OTL. O84. 19¢L.- L008. CGG6- OS¢L-1 C6LT-T POLE 1 169¢-T 6606-1 9998.-T 600S-T 6608-1 "IL0D ‘(uayoseg) soury wnijeyH—y] @1aVJ, G1Z-90E6 OFP-S8EG 6FG-1L1SG CPE. SELG C91-S0GE 670-0017 498-6617 60L-8E8T 089-1797 808-89F 676-6987 199-T17G O&T-0999 “(a1e) X *108-C89F GET -6SS7 O6@-TI7¢ “(ae) XY ~~ 8 Ne) “Ub 493 Helium- Hydrogen Series Constants. LL6- 096 116 CoG. GIO B86 996- 026 696- $96 696-0 L&O-1 + -LL9601 adi ‘9 SHON SI. ZI: 2 Os SI. VG: I78- 108 TPs: G18: 948 G6L- 868: CLL. 008. O08: 97h. LTB: + -LLOGOT cal #0) ‘Hy ELT 66-1 GPT 8G°6 96-T O86 100- 600: 600- 990+ (DE ‘soury uosoapAy— TIT G0-9. €6-9 61-8 [6-4 61.9 67g IGG: 1e¢- 61S: T8¢: 0G9- 8L8-— ‘(I-IDt 8L-L6 G6.86 PG 66 OT 4G GT -Le Bk KO BRE. c6E- cOF: SGP: CaP: en Tet O¢L-T SEL-1 668T S8rl-1 768-T QLLT 968-1 968-T is 680-6 aL +-L9G60T ‘d ‘0 X(F— et) ZuPp WTaAv J, FL: GE-RS 12: EL-66 88: GP-0E Sg. KG*hO or. £G-40 77. LE 86 1¢0-688 CL0-0L6E GE4T SEL LOTF 697-0 LOV-OFSP 968-1 9ZE-LI8P L646 $61-29E9 "uayOstd “su.ng “(arw) YX NGE-G8EG 681-SELzG 696-2068 816-6004 98G-8EEP 967 -I7G7 494 Dr. S. R. Milner on the snternal never remove the turning value, although it might dis- place its incidence. Curtis found the Ritz formula v=N(4—1/(m+p)?), w=74x107°, to best fit his obser- vations, but a simple rearrangement throws this into the form We Am?v N 2p (; il ), = ie + —. m? — 4 m?>?\4.° m? whence we see that the correction is a good deal smaller than the relativity one and vanishes so rapidly as not to affect the terms where the difficulty occurs. The fainter unresolved lines in both Helium series, as shown in Table I., had to be discarded for the same reason. These wave-lengths being presumably between I and II are too small, and we have an additional quasi-relativity correction to subtract from N. Paschen estimates the correction at #(1[—I)=°122 ecm.7!.. Furthermore, the bordering lines on the long-wave side shift the observations that way. Paschen adds on a correction of } of the ‘“‘spread”’ of the first component I. Both corrections are tabulated in our scale. The resultant values of N after the three corrections are not constant. Paschen’s values would seem to have been over-corrected. Taking +(I1I—I) instead of 4, we get the last N'g. columns, where his first three N’s have become constant. It seems safe to adopt Nae == 10907 7-9: Paschen gave (older value for refraction) 109677°691+0-6. Curtis gives as a mean of the various formule tried 109678°3. It is evident that we need interferometer measurements on the earlier hydrogen lines. London, May 1920. LV. The Internal Energy of the Lorentz Electron. By 8. Rh. Minner, D.Se.* HE Lorentz electron, as has long been recognized, is not a purely electromagnetic system. In order to make it a system amenable to the laws of mechanics it is necessary to ascribe to it a certain amount of energy in addition to that which is accounted for by the electric and magnetic forces of the field. This energy, apparently non- electromagnetic in character, is assumed to be located in * Communicated by the Author. Energy of the Lorentz Electron. 495 66 what may be called the “‘ nucleus” of the electron (to dis- tinguish it from the surrounding electronic field), and for an electron of char ge e and major semi-axis a, moving with velocity v= Be, is of the amount e” Bee (emma dienes Bo api, + (1) While the physical character of this internal energy largely a matter of conjecture, the postwlation of its presence ix essential for two separate reasons. In the first place, if we consider the electron to be a system capable of being set in motion by a mechanical force we must have simul- taneously satisfied two mechanical conditions: (1) that the force is measured by the rate of increase of the momentum of the system, and ‘2) its activity by the rate of increase of the total energy. But this fundamental law of mechanics is not satisfied unless to the energy of the electronic field the internal energy (1) is added In the second place, the presence of the same internal energy is required to make the electron fit into a relativistic scheme, viz., to make its total energy at any speed equal to ¢* times its mass. The object of this note is to point out that there is another reason for postulating the existence of this internal energy, and it is one which has a more clearly defined electro- magnetic character than either of the above. It comes from a consideration of the Poynting flux of energy in the field of a uniformly moving electron. At a point whose polar coordinates are 7, 9 with reference to the centre of the electron, the electric and magnetic forces are age 1—(? ye (1—? sin? )3 . . . (2) radially, and Eb AisimiGE wis bial oft lo Tedd) along the circles of latitude (calling the line of motion of the centre of the nucleus the polar axis). ~ The energy density is w= = (B+ B), Biansenaus: HIG) and the vector flux of energy through a plane fixed in space 1s = ;_ [EH]. Ae Aah gy 659 * Lorentz, ‘Theory of Electrons,’ p. 213 (1909 edition). 496 Dr. S. R. Milner on the Internal The net effect of the flux must of necessity be exactly such as will transfer uniformly forward, alone with the nucleus, the electromagnetic energy associated with the moving electronic field. In fact, at any point ahead of the equatorial plane the convergence of the flux will express the rate of increase there of the density of the energy, which is continually increasing as the nucleus of the electron gets nearer to the point. But the actual Poynting flux is not equivalent simply to a bodily transference of the energy of the field in the direction of motion of the electron. At the point 7, @ the flux is directed along the instantaneous meridian of longitude, and, except in the equatorial plane, it has a component perpendicular to the axis. Poynting’s theorem, although, of course, it does not require that electro- magnetic energy should be a material thing which moves bodily from one place to another of the field, does demon- strate that the local variations of energy are mathematically the same as if this were the case. he flux can therefore be expressed as the product of the energy density of the field and the velocity with which it is (virtually) moving. From this point of view the total flux (along the meridian of longitude) must be decomposable into two—one.wv which transfers the energy bodily forward in the direction, and with the velocity v, of the electron’s motion, and the other wu such that wu=P wv. od! &)- ao Usa Since div wy= — ge == Give. at the wu flux has no divergence anywhere ; from the symmetry of the field it must pass through the nucleus of the electron. Poynting’s theorem thus requires that the electromagnetic energy of the field, in addition to taking part in the forward motion of the whole system, 1s also travelling round and round through the field, passing on each journey into the nucleus at one face and out of it at the other. The magnitnde and direction of the circuital flux wu is easily determined by substituting in the vector equation (6) the values of w and P given in (4) and (5). The lines of flow are portions of ellipses of eccentricity 8 which have their major axes in the equatorial plane, and all pass through the centre of the nucleus. They are illustrated in the accompanying figure, in which O is the centre and BAC the upper half of the surface of the nucleus, moving in the Energy of the Lorentz Electron: 497 direction X. (The figure is drawn for (1—6”)?=#4, and } of the total flux lies between successive lines.) The velocity w with which the energy travels is given by 28 sin?O \? 2 U= {1 amet f oe . : . ( ~] —— wu is the velocity relative to the moving nucleus. When 8 is small, w=v practically, and the ellipses become circles. Thus at small velocities the energy of the field, viewed from the nucleus, goes round in Bhculas paths wile a uniform velocity equal to that of the forward motion of the system. The actual veloc ‘ity In space (or relative to the observer) is the resultant of wand v; this is 2v in the equatorial plane and diminishes gradually to zero as the axis is approached. It is the same as if the (completed) ellipses rolled forward on the axis without changing their shape as the nucleus advanced. Consideration of what happens in the space occupied by the nucleus presents the pure electromagnetic theory of the 498 The Internal Energy of the Lorentz Ilectron. electron with a dilemma, in that the idea of a continuous flux of ener gy is Se pmAisconit with the view that the energy of the he e everywhere electromagnetic (2. @., is of the 2 form TT For, unless denial is made of the con- ay tinuity through space of the flux, the circuit must be imagined to be closed through the nucleus, e.g., as indicated by the dotted lines. We can. exclude, as practically iden- tical with that of the discontinuity of the flux, the supposition that it traverses the nucleus with an infinite velocity ; conse- quently there must be a finite energy density in the nucleus, in spite of the tact that Ei and H are zero there. Thus accepting the continuity of the Poynting flux it is necessary to postulate the presence of non-electromagnetic energy in the nucleus. The equations (4) and (5) of an electro- magnetic field are in this respect inconsistent with each other inside the nucleus, and one of them must be supple- mented or modified. There is, however, no advantage to be gained by sacrificing both. If we make the supposition that Poynting’s expression forms a valid measure of the energy flux in every part of an electromagnetic field, we not only retain as much of the original theory as is possible, but we can deduce a value of the internal energy of the electron which is in compiete accord with that necessitated by the mechanical theory. The supposition implies that in the interior of the nucleus, since P vanishes along with Hi and H, the energy present (whatever its nature) is in a condition of zero flux through a plane fixed in space—or, in other words, is (virtually) at rest *. It is evident from the figure that the whole wu flux which passes through the moving equatorial section of the nucleus must be equal and opposite to the whole flux through the moving equatorial plane of the external field, or to . CEI ee eRe which reduces to v. = f E ! a with an atomic charge. The value of P in terms of V’ is 2 7 . given by the relation uae =27 xP, and the equation to determine P becomes “0 | 81xP aWrse % (14 a) . u 2k The values of P thus obtained are 23°7, 26, 28, 29 volts from the experiments with the smaller forces in which the values of Z/p were 40, 50, 70, 90 respectively. It might seem that this method of finding the potential P would be inaccurate, as the errors of several experimental determinations would be involved. But the values of P as given by the above formula are only slightly affected by large errors in the values of «, /, and W, and the quantity k is the only factor which need be known accurately. ‘The principal error in the calculation is probably due to taking Maxwell’s law as giving the distribution of the velocities of electrons acted on by an electric force. It should be remembered that the potentials P thus found are average values, and with the definition of a collision that has * Phil. Mag. (6) vol. xxvii. p. 269 (1914). D10 = Collisions of Electrons with Molecules of a Gas. been adopted it is most improbable that ionization takes place in all collisions in which the veiocity exceeds the value corresponding to the voltage P, or that ionization does not take place in a collision in which the velocity is less than the value. The potentials P do not therefore represent a critical potential, and the minimum potential required to ionize a molecule when the collision occurs under the most favourable conditions should be less than the above values. In air, the minimum ionizing potential as found by Lenard* by a different method is 11 volts ; bnt it may be concluded that ionization is produced in a comparatively small number of cases when electrons collide with a velocity corresponding to this voltage. In air at one millimetre pressure the observed increase ot conductivity due to the motion of electrons would be accounted for if ionization took place in about 2 per cent. of all the collisions in which the velocity was greater than that corresponding to 11 volts, when Z is 100 volts per centimetre, and in about ‘4 per cent of these collisions when Z is 50 volts per centimetre. 6. The numbers obtained for the velocities W and the factor & provide a simple means of measuring the proportion of the energy of an electron which is lost on colliding with a molecule. Thus with air at a pressure of one millimetre, and a force of 20 volts per centimetre, the value of k is 56:3, so that the mean final velocity of agitation corresponds to a potential of 2:1 volts, and in passing through a distance of one centimetre in the directicn of the force the electron makes 290 collisions with molecules. Thus in the final steady state of motion the energy corresponding toa potential fall of 20 volts is dissipated in 290 collisions, so that the average loss of energy at each collision is 1/14°5 volt. Thus about 3 per cent. of the energy is lost on each collision, or about 1°5 per cent. of the velocity, when collisions take place with velocities of 8°5 x 107 centimetre per second. The exact value of the elasticity has been found by Pidduck+ taking into consideration the distribution of the velocities. If the collisions are perfectly elastic the following relation is obtained between the values of W and &: 2 page Se . Q being the mean velocity of agitation of the molecules of the gas. * P. Lenard, Ann. der Phys. (4) viii. p. 194 (1902). + F. B. Pidduck, Proc. Roy. Soc. A. Ixxxvil. p. 296 (1918). On ihe Pressure on the Poles of an Electric Arc. 511 IE € be the coefficient of restitution, and f=4(1+e), f is unity when the collisions are perfectly elastic, but a very small reduction in f below this value has the effect of making a large reduction in the velocity of agitation. The following are the values of / found by Pidduck corresponding to the observed value of 4:— iy lng Sie nh ln aa y- 2 20 150 k—1 observed... 18 20 55 210 | LEP eee ee 9993 9988 99 ‘96 Thus as the velocity of the electrons increases, the elasticity tends to diminish, but with the smaller velocities the mole- cules nay be considered to be perfectly elastic. LX. On the Pressure on the Poles of an Electric Are. To the Editors of the Philosophical Magazine. GENTLEMEN,— N his latest publication under the above title Prof. Duffield (Proc. Roy. Soc. June 1920), discussing his results, tries to show that the motion of ions within an electric are and the electric wind could not be expected to produce a pressure upon the poles of the arc. Having myself contri- buted to your Magazine a few papers on the electric wind, will you once more give me your hospitality in order to clear up some points in connexion with this question ? Iam very much indebted to Sir J. J. Thomson for having shown to me some time ago (in a way similar to that used by Prof. Duffield) that ions moving under an electric force towards an electrode (evidently of opposite sign) cannot produce a pressure upon it, the pull on this electrode during the motion of the ions being balanced by the impact of the ions on the electrode. Prof. Duffield, however, has entirely overlooked the fact that ions moving from an electrode (of the same sign) produce a reaction upon this electrede which is not compensated by any other force. Consequently, a strong pressure upon the poles of an electric arc is to be expected as a result of the motion of the ions within thearc. Moreover, when the experiments are carried out not in a vacuum (as in the case of Prot. Duffield’s experiments), the energy of the ions is partly transferred by collision to the neutral gaseous molecules, which results in the formation of a stream of the surrounding atmosphere in the direction of the electric field —a phenomenon known as the electric wind. Experiments show that only a small part of the momentum of this stream 512 On the Pressure on the Poles of an Electric Are. is imparted to the electrode, the stream expanding. under ordinary experimental conditions over a large volume. It is therefore easy to see in view of the above considerations that in this case ions will produce a reaction also on the poles of opposite sign. Whatever the theoretical considerations may be, the expe- riments carried out by Chatteck and his pupils, Zeleny and myself, undoubtedly show that such a reaction does exist in reality. The pressure of the electric wind upon a sensitive vane may be detected when the ionization current passing through the g gasisassmallas 107? amp. (Phil. Mag. November 1916), and it appears to be proportional to the curr rent. When the current through the gas 1s of the order of 107° amp. the pressure of the electric wind may impart large deviations to a vane suspended by a rigid brass wire ‘1 mm. in diameter, and is at the lowest estimate as large as 25 dynes per cm.? In an electric arc, where several amperes pass through the gas, the expected pressure should be sufficiently strong to smash at once the carbon poles of the are. I have noticed, however, long ago the remarkable effecr, that the pressure of the electric wind increases with the current only to a certain limit, and then tends to diminish. Thus, in « discharge-tube the swind pressure appears to be disproportionately small when a current of one milliampere passes through the tube and disappears completely when this current is increased to °5 ampere. Now it appears from Prof. Duffield’s experiments that also in an electric are the electric wind for some reasons does notarise, since, supposing even the whole pressure effect observed by him to be due to the electric wind, itis negligibly small compared with the effect to be expected. These results are very striking, since the electric wind originates solely in collisions between the ions and neutral molecules in gases and can only increase with the density of the current. There seems to be no other alternative to explain it than to suppose that when the current density is large, the carriers of electricity do not collide with the gaseous molecules—an assumption which seems incredible. I believe that further investigation of this phenomenon may throw fresh light on the nature of discharge of electricity through gases. T am, Gentlemen, Yours faithfully, The Physical Laboratory, S. RATNER. Owens College. Manchester, July 12th, 1920. nels +] LXI. On a Method of Finding a Parabolic Equation of the , Y roe Gees y / mm rth Degree for any Graphically Faired Curve. By T. C. TOBIN, JZ.A. a a mean curve be drawn through a series of points plotted with respect to a set of rectangular axes, then an approximate equation to the curve may be found in the form YHA tayet ... $a," by the following method, which lends itself readily to arithmetical computation. Take “n” equidistant abscissee, such that eet) ee ee, aR correspond to Y=Y1s Yo, Y3+++Yn aS measured from the curve. Then the equations of condition to determine the constants Ap, Az, Ay, &., are Yy=Ay+ OSes aa siete + Ays Ye G+ 2 Oy 2 eae... 2". G;, Yn= Ag +n .d,+N* dot ...+N". Ap. These give 1—,.Cyyy + rCoye .. - (— Yn =l—a| .Qi- otek all) Cad --a{[1 .,C,-2. ,Cot...( =o 170) — 2" no... C=)? s n? | —ay[ 1". pC — 2. n@Cgt 2.6. (—)? in | that is, LCi Calla one (n= Day -: - (1) since n”—,0,(n—1)"+ ...(—)" 11,0, =0 so long as’'n>r. This latter relation may be easily verified by considering the identity Roe) eB oo, y y+l yt+2 ytn mn! s=o y+s which may be written ey “) (14 ht -.(14 ehh sy Z ¥ Y Ss (0, [1 ES (jn) (<) | s=0 Y Since there is no power of ; on the left-hand side less * Communicated by the Author. 014. Mr. T. C. Tobin on a Parabolic Equation of the than the nth, the coefficient of = on the right-hand side will be zero if n>~,, and this coefficient is n—,C(m—1)"+ 22. (— P71"... Hence, instead of equation (1), we may write 1 —a@g=1 71 Ci +p iCoys: 2 (—) a Having thus determined ao, the original equation for the curve may be writtcn Y — & ee Uy Peres rs =A, + Av + eee + av" I. and from this we obtain, in a similar manner to the above, of 1 7 é L—ay=1—,0). m +O). 922). ( =) tee = ee Proceeding as before, we may now write the equation for the curve in the form ae) ee Qe — On Os UE ie eo and obtain il — 4g = 1—,_,C, ° ny + eee (—)"7} ° Mr ° (4) and so on. If the curve cuts the axis of y so that y9=da) is known, then a, may be determined as before from y— Yas =) + Gin ar eee +- a, 6 x’ —1, v To illustrate the arithmetical convenience of the method, let, the observed “‘y” values corresponding to equidistant abscissee be as follows :— — Y= 10045") w= Y2=1°36 ae Yy3= 170 ys= 211 J the assumed equation of the curve being Y = Ay + Aya + Aye? + azv® + aya'. The calculation of the coefficients can be arranged in tabular form (p. 515). The method may, of course, be used to obtain an approxi- mate equation to any arc of a continuous curve of one signed curvature whose ordinates can be measured at equidistant abscissee values. 5 x ed Curve. ‘ally Fav or any Graph ee ec rth Dea L. 1). 1 “1400 2330 ce PTD. TS 3 4 5 6 7 8 9 49 il y n’. ne 4 4 “A600 ‘O055 +3 “OL65 ‘O246 ye ‘J492 ‘0037 = —a \ —6 — 10800 0228 —o — ‘0684 ‘0209 —1| — ‘0209 44 9320 (328 aa ‘0328 dg = +'0283 4 — ‘2775 a,=—0191 gg ote eet Bim ae Column 1 gives the abscissx values. 99 Columns 38 (Yond Ay 2 gives the values of ~——, v yl +1345 @— 0191 a? + 0283 2° — 0037 a". SS 3 is derived from Col. 2 by subtracting a, from each teri ; and similarly Col. 8 from Col. 5; and Col. 11 from Col. 8. , 6, 9 give the Binomial multipliers with appropriate sign as occurring in Equations (2), (3), (4)... ., 4,7, 10 are derived from Cols, 2, 5, 8 by using the corresponding multipliers in Cols. 3, 6, and 9, The coefficient “a,” is the algebraical sum of the terms in Col. 4, and a@,, ds, @, evolve successively as shown, and the equation becomes r 516 |] LXI. On the Dimensions of Atoms. By A. O. RANKINE, D.Se., Professor of Physics in the Imperial College of Science and Technology™. 1. JN the August number of this Magazine W. L. Brage Tt has placed on record estimates of the sizes of the domains occupied in crystals by the atoms of various elements. He has also examined these data, which cover about haif the known elements, in terms of Langmuir’s } theory of atomic structure, and has shown in a convincing manner that definite values may be assigned to the diameters of the atoms in a limited number of cases. By the diameter of an atom in this connexion is meant the diameter of the outer shell of electrons as defined by Langmuir. It is inter- esting to compare these values with those derived from the well-known and quite different method based on the kinetic theory of gases. It will be seen that the data obtained by calculations from viscosity afford general support to the conclusions which W. L. Bragg has arrived at in the paper referred to. The number of elements to which this procedure can be applied is small, being lmited to cases where the element is in the easeous state at temperatures sufficiently low for the convenient measurement of its viscosity. But it happens, fortunately, that the elements in question are just those for which Bragg does interpret the atomic domain as being nearly coincident with the outer electron shell. It is thus possible to make comparisons for nine elements, including four inert gases, three halogen gases, and oxygen and nitrogen. 2. The inert gases.—According to Langmuir’s theory these elements constitute the most stable arrangements of electrons round positively charged nuclei. They are monatomic and do not enter into chemical combination. A definite physical significance can thus readily be. attributed to what is often vaguely called the atomic diameter by regarding it as the diameter of the outer shell of electrons. It is true that no direct calculation of these diameters can be made from crystal measurements, since the inert gases form no com- pounds, and have not yet been obtained themselves in erystalline form. For reasons which appear to be adequate§, however, Bragg has been able to assign probable values, and these are reproduced in column 2 of Table J. The numbers in column 4 of the same table have been calculated from * Communicated by the Author. + Phil. Mag., Aug. 1920, vol. xl. p. 169. { I. Langmuir, Journ. Amer. Chem. Soc. 1x1. p. 868. § Loe. cit. On the Dimensions of Atoms. 517 the author’s measurements of the viscosities previously recorded in various papers*, and represent the distances of nearest approach of the centres of two atoms during a collision, the Sutherland correction having been applied. These distances it has been customar y hitherto to call the diameters of the atoms. In order to facilitate comparison the increments of diameter in passing from one element to the next have been placed in columns 3 and 5. All the dimensions are given in Angstrém units: (A= 10 sem). TABLE I. From Crystal measurements. | From Viscosity measurements. Element. Atomic ear Atomic Pen t | diameter. | ’ |. diameter. ese Boom ....:.-.. | oOo. 5 | Py 02 | an ee | 0-54 Argon ..i.... 2‘05 | 2°56 | | 0:30 \ 0-20 Krypton .... 2°35 2°76 } | | \ 0°35 He, Site a's Xenon ....... 2-70! 3:06 | With regard to these figures, it should be borne in mind that whereas those in the second column represent the actual diameter of the outer electron shell, those in the fourth cannot be regarded as such. It is quite improbable that even in a Aivect collision between thermally agitated atoms, the approach is so close that the electrons of the atoms interminole. The mean distance between atomic centres during collisions may therefore be expected to be in excess of the diameter of the outer electron shell of each participating atom. The above figures support this view, all the numbers in column 4 being in excess of the corresponding numbers in column 2 -althou gh the order of magnitude is the same. More- over, although there i is no aeeaca of strict proportionality ; which, quite possibly, we are not entitled to expect, there is eeneral accordance between the increments of diameter in passing from one electron shell tothe next. Thus , according to both estimates, there is a comparatively large increment in passing from neon to argon, followed by one fess than half as large in the step baineen argon and krypton, and, finally, _a somewhat larger one between krypton and xenon. This is fairly good agreement, considering the wide difference between the methods of estimation. 3. The halogen gases.—In these cases the comparison * AO. Rankine, Phil. Mag. vol. xxii. p. 45, and vol. xyix. p. 552 518 On the Dimensions of Atoms. contemplated is complicated by the fact that in the gaseous state the atoms combine into pairs forming molecules, so that we are no longer able to regard the molecules as spheres, and apply the kinetic theory ‘with the confidence permissible in the consideration of monatomic gases. According to Langmuir’s theory we should picture a chlorine atom as identical with the argon arrangement except that the nuclear charge is 17 instead of 18 and that there is one less electron in the outer shell. In other words, we should expect the diameters of the outer shells of chlorine and argon atoms to be practically the same. In the gaseous state, however, the unstable arrangement of the chlorine atom is avoided by the junction of the atoms in pairs, so that each pair shares two electrons, and thus forms a stable molecule. The fact that two electrons are shared implies intimate contact between the outer electron shells, and, as far as dimensions are con- cerned, therefore, a chlorine molecule is almost identical with two argon atoms in contact. Similarly, a bromine molecule has the size and shape of two krypton atoms in contact, and an iodine molecule is similarly related to two xenon atoms. According to this view, the molecules of chlorine, bromine, and aa Pane have, respectively, twice the volumes of the atoms of argon, krypton, and xenon. This is in complete agreement with the conclusion arrived at by the author* in 1915, from deductions from the viscosities of the gases in question. The procedure by which this result was obtained was, of course, only approximate, for it’ was based on the assumption of spherical molecules, according to the usual statement of kinetic theory. To extend this theory so as to include diatomic molecules of the shape which Langmuir’s views lead us to infer, would be a very com- plicated matter. But in so far as it is justifiable to treat such molecules as colliding with the same frequency as they would, were they spheres of the same volume and mass, the diameter of each atom can be calculated for comparison with the values obtained by W. L. Bragg. The results are shown in Table IL., the arrangement of which is the same as for Table I. It will be seen that here again the “‘ viscosity ” estimate of the diameter is in excess of the ‘‘ crystal ” estimate, and that the increments are general accordance. A comparison between the 4th columns of the two tables will show how nearly the diameters of the atoms of argon, krypton, and xenon are equal respectively to those of ehlomne: bromine, and iodine. * A.QO. Rankine, Phil. Mag. vol. xxix. p. 554. Double Solenoid for Uniform Magnetic Fields. 519 TABLE I]. | From Crystal measurements. |From Viscosity measurements. Element. “FH Vil Bd L | a Way i | Atomic Eee Atomic 7a diameter. SHOPRRETY! diameter. Bore RAL. | Chlorine...... 2°10 | 2-54 | | bishit sina \ 1 aes Bromine...... 2°33 a7. 49 Vu Mara kiet | Todine ..... | 2°80 2.98) it ‘aueigenkc an 1 Ni prober Le 1S W all iS cieal that thes leave theory has indicated that the molecules of these two eases are of approximately the same dimensions. The sicher figures are somewhat uncertain, owing to inconsistencies in the values of the viscosities obtained by different observers. Approximate calculation based on the assumptions indicated in the last paragraph indicate that the molecules of both nitrogen and oxygen resemble dimensionally two atoms of neon in contact. ‘This agrees with W. L. Brage’s mea- surements, and affords additional justification for his as- sumption that the diameter of the outer electron shell of neon is practically identical with those of the incomplete outer shells of nitrogen and oxygen. LXNIIL. A Double Solenoid for the Production of Uniform Magnetic Fields. By 8S. J. Barxerr* HE universal use and great importance of the solenoid in magnetic research have led me to believe that a brief account of a method of construction devised here will be useful. So far as my knowledge goes, it is the first description of a oles method of winding a solenoid of more than one layer, and at the same time “eliminating the troublesome effects of leads and interconnexions. Probably the best way in which a single-layer coil can be constructed with great precision is by winding uniform round wire in a spiral groove accurately cut in a circular cylinder, as first suggested by Viriamu Jones and as exem- plified in the work of the National Physical Laboratory. If the cylinder is conducting and leng, as in the case of a solenoid, the effect of the leads may be practically eliminated, as they often are, by connecting one end of the coil to the cylinder and using the other end as one terminal ; but such a coil cannot be used satisfactorily with alternating currents. * Communicated by the Author. 520 Double Solenoid for Uniform Magnetic Fields. In order to produce tields of moderate intensity in ordinary circumstances, a solenoid must have at least several layers. Such a coil may be constructed with precision, as Bestel- meyer * has pointed out, by winding one layer as described above, and with pitch considerably less than twice the diameter of the wire, winding a second layer in the same direction in the depressions between the first, and so on. A serious defect of this arrangement, however, is due to the long conductors necessary to connect the far end of each coil with the near end of the next, which may interfere greatly with the direction and uniformity of the field due to the spiral windings. All the advantaves of precise winding may be obtained, and at the same time the (small) effect of spirality on the uniformity of the field may be largely eliminated and the trouble due to connectors and leads avoided, as follows. Two solenoids are constructed with the same piteh and number of layers, and with practically the same length, but with somewhat different diameters, so that one may be placed inside the other. One coil is wound in left-handed spirals, the other in right-handed spirals, and the two are mounted coaxially and concentrically. The construction necessitites that the successive jayers in each solenoid start from points 180° apart. When each coil has two or more layers it is probably best to wind all layers of one coil with the same integral number of turns, and all layers of the other coil with this same number of turns plus or minus one half-turn. It is then clearly possible to join successive layers of the two solenoids systematically in series by very short con- nectors at the ends, in such a way that the current goes alternately up a layer of one solenoid and down a layer of the other, and that the small effects of each pair of short connectors are practically cancelled near the centre of the field. Difficulties are encountered in making the windings precise close to the ends, but slight irregularities there are of little consequence. The same general method of construction, of course, applies to solenoids of all dimensions. In a large double solenoid constructed in this laboratory the coils are wound on tubes of bakelite-dilecto, which is known to have excellent mag- netic and mechanical properties, and to insulate so well that any kind of current may be used. It can be worked readily with a diamond tool. Department of Terrestrial Magnetism, Carnegie Institution of Washington. * Phys, Zeit. 1911, p. 1107. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOLENAL OF SCLENCE. fe ' Rt (ov \ 6 (SIXTH SERIES. ] NOVEMBER 1920. LXIV. Lhe Torsion of Closed and Open Tubes. By Joun Prescort, M.A., D.Se., Lecturer in Mathematics in the Manchester College of Technology™. | aaa object of the present paper is to deduce, from St. Venant’s taeory of torsion, approximate formule for the torsion of closed tubes, and of rods whose sections are long thin strips, such, for example, as split tubes. Although the results are only approximate the percentage errors they contain approach zero as the ratio of the length to the breadth of the section approaches infinity. The results have thus the same sort of limitations as the ordinary beam theory, which applies accurately only to rods whose lengths are infinitely greater than their lateral dimensions. Axes of x and y are taken in the plane of one normal section of the rod, and the z-axis along the untwisted axis. The displacements of a particle parallel to the three axes are u, v, w; the three shear stresses are §,, S., 8; ; the first of these acts on ay planes and ca planes, the second on yz planes and ey planes, the third on zr planes and yz planes. Fig. 1 shows the relation of 8; and 8, to the axes of w and y. The usual assumption in St. Venant’s theory is that the section at z is twisted through rz, so that 7 is the angle of twist per unit length. This gives, as the component * Communicated by the Author. Phil. Mag. 8. 6. Vol. 40. No. 239. Nov, 1920. 2M 522 Dr. J. Prescott on the Torsion of displacements parallel to the axes of w and y of the particle originally at (a, y, 2), (1) The displacement w is unknown but is assumed to be a function of « and y only. This assumption means only that each similar element of rod has the same strains. If n is the modulus of rigidity the shear strains are s=n(57 + 5s) =n(S? +0). EA a (2) Oui 6 itl ogg: Ow == 7) ae —ry). / 2s) Vaan It is next shown that there is a function yr of w and y such that Ov ow Oy — Oa’ e . ° e ° ° ° (4) ov ow Da a Oy ; : : . : ; A (5) whence S,;=—x2 (se —re) 1, 2 oe rn =n (SY -1y). ee Closed and Open Tubes. 523 The condition that there should be no stress at the surface of the rod is now, in terms of , 372" -+-y") constant,, «x: . | (8) this equation being true all along a boundary of the section. If the rod is a tube, so that the boundary of the section consists of two closed curves, the constant in (8) is a different constant along each closed curve. The equations of internal equilibrium give Ow dw ee F Oy? = (), eee ee 0) and equations (4) and (5) give also Ov Oh __ a on. Se ery ey CL) It follows from (4) and (5), which depend upon (9), that Ua ae Ie Oe) a hose, fot wvuss’ Jy hoeat| Meled ) where 2 denotes /—1. It is convenient to introduce a function & such that E=Wv—47(0?+y"), ° - . : “ (12) so that the boundary condition makes Gconstamin aa) eh Of Eh, CR) along each closed boundary of a section. The torque on a section is Q=\\(@8i-yS.)dady, . . . . (14) the double integral extending over the whole area of the section. Now s=—2n ( —re) = =n 88, cho (eee hs Ly (3% = ry) =n Pee Moment (nel 8) he Therefore Qa an { (ode +88) dedy. Wane) 2M 2 3bd, the torque is approximately Q=hurbta( Wage ere =) =Anrl,(1=0-630°). LES See oS) If the width 6 is so great as one-third of the length, the formula (57) is wrong by a little over 20 per cent. But if 4 is one-tenth of @ tien the result is wrong by just over 6 per cent. If the rectangular section were bent into any curved shape, the lines of shear stress would be distributed in the section in almost precisely the same way as when the section was rectangular. Consequently there “would be just about the same proportional error in equation (57) for a curved section as for a straight section, always assuming that the thickness is small compared with the radius of curvature. To take another example. The accurate torque in an elliptic section with semi-axes a and b is al? ’ Qari tS ee (59) Phil. Mag. Ser. 6. Vol. 40. No. 239. Nov. 1920. 2N 538 Dr. J. Prescott on the Torsion of Suppose b is small compared with a. Then we get | ? Dy od Q= mur bPa( 1 ao = =Anely(1~ ia t-- aon, aah I, being the moment of inertia about the major axis. The proportional error in equation (57), when applied to this »)* Ole error of about 10 per cent. when a=36, and about 1 per cent. when a=10). For this section then our formula gives more accurate results than for the rectangular section. We may expect in general that (57) will give better results for a strip with tapered and rounded ends than for one with rectangular ends. : b? section, is therefore approximately —,. Thus there is an Uniform Circular Tube split Longitudinally. If r is the mean radius of the tube and ¢ the uniform thickness, equation (56) gives , Cauley. Q=lnr tds e/ 0 sinte x20? .5 2 « . Le 2 oO The ratio of the tOrque in the split tube to that in the 9 } a lets : complete tube is = for the same angle of twist. But this or” E is not a fair comparison because the maximum shear stresses are not the same in the two cases. The maximum shear stresses are, for the split tube ntt, and for the complete tube nt(r+tt). If we make these shear stresses equal by giving different values to 7 in the two cases, then the ratio of the torque in the split tube to that in the complete tube is ilGese see dw ate 5g + ig ~ mearly: eS 2 The split tube is therefore much weaker than the complete tube under torsion, and very much less rigid. [t is interesting to find the relative axial displacement of the open ends of the split tube. [or this purpose we may use equation (33) and regard 8 in that equation (which is. the mean shear stress across the section) as zero for the split: tube. Then w=—1\pds = — 27 | dA == 27 A, ° . e 0 p 6 ° (63): Closed and Open Tubes. 539 A being the area enclosed by the central line of the section. For the split circle we get = ae. oe ys, wy, (64) Equation (63) could be applied to any split tube of variable or constant thickness. If it is applied to a section whose open ends do not come close together, then A is the area enclosed between the curve of the central line of the section and the radii drawn from the axis of twist to the two ends of that line. For example, if it were applied to a section with a straight central line which is intersected by the axis of twist the area A would be zero, and therefore w would be zero. The difference in the behaviour of a split and a complete tube is due to the freedom of the open ends of the section of the split tube to move axially relatively to each other. If this freedom is constrained in any way the split tube does not differ from the complete tube. If, for instance, a short split tube.is gripped at the ends so that no relative motion of the particles in the end planes is possible, then the tube behaves almost exactly like a closed tube. The rules for the split tube apply to a tube twisted in such a way that the end — sections have perfect axial freedom, or to sections in a long tube at great distances from the ends. In actual practice a split tube welded into brackets at its ends would behave very nearly like a complete tube because the axial displacement represented by (63) would be so great that the effect of preventing all this displacement at the ends would be considerable even at the middle of the rod. The rules that we have given for split tubes will apply with fair accuracy to any thin portion of a section of a twisted rod where the shear lines must all go forward and return across the same normal. For example, the I-section shown in fig. 8 may be treated as if the two end-pieces and Fig. 8: the backbone were put end to end to form a continuous strip with open ends. That is, if the two ends be supposed to be cut off along the dotted lines, then the torque in any 2N 2 ’ 540 Dr. J. Prescott on the Torsion of one of the three strips is and the total torque is the sum of the torques due to the three strips separately. Again consider the box section with projecting pieces shown in fig. 9. The shear lines in the projecting pieces Fig. 9. A A's JI © Cc go and return along the same strip, whereas all the lines that run along A’A also run along AC. Thus the torque in the part ACC’A' may be obtained from the rules for a closed tube, whereas the torque in the projecting pieces must be obtained from the rules for open tubes. Suppose the width of the strip is constant everywhere and has the value ¢, and suppose AA’=a, AC=c+t, A’B=d, and let all the other projecting pieces have a length 6. Then the torque in the closed tube is, by (35), Anrazct (Q), — 2(at+e)?’ e ° e e ° e (65) and the torque in the four projections is Qe nT ORS cae) ae (66) The total torque is therefore Q=Q + Q, we nmancd > Paw As == ales. + itt Ds 6 is . 5 (67) It is obvious that the torque due to the projections is negligible compared with that due to the tube. In fact (, is itself of the same order as quantities neglected in Closed and Open Tubes. 541 finding Q, by our approximate theory. Then, to our order of approximation, 2nta7c7t a ee ee eed rene (68) It is usual to make box sections by rivetting plates together. Such rivetted girders may be treated as if they were solid for the purpose of torsion provided the rivets are near enough and strong enough. There may be, of course, considerable shear on the rivets. The formule obtained in this paper are suitable only for long thin sections in which the shear lines are nearly parallel curves over practically the whole of their length. A rod whose section is a complete circle could not reasonably come under either the rule for a closed tube or the rule for a long thin unclosed section. Nevertheless we shall, just for the sake of noting to what extent the formule fail, treat the complete circle ~ (1) as a closed tube with no central hole, (2) as a long thin section whose central line is a diameter. In the first case the central line is a circle of half the radius of the section. If r denotes the radius of the section then equation (35) applied to this section gives . Ant(4a7?)?r a = Si) (rr a ee LOOP, Again treating the section by the second method, equation (57) gives Q=4nt (iar) =—antr". ~ - Z 7 (70) The correct result for this section is ed atiage os jeroden yer) ste -y (EL) +e Of the two results in (69) and (70) one is half the correct result and the other is twice the correct result. It is not intended that the formule shall be used for such extreme eases, and they are given here only to indicate that fair accuracy can be expected from the formule for all reasonable cases. LXV. On the Variation of Thermal Conductivity during the Fusion of Metals. By Se1pe1 Konno*. § 1. Introduction. era to the electron theory the ratio of the thermal and electric conductivities of metals is in- dependent of the nature of the materials and proportional to the absolute temperature. The experimental investiga- tions | of different steels and iron alloys—such as carbon steels, iron-nickel, iron-cobalt, and iron-manganese alloys—at ordinary temperature shows that the said ratio is roughly the same for these steels and alloys, notwithstanding a large divergence in the respective values of the two conductivities. That is, the variation of the thermal conductivity of iron along with the concentration of the other components is similar to that of the electric conductivity, so that we can approximately expect the magnitude of one variation from the other. It was also found that in the case of carbon steels the above relation holds good for different high tem- peratures up to 900° C., and also that the proportionality between the said ratio and the absolute temperature is fairly well satisfied. It has been shown by KE. F. Northrup { and H. Tsutsumi§ that the electric conductivity of different metals makes a conspicuous abrupt decrease during melting, with the exception of bismuth and antimony, in the case of which an abrupt increase of conductivity is observed. It was therefore thought very desirable to see, whether up to a temperature beyond the melting-point, a similar relation exists between the two conductivities. ‘The present investi- gation was carried out on the one hand to test this relation, and, on the other hand, formed a preliminary experiment for studying the conductivity of molten steel, which has avery great importance in connexion with the metallurgy of iron. _ As to the change of thermal conductivity of different substances during melting, a few experiments have been made. DBarus || first made an experiment with Thymol, and found that the thermal conductivity of the substance * The thirtieth report of “The Alloy Research Institute.” Communi- cated by the Author. + sci. Rep. vi. (1917) ; yar(1918). ¢t Jour. Frank. Inst. clxxv. p. 158 (1918); clxxvii. pp. 1, 287 (1914) ; elxxvill. p. 85 (1914). § Sci. Rep. vii. p. 93 (1918). || Phil. Mag. xxxii. p. 431 (1892). Variation of Thermal Conductivity of Metals. 543 abruptly decreases by about 13 per cent. during the melting (at 13° C.). The same experiments on Na,HPO,+12H,0, Paratoluidine, Naphthylamine, and CaCl,+6H.0 were also made by C. H. Lees*; but there was no apparent break in the regularity of the change of thermal conductivity at the melting-point in the case of the first three substances, and a decrease of about 20 per cent. in the case of CaCl,+6H,O. A. W. Porter and F. Simeont made a similar experiment with mercury and sodium, and showed that the abrupt change in the thermal conductivity during melting is of the same order of inagnitude as the corre- sponding change in the electrical conductivity—that is, the ratio of the thermal conductivity for the solid state to that for the liquid is 3°91 for mereury and 1°31 for sodium. The only experiment at high temperatures was that of Ki. F. Northrup and F. R. Pratt { tor two typical metals— tin and bismuth, for each of which the conductivity changes in the opposite direction during melting, but their result is rather of a qualitative nature. They concluded that the characteristic change of thermal conductivity for tin and bismuth during melting is substantially the same as the corresponding change for the electrical conductivity, and that Wiedemann-Franz’s law holds good, at least qualita- tively, through the change of state of these metals. So far, the literature regarding the abrupt change of the conductivity during melting has been very meagre. Hence, to determine as accurately as possible the thermal conductivity of different metals at temperatures both below and above their melting-points was a very important work, -and, therefore, I began—as a preliminary experiment—with some fusible metals. The following pages contain the method and the result of the present experiment. § 2. Description of the Apparatus. The method used in the present experiment is to measure the conductivity of metals at different high temperatures relative to the values at room-temperature. A vertical section of the main part of the apparatus is shown in fig. 1. A, B, D are short cylindrical pieces made of a very low earbon steel, and C is the specimen to be tested. The iron cylinders A and B are each 2°5 cm. in height and 2°2 cm. in diameter; between these two pieces there is a flat * Phil. Trans. Roy. Soc. A. exci. ps 399 (1898). + Proc. Phys. Soc. xxvii. p. 807 (1915). { Jour, Frank. Inst. elxxxiv. p. 675 (1917). D44. Mr. Seibei Konno on the Variation of cylindrical cavity 5 mm. high and 20 mm. wide, in which a small, flat heating-coil is placed. On each side of the cavity two narrow holes a, 6 and c, d, each 1 mm. wide and 11 mm. deep, are radially bored at a distance of one centimetre to receive the two junctions of a differential thermocouple, which measures the temperature-gradient along the axis of the cylinder. The two thermocouples consist of iron and nickel wires in the case of the investi- gation of the metals having relatively low melting-points ; but in the case of aluminium and antimony, iron wire is replaced by a platinum wire to avoid the disturbance caused by an abnormal change of thermoelectromotive force due Fig. 1. Fig? 2 37cm 2 e fa f (a) Ph Aire to the A, transformation of iron. e, f is a duplex thermo- couple, consisting of platinum and platinum-rhodium wires, by which the temperature-gradient in the specimen C, and at the same time that of the specimen at point £ can be measured. Its connexion is shown in fig. 2. The two junctions e, / are coated with thin asbestos paper in such a way that the insulation between the junctions and the specimen, both in the solid and liquid states, is perfect, the horizontal portions of the wires being protected by thin porcelain tubes. Its cold junction is placed in a water- bath, whose temperature is read by a thermometer. R is a thick porcelain tube, and the interspace between the tube and the steel pieces B and D is tightly packed with caolin kit. The two junctions e, 7 go out through the wall of the tube, and are so firmly fixed to it that the distance between Thermal Conductivity during the Fusion of Metals. 545 e and 7 does not change in any appreciable degree during the melting and solidification of the specimen. R’! and R” are also similar porcelain tubes ‘protecting iron cylinders Aand D. Two narrow holes a’, b' are also radially bored to receive the two junctions of a differential thermocouple, as in the iron piece A. G is a thick iron tube, which serves as an equaliser of temperature, and the interspace between the tube and the porcelain tube R is packed with caolin powder in order to avoid the lateral loss of heat as much as possible. The small heating-coil in the cavity between the iron cylinders A and B consists of a nichrome wire 0°5 mm. Fig. 3. thick and about 30 em. long, wound in a spiral form and well insulated with caolin kit. To the terminals of the coil are fused thick copper leads /, 2’, which are used as the current and potential leads. ‘The terminal volt and the current passing through the heating-coil are measured by a voltmeter and an ammeter (made by Siemens & Halske); the heat generated in the cavity per unit cf time can thus be evaluated. The whole arrangement is vertically supported by a porcelain tube R”’, so as to lie in the middle portion of a 546 Mr. Seibei Konno on the Variation of vertical electric furnace. To avoid the convection-current in the furnace as much as possible, the free space in the furnace is loosely filled with asbestos fibre. The furnace consists of a porcelain tube 5°5 cm. wide and 45 em.’ long, a nichrome wire being uniformly wound round it and the exterior thickly covered with asbestos paper. The whole arrangement is shown in fig. 3. The terminals of four sets of the thermocouples ab, ca, ef, a'l! are dipped in the eight mercury cups on a paraffin block, as shown in fig. 4. Two copper wires connected to Fig. 4. a wall-galvanometer (made by Leed & Northrup), is led into a glass tube H and fixed by paraffin, the two terminals pp protruding from the lower end of the tube. The ter- minals can be put in any pair of the mercury cups by holding the glass tube in the hand. In this way the differ- ence in the temperatures between two junctions in any pair of the four thermocouples can successively be observed with the single galvanometer. § 3. Method of Observation. Let Q be the quantity of heat generated per unit of time in the heating-coil, and let g, t, and K .be the quantities of heat flowing per unit of time, the temperature differences per unit of length and the thermal conductivities respec- tively, subscripts 1, 2, 3 and 4 referring to the iron pieces A, B, C, and D respectively. When the stationary state is attained, the lateral loss of heat is always very small in comparison with g, and we have Q=KySyth, qo=KeSot,, ga=KsSsts, qa=KSyt,, where § is the sectional area of the iron pieces. Since A, B, D are made of the same material, we have Ke KO KGe ene) IK, = Ke Thermal Conductivity during the Fusion of Metals. 547 Under the supposition that the lateral loss of heat is negligibly small, we have Q=q+q.=K'S(t, +t) : rx ty he ty Waar eee Piel erie In an actual ease, qg; is slightly less than go, and gy less than g3 by nearly the same amount; hence we may put as the first approximation to +t, a= KSts= 3(go+ 94) =2Q; it by Kat Q tt i 2 Sts t, +t, Since the distances between two junctions of each of the three differential thermocouples ad, cd, a'b' are equal to each other, ¢,, t, and t, in the ubove relation may be re- placed by the corresponding deflexions of the galvanometer 6;, 05, and 6, respectively. Moreover, if the distance between two junctions of the differential thermocouple ef and its difference of temperature be respectively denoted by s and At, we have b) or At (i;=—, s rae Qs(d.+ 64) ~ 2 SAt(d,+ 65) ° In the above relation the determination of the quantity Q is somewhat uncertain. For the heat generated per unit of time in the heating-coil F is accurately known, but a part of it is lost by lateral conduction before flowing into the specimen. It is very difficult to ascertain exactly how much of the heat is thus lost. Hence the absolute values of thermal conductivity obtained by the preseut method may be somewhat uncertain; but if we assume that for all temperatures, the above loss is always the same fraction of the total heat generated, the relative values of the con- ductivity are pertectly correct. Hence in the present investigation only the relative measurements were made. The observation was conducted in the following way :— A constant electric current of about 2°7 amperes was passed through the heating-coil. After an interval of about one hour, when an approximate stationary state was attained, the temperature of the specimen was first observed, then the readings of the galvanometer corresponding to the junctions and therefore 048 Mr. Seibei Konno on the Variation of cd, ef, a'b', and ab were taken; and again the same cbser- vations were repeated by reversing the poles of the galvanometer leads pp, by turning the glass tube H, the disturbing effect of the thermoelectric current in the circuit leading to the galvanometer being thus eliminated. The difference between each pair of these readings corresponds to 26 in the above formula. The heat generated in the cavity EF per unit of time was then measured by observing the current passing through the heating-coil and its terminal volt. Finally, the temperature of the specimen and its gradient were re-determined for confirmation. Next, the current in the heating-coil was broken, and a current passed through the furnace. When the temperature of the furnace was raised by a certain amount, another current was passed through the heating-coil F ; after the temperatures had become stationary, the observations were repeated as above. In this way the observations at gradually increasing tem- peratures were made step by step. It was found that the interval required for the approximate stationary state oradually decreases with the rise of temperature, the smallest, however, being 30 minutes. S84. Results of Experiments. In order to see how much of the heat generated in the coil is lost before entering the iron piece D, the specimen C was removed and the pieces B and D were brought in direct contact, and the temperature-gradients in the two junctions cd and a'b' were observed. The gradient in the second junction was about 30 per cent. less than that of the first junction. With the specimen C in its proper position, this difference was not in any case greater than 40 per cent.; hence the portion of the loss due to the lateral conduction through the specimen is of the order of magnitude of 10 per cent. It is, however, a merit of the present method that a fairly correct value for the conductivity is always obtained. In order to show this, the absolute determination of the thermal conductivity of a carbon steel containing 0:35 per cent. of carbon was measured. As the results of the experiment we obtained the following values: 0-096 at 128° ©., 0:088 at 305° C., 0:072 at 473° C. These values - agree very satisfactorily with those obtained by Prof. Honda and Mr. Shimizu*. In the actual case the determination of the conductivity was made relative to the conductivity at. room temperature. “1 Sei, Rep. vi. p. 2a (1on7 ). Thermal Conductivity during the Fusion of Metals. 549 The values of the latter quantity for different metals were taken from Landolt and Bornstein’s table. The results obtained are given in the following tables In these tables, ¢ is the mean tem- perature of the specimen. the thermal conductivities in solid and liquid states of different metals, expressed relatively to the known values at ordinary temperature. and in figs. 5 and 6. iin: Temp. Ks. 18 0°157 (1) 108 0-151 125 0-149 209 0148 Temp. Ky, 292 0-081 417 0-079 498 0-078 In (1), (2) Lees’ values for Ks at 18° C. were assumed. 7s - Lead = Temp. 18 108 222 298 326 Temp. 359d 447 531 601 Ks. Temp. 0083 (2) 18 0-080 89 0:077 160 0-074 222 0-069 233 256 Kir: Temp. 0°039 298 0:038 286 0:037 376 0:037 484 584 K, and K, are respectively 3. Bismuth. Ks. 00194 (3) 00181 0:0170 0:0177 0:0177 0:0183 KE 0:0418 00400 0:0378 0-0372 0:0369 In (3) Jaeger & Diesselhorst’s value at 18° C. was assumed. 4. Zine. Temp. Ks. 18 0'268 (1) 97 0:263 129 0°262 242 0-250 280 0-246 313 0-241 362 0:233 400 0220 Temp. Kr 460 0°140 537 0-138 578 0°137 In (1), (2) Lees’ values for Ks at 18° O. were assumed. In (3) Lorenz’s value for Ks at 0° C, was assumed. 5. Aluminium. Temp. 18 116 273 324 430 470 605 Temp. 675 800 Ks. Temp. 0-504 (2) 0 0°490 113 0-471 182 0°454 344 0-425 469 0394 557 0° 360 610 Kz. Temp. 0°223 692 0-214 6. An timony. Ks. 00442 (3) 0:0401 0:0386 0-0414 0-0456 0:0510 0°0575 Ki. 0°0503 Tin. Zine. Lead. 500) Mr. Seibei Konno on the Variation of Thus the thermal conductivity of tin, lead, zinc, and aluminium rapidly decreases with the rise of temperature. At the melting-points a discontinuous decrease of the con- ductivity is observed, after which they diminish slightly. Fig. 5, | SS | Re ET... | BaaeaE a Sasa a = pe 0:30} ; 30 ] \ | A | H \ | | ais | aF a | i \ ) 0°25} BES tte ae 25 | | tae a K | 0:20-0:50} Cele | } 0-45 THs ee =r \| 0-10 | iS x 5 by a g = Xt 0-05 r0) These changes of conductivity are similar to those of the electric conductivity, as observed by Mr. H. Tsutsumi. For the, sake of comparison, his results are reproduced in figs. 5 and 6 with dotted curves. In order to see whether aao0)) =|00 ) 100 200 300 4090 500 600 700 800 900 etre. Llectrical conductivity m reciprocal ohms per cub.cm. a: Thermal Conductivity during the Fusion of Metals. 551 Lorenz’s law—that is, that the ratio of the thermal and electric conductivities is proportional to the absolute tem- perature—is valid, the calculation was made by combining -200 -i00 6) 100 200 300 400 500 600 700 800 900°C. my result with that of Mr. Tsutsumi, and the ratio thus found graphically given in fig. 6. Below the melting-points the law is fairly well satisfied, but in the liquid state of these metals the applicability of the law completely fails. oo2 Variation of Thermal Conductivity of Metals. The electrical conductivity of bismuth and antimony shows an abnormal change during melting—that is, the conductivity increases by melting. It is ‘therefore very interesting to see whether the thermal conductivity of these metals, unlike other metals, increases during . melting. From fig. 6 we see that the thermal conductivity of bis- muth at first slightly decreases and then increases: during melting the conductivity considerably increases in the same way as the electric conductivity does. ‘lhe conductivity of liquid bismuth slightly decreases along with the rise of tem- perature. In the case of antimony the variation of the conductivity is similar to that of the conductivity for bis- muth, but there is some difference in the quantitative respect. In the case of these metals Lorenz’s law holds good only below 100°, as seen from fig. 6 Summary. The results of the present investigation may be sum- marized as follows :— 1. The thermal conductivity of tin, lead, zine, and aluminium decreases with the rise of temperature up to their melting-point. 2. The thermal conductivity of these metals decreases abruptly during melting. 3. The thermal conductivity of bismuth and antimony slightly decreases at first, and then increases a little. 4. During melting the conductivity of bismuth con- siderably increases, and that of antimony seems to increase only slightly. ). The thermal conductivity of all liquid metals here investigated decreases but slightly with the rise of tem- perature }. The above changes of thermal conductivity are similar to ae of electric conductivity for the same metals. In conclusion, I wish to express my hearty thanks to Prof. K. Honda, under whose direction the present experi- ment was carried out. [ 553 ] LXVI. Lonizaiion and Production of Radiation by Electron Impacts in Helium investigated by a New Method. By K. T. Compron, Ph.D., Professor of Physics in Princeton University”. Introduction. HE recent studies of the ionization of metallic vapours and the excitation of their spectra by electron impacts have established relations between the absorption spectra of the unexcited vapours and their resonance and ionization potentials which are in accord with a theory of radiation and atomic structure such as that proposed by Bohr. Briefly stated, the results are as follows :—lThe atoms gain no internal energy as a result of electron impacts unless the kinetic energy of the electrons exceeds the amount hy,, where vy is the frequency of the first (longest wave-length) member of the absorption series of the vapour. Ifthe energy exceeds hy, the atom may absorb this amount of energy, which it subsequently re-emits us radiant energy of this frequency. If the energy of the impinging electron exceeds hv;, where py; is the convergence frequency of the absorption series of which y, is the frequency of the first line, the atom may be ionized. The potential differences V, and V; through which an electron must fall to acquire the energies hv, and hy; respectively, are termed the Resonance and [onization Potentials, and are given by the quantum relation eV =hv. There is evidencet that there may be two resonance po- tentials, corresponding to two different absorption series which have the same convergence frequency, but no one has yet proved the existence of additional resonance potentials corresponding to intermediate members of a given absorption series. These phenomena, together with phenomena of dispersion of excited and unexcited gases, seem to prove the existence within the atom of series of related states or orbits in which electrons may exist in more or less stable equilibrium. Energy derived from absorbed radiation or from impact by * Communicated by the Author. + McLennan, Proc. Phys. Soc. Lond. xxxi. p. 1 (1918); Tate and Foote, Phil. Mag. xxxvi. p. 64 (1918); Foote, Rognley, and Mohler, Phys. Rev. xiii. p. 59 (1919) ; Foote and Mohler (in press); ete. { Davies and Goucher, Phys. Rey. x. p. 101 (1917); McLennan, loc. ett. Phil. Mag. 8. 6. Vol. 40. No. 239. You. 1920. 20 554 Prof. K. T. Compton on Jonization and Production an electron may displace an electron from a more to a less stable state; whereas, if an electron falls directly from a less to a more stable state, it emits radiation whose frequency equals the difference of energy in the two states divided by h. Bohr’s theory of atomic structure is based on these consi- derations, and, together with its modifications by Debye and Sommerfeldt, has been remarkably successful in accounting quantitatively for the spectra of systems constituted of a single electron and nucleus. The theory has not been developed to account quantitatively for the spectra of more complicated systems, although, if certain assumptions are made, the convergence frequencies may be calculated in some cases. Itis obviously of the greatest importance to secure definite information regarding the spectra and energies of formation of those atoms (other than atoms with a single electron) for which the theoretical assumptions may be most easily put into quantitative form. Of these cases, the simplest is the normal helium atom, with its single nucleus and two ~ outer electrons. In Bohr’s model of the normal helium atom the two electrons are symmetrically located in the same orbit, and their com- bined energy is equal to e x 83 volts/300. If one electron is removed, and the remaining electron takes its most stable position, the energy equals e x 54°3 volts/300. The difference gives 28:7 volts as the ionizing potential ; and the corre- sponding frequency gives 430 A as the convergence wave- length of the fundamental series. The frequencies of the other lines in the series cannot be calculated without a knowledge of the behaviour of one electron while the other is being displaced from orbit to orbit. Analogy with hydrogen would suggest that 3/4 of 28°7, or 21:5 volts should be the resonance potential of helium. There is no reason, however, for believing that hydrogen and helium should have homologously spaced series. Evidently quite different results from those obtained by Bohr might be obtained by assuming a different configuration of electrons in the normal atom*. Early experimental determinations of the minimum ionizing potential of helium J indicated a value between 20 and 21 volts. Recent discoveries, stimulated by applications of Bohr’s theory, suggest that this is really a resonance potential, the effects previously attributed to ionization being accounted * Landé, Phys. Zeit. xx. p. 228 (1919). — + Franck and Hertz, Verh. d. D. Phys. Ges. xv. p. 34 (1918) ; Pawlow, Roy. Soc. Proc. A. xc. p. 898 (1914) ; Bazzoni, Phil. Mag. xxxii. p. 566 (1916). ; of Radiation by Electron Impacts in Helium. DDD for by the photoelectric effect on the electrodes produced by the resonance radiation. This has been confirmed inde- pendently by Horton and Miss Davies* and by Franck and Knippingt. These investigators used the well-known method of Davies and Goucher{ for distinguishing between the effects of ionization and of resonance radiation, while Franck and Knipping also used the total and partial current method of Tate and Foote§. Both sets of investigators found the true ionization potential to be at about 25°5 volts, the actual values published being 25°7 and 25°3 volts respectively. Horton and Miss Davies further concluded that the effect at 20:4 volts is one of pure radiation—. e., there is no accom- panying jonization. | On the other hand, Franck and Hertz|| have pointed out that the successive maxima and minima observed by them at multiples of 20°4 volts cannot be adequately ex- plained by a photoelectric effect of the radiation on. the electrodes, but prove an actual production of ions in the gas in the region where the collisions become inelastic. Furthermore, Rentschler {] has obtained results which cannot reasonably be accounted for by a photoelectric effect setting in when the accelerating potential exceeds 20-4 volts, but indicate an actual ionization at this voltage. Rentschler attributed this ionization to ionization from the surface of his gauze electrode, rather than to a true ionization of the gas, and concluded that helium exhibits no reso- nance at 20 volts, but only ionization at 25°5 volts. Finally, the experiments made by Benade and Compton **, and espe- cially by Bazzoniff, in which the electron current between two electrodes was found to increase abruptly as the accele- rating potential reached multiples of 20 volts, can only be accounted for on the assumption of actual ionization of the gas at 20 volts, since the effects observed were entirely too large to be accounted for as a photoelectric effect by that portion of the resonance radiation which could have fallen on the emitting cathode. The present research was undertaken to investigate the resonance and ionization potentials of helium by a method * Roy. Soe. Proc. A. xcv. p. 408 (1919). + Phys. Zeit. xx. p. 481 (1919). +t Loe. cit. § Phys. Rev. vil. p. 696 (1916) ; x. p. 77 (1917). \| Phys. Zeit. xxx. p. 132 (1919). q Phys. Rev. xiv. p. 504 (1919). ** Phys. Rey. xi. p. 184 (1918). ++ Loe. cit. 202 996 Prof. K. T. Compton on Jonization and Production ‘ different from any hitherto proposed, and was In progress when the papers of Horton and Miss Davies and of Franck and Knipping were published. ‘The results corroborate the work of these investigators, and, in addition, explain the apparent discrepancies in conelugions which have been drawn from the various experiments on the effect at 20 volts. It appears that an atom may be ionized by first absorbing a quantum of energy of the resonance radiation from neigh- bouring atoms, and then being struck by an electron whose kinetic energy might be insufficient to ionize an atom in the normal state. In other words, the energy of ionization is supplied partly as radiant energy and partly as energy of impact. Jonization at 20 volts 1s therefore a secondary eifect, which may or may not be important, depending on conditions. The possibility of ionization by this process is obviously suggested by Bobr’s theory. It appears to have been first pointed out by Richardson and Bazzoni*. ‘The proof of its existence in the present experiments is due to the use of an apparatus which permits not only the differentiation of effects due to 1onization and to radiation, but also the estimation of the proportion of the effect due to either when both are present. Apparatus and Method. The ionization chamber (fig. 1) is similar in principle to that used by Franck and Hertz. A tungsten filament F of 0-8 cm. length and 0°08 mm. diameter serves as a source of electrons. These are drawn toward the platinum gauze G by an accelerating difference of potential V,. Those which pass through the openings in the gauze encounter a retarding difterence a potential V,, which is enough larger than V, iG prevent any of the electrons from reaching the electrode EH. This electrode is connected with a quadrant electrometer, and it may gain a positive charge either from positive ions pro- duced by ionization of the gas or as a result of electrons emitted photoelectrically from EH by ultra-violet radiation set up by impacts of the electrons. The distance between F and G is 3°5 mm., and that between G and EH is 7 mm. In order to distinguish between these two causes of electro- meter deflexions, the electrode Ii is constructed so that the area presented to the radiation may be varied without altering the geometrical relations “ over all.” This is done by closing * Nature, xeviii. p. 5 (1916). of Radiation by Klectron Impacts in Helium. Dot one end of a hollow copper cylinder, shaped like a napkin- ring, with platinum foil fand covering the other end with platinum gauze g. The cylinder is suspended by a platinum wire ending in a swivel, so that it can be rotated about its Fig. 1. a oe vertical axis by the aid of a horseshoe magnet acting on a soft-iron block I, which is rigidly attached to the cylinder by a light brass rod. Evidently it is a matter of indifference, as regards the receiving of positive ions, whether the foil end fa or the gauze end g of the electrode E is turned to face the filament. But the two ends are differently affected by ultra-violet radiation; for, when the gauze side is exposed, a large part of the radiation passes through into the cylinder, and does not result in the loss of electrons from the cylinder. Thus the ratio R= Hy/E, of the electrometer deflexion with the foil end exposed, and the deflexion with the gauze end exposed, determines the proportion of the total observed effect which is due to true ionization or to ultra-violet radiation. The calculation of this proportion is made very easily as follows:—Let i and x be the rates at which the electrode EH gains a charge due to ionization and to photoelectric effect 558 Prof. K. T. Compton on Jonization and Production respectively, and let ¢ be the ratio of closed area to total area of the gauze end. Then R= gr it rer whence i ino 7s Ta Sil The ratio ¢ is a constant of the apparatus, and the ratio R is found by comparing the electrometer deflexions obtained with the foil and gauze ends respectively of the cylinder turned to face the filament. For pure radiation R should equal 1/c, while for pure ionization R should equal unity. Any change in the type of the etfect due to the electron impacts should be indicated by a change in the value of R. The value of the ratio c was found to be 0-495 when calcu- lated from geometrical considerations. This value is subject to small corrections because some electrons, photoelectrically excited on the inside of the cylinder, may escape through the gauze; because also the platinum surfaces of the foil and gauze may be intrinsically different in photoelectric sensitivity owing to different modes of manufacture; and because the surface in one case is curved and in the other case flat. Therefore, an independent experimental determination of ¢ was made as follows :—The ground-glass cone J which sup- ported the filament was removed and a quartz window sealed over the end. After thorough evacuation of the apparatus, a quartz-mercury are was placed before the window and the photoelectric currents from E were measured with a field between Hi and G of the same general magnitude as in the ionization experiments. The ratio of these currents for the two positions of E gave R=1/c=1°82, whence c=0°55. This value agrees as well as could be expected with the calculated value. The remaining features of the apparatus require little explanation. A large tube of coconut charcoal is attached directly to the ionization tube. The helium is stored in a reservoir, from which it can be pumped into or out of the ionization tube through two U-tubes. The U-tube nearest the ionization chamber contains coconut charcoal, while the other serves asa mercury trap, into which the mercury vapour can be condensed by applying liquid air before the liquid air of Radiation by Electron Impacts in Helium. 559 is applied to the charcoal tubes. The pump and the copper- oxide spiral for removing traces of hydrogen have been previously described*. The spectrum, when excited by a discharge between incandescent tungsten electrodes, showed no impurities in the helium except a trace of neon, whose strong lines appeared faintly when an intense are was set up between the electrodes. These neon lines, three in number, were of about the same intensity as the individual lines into which the band-spectrum of the helium could be resolved, and were therefore extremely faint. This, in connexion with the well-known etfect of helium in accentuating the spectra of any impurities which may be mixed with it, shows that the helium was very pure. Its spectrum excited by a Geissler-tube discharge in a baked-out tube showed no lines except those of helium. A two-stage diffusion pump, with an oil-pump backer, is used for the initial exhaustion of the apparatus. The electrometer arrangement is so satisfactory as to warrant particular mention. The electrometer sensitiveness may be varied quickly from zero to 10,000 mm. per volt by varying the potential of the needle. The electrometer key is designed either to insulate the quadrants and electrode H, to earth them directly, or to earth them through india-ink resist- ances of 3°15 (108) ohms, 3°75 (109) ohms, or 2°37 (10) ohms as desired. It is therefore possible to pass instantly from one sensitivity to another, and to measure al] currents by the steady deflexion method. The india-ink resistances are found to be constant and to show no polarization. A galvanometer is introduced in the connexion to the gauze G in order to measure the total electronic current I. In preliminary experiments large ionization currents setting in at about 11 volts, and more strongly at about 15 volts, masked the effects due to helium. These were caused by water-vapour, which was slowly given off by the glass. This disturbing effect was eliminated by surrounding the ionization tube and the two charcoal tubes in electric heaters and main- taining them at 300° to 350° C. for about a week with the vacuum-pump in operation. The ground joint J was sealed with Khotinsky cement and was water-cooled during the “baking-out ” treatment. During the later stages of heating, the filament was kept at a white heat to drive off occluded gases. Liquid air was placed around the mercury trap before these heaters were removed, and was kept there continuously throughout the course of the experiments. * Benade and Compton, lve. ci. 560 Prof. K. T. Compton on Lonization and Production Haperimental Results. Figs. 2-5 show the results of several typical series of measurements of the rate of charging of the electrode EK as a function of the accelerating difference of potential Va at various pressures p and for various thermionic currents I. Fig. 2. p=0-0005 mms; I,.=1-75(10)~° amp. false as Lidl 5 «ee AM ee See 250 | biel 7 (i MA el cade Ba Seale [fof ema fl alia etl el aoe mee 2 Se ce Rao inc fo ef poe ae a a Sore i ae ae anaes fen” syle fx ile meee... Pog of Sie Zao PRs ere Ce ae 0 22 24 26 30 32 These thermionic currents I were determined by the filament temperature, and are specified by Io, the total current to the gauze G with an accelerating field of 20 volts. To the applied value of V, there is to be added a small correction due to initial velocity of emission, to contact difference of potential between electrodes, and to the aggregate loss of energy of the electrons due to the translatory motion im- parted to the helium atoms at the many elastic collisions*. The correction due to the first two causes was not more than a fraction of a volt, except when the tungsten was heated near * Benade and Compton, Joc. crt. of Radiation by Electron Impacts in Helium. 561 its melting-point, while that due to the third cause was not important except at the two higher pressures. Although the corrections were actually measured in some cases, it was found much easier and just as satisfactory to simply assume that the correct value of Vq for the first break is 20°2 volts *, and to adjust every curve to this point. Fig. 5. 9=0'05 mm: Fo = 15 (1G)>" amp. 350 3 Jo eS eine an ie a el ak ei ti avr Se i a emia hh i — Sec Ae jeceeeene7-uens , ard a il PCE ESSER S) Se SRR Se ee se ies a eed I i 4 Jak Re Siisiee 20 22 24 2 28 30 32 Va y (Volts) Figs. 2, 3. and 4 show that there are two “ break” points in the H-V, curves, indicating two critical potentials or energies. The values of these critical potentials are 20°2 volts and 25°5 volts respectively. The R-V, curves show that between 20:2 volts and 25°5 volts the effect is due largely to radiation, whereas the effect at 25°5 volts marks the setting 1n of intense ionization. These results are, therefore, in entire * This value is chosen as a mean between the values given by suc- cessive ‘‘breaks” in two electrode tubes, and by the corrected first “ break” in tubes with three or more electrodes. 562 Prof. K. T. Compton on Lonization and Production | confirmation of the recent work of Horton and Miss Davies and of Franck and Knipping. | Even with intense ionization, there still remains an appre- ciable photoelectri ic effect on the electrode, evidenced by the fact that the ratio R= H¢/Kg approaches a value not far from 1:06 instead of unity as vi is indefinitely increased. This is exactly, as would be expected, due to the continued presence of 20-volt radiation and other radiation which may accompany recombination. Fig. 4. p=017 mm.; L, »=0°6 (10)~° > amp. BERESREMAERS BREE ERENERSS CECE EEA Seon ee 700 ee, a A ae cae : NSE NSECEE ae CHE asee XC (7 SRE A 26 2 | (Volts Between 20°2 and 25:5 volts it is seen that the ratio R decreases from 1:9 to about 1:08 as the gas-pressure is decreased trom 0:0005 to 8 mm. ‘This means that, as the pressure increases, the proportion of ionization increases, as shown by Table I. Here 2/r is the ratio of the effect due to ionization to that due to radiation. The exact values of i/r are of no particular significance, since they will vary according to the construction of the apparatus, but the variation of 7/7 with p is important. 1.00 wW of Radiation by Electron Impacts in Felium. 563 Fig. 5. p=80mm.; I,.=0'85 (10)~° amp. 280 “ATE ee sb | tyme islekas = Ea isiete psd = a Posie oe a 200 eee oa ied be ell bare OL = CS ce ae Pe ere ae ee | ° Babedimlceadarse. | J ss Palas tie] Utena tar | pe CREE eee veka 2S eee mee ey ef roe | sft “pod of af air A 24 2 6) 2 5 30 oe 34 a Volts) TaB.e [ c=0°50 p (wmm.)., R. a/?. 00005 1:90 0055 0-001 1°81 0-116 0°003 1°74 0°176 0'012 1°60 0°333 0-015 AMF 0°3878 0-044 1:55 (410 O17 1°40 Ofe 1:00 1-19 2-22 8:00 1:07 6:3 25-00 1-04. 11-4 The effect between 20°2 and 25°5 volts apparently ap- proaches one of pure radiation as the pressure is reduced. It seems obvious, therefore, that this effect must be one of pure radiation at any pressure. For in an atomic catastrophe so intimately related to the inner structure of the atom it is = 964 Prof. K. T. 0 Compton on Lonization and Production inconceivable that other atoms at the large relative distances involved in these pressures could change the entire nature of the effect of an electron impact. The observed ionization must therefore be a secondary effect due, at least in part, to the radiation, or it must be due to electron impacts against atoms in such rapid succession that the energies of the tee are additive in their effect. The latter possibility may be discarded, owing to the extremely long time during which an atom would have to retain energy from an impact in order to account for the observed results at these low pressures and current densities*. No secondary effect of the radiation which has been suggested, such as lonization by electrons photoelectrically emitted from the filament or the electrode E, seems at all adequate to account for the ionization and the variation of the proportion of ionization and radiation with pressure, except that which ascribes the ionization to electron impacts against atoms which, at the instant, are in a relatively unstable condition because they have absorbed radiant energy coming from neighbouring atoms which have been previously struck. In fact, some such phenomenon must occur, since the energy radiated is strongly absorbed and re-emitted as resonance radiation, aud is thus passed on from atom to atom. At the higher pressure it is obvious that no appreciable ionization can occur asa result of single impacts, since the electrons, however and wherever emitted, would collide so frequently while gaining the energy between 20°2 and 25°5 volts that they would certainly collide inelastically and produce 20°Z-volt radiation before acquiring the 25:5- volt energy necessary for ionization. Thus practically all ionization observed at higher pressures, whatever be the applied voltage (unless this is so large as to give a potential drop of the ‘order of magnitude of a volt in a mean free path), must be due to a secondary effect involving combined action of radiant energy and direct impact. This point is well illustrated by fig. 5, for which the pressure was 8mm. Here neither curve shows any indi- cation of a break at 25:5 volts, while the R—V, curve shows that the effect from 20-2 volts up is very largely due to ionization. ‘hus this entire curve, in which the actual ionization currents were very large indeed, seems due to 20:2 volt impacts. The shapes of these curves differ from those for lower pressures. The maximum and succeeding decrease in the H—-V, curve is due to the fact that, as Va increases, more and more of the ionization occurs on the * K. T. Compton, Phys. Rev. (in print). of Radiation by Electron Impacts in Helium. 565 filament side of the gauze, and the ions do not reach the electrode E. The rise in the R-V, curve as V, increases is due to the fact that, whereas fewer and fewer ions reach the electrode Hi, the radiation from impacts and recombination still strikes it. This risein R is therefore due to a peculiarity of the apparatus, and does not indicate a real increase in the actual proportion of radiation. If the curves were extended to higher values of V,, the successive maxima at 20-velt intervals would be shown. The question of the effect of radiation or ionization pro- duced by impacts against the gauze has frequently been raised. Horton and Miss Davies* concluded that the platinum gauze emits positive ions, this emission beginning at about 13 volts and increasing as V, is increased. This point was tested by thoroughly baking out and evacuating the apparatus, and making tests similar to those made in the presence of helium. Since the effects were exceedingly small, the thermionic current was increased to the limit of safety for the filament. The results, after correcting V, for emission velocities, are shown in fig. 6. There is a very Fie. 6. p=0-0 mm. ; I,,=205(10)7° amp. 24 2 . Va (Volts) small “effect beginning at about 15 volts and increasing considerably beyond 20 volts. The effect is seen to be almost entirely one of ivnization. These ionization currents are so small in comparison with those observed with helium in the apparatus that they are entirely negligible except in the work at 0:0005 mm. and 0:001 mm. pressure. In these cases corrections to the observed values of E were made by * Roy. Soe. Proc. A, xevii. p. 23 (1920), 966 Prof. K. T. Compton on Jonization and Production subtracting the corresponding values of E for zero pressure, allowance being made for the difference in total currents [in the respective cases. If ionization between 20:2 and 25:5 volts is to be aseribed to the combined effect of radiant energy and energy of impacts, it is to be expected that, as the thermionic current I is increased, the amount of ionization should increase relatively more rapidly than the amount of radiation. ‘That’ this is true is shown by fig. 7: The rate of variation of R- with I, however, is not as greatas might be expected. Fig. 7. p=015 mm.; Curve l, Boa 15 (10)~° amp. ; Curve 2, I, eae: 5 (10) ” amp. 50; 82 24 Semis sso 9132 1.00 Va (Volts) &% The curves of figs. 2, 3, 4, and 7 show a fairly constant value of R between 20°2 and 25°5 volts. This is not generally the case, however. Owing to the increasing values of I as V, is increased, the value of R tends to diminish. On the other hand, owing to the increased proportion of the effective collisions which occur on the filament side of the gauze G as V.is increased, the proportion of ionization (as. measured by I) tends to deer ease, and consequently the value of R to increase. Neither of these variations is very large, and it was always possible to find a value of Lat which they practi- cally neutralized each other. Table I. is based on valjies of R taken from such cases where R remained practically uniform within the specified range. An attempt was made to express analytically the proportion i/r to be expected on the theoretical grounds outlined above, but there are too many uncertain factors to make a satis- factory analysis possible. There are involved the coefficient of absorption of the resonance radiation in the gas, the time during which radiant energy of impact is retained by an atom, the probability of ionization or inelastic impact at of Radiation by Electron Impacts in Helium. 567 collisions of various speeds, ete. Itis probable that the low- voltage arc in helium, which is now being studied, offers a better opportunity to put the theory in suitable form for tests of a more quantitative nature than can be applied to the experiments of the present paper. Can the Observed Ionization between 20°2 and 25:5 volts be due to Impurities in the Helium ? Franck and Hertz have accounted for ionization between the resonance and ionization potentials by a photoelectric effect of the 20-volt radiation on gaseous impurities. The only impurity detected in these experiments was a slight trace of neon, except for the possible presence of minute quantities of water-vapour, which were shown to be too small to affect the results. Neon is the only substance which could complicate the results without its effect being easily separable from that due to helium, since its ionization potential, 19°5 volts, is so near the resonance potential of helium that the two effects may merge into one another. Unfortunately, neon is that impurity most difficult to remove from helium. The helium used in this experiment was originally supposed to be very pure. It was subsequently, in the course of this and other researches, purified by several hundred fractionations from coconut charconl'i in liquid air. No neon lines could be observed in its spectrum except with a very intense are main- tained by a thermionic current from a tungsten electrode. There are three considerations which seem to disprove the appreciable influence of neon in the above experiments :— (1) There is no evidence of radiation or of inelastic impacts due to collisions with neon atoms at velocities below 20 volts. If neon were present in sufficient quantity to account for the observed ionization, we should expect to detect its presence by the effect of inelastic collisions at its resonance potential below 20 volts, especially at higher gas-pressures where the chances of collisions with neon atoms are greatly enhanced by the enormously increased paths of the electrons resulting from the elastic impact w ith helium. (2) Since the ionizing potential of neon is a little less than the resonance potential of helium, we should expect the first effects detected as V, is increased to be due entirely to ionization of neon, and therefore to find the value of R near to unity for the first measurable currents, increasing when V, passes the reso- nance potential of helium. Such a variation of R has not been observed. (3) No variation in the gas-pressure should alter the proportion of the effect due to ionization of neon in comparison with that due to radiation from helium so long . 568 Lonization by Electron Impacts in Helium. as the electronic mean free path is greater than the distance between electrodes, or even if it is considerably smaller, It is only when an electron is likely to collide while its velocity increases by an amount equivalent to the difference between 19°5 and 20-2 volts that the influence of the neon should increase in proportion to that of the helium as the pressure increases. In the actual case, however, the ratio z/r changed more than tenfold with pressure—at pressures where the difference in probability of collision at 19°5 volts and at 20°2 voltsis absolutely negligible. EFurthermore, Richardson and Bazzoni have shown that the amount of ionization at multiples of 20 volts is about what would be expected if every electron, colliding againsta helium atom with energy greater than 20 volts, should directly or indirectly liberate an addi- tional electron. This amount of ionization could not, therefore, be attributed to collisions with atoms of an impurity present in small quantity. The first two objections above apply equally well to disprove a direct photoelectric action of the helium-resonance radiation on neon atoms in suificient amount to explain the results. Richardson’s and Bazzoni’s work, together with the observed passage of resonance radiation through the gas to produce a photographic effect on the electrodes, also proves the inadequacy of an expla- nation based on a photoelectric emission from neon. Note added to proof.— More recent experiments have been made in helium so pure as to show no trace of impurities under any conditions of exciting the spectrum. These experiments differ in no way from those described above. Summary. 1. An experimental method is described for distinguishing between ionization and radiation and of estimating the pro- portion of either when both are present. 2. Resonance radiation sets in at 20:2 volts and ionization at 25°5 volts. 3. lonization is observed between 20:2 and 25:5 volts, in proportions increasing with tne gas-pressure and with the bombarding current density. Hvidence is presented to show that this ionization is a secondary effect, due to impacts against electrons which contain absorbed radiant energy of the resonance radiation from neighbouring atoms. ‘This method of ionization appears to be very important at high gas-pressures. Princeton, N. J., U.S.A. April 1, 1920. [ 569 ] LXVIT. The Rate of Chemical Action in the Crystalline State. By C.N. Hiysuetwoop and E. J. Bowen, Balliol College, Oxford*, ITH regard to the study of reactions in the solid state, a number of interesting observations on chemical changes produced by the action of light have been recorded, e.9., Lobry de Bruyn, Ree. Trav. xxii. 298, and Padoa, Atti R. Accad. Lincei, 1919 (v.) 1. 372, but these do not lend themselves to exact met ek and, moreover , the intensity of the active light falls off rapidly in the interior of the erystal. Apart from these investigations little is known as to the mechanism of chemical reactions in the solid state. Some previous experiments (Trans. Chem. Soc. 1920, exvil. 156) on the rate of decomposition of malonic acid in the solid and supercooled liquid state, showed that the decomposition was much more rapid in the supercooled liquid than in the solid at the same temperature, and that it was found difficult to obtain a definite value for the rate of reaction in the solid state, large divergences being found among specimens of apparently equal purity. As these discrepancies might have been connected with the presence of small quantities of liquid, the subject has been investigated further by studying the decomposition of various cry stalline substances which do not melt, or at temperatures far below their melting points. Irreversible reactions were naturally chosen. The method of measurement used consisted in the determi- nation of either the volume or the pressure of the gaseous products. The substances were enclosed in bulbs connected with gas-measuring apparatus and heated at constant tem- perature in vapour “baths. 1. Evolution of Oxygen from Potassium Permanganate. The curves given in figures 1 and 3 show that the initial rate of reaction is determined by the state of subdivision of the potassium permanganate, being greater the finer the state of subdivision. In the case of the lar ge crystals the curve shows a marked acceleration owing to the disintegration of * Communicated by the Authors. Phil. Mag. S. 6. Vol. 40. No. 239. Nov. 1920. ag Per cent. decomposed. 570 Messrs. Hinshelwood and Bowen on the Rate of the crystal and consequent increase of surface as the reaction proceeds. Clearly, then, the reaction is a heterogenous one and takes place only in a zone near the surface. The final retardation is of the nature of a discontinuity due to the exhaustion of the reserve of material ; but is rounded off owing to the earlier extinction of the smaller particles. Experiments VIII. and IX. show that manganese dioxide exerts no catalytic action to account for the acceleration, which must be due entirely to the physical cause. to 20 30 40 50 GO MINUTES. I. Finely ground KMn0, (agate mortar) at 240° C II. Finely sround KMn0, (porcelain mortar) at 240° C. Ill. Large crystals KMn0, at 240° C. 2. Evolution of Oxygen from Solid Solutions of Potassium Permanganate in Potassium Perchlorate. When potassium permanganate is dissolved in solid potassium perchlorate the rate of reaction is lowered. Figure 2 shows the results obtained with mixed crystals containing 82 per cent. and 26°8 per cent. of potassium Per cent. decomposed. Chemical Action in the Crystalline State. STE permanganate respectively. Crystals containing only 6:7 per cent. of potassium permanganate gave no measurable evolution of oxygen under the same conditions. ‘The solid solution crystals were in each case fine and of uniform grain. 160 MINUTES I. Large crystals KMnQ, at 239° C. II. | Solid solution of KMnQO, and KCI1O, Ul. ( (82% KMnO,). Small crystals at 239° C. IV. Solid solution of KMnO, and KCIO, (268% KMnO,). Small crystals at 239° C, 3. Decomposition of Ammonium Bichromate. Figure 4 shows the type of curve obtained with finely ground ammonium bichromate at 219°C. In connexion with ammonium bichromate a further point of interest was noticed. When a large erystal was heated at 212°C. it did not disintegrate, but the chromium oxide formed remained as a coherent film, and when the whole surface was covered the reaction ceased or became very slow indeed. ‘The interior of the crystal was unchanged. (C/. an observation of Ball, Trans. Chem. Soc. 1909, xev. 87.) 2B 2 Per cent. decomposed. Per cent, decomposed. 72 Messrs. Hinshelwood and Bowen on the late of 200 250 MINUTES, T Finely ground KMn0O, + Mr; at 220° 0. III. Finely ground KMn0O, at 220°°5 C. IV. Large crystals KMnO, at 217°C. Fig. 4. 350 MINS, Ammonium Bichromate at 219° C. Chemical Action in the Crystalline State. 573 4. Decomposition of Tetranitroaniline. The similarity of the curves in figure 5 to those previously given is apparent. Experiment XIV. (Curve I1.), tiie most fortunate of several trials, was sufficiently precise to allow not only a ¢, z, curve to be drawn, but also an accurate ~ Fig. 5. 50 loo 150 200 “£50 3oo S2 Hours. L. Tetranitroaniline at 140° C. Il. Tetranitroaniline at 120°-0 C. ax UII. Derived curve from II. Ordinates ai (different scale). Frmeren ke Small crystals. derived velocity curve, since the lower temperature, 120-0. was more easily controlled, and in this particular experiment did not vary by more than ,\,°C. In the figure the derived curve (Fe e) is plotted on a different scale from the other two curves. It will be observed that the velocity increases steadily almost throughout, falling away sharply near the end . 074 Messrs. Hinshelwood and Bowen on the Rate of of the reaction. This illustrates the point mentioned above that the final retardation is virtually the result of a dis- continuity due tv the exhaustion of the material. The Form of the Curves. The equation of the change may be expressed thus : dx , an k fa + f(a@)f. a+ f(#) represents the magnitude of the zone in the crystal where decomposition takes place. It will depend presumably on such factors as tle nature of the space lattice and the closeness of packing. & is a factor representing the probability of a molecule in that zone acquiring sufficient energy to break up. The expression differs from the usual type in containing no term (c — «) to represent the decrease in active mass with time. The actual form of the function /(#) cannot be arrived at theoretically. Nevertheless by putting f(#) = ba”, we may determine how n varies with # and thus form an idea of the law of acceleration. Empirically it is found that nm is Frequently almost constant. lf then dx da : log cas eal i =n log «+ tog B. Plotting the values of log w and log {7 -(¥) : | i Fr | from the derived curve for tetranitroaniline a straight line was obtained, the slope of which gave n=0°85, so that dx — =A + Br, dt * A few other cases were tested. The derived curve was not actually drawn, but an empirical equation was fitted to the accelerating parts of the ¢, 2, curves, the tangents at various points were calculated and then the same procedure Chemical Action in the Crystalline State. 575 followed as above. TF'rom experiments V.and VI. the values of » for the 82 per cent. solid solution were 0°62 and 0°64. The large crystals of potassium permanganate give values of n which vary considerably with w, n being greater than unity for small values of # and diminishing as x increases. Influence of Temperature. The influence of temperature is complex since both & and a+j(wv) vary. The variation of k is the true temperature coefficient in the ordinary sense of the term. If for the same substance in the same state of subdivision a+/(#) could he assumed not to vary with temperature, & could be directly determined. In this case the curves for two temperatures should be similar. But the figures given below show.that this is not so. Potassium Permanganate. Tetranitrosniline. zw. bo40° borg: Ratio 2 wv. t, 10° tioo: Ratio 120 | 240 ; L140 5°45 14 50 3°6 79 19:1 94 4°9 15°4 20°3 79 3°9 12°1 22°2 118 a3 26°2 26°7 110 4-1 24:4 27 167 62 40-0 30 140 49 43°5 32 214 67 56°0 33°95 170 51 57:8 35°5 239 67 65°0 39°9 185 52 73°3 39°5 262 6°6 85°0 41:2 230 56 Extrapolated toz=0 34 Extrapolated to r=0 4:4 If we could assume that a were constant, then the above ratios extrapolated to z=0 would serve to measure the true temperature effect. Conclusion and Summary. When reactions take place in the crystalline state the change appears to be confined to those molecules in the neighbourhood of the surface. Where a progressive dis- integration of the crystal structure takes place the change is strongly accelerated. Solution in another solid causes a reduction in the rate of reaction. It is clear that the molecules in the interior are under some kind of restraint. This may be connected with the fact that in the interior the molecules are bound by valency forces on all sides, or it may be referred to the internal pressure. The two points of view are probably equivalent. We have pleasure in thanking Brig.-Gen. H. Hartley for the interest he has taken in these experiments. D576 ~=Messrs. Hinshelwood and Bowen on the Rate of A. Potassium Permanganate at 240°. ; Il. | III. Very finely ground in agate Ground. | Large crystals. mortar. ‘Temperature 240°. Temperature 240°. | 240° Time in Per cent. |Timein — Per cent. | Time in Per cent. minutes, decomposed. minutes. decomposed. minutes. decomposed. : iL: | t. iti | t. De 4 161 | 4 11-5 | 5 LO 7 23°0 6 166 10 2°4 12 30°6 10 24-4 Pale 6°3 17 472 14 32:0 | 20 116 22 59°8 20 41:5 M5) 20°2 27°5 72°5 27 54:2 30 40°9 33°0 83:9 30°D 69°2 | ob 63:0 40 94:1 43 92:0 40 81:8 47 98°9 50 99-0 | 45 92°4 50 98 55 99 See Fig. 1, Curve I. | Fig. 1, Curve IT. Fig. 1, Curve III. B. Solid Solutions of Potassiwm Permang ganatein Potassium Perchlorate at 239°. IV. Ve Large crystals of KMnO, | Solid solution with 82% (for comparison). KMn0O,. Time in Per cent. Time in Per cent. minutes. decomposed. minutes. paper t. x. c. 10°5 50 9° 9: 6 19 12:2 18 6°38 28 25°0 28 13°5 33 36°4 51 32°8 37 496 60 409 39 55:2 70 52°2 44°5 1158 82 68:9 49-5 83:0 94 83:0 o4 93:0 104 91:0 60 968 128 97-7 73 99°3 160 99°5 Fig. 2, Curve I. Fig. 2, Curve II. 3 VII. Solid solution with 827 Solid solution with 26:8 MnO,. KMnO,. Time in Per cent. | Time in Per cent. minutes. decomposed. minutes. decomposed. t. x t. ae 9 2:2 100 9:5 21 5:1 160 14-7 32 11-0 210 20°6 39 16°6 | 250 25:2 47 22:0 300 321 61 34°9 360 374 71 46°6 700 873 82 61:5 98 80:0 110 92:0 130 97:0 145 98:0 Fig. 2, Curve IIT. Fig. 2, Curve IV. Chemical Action in the Crystalline State. C. Experiments at lower temperature. ay: VELL Potassium permanganate finely ground with MnO, (220°). Time in Per csnt. minutes. decomposed. t. x. + 8°9 9 16:6 17 25°5 27 33°6 39 42-9 52 51-2 67 61-4 82 viel 95 78°4 115 86°9 138 93-0 166 97-7 Fig. 3, Curve I. X. Finely ground KMnO, (220°°5). Time in Per cent. minutes. decomposed. t. x. 11 18:1 30 282 40 35°2 52 44:3 63 50°5 75 56'3 98 66°6 Nit 720 125 781 152 90-2 182 96°35 225 99-0 Fig. 3, Curve ITI. IX. Similar to VIII. Time in Per cent. minutes. decomposed. a ibe 7 86 19 19°4 35 29:0 45 ODS 5) 41°2 67 49-4 75 559 87 65'0 105 782 135 93°3 150 93:0 Fig. 3, Curve IT. XI. Large crystals of KMnO, (CAW Ep Time in Per cent. minutes. decomposed, t. Z., 15 14 40 35 50 5h Ae 15°4 95 20°3 110 26°2 130 348 140 40°0 150 44:0 170 56°0 185 65:0 200 743 230 85-4 243 91-6 Zoo 99°6 Fig. 3, Curve IV. Ammonium Bichromate at 219° C. Time in Per cent. minutes, decomposed. 2-5 x XII. Time in Per cent. minutes. decomposed. tf . 127 67:2 136 72-0 149 19°7 164 82°7 194 89°5 219 ie Pay 94-6 307 96°1 332 97°9 352 98°5 Fig. 4. - _ rv] i | DTS Mr. H. D. Murray on the K. Vetranitroaniline. XIII. XIV. XY. At 140°. At 120:0°. Time in Per cent. | ‘Time in Per cont, | Derived curye from X1V- hours. decomposed.| hours, decomposed.) t. x, t. x. v dat 3 1:0 Zi 0:93 < dt” 5 15 45 2°29 D "044 21 10°5 70 4°56 16 057 24 160 04 7:89 | o'4. ‘091 28°5 29:5 118 12°12 6:2 139 43, 87-0 142 17-27 10°0 "176 49 95°5 167 24:40 159 214 6 99:0 189 32:08 20°38 286 725 100 214 43°47 28°2 "349 79 100 239 57°76 37°8 "456 262 73°29 50°6 ‘572 287°5 94:3 65:5 :- “672 310°5 98°4 83°6 "824 330°5 99°1 96°4 179 337 100 99-6 ‘04 3545 100 | 100 0 Fig. 5, Curve Fig. 5, Curve IT. Fig. 5, Curve IT. LXVIIIl. The Precipitation of Colloids. By Humpurny D. Murray, Hehibitioner of Christ Church, Ouford *. HH author has recently attempted to obtain a general expression for the precipitating effect of ions on colloids. Whilst the attempt has not at present been successful in discovering a general relationship, a special case has been found for which the expression reduces to a simple equation. ‘This, as derived, is applicable only to the effect of univalent cations with the same anion upon the same colloid under equal conditions of temperature and concen- tration, the colloid carrying a negative charge. It relates the concentration of the cation in gm. atoms required to precipitate the whole of the colloid in a given time, with the atomic number of the cation, and is of the form C= lo. N®: where C represents the concentration of the cation, N represents the atomic number of the cation, nis constant for the colloid at that particular con- centration, K is a constant depending upon the nature of the colloid and the anion. * Communicated by the Author. Precipitation of Colloids. 579 In order to test the validity of the equation it was necessary to plot values of log N against values of log ©. It was difficult to find a series of concentration values of sufficient length to make an experimental verificatien possible. Failing an independent determination of a series of values for which the author had not sufficient time, an attempt was made with some results of Oden’s, given below, on the precipitation of colloidal sulphur, the latter being negatively charged. ‘he left-hand column shows the minimal concen- tration of cation in gm. atems per litre required to effect precipitation in a given time. The anion in every case is Cl’ :— TABLE I. Colloid :—Sulphur 18°-20° C. Minimal ras Minimal ae . concentration | fen concentration Atomic Ents Cation. | ‘ 4 Minimal : ; Atomic | with Cl ne te with NO,’ | number. hae | anion. \Speven ration. anion. humoer. / eer. PS 6:0 le 0-778 == Mee ee a ee ee ‘913 —-1-960 = a ee Li | ees S153 1-185 163 11 1:041- | er ‘0044 3643 ae eens 1-114 ee 021 2322 022 19°} 1-279 a eae “0041 3613 — 20.) 130h~] ere 008, /| . 3908 ~ 29 1462 | ee ‘016 [> _ #90# ‘0175 37. | «1:568 Bis ies Ve: ‘009 | 3-954 ‘0096 55 | 1740 fae i yok. de: 0021 3°322 — 56 | 1-748 | On plotting the values of log minimal concentration of the cations associated with Cl’ anions (C) against log atomic number (N), it will be seen that the values of the univalent ions, with the exception of that of potassium, lie approxi- mately upon a straight line to which the equation is O,=5:°9N7™. In the case of the divalent ions the values are too few to be of any use in deducing a relation. If, however, we assume that the equations will be of the same form, 2. e. that the expression N~'™ is constant for sulphur, and that barium 580 Mr. H. D. Murray on the and aluminium are typical divalent and trivalent ions, we obtain the equations Oo = 46 Nea CO, = roNT where C, and C3 are the minimal concentrations of divalent and trivalent ions respectively. We shall return to these Fig. 1. eRa" . Hi 2:0 | 1-0 0-8 Log C results later. The concentrations of cations with NO,’ anions are given in some cases to show that they follow approximately the same rule. Precipitation of Colloids. 581 The next series of concentration values were taken from some data by Freundlich, corrected for dissociation, so that, as before, the minimal concentration of cations per litre is shown required to precipitate colloidal As.S;, at a concen- tration of 007539 mols. per litre. The anion in every case is Cl’. The colloid is negatively charged. TABLE IT. | | | Cation. | Pee cris, Minital Boers | Atata concentration. | number. BA nds. | 029 3462 | led diaiee ( ii Soe PinOBI SN, aha deh BIRO. piadhyh Bole. Eyer AIT : the current is limited by the space-charge. If V is increased sufficiently, (fered CAS —_ Bure Disappearance of Gas in the Electric Discharge. 591 the current becomes saturated, and reaches a value (i)):deter- mined wholly by the thermionic emission and not by V. The relation between 7 and V in this condition is shown by curve I. in fig. 2, where 7° is plotted against V. The curves Fig. 2. L% Relation between current (i) & potential (v) at different pressures Vv in this figure are not drawn accurately, but merely represent qualitatively the charges that occur. If a little gas is present, curve II. is obtained. It leaves curve I. at the ionization potential of the gas (Vo), and attains the saturation current at a value of V less than before. But the saturation current reached is the same, because the current conveyed by the ions formed from the gas is an inappreciable fraction of that conveyed by the thermionic electrons. No luminosity appears in the discharge-tube before the saturation currrent is attained : if it appears at all, it is at potentials such as are used with a hard X-ray tube, which are much higher than the highest used in these experi- ments (600 volts). The pressure at this stage will be less than ‘001 mm. If the pressure is raised to (say) ‘002 mm., a new feature appears (curve III.). Above the ionization potential, the current increases still more rapidly with the potential, but before the saturation current is attained, a potential is ot 592 Research Staff of the G. E. C., London, on the reached at which there is a very sudden increase in the current. The change in the current at this point is discon- tinuous ; 7 increases from a value which is a small fraction of the saturation current immediately to the saturation current. At the same time, luminosity appears in the discharge-tube. The potential at which this discontinuous change occurs, accompanied by the deve'opment of the glow, will be termed the glow potential, Vg. Such discon- tinuous changes are obtained when the pressure is still so low that the ionization is an inappreciable fraction of the thermionic current and the saturation current is practically the same as in the highest vacuum. It has been said that the change is discontinuous. The best proof of this statement is that, on reversing the changes of potential, the changes of current are not reversed. If, after the potential has been raised above Vg and the glow started, the potential is reduced, the alae does not cease - im mediately, nor does the current fall below saturation. The dotted curve IIT.’ is followed. When V has been reduced to a lower value Vg’, tha discontinuous change is reversed, and the current falls once more sharply to the value which it had at the same vaine of V when the potentials were increased. When it is necessary to make a distinction in words between Vy and Vg’, they will be called the rising and falling glow potentials. If the pressure is increased once more to ‘05 mm., curve IV. is obtained. It now appears at first sight as if the current rose continuously with the potential, until a value greater than the vacuum saturation current was obtained ; it then continues to rise yet further, but more slowly. The rise of the current above the saturation value in a vacuum is, of course, due to the occurrence of ionization producing a number of eleotrons comparable with those emitted from the filament. At some point before the final slower rise sets in, a glow appears in the lamp, which is usually fainter and much more difficult to see than that characteristic of curve III. It was thought at first that this glow entered without any discontinuity in the curve, but later observations indicate that here also there is a point of definite discontinuity, and that it is at this point that the glow enters. In any case, the glow appears at that part of the curve where the rate of increase of 2 with V is greatest; the potential at which it appears can be determined with somewhat less accuracy than in curve IIL. ; it will be denoted again by Vg. When there is no appear ance of discontinuity in the curve, there is also no difference between Vg and Vg’; if the potential is reduced, the curve - INsappearance of Gas in the Electric Discharge. 593 appears to coincide with that obtained when it is increased. But once more there are indications that Vg and Vg! are not precisely the same, but that it is merely very difficult to detect ihe difference between them. As the pressure is decreased, the curves of type IV. shade continuously into those of type III. Observations on the Glow Potential. 6. Many of the facts just describedare well known. Several observers have noted that the glow enters suddenly at a definite potential accompanied by a large increase of current. In particular, Horton and Davies* have described the phenomenon in a recent paper. But it was found that the glow potential was so intimately connected with the disappearance of gas under the discharge that a large number of observations on it were made in vafious cireum- stances. A full account of the results and the conclusions to be based on thein is reserved for a later paper; here only the facts that are immediately relevant to our main purpose will be recorded. A few remarks may be made on the method of determining the glow potential. It has been said that the appearance of the glow and the sudden increase of the current occur at the same potential, and all experiments indicate that this state- ment is accurately true. But it is usually very much easier to observe the increase of current than the occurrence of luminosity, for (especially in hydrogen) the glow is often so faint that without special optical arrangements it is difficult to see it in the neighbourhood of the incandescent filament ; moreover, there is always the chance that the glow may occur In a region of the spectrum which does not affect the eye. Accordingly in all observations it is the increase of eurrent and not the luminosity that has been observed when accurate measurements are taken. At low pressures, corre- sponding to curve III., there is no difficulty in observing this increase in the form of a ‘“‘ kick” of the microammeter A, but there is an even more convenient method of deter- mining Vy. Since V was supplied through a_ potential divider of high resistance, the rapid increase of current causes a fall of potential across the electrodes s, and, owing to the difference between Vg and Vg’, this fall does not cause the glow to cease. Accordingly the method adopted we to increase V regularly and to watch the voltmeter; the pointer rises to a definite value and then drops back ; the * F. Horton and A. C. Davies, Proc. Roy. Soc. A, March 1920. D94 Research Staff of the G. E.C., London, on the maximum value attained is V *. Similarly, Vo’ can be determined by decreasing the voltage regularly and noting the minimum reading recorded, If the pressure of the gas remained constant during the observations (a condition by no means always fulfilled), Vg could be determined as. accurately as the voltmeter could be read. The deter- mination of Vy’ was somewhat less accurate and consistent. When the pressure is higher and the condition is that represented by curve IV., the determination of Vg is more dithicult ; the increase of current associated with the glow is much less. If the glow is visible it is best to watch ‘for it ; : if it is not, rough measurements of ¢@ have to be taken in the neighbourhood of Vg. It is not pase in these circum- stances to determine Vg to less than 4 volt, and it is rather surprising that the consistency of the result shows that even this accuracy can actually be obtained. It should be pointed out that the values of Vg given are always those between tne anode and the negative Koad of the filament when the glow appears. The remainder of the fila- ment differs less in potential from the anode ; and the true value of Vg, or that which would be found if the filament were all at the same potential, may be slightly less. Butthe difference must be a small fraction of a volt; for the true Vg will be the potential difference between the anode and that part of the filament, hot enough to give a thermionic emission which falls within the range where Vg is indepen- dent of that emission, which is nearest to the negative end. Accordingly the observed values have only to be corrected for the potential drop along the portion of the filament which is cooled by the leads. The whole drop along the filament was usually 8:4 volts, and since it never varied by more than 4 volt, the values obtained must be accurate within the error of observ: ation, which was never less than 0-1 volt. The first observations proved that the glow potential is independent of the temperature of the filament and of the thermionic emission froin it within wide limits. If there is no thermionic emission, the glow does not appear, of course, until the spark potential of the gas is reached, but the change from this condition to that in which the olow occurs at the very much lower potential obtained with a hot cathode is so rapid that it could not be determined certainly whether the change was continuous or discontinuous. Once the lower * V, determined in this manner is doubtless not exactly the same as that determined from the “ kick” of the current, unless the resistance in series is practically infinite. But the resistance was sufficiently great to. make any difference between the two values inappreciable to experiment, Disappearance of Gas in the Electric Discharge. 595 value is obtained, no further change occurs even if the thermionic emission is increased 100-fold. There is no evidence that the glow potential depends at all on the ther- mionic emission, so long as it is great enough to give at all a glow potential distinct from the spark potential. On the other hand, the glow potential depends greatly on the pressure, as has been indicated already, and on the nature of the gas. Fig. 3 shows the variation of Vg and Vz! with Fig, 3. : 2 a “ 180 : | Vg &Vg' for Argon | 160 140 Vg & Vg! volt 120 100 -——-—- 12) -005 N “Ol “015 p (mm) the pressure in argon, containing about 5 per cent. of nitrogen. (This gas is chosen because the difference between the rising and falling glow potentials is here measurable even at the highest pressures investigated. In the other gases it becomes inappreciable at pressure above 0-02 mm.) It will be seen that Vy, Vy’, and the difference between them all increase rapidly with decrease of pressure at the lowest pressures; at the higher they vary but little with the pressure. It is better, therefore, to plot 596 Research Staff of the G. E.C., London, on the the glow potentials against the reciprocal of the pressure (which is proportional to the mean free path) ; this method is adopted in fig. 4, which shows Vg for various gases. It is to be observed that the glow potentials of hydrogen are much greater than those ae nitrogen, carbon monoxide, or argon, and that the order of the glow potentials is roughly the inverse of the order of the molecular weights ; if the glow potential were determined only by the molecular weight, nitrogen would have the same glow potential as carbon monoxide : ; actually the glow potentials are not very different. At the lower pressures all the curves seem to be- come straight lines, and at the higher all appear to tend towards nearly the same limit. The observations shown on one curve indicate the accuracy obtainable in the best conditions; the irregularities appear to be due to errors in the measure- ment ae the pressure rather than to errors in Vg; if the pressure was constant, the determination of Vg could be repeated to a few fanaa 2 volt. Lastly, the glow potential depends on the form of the electrodes and of the vessel. A full consideration of this influence is reserved for later discussion; but it may be recorded here that the phenomena are changed completely if there is substituted for the wire anode a cylinder sur- rounding the filament, and that—contrary to what might have been expected—the glow potential at the lower pressures is apparently independent within wide limits of the size of the vessel, whereas at high pressures it depends upon that size. liffect of Impurities. 7. If the eases are not pure very different results may be obtained. The “ pure” hydrogen was pr ae ed by allowing the vapour of water (that had been boiled {o one-tenth of its volume in a vacuum) to act on ene sodium in a racuum ; the measurements were made after all possible precautions had been taken to free the apparatus from residual gas. ‘“‘ Hydrogen 99 per cent.” refers to gas taken from a cylinder; it had been prepared from water-gas, and contained about 1 per cent. of impurity, chiefly methane, carbon monoxide, and nitrogen; the remaining curve to the pure gas in the presence of the vapour of special “‘ vacuum ’ Wax. It is clear that the glow potential of hydrogen is very greatly affected by the presence of small traces of other gases. On the other hand, no change in the glow potentials Disappearance of Gas in the Electric Discharge. 597 of carbon monoxide or nitrogen could be made by special efforts at purification; atmospheric and chemical nitrogen agreed perfectly. The argon used contained about 5 per cent. of nitrogen ; pure argon has not been examined. The effect of mereury vapour, Ww hich was absent from all the gases of fig. 4, is especially remarkable. If the lamp is Fig, 4. 180 140 120 eee tt VL Lp al I Pressure & glow ren 20 o 100 200 300 400 S00 600 700 300 300 1000 1100 1200 Yp(mm—') p=co O-Olmm 0-005 0-002 0-00! completely evacuated of all permanent gases while mercury vapour is allowed to remain, the glow potential is found to be 32°5 volts. The pressure of this vapour is known to be about 0:002 mm.; so that the glow potential of mercury is very much lower than that of the other gases at the same pressure—a difference to be expected from its greater molecular weight. If, now, one of the permanent gases is admitted, there is no change in the glow potential ; so long as mercury vapour has access to the Jamp, the glow potential is always 32°5 volts, whatever the nature of pressure of the gas with which it is mixed. Such a result might be antici- pated. Since the glow potential decreases with an increase of pressure, it is not to be expected that the addition of any 598 Research Staff of the G. E. C., London, on the gas would raise the glow potential above that which would be obtained if the gas were not added. But it is important to notice that an impurity consisting of a gas or vapour, which is characterised by a low glow potential throughout the curve, may reduce the glow potential of a gas with which it is mixed far below that which the impurity would have in the absence of the gas ; the glow potential of a mixture may be lower than that of either of the constituents. Thus, if all the hydrogen were removed from the impure hydrogen from the cylinder, the residual gas, which would have a pressure only 1 per cent. of that of the original hydrogen, would have a glow potential a great deal higher than that of the mixture. Again, if mercury vapour in contact with frozen mercury (234° K. i. and not with re y at room temperature, is allowed access to the lamp, the glow potential, when all permanent gases are evacuated, is greater than 200 volts. Nevertheless, this same quantity of mercury vapour, when mixed with nitrogen at a pressure of 0-01 mm. having a glow potential of about 45 volts, causes a notable diminution of its glow potential; it makes the glow potential of the nitrogen almost identical with that of carbon monoxide (see fio. 4). Even when the temperature of the mercury was reduced to 200°K., its effect could still be detected in a ‘slight. but consistent diminution of the glow potential of nitrogen at low pressures. It may be added that the wax vapour which has so marked an effect on the glow potential of hydrogen had a glow potential greater than 600 volts in the absence of other gas. Since, as we shall see, the glow potential is a very im- portant factor in determining the disappearance of the gas, these observations, proving the large effect of very minute traces of impurity, are of great importance. The impurities that are important are those which, like mercury, are characterized by low glow potentials at all pressures; and the effect of the impurities is greatest in gases which, like hydrogen, have a high glow potential. The glow potential of hydrogen is probably one of the most delicate methods for detecting any common impurities, which are usually characterized by low glow potentials. It is not certain, of course, that the hydrogen of fig. 4 is absolutely pure ; all that can be said is that two samples agreed in giving the same glow potentials, and that these are higher than those of any other sample examined. The reduction of the glow potential by impurities is associated with the development of the spectrum of the impurities in the glow ; such an association is natural, for the Disappearance of Gas in the Electric Discharge. 599 low glow potential obviously indicates that the discharge is passing through the impurity as well as, or rather than, through the gas. Thus it is well known that the presence of mercury vapour is apt to mask the presence of all other gases, and that hydrogen is often concealed by CO. The spectroscopic observations in these experiments have been made so far with no instrument more effective than a small direct-vision prism pointed at the main body of the lamp ; but the great changes in the spectrum are visible to the unaided eye. Thus the glow in the purest hydrogen is almost invisible, much fainter than in any of the other gases examined. The minutest trace of impurity at once brightens the glow, and at the same time the spectrum of the impurity (mereury or carbon monoxide) appears. ‘The spectrum of carbon monoxide is not easily suppressed ; ; it is clear even when mercury at room temperature is present, though the mercury lines are also bright. On the other here such mercury entirely suppresses the nitrogen spectrum, so far as could be seen; and the mercury lines could be seen in the nitrogen spectrum even when the mercury was at 200°K. As the mercury is allowed to warm up, the eudien cessation of the red nitrogen glow and its replacement by the mercury glow, much fainter (to the eye), are very striking. The Glow and Tonization. 8. The facts that have been stated do not provide a complete theory of the physica] meaning of the glow potential, but a few obvious conclusions may be noted. There is no doubt that the appearance of the glow is accompanied by an increase of the ionization of the gas, but it should be remarked that the sudden increase of current, at the pressures of which curve III. is characteristic, does not represent merely the addition of current carried by ions from the gas. By far the greater part of this increase is due to the neutralization of the space-charge and to the passage between the electrodes of a greatly increased proportion of the electrons liberated by the thermionic emission of the cathode. That the glow is closely connected with the neutralization of the space-charge is indicated also by the relation between the molecular weight of a gas and its glow potential at a given pressure. A given number of ions will be the more effective in neutralizing the space-charge the less is their velocity : their velocity with a given potential will be the less the greater is their molecular weight. It is doubtless 600 Research Staff of the G. EH. C., London, on the for this reason that the gases of greater molecular weight have the smaller glow potentials, for the smaller potential doubt- less corresponds to the smaller number of ions. But as yet no simple nuinerical relation has been found between the molecalar weight and the glow potential at a given pres- sure: doubtless the mean free path and the ionization pobes nal of the gas are also effective in devel that relation. However, though it is not permissible to conclude that, if the current increases 10- or 100-fold at the moment the slow starts, the ionization increases in the same proportion, the - soe ne when there is a glow must be greater than when there is not. For the aeiriealtnersion ale the space-charge implies the emergence of more electrons from the cathode ; and, unless the number of ions produced by each electron decreases very greatly, the greater number of electrons must produce a greater number of ions. Such considerations are of importance if it is asked whether the appearance of the glow is merely an indication of a great increase of ionization or whether it represents the entry, of some new form of ionization. If the first alternative is adopted, it must be concluded that there is a glow even at potentials lower than the glow potential although it is too faint to be seen ; for at such potentials there is certainly some ionization. The matter can only be decided definitely ky determining how great is the increase of ionization, and whether it is great enough to account for a change from a state in which the glow cannot be seen with the most elaborate precautions to one in which it can be seen in bright daylight. The fact that the increase of current is not a measure of the increase of ionization shows that much more inquiry must be made before the matter can be decided ; but all the evidence that has been accumulated so far tends to show that the appear- ance of the glow is not merely due to an increase in the number of ions produced. On the other hand it is clear that the glow potential is not, like the ionization potential, a direct property of the individual atoms of the gas; it must also be a function of their mode of reaction with each otber or with the walls of the vessel ; for the glow potential is not, like the ionization potential, independent of the pressure. ‘There is no evidence, therefore, that the glow represents a new form of ionization of the individual atoms. Disappearance of Gas in the Hlectric Discharge. 601 Some Consequences of the Observations of the Glow Potential. 9. Before observations of a different nature are described it will be well to point out in what manner the giow potential, and in particular the difference between the rising and falling glow potentials, may affect the changes that are the main subject of investigation. It will be seen later that there are circumstances in which gas disappears in the discharge if V is greater than Vg and the glow is developed, but not if there is no glow. Suppose, then, that in these circumstances we are raising V so slowly that the gas has time to disappear under the discharge. If the initial pressure is represented by A (fig. 3) and V is raised from zero along the line NA, then, when A is reached, the glow starts. The gas begins to disappear and the pressure to fall; but the glow will not cease immediately. It will continue, while the pressure falls along the line AB, until B is reached and the potential is no longer able to maintain the discharge. Jf the potential is raised once more along BC, the glow will not start again until © is reached, and till C is reached there will be no disappearance of gas. But when C is reached the gas disappears once more, and the pressure falls along (D until the glow ceases at D. It will not start again till V rises to H, and so on. In suitable circumstances this process can be easily traced experimentally. As the potential is raised slowly, the glow in the Jamp flickers in a way familiar to all who have watched the “cleaning-up” of a lamp; and by measuring the pressure every time the glow ceases, it can be established that the points lie as they should on the full and dotted curves of fig. 3. The flicker also appears when the potential is kept constant, if gas is slowly leaking into the lamp from the walls or along a narrow connecting tube. The glow appears when the pressure rises to the point at which V is equal to Vg; it falls until V is equal to Va’, and the glow does not start again till the pressure has risen once more. A further consequence of the difference between Vg and Vo’ should be noted. The potential available for causing the discharge which produces the disappearance of the gas may be limited—for example, in an ordinary lamp it is limited to that which can be applied to the ends of the filament without burning it out. If the maximum potential is C (fig. 3), then so long as the initial pressure in the lamp is greater than that corresponding to C, the lamp can be Phil. Mag. Ser. 6. Vol. 40. No. 239. Vov. 1920. 2K - 602 Research Staff of the G. E.C., London, on the ‘“‘cleaned-up” by the discharge so completely that no further discharge will occur until the much greater poten- tial His applied. On the other hand, if the initial pressure is slightly less than C, the available potential will not start the discharge, and no “clean-up” can be obtained ; if at any subsequent time the potential is raised slightly above C the discharge will occur. It may easily happen that too low an initial pressure in the lamp is prejudicial to the obtaining of a complete “ clean-up,” and that the clean-up is more complete if initially a little extra gas is introduced. CHEMICAL KFFECTS OF THE DISCHARGE. The Disappearance of Carbon Monoxide. 10. The changes just described will not occur unless the circumstances are such that the gas disappears only when the glow discharge passes, and not when the potential between the electrodes is insufficient to produce the glow. They would not occur if the gas disappeared when there was no potential between the electrodes, or when the potential, though sufficient to produce some ionization, was less than Vg. There undoubtedly are such circumstances ; there are conditions well known in technical practice in which the “clean-up” of the lamp is accompanied by, and inseparable from, a marked glow, and the experiments of Langmuir give clear evidence of their existence; for he shows that in some gases—notably nitrogen and carbon monoxide—when the temperature of the filament is below that at which notable vaporization occurs, the gas disappears only * elec- trically ” and in consequence of the passage of a discharge. It should be observed that in all his experiments there must have been some “discharge,” for there was a difference of potential between the ends of his filament; and if that potential exceeded (as it often did) the ionization potential of the gas, it must have been accompanied by some ioniza- tion of the gas, and cannot have been a pure electron current. The fact, therefore, that he distinguishes between conditions in which there was a discharge and those in which there was not, shows that it is only some special form of discharge that is effective: the experiments described here, in con- junction with his observations, leave little doubt that this form was the glow. The matter has been studied more closely in the case of carbon monoxide, nitrogen, and hydrogen, both in the presence and in the absence of phosphorus vapour. It will Disappearance of Gas in the Electric Discharge. 603 be well to consider first carbon monoxide in the absence of phosphorus, for here the most definite and illuminating results have been obtained. The carbon monoxide was prepared from sulphuric and formic acids and stored over water ; it probably, therefore, contained a trace of air, but other likely impurities would be removed by the liquid-air trap T;. When the apparatas was arranged as in fig. 1, it was found that practically no change of pressure occurred when the filament was heated to 2000° K. so long as V was less than Vg. But as soon as Vg was exceeded and the glow appeared, the pressure began to decrease, if it lay initially between 0-1 and 0-001 mm., und the gas disappeared. A large number of observ rations were taken on the relation between the rate of decrease of pressure and the values of V andz; but, for a reason which will appear presently, it is not proposed at present to consider them in detail. In general, it may be said that the rate of decrease increased, as might be expected, when i was increased, either by increasing V or by increasing the temperature and thermionic emission of the filament. Further, an increase in V in general increased the rate of disappearance even if 2 was unchanged, though there were some exceptions to this rule; sometimes there was an optimum value of V at which the rate of decrease was a maximum. With the very crude spectroscopic arrangements employed, no lines except those of the ordinary carbon monoxide spectrum could be detected. The disappearance of gas continued in all cases until the pressure became so low that the potential applied could no longer maintain the discharge. V did not usually exceed 300 volts, and with this potential the glow ceased at about 0:0008 mm. If the potential were maintained after this limit was reached, a further slight decrease to 0°0006 mm. occurred, the ultimate limit depending on the potential ; the lowest limit was reached when V was about 200 volts, slightly higher limiting pressures being reached with either higher or lower potentials. Since some of the gas which disappears is undoubtedly contained on the walls of the lap, it is thought that this limiting pressure represents a balanee between the disappearance of gas and its evolution from the walls by the action of the discharge. G By admitting fresh gas to replace that which had dis- appeared, the process could be continued apparently without end and without any marked change in the rate of dis- appearance. A slight wastage of the filament occurred and the walls became slightly blackened, but no simple relation 2K 2 : 604 Research Staff of the G. E.C., London, onthe between the wastage of the filament and the disappearance of the gas could be traced. A fraction of the gas could be liberated again by baking the lamp, but the amount thus liberated appeared all to be absorbed in the early stages of the disappearance, and did not increase greatly with the quantity of gas that had disappeared. lt may be observed that there was no evidence of the accumulation of gas which would not disappear, and accordingly any impurities that the gas may have contained must have disappeared with the carbon monoxide. lf after a quantity of gas had been caused to disappear 1m the discharge the liquid air were removed from the trap T and the trap warmed to room temperature, a marked increase of pressure occurred, and much of the gas that had disappeared seemed to reappear. Experiments showed that the gas thus. reappearing was about half of that which had disappeared. The following table gives some of the results :— TABLE I. Decrease of pressure Increase on Rati a : atio, under discharge. warining, °0197 mm. 0116 mm. 0°59 “1258, ODO ay 0°47 0323" ,, 0168 __s,, 0°52 ‘0780 _,, (05500 0-42 "0445 _—,, 265), 0:57 0325" 4) 70200 __,, 0°62 "1020 _,, ‘0560 __,, 0°55 0:4348 mm. 0:2236 mm. 0°515 The first column gives the total pressure which would have: been exerted by the gas that had disappeared if it had been present in the lamp at the same time—that is to say, it 1s the sum of the decreases of pressure produced by the discharge when several doses of gas were admitted succes— sively ; the second column gives the increase of pressure that occurred on warming the trap T (or the side tube A ——see fig. 1) ; the third column gives the ratio of the second to the first. It is not certain whether the variations in this ratio represent experimental error or indicate real variations, but they justify the statement that about half the gas reappears. If the trap was cooled once more, the gas that had re- appeared disappeared once more completely. Accordingly it is clear that by the action of the discharge the carbon monoxide had been converted into some gas that condenses Lnisappearance of Gas in the Electric Discharge. 605 at liquid-air temperature, Carbon dioxide was naturally suspected. Accordingly the apparatus was rearranged so that the cooled tube in which the gas condensed could “be sealed off from the ioe of the appar: atus. Glas to the total pressure of 1:02 mm. was then caused to disappear in the discharge, representing a laid at N.T.P. of 1:34.¢.c. The sealed-off tube was opened under mercury and 0°74 ¢.c. of gas found to be present init. All but a few per cent. of this gas was proved to be carbon dioxide by letting up a solution of harium hydroxide and showing by chemical tests that the white precipitate formed when the gas was absorbed was indeed barium carbonate. The result was surprising. If a tungsten filament is heated in carbon dioxide, it is well- known that it absorbs oxygen and reduces the gas to the monoxide ; it could hardly have been expected that in the presence of so powerful a reducing agent as incandescent tungsten this action would be reversed. If oxygen had been present, the formation of some dioxide by combustion of the monoxide might have been anticipated ; but there was no free oxygen present in sufficient quantity, and the ratio of dioxide to monoxide indicates that the change is effected, not by the addition of oxygen, but by the abstraction of carbon: it appears to occur according to the equation 2nCO=nC+nCO,*. The destination of the carbon abstracted has not been traced completely. It is almost certainly not taken up by the tungsten, for the filament showed none of the known charac- teristics of a carbonized filament. It may be deposited on the walls and give rise to some of the blackening, but it is suspected that it passes io the nickel of the anode; for the anode was found to be blackened and the surface layer undoubtedly contained carbon, detected by burning it. Unfortunately, however, it was Soane later that the metal itself contained some ec: cba ; and the examination was not sufficiently accurate to determine whether the carbon found was in excess of that contained in the original metal. The loss of weight of the filament during the disappearance of the uantity of carbon monoxide mentioned was 0°45 mom. ; it is thought that this loss is merely due to the bombardment of the filament by the positive ions, or, in other words, merely represents cathodic spluttering ; as has been said, it * Tt is not easy to explain the occurrence of numbers greater than half in the third column of Table I.; those less than half may be due to gas absorbed on the glass. But it is not yet Certain that the deviations from half are real. 606 Research Staff of the G.E.C., London, on the did not seem to bear any simple relation to the carbon monoxide absorbed. } An attempt was made to cause the gas to disappear without the trap T cooled, but it was found that in these circumstances the discharge caused little or no decrease of the pressure, except perhaps when the pressure was less than 0°002 mm. But if the trap is not cooled, mercury vapour has access to the lamp, the glow potential is depressed, and the spectrum of mercury is well developed: as was said on p. 599, it appears that the glow is carried rather by the mercury than by the carbon monoxide, so that the absence of action on that gas is intelligible. Steps were therefore taken to exclude mercury vapour otherwise than by the cooled trap, but it was not easy to attain that condition. Gold-foil placed in the tube between the lamp and T' was quite ineffective in excluding mereury vapour*. Cooling the trap to some temperature above that necessary to condense CO, was an obvious course, but the experiments on nitrogen to be described later make it doubtful whether even at 200° K. the vapour-pressure of mercury is inappreciable for the present purpose. Moreover, it was decided to avoid having any part of the apparatus appreciably colder than the rest. Some kind of stop-cock seemed necessary, although it would involve the presence of wax or grease vapour. A pinch-cock of rubber tube with walls 1 em. thick was actually used between the lamp and T ; but it was necessary to cement the ends of the tubing with wax, which must have introduced some vapour. This vapour (see p. 598) was sufficient to change somewhat the glow potential of hydrogen, but it made no measurable difference to the glow potential of UO. Mor eover, it is known that the discharge through wax vapour always produges CO, so that it is less likely on this account that the vapour had any prejudicial effect on the observations. When the absence of mercury vapour was thus secured without giving the gas access to liquid air, it was found that under the glow discharge, carbon monoxide would still disappear (though less rapidly than before) at the higher pressures. But when a _ pressure between 0°008 and 0-009 mm. was reached, the decrease of pressure ceased although the glow was continued. If fresh gas was admitted * It seems that gold-foil excludes mercury only when the gas is at considerable pressure. Probably the mercury molecules stick to the gold when they strike it, but they also tend to evaporate again. The high pressure reduces the ev aporation. It is not asserted that the gold did not reduce the vapour-pressure somewhat, but only that it did not. reduce it enough for the present purpose. INsappearance of Gas in the Klectrie Discharge. 607 without pumping out the residue of the previous charge, the same limit was reached once more. The side tube A of the lamp was now cooled in liquid air in the expectation that CO, would have been formed and that much of the gas would condense. But it was a that there was no decrease of pressure due to the cooling, rather none that would not follow from the mere ihe e change of incondensible gas. Though the carbon monoxide had disappeared under the discharge it had not now been converted in any appreciable quantity into the dioxide, With A still cooled, the discharge was started once more, and now, as before, the pressure fell rapidly to 0-0008 mm., when the glow ceased. If the liquid air was now removed from A and the tube allowed to warm up, some of the gas was restored and could be condensed again by cooling A once more. But at the first trial the restored gas was found to be markedly greater than half that which had been removed in the presence of the cooled tube. The decrease in pressure during the discharge was 0°007 mm., the increase on warmin the tube 0:006 mm. Further investigation showed that We quantiry restored on warming A depended on the time that the discharge was continued while A was cooled, and that in suitable circumstances the gas resiored might be as great as _ half the total that had disappeared both with and without the side tube cooled. That is to say, if we start with gas at a pressure of p;, and, by passing the discharge with A warm, reduce this pressure to p,; further, with A cooled, mendes the pressure to p3; then the gas restored on warming A again may beas great as 4 (p;—p3) and not merely 3 (~.—ps) as would he expected. ‘The discharge can convert into carbon dioxide, so long as the tube A is cooled, not only the gas that disappeared while the tube was cool, but also the gas that disappeared while it was warm, although the dis- charge while A was warm did not at the time convert the gas into dioxide. Ceplanation of the Observations. 11. A very simple and, to our minds, plausible theory wi ill explain these facts. We have only to suppose that the effect of the discharge is to cause several reversible chemical actions to take place between the gas and the other materials in the lamp. Of these actions, one consists of the abstraction of carbon from the monoxide, resulting in its transformation into dioxide, together with its reverse, the combination of the dioxide w nif ‘carbon to form aninosidd: Another of 608 Research Staff of the G. E.C., London, on the these actions results in the production of some compounds of the carbon and oxygen, which are solid and are deposited on the walls or electrodes; it may be the formation of carbides and oxides of the metals of the electrodes, or it may be the formation of the compound WCO which Langmuir has detected. And this action again is accompanied by its reverse. The ultimate result of this complex of reversible reactions will depend, according to the accepted doctrines of chemical theory, upon which of the products of the reaction is being removed from the scene of action. If CO, is con- tinually being removed by access to a cooled tube, then the action in which CQ, is involved will proceed to completion in one direction, and all the carbon monoxide will ultimately be converted into CO,. If it is not removed, then, since the equilibrium concentration of CO, is very small, no appreciable amount of the monoxide is converted into the dioxide. On the other hand, the equilibrium concentration of the solid products is greater, and much of the gas is converted into them ; but since they are not effectively removed from the scene of action, but remain on the walls and electrodes, where they are still influenced by the discharge, the pro- duction of these products does not proceed beyond the equilibrium concentration; and if the removal of CO, is resumed, these products are decomposed once more and the ultimate complete conversion into CO, is again attained. Nor is there anything extravagant in supposing that such reversible reactions are proceeding concurrently ; for the evidence of positive ray analysis has shown that, when a molecule is ionized, there are usually or always produced free atoms of all the elements contained in it, bearing electrical charges of both signs. Accordingly in an ionized gas there are always present positively and negatively charged atoms of all the elements on the scene of action. In these circumstances it is only reasonable to suppose that there will be formed in some quantity any chemical com- pound that can be formed from any grouping of those elements ; some of the compounds may be formed in very small amount, and all of them, like the original compounds, will be broken up again after a short life. But this is precisely what is asserted when it is said that several reversible reactions are proceeding in the gas. If the discharge, when accompanied by any appreciable ionization of the gas, produces any chemical changes whatever, the changes that are produced would be expected to be precisely of the nature that is necessary to explain the observations which have been described. isappearance of Gas in the Electric Discharge. 609 The only objection to this view that can be suggested is that if ionization is all that is necessary to cause these actions to proceed, the gas should disappear before the glow discharge starts ; for, as has been insisted alreacy, there is ionization before the glow. But the difference may be merely one of degree * : the actions m: vy proceed before the glow, but so slowly that they are inappreciable. For, in order that the compounds may be formed, the charged atoms, which are a small fraction of the ionized molecules, have to meet. If n of them have to meet to form the com- pound, the rate of the reaction will vary as the nth power of the concentration of the ions. The greater concentration of the ions in the glow discharge would account for the far greater rate of chemical combination. Indeed, in order that the view offered should explain the facts, it is necessary to suppose that the actions do proceed to some extent even when there is no glow; for the conversion into carbon dioxide of the gas originally disappearing, when the tube A was warm, is effected by the continuance of the discharge at a pressure too low for the applied voltage to cause the glow. We must suppose that this discharge, in which the thermionic current is ver ¥ much greater than the ionization current, is capable of causing the pened to proceed, and of maintaining the change whereby the solid products are converted reversibly into CO,, which is condensed in the liquid air. The reaction proceeds at an appreciable rate because a high potential can be applied; at higher pressures it is not possible to apply so high a potential without causing the glow. Two further observations may be quoted in support of the view that the chemical actions involved are essentially reversible :—First, if, after the gas has been rest: red to the lamp in the form of CO, by warming the previously cooled tube A, the discharge is passed (of course, with A warm), its first effect is to increase and not decrease the pressure ; it is only after the discharge has lasted some time that the pressure begins to decrease again. The increase of pressure doubtless represents the conversion of the CO, back into CO and the tendency towards the equilibrium. concentration. Second, the rate at which the gas disappears in the presence * if (the hypothesis is not thought probable) the glow represents the incoming of some new form of ionization, it is just possible that it is only this form of ionization which is effe tive in inducing chemical changes. But the idea is not plausible, for many lines ot arcument— especially that based on work on “ delta rays ”——show that the ionization of a molecule is of the same nature by whatever agent it is effected. 610 . Disappearance of Gas in the Electric Discharge. of a-cooled tube depends very greatly on the ease of access to that tube. If A is cooled, the rate of disappearance in given electrical conditions is much greater than if T is cooled (actually T was separated from the lamp by about 40 cm. of tubing); and if the tube between the lamp and T is constricted, then (with A warm) the rate of disappearance is very much decreased. The rate of resultant reaction depends on the rate of removal of the product. It is because the rate of disappearance depends so intimately on the ease of access to liquid air that it is difficult to interpret at present the observations on the relation of that rate to the electrical conditions and that it is useless to give them in detail. In order to study this relation, conditions must be found in which this complicating factor is not present. Summary. 1. The research is an attempt to determine the nature and cause of the disappearance of gas under the electric dis- charge at low pressure. 2. A brief summary of Beta knowledge is given. It is not yet known what parts are played by absorption on the walls and by true chemical combination ; nor is it known precisely what electrical conditions are most favourable to the disappearance. 3. Though an incandescent tungsten filament forms part of the discharge vessel, the changes investigated are not those studied with oreat care by Lanemuir. 4. The apparatus is described. 5. Preliminary observations showed that the disappearance ip gas was closely connected with the appearance of the glow in the vessel. The electrical conditions in which the glow appears are described briefly. In any given state of the discharge vessel it appears sharply at a definite potential difference between the electrodes, called the glow potential. 6. Observations are described on the relation between the glow potential and (1) the thermionic emission, (2) the pressure of the gas, (3) nature of the gas, (4) the form of electrodes. | 7. Small amounts of impurities change very greatly the glow potential, especially in hydrogen. 8. The theory of the glow is deferred for later discussion ; but the general connexion between the appearance of the glow and the inerease of ionization in the gas is considered. 9. The great importance of the glow potential for inter- preting the rate of disappearance of | gas is pointed out. Triple Pendulums with Mutual Interaction. 611 10. Experiments on the disappearance of carbon monoxide in the glow discharge are described. 11. The results are discussed and explained by supposing that the glow causes a chemical change in the gas, which is reversible. The recognition of this reversibility seems necessary to explain the phenomena. The disappearance of other gases, illustrating other types of action, will be discussed in the sequel. Research Laboratories of the General Electric Co., Ltd., Hammersmith. April 16, 1920. LAX. Triple Pendulums with Mutual Interaction and the Analogous Electrical Circuits.—I1. By Prof. H. H. Barton, F.RS., and H. M. Browntine, J.Sc.* (Plates XI.-XII1] I. Inrropucrion. ECHANICAL analogies to two circuits coupled together have been used to show the type of vibrations set up by the latter. (See “ Coupled Vibrations,” Phil. Mag. Oct. 1917, Jan. 1918 and July 1918.) The mathematical theory of these circuits is comparatively simple, but the visual results of the mechanical analogy tend to elucidate and interpret the theory. In the case of three circuits or three pendulums which are arranged to act on one another, the theory is more com- plicated, although in certain cases it can be resolved into a somewhat simpler form. The experiments were carried out with a modified form of the apparatus shown in figs. 1 and 2 of *‘ Vibrations under Variable Couplings ” (Phil. Mag. Oct. 1917), and fig. 13, Plate V. “ Coupled Vibrations” (Phil. Mag. July 1918). An extra bridle and pendulum were added, and three con- nectors substituted for the single one, each of the three connecting two of the pendulums. Later it was thought that two connectors were sufficient, and the apparatus was modified to obtain a result which might be more amenable to analytical treatment. From the mathematical theory of three circuits coupled each to the other two, it was expected that the mechanical arrangement would give for each pendulum three superposed * Communicated by the Authors. 612 Prof. Barton and Miss Browning on vibrations. The periods of these vibrations were expected to be the same for each pendulum. The phases and ampli- tudes might, however, differ for each pendulum, thus giving very different resultant vibrations. The results obtained with the first form of apparatus were even more complicated than was expected from three super- posed vibrations, and on reflection this was seen to be due to the special arrangement in use. So it is proposed in this paper to describe this apparatus and give the figures obtained without dealing with its mechanical theory. Results obtained with a modified apparatus will be dealt with later. Il. THEory oF THREE MUTUALLY-INTERACTING ELECTRIC CIRCUITS. The electric circuits are supposed to have self and mutual inductances, and capacities with resistances negligible. Let L,, L,. Ls be the self-inductances of the separate circuits, 8,, S., S3 their capacities, and M,, M., M3; the mutual inductances between the seeond and third, the third and first, andthe first and second circuits respectively. Then the simultaneous equations of motion may be written as follows:— ad?x u a” as BO ay Evel 0 se Ng Oe al) ae S.. (dé ee Se dy y dz anu dee IS) ‘dt * dt?’ 2 2. z ome 2 d ww ef Kh M al Lb M d*y (3) “dt? where 2, y, and z are the charges on the respective condensers. The three couplings are given by coupling y toz, #= | u ztow, B= rt, Se RCO | (4) 2tOY, Y= Lal 4s ar eee Suppose as solution that ge Aer, Bee (ae hee oe eer” Triple Pendulums with Mutual Interaction. 613 Substituting these in (1), (2), and (3), we obtain 1 A ? a be: — Typ?) .« +p’ May +p?M,-=0, | pM; ot (5 —Lap*)y +p Mees Ue pes Pe (CG) pe Mer +p? My + (= —Isp?)e=0, ) &% Eliminating wv, y, and z from (6) by the method of deter- minants, we have ( = — Lp"), pMs, p’M2, | ee | | f / it s | pM, (<—lap), pM, |=0. © (1) | | pM, pM, (= —lnp’). This gives an equation of the sixth degree in p, viz. p®(2M,M.M;— L,L,L;+ L,M,?+ L,M,? + L,M,? ie 4 rL,L,—M,? LL, aa: M,” L,L, — M;? ie, a | Joba, US : ] are eles Pees 3 ak cee) «(SSS This is a cubic equation in p?, of which the roots may be written, p?=p,", po”, p3”. . Then p= ae + Po, F ps 5 SNE cinta) Aantal) but the negative signs may be disregarded as they introduce nothing new. Hence vibrations of three periods are set up. Thus the general solution of the equations may be written «=H, sin (p,t+e) + Fy sin (pot +61) + G, sin (pst +1), (9) and similar equations for y and <¢. If the circuits were vibrating each isolated from the others, then their vibrations would be proportional to sin lt, Sime and simnt, .<+..). (10) where ae cee aia ies ee ere) tea eee ‘ee ‘614 Prof. Barton and Miss Browning on Then using equations (4) and (11), equation (8) may be written ps(l— 2? — 8? - 9? — 2aBy) —pt| PO — 2) +m(1—8") +n?(1—y’)] + p?(?m? + min? + nl?) —Pm?n? =0.. (12) Thus it is seen that in general the vibrations after coupling the circuits differ from the free vibrations of the separate systems. The less the coupling the more nearly do the superposed vibrations approximate to those of the systems when free. III. ARRANGEMENT OF THREE CONNECTED PENDULUMS. The apparatus shown in fig. 13, Plate V. “ Coupled Vibra- tions ” (Phil. Mag. July 1918) was adapted for the tripled pendulums. Figs. 1 and 2 give the side and end elevations Fig. 1.—Side elevation. P(A) of the actual arrangement used. The masses are confined as far as possible to the bobs P, Q, and R, which. are heavy carriers with funnels containing sand. Triple Pendulums with Mutual Interaction. 615 In fig. 1 only two pendulums are visible, the third is hidden by FP. The positions of the bridles are determined by the hie light connectors BB’, CC’, and DD’. The normal oscillations occur in the plane of fig. 2, and are recorded by sand traces on a black-board drawn perpen- dicular to the direction of vibration. Fig. 2.—End elevation. _ Al) Ale) __ Ale? The trace from ( is received on a narrow black-board. The pendulums are started when all the bobs are just above the boards, and afterwards the board containing the trace from Q is moved until the three initial motions, shown by the traces, are in the same straight line. If the advantages and disadvantages of this arrangement are considered, it is seen :— (1) That although the three connectors make for gene- rality, yet if DD’ is eliminated, the action of the pendulum-bob 616 Prof. Barton and Miss Browning on P may still be transmitted through the connectors BB! and CC' to the pendulum-bob R. Further, the connector DD’ may tend to force in extra vibrations, so that four, or more vibrations may be felt by each bob. (2) That the bridle droop being different for one pendulum may also complicate matters. The bridle droops do not enter into these experiments in exactly the same way as in those of the Double-Cord. Pen- dulums, ‘‘ Coupled Vibrations”? (Phil. Mag. Oct. 1917); for in that the bridle droop directly determined the length of the oscillating pendulums. Suppose the pendulum P is considered, and the one con- nector at BB’. Then the two oscillations possible to this pendulum, if others are not forced upon it, are :— (a) One about AE ; (6) One about PHar Bis held stationary. Now if the bridle AFH is altered, but the connector B is so adjusted that BE makes the same angle with the horizontal, then the period of the oscillation of the pendulum about this position is unaffected. For small oscillations, the effective length would be found by producing PF upwards until it met the line joining BE in Bo. For this arrangement it is not easy to write down definite values for the couplings. However, from general consi- derations, it is seen that, if BB’ and CC’ are near AA", and DD’ to EE”, the couplings will all be small. The coupling between two only will be large if those are connected near F, and the others near A or E. The early experiments were done with the apparatus as described. Later the connector CC’ was removed, and results obtained for certain couplings were found to be similar to those obtained when using three connectors. LV. PHorocgrapHic REcOoRDs OF VIBRATIONS. Nine photographs of traces were taken with the three connectors in use (Pls. XI. & XII). Six others with only the two connectors BB’ and CC’, but otherwise the same arrangement (Pl. XIII.) For details of the lengths mentioned in the table reference must be made to figures 1 and 2. The masses of the bobs in these experiments were all the same and equal to 660 gms. In all the experiments the total depth of the bobs below AH was kept constant and equal to 140 cms. The distance between AE was 180 cms. The positions of B, 0, D, F, and G will now determine the whole of the system. Triple Pendulums with Mutual Interaction. 617 Table of Experimental Details. Photograph Length Length Length Length Length Length Letter. AB. A'C. AF. AG. FE. Di. \ cm, em, em. cm, cin. cm. EU G8 oss. 52 71 84 100 98 SI meré | Band By... 52 71 84 100 98 45 Br canats ) Pee 18 71 84 L100 98 92 : | Dand E...- 18 71 86 100 106 68 \F,G,andH 18 mE 86 100 106 30 "ey Al ee ae 18 ‘Gk 86 100 106 Two 4 4 ghee tn ee 32 75 86 100 106 Connectors.} KandL... 31 31 86 100 106 \MandN... 41 41 86) {100 +. 106 The table indicates that for photograph A (PI. XI.), the coupling due to the bridle droop is small, but, for this droop, that due to the connectors is large. The vibrations were produced by drawing aside the pendulum P at the point F. The traces from the three bobs P, Q, and R all show that there are more than two simple harmonic motions combining to produce the curves. For the photographs B and B, the coupling is still less between the pendulums P and R. This is shown both in the table and in the traces. For photograph B the pendulums were started by drawing Q aside at G, and for B, by drawing P aside at F. The trace of P in photograph "B does not indicate more than two simple harmonic vibrations, nor are the traces in photograph B, very striking. For photograph ©, the ecuplane between the pendulums P and R was considerably increased, but that between P and Q was decreased. The resultant effect was to materially increase all the couplings as P acted on Q by means of the connectors DD’ and CO’", which were both tightly coupled. All these traces can be seen to contain more than two simple harmonic motions. For the photographs D to H inclusive the droop of the bridle wasincreased. For traces shown on photographs D and E, the couplings between all the pendulums due to the position of the connectors were decreased, but this was compensated by the increased droop. With P pulled aside (see D), the traces _ from P and Q only appear to be the resultant of two simple harmonic vibrations, but the beats in Rare slower on reaching their maximum than in falling to their minimum. All the traces with Q pushed aside (see E) show more complicated curves. These could be obtained by compounding three simple harmonic vibrations of the proper phase and amplitude. Phil. Mag. 8. 6. Vol. 40, No. 239. Nov. 1920, 258 618 Triple Pendulums with Mutual Interaction. For F, G, and H (Pl. XII), the coupling between pen- dulums P and R was decreased by moving the connector DD! nearer to Hi. The curves are very similar, F was obtained with () pushed aside, and G with R pushed aside. From the photographs it is seen that the couplings are small between P and each of the other pendulums, but that between Q and ois very much larger. The interaction between the pen- dulums Q and R does not seem to be communicated to P at all, although the reaction of the slow waxing and waning of the amplitude of P is clearly seen in the traces of both the other pendulums. ) Photograph H was obtained with an arrangement of pen- dulums like those used for F and G. R was started and after three complete cycles of change was held stationary; the pendulum then settled quickly toa state showing two superposed simple harmonic vibrations, as in coupled pendulums. At this point it was thought probable that the two con- nectors 5 and C would be sufficient. D was removed. Everything else was kept the same. The photograph I (Pl. XIII.) shows traces with the couplings between P and mot of the other pendulums considerably meshneatl but other things much the same. The connectors were then lowered considerably, and curve J was obtained by drawing aside R. This was found to be almost identical with G, which shows that connector D was superfluous. Photographs K and L were obtained with the two con- nectors at the same position on the cords. The photographs show which bobs were drawn aside, and the resultant effect was that of a light bob driving one heavy bob which is com- posed of the other two. ‘There is little to indicate more than two harmonic vibrations in the curves. M and N were obtained with the connectors together on the cords, but with the coupling increased. Studying the four last photographs carefully, it is seen that the two pendulums which are the driven ones to begin with do not give exactly identical curves, the distance between the nodes is gradually increasing. This points to the fact that there are more than two simple harmonic vibrations involved. Other experiments have already been carried out with a simpler arrangement of three interconnected pendulums, but these are reserved for a separate paper. Nottingham, df uly 14, 1920. [ 619 ] LXXI. On the Deduction of Rydberg’s Law from the Quantum Theory of Spectral Hmission. By SATYENDRA Natu Basu, M.Se., Univer es Lecturer in Physics, University College of Science, Calcutta * T is well known that Rutherford’s model of the atom has been fruitful in explaining many facts connected with atomic radiation. In the simplest case of hydrogen, with a nucleus consisting of a single positive charge, ‘and an electron, Dr. Bohrf has successfully applied the quantum theory to explain the Balmer series of hydrogen spectra, The mathematical problem of finding the spectral series for any atomic system has since been ‘clearly formulated by Sommerfeld {, and the quanta condition has been gene- ralized in a form suitable for systeins with any number of degrees of freedom. If q1, ga, G3, -.. Yn are co-ordinates to fix the position of the electron responsible for emission, and Pis P25 P3.--. Pn are the corresponding generalized momenta, any statical path, according to Sommerfeld, is characterized by the conditions { prdqi=mh, \prdq2=nzh, § Pad gn = ty h, where n’s are whole numbers and h is Planck’s constant, the integral being extended generally over the complete orbit. The radiation js supposed to take place when the electron jumps from one: statical path to another. The difference in energy, at the same time, flows away in the form of a homogeneous radiation of frequency v, which can be calculated from the Bohr’s equation hy= W,—W5,. Sommerfeld has successfully applied this conception in explaining the fine str ucture of hydrogen lines. It is clear, however, that the problem of theoretically calculating the spectrum of any atom other than hy drogen i is beset with difficulties of a formidable ae It is exactly analogous to the dynamical problem of ‘‘n’’ bodies, where only in favourable cases we are able : find approxi- mate solutions. Nevertheless, from a purely experimental standpoint, we know that the visible radiation from any element can be classified in definite series. The frequency of any line in the series can be expressed as a difference of N two terms, each of which has the form ~---——_,.,,, where (m +a+ £) m is a whole number and e and 8B are two constants * Communicated by the Author, + Bohr, Phil. Mag. July 1913. { Sommerfeld, Ann. der Physik, li. (1916). 282 620 Mr.8.N. Basn on the Deduction of Rydberg’s Law depending upon the element and the nature of the series. So that if we are to explain the formation of the series from thecretical considerations following Bohr and Sommerfeld, we must look upon each member multiplied by “/” as giving the energy of the atomic system when the radiating electron moves in a definite statical path. The complexity of the inner atomic field under which the radiating electron moves is to be looked upon as bringing in the terms involving a and ~. No it seems interesting to see what will be the corresponding expression for energy in a system by which the complex nature of the internal field may be approximately represented. In the case of any atom we have, in general, a condensed nuclear charge of +-ne (where n is the atomic number) surrounded by rings of electron at different distances. The number of electrons in total must be also equal to n in order to secure that the atom is electrically neutral in the ordinary state. In X-ray emission the electron displaced comes from the inner-rings ; in the case of visible radiation, however, we have reasons to think that the displaced electron responsible for radiation comes from the outermost ring—the valency electrons, as they have been designated by Sommerfeld. When excited for radiation, we can suppose that the electron in the outermost ring is removed to a greater distance from the centre than the others, so that the force acting upon it may be regarded as the resultant of the various forces exerted by the central charge and the remaining electrons. The potential at any point can be regarded as given by 2 ——-+e? & —, where r is the distance from the centre . s=n-1"'s and 7, is the distance from the s-th electron. If we neglect the influence of the moving electron upon the arrangement of the others surrounding the nucleus, it is clear that the 2 potential can be approximately represented as =. + ee o The resultant field might be looked upon as due to a single positive charge, together with a doublet of strength L in a certain fixed direction, which we take as our Z-axis. If we neglect the disturbing effect of the outer electron, L may be taken to be approximately fixed in direction and in magnitude in the small interval of time during which the active emission takes place. | We may, therefore, take as our model a system consisting of a positive charge and a doublet of strenoth L. We proceed to caléulate the energy in a statical path on the above simplified hypothesis, from the Quantum Theory of Spectral Emission. 621 The kinetic energy of the moving electron is obviously 4 DT = $[mr? + mr? +7? sin? 06" | ; the potential energy 9 e’ elLeosdé Wee ee fh li Two integrals can be at once written down mr? sin? 0d = ¢, . * . D 52 Alf DEMS Q <6 ‘ : : ae 41ué€ COS ml 7? + 7°67 +r? sin?Og¢? | —— + ——W. - , Fe To get another integral, we write d oF i ely sin 0 OYE b weet ee ae oe am 0) — mr? sin @ cos 0¢? = ae or c,2cos 0 3 l mr do — mr26 —_— ai sin® @ —meLsin@ = 0. Integrated, it srs + 2meL COS.0 =6,. The expressions for three impulses m7, mr°0, and nmr sin? Ad can now be written down: in terms of the constants of integration we have mr sin? 6 = «a, 2 9 , Cy > mrog = Nhe — .5—2mel cos 0 2 sin? 6 ; mi = — f — Wmr? 4- 2me?r— cy. of he The quanta conditions can be written down as i mr sin? Opdb= nh. . |. » « QQ) i) mi?6 dé ee eae ere Og \ mdr W/E) ae eke ol —W being twice the total energy of the system. The integrals are to be extended over the whole range within which the expression within the square root remains positive. 622 Mr.8.N. Basu on the Deduction of Rydberg’s Law The integration. From (1) we have obviously nyh 1S Dee? of the two remaining expressions, (3) can be integrated most easily : in fact, a) Ih - nah = — / — War? + 2mer— co dr ? gives after integration Ty gy ait he $0 ON sk au ngh = Qa l¢ (Ww! Co i The second integral can be written as a i a —-[= = (ve@=a= ) = Cv? —-2melLa(1 a OU Py) {=e ‘by putting cos 0=w. The right-hand side is to be integrated throughout the region, when the cubic remains positive. It cannot be integrated in finite terms; an approximation suitable for our purpose can, however, be made, assuming 2meL to be small compared with (c,—¢”)=A. To see what this means we are to remember that c.—c,? is of the dimension of h?; so that 2meL must be small compared with h?, or 2 L must be small compared with toe Now, if a, —6, and y m are taken as the three roots of the cubic, the cubic can be written down as D(y—a)(e—-#)(#+ 6), where y is the greatest of the positive roots and D=2meL. The limits of the integral are obviously « and —§8, I =| VA —¢,2?— Da(1— 2”) a Hence Olin if" — wide OD “Jn V DIG=2)@—a)e+ 8)" or ol 1 Ky (y—x) dx DL) _s@=aynes ay pe da - |" (y— vw)? (a— wv)? (@+ 8) rik - — from the Quantum Theory of Spectral Emission. 62: Supposing a—wv = (4+ 8) cos? 9, +x = (a+ 8) sin? 8, co SO a ain? O12 Sele TD [24 [y+ B—(a4+ 8) sin’ @]'?dé we get » 2 dé = tolerannly / 28) —eataye® (Gov/ St) | where E and K are the usual elliptic integrals, defined by Or/2 Ei(1/2, &) he /1—Kk? sin? dé, dé K (2/2, k) = Gal V1i—ksin’ 0 On the assumption that D is small, we have _ & , D(A) 2A sD) ar 4 gra 8 a — eS. A), eer 8. So that, expanding E and K and making necessary approximations, we have finally ol aD | 4 = = SS ——— ce ie Pe) D Ac,°/? c v4 624 Mr.S8.N. Basu on the Deduction of Rydberg’s Law Now, collecting all the quanta conditions, we have nh (mel)? Noh age, ee ore m \iV2 noah e(y —cl2?= =, a 4 2a So that we have 24 ee = ) — (m+ 12)h Ge —e,+ 2 Aa a Qa and Assuming Lalita 2 on we have gh (meL)? (27)? tea _y¥l? i _ (my +72)h 27 AyPh? An? Aq? | Dp ed or 2 L\? y+( su as (On —y’) = m+n. Calling 27r?meL hi? = 7AR? we have ray YE ] o(G)"= Lows where utes (dn? —y?) = m+ 5 we have approximately aff DN es pada) a (ny; ‘i + n.)° ae So that Aqr2e*m oy ee sea A ei iL fa ae Davee 2 Del oho ate) er ne RE ) h? | ny Ang +n3+ Gs ep (1 Cen: So that the energy in the statical path Qar2e*m = i F 4 i ao (1 Oba Fh } 7 ? > EC = Se eo E ee (m1 + ng)? (my + ng)? from the Quantum Theory of Spectral Emission. 625 If we suppose that the spectrum is due to ionized atoms in which the field can be approximately represented as a central charge of H=pe and a doublet of different strength L’. We have, by a similar reasoning, energy in a statical path 2a? e? H?m ‘a Lo + 2g+ apart de (i- 3m ) Ce es (n1 + 29)? (m+ 22)?/ J —p Nh | ARR Ml PEIN. AD a ae (ny +)° (24+)? where N=Rydberg number. As a result of numerous investigations on the nature of the spectral series, it has been shown that for many elements the different series can be grouped according to the following scheme :— ( P-Series: v= Cl Ss) mn py) ) me 2,3, 2 —(m, pz) II. Subordinate Series : v = (2, py) —(m+'5, s) » = (2, ps) — (m5, 8) 1st Subordinate Series: _ vy = (2, pi) —(m, d). Companion : (2, pi) —(m, a’) m = 3, 4, 5. v = (2, po)—(m, a"). Wi =) Daye AS [ese Dinca aa nas V N Shot. f 7c tea EL ADSp ihe yi eo ae ie Sym ee stands for tae iccording to Rydberg, and ————— $\? according to Ritz. (m+7+5) Mt The frequency of the lines emitted thus appears as the ditterence of two terms, each of which is to be regarded as corresponding to the energy in a definite statical path, on the Quantum theory. Sommerfeld* has recently given reasons for assuming that in any series of statical orbits corresponding to the different kinds of s, p, and d terms the azimuthal quanta generally preserve a certain definite * Sommerfeld, Verh. d. Phys. Ges. May 1919. 626 Mr.S.N. Basu on the Deduction of Rydberg’s Law value, whereas the radial quanta can have all values from 0 to 0; he thus shows that in s, p, and d terms the azimuthal quanta generally have values 1, 2, and 3 respectively. Making the above assumptions in our formula, we see that Nh n, +n,+n3+ A)? in the same form as required by the Rydberg formula. The constant A, however, depends only upon n, and ng; it diminishes for increasing values of the azimuthal quanta ; so that they decrease progressively in the s, p, and d terms. Moreover, our form shows that A depends upon 7, and ng separately, so that for the same value of njy+n, we may have different values of the constant. Thus, if we suppose n1+n,=1, we have two values corresponding to the values 1,0 and 0,1; for ny +n,=2 we have three values ; and so on. Thus we see, even on Sommerfeld’s assumption, for the con- stancy of the azimuthal quanta we shall have two different s, three different p, four different d terms. At least two dif- ferent values of p and three different values of d seem to be required by the series formula, which is essential for the explanation of doublets and triplets of constant frequency difference *. We thus see that our model serves at least as a qualitative explanation of the following facts :— the expression of the energy comes out as — (1) The progressive decrease of the characteristic numbers in the s, p, and d terms. (2) The existence of different sets of s, p, and d terms for the same element. It is clear, however, that our simplified assumption will not fit in any actual case exactly. The complex nature of the internal field can in no case be properly represented by L cos 0 a simple term, ——;—, in the Potential. Moreover, we Gf have reason to believe that the internal arrangement of the electrons itself will be influenced, in a large measure, by the motion of the outer electron, which we have neglected in our formula. In fact, Landéf has tried in a recent paper to take account of this disturbance in the comparative simple case of the helium series. But at the same time, it is hoped that the calculation, in this comparatively simple * If we exclude the case n,=O0—. e., if we assume that the motion in a plane containing the axis of the doublet is excluded—we get the proper number of s, p, and d terms as observed in the case of the alkali metals and the doublet system of alkaline earths. + Landé, Phys. Zedt. 1919. rom the Quantum Theory of Spectral Emission. 627 t a) Y¥ yi case, will serve to illustrate at Inst some general principles at which we have arrived by an experimental study of the spectral series. Summary. In this paper an attempt has been made to deduce the laws of regularity in the spectral series of elements on the basis of Bohr’s quantum theory of spectral emission. Starting from Sommerfeld’s assumption that the ordinary line-spectra of elements are due to the vibration of one outer electron (the valency electron), it has been shown that the field of the nucleus and the remaining (n—1) electrons may be represented by the Potential y ee TISOn 2 ? liad t.e., the field due to a single charge plus a doublet of strength L. The axis of the doublet is variable, but the emission is supposed to take place so quickly that in that short time the axis does not appreciably change. The quanta conditions have been applied according to Sommerfeld’s rule, nh = \ p, dq,, and the energy of. the system has been eBaGoed to the quanta numbers. The energy comes out in the form Nh en ee Re ee = 7 J > eengncey® n+z=y In tne paper, where ng is the radial quantum, n is the azimuthal quantum, and zis given by an equation of the sixth degree, involving only the azimuthal quantum, and is a function of n only. It has been next shown that if, in accordance’ with Sommerfeld’s principle, we assume ee ie s-orbits, n=2 for the p-orbits, n=3 for the d-orbits, n=4 for the b-orbits, then, with a very simple assumption, we obtain a single value for the energy of the s-orbits, a double value for the energy in the p-orbit, a treble value for the d-orbit. Then, applying Bohr’s law hv=Wi— W, we arrive at Ry dberg’ s laws of the regularity in spectral series, in the wage of the.alkali metals. Exact calculations are not tried on account of the uncertainty of the value of L.; but it has been pointed out that the values of s, (p1, po), (di, do, d3) progressively decrease, as is actually the case. If the value of L be supposed to vary with ns, the radial quantum, then probably the above calculations would lead to Ritz’s law. [ 628 ] LXXIL. Vhe Mass-Spectra of Chemical Elements. (Part 2.) By ¥. W. Aston, .A., D.Sc, Clerk Maxwell Student of the Unwersity of Cambridge* . [Plate XIV.7 i a previous paper (Phil. Mag. xxxix. May 1920, p. 611) the apparatus for obtaining mass-spectra was fully de- scribed and the results of analysis of eleven different elements tabulated. The following paper deals with the analyses of some additional elements for which the same apparatus and method was used. Boron (At. W. 11:90). Frivorrine (At. W. 19°00). SILIcon (At. W. 28°3). It will be convenient to treat of these three elements together. The atomic weights of boron and fluorine have both been recently redetermined by Smith and Van Haagen (Carnegie Inst. Washington Publ. No. 267, 1918), with the above results. On the atomic weight of silicon there is some divergence of opinion. The international value is quoted above, but Baxter, Weatherell, and Holmes make it nearer 28-1 (Journ. Am. Chem. Soe. vol. xlii. p. 1194, June 1920). After a failure to obtain the boron lines with some very impure boron hydride, a sample of boron trifluoride was pre- pared from boric acid and potassium borofluoride, and this gave good results. Following the usual practice, it was mixed with a considerable quantity of CO, before intro- duction into the discharge-tube. Very complex and inter- esting spectra were at once obtained, and it was remarked that this gas possessed an extraordinary power of resurrecting the spectra of gases previously used in the apparatus. Thus the characteristic first and second order lines of krypton were plainly visible, although the tube had been washed out and run many times since that gas had been used. This property of liberating gases which have been driven into the surface of the discharge-bulb is doubtless due to the chemical action of the fluorine, liberated during the discharge, on the silica anticathode and the glass walls. After running some time the corrosion of the anticathode was indeed quite visible as a white frost over the hottest part. After several successful series of spectra had been secured, the percentage of boron trifluoride in the gas admitted was increased as far as possible, until tie discharge became * Communicated by.the Author. The Mass-Spectra of Chemical Elements. 629 quite unmanageable and the tube ceased to work. Just before it did, however, i it yielded two very valuable spectra which confirmed the isotopie nature of boron. These are reproduced side by side as they were taken (Spectra I. & IT.). The lines at 10 and 11 are undoubtedly both first-order lines of boron. The hypothesis that these might be due to neon liberated by the action mentioned is not tenable, both on account of their relative intensities and the absence of strong neon first-order lines. Even if it were, it could not explain the presence of the well-defined lines at 5 and 5°5 which had never been obtained before at all, and which must be second- order lines of boron. ‘This element therefore has at least two isotopes 10 and 11. The relative photographic intensity of the lines 5 and 5°5 does not agree well with an atomic weight as high as 10°9, and the discrepancy mievht be ex- plained by the presence of a third isotope at 12; which would be masked by carbon, for it has not yet been found practicable to eliminate La ae from the discharge. But Spectrum IV. contradicts this suggestion for, as will be shown later, the line at 49 is mainly if not wholly due to BUMF,, so that Ghore should also be a line at 50 for BYF,. The line at 49 is very strong, but at 50 any small effect there may be can safely be oneal to the fourth order of mercury. The evidence is clearly against the presence of a third isotope of boron. The exceedingly accurate whole-number value for the atomic weight of fluorine suggests the probability of this element being simple. This conclusion is borne he by the strong line at 19:00 with second-order line at 9°50. The accompanying line at 20, very faint in Spectrom IT., is no doubt HF, though it may be also Ne”? or second-order A”. As there is no evidence whatever to the contrary, fluorine is taken to be a simple element with an atomic weight 19. Having adopted these values for boron and fluorine, we may now apply them to Spectra III. and IV. taken with boron trifluoride. Consider first the group of three very strong lines 47, 48, and 49. The last two are to be expected as being due to BYE, and B" FB, respectively, but since there is no evidence of a boron 9 or a fluorine 18, line 47 cannot be due to a compound of these elements. But line 47 only appeared when BF’, was introduced, and so must be due to silicon fluoride educa by the ee of the fluorine on the glass walls and the silica anticathode. To test this the BF’; was washed out and replaced by SiF,, which has been made by the action of sulpburic acid on caicium fluoride and silica in the usual way. ‘This greatly 630 Dr. F. W. Aston on the reduced the lines 48 and 49, and so they must be attributed to boron compounds. At the same time line 47 remained very strong, and was evidently due to a compound Si*F, so that silicon has a predominant constituent 28. This con- clusion is further supported by the presence of very strong lines at 66, Si? F,, and 85, Si Fs. The chemical atomic weight shows that this cannot be its only constituent. Lines at 29, 48, 67, and 86 all suggest a silicon of atomic weight 29. Practically conclusive proof of this is given in Spectrum V., which shows its second-order line unmistakably at 14°50. The only other reasonable origin of this line, namely second-order B!°F, is eliminated by the fact that there is no trace of a line at 10 in this spectrum. The evidence of a silicon of atomic weight 30 is of a much more doubtful character. Its presence is suggested by the lines 30, 49, 68, and 87, but the possibility of hydrogen compounds makes this evidence somewhat untrustworthy, and no proof can be drawn from a second-order line 15, as this is normally present and is due to CH;. On the other hand, if we accept a mean atomic weight as high as 28-3, the relative intensity of the lines due to compounds of Si and Si in- dicates the probable presence of an isotope of higher mass. These considerations taken with the complete absence of any definite evidence to the contrary make the possibility of Si worth taking into account. Molecular lines of the Second Order. The work of Sir J. J. Thomson on multiply-charged positive rays showed very definitely that molecules carrying more than one charge were at least exceedingly rare (‘ Rays of Positive Electricity, p. 54), for not a single case was observed which could not be explained on other grounds. Up to the time of the experiments with the fluorine compounds the same could be said of the results with the mass-spectrograph. This absence of multiply-charged molecular lines, though there is no particular theoretical reason for it, has’ been used as confirmatory evidence on the elementary nature of doubtful lines. The spectra obtained with BI’; show lines for which there appears no possibility of explanation except that of doubly- charged compound molecules. The two most obvious of these may be seen on Spectrum III. and at the extreme left- hand end of Spectrum IV. They correspond to masses 23°50 and 24°50, the first being quite a strong line. Were Mass- Spectra of Chemical Klements. 631 there no lines of lower order corresponding to these, the whole- number rule might be in question ; but all doubt is removed by the fact that the lines 47 and 49 are two of the strongest on the plate. A comparison of several spectra upon w hich these lines occur shows a definite intensity relation which practically confirms the conclusion that the first pair of lines are true second-order lines corresponding to the first-order lines of the second pair. Now lines 47 and 49 cannot by any reasonable argument be elementary, they must in fact be due to compounds of fluorine with boron ee ih Ox silicon Si, or due to both. Further evidence of the rapability of fluorine compounds to carry two charges is offered by line 33°50, which is undoubtedly the second-order line corresponding to 67, i. e. BF, or SiF,. So far as results go, fluorine appears to be unique in its power of yielding doubly- charged molecules in sufficient number to produce second-order lines of considerable strength. Bromine (At. Wt. 79°92). The results with this element were definite and easy to interpret. Its chemical combining weight is known with ae certainty, and is very nearly the whole number 80. It vas rather a surprise, ther efore, that it should give a mass- a which showed it to consist of a mixture of two isotopes in practically equal proportions. Methyl bromide was used for the experiments, and one of the results is reproduced in Spectrum VI. The characteristic group con- sists of four lines at 79, 30, 81, and 82. 79 and 81, appa- rently of eqnal intensity, are much the stronger pair, and are obviously due to elementary bromines. This result is practically confirmed by second-order lines at 39°5 and 40:5 too faint to reproduce, but easily seen and measured on the original negative. The fainter pair 80 and 82 are the expected fines of the two corresponding hydrobromic acids. The same difficulties as were discussed in the ease of chlorine in the previous paper prevent the attainment of absolute certainty in determining the composition of bromine, or indeed that of any eeip capable of forming hy drogen compounds; but the conclusions stated above may be re- garded as having a high degree of probability. SuLPHoR (At. Wt. 32:06). Spectra VII. and VIII. show the effect of the addition of sulphur dioxide to the gas in the discharge-tube. Above each is a Comparison spectr um taken immediately before the 632 Dr. F. W. Aston on the gas was admitted, on the same plate with approximately the same fields. The very marked strengthening of lines 32 and 44 is no doubt due toS and CS. New lines appear at 33 SH, 34 SH,, 60 COS, 64 SO, or 8, and 7608,. It may be noticed that lines 32, 60, and 76 are accompanied by a faint line one unit higher anda rather stronger line two units higher. In the first case it is certain and in the others probable that these are, at least partly, due to hydrogen addition compounds. If a higher isotope of sulphur exists, as is suggested by the chemical atomic weight, it seems unlikely that this should have mass 33, for this would have to be present to the amount of 6 per cent., and should give a line at 35 one-thirteenth the strength of 34 (normal SH,). No such line is visible. A sulphur of atomic weight 34 present to the extent of 3 per cent. is more likely, but there is hardly enough evidence as yet to warrant its serious consideration. PuHospHorus (At. Wt. 31°04). Arsenic (At. Wt. 74°96). The gases phosphine PH; and arsine AsH; were used in the experiments on these elements, and the results were of notable similarity. The mass-spectrum of each gas was characterized by a group of four lines. The first and strongest doubtless due to the element itself, the second rather weaker due to the monohydride, the third very faint to the dihydride, and the fourth fairly strong to the trihydride. The spectrum of AsH; is shown in Spectrum I[X.; that of phosphorus is similar but its lines are weak, and therefore unsuited to reproduction. Both elements appear to have no isotopes, and neither give visible second-order lines. Lines of unknown origin. During the experiments with mass-spectra, lines have appeared from time to time to which it has been difficult ‘to assign an origin with certainty. Three of these seem worthy of special note. On one of the spectra taken with boron trifluoride there appeared a faint but unmistakable line at 5:33. The accuracy of its fractional value seems to insure it being triply-charged 16. If the source of the mass 16 is oxygen, it is somewhat odd that the line never appeared when oxyger was present in much greater quantity, but it may be possible that the loss of three charges only takes place when fluorine is present. An even more baffling line is one at 6°50 which has Mass-Spectra of Chemical, Hlements. 633 appeared on a chlorine plate, and also on one taken with BF,. This is naturally put down to doubly-charged 13. If the source of the mass 13 is put down to CH, it is very surprising that the line should not appear more frequently, also that it should not be accompanied when it did appear by a line at 7°50 due to CHs. The third line of doubtful origin is a very faint one at 13°50 which appeared when chlorine compounds were present. The most likely explanation of this seems to be doubly-charged aluminium derived from small quantities of aluminium chloride formed by the action of the gas on the electrodes. General Remarks. All the seven elements whose analyses form the subject of this paper obey the “ wnole number ”’ rule within the accuracy of experiment. They may be said to conclude the list of those which can be easily introduced into the apparatus in the form of gases or gaseous compounds. The following table gives a complete list of all the elements so far analysed. Some very clear results recently obtained with the mereury group of lines, e. gy. Spectrum VIII., show that its isotopes 202 and 204 need no longer be regarded as doubtful. ‘Table of Elements and Isotopes. lemon Atomic Atomic eae Masses of Isotopes in * Number, Weight. I order of intensity. sotopes. 5 ee il 1:008 1 1:008 te... Z 3'99 1 4 tae ee 5 10°9 2 LEE 5) See 6 12:00 [ 12 i eee nT 14:01 1 14 i eee 8 16°00 1 16 ip ot 9 19:00 1 19 Ne 6; 10 20°20 2 20, 22, (21) Siti haa ints 14 _ 28:3 2 28, 29, (80) LEY a 15 31:04 1 il Bids ue 16 32°06 1 32 2) eae 17 55°46 2 30, 37, (39) ye ae 18 39°88 (2) 40, (36) PGi Ne Oe 33° 74:96 i 1D Be 42-3 30 79°92 2 79, 81 4 eee: 36 82°92 6 84, 86, 82, 83, 80, 78 h. Sees 54 130°2 Gh (128, £31, 180, 138, 135) Eig. si. 80 200°6 (6) (197-200), 202, 204 In conclusion, the author wishes to express his indebtedness to the Government Grant Committee of the Royal Society for defraying the cost of some of the apparatus employed. Phil. Mag. S. 6. Vol. 40. No. 239. Nov. 1920. 2T 634 Prof. R. Whiddington on Summary. Further experiments with the mass-spectrograph yielding provisional analyses of the elements B, F, Si, P, S, As, and Br are described. Of these B, Si, and Br are definitely complex, the others apparently simple. | The atomic weights of all conform to the whole-number rule. Some anomalous fractional lines are mentioned and their possible origin discussed. Cavendish Laboratory, August 1920. LXXIIL. he Ultra-Micrometer ; an application of the Ther- mionie Valve to the measurement of very small distances. By R. Wuipprneron, M.A., D.Se.; Cavendish Professor of Physics in the University of Leeds *. [Plate XV.] oe the discovery and application of the various forms of interferometer the direct and accurate measurement of small distances has been regarded, with some reason, as a problem solved. All interferometer methods, however, are limited in accuracy of measurement by the wave-length of the light used in the production of the fringes. The-present paper is an account of a new method which is not limited in this way, and which is easily capable of measuring small distances of the same order of magnitude as the atomic diameter. The theory of the method may be put very simply. If a capacity C be connected to an inductance L, the frequency of oscillation N natural to the circuit is given by a ae 2a V LC If the condenser be composed of two parallel plates area A, separated by distance w, speeN ia Whence BY /CARINe Ws ny F Clearly a change in # involves a change in N, so that we _may take the change in N as an indication of a change in w. * Communicated by the Author. the Ultra-Micrometer. 635 For a sensitive indicator aN: 2: I: dn aCAR should be as large as possible, that is, the product LAw oD should be small. F , iC In order to get some idea of the actual value of — to be Ax expected under practical conditions we may conveniently rewrite the above in the form aN. 2 IN ade Qe A value of N easily obtained under conditions shortly to be described is 10°; assuming «= 9 inch, say, we have dN ibs 10° du Dah By a beat method—explained later—it is possible to observe a change in N of less than unity—assuming 6N=1 we get the corresponding value of 6 = 2 x 10-° inch, =X 10>? em, _ With this introduction the following description of an apparatus built along such lines is put forward. Fig. 1 shows the apparatus in diagram. A is an oscil- lating valve circuit involving the parallel-plate condenser (P) fo) discussed above; T is a loud-speaking telephone shown for Pig. 1. ; a simplicity directly inserted in the. valve anode circuit, although in actual practice a three-stage transformer am- plier intervened, to magnify suitably the currents passing through T. 2T2 : 636. Prof. hk. Whiddington on The values of the coils in the grid and anode circuits of this part of the apparatus were so chosen as to produce oscil- lations of about a million frequency (N). * In order to make obvious any change in N a second valve circuit B was set up close to the one just described and shown on the left of fig. 1. The frequency of this cireuit could be adjusted by means of the condenser CU, so as to be nearly, but not quite, equal to N, so producing a loud audible note in T, the frequency of which could be adjusted to any desired value by a suitable choice of C. In order to provide an unvarying standard of pitch to which the note could be adjusted, another valve circuit was set up (inducing into the amplifier), with capacities and inductances so large as to produce an audible note in T*. This note was usually kept constant, and the heterodyne note produced from the N frequency oscillation tuned to it by the condenser UC. The musical note circuit is not indicated in fig. 1, but is shown on the extreme right of the photograph of the ex- perimental arrangement (PI. XV.). The two large coils and standard condenser box (maximum value about 1 m.f.d.) are here clearly shown, the valve being hidden behind them. On the extreme left of the bench is shown the metal lined box con- taining a geometrical slide carrying two stitf vertical insulated rods and the attached condenser-plates P, one on each. These plates were of polished steel about 5 mm, thick and about 17 sq. cm. area each. They were set as nearly parallel as possible by an optical diffraction method, although great care.in this respect is not essential to the ultimate working of the apparatus. Just to the right of this condenser-plate box is a second box, also metal lined, containing two stiff paper bobbins (each carrying two small inductance coils) and mounted side by side, the left-hand one associated with the parallel-plate condenser P, and the right-hand one with the variable con- denser Cf, here shown witha pointer moving over a graduated scale. On the top of this box are seen the two thermionic valves. which perform the duty of maintaining the oscillations in these high-frequency circuits. In the left top of the photograph is the trumpet of the loud-speaking telephone, while in the centre top are shown the 100-volt lead accumulator units supplying potential to the various valve circuits. * The frequency of this audible note could be kept constant to about 1 part in 100,000 over a period of several hours. + There was also another auxiliary condenser, not shown, of which C was really a fine adjustment. the Ultra-Micrometer. 637 It was very soon found that the apparatus just described was most extraordinarily sensitive to small changes in the value of P. Although, in anticipation, the whole arrangement was placed in a solid basement room of the University, the note emitted from T instead of being pure was rough, owing to the continuous small vibrations of the buildin’. ih It was only, in fact, in the small hours of the morning that the purity of the note could be maintained, and even then distant traffic proved troublesome at times. Before explaining the methods used for determining the sensitiveness of the instrument, it may be mentioned in passing that the bending of the solid looking table shown in the photograph, produced by a penny placed on its edge, was clearly indicated by a change in the note from T. A series of experiments was now carried out to find what was the smallest change in w that could be detected by this arrangement. This involved preliminary experiments on the application of comparatively large known bending moments to the — left-hand rod in the photos raph*, and measuring by means of a micrometer the consequent lateral shift of the plate. A micrometer capable of measuring to 107* inch ‘was used, contact with the plate being determined by use of a sensitive galvanometer and battery. Two ditterent mechanical schemes were tried shown in fig. 2; of these, the one marked (a) was very soon abandoned as it was found that when ver y small weights were used the law of proportionality between shift of plate and applied load broke down. This was doubtless due to the frictional forces introduced at the junction J even when the finest unspun silk was used. A method free from this defect is that of fig. 2 (0), in which the bending couple is applied by placing suitable weights on the graduated quartz rod LM. As a result of a large number of experiments it was found that Hooke’s law was “obey ed to the limits of accuracy set by the micrometer. By extrapolation it was concluded that 1:4 grams placed one inch along the rod would produce a lateral shift at the centre of the plate of exactly 107° inch. This was a convenient number to remember from which to calculate the shift produced by any other weight at any distance. _ After a long series of trials it was found that the smallest * [ am indebted to Mr. J. Gilchrist, M.Sc., for great help in this part of the work. 6358 Prof. R. Whiddington on change that could be detected with certainty was that pro- duced by 1 m.gm. at 5 inches—the number of beats (about 2 per sec.) changing perceptibly and by a constant amount. Bio a2, Two. METHODS USED FOR MAKING SMALL CHANGES IN X SCALE - % FULL SIZE (EXCEPTING 20) (0) This shift is approximately ioe XOX 10° immeh==4-3 x 105? inch — 105° em. (appro or about 54, millionth part of an inch. It will be noticed that this figure is not so small as that estimated in the introduction (2 x 10-*)*. * Probably due to the plates not being so close as yo\55 inch, the value of a2 assumed in the initial calculation. It is to be noted in this con- nexion that when oscillations occur a voltage may be set up between the plates whose mean square value can easily be of the same order as the constant voltage supplied by the anode battery—in this case 50 volts. This faet in itself sets alimit to the pessible closeness of approach of the plates. the Ultra-Micrometer. 639 There is not the slightest doubt that an apparatus built on these lines could be made to indicate much smaller changes than that experimentally obtained here. | have not so far attempted to push it to its practical limit, as for the purpose I have in immediate view its present sensi- tiveness is sufficient; there seems little doubt, however, that a sensitiveness of 100 times that just attained could be got without much difficulty. In the previous paragraph it has been suggested how the apparatus may be used to detect extremely small distance changes, of the order of ;}, millionth of an ineh. We there kept every thing constant except the variation 1p wv and observed the consequent change in the number of beats. When, however, we are dealing with a much larger shift in #@—say ete part of an inch or so—it is possible to adopt a rather simpler scheme. With such comparatively large changes in wa very large change in N occurs, so that a considerable change in C must be made to recover the standard note issuing from T. With the actual apparatus shown in the photograph it was found that over quite a large range of the condenser scale —viz. from 20 deg. to 60 deg. pointer readings—a change of 1 deg. meant a change in ee ean web: 2. . As at was ppssibls to read 75 fierce quite easily, this scale-reading allows of a direct determination of a change in 2 within one three-millionth of an inch. It is hoped to apply this extremely delicate method of observing small distance changes to various problems in the near future. Summary. If a circuit consisting of a parallel-plate condenser and inductance be maintained in oscillation by means of a therm- ionic valve, a small change in distance apart of the plates produces a ‘change i in the frequency of the osciilations which can be accurately determined by methods described. It is shown that changes so small as ,4, millionth of an inch can easily be detected. ‘The name “ ultra-micrometer” is tenta- tively suggested for the apparatus. The Physics Laboratories, The University, Leeds. 1640). | LXXIV. The Directional Hot-Wire Anemometer: Its Sensi- tavity and Range of Application. By J.S. G. THomas, M.Sc. (Lond.), B.Sc. (Wales). A.R.C. ‘S., AGS Senior Physicist, South Metropolitan Gas Company, one Hi, INTRODUCTION. N a recent communication f the author described a type of hot-wire anemometer, which in addition to affording a ready indication of the direction of flow of gas in a pipe, was, on account of the almost complete elimination from the observed effect, of the influence of the free convection currents arising from the heated wires, especially applicable to the determination of the velocity of a very slowly moving current of gas. The directional hot-wire anemometer consists of two fine platinum wires mounted parallel, and one belfind the other in close juxtaposition, transversely to the direction of flow of the gas in the pipe or channel. The wires constitute two arms of a Wheatstone bridge (see fig. 7 later), the remaining arms of which are formed of a resistance of 1000 ohms and an arm capable of adjustment. A constant current is maintained in the bridge, and the battery terminals are connected through a rheostat to the appr opriate ends of the platinum wires so that the maximum heat is developed in these wires. A galvanometer is inserted in the bridge in the usual manner. Further details concer ning the method of mounting the wires ete. will be found in the papers referred to, wherein it is shown that as the velocity of the gas-stream to which the heated wires are exposed is gradually increased from zero, the galvanometer deflexion increases until a critical value of the velocity of the gas stream is attained. With further increase in the impressed velocity of the gas stream, the galvanometer deflexion very slowly diminishes. The direction of the galvanometer deflexion is reversed on reversing the direction of flew of the gas current, and the instrument affords an indication of the direction of flow even for extremely high values of the velocity of the gas stream. In the papers “referred to, the maximum jane of the galvanometer deflexion corre- sponded with a mean velocity of the gas stream equivalent to about 5 cm. per sec. The directional ty pe of hot-wire anemometer possessing certain advantages in the region of low velocities—greater sensitivity, more complete elimi- nation of the effect of the free convection current, greater . * Communicated by the Author. + Phil. Mag. vol. xxxix. pp. 525-527 (1920). See also Proc. Phys. Soe. vol. xxxii. Part 3, pp. 196-207. ~ On the Directional Hot- Wire Anemometer. 641 stability of zero—over those due respectively to Morris * and to King +, it appeared desirable to examine the behaviour of the instrament further. Experiments were accordingly made to determine (1) the relative sensitivity of the directional hot-wire anemometer with the wires mounted vertically compared with the sensitivity when the wires were mounted horizontally ; and (2) how the impressed velocity of the gas stream ‘at which the maximum galvano- meter deflexion oceurred—which may be termed the critical velocity—was dependent upon (a) the heating current employed, and (b) the diameter of the wires. EXPERIMENTAL. The general procedure in all the experiments was as follows :—The two anemometer wires were cut from the same sample of pure platinum wire supplied by Messrs. Johnson & Matthey. In experiments connected more particularly with the dependence of the sensitivity of the instrument upon the inclination of the wires, it is desirable that the resistances of the wires should be as nearly possible the same, in the absence of flow. This adjustment to equality im situ is a matter of some difficulty and, in general, was only achieved after a number of trials. The following procedure was finally adopted. The sample of wire was aged by the passage of a current sufficient to raise it to a bright red heat for about two hours. Two portions, as free as possible from flaws and pittings, and of about the same length, were cut from the sample, and attached to copper leads of diameter 0°82 mm. This was done by filing from the end of the copper lead a quadrantal segment about 3 mm. iong. One end of the platinum wire was inserted into the segment and joined to the copper lead by means of the minimum amount of silver solder affording a sound junction. ‘he remaining lead similarly treated was set up at a distance equal to the diameter of the flow-tube from the end of the first lead to which the platinum wire was soldered, and the platinum wire affixed thereto under a tension sufficient to maintain it quite straight. A second wire was prepared in an exactly similar manner, and the two wires inserted into the flow-tube as previously described. The resistances of the wires were compared. Employing a small current for this purpose, they were found usually to be equal to within 0°3 per cent. When the current employed was such that the wires were raised to a bright red_ heat, their resistances were found to be, in general, ‘within 0°5 per eent. of one another. Occasionally agreement to within * Morris, Eng. Pat. 25,923/13, + King, Eng. Pat. 18,563/14. 642 Mr. J. 8. G. Thomas on the O-l per cent. was found. For most purposes, equality to . within 1 per cent. is amply sufficient; but when it is desired to obtain a closer approximation in the adjustment to equality, a current, sufficient to raise the wire of lower resistance to a bright red heat, was passed through this wire alone for the necessary period—extending to hours or days—an air-stream being passed over the wires at half- hourly intervals for about 10 minutes. In this manner the resistances could be adjusted to equality when heated to within O-l per cent., and thereafter both wires were aged by the passage for about 2 hours of a current sufficient to raise them to a bright red heat. The ratio of the resistances remained remarkably constant for a very considerable period after such treatment: thus the ratio in the case of anemometer D4 showed no greater variation than O°1 per cent. while the anemometer was employ ed in determining about 50 calibration curves, employing various values of We heating current. In all probability, owing to the wires being throughout their use subjected to pr actically identical conditions, no calibration would be necessary until the eceamicn: had been employed in the measurement of con- siderably more than the 1000 velocities recommended by King* in the case of the non-directional type of instrnment. Tha srhed having been inserted in the fiow-tube were introduced into the Wheatstone bridge arrangement as described. The wires could be placed horizontally, or at any inclination to the horizontal transversely to the direction of flow of the air in the flow-tube which was horizontal. A millivoltmeter could be intreduced across the ends of the second wire of the pair so that the resistance of the second wire could be calculated from a knowledge of the observed drop of potential and the current in the bridge, this being maintained constant by means of a rheostat inserted in the battery circuit, and its value ascertained by means of a Siemens and Hal: ske milliammeter provided with suitable shunts enabling current determinations to be made correct to 0-001 amp. One of the two remaining arms of the bridge—that connected to one end of the second anemo- meter wire—was made equal to 1000 ohms and _ the resistance of the first anemometer wire, when any gas- flow was established in the pipe, was caleuleteds avon dhe of the second wire of the pair, by determining the resistance necessary in the fourth arm of the bridge to restore the balance of the bridge. ‘The galvanometer deflexion corre- sponding to the flow of air was read prior to restoring the * Phil. Mag. vol. xxix. p. 573 (1915). Directional Hot-Wire Aneniometer. 643 balance of the bridge. The current of air was derived from a weighted gas-holder of 5 cubic feet capacity, pro- vided with automatic pressure compensation, and was adjusted to any desired value by means of a micrometer cock. The air was dried by passage through a column of calcium chloride. The magnitude of the air-flow was ascertained by means of a wer gas-meter, by Sugg, of 1/12 cub. ft. capacity, this, being standardized by means of the 1/12 cub. ft. bottle prescribed by the Metropolitan Gas Referees. ‘The pressure at the meter outlet did not exceed 0°15 in. water. Further details of the experimental method may be derived from the papers already referred to. (1) Comparative sensitivity of the directional hot-wire anemo- meter when the wires are mounted (a) horizontally and (b) vertically in a horizontal flow-tube. The horizontal or vertical condition of the wires could, if desired, be very accuratety ascertained by the electrical method devised by the author*. In general the horizontal position was sufficiently accurately determined by the hori- zontality of a generating line forming a continuation of the line joining in eee of the holes through which the wires were inserted into the flow-tube, and filed on the tube. The vertical position was obtained therefrom by rotation of the flow-tube through 90° as indicated by an index moving over a scale engraved on the circumference of the flow-tube. Results and Discusston.—The following particulars refer to directional anemometer D 4a used in these experiments :— Digeeter.OmcUDE res > ass. “is, 2° 0034 em. Diameter of wire employed amen of 7 readings, differing by not ? 0°101 mm. more than 0: “002 TUT Voss ee °c). Distance between wires . . . . O*l cm. Temperature coefficient of wires . 0°003253 R,, 1st wire (current 0-01 amp.) . 0°2556 ohm. R,, 2nd wire (current 0°01 amp.) . 0°2547 ohm. Heating current in bridge . . . 1:2 amp. The respective volumes (at O° C. and 760 mm.) are converted to the equivalent mean velocities of the air- stream (calculated at 0° C. and 760 mm.) by multiplying the former, as given in cubic feet per hour, by 2°374. The galvanometer sensitiveness was suitably reduced by shunting with 6 ohms. The results are set out in Table I. * Proc. Phys. Soc. 1920, vol. xxxii. Part 5, pp. 291-314. 644 Mr. J. S. G. Thomas on the TaBLE [.—Directional Anemometer D 4a. Inclination of Wires. Horizonrau, V GRTICAL. Temperature. | eo 6a Barometer Raye | (ins. ). Stream. | Meter. ee meats (88) ne) 24-0 751 29°99 24-0 To 29°99 24:0 Tol 29:99 24:0 715°2 29:99 24:0 oa 29°99 24-0 jou 29°99 TAO Oo HD 29-99 24:0 Tou 29°99 24-0 Tou 29°99 240 CON 29°99 24:0 75°2 29°99 24:0 78:2 29:99 24:0 19°2 29°99 24°5 70:2 29:99 245 oe 29°99 25°0 (o:2 29:99 25-0 TO 2 eee 2O209 27:0 royal 29°99 222, Ye? 29°98 22:2 72:2 29°98 22:0 72°3 29°98 22-0 72:2 29°98 22:0 G21 29-98 22'0 Ma, 29°98 22°5 72:2 29°98 pty (21 29:98 22°5 72:0 29°98 230 72:0 29°98 23°0 730 29°98 23:0 73°0 29°98 23°0 730 29:98 23°5 73°0 29°98 23°5 | 29:98 (mins.) |(cub. ft.).' Time | Uncorr. | volume | Observ. | per hour, ‘Deflexion. | Left. | Right. eee ee nePWVwnDnnwnhwnwnvbdy wo | Lo bo FIN Oo HS oT SeoeS o> He ~I lo ee el CO CO OO w D (==) 272 289 310 330 373 401 425 446 457 464 462 459 441 423 410 396 389 376 271 302 don 362 393 422 446 463 467 463 454 431 422 400 388 227 209 190 164 128 100 78 54 “1S BOD We “Tho hb © | Directional Hot-Wire Anemometer. 645 Heating Current 1*2 amp. Balance | Drop of Potential | Corr. vol. | | Resistance of Wires (ohm). Resistance | across second wire of _ (dry) Equiv. | F aCe aig ees By ae Rohisa | oh U° C. ent | Zero flow. | With flow. / and -| (ems. é "| ee Zero | With | Zero | With | 760mm. per sec.), First | Second | First Second flow. | flow. | flow. | flow. |; (cub. {t.). | Wire. | . Wire. | Wire... |~ Wire. 1001 988 O771 0-772 O42 1 1-00 0-648 0°642 0-635 0643 977 0781 0-617 1-46 0°636 0651 967 C790 0823 1-95 0636 U°658 952 0-796 1:07 2°54 0-651 0°663 935 0809 1-40 332 | . 0°629 0-674 919 0-819 1-64 3°89 0627 0°682 909 0°829 1:78 4°23 0621 0683 896 0°825 2°00 4°75 0-616 0°687 889 O'8l2 ~~ 2-86 6-79 0602 0-677 1001 880 0771 0-770 4°17 9°90 0°643 0642 0°565 0642 884 0°792 3°46 8:21 0-583 0-660 878 0:750 5'U3 159 0:549 0-625 879 0-681 8:32 19°8 0-499 0568 885 0647 = 11-09 26°3 0477 07539 886 0°620 13°43 319 0°458 0°517 892 0593 16°73 39°7 0-441 0:494 894 0581 18°52 44-0 0°435 0484 ‘1001 901 O771 0560 22:10 52°5 0643 0642 0-420 0467 1005 992 0°838 0-842 O71 0-41 0:700 0-698 0696 0°702 976 0-849 0°388 0:92 0°690 0°707 960 0-851 0-608 1°44 0681 0°709 945 - 0°850 0-892 2°12 0°669 0°708 20 0840 141 3°35 0649 0-700 912 0-828 2°04 4°84 0-629 0690 897 0810 2°62 6°22 0°605 0675 1003 887 0°838 0-790 341 8:10 0'700 0695 0584 0°658 881 0-770 4°12 9°78 0:566 0642 876 O73 5°66 13°44 0'533 0-609 876 0°699 7°20 17°09 O'511 0583 881 ; 0644 10°97 26°0 0°473 0537 884 0-629 12°50 29°7 0°4685 0-524 892 0587 Ly Ee) 42°0 0°437 0-490 1003 896 0°838 0°567 22:0 52°2 0°700 0'698 U'423 0°472 646. Mr. J. S. G. Thomas on the A similar set of readings was made using platinum wires of diameter 0°202 mm. and temperature coefficient 0°003434, installed as already described in the flow-tube. In this case, owing to the smallness of the resistances of the wires, it was not so easy to adjust the separate resistances to equality when zn situ and heated by a current, as was the case with the finer wires.. The values of the resistances, the wires conveying a current equal to 0:01 amp., were 0:0753, ohm and 0°07563 ohm for the first and second wires respectively. The curves in fig. 1 show how the galvano- meter deflexion depended upon ‘the air current in the case of both thicknesses of wire. For parposes of comparison, the respective heating currents, in the cases of different thicknesses of wire, were chosen so that the wires were in the two cases, in the absence of flow, heated to approxi- mately the same temperature. Table ie gives particulars ot the heating currents employed, and the temperatures of the wires etc., in the absence of flow. Fig. 1 also gives a calibration curve for the case of the thicker wire (diameter 0°202 mm.), employed vertically and heated by a current (3°15 amp.) so that in the absence of flow, the temperatures of the vertical wires were as nearly as possible equal to their respective temperatures, with zero flow, when they were installed horizontally in - the flow-tube aid heated by a current equal to 3°3 amps. Fig. 1 likewise gives the form of the calibration curve obtained when the anemometer wires (diameter 0°202 mm.) were employed inclined at an angle of 45° to the vertical and heated by a current of 3:3. amps. Considering the curves obtained when using the finer wires (diameter 0101 mm. and heating current 1:2 amp.), it will be seen from fig. 1 that the sensitivity of the arrangement is, for very small velocities of flow, considerably greater when the wires are installed vertically than is the case when the wires are employed horizontally. Corresponding to a mean impressed velocity of about 4 cm. per sec. indicated by the point P, the same deflexion is afforded by the two arrangements. Over the region of velocities embraced between values corresponding to those represented by the points P and Q, the horizontal arrangement affords the greater sensitiveness. Corresponding to values of the impressed velocity greater than that represented by Q, the sensitivities of the two arrangements are, within the limits of experimental error, equal. With regard to the greater sensitivity of the vertical arrangement for very low velocities (region embraced between the origin and P), Directional Hot- Wire Anemometer. 647 it may be remarked that when the wires are installed vertically, they are each laved by their respective free 20 oe. M WIRES VERTICAL o) WiRES INCLINED AT 7 45° TO HORIZONTAL WIRES VERTICAL. ‘i WIRES HORIZONTAL. DIAMETER OF WIRES 0:10} mx + — +» ——+-+ DIAMETER OF WIRES O-202mmM. / pea Ces fo) VELOCITY (CMS PER SEC VOLUMES REDUCED TO O°C #760 mm) Le.) 500 DEFLECTION. convection currents, and as shown in Table I. the wires, in the absence of any flow, are at a considerably higher temperature than that to which they are raised when used horizontally and heated by the same current. The Mr. J. S. G. Thomas on the 648 ' 18h OLY = ez: OFG CSG GLY 69P ecg eg¢ IL? OLF ‘OTL AA PUG | “OAT MA JST ‘OL AA PUG ‘OLLAA IST "[VOT}AVA SOIL AA "[B}UOZIAOY SAL AA ‘CO 9) MOP O.toz YIM orteydsowye GAOGL SIT JO a.inyetedwue4 ssooxay 4861-0 &96I1-0 Me ea IG1é-0 3606-0 9461-0 O96T-0 869-0 002-0 Gr9-0 &P9-0 OAL PUZ “OULAA IST | "TVOTZTOA SOALAA ‘AIM PUg | "OLA IST ‘[VJUOZIAOY SOIT MA ‘(WLYO) MOP O10Z YIM JUatind ospraq Aq poyVoy Ud SalTM JO voULYSISOI CLE ‘9G40-0 ‘S¢10-0 FSFE00-0 GG = 90/00. e0-0 FEF800-0 G1 LEGS-0 9GEZ-0 SGE00-0 OITA PUB] “OTL AA 9ST | ee fx | Cece) | | as _yueat.ind ‘(uTo) qutaloyyaoo (suey -duv 19.9 = quetano ainyzeied way, “J > 4v 90UBISISENT a Clay |, ‘(utut) SOTA jo : UreL Directional Hot- Wire Anemometer. 649 greater sensitivity of the vertical arrangement might con- ceivably be due to the higher initial temperature to which the wires are raised by the bridge current when installed vertically. It will, however, be clear from the. sequel, and may be seen from fig. 1, that very little variation of sensitivity in the region of ‘low velocities accompanies a change of the initial temperature to which the wires are raised. Thus in fig. 1, in the case of wires of diameter 0-202 em., it will be seen that when the wires are vertical, the calibration curves in the region of impressed velocities between 0 and 3 ecm. per sec. are practically identicai when heating currents equal to 3°3 amps. and 3°15 amps. are employed, the initial temperatures of the wires in the two cases being 540°C. and 480°C. approximately. The initial greater sensitivity of the vertical arrangement, com- pared with the horizontal, is to be attributed to the relative disposition of the free convection currents to the wires in the two cases. With the wires vertical, the hot free convection current from each wire completely surrounds the same, whereas, with the wires horizontal, the heated free convection current rises directly away from each wire. In the case of the vertical arrangement, an impressed stream of low velocity would disturb the free convection current so that the central portion of the wire would no longer be completely protected by the hot ascending current of air. With increase of impressed velocity, the shielding effect of the free convection current would be reduced still further. The transfer of heat from the wire to the stream being in this case initially effected through the surrounding ascending free convection current, in which, moreover, the radial temperature gradient for some distance —up to something of the order of three times the radius of the wire*—is small, it is to be anticipated that any disturbance of the protecting free convection current by means of an impressed air-flow would produce a com- paratively considerable reduction in the temperature. of the first wire of the vertical pair. In the case of the horizontally disposed pair of wires, the heated free con- vection currents pass transversely over the wires, and it is evident that the first wire of the pair would experience a smaller cooling effect due to a slow air-stream than would be the case with the wires mounted vertically. The various effects can be readily discussed by the aid of fig. 2 * See Thomas, ‘An Electrical Hot-Wire Inclinometer, Proc, Phys. Soc. 1920. Phil. Mag. 8. 6. Vol. 40. No. 239. Nov. 1920. 2U 650 Mr. J. S. G. Thomas on the and fig. 3. The former shows how the resistance of each wire of the pair depends upon the impressed veloeity of the air-stream in the case of wires of diameter 0'101 mm. 075 | aS Pi Ses | is WiRE © WIRES VERTICAL 2-4WIRE X ae Oo 2rewiRE + ES, O-1OL MM, WIRES HORIZONT: RESISTANCE (oH™.) a 0-50 0-45 cen : oe 0:40 { | ] : — 0 10 20 30 40 50 VELOCITY. (cms. PER.SEC. VOLUMES REDUCED To O°C & 760 MM.) Fig. 3 shows the corresponding results for wires of dia- meter 0'202 mm. In each case, results are given for wires mounted both vertically or both horizontally, the experi- ‘mental points in each case corresponding with those given in fig. 1. The main features of the curves shown in fig. 2 RESISTANCE (OHM) Directional Hot- Wire Anemometer. G51 are as follows:—With the wires mounted horizontally, there is, with increasing impressed velocity of air-stream, a comparatively large increase in the resistance of the second wire. The resistance of the first wire meanwhile initially falls off comparatively slowly. On the other hand, VELOCITY (CMS. PER.SEC., VOLUMES REDUCED TO O°C & 760 MM). with the wires mounted vertically, with increasing velocity of the air-stream, the initial increase of the resistance of the second wire is comparatively small, while there is a very considerable decrease in the resistance of the first wire. ‘The sensitivity of the arrangement employing hori- zontal wires is due, therefore, principally to the increase in the resistance of the second wire, whereas that of the arrangement employing vertical wires originates principally in the decreased resistance of the jirst wire. The greater initial sensitivity of the latter arrangement is now readily 2 U.2 652 ~ Mr. J. S. G. Thomas on the explainable in the following manner. The passage of a slow stream of air over the wires is accompanied, as already explained *, by the convection of heat from the neigh- bourhood of the first wire to that of the second, whereby the second wire attains a higher temperature than in the absence of the impressed flow of air. The stream of air is, however, owing to imperfect shielding of the second wire by the first, operative in producing some cooling effect upon the second wire, and the resultant temperature of the second wire is determined by the net effect of these opposing tendencies. An impressed velocity can be imagined of such magnitude that with suitable disposition of the wires the temperature of the second wire remains practically unafiected by the stream. Such a condition of affairs is represented by the point R in fig. 1, and a number of other illustrations are to be seen in sub- sequent curves (see fig. 5). The initial difference in the temperatures of the two wires when mounted horizon- tally—an arrangement in which the sensitivity depends mainly upon an increase of the resistance of the second wire—is therefore small, compared with the corresponding difference in temperature of the wires for the same impressed velocity, when mounted vertically, as in this latter case any reduction of temperature of the jirst wire, due to cooling by the stream of air, is not diminished by a heating effect operative in the counter direction. In the absence of such considerations, considering merely heat convected by the stream from the first wire to the second, one would anticipate a greater difference of temperature between the wires for a given low value of the impressed velocity of the air-stream, when such difference of temperature is attributable principally to cooling of the first wire, than when it is mainly due to heating of the second wire ; for in the latter case, owing to loss of heat by the stream of fluid during its passage from the first wire to the second, and owine to the fact that the whole of such residual convected heat is not transferable, under the conditions of the expe- riment, to the second wire, the rise of temperature of the second wire due to this cause 1s necessarily less than the fallin temperature of the first wire, the wires being assumed similar in all respects. Now it will be remarked from fig. 2, that when the wires are mounted horizontally, corresponding to impressed velocities of the air-stream of from 1-6 cm. per sec. * Proc, Phys. Soe, vol. xxii. Part 3, pp. 196-207 (1920). Directional Hot- Wire Anemometer. 653 to 6'4 em. per sec., the resistance and consequently the temperature of the second wire is increased by more than that of the first wire is diminished by the same impressed velocity of the air-stream. It is clear, therefore, that over this region of impressed velocities, the whole heating effect experienced by the second wire cannot originate entirely in heat convected directly from the ae wire by the stream, but is partly due to heat derived from the free convection current arising from the first wire, and partly to the altered thermal conditions in the neighbourhood of the second wire arising from the dis- turbance of the free convection currents from the two wires by the stream of air. The deflexion corresponding to a velocity of about 4 cm. per sec., indicated by P, fig. 1, in the case of a pair of wires of diameter 0'101 mm., is seen to be the same whether the wires are mounted vertically or horizontally. The corre- sponding values of the respective resistances are indicated iyethe points 2), PF, Po, Ps, im)fig. 2, the points P,, P, referring to the vertical arrangement, ane Poe towthie horizontal arr rangement. The equality of deflexion in this case arises owing to the very approximate equality of the difference between the resistances of the first and second wires in the two arrangements (P), P, = Py, P,). Equality of deflexion corresponding to a velocity of about ie cms. per sec., indicated by ( in fig. 1, is seen from fig. 2 to be due to actual equality of the resistances of the respective wires in the two arrangements—i. ¢., the resistance of the first wire mounted either vertically or horizontally is the same when subjected to the cooling effect of a horizontal stream moving with a mean velocity of about 8 cms. per sec., and similarly for the second wire. In fig. the resistance of tle second wire, mounted horizontally or vertically, corresponding to an impressed velocity of the stream of from 8 to 16 cms. per sec., is shown as slightly different in the two cases. It is clear, however, that the experimental results would, within the limits of experimental error—estimated at 0°3 per cent.—be equally well represented by equality of the resistance in the ti cases, as shown by the dotted portion of the curve in fig. 2. It will be seen from fig. 2, that whereas when the wires are mounted horizontally, the resistance of the second wire increases very much more than is the case when the wires are mounted vertically, the maximum resistance attained 654 Mr. J. S. G. Thomas on the by the second wire in the former case is considerably less than in the latter case, and corresponds moreover to a considerably higher value of the impressed velocity. Pig. 3 shows analogous results to those given in fig. 2, employing platinum wires of diameter 0°202 mm. The corresponding deflexion-velocity curves are given in fig. 1, together with the calibration curve for the same wires when mounted at an inclination of 45° to the horizontal. The values given in Table III. are deduced from the results represented in figs. 2 and 3. Taste ITI. Soa Sy \ a SZ = ro cS) = aw | Syn eee Ree reese | Sea g5 | BF age | < oo Oa ey) Sa Ratio: | Cea eas Diameter | Ew POSS se STS | |r Some bee | ee is ¢ | Max Re Wires | a eae lines ar S paleetS So Initial Res. | S aise (mim) fT eee Se ee eles (£88a8 | Se | EN | SES! Se EF | 2 ae | — | Near! ba 5 si he | (es i | st ‘ 0'101 Horizontal. 1:2 470 0-642 0:686 1068 5:2 Vertical. ez 555 0698 0-710 1 Oni 18 0-202 Horizontal. 3°3 A 2 SOLS s020252" StOre 59 Vertical. oO 538 0:2121 0218 027 2:0 (2) Dependence of the critical velocity—the velocity at which the maaimum galvanometer defleaion occurred—upon (a) the heating current employed and (b) the diameter of the wires. In order to determine how the critical velocity—. e., the velocity at which the galvanometer deflexion attained its maximum velocity-—depended upon the heating current employed in the bridge, a number of calibration curves of directional anemometer D4 (diameter of wires 0:101 mm.) were obtained, employing heating currents in the bridge ranging from 0°7 amp. to 1°5 amp., the temperature of the wires being thereby raised to temperatures ranging from °C. to 842°C. The various calibration curves obtained Directional Hot- Wire Anemometer. 655 employing throughout a galvanometer -shunt of 4+ ohms are shown in fig. 4, and the values of the resistance of the 45 : Pete eal eae Se a 40 35 30 nN Ww ny ° a VELOCITY (CMS.PER SEC, VOLUMES REOUCED To OC ANo760MM™) 5 DEFLECTION. second anemometer wire, determined in the manner already described, when heated by various currents and exposed 656 Mr. J. S. G. Thomas on the to various rates of air-flow, are shown in fig. 5. The calibration curves shown in fig. 4, in the region of low 1:0 DIAMETER OF WIRES, O-10llMM Cc =0°8 AMP Ce OuWAME RESISTANCE. OF SECOND Wire (OHM, ae 7 en 02 4 6 8 10 i2 IA 16 18 20 22, 24 26 28 30 S2 34 36 38 40 42 44 46 48 50 VELOCITIES (cms. PER SEC.,voluMes REDUCED To O°C & 760mm) 03 02 VARIATION IN RESISTANCE. OF SECOND WIRE FOR VARIOUS VALUES OF HEATING CURRENT. velocities, are markedly different from the type of eali- bration curve afforded by the Morris type of anemometer. Attention has been drawn to one feature of the calibration Directional Hot-Wire Anemometer. 657 curves in a previous paper, indicating the more complete elimination of the effect of the free convection current in the directional type of instrument. It has likewise been shown + that in the type of hot-wire anemometer devised by Morris, the calibration curves obtained by using various heating currents in the bridge intersect in the region of velocities of the air-flow, where the velocity of the free convection current is comparable with the impressed velocity of the air-stream. The curves shown in fig. 4 are characterized by absertce of such points of intersection. ie) Cc « 760 Mm.) » S tr u a ait oa EC, VOLUMES REDUCED To O” a LS =R ity " i (cms. P Od +0 DO UBOX MOKA Ne {OV @ OKO @V(@) oO VELOCITY ) 50 100 150 200 250 300 350 400 450 500 DEFLECTION Fig. 6 shows the form of the calibration curves in the region of low impressed velocities, employing the galvano- meter at its maximum sensitivity. It is clearly seen that any intersections of the respective curves, if occurring at all, must occur for values of the impressed velocity of the air-stream less than about 0°75 cm. per second. The respective values of the impressed velocities of the * Proc. Phys. Soc. vol. xxxii. Part 3, pp. 196-207 (1920). 7 Phil. Mag. vol. xxxix. pp. 515-516, pl. x. fig. G (May 1920). 5 -~ 658 Mr. J. S. G. Thomas on the air-stream corresponding to the maximum galvanometer- deflexion when various heating currents were employed in the bridge could not be accurately determined from the curves shown in fig. 4, on account of the extremely small variation of such maximum deflexion accompanying alteration in the value of the impressed velocity of the air-stream in this region. The value of such impressed velocity, corresponding to the maximum deflexion, could, however, be very accurately determined by increasing the sensitivity of the galvanometer employed, whereby the rate of variation of the deflexion in the region of its maximum value could be increased as desired. For this purpose, therefore, the suspended coil galvanometer employed was substituted by a similar galvanometer of equal resistance whose sensitivity could be suitably adjusted by means of a shunt. Hmploying a shunt of 4 ohms, and a definite heating current in the bridge, the approximate value of the impressed flow of air corresponding to the maximum deflexion was determined in the ordinary manner. Torsion was now applied to the suspension of the galvanometer until the spot returned to the zero of the scale. (The desirability of substituting the galvanometer employed for measuring comparative deflexions by another, arises from this necessity for subjecting the galvanometer suspension to torsion.) ‘The sensitivity of the galvanometer being now suitably increased by employing a suitable galvanometer shunt, and a constant predetermined current being main- tained in the bridge, the flow corresponding to the maximum deflexion could be accurately determined. The bridge arrangement employed is such that no appreciable alteration in the heating current in the anemometer wires accompanies the alteration in the equivalent resistance of the galvanometer. This is readily seen from a general consideration of the relative magnitudes of the various resistances in the bridge. Fig. 7 shows the bridge arrange- ment employed, the values of the respective resistances and currents being as indicated, and the first and second anemometer wires denoted by the ordinal numbers within brackets. It is well known that __1G(at+P)t+alatP)}o 4 Gatb+eat@)t(b+a)(a4+ ey and as a and 6 are throughout, in the present experiments, Directional Hot- Wire Anemometer. 659 necessarily relatively small—of the order one-thousandth, at most---compared with « and 8, and as moreover no + 8, Fig.7, measurable alteration in their respective values accompanied an alteration of G from 0 to ~, we have dG), {G(atb+a+f)+ (b+2)(a+8)}” and Ay the alteration in the current through the leading anemometer wire accompanying an alteration of the galvano- meter resistance trom an infinite value to the value G is given by ie | SUE Gea © Lie © (atb+a+ Bi {G(atb+a+) + (b+) (a+B)} and is readily seen to be very small for all values of G, under the conditions of the present experiments. It may also be readily shown that under the conditions of the present experiments, in which «£1000, 8B=1000, and a~0 is at the most of the order 0:1 ohm, the value of the galvanometer current alters only between the limits 0 and 10-4 when the galvanometer resistance is altered from « to zero, so that, to within 1 part in 10,000 at least, the current passing through both anemometer wires is the same. The values of the impressed velocities at which the respective maximum (3 | {(bB—aa) (a+ f)}a ~ 660 Mr. J.S. Gt. Thomas on the deflexions occurred when various heating currents were employed are shown by the broken curves in figs. 4 and 8 in the cases of directional hot-wire anemometers D4 and D5 respectively, employing wires of respective diameters 0101 mm. and 0°202 mm. In addition to determining the velocity. corresponding to the maximum deflexion, it appeared desirable to ascertain the values of the velocities at which the resistance of the second wire attained its maximum. value in the case of each heating current employed in the bridge. In the ease of wires of diameter 0:101 mm., this was readily done from the results plotted in fig. 5. The experimental points in the neighbourhood of the respective maxima were plotted on an enlarged scale, and the maximum point accurately determined by joining the mid points of chords drawn parallel to the axis of velocities. The results obtained in the case of wires of diameter 0°101 mm. are set out in Table IV. herewith. Referring to Table IV. and fig. 4, it will be noticed that the possible range of velocities,—determined by the impressed velocity of the stream corresponding to the maximum deflexion— within which the directional anemometer may be employed in a quantitative manner can be very considerably extended by increasing the initial temperature to which the wires are heated. Thus, while with wires of diameter 0'101 mm., employing a heating current equal to 0:4 amp., the maximum deflexion occurs at an impressed velocity of the air-stream equal to 2°57 cm. per sec., the maximum deflexion occurs at an impressed velocity equal to 27°7 cm. per sec. when the heating current is increased to l‘i5amp. The initial tem- peratures of the wires in these two cases were respectively 36°C. and 826° C. in excess of atmospheric. From fig. 8 it will be noted that the range of application can be still further extended by employing thicker anemometer wires. Thus in the case of wires of diameter 0°202 mm. the velocity at which the maximum deflexion occurred was increased from 4°95 em. per sec. to 30°2 cm. per ‘sec: when the heating current was altered from 2:0 amps. to 3°8 amps. The corresponding initial temperatures of the wires were 108°C. and 735°C. respectively above atmo- spheric. The sensitiveness of the anemometer employing the finer wires is somewhat greater than that of the arrange- ment employing the thicker wires, the wires being in each case raised to the same initial temperature above atmospheric temperature for purposes of comparison. In the case of the finer wires (diameter 0°101 mm.) the initial excess tempe- rature is given with considerable accuracy by the relation 661 Directional Hot- Wire Anemometer. Directional Anemometer D4. Galy. shunt 4 ohms. Res. @ (see fig. 7)=1000 ohms. Ry (Ast wire) =0°2583 ohm. Rp (second wire) =0°2547 ohm. TPA BILE Ws Diameter of wires 0'101 mm. Temperature coefficient of wire 0°:003253. Balance | Temperature Mean | Mean | Excess resistance Resistance Maximum | of 2nd wire Temperature} impressed , impressed | temp. of Heating (ohms). of 2nd wire. resistance ELON, of Ist wire velocity | velocity Max. 2nd wire current of r at max. |corresponding corresponding galvanometer above (amp.). 2nd wire | | deflexion | to max. res. | to max, deflexion alinosphere Zero | At max. | Zero | At max. | (ohm), | Zero | At max. (CL) of 2nd wire galv. defl. (mm.). Zero tlow flow. piesa) flow. | deflexion. flow. deflexion. (cms./sec.). | (cms./sec.). (Os / | 15 1015 876 0875 O778 0924. 842 86689 533 6:40 ied 538 826 1-4 1014 883 O767. Ose 084-4 708 = 638 495 6°10 9:2 448 692 13 1016 894 0-703 0°695 0-755 585 573 456 O76 129 O78 569 12 1015 903 0633 0642 0-675 487 499 400 5°56 9°28 299 iat 1] 1014 915 0558 0°555 0596 384 380 307 5°00 778 223 368 10 1016 951 04900 0°507 0°525 293 315 260 4°60 6°88 159 277 09 1017 945 0°4380 0°452 0'466 226 244 203 4:00 5:99 108 210 08 1016 9638 03975 0401 0415 Iie elo 153 3°40 5 07 70 157 O7 1016 982 0°3600 0°367 03738 127 137 121 2'96 4:17 43 113 06 1016 996 0°3333 0°340 — 95 103 95 3°50 30 79 O-4 1014 = 1008 0:2980 0°300 - 52 5d 52 -— 2°57 18 36 662 Mr. J. S. G. Thomas on the 6 = 289 02557, where © is the heating current measured in amperes. I'he corresponding relation in the case of the thicker wires (diameter 0°202 mm.) is 6=12°8C7%, . (cms.P VELOCITY 50 EEE ET Te zs ek 2 he ee ere | 45 J 40) ees ok : : be tS > k, City a | r : : Ve a = B might, in spite of its greater extension, equal or exceed that in the stream. The stream would in ¢ any case draw the slower moving charges sideways from the ‘“ atmo- sphere,” and be itself slightly expanded sideways, until within the enlarged stream the volume-density of charge was equal to that of the “ atmosphere ” outside—the stream would then be effectively neutral and would still convey charge mainly of one sign to the earth without having suffered any serious lateral dissipation. 5. This establishes the possibility of the arrival of streams such as I had contemplated : their origin, even their existence, are matters concerning which little or no evidence is yet afforded by our imperfect knowledge of solar physics. In this respect the neutral clouds with which Prof. Lindemann deals are on much the same footing. ‘The real evidence for either must at present be inferential, e. g. derived trom the study of terrestrial magnetism. Considered from this standpoint, the stream hy pothesis seems to possess advantages over the cloud theory. The particles of the two signs in the neutra’ cloud are supposed to have the same velocity along the stream: on nearing the earth they will be deflected in opposite direc- tions in the earth’s maonetic field. According to Stérmer’s calculations re such corpuscular paths, the two sets of particles should penetrate the earth’s atmosphere (presumably ‘to dif- ferent levels) within two zones, one round each pole of the miugnetic axis, but so that the heavier particles fall much nearer the poles than the lighter: also, while entering the atmosphere all round either pole, the two sets of par ticles falling near each pole will fall preferentially on opposite sides of that pole. The phenomena of aurore and magnetic disturbance are not yet sufficiently well observed in polar regions to allow any decision to be arrived at, as to whether these two kinds of effects, corresponding to the two kinds of particles supposed present in equal numbers, actually occur ; but until they have been observed it seems more in accordance with scientific procedure to adopt an hypothesis in which the phenomena to be explained depend on one set of particles 2 te 2 668 Prof, 8. Chapman : A Note only, provided the explanation attorded is equally good in rere of points which can be compared with observation. On my stream hypothesis the particles entering the ee s atmosphere are mainly of one sign of charge: those of the opposite sign which also enter will be neutralized by the flow of the most mobile ions present, but this I regard as probably a secondary and unimportant part of the phe- nomena, and as detracting, though only slightly, from the storm-producing capacity of the preponderant charges. The latter give rise to the storm in the course of their escape from the earth: their motion would be radial, under the influence of their mutual electrostatic repulsion when brought to rest in the earth’s atmosphere, did not the earth’s mag- netic ficld deflect them on their escape as at entry. Bnt the inward and outward paths differ, because of the difference in distribution and velocity of the charge on entry and at escape. The charge spreads over the earth very rapidly and nearly uniformly, and is deflected sideways round the earth: and the magnetic storm I regard as due to this horizontal component of the outward flow of electricity. On the cloud hypothesis the two kinds of charge need not escape, but have simply to coalesce : and since both sets enter the earth near the poles, their mutual approach and neutrali- zation would seem to be a simpler task, and one less likely (on consideration) to produce the world-wide magnetic field of a magnetic storm, than in the alternative case, where the escaping particles must redistribute themselves and also travel outwards and round the earth for great distances while making their escape. These considerations (to be supplemented by detailed caleu- lations later) are the chief ones which, though not urged as conclusive against the cloud hypothesis of Prof. Lindemann, yet seem to me to render that hypothesis less probable than the one I originally proposed. 7. In conclusion, though the matter has no direct relation to the foregoing, I will indicate briefly my reasons for haying come to the conclusion that magnetic storms and auroree are produced by negative charges. When writing my “ Outline of a Theory of “Magnetic Storms,” I was of the opinion that charges of either sign would serve equally well to explain those phenomena, considering that, when they became en- tangled in the atmosphere, the char ged layer -of air rose bodily, and produced horizontal H.M. Fs and electric currents by thus cutting across the earth’s horizontal magnetic field. The mean free path of an ion at auroral heights is, however, such that the ions have far less “ grip” on the air molecules on Magnetic Storms. 669 in general than this view supposes: the ions have, indeed, considerable freedom to follow their own courses under the electrostatic and electromagnetic forces affecting them. Consequently, under the radial electrostatic force resulting from the mutual repuision of the injected charges, the e positive and negative ions present will at first tend to move radially in opposite directions: the vertical electric current thus pro- duced will be unidirectional, and so also will be the horizontal electromagnetic force tending to deflect them along the circles of latitude, owing to their motion across the earth’s horizontal magnetic field. This deflecting force will thus move both positiv e and negative ions in the same latitudinal direction, but the more mobile electrons will have much the oreater velocity in this direction, both because of their smaller mass and of the greater deflecting force on them (proportional to their greater radial velocity under the electrostatic force). The horizontal current, not unidirectional for the two sets of ions as is the vertical current, will thus be directed opposite to the direction of motion of the electrons (opposite, because of their negative charge). Now the direction of this. horizontal current is definitely known from the characteristic ' diminution of the earth’s horizontal magnetic force during a magnetic storm: it is towards the west. Hence the motion ° of the electrons must be easterly, which (considering the direction of the horizontal magnetic field which « leflects the electrons to the east) indicates that the radial motion of the negative electrons ue be upwards. Consequently the electrostatic field impels electrons up- wards, and positive ions downwards: it must therefore be due to a negative charge on the earth. This simple argument is supported by another which, though less weighty than the former, seems yet to have more force than most of those on which attempts to decide the sign of the injected charge have been based. The effects of magnetic storms in the middle belt of the earth are definitely greater over the afternoon than over the torenoon hemisphere of the earth, and this suggests that the injection of corpuscles is the more intense over the former hemisphere. Now the simplest paths of the particles are likely to be the more numerous, and these (as calculated by Stérmer) correspond to entry of the particles on the afternoon side of the auroral zones, if the particles are negative, and on the forenoon side if positive. The circumstance mentioned is thus favourable to the view that the particies are negative. Oct. 11, 1920. i 6FO. LXXVI. Advance of Perihelion of a Planet. To the Editors of the Philosophical Magazine. (GENTLEMEN, — AM much obliged to Mr. ee for pointing out an error in my paper in the May number of the Philo- sophical Magazine. If we take the expression yd? — de? — We for ds? 4 and use the transformation — we get for the new ds? (es) i) (4431) These gravitation potentials satisfy Hinstein’s equations, and there is no restriction as to the magnitude of m. It is therefore quite legitimate to use this expression for ds’, and it makes the velocity of light at any point in the sun’s gravitational field independent of direction. Using the method of the Caleulus of Variations, I find that the differential equation of the path of a planet without any restriction as to the magnitude of m is ee (14 x) (dP Pde). (14 2 meee : a a a cae - : = Jha yeas G))- — R= a@? he __ mu in "ele 4. where h isa ae If now we make m small, we have as an approximate solution u=(l+ecos@}/L, where L=h?/m. Hence the equation for a second approximation, if small terms are neglected, 1s du i: 1 Leah 6me cos 0 U= = —, — dé? eas L? which gives the correct amount for the advance of the perihelion of Mercury. Recess, Yours faithfully, 4th Sept., 1920. ALEX. ANDERSON. (1+ e?) + rea} LXXVII. Note on the Theory of the Velocity of Chemical Reaction. By F. A. Linpemann*. OME years ago Professor W. C. McC. Lewis put forward an interesting theory in which he endeavoured to prove that the velocity of a chemical reaction was deter- mined by the energy density of radiation of a certain frequency. ‘This theory, if true, would be of such funda- mental significance in all future work on the mechanism of chemical combination, that it requires most careful con- sideration; and it may therefore be worth while to cail attention to a difficulty which appears to be fatal to the whole theory. According to Arrhbenius’s well-known empirical relation, the temperature coefficient of the reaction velocity may be written A/T’, A being a constant and T the absolute tempera- ture. If v is the reaction velocity therefore, 1022 NTE AdT , : or dlogv= See whence v=Be-4". It is clear, therefore, that any theory on which v is proportional to e~** will agree with the observed facts. Marcelin and Rice assume this to be the case because only molecules whose kinetie energy is greater than =a react. The number of these is proportional to e~4/, so that Arrhenius’s relation is satisfied. Prof. Lewis assumes that the reaction velocity is proportional to the radiation density Sarhv? it Cas 3 hy 9 eft —] which may be written hv 8rhv® if hy is large compared to kT, so that again the Arrhenius relation is bound to be found. The only check obviously would be to establish some relation between the value of v derived from the observed value of A by Prof. Lewis and the optical properties of the * Communicated by the Author. 972 Prof. F. A. Lindemann on the reacting substances. The temperature coefficient of a re- action is usually given by chemists as the factor n by which the velocity is increased when the temperature is raised 10°. Hence 1p =n—1, so that PT (nt/10— 1) and an absorption-band would be found at a wave-length ch op) A= sas as = = kT?(n¥}0—1) T? (n¥/10— 1) 9 where ¢,=1'46 is the constant in Planck’s radiation law written in the form C1 1 DS oe D PY erAT _] nechegease of elie: inversion of sucrose by 0°9 N. hydro- chlorie acid, Lewis finds n=4°13 between 25° C. and 35° C. - Therefore hy kL The solution does absorb radiation between lw and 10, so that Lewis concludes that his theory is supported by the facts. The fundamental difficulty to which this note refers arises from the fact that the radiation density may be profoundly modified by exposing the reaction to some external source of radiation. If Prof. Lewis’s theory were true, this should completely change the velocity of reaction, but no such phenomenon has been observed. As an example, one may consider the change in radiation density produced by ex- posing a reaction to sunlight. If a is the apparent angular semi-diameter of the sun and T, its temperature, the energy density in sunlight is given by = 303(1:152-1)=46 and rA=L-05 p*. e hts BAS * Lewis finds a somewhat different wave-length, A=1:23y, pre- sumably because he uses Perrin’s value 6°85 .10°° tor N; which figure one takes does not materially affect the argument. Theory of the Velocity of Chemical Reaction. 673 The radiation density in the dark at temperature T is given by = Sarhv? 1 so that the ratio of the radiation density due to sunlight to that in the dark is given by hv us pee | ae hv 4 u sides ' ekto — | In the above instance it has been shown that for T=303°, hp qe ai =46, so that putting the sun’s effective temperature T)=6000° and its apparent semi-diameter a= 16'=4°65.1073, one finds ! 4645 Hal Sa ape 2 5-8 1018. U, e233 __ | In other words, the energy density of the radiation which is supposed to determine the reaction velocity, namely that of wave-length 1:046u, is some 10” times greater in sunlight than in the dark. Yet the reaction proceeds at appreciably the same rate whether it is exposed to sunlight or not. The obvious conciusion would appear to be that the rate of reaction is not affected by the radiation density. There are, of course, various ways in which one may attempt to escape this conclusion, but a moment’s consideration shows them to be untenable, and it is therefore scarcely worth while to enumerate and disprove them here. But until and unless the above difficulty can be disposed of, it would seem to constitute a grave objection to Professor Lewis’s theory. Clarendon Laboratory, Oxford, Sept. 19th, 1920. Pe ene aT LXXVIII. The Modification of the Parabolic Trajectory on the Theory of Relativity. By W.B. Morton, M.A., Queen’s University, Belfast*. HE simplest case of accelerated motion, that in which a particle moves in a straight line with constant “ rest- acceleration,” was investigated by Bornt who gave it the name of ‘‘hyperbolic motion” because the distance-time graph, or “ world-line,” has that form. The next in order of simplicity is that which corresponds to the parabclic path under gravity—i. e., when the particle has rest-acceleration constant in magnitude and direction, but possesses a velocity- component perpendicular to this direction. Let the origin be taken at the point where the velocity is perpendicular to the acceleration. It will be convenient to speak of the tangent to the path at this point as “ hori- zontal ”’ and the direction of the acceleration as the down- wards vertical. Let the axes of « and y be taken in these directions respectively, wu is the velocity at the origin, v the velocity at any point, and ¢ the speed of light. The in- clination of the tangent to the horizontal will be denoted b the rest-acceleration is derived from the ordinary acceleration as estimated from a “fixed” standpoint by multiplying the tangential component by y’ and the normal 2\3 component by 9? where y= (a — =) : . 3 vdv ds along the tangent and y “along the normal .is in the direc- Therefore in the present case the resultant of yv tion of Oy and has the constant magnitude f say. The equations of motion are vad 7° “cos b— 7 sin $= (eae: (1 | ue and ee SP COS Oe) bore em Equation (1) gives vip = tap: Put v=csin @ so that y=sec 0, then cosec 0d? =tan odd, giving the relation tan 10 . cos @=const.— tan 36), say, - - (a where sin 0) = u/c. * Communicated by the Author. | ~ Born, Ann. d. Phys. xxx. p. 1 (1909). Parabolic Trajectory and Relativity. 675 When @ is small this equation obviously reduces to v cos @=const. in accordance with the elementary theory. The subsequent analysis is made much neater by the introduction of an angular parameter « defined by | sin e=tan $0)={e—(c?—u’)?}/u. Using this equation (3) yives the connexion between magnitude and direction of v in the form v/e=sin 0=2 sin a2 cos ¢/(cos?@+ sin? a). . . (4) From equation (2) p=c? tan? @ sec d/f. The linear seale of the path may be expressed in terms of the parameter of the parabola which would be described on the old theory, a=u7/2 f=2e sin? af f (1+ sin? a)’. Using this, and the relation (3), we express the radius of curvature in terms of ¢ and the constants a, a, p=ds/db=2a(1+sin? «)” cos ¢-/(cos’h — sin’ a)’. The integral of this is the intrinsic equation of the path a(1+sin? «)? 2 cos asin Fajiiele +sin gd (5) s — x . roy ye . s e e F 2 cos? x cos (ob +a) cos (6 —a) ©cosa—sin d ; Integrating dxl|dp=cos ¢ . ds/db and dy/dp=sin¢ .ds/dd, we obtain fer the coordinates of a point on the path the expressions eee kN) ( sin 2¢ 2 5» COS (p= iG (6) 2cos*« Lcos(d+a) cos Gaus. sin 2a >’ Gos (pb+<) y=a(1 +4 sin’«)’{sec (d+) sec (P—a)—sec’al. . 2. 2. . . (7) It only remains to find the relation between # and the time. This can be obtained from (4) by writing v=ds/dt=ds/dd x dd/dt, leading to | cdt/dp=a(1+sin? a)?(cos? d+ sin’ «)/sin 2(cos? ¢ —sin? a)’, the integral of which is a(1+sin? a)? sin 2 ees t= 5 3 oe = 9 sina cos? « | cos (p+) cos (p—a) hase a log. cos (b+a) J” (8) The rie properties of the motion are seen immedi- ately from the equations. In the first place as ¢ approaches T ° . the value @ —«), x, y, and ¢t become infinite and v approaches c. That is tosay, the direction of motion, instead of swerving 676 Prof. W. B. Morton on the Modification of the through a right angle and approaching coincidence with the acceleration, as in the parabolic case, is turned through (3 -) and the particle approaches a steady motion at the speed of light in a direction inclined at angle a to that of the constant “ rest-acceleration.”’ The following table gives the values of « for different values of u/¢ :— 1/C. a. A cece nae OF ope APs! Din Mee Ree oP 43" Drees eee 8° 50’ SEL er BoP ae Diviss oeirbak wee 15° 33! Dice tae 19? 28! gaan the ses D4o Gy CeCe een a0: 0h a Laer c: 38° — 49! 10 Ute eee : 90° For u/e=:8, sin a= {e—(c?—w?)2 bw, To gain a clearer view of the terminal conditions, which have a rather paradoxical appearance at first sight, it is worth while to evaluate the components of the (ordinary) acceleration. We find, reinstating f, vdv/ds =f sin $(cos” 6—sin? «)3/(cos? g + sin? #)?, w/o =f cos }(cos? d—sin? «)?/(cos? d+ sin’ a)’, and y= (cos? + sin? a)/(cos? 6—sin? a), so that y’ times the former component and y? times the latter combine to give f. This verifies the analysis. Now put @=a7—a—y where y is small, udojds—F cosia = cob a, yP, 9 44 } —=/fsina.cot?a. x’, y=tan a/y. The tangential acceleration (ordinary) is small compared with the normal, but in passing to the rest-acceleration it is multiplied by a higher power of the large quantity y, so that the modified components are of the same order of magnitude and have the finite ratio tan a. I have taken the case w/e="8, «=30°, and have calculated wx, y for different values of @ and plotted the path shown on the diagram. The parabola with the same ‘‘a” is shown for comparison. The figures marked above the path give the values of v/c at the corresponding points, and those below give the time, the unit in the latter case being the time required for light to describe the parameter ‘ a.” These points were obtained by calculating the values of v/c Parabolic Trajectory on the Theory of Relativity. 677 and ct for each @ and plotting against # (shown on the other curves). ‘The points on these curves corresponding to even values of v/c and ct could then be projected on the path. Fuge 1. ‘80 x “95 5 15 19 et 20 iS ag It will be noticed that the graph for ct against #@ is nearly straight, showing that the equable description of horizontal distance is not seriously departed from even at the high velocities taken. The analytical reason for this is seen on inspection of the formulee (6) and (8) for # and ct. It will be seen that the expressions are built up of the same two functions of @ with different constant multipliers. The first term in ct is cosec @ times the term in a, whereas in the second term the ratiois sin «. The two functional expressions are roughly of the same order of magnitude in w for the smaller values of @, but as the limiting value is approached, the second term becomes insignificant compared with’ the first. Hvidently the ratio of cf to vis mainly determined by the first term and approaches the constant value cdsec a. [ 678 |} LXXIX. Notices respecting New Books. Thermodynamics for Hngineers. By J. A. Ewine, Cambridge University Press. Pp. xin+383. Price 30s. net. puss book is largely based upon Sir J. A. Ewing’s two well- known books on applied thermodynamics, ‘ The Steam Engine’ and ‘The Mechanical Production of Cold.’ The chapter on ** First Principles ” follows closely the second chapter of ‘*The Steam Engine,” but has gained in clarity in rewriting, and the general treatment in the book before us of the Theory of the Steam Engine (Chapter III.), the Theory of Retrigeration (Chapter LYV.), Jets and Turbines (Chapter V.), and Internal Combustion Hngines (Chapter VI.) does not differ essentially from the exposition given in the books mentioned. We notice that the author restricts the term adiabatic to reversible processes: the general practice nowadays is rather to indicate by adiabatic a process in which, no heat is allowed to enter or leave the substance, and to specify further whether the process in question is reversible or irreversible in the case considered. The last two chapters, devoted to general thermodynamic relations and their applications to particular fluids, contain much new matter. A list of thermodynamical formule is given which is very convenient for reference. The porous plug experiment, which is often so ineffectively tandled, receives adequate treatment, and Callendar’s equation, now so widely used, is discussed in considerable detail, the author being careful to draw attention to its limitations. The variations of the specific heats of saturated vapours, the subject of recent papers by Sir J. A. Ewing and Professor A. W. Porter in this magazine, are not considered. ‘The book is illustrated by excellent diagrams, some of them, such as that exhibiting Witkowski’s isothermals for air, being not usually found in text-books. For the physicist who wants to study thermodynamics from the point of view of the engineering, rather than the physico-chemical, applications, as for the engineer who wants to grasp the theoretical foundations of all engine-efficiencies, the book is in every way excellent, com- bining as it does sound theory and practical knowledge in a way that is only too rare. Life and Works of Sir Jagadis C. Bose. By Parrick Guppxs. Longmans, Green & Co. Pp. xi+259. Price 16s. net. Harty in hfe Sir J. C. Bose set himself the task of winning a place for the native of India in modern science, and this task he has pursued with a steadfast and unselfish devotion in the face of very great difficulties. The foundation of the Bose Research Institute and the somewhat tardy award of the F.R.S. may be said to mark the final success of his endeavours. Some twenty-five years ago his name became known to physicists in connexion w ith his work on very short electromagnetic waves, an investigation w hieh he carried out with the utmost experimental skill ; and although his Notices respecting New Books. 679 more recent researches have dealt almost entirely with plant-life, his methods have always been those of the physicist, Perhaps the fact that he is a physicist rather than a physiologist in training and language, may account for some of the opposition which his work and views have aroused in certain quarters. His researches on complex strain effects in metals, which he included under the title of ‘‘ Response in the Non- -Living,” led hum to extend con- siderably the analogy between the living and non-living which had already been expressed by the use of such terms as “fatigue” for describing non-organic phenomena. From this he passed on to the fascinating problem of plant response and irritability. Many of the astonishing results obtained with the help of his various ‘ erescographs,” or instruments for magnifying the small move- ments of plant-growth, were demonstrated in London early this year, and vindicated against the critics. A magnification of movement approaching one hundred million times has been obtained with the magnetic crescograph, the instrument repre- senting the culmination of a long series of experiments. His main discoveries on plant-life have been recently discussed in the scientifie press, and are still fresh in the minds of all. In the book before us Professor Patrick Geddes presents us with very sympathetic study of the career, personality, and ideals of Sir J. C. Bose, and summarises in some detail his scientific work. He comments without bitterness on the difficulties placed in the way of a sincere and gifted student of nature, and leads us to rejoice with him in the ultimate recognition of Sir J. C. Bose’s claim as a citizen anda scientist. The book is excellently pro- duced and illustrated. Traité de La Lumiére. Par Curist1an Huyerns, Gauthier-Villars: Paris. Pp. x+155. Price 3 fr. 60, Tus reprint of Huygens’ famous treatise is one of the first volumes of a new series, appearing under the title of “Les Maitres de la Pensée Scientifique.” Some sixty volumes are already announced, other classical works on light included in the list being those of Newton, Young, and Fresnel. The biological sciences are represented, but mathematics and physics claim the majority of the volumes of the series. In England we are not well provided with reprints of the classical papers. In the par- ticular case of Huygens’ “Treatise” we have, it is true, Silvanus Thompson’s beautifully printed translation, which, however, is comparatively costly. The little book before us is quite well produced, and very moderately priced, and we think that the series should enjoy considerable popularity among English readers. We hope to see the list extended until there is no need to go to “Ostwald’s Klassiker” for the original authorities. Apart ‘from sentimental reasons French is a language more accessible to most 3ritons than German, and the page “of the French series is more restful to the eye than the German page. We wish the new yenture every success. 680 | Notices respecting New Books. The Elementary Differential Geometry of Plane Curves. By R. H. Fowrrr, M.A. Cambridge University Press. Cambridge Tracts, No. 20. 1920. 8vo, pp. vilit105, Price 6s. net. THis tract gives a concise but clear and rigorous presentation of the differential properties of plane curves. Tangents and Normals, Curvature, Theory of Contact, Theory of Envelopes, Singular Points, and Asymptotes of plane curves are the subjects of its chapters JI. to VII. The first chapter gives a good analytical Introduction to the subject. This excellent tract deserves the attention of all students of mathematics. Lectures on the Theory of Plane Curves, delivered to post-graduate students in the University of Calcutta. By SurmmMpRANSHAN Ganeanul, M.Sc, Published by the University of Caleutta, 1919. 8vo. Pp. vit350+17 (diagrams), in two Parts. Contents: Introduction (on homogeneous coordinates, the line at infinity, line-coordinates, circular points at infinity, projection). Theory of plane curves. Singular Points. Polar Curves. Co- variant Curves; the Hessian Polar Reciprocal Curves. Foci of Curves. The Analytical Triangles; Asymptotes. System of Curves (pencil of curves, envelopes, etc.). Cubic Curves (Chapters | X.-XV.). Curves of the 4th order (Chapters XVI.-XX.), Appendix I, Notes on the Bicircular Quartic. Appendix IT, Note on Trinodal Quartics. This handy and easily readable work will be particularly helpful to those interested in erbics and quartics, which are treated very fully in the last eleven chapters. EprroriaAL Note. ph the February issue of the Philosophical Magazine we pub- lished a paper by Mr. Norman Campbell on “‘ The Adjustment of Observations.” That paper was intended to be followed by two others in which the reasons for the contentions advanced in the first would be given. We now discover that the second paper of the series, which Mr. Campbell imagined to be in our hands, has gone astray and cannot therefore be published. Mr. Campbell asks us to state that the substance of the second and third papers is to be found in his book ‘Physics—The Elements’ just pub- lished by the Cambridge University Press. It will be seen from that book that he does not accept the assumptions on which Dr. Meldrum Stewart, in the April issue of this Journal, based a criticism of the first paper. W. FRANCIS. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE > AND fF OO [SIXTH SERIEB.] . & SS Ss, PATENT St*" DECHMBER 1920. vee ad LXXX. On the Closure of Small Cavities in Rocks exposed to High Pressures. By J. Jouy, F.RS., &.T.C.D.* [Plate XVI.] JT is now nearly eight years ago since I commenced expe- riments on the behaviour of rocks under high pressures, hydrostatic in character ; the object being, more especially, to determine the stress under which cavities in these materials will close. Data bearing on this point have many appli- cations in Geological science. These experiments extended over a period of nearly two years, from Dec. 1912 to Sept. 1914. They were all carried out at atmospheric temperature. It was intended to modify the apparatus so as to permit of the temperature being raised and maintained at moderate heights, but the outbreak of war rendered any development in this direction impossible. It appears desirable now to publish an account of what has been accomplished. The method employed seems to possess advantages on the score of directness, sensitiveness, and simplicity over any work of the kind described before or after these experiments by other observers. The rock specimen under examination is, as truly as may be, spherical in form. But the sphere is formed of two hemi- spheres accurately fitted together on a plane snrface. A small hemispherical cavity is ground centrally in the flat * Communicated by the Author. Phil. Mag. 8. 6. Vol. 40. No. 240. Dec. 1920. a DEC 13 1920, ‘ 682 Prof. J. Joly on the Closure of Small. face of one of the hemispheres. The hemispheres may be cemented together by a thin varnish of Canada balsam or simply laid together. The diameter of the sphere so formed is closely 2.cm. It is highly polished on the outside as well as on the flat meeting surfaces. The sphere of rock is enclosed in a lead cylinder. having an outside diameter of 2°5 em. This cylinder, which is about 3°4 cm. in length, is composed of two short cylinders in each of which a hemispherical cavity, having the same dimensions as the hemispheres of rock, is formed. The lead cylinder containing the sphere of rock is pushed into the crushing mortar, and exposed to pressure transmitted through a plunger. In this manner a hydrostatic pressure acts upon the sphere of rock, the lead flowing freely. The arrangements permitted of the pressure being maintained practically con- stant for several months at a time. The crushing mortar is shown in section in fig. 1, to a scale of one half. Itis made of vanadium steel, the outer strengthening ring being shrunk on. Messrs. Amsler of Schaffhausen were the makers, and it would be difficult to find fault with the manner in which the apparatus behaved. Tt will be seen that the lead cylinder is contained between a shorter plunger below and the npper plunger. The latter carries a movable cap resting on a smooth polished surface, to secure as far as possible a truly axial direction for the applied compressive force. This mortur is specified to take on the plunger one hundred tons per square inch. Jn order to prevent leakage of the lead past the pistons (which fit very accurately) copper washers—turned over on the edge—are provided. One of these is fitted to each end of the lead | eylinder. in The shaping of the rock material to the form of two aceu- ‘ately fitting hemispheres was undertaken by Dr. Krantz of Bonn. This work was very beautifully and accurately carried out. The internal cavities were ground in the laboratory, using asmall spherica: steel ball (intended for a ball-hearing) to carry the abradent (very fine carborundum). This ball was mounted in lead, projecting just a very little more than its radial dimension, and spun in the lathe against the central point of the plane surface of the rock hemisphere. The cavity so produced had a diameter of closely 6:2 millimetres. During this operation the rock hemisphere was held in a lead cup, externally cylindrical, in which a cavity of the shape of the hemisphere had previously been formed by a special tool. Special means were employed to secure the central position of the cavity ground in the rock hemisphere. Cavities in Rocks exposed to High Pressures. 683 At the pressures finally found requisite to break down the resistance of the rock, there was no reason why two spheres could not be dealt with at the one time. In this case two spheres, as described above, were each enclosed in a separate ‘ lead container having the dimensions already given, and these containers placed in the crushing mortar. The copper flanges were applied, of course, to one end only of each cylinder. ” ; 22 684 Prof. J. Joly on the Closure of Small The hydraulic press used wasa Tangye Press, with Bourdon ouuge attached, and capable of affording a crushing force of 400 tons : ; the water being pumped in by hand by two pumps, a fast low-pressure pump and a slow high-pressure pump. The crushing mortar is placed centraily on the platten of the press, and this is up-lifted by the operation of the pumps till the p'unger of the mortar comes against the ceiling of the press. Further pumping now forces down the plunger. In order to counteract the slow leakage of water around the leather collar of the press, a very simple and effective device was adopted. In the cellar beneath the laboratory a steel gas-cylinder of the usual type, filled with compressed air, was installed. This was supported in an inverted position, 2. e. nozzle downwards, and a high-pressure copper pipe brought up from it through the floor and tapped into the high-pressure water-chamber of the press. A screw cone-in-cone stop- Lee served to open connexion between the tube and the press. The action was as follows:—When the hydraulic pressure had risen above a certain point, the stopcock was opened and water from the press allowed to flow into the gas-bottle, raising the pressure in the latter. The pumping was then continued till the required pressure was attained. If, now, the whole system was left to itself, any small leakage of water was made good by water fed by air pressure from the bottle. In short, the air acts as a spring, storing energy and replacing its loss by supplying water practically at the original pressure. This acted very successfully. For months the. pressure was maintained constant with no loss detectable by the manometer. The temperature in the cellar was very uniform. In two of the experiments the rock hemispheres were used without any cavity being formed in either of the hemi- spheres. A cavity common to both was provided by the device of inserting a thin steel] washer between the hemi- spheres: this washer having a central circular opening of the same diameter as that of the cavily as formed in the hemisphere. In this case care was taken to grind the washer to a uniform thickness of about 4a millimetre. The washer may be cemented between the hemispheres with hard Canada. balsam or used without cement. It will be evident that by ordering the experiments as described above, directness and sensitiveness are the objects in view. The sphere is the form of highest symmetry, and the mathematical treatment of stress and strain in the deter- mination of conditions of rupture is simpler than in any other Cavities in Rocks exposed to High Pressures. 685 vase. Again, the form given to the specimens under expe- riment is such as to enable the first beginnings of yielding to be detected. There is in every case a_ polished > flat surface, unsupported, ina small central area. Examination of this by a bright reflected light affords a ios sensitive test of distortion. And, in addition to this test, examination with the lens or microscope may readily be applied to this polished surface and minute cracks detected in an incipient stage. When a cavity in one hemisphere is provided rapid yielding reveals itself in the debris which spalls off and _ accumulates in this cav ity. The importance of securing the means of sensitive obser- vation arises from the fact that in Nature time for the accummlation and development of effects exists in a far greater degree than can ever prevail in the laboratory. Our only chance of detecting such effects must be such conditions of sensitiveness as will enable their first beginnings to be observed. | Observation soon revealed the fact that while rapid yielding of the material to the external hydrostatic stress was shown by the accumulation of debris in the cavity formed in one ef the hemispheres, less intense stress might show a distinct effect in the optical distortion of the flat surface vis-a-vis to it or by the development of cracks in the same. It is very improbable that such signs of yielding could arise without ultimate breakdown of all resistance and closing of the cavity. And when we further bear in mind that the effects of raised temperature which must prevail in Nature are here absent, we seem justified in treating such stresses as produce these first s signs of yielding as probably exceeding those which in Nature would result in the closing of cavities. The use of a flat thin washer between the hemisph eres was dictated by this consideration. And although it possessed the disadvantage of slightly disturbing the truth of the spherical fori in the specimens as prepared { or these expe- riments, it increased the number of observations possible with the available number of specimens. War-conditions had put an end to all possibility of obtaining others. The rocks dealt with were of four markedly different kinds, and all of widespread importance :-—Granite, basalt, pista. and Solenhofen lithographic limestone. ‘The material has necessarily to be fine-grained. The granite was from Selb, Bavaria; the basalt fro om Jungfernstein, Siebengebirge ; se obsidian from Iceland; and the lithographic limestone fragn Solenhofen, Bavaria. The resistanee to crushing of cubes of the granite, basalt, und lithographi¢e slate was made the - 686 Pivot Joly. on the Closure of Small subject of careful experiment; and the densities of these materials were determined by accurate measurements of the dimensions and weight of the cubes. The cubes were cut by Krantz, special ‘precautions being taken to secure the parallelism of two special faces intended to take the pressure of the hydraulic press. The volume of the granite cube was 5:07° cm.==1380°299, and its weight 338°744 gram, Bane a denen — 2-600. A pressure of 20 tons was applied for 20 minutes with no apparent effect. The pressure was then raised to 30 tons and applied for 70 minutes withcut visible effect. Raised now to 35 tons one corner split off, but the cube remained otherwise sound. Next day at 36 tons it exploded violently. This is 9 tons or 20,160 lb. per square inch. This appears to be a fairly high result. The strongest mica granite cited in Merrill’s tables* gave 23,358 Ib. per square inch, and of 59 such granites cited only 7 exceed 20,000 lb. per square inch. These tests were mostly made on 2 in. cubes as in the present case. The volume of the basalt cube was 5:05° em.=128,775 c.c. Its weight was 374:194 grams. Hence density= 2°871. 20 tons applied for 18 minutes produced no effect. At 27 tons a small crack near one side appeared, and a piece flatted out at 37 tons. Next day the pressure was applied anew and slowly increased to 59 tons, when there was a violent explosion and the cube was scattered. This is 14°75 tons, or 33,000 Ib. per square inch, and compares favourably with certain. similar rocks, diabase, cited by Merrill, oc. ct., the strength of which in no case reached 27,000 lb. per square inch. In the case of the lithographic limestone the cube volume was the same as in the case of the basalt, and the weight 343°950 grams: giving a density of 2°639. Raising the pressure at intervals approximating to 10 minutes from 20 tons, as much as 50 tons was reached by increment of 10 tons, when a small chip broke away at one corner. At 53 and 54 tons the cube began to chip rapidly — finally cracking vertically through the middle. This is 13:5 tons=30,240 lb. per square inch. Adams applied similar tests to this rock, his results ranging from 28,000 to 40,000 lb. per square inch f. The obsidian was not tested for crushing strength. Its specific gravity, determined by weighing one of the hem1- spheres in water, was found to be 2°380. 5) ea for Building and Decoration, New York, 1897. nae ce . Adams and L, V. King, The Journal of Geology, vol, xx, Cavities in Rocks exposed to High Pressures. 687 The Experiments. ae 7 wo spheres were dealt with: one of basalt, the other granite. The hemispheres were cemented by bard: Canada balsam. The pressure was raised little by little to 50 tons read on the manometer. This was maintained constant from December 26th, 1912, till March 4th, 1913, 7. e. for 68 days. During this time the plung ger sank in about $ mm., or possibly a little more. When pressure was tolioved ‘the lower plunger under the elastic recovery of the lead within the mortar protruded about 3 mm., lifting the heavy crushing mortar by this distance. On forcing out the lead cylinders the copper washers showed considerable flowage, where they were pressed against the plungers. The lead ‘cylinders were easily parted, and by gentle warming the hemispheres opened. Both basalt and granite had yielded in the same manner. The cavity held a small quantity of powdered rock— considerably more in the granite than in the basalt sphere. The debris in the basalt weighed 0°100, and in the granite 0-170 gram. Under the microscope these powders were in- distinguishable from rock powdered—not very finely—in an agate ‘TMmortar. “The flat polished rock surfaces covering the cavity had bulged downwards. The amount of bulge in the case of the basalt was estimated to be not more than0-1mm. The bulge on the granite was rather more. It is dificult to reproduce the appearance of the specimens by photography— but the photos figs. 1 & 2 (Pl. XVI.) show that the effects are quite conspicuous. The manner in which the ony in the basalt has been extended is remarkable. The bulge on the flat covering surface faithfully follows the outline shape of the cavity. Examination with a strong lens of the flat, bulged, surfaces showed that the distortion was largely due to the “relative dis- placement of the individual mineral particles. This points to the risk of error which may obtain when experiments of this nature are carried outon simple minerals only. It may be safely inferred from this result that the pressure of 50 tons prolonged over the duration of the test is con- siderably in excess of what would suffice to close the cavity. (II.) As in (I.) granite and basalt spheres were dealt with—fresh specimens being, of course, taken. The granite in this case, although from the same locality, has a felspar which is of a pale pinkish tint, and in that respect seems to differ from the granite of (1.) and as used in the crushing 688 Prof. J. Joly on the Closure of Small test: the grain is alike. It is probable the difference is quite unimportant. In all particulars the arrangements were same as in (I.). The test extended from March 6th, 1913, till June 24th, 1913, 7. e. for 110 days—during which time a pressure of 30 tons on the plunger was steadily maintained, On relief of pressure it was found that there was no debris in the cavity in the basalt, but reflected light revealed a faint but unmistakable distortion of the flat covering surface. The appearance is that of a thin protuberant ring ; but close examination shows a raised circular area of the diametral size of the cavity. The cavity in the granite sphere contains debris, but the breakdown is less than in experiment (I.). The flat surface covering the cavity shows signs of cracking: Photos of these effects are given in figs. 3and 4 (Pl. XVI). We must conclude from this result that 30 tons pressure, even when exerted over the brief period of 110 days, has broken down the granite and produced a distortion of the basalt which remains as a permanent set. In short, there has been something like flowage. Prolonged over years or centuries—even at these low temperatures—such distortion would very surely ultimately close the cavity. gement was somewhat (IIJ.) In this experiment the arran different from that of (I.) and (I1.). Two spheres were dealt with, A and B. A consisted of a hemisphere of basalt and one of granite separated by a thin ground-steel washer, closely 0°53 mm. in thickness. B was similarly constituted. Canada balsam cemented the hemispheres together in both cases. It will be understood there is no cay ity in any of the hemispheres. The opening in the washer is of the same diameter (6°2 mm.) as the cavity existing in (1.) and (II.), and the observation is confined to effects of edi of the flat surfaces covering this opening at either side. A pressure of 20 tons was applied from October 13th 1913, till Mareh 21st, 1914, 7. e. for 150 days. Tt was then found that both the hemis +pheres of A showed visible but not very definite effects. In the case of the granite there is some fine but distinct cracking around the margin of the central circular area of relief of pressure. ‘The basalt showed one fine crack touching this central area tangentially, Cavities in Rocks eaposed to High Pressures. 689 In the case of sphere B there was no visible change pro- duced in either hemisphere. It seems as if we must conclude that at normal tempe- ratures the pressure of 20 tons on the plunger must be near that critical pressure which will just determine the closing of small cavities. This load on the plunger, the area of which is 0°775 square inches, produces a pressure of 99,145 Ib. per square inch, or 4167 kilos per square em, The depth of the earth’s surface, assuming the crust-rock to possess a density =2°8, at which such a pressure might prevail, would be about 9 miles. The temperature prevailing at this depth might be about 450° C. (Adams, loc. czt.). It would be desirable. to repeat the experiment at this tem- perature. It seems probable that this pressure would, under the high-temperature conditions, ultimately close every cavity. (1V.) In this experiment the granite and basalt hemi- spheres used in (I1I.), which showed no visible injury, were re-arranged as before. No Canada balsam was used. The second sphere was composed of a hemisphere of Solenhofen limestone and one of obsidian. No balsam was used. The washers were re-ground. A pressure of 22 tons was applied from March 21st to March 23rd, 1914, 2. e. 2 days, and then it was increased to 30 tons and left ol March 27th, 2. e. 4 days more. The results were:—A faint raised Since area in the case of the basalt—so faint that it 1s not easily seen without breathing on the polished surface. The granite reveals no definite effect. The Solenhofen limestone shows a quite distinct raised area corresponding to the opening in the washer. A straight-edge applied across the stone showed the slight central elevation. The obsidian showed a similar effect but very much fainter. Comparing these results with (II.), we are entitled to conelude that what difference there is was due to the lesser duration of the stress in (LV.). (V.). It was now resolved to apply the procedure of (I.) and (II.) to the limestone and obsidian, and to seek fora definite positive result at 30 tons. Two limestone hemi- spheres were placed vis-a-vis. One of these was that which had been used in (LV.), and which exhibited a small but distinct bulge. The other carried the cavity, and had not been previously used. The obsidian sphere was prepared in the same manner, one of the hemispheres being that which was used in ([Y.). No balsam wis used in either sphere. 690 Prof. J. Joly on the Closure of Small From March 27th to March 28th, 1914, 20 tons were applied, and from March 28th to Sept. 19th, 2.€. 175 days, 30 tous. The results were a small collection of debris manifestly spalled off the walls of the cavity, in the case of the Solenhofen rock. The covering surface was cracked and broken out- wards round the edges of the circular area of no pressure. In the case of the obsidian there was a small amount of debris in the cavity, and the flat surface was very faintly bulged outwards over the central circular area. This experiment, taken along with (II.), clearly shows that in 80 tons onthe plunger, that is 38°70 tons or 88,717 Ib. per square inch (=61(00 kilos per sq. em.), we have a pressure which is certainly sufficient to close cavities in granite, basalt, obsidian, or limestone. Here the experiments had to come to an end, the prepared material being exhausted. It had been intended to apply to Dr. Krantz for additional spheres, but, of ccurse, war conditions rendered this impossible Modifications of the crushing apparatus permitting of prolonged application of temperatures up to about 500° C. were designed, but there _ was no possibility of having these carried out. So far as they go, the experiments show that for the four different varieties of rocks tested the pressure of 88,700 lb. per square inch, 2. e. 6100 kilos per square cm., must even in the cold close all cavities in the rock, and at the probable tempe- ratures attending such pressures (between 800° and 900° C.) must be considerably in excess of the critical pressure. And further, that the signs of yielding in the cases of ae und basalt at the pressure of 59, 100 Ib. per square “inch, a. 4067 kilos per sq. cm., are » sufficient to justify the seni that at the probable corresponding earth temperature (450° C.) this pressure must be near the critical pressure for these materials. Microscopic examination of the granite and basalt shows that these rocks are quite typical ve irieties. The granite has two micas, a brown mica partially chloritised (and containing abundant haloes) and a limpid mica (muscovite). Most of the felspar is plagioclase; a part is orthoclase. All the felspar is fresh and limpid. There is abundant quartz showing the usual strings of fine bubbles or cavities. The structure is typically granitic. The basalt has fine olivines, augite, and basic felspar. It is very fine-grained. The phenocrysts are augite mainly. No glass was detected. The obsidian was not examined microscopically. It was obviously a, very homogeneous glass. A few very minute bubbles or Cavities in Rocks exposed to High Pressures. 691 cavities appear on the polished surface. The Solenhofen rock is too well known to require description. lt is hoped to continue the experiments at the earliest opportunity. The elastic theory is readily applied to these experiments, the requisite equations having already been developed by Love and Williamson. Taking the case of a sphere of homogeneous material with a small concentric spherical cavity, it is shown that if the sphere is externally submitted to a hydrostatic pressure p, then any diametral plane is a plane of principal stress ; so that if the sphere were composed of two hemispheres merely laid together, the case is theoretically the same as that of the undivided sphere. The radial stress diminishes, of course, from p on the exterior to zero at the interior surface. The perpendicular stress (e.g. on the plane surface of each hemi- sphere) is slightly greater than p near the outer surface, and near the inner surface (¢. e. the cavity) it increases to nearly =p. Rupture will tend to begin at the surface of the cavity and the planes of greatest shearing stress are at 45 to the radius. ! The actual conditions of experiment differed from this ideal case in that the interior cavity was hemispherical instead of spherical. In such a case we should expect rupture to begin in the plane surface of the cavity where this first loses support from the opposed flat surface, 7. e. at the corners where the cavity meets the overlying plane. The elastic theory shows that for a spherical cavity, if P and () are the stresses (measured as pressures) respectively radial and at right angles to the radius at a distance 7 from the centre, where 7; and r,are the radii of cavity and sphere : 7 Q 11? Po ( il iL )p To° as r° re 23 I desire to express my thanks to Mr. J. R. Cotter for a helpful discussion of the theoretical aspect of these experiments. Iveagh Geological Laboratory, T. C. D. Sept. 25th, 1920, ge ess LAXXT. Convection of Heat and Similitude. By A. H. Davis, B.Se:* CONTENTS. INTRODUCTION. 1, NaruraL ConveECcrion. Formula. Examination of Formula. 2. CONVECTION FROM A Bopy IN A STREAM OF FLUID. Formula. Ixaminatioa of Formula, (1) Thin cylinders. (2) Spheres and thick cylinders. (3) Long cylinders: thick and thin compared. SUMMARY. INTRODUCTION. NHE heat-losses from a hot body may be due to con- duction, radiation, and convection. While the two former may be calculated, if the constants of the materials and surfaces be known, the loss by convection is com- plicated by its dependence on the geometrical form of the surface. Although for the simplest forms (spheres, cylinders, etc.) the effect mav be calculable, it is obvious that in general it can only be found by experiment, using either the actual object or an object of similar form. It may be remarked here that, mathematically, convection is a combination of hydrodynamics with the Fourier equations for heat-flow, and that in the solution of the purely hydro- dynamic problems presented by ships and by aircraft the value of experiments with models has been proved. One is led, therefore, to derive relations applicable to convection from the standpoint of models. L. NarurAL CONVECTION. Formula. Boussinesq has given a mathematical solution of the problem of heat-ioss by convection, natural f and forced {. The desirability of conducting experiments and expressing experimental results in a form applicable tu models is further indicated by the fact that study of his analysis shows that he is led to consider bodies of similar form. ~* Commmnicated by the Author. + Boussinesq, Comptes Rendus, cxxxii. p. 1882 (1901). { Ibid. exxxiii. p. 257 (1901). Convection of Heat and Similitude. 693 The equations required for this paper may be deduced from his. However, they may also be obtained ‘from the principle of similitude. Rayleigh * has pointed this out for the case of forced convection. It is instructive to work out the ease for natural convection from this alternative point of view. Let the fundamental units be those of mass (M), length (L), time (T), and temperature (@). The derived units are given in brackets below. For an inviscid fluid, Boussinesq’s equations show that the mean heat-loss “hk” (MT~*) per unit area of a body per unit time depends upon k, (MU-"LO-!) ~~ thermal conductivity of the fluid. ce, (MT~*L-1®~-1) capacity for heat per unit volume of the fluid. 0, (O) temperature excess of the body a, (@-') coefficient of density reduction of the fluid per degree rise of temperature. 9. (LT *) acceleration due to eravity. l, (LL) linear dimensions of the body. Sines we deal with the capacity for heat per unit volume of the fluid, reflection shows why the density (p) of the fluid does not enter. The convection currents, essentially gravity currents, depend upon the density of the heated fluid relative to that of the cool fluid, and not upon the absolute density. Boussinesq regards the effect of the thermal expansion of the fluid as negligible except in so far as it alters the weight of unit re of the fluid—that is, except in so far as “a” and “g” occur as a product. C onsequently, let Pears tern ier te te a tC YO) then we readily derive : By mass = ae, By length O0O=u—-v+ete, By time —3 = —3u—2v—-22, By temperature 0 = —uw—vt+w—a2; whence j=—t—27, w=lte, v= de-l, v=2er. . (2) * Rayleigh, ‘ Nature,’ xcy. p. 66 (1915). GOAe as Mr. A. H. Davis on Substituting in (1) we obtain h = (hO/ (gladly <2 ey Since “‘ 2” is undetermined, we must write h = (hO/l) P(e’gab/k?), 2. ae Let us consider the assumption that expansion effects of the fluid are negligible except in so far as they alter the weight of unit volume and so set up gravity currents. With large bodies this will be fairly true. But in an extreme case of a hot thin wire, the mere volume changes of the air near the wire may be far from negligible. Indeed, observation of smoke near a hot thin wire shows that the effective diameter of the wire seems several times its true diameter. In this case we shall not expect (4) to hold. Convection data for wires, given by Langmuir *, show on test that the relation (8) based on (4) does not hold ; and in fact Langmuir, in his theory, supposes the wire sur- rounded by a stationary film of fluid, of thickness several times the actual diameter of the wire. For large bodies at moderate temperatures our formula should be satisfactory. For a viscous fluid we find, on introducing a term yt (v being the kinematical viscosity coefficient with dimensions ans h = (kO/l) (PgPab/k? )"(ep/k)y. . . . . ©) For gases, by the kinetic theory, (v/k)=const. Actually it seems constant for a given kind of gas, and varies but moderately from one kind to another. If, analogous to Rayleigh’s + treatment of convection in a stream of uid, (cv/k) be supposed constant for all fluids, the result reduces to (3) as before. Viscosity, therefore, has less effect than one might expect. On reflection this result may be made more or less clear. The kinetic theory of gases shows how increasing the viscosity (v) of a gas is eqnivalent to increasing its thermal diffusivity (A/c) in the same ratio. Consequently, although increase in viscosity may decrease the speed at which cold gas sweeps over the heated body, still at the same time the thermal diffusivity is increased, and the heat escapes laterally by this means. * Langmuir, Phys. Rey. vol. xxxiv. p. 416 (1912). + Rayleigh, loc. ct. . Convection of Heat and Similitude. 695 Examination of Formula. ae UGG gim OMe yt els ss | (4)" The form of the function F may be determined experi- mentally by finding its variation with “6,” say, the other quantities remaining constant. Dulong & Petitt, Pécletf, and Compan § have given formule in which, for the same body, hac 6'?3. Hence BP (c?ql*a0/k?) — fe"alene en)? 2°. whence (5) becomes ) he oe O1 233 e534 o'465 9293 g'283]-"901, | (6) Now we may put c=c,p, where ¢, is the specific heat of unit mass of the fluid, and for air is known to be practically independent of the density ‘“‘p.’ Hence, for any given Peet ok. = od. andes. g~ do: not alter, hoc pi 0! ®, In excellent agreement with this, Dulong & Petit found freair hocp= 023 the pressure “p’’ being of course proportional to ‘‘p.” Further, in one experiment with a very small cooling-chamber, Campan || found ha @'}4, Similar reasoning in this case gives ha p®@'l54, and Compan found experimentally h&p°° as his mean for this series. This is excellent agreement. Let us return to the question of models. If, instead of attempting to determine the form of the function in (5), we so choose our models that its value is constant, * Equation (4) is present in Boussinesq’s paper in the form pat Rea i bale aie) 2. 2). (4a) which is seen to be equivalent to (4) for kejix O?(ke2a)"? when 1x (e?a6/k?)~V, From this Boussinesq obtains an equivalent of (6), but limited to the same body so that “7” does not occur. He does not further test the equation. + Dulong & Petit. See Preston’s ‘ Heat.’ t Péclet. See Pauldiny’s translation, ‘Practical Laws and Data on the Condensation of Steam in Covered and Bare Pipes.’ C. P. Paulding. Van Nostrand Co., 1904. § Compan, Ann. de Chimie et de Physique, xxvi. p. 482 (1902). | Compan, Joc, cit. 696 ) Mr. A. H. Davis on we have eoPabik? = constant, .... 3. and then for models so chosen in size (J) hak. 9. 9 If only the temperature and size of the model are varied, c,g, a, and k, the gaseous constants, remaining the same, (7) becomes:0/°? = constant:” > 2 a eC Data are available for testing whether hAok@/l when él’ = const. H. Péclet has made, perhaps, the most complete experiments on heat-losses from surfaces (Traité de Uhaleur, 3rd ed., 1860), and he measured the rate of cooling of metal cylinders and spheres filled with water. He used bodies ranging from 5 to 30 cm. in diameter and 5 to 50 cm. in length, placed in a large water-jacketed metal cylinder about 80 cm. in diameter and 100 em. high. He worked at various values of temperature excess up to 65°. He found for these bodies hx AGP, where A has the value for spheres... f= 2 Ay Se for horizontal cylinders. A = 2°058+0:0382/r, “>” being radius in metres. Also a formula is given for vertical cylinders from which the values of A can be calculated for the similar bodies we require of this shape. All spheres are similar: and in a long horizontal cylinder it is obvious that the length does not aftect the heat-loss per unit area, so the formula is applicable to models, even although in the actual expe- riments the ratio of length to diameter may not have been constant. From Péclet’s formule heat-losses have, therefore, been calculated for similar bodies in the cases mentioned above. Following (7a) the models have-been so chosen that 6? = const. = 6x 10%. The result) is given in “Lables This choice is such that the first seven models roughly range in temperature excess and size over the region in which Péclet’s experiments were confined. The eighth model, working at 480° C. excess, is somewhat theoretical. Convection of TTeat and Similitude. 697 TABLE I. Heat-Loss from Models caleulated from Péclet’s Formule. (Test of relation h «@// for a series of models in which é/? = constant.) Model. a Relative value of h-- (0/7) *, i. area. 7 SD. Sat a EER : a irae - 26s ny "ey dimensions, Sphere. ate Bes Mean , Z (cm.). } : ylinder tf. 60 10:00 1:00 1-00 1:00 1-00 50 10°62 0-99 1-01 0-99 1:00 40 11°45 0-99 1-02 0°99 1:00 30 12°60 0:98 1-04 0:98 1:00 20 14-42 0-97 1:06 0:97 1:00 10 1817 0:97 lll 0:96 1-01 75 20-00 097 1:13 0:96 1:02 480 . 5-00 1:16 0-94 1:09 1:06 0-94 40-00 a es 0-97 0:97 * Value given, for each shape, relative to value for first model of that shape. Bo spheres and horizontal cylinders “/” is the radius, for vertical cylinders it is the height. t Vertical cylinders have height=5 x radius. N.B.—In any test of data obtained from empirical formule it should be borne in mind that the formula will tend to be most reliable in the central region of the experiments on which they are based, and less reliable at the limits. This table indicates that instead of the arbitrary form ha AO'233, where A has complicated empirical values, Péclet’s results may possibly be presented in the form suggested by (5), h a (0/l) F(@l3). This could only be decided by a complete study of the whole of his experimental results. However, the above shows that it is fairly true for F(.r)=2«°?*, In any case, it would be valuable to express results in a form applicable to models, and if this were found impossible the fact is worth stating. An alternative way of testing Péclet’s data is to plot a curve with h+@/l as ordinate and @/? as abscissa. Of course, only values of / and @ are taken which fall within the range of the actual experiments. Phil. Mag. 8. 6. Vol. 40. No. 240. Dee. 1920. 22 698 Mr. A. H. Davis on 2. CONVECTION FROM A Bopy IN A STREAM OF FLUID. Formula. Convection from a body in a stream of fluid moving with velocity v (natural convection being negligible) has been determined by Boussinesq, == (kO/D) Bilvefk), - . s,s ae and Rayleigh has shown the derivation from the principle of similitude. Since from the kinetic aS of gases, as mentioned above, cv/k is constant, (9) becomes h=(/0/l) F(lv/v), and ‘lv/v) is the well- inca variable In experiments on fluid resistance and flow. This raises the possibility of a useful relation between the thermal and dynamic effects of a fluid stream. Jt is important apart from this, however ; for if heat-loss is affected by change from stream-line motion to turbulence, it is satisfactory to find the equation for heat-loss contains the variable (/v/y) which determines turbulence. A formula given by Russell* for stream-line flow past a cylinder at right angles to the stream is readily seen to be equivalent to (9), where F(lvc/k) x (lve/k)¥*. Although such special approximate forms of the function F calculated for stream-line flow might cease to hold when turbulence set in, the formula (9) would still be satisfactory. One recalls that in dealing with purely hydrodynamic problems of the flow of viscous fluids in cylindrical pipes, Stanton & Pannell + investigated the resistance to flow from the stand- point of similitude. Their results show that although the relation analogous to (9) has a special form for str eam-line motion, which apparently alters when turbulence sets in, the general relation holds even for turbulent motion, and is continuous through the transition region. Eevamination of Formula. Experimental work on the heat-losses from bodies in a stream of air has been carried out chiefly on thin wires, but some data are available for larger cy linders. Let us consider these two cases. * Russell, Phil. Mae. xx. p. 591 (1910), t Stanton & Pannell, Phil, Trans. ccxiv, p. 199 (1914). Convection of Heat and Similitude. 699 (1) Thin Cylinders. Kennelly * and Morris ¢ have investigated the cooling of thin wires in streams of air. King ¢ has given, perhaps, the most comprehensive investigation, working over a wide range. He used wires ranging in diameter (d) from 0:003 em. to 0-015 em. He worked at velocities (v) of 17 to 900 em. per sec., and at temperature excesses (@) (wire over air) of 200° C to 1200° C. Asa result he gives a formula, which may be written a= bee Oe. oe - 60) H is the heat-loss per cm. of wire, B and C are practically constant. B has a slight coeficient depending on the temperature excess of the wire; C has a larger coefficient, and also depends somewhat on the diameter of the wire. We may note that, in (9), “h” refers to unit area of the body. If the body is a long cylinder, the length obviously does not affect “A.” It readily follows for long cylinders (diameter “‘d”’) that the rate of loss of heat “ H” per unit length is given by H = (k0)F(vde/k). 2 2. . . (AD Thus King’s results show that the formula (11), which for these experiments reduces to El eBay oe a ad oe ELD) is very satisfactory, but slight corrections seem necessary. The results are in a form which readily shows this, being essentially the evaluation of the constants B and C in his theoretical formula. (2) Spheres and Thick Cylinders. Compan §, using a sphere, verified Boussinesq’s approximate equation within narrow limits of temperature and air-velocity, but he did not use bodies of different size. However, Hughes || has investigated the cooling of cylinders (0°5 to 15 cm. diam. ) over a range of air- -velocity from 2 to 15 metres per sec. The experimental data are given in a full table in his paper, and curves are given hetween the heat-loss “H” * Kennelly, Trans. A. I. E. E. xxviii. p. 363 (1909). + Morris, ‘ Electrician,’ Oct. 4, 1912, p. 1056. { King, Phil. Trans. cexiv. p. 378 (1914), § Compan, loc. cit. || Hughes, Phil. Mag. xxxi. p. 118 (1916). 2242 700 Mr. A. H. Davis on in calories per cm. of the cylinder and “wv” the air-velocity : one curve for each value of the diameter (d). They have been re-drawn in fig. 1 in c.G.s. units. The relation between the four curves is not obvious. However, formula (12) Riera: (e) 2:0 Vo er cm. per sec, P P 05 suggests that they can be replaced by one curve if “ H” is plotted against “vd.” In fig. 2 this has been attempted from the experimental data*. It is seen that the result is very satisfaetory. It is excellent for the three smaller eylinders (0 43-1:93 em.) and quite good for the fourth (5°06 em.), although the points of the latter all seem displaced somewhat from the line of the former. Possible sources of this displacement may be mentioned. In the “vd”? term there is possibility of error in the velocity “v.’? Hughes’s method of allowing for lack of uniformity of distribution of the air-stream in his channel is open to criticism from a precision point of view. A * The points of fig. 1 are too numerous to plot individually in fig. 2, and advantage has been taken of the natural groups (of from 2 to 6 points) which occur, and mean values have been plotted. Convection of Heat and Similitude. 701 Pitot-tube reading is proportional to v?, whereas the heat- loss from small and large cylinders is proportional to v° and wv? respectively. So a value of average velocity, as deduced by Hughes, from the mean square velocity obtained Fig. 2. 2:0 4° a @ ey 1:5 = : oa WM S. - = 1040 0-4 —- 2cm, diam. ( Hughes.) 506 " " ( " ) * 5:08 » “ (Carpenter.) © 0-003 -0-015 » (Formula, Kirg. ) from Pitot-tube readings, is not strictly applicable. Turther, the particular average required depends on which cylinder is in use. It will be noticed that the double series of points on the graph of fig. 2 come where the high-velocity values of the smaller cy ‘linders are plotted near the low- velocity values of the 5°06 cm. eylinder. One may mention also that experience with wind channels shows that rectilinear stream- line motion is not obtained unless special precautions are taken. ‘The entrance must be of stream-line shape, and a special metal honeycomb should be used at the beginning of the channel proper to break up vortices and ensure recti- linear delivery, Even if there is actual discrepancy, it is obvious that the law Ha (cd) is approximately true. Unless a better theoretical formula can be advanced, this point is worth emphasis in the form in which results are presented, even 702 Convection of Heat and Similitude. although a slight correction factor be introduced for deviation with diameter (d). Finally, Langmuir *, in his study of convection of heat, quotes from Carpenter (‘ Heating and Ventilation of Buildings,’ Wiley & Sons, 1903) values for a 2 inch steam pipe, sensibly the same size as Hughes’s largest cylinder (5'06 cm.). These convection values, caiculated for a temperature excess of 85° C., are plotted as asterisks in fig. 2, and are seen to lie higher than the mean curve, although the points given by Hughes lie lower. (3) Lonug Cylinders: thick and thin compared. Hughes gave the results for his cylinders in a form Had*"y", where n varies with diameter from 0°55 to 0°7, and shows signs of depending also upon the velocity itself. Obviously, this is not a suitable formula for extrapolation. King, working over a wide range of conditions, gave the formula (10) above. It is better adapted for extrapolation. Particularly, there is no sign of variation with velocity apart from that under the root sign; so we can work with a wire of diameter within King’s range, only extra- polating the velocity. Let us take, then, two wires approximating to the smallest and to the largest used by King, of diameter 0°003 and 0-015 cm. respectively. As with Hughes’s cylinders, let them work at 100° C., in air at 15° C., this being appa- rently the temperature (deduced from his radiation data) at which Hughes worked. ‘Then, if we choose velocities to give values of “vd” similar to those in Hughes’s experiments, we may calculate from King’s constants the appropriate heat-loss (H). We find for both wires when vd = 500, H = 0°46, 5 Cl = 1000; EH = 0:66, These points have been plotted on the graph of fig. 2, where remarkable agreement is seen with the curve obtained for cylinders 30 to 1600 times as thick. The extrapolation is considerable, but we recall that the most satisfactory part of the most satisfactory formula was chosen to stand the strain. It would not seem possible to conduct experiments to test this particular extrapolation, as the velocities are so great that the fine wires would oscillate or break under the mechanical strain. * Langmuir, Trans, Am. Mlectrochem, Soe. xxii. p, 324 (1918), s Space-Time Manifolds and Gravitational Fields. 703 SUMMARY. 1. The hydrodynamic basis of convection of heat suggests ey by means of models. - We have the formule :. Natural convection . . Ah = (kO/l) F(c?gl?ad/k*), Forced convection . . h= (kO/l) F(lve/k). Graphs drawn with h=+(h@/l) as ordinate and either (eylab/k?) ov (lve/k) as abscissa should be independent of the size of the object; consequently, to ascertain the heat-loss for any particular body it should be necessary only to perform the appropriate experiment with a model. 4. For natural convection Péclet’s data have been analysed with promising result. The formula would not be applicable to bodies where the fluid expansion caused was no longer negligible as a mere volume change. ). For forced convection the formula, texted by data given by Hughes, is very promising. The cooling fluid is not heated so much as in natural convection, and can still be regarded as incompressible for smaller bodies at higher temperatures. The formula is good, even for thin wires, and it is satisfactory to trace in it the hydrodynamic variable determining turbulence. 6. Hyvidence available in published data indicates that, for heat-loss from a body, an excellent first approximation can be obtained from experiments with a model. The principle of similitude affords a convenient method of expressing experimental results. March 1920. LXNXXIT. Space-Time Manifolds and corresponding Gravita- tional Fields. By Witrrip Witson, B.Sc., Northampton Polytechnic Institute *. HE main purpose of the present paper is the investigation of the gravitational field of an infinite uniform recti- linear distribution of mass or, more precisely stated, the determination of thé equations of the geodesics in a space- time manifold in which the square of the element of length has the form ds? = — fdr? —fydz? — fsr'dd? + f,dt?, pe CE) where the /’s are functions of 7 only f. * ie es by Dr. Wm. W ilson. + When f,=f.=f;=fi=1, 1 # and @ are the ordinary cylindrical space co- ae 704 Mr, W. Wilson on Space-Time Manifolds — So far as the writer is aware, the only gravitational field which has been investigated from the point of view of Hinstein’s theory is that of a single particle or of a number of isolated particles. On Newton’s theory the intensity of the field in the neighbourhood of such an infinite rectilinear Agee ae : 2m ; distribution of mass is equal to —, where m is the mass r per unit length, using gravitational units. The following investigation shows that the intensity as given by the general theory of relativity is, to an exceedingly close approximation, equal to the Newtonian result. Before proceeding to the actual investigation it will be well to study the following simpler types of manifold in which the square of the line element has the forms :— ds? = —da*—dy? —dz* —2atdadt+ (1— at?) dt?, . (2) ds* = —dr*—dz* —1°dd? —2ar'ddpdt+ (1—r’w’) dt?, (3) ds? = —Adr’—dz*—r'd¢'+ Bat, os a, 7 where «, w, A, and B are constants. : In the relativity theory of gravitation the general form of the square of the element of length is 1, 2, 3,4 Gstce = Gxt dacdar. KT The potentials 9,, satisfy the equations Sug = Oy a er where ean S ae ey So" Ose Gyr = > oe. Oxo i ae o_ logVy Hele eee o logV9) and 1,2,3 fe b> J ge (oe M80 , Oye - 221) BY 22, On, * Bie, 0X5 In some cases the oatenala 3 Jer may also satisfy the equations ee a ep pve * Hinstein, dnn. d. Phys. xlix. p. 769 (1916). Hinstein only uses co-ordinates for which g=1, See also Hddington, ‘ Report on Relative Theory of Gravitation,’ where 0 ey eae 6 ge - Abie = ss = > Dr, as ore se I €0 : Ou, et . fio fo) US pv The latter equation expresses the necessary and sufticient condition that, by a suitable choice of coordinates, the square of the element of length can be put in the form :— Eee 8.4 a Sealant ees oe a (7) s and the gravitational field made to vanish everywhere. Such a field may conveniently be termed a non-permanent one. The manifolds (2), (3), and (4) furnish simple illustrations of such fields. In the manifold (2) the values of the ger and g*t are :— er =—] : i= pepe: | gm =—1 |g? =-1 | 9Js=—1 ee i el | Jas = 1-2?" Role = 1 Lodi = —at git =—at, and erent where g is the determinant of the ger. The Christoffel expressions I By all vanish with the exception of T'4,, which has the value R= a: The equations 02, ee o Oe 32, ‘ eee ~ Py.55 35 2 7=12,34 -. (8) a of the geodesics then give the following equations of motion :— i ; ds? ag d*y ay = U, ds 9 d*z —— ds” ‘ Og 0, 706 Mr. W. Wilson on Space: Time Manifolds where the constant ¢ is given by at - = Ce ds The first of these equations may now be written 2x ——- = — & dt? 4 showing that the manifold (2) corresponds to gravitational field in the « direction. In the manifold (3) we have a uniform (gy =—1 (ge il fu g [on =—1 |g? =-1 1 ce are aS UAE ae { 933 = —1 oe oe Oe ate ee — pee 34 —= 2 NACE Disco are Cte ams and g = fare Pr e The non-vanishing Christoffel expressions [%, are : i Ui 2 IY;3=—?, Tis —, 7 oy) Vu oF, Y= pe My =—@'r ; and the equations of motion (8) are : a. = ; fa) : ye a } = 0, ads” ds GG) ean: a(t ot) i ds? rds ds : (alae j ads? = 0, a*t ds? = 0, The last equation gives dt oe Ae . and corresponding Gravitational Fields. 707 it . . . * . ( . where ¢c is a constant. Substituting this value ot aa in r the first equation, we obtain ar ( + o) — ). dt? aye dt and taking @ to be an angular velocity we have the equations of motion in a centrifugal field. In the manifold (4) the non-vanishing Christoffel ex- pressions are i r 3 3. r= 2 / and E 33 = Bans The equations of motion (8) become ar ir (dd\? eee | 2 a LO 2 eree — 6: | ds? — rds ds Pe Siac Lines eel onthi stem top, eee at | fae If, instead of @ we use ¢’, where $= VA, the equations (9) become re (2 ) Ly ds? ds d°g' 2 dr dq - — () ds? ‘rds ds : dz = ) ds? ; d’t = (). ds? We may therefore regard equations (9) as the equations of motion of a particle moving with uniform velocity ina straight line with respect to an inertia system (r, 2, p); ~ 708 Mr. W. Wilson on Space-Time Manifolds _ viewed from a system (7, z, @) rotating in the inertia system with angular velocity (1— WA) times that of the particle in the inertia system. We now proceed to the investigation of the manifold (1), where ds* = — fdr? = fydz? —/37° dd? + f,dt?, and the functions SS a eR se ie) must be solutions of the equations (9). lor our purpose there is no loss in generality in using any function of r in place of r. We therefore write —rf3(") = —r? oor 2 ee and obtain for the square of the line element the form : ds? = —f,dr?—fyde’—r'd¢*+fi,di?, . . (11)* where the aceent has been dropped after making the sub- stitution (10). It is easier to deal with the equations (5) if we write ds? in the form ds? = —eddr*—e'd2—rdd?+edt?, . . (12)t where e’=/\(7) etc., and A, mw, and v are functions of which have to be determined. The Christoffel expressions [y are then found to be - (My = 43N, (V4. = 3h, My =— He’, 4 Tea I y 1 yv aes ame ea OY mea [araee OS ren setae cane tees ee i, CW eh where x! = O% ij. OF OP SS Cr’ | Or” Ds On" Substituting these expressions in the equations (5) the * T am indebted to Dr. Wilson, of King’s College, for suggesting the investigation of a line element of this type. i Kddington, Report. and corresponding Gravitational Fields. 709 latter become : sy — IN +h? = ~p'aaX + 2/y' —1y"?, G3) ie Fhe dh — eee ap tte wales = (14) Re ets Pe ween net cne as CLD) ae duly = — a) EIN ave (16) Substituting the value of X' given by (15) in (13), (14), and (16), we obtain the equations : oe a — +, 7 sees ee (Lh) / ee ee (18) - Vv ae , = 0. . ° e ° . (19) From (18) and (19) we find pl am Pere sag oo ; a p=: ” 3 and therefore from (15) 4m + a b) N= fe where 4m and a are constants of integration. Substituting these values of pw’ and vy! in (17) we see that . 4m Ol = Ln’ whence Sm? —4Im+a = —— ] — Jn’ and therefore 3 , Am —_—— ihe = a, 4m if 1~—2m° 7’ eee Sim? ] 710 Mr. W. Wilson on Space- Time Manifolds ‘Integrating these last equations, we obtain =—4mlogr+A, ) | yp ee. 8m? Ne One ty where A, B, and C are constants of integration. We shall see, when the equations of motion are written down, that m can be identified with the mass per unit length of the z axis. When m is zero the square of the element of length will take the form ds? = —dr*—dz*—1r'dd"*+dt’ ; 1.¢., when m=O, w= 0, w= 0; v= 0. Therefore A= 0.0 B= 07.0 = 0, and the values of A, », and v are po — — 4m lower, VY = 4m i : 1—2m Te 19 8m? = oe log’. The square of the element of length (1) is therefore aC 4m ) dr? — 7-4" dz? — v2 dd? +r 17 on ge (1') The Christoffel expressions which do not vanish are :— = Hel ree —2m TP 2a ar OER Ta ee 2m A rl il {fei oe be 1—2 Deg Wee if 22 aa a a 0 event 5) 7 r (g- 8m? ) 9 il ‘it er 1—2m he of mn a BR} b) eh as 1 9 eo. —Zm LY 2m (4m-1) Bae e tie and corresponding Gravitational Fields. ie | Substituting these values in the equations (8), we obtain for the geodesics the equations ee pees d*r Am? \ 1 =) . = 1-_ “Gy Tat Tale (G + 2m! ds. MEH) 62 (Hho, cy "sy ce eM hae! eee 0, cs eH) Fa =e C3 We may interpret these as the equations of motion of a particle in the gravitational field of an infinitely extended uniform rectilinear distribution of mass alone the z axis. From (25) we see that if the particle is moving initially in the plane z=constant, it must remain in this plane ; i.e., we have always dz / aa 0 (25°) From (24) we have ne =i fae tint Gade) where / is a constant of integration. This equation simply states that a radius vector sweeps out equal areas in equal times. From (26) we get — 47 dt a) = (Pp 1—2m Bi ale Seat CCB GN) where ¢ is a constant of integration. dz lt Substituting these values of aa ae (25') and ds ds (26’) in (23), we write (23) and (24) in the form: eee (tm eS) l ds? ds ? —2m ee ({) ben € : pe As. (88) 712 = Space-Time Manifolds and Gravitational Fields. : The Newtonian equations of motion for the type of field we are dealing with are: | ee o (Z) ee dt? \dé we? = i, te ae ee ree Comparing these with (27) and (28) we notice they are identical for a sufficiently small m. Since we are using gravitational units, m is, for any solid cylinder of laboratory dimensions, negligible compared with unity and we see that (27) takes the form 2, ld 2 2 dy (B= a a d= \ds J ~~ > But for small m, (26') gives dt ds eae and substituting _ for c in (29) we obtain the Newtonian equation (27’). When m, however, is very great the equations of motion are more complicated. It is instructive to put equation (27) in the approximate form obtained by neglecting small quantities of the second order. On eliminating ds by means of equation (26') we obtain dy apy? = 2m dt? ; dt pl = 4m? 2 nevlecting quantities of the order of m*° and assuming the radial velocity component S to be small or zero. t To this order of approximation therefore, and with the assumption just mentioned, we may take —3gu to be the gravitational potential of the field we have been investigating, since 0 a — ie 0 = ec rs Ai. ( D gives for the intensity of the field the expression obtained above. Northampton Institute, EC: 20th July, 1920. porary] LXXXIIIL. On some Optical Hfects including Refraction and Rotation of the Plane of Polarization due to the Scattering of Light by Electrons. By Sir J. J. Toomson, O.M.,, J gel oer Mig W* begin by considering the radiation along the axis of w emitted by the electrons in a very thin slab bounded by planes at right angles to this axis; all the elec- trons in the slabare supposed to be moving in the same way at the same instant. If an electron O has an acceleration f in a direction at right angles to OP the electric force at P at the time T due to this acceleration is parallel to fand equal to —ef,/c? . OP, where /; is the acceleration at the time T—OP/c, e being the charge on the electron and ¢ the velocity of light. Let P be a point in front of the slab on the axis of 2; if the acceleration parallel to y of an electron Q in the slab is @?y/dt?, the electron will produce an electric force parallel to y at P at the time T equal to e d*y cee pola ), 80? pfovided PQ is at right angles to y. Since the optical effects at P of a plane wave arise from the parts adjacent to O, where O is the foot of the perpendicular let fall from P on the plane of the wave, we may make this supposition without loss of generality. If D is the thickness of the slab, o the density of the electrons, Y, the electric force parallel to y at P at the time T is given by the equation ‘ e.oD(1/d?y —— { (iz) ,_ <2 edb , z where pP=OO. r= Now 7? =p? + OP2, hence rdr=pdp, anid therefore 2 */ qa Re ema) { (5) ad Cc dt T-* * Communicated by the Author. Phil. Mag. 8. 6. Vol. 40. No. 240. Dec. 1920. 3 A. 714 Sir J. J. Thomson on some Optical fects Let o=t— set ? as throughout the integration t is constant dr=—c.do. Since at the same instant the accelerations of all the electrons in the slab are equal, y=), (Gi), a ee hence y= 2 eep| Be da ie es ae kel ee @}] ea x R The limits of w are fae where v=OP, and t— 1 where R is the ereatest value of PQ; when R is so large that t—R/c corresponds to a time before the electrons began to be affected by the light wave, so that Y joo =F eg) (4 where v a is the velocity of an electron parallel to y at the time t—a/c. If k=" oD, we have eg nee a Oe Si c Similarly Z, the force parallel to z will be given by L,=—kw aed pie c where Ws is the velocity of an electron parallel to z at the e x time t— —. C due to the Scattering of Light by Electrons. 715 Let us now consider a plane wave travelling in the positive direction of the axis of w, let f(a, t) be the value of the electric force parallel to Z at the place w and the time ¢. Consider now how the force at the point #+ 6a, at the time t+6z/c, differs from that at v, and ¢,. The force which is at « at the time ¢ will be found at «+dz, at the time ¢ + 82/c, ‘and in addition to this force there will be found at # + 6x the force due to the secondary waves which come from the electrons in the slab 6a. This force, as we have just seen, is equal to Qa dz — eC (S Ou. C at Hénce we have f(@+ 6x, t+ 60/c) = f(a, t)— EL eo (5) On. This equation is a ates to yee NS eee 5 «) Pree ae es 6 on dt If the equation of motion of an electron in the slab is (2) and if 7 varies as e'?*, then and equation (2) becomes Opi Lay me 2a ea df ee ne a or Bie ay Pikeman went ate (3) Gta Ce 1 fe Pa ag eee ye: it — ——- . . . . . 4 : Fae Ree n?— pp? oe. The solution of (2) is ee akae ‘ . . : E representing a disturbance propagated with a velocity c’. If w is the refractive index, pele, 2a ze o (=p) hence by (4) po=l+ —— 3A 2 ~ 716 Sir J.J. Thomson on some Uptical Effects This is the well-known expression given by Lorentz. If the electrons have not all the same frequency, then if oy is. the density of those with the frequency n,, we can show without difficulty that Tones We Soc ne — p : This expression has been used by Drude and others to determine the number of electrons in the atoms. In using it for this purpose, however, it is necessary to take account of some considerations which may be illustrated by the following example. Let us consider the case of an electron in an atom of such a character that the displacement of the electron by an electric force is not in general in the direction of the electric force. There will be three directions fixed in the atom such that a force along one of these directions. produces a displacement in the same direction. If these directions are taken as the axes of &, 7, €, the equations of motion of the electron may be written m (a5 ae +n) =el’,, m (a ate ile 'n) =el’ , m (5 oy n,tt) =cl,, where I’,, F',, F, are the components of the electrie force in the directions &,, € respectively. Letthe axes of &, 7%, & cut a sphere at the points A, B, C, and the axis of « at the point Z; then, if 2 ZC=9@, pnd f is the angle between ZC and CA, an electric ee Ziinivonine light wave will be equivalent to the forces P,=Z cos 8, Pi=Z sin @ sin ¢, F,= —Z sin 0 cos ¢, and if Z varies as e”, we get from the equations of motion of the electron e Lsin@cos _ ¢ Asin Osin fb Sm n—p pe a= ea 1 P DL Ns =P ¢ e Loos @ mn; — p? The displacement resolved along the axis of z is thus ay, ( cos?@ sin? @ sin? hi. sin ae cos? one’). Ne? — ne? — p 0 due to the Scattering of Light by Electrons. fay If the atoms are orientated uniformly, the mean value of this expression is Le uF 1 Ti . (a0 RO CaO elke om ‘) eo) D My aay Ilo ae eal! Ig “ea So that if o is the density of the electrons, the value of yu the refractive index will be Pre as. I iL 1 1 (tg ta) sO) d3 m\n?—p* ne—p* n3’—p" Thus if no and ngs were very large compared with 1, HOT OR ee thus the true value of o would be three times that obtained by the usual formula p=1+20—-—— _The usual formula‘is only true when the displacement of the electron is always in the direction of the resultant force ; when this is so ny=ny=n,, and the expression for mw takes the usual form. Conduction in Metals. If we suppose that the velocity of the electrons is pro- portional to the electric force, then | de Cf ati wea’ where ( is the specific conductivity of the metal; for eo is the current through unit area. Using this value of ad dz/dt equation (2) gives ee. Bah Hag Oe i Rape da c dt rs The solution of which is ae eo a Cth =e), Since the intensity of the radiation is proportional to /’, the rate of decay of this intensity is 4nrC Er Pe os so that the coefficient of absorption is equal to 47 /e. . 718 Sir J. J. Thomson on some Optical Effects - Changes produced by the Klectrons on the type of Polarization. We see from equation (1) that if the electrons acquire from the primary wave a finite average velocity in any direction, secondary waves with the electric force in that direction will be emitted. Thus unless the displacement of the electrons due to the electric force in the primary wave is in the direction of that force, the electric force in the secondary waves will not be in the same direction as that in the primary, and the mixture of primary and secondary waves will differ in the state of the polarization from the primary light. The change may be one or other of two types: (1) the mixed light may differ from the primary by being elliptically polarized; or (2) it may still be plane polarized, but the plane of polarization may be rotated. Y the force in the scattered light is by equation (1) proportional to dy/dt: hence, if y is in,the same phase as Z, the force in the primary light, dy/dét and therefore Y will differ in phase from Z by a quarter of a period, so that the mixture of primary and scattered light will be elliptically polarized. If, however, dy/dt is in the same phase as Z, Y and Z will be in the same phase and the mixture of scattered and primary light will be plane polarized, though the plane of polarization will not coincide with that of the primary light. Unless the electrons in an atom are distributed in an exceptionally symmetrical way, a force parallel to z will produce a displacement parallel to y, but in cases similar to that discussed on page 716 the y displacement will be in the same phase as Z, and dy/dé will differ in phase from Z by a quarter of a period and the light will be elliptically polarized. The amount of ellipticity in the polarization will depend upon the orientation of the atoms or molecules, and in non-crystalline substances there may be as many molecules producing a positive effect as there are producing a negative one, so that the aggregate effect may be nil and the light will ‘continue to be plane polarized. Thus in the case considered on page 716 the displacement ao to ¥ produced by the force Z is £(cos O sin d cos r+ cos ¢ sin W) | +n(— cos @ sing sinw-+ cos ¢ cos ~)+ (sin 6 sin f), where @ and ¢ have the same values as before, and y is the angle between ZC and ZX. Substituting the values of due to the Scattering of Light by Electrons. (ib he &,7, € given on page 716, we find that the displacement parallel to 7 is Z : pt ae ‘ a sin 6 sin y(cos @ sin g cos + cos d sin Yr) m (n;*— ne a sin 6 cos w(— cos @ sin ¢ sin ~+ cos ¢ cos fy) + — cos @ sin @ sin ot nr 3 If there is no order in the arrangement of the atoms or molecules, we see that the average value of each term is zero, so that the light would not be elliptically polarized. Magnetic Rotation of the Plane of Polarization. An important case when an electric force acting on an electron may produce motion at right angles to itself is when the electrons are exposed to a magnetic field parallel to the direction of propagation of the light. Let H be the magnitude of the magnetic force, y and z the displacements of the electron parallel to y and z respectively, Z the electric force in the light wave. The equations of motion of the electron will be of the form at NO RA dy m (Ge +n 2) = Ze +e = m (2 +n) — He : Hence, neglecting terms in H’, we find if Z varies as e?", oR dt mF (n? =p")? We see from equation (1) that this involves the production of a wave in which the electric force is parallel to y, and that Y the magnitude of this force for a slab of thickness D is given by the equation V2 2p eS RE Qarae® p? me . . (1? —p?)? ° H . Lis In traversing the distance D the plane of polarization is twisted through the angle Y/Z, or ane ep 3 eH. D.. m2 5 (n?— py? 720 Sir J. J. Thomson on some Optical Effects If w is the refractive index, | 2me-o Lee mn? ay Py? hence the angle through which the plane of polarization is twisted is equal to R= preh te) Oe pa (6) ( mn?— p this can be written as p é dw 2¢m ipo (oho) which agrees with the expression given by Becquerel. He/m is the angular velocity with which an electron would describe a spiral round the line of magnetic force H ; let this be denoted by , then if 7 is the time the light takes’ to pass across the slab D, the angle through which the plane - of polarization is rotated in this time is 2 pane (u—1) ep Thus when the atoms have only one intrinsic frequency the rotation is given by a very simple expression. If n is large compared with p, the rotation is proportional to (u—1)p?/n’, and is thus proportional to the square of the frequency: a result which is a rough approximation to the truth in a considerable number of cases. If, as indicated by equation (7), the rotation can be expressed as a function of the dispersion, it follows that no information as to the structure of molecules can be obtained by experiments on magnetic rotation which cannot be obtained by experiments on dispersion. 2 OT. Rotation of the Plane of Polarization by substances such as quartz or sugar solution. We cannot explain this rotation if we consider isolated electrons in an atom, but we shall see that we could account for it by a system of electrons held so firmly in position that they act somewhat as a rigid body, a force acting on one electron displacing the whole system of electrons. We shall suppose as before that the primary beam is travelling along the axis of «, that it is plane polarized, and that the electrical force in it is parallel to z. Let the co- ordinates of an electrical charge, electron or positive particle, be as, Ys, %. If Z, is the ‘electrical force in the primary wave at this point, the moment about the axis of w of the forces acting on the electrical charges in the atom is LeZ,ys. due to the Scattering of Light by Electrons. 72] If the dimensions of the molecule are small compared with the wave-length of the light, we have, very approximately, IZ, 4.= Zot i Dats du where Z, is the value of Z at the origin O a point inside the molecule. Thus the moment of fhe forces about the axis of w is Vp , dZo 4 , LOY + reo AOU GY 50 av Similarly there is a couple around the axis of 7 equal to SE Lo dets— gan 7 Rae Dex,”. These couples acting on the ae of electrical charges considered as a rigid body will cause it to rotate, and thus move the individual charges. If the system is not symmetrical the average velocity of the charges parallel to may be finite, and hence, by equation Cine give rise to an electric wave in which the electrical force has a component paralle] to Y. ‘The phase of this force is the same as that of ee The amount of rotation of the plane of rotation will depend entirely on that part of the force Y which is in the same phase as Z. There may be other parts differing in phase from Z by a quarter of a period; these will affect the ellipticity of the polarization, but not the rotation. Though the values of dy/dt for the different electrical charges may be all in the same phase, yet, since the charges are not all i in the wave-front, the secondary waves from fem will not, when they arrive at a point, be all in the same phase. Thus, if x, be the x coordinate of one of the particles, the phase of a vibration due to this particle relative to one ; : + - d Zz starting from the origin will be accelerated by i so that, if the velocity of this particle were representéd by ats x i-# vo) a the electric force due to it ae have the phase corresponding to . 277 C08 | (vt — y+ Ws), and would be represented, since w,/A is small, by 20 , ’ 2m. 2 cos —({ vt — 4) —#,— Sin — (vi— Xp). + v a » ( ») 722 Sir J. J. Thomson on some Optical Effects dy sen . 29 Similarly, a term in a represented by sin > (vt — vo) ¢ would give rise to vibrations represented by j 2a 2a sin ~ (vt — vp) + & cos —(vt — 2). 7 none r vs 0) Treating the collection of electrical charges as a rigid body, we proceed to calculate the value of Se due to the rotation of the molecule produced by the couples arising from the electrical forces in the light wave. Let the principal axes of etn of the molecule be taken as the axes of a’, y', 2’; let (1,, m1, 71), (lo, m9, M2), (Ig, 73, 23) be the direction cosines of those axes with respect to the fixed axes @, y, 2. Then the moment of the forces about the axis of a” is de (4+ : v) (sy! = 12") Chee Now whe’ + loy' +152’, hence the moment of the forces about a’ =Z,(n3dey' — ngdez') + oe (ng —n,N). ih=Se(ha'la! + ley! + lna'2"), M =e (lw’y! + loy'y’ + lsy’e"), N =e De (da'e" + Loy'2! + lag 2’). Similarly the moment of the forces about the axes of y' and 2' are respectively F VAS Zo(ny Bez’ — ngBew") + = (2,N —nghL), Zy(ngdev’ — ny dey’) + Soa —71,M). Let Ly = ake - ~~ (t— e, 726 Sir J. J. Thomson on some Optical Hffects where wv, y are the coordinates of the centre of figure referred to axes through the centre of mass wpe = Deu! yre= dey! hence we see Ley’ Dex 2! = 2 y e( ze) = Dew yey 2 = Desk (5) if there is a plane of symmetry in me molecule : for if this plane be taken as the plane 2’=0, Pee 0, ere — 0), Leyla'=0, and hence Q=0. It follows from this that if ail the atoms in a molecule are in one plane, the molecule cannot when in solution give rise to optical rotation. _ It should be noticed that molecules when in a crystalline arrangement could produce optical rotation when they could not do so in solution, it being assumed that the molecules are not distorted by solution. We have seen that when in solution, ether dynamical or electrical symmetry is fatal to rotation. Whereas in the crystalline arrangement rotation would in general exist, unless the molecule was symmetrical dynamically as well as electrically. For rotation to be absent in the crystalline arrangement the coefficients of 1,7, 157, U3”, Uyl, dyl3, Uof3 in the expression for Q (p. 725) must all vanish. ‘T'wo important cases in which this condition is fulfilled are (L) when the centre of the electrical charges coincides with the centre of mass ; (2) when the molecule is symmetrical about an axis. For the present I shall confine myself to the case of rotation in solutions, as this is the one to which the attention of investigators has in the main been directed. As most of these have occupied themselves with substances which contained asymmetric carbon atoms, it is interesting to com- pare the value and sign of Q for two molecules (.) amd aye where (ii.) is such that its masses and electrical charges occupy positions which are the images of their positions in (i.) in a mirror placed in one of the principal planes of (i.). Let us take this plane as the plane z’=0. The coordinates of the masses and charges on (il.) will be those of the corresponding mass of (1.) with the sign of the 2’ coordinate reversed. Thus Sew’, Sey' will be unaltered, while Sez’ will change sign. Again, Seu'y' will be unaltered, while Sex’s! and Yey's’ will change sign: hence we see that Q will have due to the Scattering of Light by Electrons. 727 the same magnitude in both (i.) and (il.), but if it is plus for one it will be minus for the other. So that if (2.) rotates the plane of polarization in one, (ii.) will rotate it in the opposite direction. Thus a molecule represented graphically by fig. 1 would rotate in the opposite direction to one represented by fig. 2, or by fig. 3. sei L, Fig. 2. Fig. 3. A ee D ; 2 Cc C B ea ee Dx 7 The geometrical meaning of () is interesting. The term Ley’ Yew! z! —Xez'Yex'y’ may be written de peq (tp — @) Zeula = 24) where wp, Yp, =p 3 Vg, Yq, 2q are the coordinates of two points P,Q at which the charges are e,, e,; the expression may be oe ANS ps &g put in the form DepegPQ cos P. PQ sing . p, where @ is the angle PQ makes with the axis of x, and p the shortest distance between PQ and this axis. Since PQsing.p is the moment of PQ about-the axis of «, and PQcos¢@ the component of PQ along the axis, the term is equal to the product of the charges multiplied by the product of the component of PQ along and its moment about the axis of zw. Thusit vanishes if PQ is parallel to the axis, or if it is at right angles to it, or if it intersects it. The term is equal to é,e,6V, where V is the volume of the tetrahedron whose corners are P, () and the projections of P and Q on the axis of «. The term is to be taken as positive or negative according.as the component of PQ along and its moment about the axis of w are related lke translation and rotation in a right or left handed screw. The hypothesis we have made is that the whole molecule rotates like a rigid body under the influence of the electric forces in a light wave, so that there is no displacement of the electrons relative to the atoms in a molecule. We shall now consider whether on any reasonable supposition as to the values of the quantities involved, the expressions we have found would give values for the rotation of the same order as those observed for optically active substances. 128 Sir J. J. Thomson on some Optical Liffects Consider first the value of Q: the terms 1/A, 1/B, 1/C, will be of the order 1/Md?, where M will not be less than the mass of the smallest atom in the molecule, nor greater than the mass of the molecule itself, dis a length comparable with the radius of the molecule. The terms Yev'y', Yew", may for very unsymmetrical molecules be of the order ed? and ed respectively, hence () will be of the order e#d/M: hence we see from equation (4) that the angular rotation in circular measure per centimetre will be of the order Ne?d e7d 2M oO NM. 2ue When there is one gramme of the substance per cubic centi- metre NM will be less than unity, hence the specific rotation will be less than e?d/c?M?. Now there are many active substances in which the lightest atom is heavier than OH. If we take M to be the mass of this atom, then since e is expressed in electrostatic measure, e/-M=10°/17, so that the specific rotation per cm. will be less than d x 108/289, or taking d=10-* less than 1/289, 2. e. less than 12’. Many optically active substances have specific rotations greater than 10°, so that our expression for the rotation only accounts for a small fraction of that observed. Nor is this all, the expression we have obtained does not depend upon the frequency of the light, whereas the actual rotations are approximately proportional to the square of the frequency. This discrepancy is due to the fact that we have regarded the molecule as a free system, uninfluenced by other molecules. If the influence of the other molecules is such as to make it set in a definite position and vibrate about this position with a frequency n, the displacements and velocities will be less than when the system is free in the proportion of p? to p?—n?, where p is the frequency of the light. Thus when the influence of other molecules is to be taken into account, we must multiply the expression we have obtained by p?/p?—n?. If n were large compared with p, this factor would be p?/n?, and as this is proportional to the square of the frequency, we should get the correct variation of the rotation with the frequency of the light. There are, how- ever, two very serious objections to the modification of the formula in this way. In the first place, it is very unlikely that the natural frequency of the motion of a heavy molecule as a whole should be large compared with the frequency of the light ; and secondly, even if it were, since the factor p?/n? would then be small, the rotation, as calculated by the due to the Scattering of Light by Electrons. Ped modified expression, would be mueh smaller than that calculated by (8) which, as we have seen, is already far too small. Again, from this formula the rotation ought to tend to be smaller for heavier molecules than for light ones ; there is no indication of this in the very numerous Pdofennibations which have been made of specific rotation. Thus we do not get sufficient movement to account for the rotation produced by optically active substances if we suppose the whole molecule to rotate as a rigid body under the electric forces in the wave of light. Wed may, however, regard the molecule as made up of two parts, the first part consisting of the atoms in the molecule, the second of the electrons which bind these atoms together. Thus, with a tetrahedral arrangement of the atoms round the carbon atom, the first system will consist of the carbon atom at the eentre O and four other atoms or radicles, A, B, C, D at the corners of the tetrahedron (fig. 4). If we make the Fig. 4, usual assumption that A, B, C, and D are bound directly to the carbon atom, then between O and each one of these atoms there must be electrons whose attractions on the positively electrified atoms will bind them together. These structural electrons will form a tetrahedral arrangement which will occupy a definite position relative to the tetrahedron A, B, C, D, and if disturbed from this position will vibrate about it with a definite frequency. When the light waves fall upon the molecule what I imagine to happen is, that in consequence of the relatively enormous mass of the atoms themselves there is but an insignificant displacement of the atomic tetrahedron, but that the much lighter tetrahedral arrangement of the electrons suffers a much greater amount of rotation, and that the motion of the electrons consequent upon this rotation gives rise to the rotation of the plane of polarization. We have thus to regard the system of electrons as that to which we must apply the preceding analysis. It may be urged against this that since the electrons are all of the same mass, and the electrical centre of the system coincides with the mass centre, the analysis shows that the system would be far too symmetrical to give rise to optical rotation. This Phil. Mag. 8. 6. Vol. 40. No. 240. Dec. 1920. 3B 730 Sir J. J. Thomson on some Optical Effects would be a valid objection if all the electrons were quite free, or even if they were all under similar conditions of restraint. The electrons, however, are not free, they are acted upon by forces which tend to drag them back to the places from which they start. If these forces are different for the different electrons, the behaviour of one electron will be different from that of another, and this may make the system so unsymmetrical that it may be able to produce rotation of the plane of polarization. We may make this clearer by considering a very simple case, that of four electrons a, 8, y, 6 at.the corners of a tetrahedron, and suppose that the restoring forces for two of the electrons, say « and #§, are very large compared with those for y and 6. The effect of this will be much the same as if « and 8 were fixed, so that the tetrahedron would, when acted upon by the electrical forces in a wave of light, rotate about the line 28, the two electrons y and 6 would rotate about this axis. We see from the considerations given above that each tetrahedron will contribute to the expression for the rotation of the plane of polarization a term equal to 27 e? [| PQ] Boe on ma . where A is the moment of inertia of the two eJectrons y and 6 about the axis «8, and | PQ] is six times the volume of the tetrahedron whose corners are y and 6 and the feet of the perpendiculars let fall from y and 6 on «8. Let us consider the numerical magnitude of this term. If d is a measure of the radius of the molecule, [| PQ] will be of the order ad? and A of the order md?, where m is the mass of an electron. Thus the contribution to the rotation will be of the order ed cm’ and if there are N molecules per c.c., the rotation will! be of the order If M is the mass of the molecule, then when there is ‘one gramme of the active substance per c.c.. NM=1, so that the intrinsic rotation is ay At aD (a att Now © ud 2, 4, uns . 7 2 — —8 — e*/c?m? = (1°8 x 10)?, d=107°, and m/M= seo? where W is the molecular weight of the substance. Thus due to the Scattering of Light by Electrons. Rat 1°83 x 10? the intrinsic rotation would be Sar | radians or about 104 cee al} Ww degrees. This has to be multiplied by p?/n?, where nis the natural frequency of the vibration of the tetrahedron about the axis 28. But we see that,-even allowing for a small value of p?/n?, the system we have considered is able to produce rotations comparable with those excited by optically aetive substances. If the specific rotation is multiplied by the molecular weight of the active substance, the product is a measure of the rotation due to a single molecule, a quantity which is much more likely to throw light on the properties of the molecule than the rotation due toa gramme of the substance. The volume of the tetrahedron which measures the con- tribution to the optical rotation of one molecule, when the electrons yd rotate round #8, vanishes in the following cases :— If yé is parallel to a8. If v6 intersects a8. If yé is at right angles to a8. From the third of these conditions it follows that no optical rotation will be produced if a, B, y, 6 are at the corners of a regular tetrahedron, there must therefore be some lack of symmetry in the distribution of these electrons. The rotation vanishes whenever y6 is in a plane at right angles to #8. Thus it would vanish if the two atoms or radicles which the electrons y and 6 bind respectively to the central carbon atom were identical; for then by symmetry «and 8 would both be in the plane bisecting yé6 at right angles. A shift in the position of one or both of the electrons y, 6 might change the sign of the optical rotation produced by the molecule. Thus, for example, when the electrons are Fig. 5. Fig. 6. eo A 6 7 } p distributed as in fig. 5, the molecule would behave like a ositive screw, while if they were as in fig. 6, it would | 3 ) . 5 ? behave like a negative one. : 3B2 132 Sir J. J. Thomson on some Optical Lfects We can explain by displacements of this kind the very interesting fact that the optical rotation by a solution of an active substance may depend not merely in magnitude, but also in sign on the nature of the solvent: thus, for example, a solution of d acetic acid in water is dextro rotator y, while when dissolved in a mixture of acetone and ether it is: levo rotatory (Landolt). If a solvent made some progress. towards ionizing the substance by weakening the attachment of one of the atoms (say D) to the carbon atom, it might make the electron 8 which binds D to the carbon atom move from 6 to a position more remote from the carbon atom. This change, as fig. 6 shows, might reverse the sign of the rotation. The general conclusions to which we are led by the preceding investigation, is that the electrical system which is instrumental in producing optical rotation are the electrons which couple the atoms in the molecule to the central carbon atom, and that the most important quantities in the a for the rotation are (1) the rigidities of those electrons, 7. e. the intensity of the forces restoring them to their nace of rest when displaced from it; and (2) the distances of these electrons from the central carbon atom. For it is on these quantities that the dynamical and geometrical asymmetries depend respectively, and as we have seen, both these asymmetries are essential for rotation in solutions. If it were not for differences in (1) there would be no dynamie asymmetry, for the electrons have all the same mass. When, however, they are pulled back when displaced from their position of equilibrium with forces of different inten- sities, the effect will be much the same as if the electrons had different masses. Thus, if the frequency of the vibrations. of the electron when displaced from its position of equilibrium is n, and p is the frequency of the hght, the behaviour of the electron under the forces in the light, wave will be much the same as if the electron were free but had a mass. (p? —n,7)/p? times itsinormal mass. When 7,’ is large com- pared with p, the effective mass is approximately ny2/p? times. the normal inass, and is thus much greater. Thus the dyna- mical asymmetry of the molecule will be measured, not as on Guye’s theory * by differences in the atomic weights of the atoms attached to the carbon atom, for these on our view have no direct bearing on the rotation, but on the differences in the frequencies of the electrons which bind the atoms. to the carbon atom. * Ann. Chim. Phys. (6) xxv. p. 145 (1892). due to the Scattering of Light by Electrons. 133 Avzain, the geometrical asymmetry which is necessary for rotation will depend on the distances of these electrons from the carbon atom. Thus, to predict the rotation produced by a molecule containing one unsymmetrical earbon atom and represented by CR, R olt;Ry, we require to know the pro- perties of the linkages GR, CRo, CR;, CR, ; 7. e., we require to know the frequency of the coupling electron in ‘the linkage CR and its distance from C. Unfortunately our knowledge on these points is extremely meagre. We might expect to get some knowledge of the frequencies by mapping the absorption spectra of various hydrocarbons. Thus, ifthe coupling C—H hada frequency 7, we might expect to find in the absorption spectrum of a compound containing this coupling a band corresponding to this frequency, modified it might be by the proximity of other couplings in the molecule. A study of the spectra of compounds containing the coupling C-OH might, in a similar way, lead to the knowledge of the frequency of the electrons concerned in this coupling. And in this way, by the study of the absorption spectra of a large number of organic compounds, we might hope to determine the fre- quency of the coupling of the carbon atom with the various atoms and radicles to which itis joined in organic compounds. Considerable progress has been made in the study of such absorption spectra. Thus Hartley (see Kayser’s ‘Spectro- scopy, vol. ii.), working with ultr a-violet light down to the wave-length 2000, studied the absorption in “the ultra- violet down to this lace. and discovered well-defined absorption bands in benzene and other aromatic compounds containing the benzene ring. Our knowledge of this subject has been greatly extended by Stark and his collaborators who, by using fluorite lenses and prisms, were abie to work with ultra-violet light down to wave-length 1850, and dis- covered, among other things, bands which ihey ascribed to the linkages C=CandC=C. Weare however not yet, I think, in a position to be able to say whatare the fundamental frequencies of the various linkages; to do so we require observations over a wider range of frequencies extending to wave-lengths considerably smaller than those hitherto st tudied, for in this, the Schuminn region, nearly all gases have great absorption, and some of it may well be selective. The experiments hitherto made indicate that the frequencies cor responding to the linkages C-H and C—OH will be very far in the ultra-violet, and consequently larger than the normal value. Now, on the view we have taken, a high frequency corresponds to a large effective mass, so that i in 734. . Mr. J. Chadwick on the Charge on the compounds of the type C.H.HO, R,, R, the system of coupling electrons will tend to twist round the line joining the electrons which couple H and OH with the carbon atom. The other condition for optical rotation is geometrical asymmetry. ‘The electrons cannct be at the corners of recalar tetrahedron, and this arrangement, which is the one usually asssumed, 1s incompatible with optical rotation. The departure of the molecule from this form, in fact the shape of the molecule, is of vital importance in connexion with optical rotation, but on this subject little, if any thing, seems to have been done. If we had a theory which gave the configuration of the molecu'e and the periods of vibrations of the electrons, we could calculate by the expression given above the value of the molecular rotation. Inasmuch as the configuration and periods enter into these expressions in a complicated way, the effect of any one period, for example, depending on its relation to each of the other periods, the periods are not easily calculated from the rotation. Thus observations on the optical rotation are more likely to be useful as a test of any theory of the configuration and structure of the molecule than as a means of discovering this structure. Since the above was written I have seen a paper by Stark (Jahr. f. Radioaktivitdt, xi. p. 194, 1914), in which the subject of optical rotation is also treated from the point of » view of the electron theory ; the treatment is wholly quali- tative, and I find it difficult to follow the reasoning ; so far as I am able to do so it seems to me to be fundamentally different from that given above. — = TSE FESS = = = = LAXXIV. The Charge on the Atomic Nucleus and the Law of Force. By J. Caapwicn, M.Sc., Wollaston Student of Gonville and Caius College, Cambridge * yd: A ya theory of the nuclear constitution of the atom, put forward by Sir Ernest Rutherford f in 1911, has been confirmed by evidence gathered from such various sources that it now forms the foundation on which the development of atomic physics is based. Qn this theory, the positive charge associated with an atom is concentrated - on a massive nucleus of small dimensions, surrounded * Communicated by Professor Sir E, Rutherford, I.R.S; + Rutherford, Phil. Mag. xxi. p. 669 (1911). Atomic Nucleus and the Law of force. T3530 by a distribution of electrons extending over a distance comparable with the diameter of the atom, as usually understood, The physical and chemical properties of an element are determined by the charge on the nucleus, for this fixes the number and arrangement of the external electrons, on which these properties mainly depend. The mass of the nucleus influences the arrangement of the electrons only to a very small degree. The nuclear charge is thus the fundamental constant of the atom and the question of its actual magni- tude of great importance for the development of atomic theory. In the paper referred to above, Sir Ernest Rutherford showed—assuming that the electric forces between the nucleus and an « particle passing close to it varied ac- cording to the inverse square law—that the « particle would describe a hyperbolic path, and obtained the rela- tions connecting the fraction of « particles scattered through any angle with the charge on the nucleus and the velocity of the « particle. From some observations of Geiger and Marsden* on the reflexion of « particles he deduced that the charge on the nucleus of an atom is roughly $A.e, where A is the atomic weight and e the electronic charge. In their experiments on the scattering of a particles, which proved conclusively the truth of the nuclear theory, Geiger and Marsden showed that the charge on the nucleus was equal to 3A.e to within about 20 per ‘cent, The large error is due to the fact that the scattered particles formed such a small fraction of the original beam that different methods of measurement had to be employed in the two cases. Later, van den Broek + suggested that the nuclear charge might be equal to the atomic number of the element, i.é. the number of the element when all the elements are arranged in order of increasing atomic weight. This proved to be in good agreement with the ex dain eT 14 E a Do SelO*, The following values of N were obtained : 46-0 (458), 41-0 (156), 44-0 (190), 44:0 (307), 45-9 (529), 46:5 (411), 49:1 (224), 466 (416), 49-1 (297). The weighted mean is 46°3 and the total number of scattered particles is 3000, giving a probable error in N of 13 per cent. The atomic number of silver is 47. Copper.—The foil weighed 4°78 mgm. per sq. cm. Hence nt = 4°64 x 101% and the average mu? —_ —f)’ 14 E ='6'60 x'LO™, The following values were obtained for N : 27°8 (125), 29°0 (340), 29-6 (369), 27°6 (212), 29°6 (178), 29°6 (364), 30:0 (377), 28:2 (243), 30-4 (351), 29-6 (302). The weighted mean is 29:3 and the total number of scattered particles 2900, giving a probable error in N of about 14 per cent. The atomic number of copper is 29. 744 Mr. J. Chadwick on the Charge on the § 5. The Law of Force. In his theory of the single scattering of « particles Rutherford assumed that the law of force between the a particle and the nucleus was that of the inverse square. It bas been pointed out by Darwin * that a direct test of the law of force is given by the dependence of scattering on the velocity of the « particle. If the force around the nucleus vary with the distance as 1/r?, then the number of scattered « particles, other conditions remaining constant, a is proportional to (4) . The experiments of Geiger and Marsden showed that the number of scattered particles depended on the fourth power of the velocity to the nearest integral power. Combined with the observed law of angular distribution of the scattered particles, this leaves no doubt as to the general validity of the inverse square law. In view, however, of the accuracy attained in the measure- ment of the nucleus charge, in which-the inverse square law is assumed, it was considered necessary to repeat this test in a stricter manner. — The same apparatus was used, but the mica sheet at O was removed and the ZnS screen waxed on in its place. This was necessary in order that short range 2 particles could reach the screen. By means of a ground-glass joint H mica sheets were brought in front of the source to cut down the velocity of the a particles incident on the scattering foil. The stopping powers of the mica sheets were measured by adjusting the pressure of air in the box until the direct pencil of « rays just failed to reach the screen. The velocity was calculated from the emergent range, using Geiger’s relation w=aR. A slight correction of the measured ranges was necessary, since the particles which hit the scattering foil travel through the mica at a small angle. The scattering foil was platinum weighing 1°58 mgm. per sq. cm. The number n of scattered particles was counted for three velocities, about 1600 particles being counted for each. The results are given in the table : Mica sheet. Relative w*. nut, On ee 1 100 Pe ea Dad) 101 it ka ehaee Doe 103 * Darwin, loc, cit. Atomie Nucleus and the Law of Force. 745 The values of nut are constant within the counting error of about + per cent. Taking the nuclear chargé of platinum as 78e, the distance of approach of the high-velocity @ particles was about #x10-" em. and of the low-velocity particles about 14x 107" em. We conclude, therefore, that in the region of 107! em. from the platinum nucleus the force varies as I/7?, where p lies between 1°97 and 2°03. § 6. Summary. 1. The charges of the nuclei of three atoms, viz. platinum, silver, and copper, have been measured by a direct method depending on the scattering of @ particles. The values found are 77°4, 46°3, and 29°3 respectively in fundamental units of charge. The atomic numbers of these elements are 78, AZ, and 29. 2. The law of force around the platinum nucleus has been tested by the dependence of scattering on the velocity of the a particle. The results show that the inverse square law holds accurately in the region concerned, viz. around 10-1 em. from the nucleus. - § 7. Discussion of Results. The good agreement between the measured values of the nuclear charges and the atomic numbers of the elements deduced from the X-ray spectra affords strong confirmation of Moseley’s generalization. Owing to the probability fluctuations, it is hardly possible to attain sufficient accuracy in the measurements to prove that the charge on the nucleus is a whole number times the electronic charge. At present this is only possible in one case—that of the helium nucleus or a particle. There can, however, be little doubt that the nuclear charge does really increase by unity as we pass from one elem:nt to the next, and that its net value is given by the atomic number. The values of the nuclear charge combined with the dependence of scattering on the velocity of the « particle indicate that the inverse square law of force holds to a high degree of accuracy in the region investigated, 10-" cm. from the nucleus of a heavy atom like piatinum. The experiments of Geiger and Marsden on‘ the’ angular dis- tribution of a particles scattered by gold between 5° and 150° show that the same law must hold for distances between 3'1 x 1072 cm. and 36 x 10~ em. from the nucleus. Phil. Mag. 8. 6. Vol. 40. No. 240. Dec. 1920. 30C 746 Dr. G. Borelius on the Further, the agreement between the experimental measure- ments of the K-series spectra and the theoretical values of Debye and Kroo shows that the inverse square law still holds at the K ring. In the case of platinum the radius of the K ring is about 107” cm. Thus, measured from any point in the region between 3x 10-12 em. and 107! em. from the nucleus of a heavy atom like platinum, the charge is equal to the atomic number and the law of force is the inverse square. We may therefore conclude that no electrons are present in the region between the nucleus and the K ring. I wish to thank Sir Ernest Rutherford for suggesting this work, and for his interest and advice throughout its progress. LXXXY. On the Electron Theory of the Metallic State. By G. Boretius, Ph.D.* § 1. Introduction. HE well-known theory, founded by Riecke and Drude, and worked out in its further consequences by Lorentz, Bohr, and others, assumes the conducting electrons to be freeiy moving among the atoms of a metal, and to have the same mean kinetic energy as gas molecules at the same temperature, These assumptions gave, as a first striking result, a deduction of the laws of Wiedemann and Franz and of Lorenz. However, other consequences of the theory cannot be said to agree very well with experimental facts. There is, for example, no probable ground for the cha- racteristic dependence upon temperature found for the electric and thermal conductivities in pure metals ; without the addition of the most improbable assumptions, it gives a wrong idea of the magnitude of the thermoelectric pheno- mena; the specific heats observed require the number of free electrons to be very smali compared with the number of atoms, whereas optical phenomena require these numbers to be of the same order of magnitude ; in the Hall pheno- menon it gives us directly only negative signs, though both signs are actually observed, and so forth, All these facts have eaused J. J. Thomsont, W. Wient, * Communicated by the Author. + J. J. Thomson, ‘Corpuscular Theory’ (1907); Phil. Mag. vol. xxx.. p. 192 (1915), .t W. Wien, Berl, Ber, 19138, p. 184. Electron Theory of the Metallic State. 747 Stark *, Lindemann f, and others to propose separately very radical alterations in the fundamental assumptions of the electron theory of the metallic state. ‘The present writer, too, has made an attempt in this direction. Last year I treated in three communications} the thermoelectric phenomena, the thermal and electric conduction and the magnetic and galvanomagnetic phenomena from new funda- mental assumptions. I here wish to give an abstract and a revision of these papers. Especially the work published during the past year by Madelung § and Born and Lande |, on the electrostatic forces in the atomic space-lattices in connexion with some results of X-ray analysis of metals, makes it possible for me now to treat some of the phe- nomena in a more concise manner. § 2. Number and Arrangement of the Conducting Electrons. We will, as a general assumption, suppose the metallic state to be to a certain extent comparable to the state of a halogen salt according to the modern space-lattice theory, the negative ions of the salt being replaced by electrons. The great dissimilarity in the behaviour of metals and salts is then chiefly due to the greater mobility of the electrons, caused by the diminutiveness of their mass compared with that of the ions. We thus, as will be seen in §4 for a simple case, at once get an under- standing of the way in which the statical equilibrium of the metallic space-lattice is built up by electric forces in full conformity with the theory of Born and Lande. In the simplest case the metal would be of the type NaCl, a simple cubic lattice with alternating Na and Cl atoms. The figure of symmetry for one of the com- ponents must then be a “face-centred cube ’—-that is, an atom at each corner of a cube and one in the centre of each of its sides. And, indeed, this symmetry is found by the X-ray analysis{] for a great part of the examined metals belonging to the regular crystal system, namely, * T. Stark, Jahr. d. Rad. ix. p. 188 (1912). + F. A. Lindemann, Phil. Mag. vol, xxix. p. 126 (1915). +t G. Borelius, Ann, d. Phys, lvii. pp. 231 & 278 (1918), and lviii. p- 489 (1919). § E. Madelung, Phys. Zeitschr, xix. p, 524 (1918). | M. Born and A. Landé, Berl. Ber, 1918, p. 1048; Verh. d. deutsch. Phys. Ges. xx. pp. 202 & 210 (1918). q W.L. Bragg, Phil. Mag. vol. xxviii. p. 355 (1914) (Cu). L. Vegard Phil. Mag. vol. xxxi. p, 88 (1916) (Ag), and vol. xxxii. p. 65 (1916) (Au, Pb). P. Scherrer, Phys. Zeitschr, xix. p. 23 (1918) (Al). A. W. Hull, Phys. Rey. vol. x. p. 661 (1917) (Pb, Fe, Ni, Na, Li, Mg, and others). : $02 748 Dr. G. Borelius on the Cu, Ag, Au, Al, Pb, and perhaps one form of Ni. On the other hand, the symmetry of the ‘centred cube” was found to exist in the case of Na, Fe, and Ni, and probably that of the “simple cube” with two atoms connected with each corner in the case of Li. Of the non-regular metals the only one examined as yet is Mg, which shows an hexagonal symmetry. We will in the following confine our calculations to metals of the simple type NaCl only. Some of our results will therefore be valid only for metals of the type of the face-centred cube. Many of them, however, will most probably be approximately true for any good conductor. For a metal of this simple type the number » of free electrons in the cubic centimetre must be equal to the number n!' of atoms, se that eee BR ee | (N=6:06 . 1072, number of Avogadro, p density, A atomic weight). Though it is not @ prior probable that this relation will hold throughout for all metals, we will adopt it for our trials, and indeed we shall find as yet no reason to alter it; some facts rather argue in favour of its general validity. After the present paper was nearly finished, an investi- gation of Haber * came to my notice, where he treats the compressibility and the ultraviolet characteristic frequencies of the metals making use of similar assumptions. The last problem was, moreover, treated already in 1911 f. § 3. Optical Phenomena. As was first shown by Drudef, the refraction and absorption of light in a metal enable us to calculate approximately the number of free electrons in it. Schuster § and later on, in another way, Drude || have calculated v n the ratio 4 for a number of metals from their optical properties. As a result of these investigations, it was found that this ratio is for all good Gonder of the * ¥. Haber, Berl. Ber. 1919, p. 506. ' Verh. d. Deutsch. Phys. Gees. ae Dat 12S GSI t P. Drude, Phys. Zevtschr. 1. p. 151 (1900). § A. Schuster, Phil. Mag. vol. vii. p. 151 Soe 125 ( || P. Drude, Ann. d. Phys. Xly. pp. 725 & 9 6 (1904). Electron Theory of the Metallic State. 749 order unity, or perhaps two or three times greater than unity. We can regard this fact as a further support for our theory. As is well known, it is a rather great difficulty for the classical electron theory. § 4. Potential Energy of the Metallic Space-lattice. In full analogy with the theory of Born and Landé; the work done in bringing an electron or an atom from infinity to their place in the lattice is —ed, or +€¢., where the potential functions he and ¢, due to the surrounding electrons and atoms are given by ‘ ax 3b emerep re ae (2') ee ep Si a ee Pepe Ba (2"") where R denotes the distance between an electron and ° its nearest atomic neighbour, so that for a metal of the type NaCl | oy A ni R — i 5 . . ¢ . ° (3) | a : The terms = arise from the forees between electrons R . b . and atoms regarded as point charges. The terms pe arise from the repelling forces on an electron from the nearest atom-ions or on an atom from the nearest electrons, due to the arrangement of a number of electrons around and at an appr eciable distance from a positive nucleus in the atom. ! The term — Re arises from the attracting forces on the positive charge of the atom-ions caused by the mentioned. arrangement of the electrons in the nearest atoms, and vice versd. For g, there is, of course, no term corre- sponding to this. A term arising from the repelling forces between the neutral parts of the atoms is to be neglected, if the atoms’ dimensions are sufficiently smaller than their distances. The constant a@ may be ecaleulated by a method of Madelung, and is for a lattice of the simple type NaCl, which we have adopted for our trial, Seige hee aie Sys. 1D) The constants 6 and 6’ are deduced from the condition 750 Dr. G. Borelius on the for a stable state and the geometrical properties of the lattice. The condition is d Le(d-—da) = min. or IR (Pe Pa) = 0), en which gives, in combination with (2’) and (2"), 2aRe7} 2b—b' = 2 Leen i (6) The geometrical properties give approximately Ob = ERED p 2°35, (7) H(R,/2)" HR’ For different symmetries of the outer electrons in the atom, w takes different values: 5,9, or more. This will be discussed further in the next paragraph. § 5. Compressibility. Born and Landé, from their space-lattice theory, found the compressibility at the absolute zero-point for salts of the type NaCl to be given by ee. 5) ( Ane Na ~ 8. 1°742e (w—1) Np ; where A, and A_ are the atomic weights of the ions. With w=9 they find good agreement with experimental results. As the atom model of Bohr would lead to w=5, they propose a cubic symmetry of the outer electrons in the atom that would, at least under certain conditions, give w=9. Our fundamental equations lead, in the way shown by Born and Landé, to the same value for w. Neglecting the weight of the electrons besides that of the atoms, we may write 16> Re -(eqst K From this equation I have calculated mw for the metals that are known from X-ray analysis to have a symmetry consistent with our premises, using values for « and p* holding for ordinary temperatures, and have found LO) iba ee marten aL 7 EN AR tae a coh) ALU re ete 12 LS ae ieee 6°D Pb epee eH NGS ae oes (8) * Tables of Landolt-Bornstein-Roth. Llectron Theory of the Metallic State. Tol With values for « and p valid for the absolute zero-point, the values for ~ would have been greater by about 1 and their mean thus equal to 9-—a value that we will use in our further caleulations. As the calculations of Born and Landé applied to the metallic space-lattice would give w=5 also for a cubical symmetry, the derivation of #=9 from the experiments seems, if our theory holds, to indicate another sy mmetry as more probable than that of the cube. Haber has calculated » in the same way also for the alkaline metals and found here much smaller values, between 2+ and 3°53. . As, however, he assumes the alkaline metals to have the same structure as their halogen DoD salts, which is, at least for Na and Li, not consistent with the results found by Hull by the X-ray analysis, these low values are not as yet in any way definite. $6. Motions and Kinetic Energy of the Electrons at [High Temperature. To get an understanding of the electric conduction, we must discuss the motions ef the electrons in relation to the aton as hitherto been left unconsidered. It follows from our general assumptions that this motion must be intimately connected with the motions of the atoms. ‘To see this, we may for a moment consider what would be the case if the atoms were at rest. From the equations of §4 it is easily seen that the energy required to overcome the hindering forces from the atoms passed by, when an electron is displaced from one point of equilibrium to another, even though all neighbouring electrons were supported in the same way, so that no forces from them were to be'added, would be more than a hundred times greater than the energy of a gas molecule at ordinary temperatures. The electrons thus would not go very far away from their equilibrium positions. In the same way, the thermal oscillations of the atoms cannot have any great amplitudes without a similar motion of the electrons. “Now the atomic amplitudes are thought to be considerable, and are, in the neighbourhood of the melting-point, of the same order of magnitude as the atomic distances, so that the atoms’ mean yelocity is of the order 2v6, if v is the frequency and 6 the atomie distance. ‘The velocities of the electrons will then be of the same order. But as the electrons are much smaller and lighter than the atoms, it seems probable that there will be an irregular transport of electrons, so that the electrons’ space-lattice Toy Dr. G. Berelius on the will be in a “ fluid ”’ state far below the melting-point of the metal. As an electron cannot escape from between the atoms without a way being opened for it by the elastic waves of thermal agitation, and as the greatest velocities for these waves are, by the theory of Born and von Karman ™%*, 2v6, we may as a limit for high temperatures write the transport velocity =p. 210, ae re where p is a constant of the order one. The kinetic energy of the electrons must be intimately connected with the transport velocity v. It will probably not tend to a limit in the same way, but increase pro- portionally to the heat content of the metal, which is at high temperatures proportional to the absolute temperature. 12 ae iy ne The correspondence of the kinetic energy with oy will probably be best in the neighbourhood of the characteristic temperature 6 (=Pv, B=4'87.10~"), where the thermal oscillations become appreciable. We are therefore led to write the two-thirds (w) of the kinetic energy corresponding to the two degrees of freedom of the motions on the (variable) equipotential surfaces as ug. (BVb) 5, . ee rs where the constants g have to be at least approximately of order one. The part of the electrons’ kinetic energy that exceeds mv j , ; : = - must, as 1s seen from the foregoing, give rise to oscillations with amplitudes smaller than 6. We have shown that it is probable that this surplus is at least com- mv? yore thousandth of the kinetic energy of a molecule at ordinary temperatures. It is perhaps worth while giving a further reason for this disproportion between the energy of oscil- lation of the electrons and atoms. If their kinetic energies - were alike, the electron, which is many thousand times lighter than the atom, would have velocities that were a hundred times greater. Now as the amplitudes must be of the same order of magnitude, or about 6, the frequency of the electron would be about a hundred times greater than * M, Born and vy. Karman, Phys. Zetisehr, xiii. p, 297 (1912). parable with which is only about a hundredth or Electron Theory of the Metallic State. 753 that of the atom. But according to the quantum theory such great frequencies rarely occur at ordinary, tempe- ratures, so that the supposition of equipartition of energy would lead to discrepancy with the quantum theory. As the theory of Born and von Karman, which we have made use of, is deduced for a simple cubic space-lattice, it is not fully clear what is meant by 6. We will, however, for our approximate calculation simply put it equal to the smallest atomic distance, or R V2. § 7. Hlectric Conduction at High Temperatures. The first effect of an electric field of force in the metal will be a displacement of the free electrons in a direction opposite the electric force; and this polarization in con- nexion with the thermal motions of the electrons will call’ forth the electric current. To calculate the polarization, we may in any definite way connect every electron with one of its neighbouring atoms and inversely every atom with one electron. The state of the cubic centimetre of the metal will be equivalent to » dipoles of a mean moment nearly equal to eR. The motions of the electrons will be equivalent to a rotation of these dipoles with an energy of rotation equal to the kinetic energy (wu) of the electrons in the equipotential surfaces, corresponding to two degrees of freedom. It is, moreover, possible that this way of description of the state of a metal will have not only a mathematical but also a physical meaning. In the statical state, dealt with in § 4, the electrons are, because of the third term in (2”), always repelled by the neighbouring atoms. But where the atoms, by their oscillations, are removed from each other, the electrons will be attracted to them; and as the electrons will seek out such places, a great number of them at least will move under the influence of a resulting attracting force from one of the neighbouring atoms. The polarization as a function of the electric force X can be calculated in conformity with the theory of magnetism given by Langevin *, and is found to be - = Paz ° ° ° ° . . (11) where P,,=neR is the maximal moment conceivable as due to polarization, and wu is given by (10). To the maximal * P, Langevin, Journ. de Phys. iv. p. 678 (1905). 754 Dr. G. Borelius on the polarization would correspond a maximal current density, Ly = NED SS rr if all the electrons travelled in a direction opposite the electric force. If now the current density i is throughout proportional to P, we must have i le —_ —-pyp- . ° . . : . . 13 Um le ( Inserting (11) and (12) in (13), and as v=2vd=yR V8, we get for the specific conductivity at high temperatures, d 2/8 nvR? el, Gal a ee eee ee (14) and because of (10) 2 eal. eee q 3 /2m yp T We thus get the conductivity inversely proportional to the absolute temperature, in good agreement with experi- mental results. Further, if our premises are valid, the constants a which can be calculated from (15), must be of order one. The values in the following table are simply caleulated from the conductivities at 0° C., though that temperature cannot, for all metals, be looked upon as a rather high one. ‘We see that the predictions of our theory are fulfilled very precisely for the best conductors in the first group of the periodic system. : is here nearly equal to one. Passing to the right side of the system, this constant becomes piacualyes greater up to about ten for the metals in group VIII. : L rae 1 inv ee eae ye VIII. Na 15 | Mg22 | Al 19 | Sn 36 | (Sb 15)-| Becwo K 1) Zao | Pb 78 | (Biss) | Cm BF Cu 16 | Ca 3:8 + | | Rh 48 us Vig elke | | Iv 492 Awe eS | | | Ni 7B | | Pa 82 | | | Pb: St Hlectron Theory of the Metallic State. 755 $8. Thermal Conduction. Thermal ‘conduction in non-metallic solids is generally thought to-day to be due to elastic thermal waves. ‘Ti e classical electron theor y interprets the quantitatively superior metallic conductivity also in a qualitatively dif- ferent manner, as due to an hypothetical electron gas. We are now forced to give up this standpoint, as neither the kinetic energy of the electrons nor their free paths are great enough to explain the great conductivity of the metals. We therefore are led to try if the conduction cannot in the metals as in all other solids be referred to the elastic waves ; and, indeed, we shall in this way find reasons enough for the superiority of the conductivity of the metals in their peculiar structure. In a salt of the type NaCl, for example, the lattices of positive and negative ions, as is known from the residual rays, have different characteristic frequencies and thus transport different elastic waves. Now as the two lattices have like energy, there will, at high temperatures, be a lively exchange of energy between them, so that the distance within which the intensity of the wave diminishes to an infinitely small part of its original value, will be but few atom distances, which is also in good agreement with the experimental values for the conductivity. In a metal, on the other hand, the energy of the electron space-lattice is relatively very small, so that the waves in the atom- lattice will be very little damped, and thus give a great conductivity. For a quantitative discussion we start with a general formula* for the conductivity 2, called forth by damped elastic waves : X= LpcwLh, tee ke Ler hy Swe Be) where p is the density, ¢ the specific heat, w the velocity of the waves, and L their mean range, defined by the decrease dK of the intensity K over the distance dS by the equation dK 2a ; eT Meee ee ot (16') We may, for high temperatures, transform this expression for X in the following manner. pc is the specific heat for the unit of volume, and as a molecule in a solid has the * P. Debye, ‘ Vortrage iiber d. kinet. Theorie,’ Gottingen, 1914, p. 50. 756 Dr. G. Borelius on the double energy of that in a gas we can write pe = 200’, 03. i a=2°06.107!° being the gas constant for a single molecule. As is shown by Born and von Karman, most of the energy in a space-lattice is transported with velocities nearly equal to the maximal velocity 2v,,6; so that we ‘ have approximately Wi = 200. oe 6 Oe ee rr vy being as before a mean frequency nearly equal to the maxima! one p,,. ~The mean range L may be calculated from (16’) as follows. Every time a wave, embracing a number of atoms, advances a distance dS=6, this wave gets into an energy exchange with a like number of electrons. Now this exchange must be effected in such a way that the ratio of mean energies of the electrons (3 corresponding to equipartition of energy in three kinetical and three potential degrees of freedom) and the atoms (2e'l’) is kept unaltered. The mathematically simplest way in which this can be done is to assume that the mean relative loss of energy of the dk . : ; ; : wave i is every time in this ratio. As the electrons are not able to transport the energy a long distance, the energy given off to them is lost for the regular transporting waves. We thus get QR ot ro aR eat. ie and from the last equation 0 Inserting (17), (18), and (19) in (16), we get the heat conductivity for high temperatures, 2 i Le Ee . oO U As u, is. proportional to T, and all other quantities are constant, we find X constant for high temperatures, in good agreement with experimental facts. The calculated value of X is also of the 1ight order of magnitude. We need not, how- ever, show this separately, as the electric conduction is already discussed, and the ratio of conductivities will be the subject of the next paragraph. Electron Theory of the Metallic State. 757 § 9. Ratio of Conductivities. From equations (14) and (20) we get, as u=N, Xr 1 J 2a? 7 aes es (21) The ratio of conductivities is thus found to be nearly constant for all pure metals (law of Wiedemann and Franz) and proportional to the absolute temperature (law of Lorenz). Also the numerical value of the constant is in good agree- ment with experimental facts. We find for 18°C. nee 1 6°9.10!8 e.m.u. Ws 2 The measurements of Jaeger and Diesselhorst* gave, as a mean for eleven metals, X = 7-1.10” e.m.u. K The agreement is good if p is, as was predicted, nearly equal to one. § 10. Thermoelectric Phenomena. In the classical theory the thermoelectric phenomena are referred to the variations of the energy of motion of the - electrons and the variations of their number with varying temperature. We have assumed this number to be constant in good conductors and thus without influence on their thermoelectric properties. But our theory gives three other origins for contributions to a specific heat of electricity or the Thomson effect o, from which the other thermoelectric phenomena, thermoelectric power e,, and Peltier coefficient II,,, of a couple of metals @ and 6 are determined. By Thomson’s thermodynamic equations, ¢ ee | ee iB Cu — 9h Cab pn Oe i, E ap ET, in which, however, the lower limit of the integral is first fixed by the theorem of Nernst or the quantum theory. The three origins are :—Ist, the variations with the tempe- rature of the kinetic energy of the electrons; 2nd, the variations of their potential energy by heating at constant volume ; 3rd, the variations of their potential energy due to * W. Jaeger and H. Diesselhorst, Abh. Phys. Techn. Reichens alt, iii. p- 269 (1990). TES Dr. G. Borelius on the thermal dilatation. We will discuss them separately in the three following sections. 1. The part of o, we may call it o,, due to the kinetic energy of the electrons is 3u n=-25(5) ogy) re when w is got from (10). It is for many ae but a few tenths of a microvolt per degree, for none of them greater than two microvolts ; and it is but a small part of the observed Thomson effects. 2. Of the atomic heat c, at constant volume, one half is thought to increase the kinetic, the other half the potential energy. If this increase of the potential energy were equally distributed to atoms and electrons, we should get the corresponding part of the Thomson heat to be c ¢; microvolt eee ee ANie es ae Go degree * However, such an equipartition of potential energy is not at all probable, for the light moving electrons will be better able than the atoms to avoid places where this potential is increased. oy» is therefore probably but a little part (z) of the value above, and we may put ¢, microvolt Oo = Aa oe Py cei . e e ry ( Cy degree 23) The calculation of z will not be possible without an intimate knowledge about the external structure of the atoms. 3. From the equations of §4 we can calculate the variations o3; of the mean potential $4, of the electrons due to the thermal dilatation. It is 1 dd. dR : 03; => a aes ae * e e ee ° » (24) The multipler 1/2 appears because, by summing up all the ¢,., every electron is counted twice. From equations (2) and (5) we find Ube _ pb’ dR Sines Llectron Theory of the Aletallic State. 759 which for ~=9 according to (4), (6), and (7) gives dp,__ 177. 1742 jee ~ 2R? tes : ee er P As further Rar the linear coefficient » of dilatation, we have "177. 1:7426¢ WHCTOVOlbo = 5 a3 > RTGS? San -7= + 11° qe “degree” (24 ) where 7’ =7.10° and R'=R. 108 are both of order one. We may at least for high temperatures sum up our results as to the Thomson heat in the formula Be) wen Ss,” (25) where the second term is a sum of o,; and gy. A and B are positive constants. This formula was deduced in an earlier paper from somewhat less definite assumptions, and was also shown to be well consistent with the unfortunately rather few experimental facts that are known to-day Agen thermoelectric phenomena at low temperatures. For metals with »«=9 we have microvolt ee ata degree R! To show the possibility of this equation, I have in fig. 1 plotted the Thomson coefficients of a number of heavy ! metals against the ratio = The line in the figure corre- ! sponds to c=11 a The experimental values are all under this line, and there also seems to be a certain orientation of them in this direction. Certainly the structure of the metals is known to be consistent with our premises only by Cu, Ag, Au, Al, Pb, and perhaps Ni; but the other metals are given because their compressibilities by means of equation (8) give values for » of the same order as those of the named metals, so that it seems probable that our calculations are approximately valid for them. ‘The alkaline metals, on the other hand, are excluded. The values for o refer to temperatures greater than 6. They are taken from measurements of o or of the thermoelectric power against Pb, ae OSH 760 Dr. G. Borelius on the and are often very uncertain. As, however, this uncertainty is of very little importance in our figure, there is no need to give an account of authorities or way of calculation. > Fig. 1. Pt@: eS 7R § 11. Specjic Heat. From (14) we are able to calculate the mean kinetic energy bss of the electrons and its contributions c’ to the ered We find for high temperatures ; ie fs > SS dr G ar lee 9 = pP- Ne“ 2. nv R? dt? Electron Theory of the Metallic State. 761 Bxg ae ; z where r=— Is the specific resistance. Putting p=1, and kK AAV Mee ee tet. 2 (26) A at degree Inserting numerical values, we find in two extreme cases for Ag c'=0:004, and for Bi c'=0:09 calorie per degree. As the atom-heat is for high temperatures almost equal to 6 calories per degree, we see that the electrons give but an imperceptible contribution to it, which is in good agreement with the modern theory of specific heat and experimental facts. $12. Emission of Electrons from Hot Metals. Measurements on the density 7 of the electric current from the surfaces of hot metals are satisfied by b ie ay ee where A, A, and b are constants. The constancy of 6 is well established. It is for some metals known with an uncertainty of a few tenths per cent. A has a similar uncertainty already in its logarithm. About we only know that it cannot be much greater than one. On the presumption that the kinetic energy of the electrons is proportional to Ty and that Maxwell’s law of distribution holds, this expression is deduced theoretically, Rodi ae whereby A is found to be one-half. 7 is then interpreted as the ratio of the work necessary to remove an electron from the metal to two-thirds of the mean kinetic energy of the electrons. Assuming the electron energy to be equivalent to that of a gas molecule at t e same temperature, and the work done when an electron is removed to be of the order €2 R agreement with the experimental values for b. We can try to calculate 2 in a similar way from the point of view of our theory. If the electrons are supposed to be emitted from the interior of the hot metal, we should have b - aoe where ¢, is calculated from (2), (4), and (5) to be 1:742 ane and wu is given by (10) or (14). In this way - Phil. Mag. 8. 6. Vol. 40. No. 240. Dee. 1920. 3 D- of magnitude of |, the classical theory comes into good 762 Onthe Electron Theory of the Metallic State. we should, however, since w is very smill, get values for b that would be about a thousand times greater than the values known from experiment. ‘This difficulty for our theory will, however, disappear if we assume the electrons to be cared from be outermost layer of electrons—an assumption that is indeed well supported ‘by the great dependance of the emission phenomenon upon small impurities of the surface. For these surface electrons, our reason from § 6 for assuming the kinetic energy of the electrons to be very little does not hold. Indeed, there will be more free impacts here with the outer atoms; so that the kinetic energy of the electrons will probably be of the same order of magnitude as for the atoms themselves. § 13. Magnetic and Galvanomagnetic Phenomena. I shall not here treat these phenomena in detail, but will only refer to the results of my earlier investigation on this subject. ‘The motion of the free electrons, as we have imagined it, will give rise to paramagnetic and diamagnetic phenomena with susceptibilities within the limits 1 Np? eres if e’N ph? 97 GC Mat A oe So) ee ey which for high temperatures must take approximately constant values. ‘This is in good agreement with the observed magnetic properties for the metals situated in the left groups of the periodic system. The bad con- ductors Sb and Bi in the 5th group make an exception with great diamagnetic effects, and the ferromagnetic metals in the 8th group, at temperatures where “they are not ferromagnetic, as well as their neighbours in the system have very great paramagnetic susceptibilities, which are, like the ferromagnetic phenomena, probably to be referred to the influence of: the magnetic field of force on the inner electron rings in the atom. Also the positive and negative Hall effects and their order of magnitude are, in a similar manner, deduced from our general assumptions. § 14. Concluding Remarks. In our investigation the phenomena of thermal and electric conduction at low and very low temperatures are as yet left aside. The reason is that the treatment of these Flectrical Method for Measurement of Recoil Radiations. 763 phenomena would necessitate the theory to take into con- sideration the quanta conditions for the energy exchange between atoms and conducting electrons, which seems to be a rather difficult task. However, there are good possibilities for a qualitative understanding of these phenomena. + Thus, for example, as we have assumed the electron lattice to be ordinarily in a “fluid” state, we have a very promising chance to interpret the transition of a metal into a supra- conductor at very low temperatures as the solidifying of the electron lattice, the critical temperature being its melting- point. We thus should adopt for the supraconducting state the ideas used by Lindemann for the interpretation of conduction generally. Lund, November 1919, LXXXVI. An Electrical Method for the Measurement of Recoil Radiations, By A. L. McoAutay, B.Se.* General, N the course of a research made with the object of investi- gating soft X radiation produced by the impact of «rays upon metal targets, it was found that there was an easily detect- able increase in ionization in a chamber beyond the range of the « particles when they were fired through hydrogen instead of through air. Various tests made to determine the nature of this effect showed that it was due to a stream of hydrogen atoms set in swift motion by collisions with @ particles. Sir Ernest Rutherford has investigated this radiation by the scintillation method (Phil. Mag. June 1919), but it was supposed that it would not be “measurable by an electrical method, owing to its smallness, and to the large y ray ionization necessarily present near the radioactive sources used. The above results, however, seemed to indicate that by making use of a balance method such recoil radiations might be ” satisfactorily investigated by the ionization they produce. An electrical method has many advantages over one based on counting scintillations. It does not involve the dark room and eye-strain inseparable from the latter, and owiny to the large number of particles effective in one observation, probability variations are negligible, and a smaller number of determinations are necessary to fix a magnitude with accuracy. * Communicated by Prof, Sir E. Rutherford, F.R.S, 3D2 764 Mr. A. L. MeAulay on an Electrical Method A research was consequently undertaken with the following objects :— (1) To develop an electrical method for the measurement of such recoil radiations produced by « rays, so far ex- clusively investigated by the scintillation method. (2) To check by ionization measurements the results given by Sir Ernest Rutherford in the paper above referred to, which show that in close collisions between an « particle and a hydrogen atom, the nuclei no longer behave like point charges. (3) ‘To see whether any difference could be observed between the curve obtained by ionization showing the ab- sorption of the radiation in aluminium, and that given by the scintillation method, due, for instance, to change in lonizing power with velocity of the hydrogen particle. Description of apparatus. Three forms of apparatus, all on the same principle. were used in succession. Only the last will be described, It consists of two main parts: the first, figure 1, is a brass box, 8 cm. tong by 2 wide by 6 deep, in which the hydrogen recoil atoms are produced. The source of rays, A generally a brass cylinder about 3 cm. long, coated with active deposit from radium, is carried on a vuleanite plug B, fitting a tube in one end of the box. In the other end, exactl opposite, fits another plug, C, carrying a hollow brass cylinder, D, in which are cut slots with aluminium screens waxed on them for intercepting the « rays. Two slots were for the Measurement of Recoil Radiations. 765 used, one open, and the other covered by an aluminium screen of stopping power equivalent to about 5 cm, of air. The effect is investigated by measuring the difference in ionization produced in an ionization chamber placed above the aluminium window I’, when first one of these slots faces F and then the other. Hy drogen from a Kipp generator is passed through the box, by way of the side tubes 1 & J, throughout the experiments. The second part of the apparatus (fig. 2), which measures the recoil radiation produced in the box, consists of electroscope, (J, ionization chamber, M, and a compensating ionization chamber, R, used to balance the current through M. P and P’ are two vuleanite slabs shaped to fit in the V of the brass box. They support the electroscope and two ionization chambers and insulate each from the others. M and part of Rare over the window F. The compensating chamber, R, is merely a cylinder of thin brass (183 mgs. per sq. cm. thick) in which slides a piston, 8, also of brass, so that the volume of R in which ionization is effective may be adjusted. ‘The ionizaticn chamber at M !s a small aluminium box made of sheet aluminium soldered into a 2 cm. cube of 6°24 mgs. per square cm. thickness. The electrode N carrying the electroscope leaf passes from the electroscope through the ionization chamber into R, being supported by sulphur insulation in P and P’. The dimensions of the electrode shown in the figure were arrived at as giving a compromise between complete saturation in the ionization chamber and a low capacity. The electroscope is somewhat similar to one designed by Trepathi, working at the Cavendish Laboratory under Mr. U. T. R. Wilson. It i is, however, a much rougher and less sensitive instrument. It is made of a brass plate about 2 mm, thick, in which is cut a semicircular hole, U, 13cm. in diameter. The sides are of thin lead clamped by brass plates of the same shape as the main plate forming the body of the electroscope, bolts passing right through from one side to the other. The leaf is about 3 mm. long and is observed through mica windows, 3 mm.x1 mm. by « microscope magnifying about 80 diameters, . 766 Mr. A. L. McAulay on an Electrical Method ~ The potentials on the various parts are as follows :— The sides of the compensating chamber are charged to —120 volts, the aluminium ionization chamber to +40, the leaf through the charger, O (fig. 2), and the electrode to —40, and the case of the electroscope is coniected to earth. When in use the whole apparatus is of course flooded with y ray ionization from the source, and the current which this carries in the compensating chamber gives the leaf a higher negative charge, while the current in the ionization chamber tends to discharge it. The volume of the former can be adjusted to give any strength of current, and thus any rate of movement to the leaf. The principal objects in view in the design were :— (i.) To keep the capacity as low as possible while ensuring approximate saturation. Gi.) To use an ionization chamber as small as possible. This is necessary to give definiteness to the absorption curve. Dimensions at right angles to the radiation must be small in order that hy drogens driven in the direction of the a particles shall not enter the chamber too obliquely, and deonsibae parallel to the radiation must be small in order that the depth of air in the chamber shall not represent more than a small part of the absorbing material necessary to bring the hydrogen particles to rest. A lower limit is set to the size of the ionization chamber by the necessity of a measurable ionization, (iii.) To make the volume of the electroscope as small as possible. This is important because the ionization due to the recoil radiation must be as large as possible compared with that due to y rays, in order to reduce fluctuations, and while the former is only produced in the ionization chamber, the latter is produced in the electroscepe as well. Method. The method employed is briefly as follows :—The brass box is placed between the poles of a powerful electro- magnet which*prevents @ rays from the source entering the ionization chamber. The « rays are alternately cut off and allowed to fall on the hydrogen by rotating the cylinder, D. In the latter case, though the « particles are absorbed in the window, F, the recoil atoms penetrate it and enter the ionization chamber, causing a slight increase in the ionization. The walls of the compensating chamber, being certainly equivalent to more than 50cm. of air absorption, are not penetrated. With the & particles cut off, the volume of the for the Measurement of Recoil Radiations. 767 balancing chamber is adjusted till the leaf is almost stationary. A reading of its rise or fall is taken, the @ particles are ullowed to impinge on the hydrogen, and another reading taken, The difference between the ionizations gives a measure of the radiation. Readings such as these are taken with different thicknesses of absorbing material, generally aluminium, between the window eh the Bavtaie of the ionization chamber. The maximum ionization due to the recoil atoms was, in the different arrangements used, from 2 per cent. to 10 per cent. of the total. Tests made to establish the nature of the radiation. Experiments were made with a gap between the box in which the recoil particles were produced and the ionization chamber. ‘This gap was in an intense magnetic field, which when varied produced no corresponding change i in the ionization. It cannot therefore be a @ radiation. The form of the absorption curves shows that it is not of y or X ray type. The following experiments were made to show that it is not a primary radiation from the source. as x § Tee > 2 5 onization S S Hydrogen, S : Ss © 2 chamber. R S 5 R wR Sereen A is between the source and the hydrogen. Screen B between hydrogen and ionization chamber. If the radiation arose at the source, though a change in A+B’ would affect the ionization in the chamber, it would be immaterial how the total thickness of absorbers was dis- tributed between A and B. It was found that the effect was reduced to less than 1/5 when A was changed from 3 cm. alr equiv: alent to 6 em. air eqnivalent, instea| of from Oem. to 3em., although in both cases A+ B had changed from 6 cm. to Vem. Results. The following are the tabulated results of the absorption experiments, plotted in curves I and II. Table A, This table gives the results from two typical absorption experiments made with an a-ray tube as source. a rays of ranges from about 5 em, in air to zero. were 768 Mr. A. L. McAulay on an Electrical Method incident on the hydrogen. The values are plotted as crosses on curve I. Absorption in path of rays (between HE Ionization ae source and ioniza- : cea Absorption in at ; . in arbitrary : ¥ lonization. tion chamber) given oe path of rays. in equivalent cms. eee ee of air. 13:1 "09 93 cm. of air. 1 14°4 "39 10°6 “75 158 "23 13°1 33 16°9 0 14°4 “41 16°9 ‘tl 18:2 08 Table B. This table gives the result of one experiment made with a later and improved form of apparatus. An a-ray tube was used as source, and the ranges of the Absorption in path of rays. Ionization. 8:7 cms, of air. 1 12°6 48 14°4 "06 18°3 ‘ll 20°1 ‘02 Curve I. fonssation. 8 l2 16 20 Absorption in eguivelent cms. of arr. emerging particles were from about 5 cm. to zero as in the experiments from which table A is derived. The values are plotted as circles on curve I, forn/sation. for the Measurement of Recoil Radiations. 769 Table C. This table gives the results from four. experi- ments made with a brass cylinder coated with the active deposit from radium as source. a rays of ranges from 7 em. maximum to about 5°8 em, hydrogen. ‘The values are Absorption Absorption minimum were incident on the plotted in Curve IT. Absorption _Toniza- Loniza- ; Toniza- : in path . in path in path tion. of rays. ee gis of sa a oe of rae 11:0 ‘97 11:0 1:00 11:0 1:08 16:2 1:04 22°8 ‘79 14:9 ‘98 21-4 1:00 24°1 ‘61 16:2 1:02 25'3 ‘53 25°3 ‘Ol 18:9 "95 26°7 "38 28 0 "25 20°1 ‘G1 28°0 "21 — a 21-4 "82 29-2 ‘06 11:0 1:00 22°8 ‘81 20-0 “95 29°3 ‘09 30°4 0 The dotted curves are replotted from figure 5, p. 550, of the paper by Professor Rutherford above referred to (Phil. Mag. June 1919). That shown with curve I. is E in fig. 5 of his paper, and that with curve II. is A. Curve II, Absorption in equivalent ems, of air. Summary of Results. The results obtained check the main points summed up in fig. 5 of the paper above referred to (Rutherford, Phil. Mag. June 1919). Curve II. is almost directly comparable with 770 = Electrical Method for Measurement of Recoil Radiations. curve A. It will be seen that the principal difference is that the ionization curve gives a greater range for the recoil atoms than the scintillation curve. This may be because different values have been taken for the stopping powers of the absorption screens used in the two cases or to the fact os the scintillation curve was made with mixed absorptions Faluminium and mica. At the same time, in the analogous case of the absorption of a homogeneous beam of @ rays, ionization is detected beyond the range of the particles as determined by scintillations, and the difference here may be due to the same causes. Curve I. is less definite than curve LI. because the glass of the a-ray tube used as the source was not of uniform thickness. Some @ particles of range just over 5 cms. were present, und probably every range was represented between this and zero. The difference between the ranges of the recoil particles given by this curve and the scintillation curve E is not accounted for by the small difference in the maximum ranges of the « particles producing them, and must be given an explanation similar to that suggested for the long range particles of curve IT. The above remarks indicate the success of the research from the point of view of two of its objects. It confirms Professor Rutherford’s scintillation experiments, which show that in close collisions between « particles and hydrogen atoms, the nuclei cannot be considered as point charges, and that most of them are thrown straight forward. It does not show any essential difference between curves which plot ionization against absorption and those which plot scintilla- tions against absorption, except the small difference in range above discussed. ‘The effect of increase in ionizing power of a particle as its velocity decreases is probably masked by differences in the experimental arrangements used to obtain the scintillation and ionization curves. With régard to the first object of the research, the electrical method with the apparatus as finally used is suitable for the investigation of recoil radiations, and should be capable of modification for use on an effect only about a fifth as great, with similar quantities of radioactive material. The maximum effect was about two divisions on the micro- scope scale per minute, for an activity corresponding to one milligram of radium. My best thanks are due to Professor Rutherford for his continued interest and advice throughout the course of this work, een: | LXXXVIL. Note on Einstein's Law for Addition of Velocities. By W. B. Morton, J.A., Queen’s University, Belfast *. FPXUE way in which the theory of Relativity has modified the law of combination of velocities can be shown very clearly by an obvious graphical method, and the curves bring out one or two points of some interest. Take first velocities in the same line. Let wu be the velocity of the “moving platform” as seen from the “ground” and v the velocity of a point on the platform measured by platform standards, the velocity of light being taken as unity. Then the velocity of the point as seen from the ground is (u+v)/(1+uv). Giving wa definite value, we may plot this “resultant” against v. This is done on fig. 1, for w="2, *4, °6, °8, these * Communicated by the Author. : 4b2 Prof. W. B. Morton on Einstein’s Law values being given by the intersections of the curves with the axis of y. The curves are portions of rectangular hyperbolas having centres at ec=—1/u, y=1/u, and asymptotes parallel to the axes of wy. They run to the terminal points which corre- spond to the velocity of light in the forward and backward directions. On the old theory the graphs would, of course, be straight lines, starting from the same points on the y-axis and making 45° with the axes. The most interesting features of the curves are in the region of negative v (backward motion along the platform). The resultant vanishes for v= —u, just as in the old theory, although now the “ equal” speeds are measured with different standards. This is merely another way of expressing the reciprocal nature of the Lorentz transformation: the ground appears to move backward with velocity wu from the platform standpoint. When v= —1 the resultant has the same value ; a light-signal travelling backward has unit velocity to both sets of observers. When w is large there is a very sudden increase in the resultant speed as v passes from —uto —1. For example, if the platform moves at °9.of the speed of light, a point moving backwards along it at this same speed will appear stationary from the ground. Butif the speed is pushed up to the full value for a light-signal, this has its full value also as seen from the ground. On the diagram this is shown by the sudden downward plunge of the curve for 8. As the value of w is increased towards unity the hyperbola ap- proaches the rectangular lines along the top and down the left-hand side of the diagram. From this consequence of Hinstein’s formula there follows a curious paradox which I have not seen mentioned, and which may be added to the many odd things which would happen if anything but light could be made to move as fast as light. First let the platform have any speed, and let a point move backwards along it with the same speed as measured on the platform. Then the point is at rest as seen from the ground. Increase the common speed up to that of light and we have a light-signal moving backward along a platform which moves forward at the speed of light and appearing stationary from the ground. But now let the backward-moving thing on the platform be a light-signal from the first. This, on the fundamental assumption of the theory, moves with the velocity of light as for Addition of Velocities. 173 seen from the ground, no matter at what rate, under light- velocity, the platform is moving. Assuming physical con- tinuity up to light-velocity, we reach a conclusion contra- dictory to the last. Mathematically the paradox has its origin in the fact that the expression (w+ v)/(1+ uv) is indeterminate for the values u=—v=l,and approaches different limits in the two ways in which we have reached these values of the variables. Physically it means, I suppose, that the measured speed of 300,000 kilometres per sec. cannot be attained by matter, and that, at speeds very close to this, there would be an extraordinary instability in the measured value of the re- sultant of opposite velocities of that order of magnitude, ,aw) Vig. 2 shows the variation of the magnitude of the resultant 1S when wv are at right angles to each other, in accordance with the formula (w?+v°—w?v?)2, The curves are again hyper- bolas with centres at the origin and asymptotes y= +au(1—w’*)?. On the old theory they would be hyperbolas, passing through the same points on the y-axis with asymptotes y= +2. In the general case, when wv are inclined at an angle y the magnitude of the resultant is (w? + v? + 2ur cos y—u?e? sin? y)!/(1 + ur cosy). 174 Ktinstein’s Law for Addition of Velocities. Two additional points may be mentioned. First :—Seeinge y = that the curves are convex upwards in fig. 1 and downwards in fig. 2, it is evident that they may become approximately straight for an intermediate value of y. The tangent to the curve at the axis of y is directed towards the terminal point (1 1) for seey=1+u. In that case there cannot be much variation from the straight line—z. e., the Speed increases ina linear manner up to that of light as vis given larger and larger values. This is seen on the diagram: (tig. 3) which is. ot) for y=45° ; there is not much deviation from straightness in the graphs on the positive side of the origin. Second :—It comes out that the condition for a resultant of minimum size, when wy are given, is identical with that found on the old theory—viz., v=—ucosy. The square of the resultant may be written in the form (1—u’) ees (1+ wv cosy)? ’ and (1—v?)/(L+wuvcosy)? is maximum for v = —u cosy To show the continuity of passage from fig. 2, through fig. 3 to fig. 1, the parts of the curves in fig. 1 which he below the axis must be reflected upwards so as to give the magnitude of the resultant without respect to its direction. 1 — PN See LXX ahae The Absorption of Light by the Goldberg Wedge. By F. C. Toy, M.Se., A. Inst. P., British P hotographic code et Association, and J. C. Guosn, )).Sc., University College, London™ . [Plate X VIT.] “pes need of light absorption are now fre- quently made by means of a thin wedge of lamp-black in gelatine contained between inclined glass plates. This wedge was devised by Goldberg f in 1910, and is now available in several commercial forms. It is usually assumed to be neutral, 7. e., to have a constant extinetion coefficient for all wave-lengths. As this constancy does not appear to have been verified over any considerable range, an investi- gation of its possible variation was made. The “ gradation ” of the wedge, 7. e. the increase of density (in ‘the photo- graphic sense) t per unit length in a direction perpendicular to the Popodnes: is its most important characteristic, and the variation of this gradation with wave-length was investi- gated. For, while this variation is directly proportional to the variation of the extinction coefficient, it cannot be directly calculated from the latter, which is therefore of lesser practical importance. For the details of the method used, the reader is referred to a previous paper§. This method is independent cf the Schwarzschild effect ||, so that greater accuracy can be obtained than in methods previously employed. The accuracy of the method allows the variation of the gradation with the wave-length to be followed very closely. — Fig. 1 shows diagrammatically the arrangement of the apparatus, The source of light Misa quartz. mereury lamp * Communicated by Prof. A. W. Porter, F.R.S. + B. J.P. lvii. p. 648 (1910). t The density is defined as —log,,T, where I’ is the photometric Intensity of Light Transmitted Intensity of Light Incident. § Proc. Roy. Soc. xevii. p. 181 (1920). | Photo. Journ. lyi, p. 11 (1916), transparency, 1. e. 776 Mr. F. C. Toy and Dr. J. C. Ghosh on the fitted with an iris diaphragm A, by means of which only a small central circular portion of the Jamp, about 1-2 mm. in diameter, is used. A lens L is placed so as to project a magnified image of this on to the slit SS of the spectro- scope. The intensity of the image is very uniform, since it is formed from such a small isolated patch of the source of light. The Goldberg Wedge, W, is placed in front of the slit and in contact with it. The length of slit used is ab ut 1°5 em., and a special wedge was made up between two fused silica optical flats by Messrs. Ilford Ltd. The intensity incident on the wedge is varied by varying the aperture of the projecting lens L by means of the adjustable stop B. By keeping the aperture small and the focal length large, so that the curvature of the lens surfaces is small, the intensity of the image is almost exactly proportional to the square of the aperture of the lens. The current through the lamp is of course kept constant during each experiment. A metal filament lamp is used for experiments at the red end of the spectrum, since there are no lines in the mercury are suitable for measurement in this region. The intensity is varied in this case by changing the distance of the light source from the wedge. This apparatus is used by taking two spectrographs through the wedge on the same plate with two differing intensities in known ratio and with the same time of exp sure. From this negative the gradation at several wave-lengths can be determined as follows. Let A, be the gradation of the wedge for any wave-length A, and the two incident inten- sities be I, and I,. Let the densities at the two points at which there is the same intensity (I,) transmitted in each case be D, and Dy, and the distances of these points from any arbitrary fixed point on the wedge be a, and a. At the images of these points at any particular line of the spectrum the same effect will be produced, so that on development the densities will be equal. This equality is independent of the exposure, the kind of plate used, or the treatment it receives. From the definition of density, Lad xa ox IO Pe, logyo L/2=D,— Dp =A, X (a — a2). Therefore 1 Cree % pet dl pce ah iia Absorption of Light by the Goldberg Wedge. 777 If y, and yp are the distances on the plate corresponding to wy and w, on the wedge, then (41 —Y2)/(@1—a)=My,. -- . « - (2) where M, is the magnification produced by the optical system of the spectroscope at wave-length 2. Whence from (1) and (2) Mos x logio iyo? pi ae see opts (3) —Y2) The magnification is ee by placing an opaque object of known length across the jaws of the spectroscope and comparing the length, at different wave-lengths, of the image obtained on the plate. If, now, y; and v3 can be determined from the plate, A, can be calculated. Now, y; and y are simply the distances from an arbitrary point, such as the image of the end of the slit, along any one line to any two points of equal density (intermediate between zero and maximum density), and can readily be measured in the following way. Big. 2 (Pl. XVII.) is a positive printed from one of the original spectrographs. It will be noticed that the change from black to white along any one line is very gradual. The steepness of this change will, of course, depend on the gradation of the wedge and the kind of plate used. The a y, and y. will be approximately as indicated in fig. 2. If a succession of transparencies of this spectrograph are now made, the change from black to white will be made sharper with each reproduction. Fig. 3 (Pl. XVII.) shows the “ second” positive obtained from the same original spectrograph by three reproductions. The change from black to white is now practically a sharp ° line, and coincides with the point of half maximum density, or any other density intermediate between the maximum and zero. The distances y, and y, can now be measured directly by means of the travelling microscope. In fig. 3 it will be noticed that towards the end of longer wave- lengths some of the lines (e.g. H, K, and M) are suf- ficiently intense to cause blackening over the whole length of the slit. These, of course, cannot be used for making measurements, but on each photograph wave-lengths can be selected at which the intensity is such that y,; and y, are both of suitable length, with the ends well clear of the end of the slit. For example, lines A, B, C, D, F, G, H, and L were used in this case. The wave-lengths of the various lines can always be ascertained at once by reference to a standard spectrograph. The value of (y;—yz) at any given wave-length for a given Phil. Mag. 8. 6. Vol. 40. No. 240. Dec. 1920. dH 178 Mr. F. C, Toy and Dr. J. C. Ghosh on the ratio of intensities incident, is determined for several times of exposure. This variation of exposure will change the values of y; and yz, but the difference will remain constant. The photograph shown in fig, 4 (Pl. XVII.) is taken over the same range as that shown in fig. 3 and with the same ratio o£ incident intensities, but with a different time of exposure. Table A is typical of the values obtained :-— TABLE A. Ratio of Intensities 23:1. Wave-length 308°8 up. Vime of exposure Peo». Deviation of single in minutes, in ems, observation from mean. 5 0:418 0:008 15 0:410 Q-0C0 15 0:395 0015 1:5 0:424 0014 3 : 0-415 0-005 5 0°398 0012 Mean 7a Se 0:410 0:009 eet! 0:009 Average deviation of Mean = ie = (004. Therefore (y1—Y2) =0°410 + 0-004 cm. TaBLE B, ° | Gradation (increase of density per cm.). | Wave-length Mercury Lamp. M. F. Lamp. | Le Eo | Tie SO: f/=230; T,/1,=310: | OOO Bats ccm eh eem ee lag Retege a kamen 1:23 oy Oram aire A eter gaa e2 MeN Wen ch Rete a 1-21 DOO cece ani Ae eghaeneme agian he siete e 1:25 | SAS pornsnetocs 1:29 LAO MRR sera akan Marian ie ca ia ae 1-28 OO) soir ach waa alin ha ae By Cea ee ia Eh cannon 1:29 | DEO Te poe ties 1-42 AOD yew Behe 141 Ae et eee 1-41 AOS ican? carat 161 1:67 So fe) Teena 161 SO ei uate hier Coewaret 171 BHO veheeeen as Lay (3) 1 SOS slam 1:80 1:82 DOD Mray se Senne 2°70 2°63 BILD 8s phew eae 3°02 3:07 BOB i ae hee 3°22 3°29 DOS anes 3°33 3°51 DOO ee neacraniee 4:06 3°86 Absorption of Light by the Goldberg Wedge. iio The values in Table B are plotted in fig. 5. The circles represent points determined with the metal filament lamp, the crosses and solid circles those determined with the mercury arc, but with ratios of intensities incident on the 3 400 500 600 r IN Me wedge of 8-1: 1 and 23:1 respectively. It will be seen that the density is very nearly constant throughout the visible region of the spectram, though there is a slight decrease from the yellow-green towards the red. In the ultra-violet, however, the density increases rapidly with decreasing wave- lengths. It was thought that this might be due to the presence of gelatine in the wedge, but no absorption by the gelatine alone could be noticed above the accepted wave- length 220 uu. The values for the gradation in the visible spectrum were verified by means of a Koenig-Martin Spectrophotometer. It is evident from the results that while this particular type of wedge is approximately neutral throughout the visible spectrum, there is a very pronounced variation of the extinction coefficient with the wave-length in the ultra- violet part of the region which is normally used in photo- graphic research. dK 2 ie 80 | LXXXIX. On the Pressure on the Poles of an Electric Arc. To the Editors of the Philosophical Magazine. G ENTLEMEN,— NSERTED in your October issue isa letter from Dr. Ratner in which he objects to the argument of Prof. Duffield that any motion which ions may acquire in the field of an electric are cannot give rise to a pressure upon its poles. It is not for me to discuss the part of his communication which deals with Prof. Duffield’s work. But he goes on to make deductions from experiments upon the eleetric wind; and as one of the investigators of that subject to whom he refers, I think it is desirable for me to draw attention to the faulty interpretations that he places upon some of the results of their work. In his own work which he quotes, the vane upon which he detected a pressure was placed behind a perforated electrode which received the ions. The vane therefore received some of the momentum of the wind which blew through the per- forations without experiencing the compensating attraction of the ions in the discharge; this was taken entirely by the electrode. He is therefore incorrect in stating that those experiments on an electric wind give evidence of a repulsion of an electrode which has an opposite sign to the ions giving rise to the wind. Secondly, in his remarks on the diminution of the wind with high values of current he ignores the fact that an electric current in a gas is notin general carried by one sign | of ion only. In discharge at atmospheric pressure from a point or a wire with small values of current, only one sign of ion is present through the greater part of the path of the discharge. Butthis is not true for all values of pressure and current, and it is very improbable that it is true in are discharge for any values of pressure and current. In conjunction with Mr. H. E. G. Beer, I have been engaged for some months on an investigation of the electric wind in the are under various conditions. We find that in general itis exceedingly minute. Its direction is from cathode to anode, a fact which under certain circumstances is in accord- ancewith the manometric observations made by Dewar many years ago and referred to by Duffield in his paper. The results are quite consistent with the view that the effect is a residual one, being the difference between the effects which the positive e and nevative ionic streams would have separately On the Alternating-Current Carbon Are. 781 produced. Owing to the great difference between the specific mobilities of positive eand 1 negative ions at are temperatures, it is only necessary to assume ‘that through the body of the are a small fraction of the total current (less than 4 per cent.) is carried by positive ions to account for the magnitude of the observed wind. A fall account of these experiments will be published shortly. I am, Gentlemen, Physics Laboratory, Yours faithfully, University, Bristol. A MoT yD Ar Oct. 15, 1920. XC Note upon the Alternating-Current Carbon Arc. By Prof. W. G. Dtrrietp, D. ek, and Mary D. WA.LER, B.Sc.* AXPERIMENTS carried out in the Physies Laboratory of University College, Reading, some years ago showedt that the amount of carbon lost from the cathode of a direct eurrent are consisted of two parts: (1) that necessary for the’ mechanism of the are, and (2) a quantity lost by evaporation or combustion, which, though possibly affecting the voltage, temperature, br ightness, and the current, could only be regarded as subsidiary. Chief interest centred about the shortest are, less than one millimetre in length, which it was found possible to maintain; for here the loss from the cathode was entirely of the first category, and such that for each carbon atom lost from that pole there was a transfer between the electrodes of a quantity of electricity equivalent to four electronic charges. Fi was farther foundt that, though in a normal arc the anode loss is considerably greater than that from the cathode, it is nevertheless possible so to cool the anode by rotating it, that its loss of weight may be reduced, not only below that of the cathode, but nearly to zero. A careful examination of the contour of the poles of a direct-current are showed that the shape of the anode re- mained practically unchanged, but that, even though pre- viously burnt to shape, the contour of the cathode ofa v ery short are went through a well-marked cycle of changes, which were repeated again and again as the expenditure of carbon proceeded. * Communicated by the Authors. t Duffield, Roy. Soe. Proc. A. xcii. p. 122 (1916). H Duffield & Waller, Roy. Soc. Proc. A. xcii. p. 247 (1916). 782 Prof. W. G. Duffield and Miss M. D. Waller on the - The present set of experiments was undertaken for the purpose of testing the behaviour of the alternating-current are chiefly as regards the above-mentioned features. Since in an alternating-current arc each pole acts for half a period as anode and halfa period as cathode, it seemed most likely that the loss from each pole would be intermediate between those from the anode and cathode of a direct-current are using the same current. This we have put to the test and shown to be the case; indeed, the loss in the case of the alternating current approximates a little more closely than might be expected to the mean of the anode and cathode rates of carbon consumption. In putting the matter briefly upon record, it is only necessary to say that the previous method of experimenting was adopted (loc. cit.), the special precautions being the burning to shape of the poles before the first weighing, and the prevention of absorption of moisture from the atmosphere. The results are best shown diagrammatically. Diagram 1 shows the relationship between the loss per coulomb and the arc length for different currents. The curves possess the general shape of those obtained with the direct-current are, rising at first rapidly, but ultimately reaching a nearly stationary value. They resemble the anode rather than the cathode curves in that they do notall diverge froma common point upon the vertical axis. Hven in the case- of the shortest possible are the consumption of carbon per coulomb exceeds 3:1x107° om., the theoretical limit; so it is clear that in all cases there is subsidiary as well as essential carbon Joss. Diagram 2 shows the loss from either pole of an alternating- current arc using 8 amperes, compared with the individual losses from the anode and cathode of a direct-current are carrying the same current, and with their mean value. The latter agrees well with the alternating-current curve, but is a little higher when the arc-gap is long. For currents of 4 and 2 amperes the agreement is not so good, though of the same order of magnitude, the alternating-current consumption being definitely less than the mean value in the direct-current curves. It thus appears that during the half-period when any given pole is acting as anode, it does not play a passive role like that which was, by artificial means, induced in a rotating anode, but that there is time for the pole to assume to a large extent the condition of the anode in the direct-current are, though there is no crater; it certainly gets hotter than the cathode, and is able to lose a consi lerable amount of material. Alternating-Current Carbon Are. Diagram 1. -5 2 Amps. in gms. xO 25 Loss per Coulomb Loss per Coulomb. Arc Length in mms rs ieee a Rae Pee Cee cera 2° a Bi Be Ie 14 The rate of carbon consumption per coulomb plotted against are length. The shaded area represents hissing arcs, Loss of weight per Coulomb. “5 Loss per Coulomb 2 in.gms x10 783 Diagram 2. Mean D.C. 2 Amps i Alternating Mean D.C. 4 Amps. L£ Alternating 7 Anode 8 Amps. — Mean D.C. 8 Amps. ~~Alternating Cathode Arc Length in mms. 7 ee (0 Comparison of Alternating- and Direct- Current consumption. 784 On the Alternating-Current Carbon Are. The wastage is increased by the fact that during the second half-period, when the same pole becomes the cathode, it is_ hotter than it would have been in a direct-current are. Successive contours of a very short alternating-current are were drawn by projecting an image upon transparent paper and tracing the outlines at short intervals of time, the are length being kept constant throughout. There is a very marked difference between the results and those obtained’ with the direct-current are already referred to; here there is no cycle of changes, but a contour of practically unvarying form. No constriction develops in the rear of the pole face when the are is short, though a tendency to form one has been observed in the case of longer alternating-current ares. No deposition of material takes place upon either pole as it does in the direct-current are; or, if deposition occurs in one half-period, a still larger quantity i is consumed in the second half-period. Three further points may be briefly mentioned :—We noticed that the upper and lower carbons were not equally consumed ; for short ares the lower, and for long ares the upper pole lost more rapidly, the value of the current not affecting the ratio seriously: the ratio of upper to.lower pole loss varied in almost linear manner from 0°8 for the shortest to 1:9 for the longest are used, 15 millimetres. This is, no doubt, due to convection currents, which have greater freedom of access to the hot under-surface of the upper carbon in a long arc. Notes are made when the are was hissing, and such obser-. vations have been distinguished on the first diagram, from which it appears possible to divide it into two regions, one cha- racterized by hissing and the other by silence. Mrs. Ayrton found that hissing seemed to depend upon the access of air to the crater of the direct-current arc ; if more than a certain amount reached it, hissing ensued on account of an automatic protective mechanism which seemed designed to oppose the access of too much oxygen. Diagram 1 seemed at first sight to offer evidence in opposition to this, since silent ares are associated with high rates of carbon consumption per coulomb; but, as we have stated elsewhere, the loss per coulomb is governed more by the time taken for the coulomb to pass between the hot poles than by the strength of the current (since it obviously takes longer with small than with large currents) ; hence the objection is not serious. If we draw out curves representing the total loss in one second against current strength for arcs of different lengths, we find that hissing arcs are, in accordance with Mrs. Ayrton’s view, associated with high carbon consumption. Discharge of Electricity through rarefied Gases. 785 In conclusion, it is of interest to note the bearing of this investigation upon the pressure observed upon the poles of an alternating-current arc, which has been shown elsewhere to agree well with that for the D.C. are. We -now find the rates of carbon loss to be also approximately equal; hence the argument used in the case of the D.C. are, that the evaporation of carbon is of the wrong order of magnitude to account for the effect, holds good in the alter- nating- current are also. The reaction in the latter arc, as in the former, is believed to be occasioned by the cathodic expulsion of electrons whose momentum is propagated across the are through the vapour to the anode, where it produces an equal pressure. In the alternating-current are the pressure is therefore made up of a reaction consequent upon electron emission during the half-period when it is the cathode, and of the pressure consequent upon the reception of the momentum from the other pole during the second half-period when it is acting as the anode. . “ < Physics Laboratory, University College, Kkeading. XCI. Experiments on the Nature of Discharge of Electricity through rarefied Gases. By 8. Ratner, Research Student, University of Manchester * 1. JN spite of the large amount of work contributed to the study of the nature of the dischar ge of electricity through vacuum-tubes, the present theories fail to give a clear and adequate representation of the phenomenon. The most obscure point is the process by which, under a certain potential difference, the current begins to flow through the gas. it is usually supposed that the stray positive ions which may be present in the tube, strike against the cathode with sufficient energy to cause an electronic emission from the surface-layer of atoms of the cathode, and that these electrons, by collision with the gaseous Moieenica. produce sufficient ions to carry the whole current. Certain experi- mental facts, however; disagree with this theory. Thus, when the vactum is sufficiently high, the potential necessary to start the discharge increases with further exhaustion of the tube, although the energy imparted to the cathode by the positive ions depends only on the potential through which * Communicated by Prof. W. L. Bragg. 786 Mr. 8. Ratner on the Nature of Discharge of they fall. There is some evidence that «-particles from radioactive substances, by impinging upon a metal target, may set free slow electrons from the surface of the target, but the energy of a-particles is enormous compared with that of the positive ions striking the cathode in a discharge-tube. It seemed important, therefore, to investigate directly the role played by the positive ions in the process of the discharge, and further to carry out experiments with a view to a direct study of the conditions under which positive ions may, by impact with the surface of the cathode, give rise to an electronic emission from the latter. 2. The apparatus used in these experiments consists of a glass tube A (fig. 1) provided with two ground joints m and n, by means of which two electrodes—an aluminium plate P and a strip of thin platinum foi] s—may be easily introduced into or removed from the apparatus. A third electrode r in the shape of a rod or a small plate is introduced in the bulb through the side-tube ¢. The platinum strip’s is coated with aluminium phosphate or calcium oxide, and when electrically heated by an insulated battery of accumulators provides large supplies of positive or negative ions, according to whether it is used a4 an anode or a cathode. The electrodes may be raised to any desired potential by means of a battery of small cells or an induction-coil provided with a rectifying valve and a spark-gap. By means of a Gaede pump the pressure in the bulb may be reduced down to ‘001 mm. of mercury in spite of some grease and sealing-wax being always present in the apparatus, and if necessary the ex- haustion is carried still further by a charcoal tube immersed in liquid air. The electric current passing through the bulb, varying over a wide range between 107° amp. and several milliamperes, is measured either by one of two galva- nometers of different sensitiveness or by a milliammeter. 3. In the first place experiments were made in order to ascertain whether the conditions of the discharge would be appreciably changed by a stream of positive ions produced in the vacuum-tube. It is well known that a hot cathode emitting negative carriers of electricity lowers considerably the sparking potential, and according to the theory men- tioned above one could expect that the same effect would be reached by using a hot anode emitting under’ an intense electric force a stream of positive ions. The experiments were carried out in the following way :—The heated strip coated with aluminium phosphate is connected with the positive terminal of tbe induction-coil, the cathode P being earthed and the electrode r insulated, and the pressure in Electricity through rarefied Gases. 787 the bulb reduced to a certain value. The potential of the anode is then gradually raised by means of the spark-gap, and the current carried by the positive ions from the strip Fig. 1 LTT ITT TT TI TTT PITTI TTT ITT TTI TIT TI TTT TT TT TTT} UT] GL] lobbies TTT TTT TTT TILT TT] LY ELT ITITITI LIT ITT TNL. TTT TT TI TIT JOCRTACAROIAAI AMARA EEE es es esses eeaeneaeesenesaa: WITT TIT TTT a i er is measured. This current was usually of the order of 10-® amp. When the potential of the anode becomes sufficiently high (corresponding to a spark-length of: say, 788 Mr. 8. Ratner on the Nature of Discharge of 15 mm.), the discharge suddenly sets in, and the current through the tube abruptly rises to the order of a milli- ampere *. The external current heating the strip is then broken, and the experiments repeated with a cold anode, other conditions remaining the same. The discharge is found to take place at exactly the same potential as when the strip was heated and the cathode intensely bombarded by a stream of positive ions. The same procedure was followed at a lower vacuum, when a potential difference of the order of 1000 y. applied to the electrodes was enough to produce the discharge. In this case the voltage was supplied by a battery of small cells, and the sparking potential could be determined with great accuracy. The results obtained were as before, and the positive emission from the strip did not lower the sparking potential by as much as one per cent. 4. These results are hardly compatible with the usual theory of discharge of electricity through vacuum-tubes, for it is equivalent to admitting that positive ions falling through a potential of more than 20,000 v. in a high vacuum do not carry with them sufficient energy to liberate electrons from the cathode on striking against its surface ; while in other exses, on the other hand, a fall through less than 1000 v. in a lower vacuum enables them to produce this effect. If we suppose that the electrons necessary to start the discharge may be bombarded not necessarily only out of the cathode, but also from the gaseous molecules by collision with the positive ions, the same difficulties will still remain in the interpretation of these experiments. One might perhaps suppose that in a high vacuum, where the number of gaseous molecules present in the tube is comparatively small, the potential necessary for the discharge is necessarily high in order to ensure ionization at every collision, but such a theory would require a gradual increase in the discharge current as the potential rises from about 1000 v. and goes on increasing. It is well known, however, that the discharge in a well-exhausted tube starts abruptly when the applied potential attains a value considerably higher than 1000 volts. Whatever the nature of the discharge may be, the expe- riments described above show distinctly that it is not caused by the impact of positive ions on the surface of the cathode. It seemed important, therefore, to carry the experiments * It may be mentioned here that the walls of the apparatus were previously carefully freed from occluded gases by electronic bombard- ment and heating of the strip, so that the pressure in the bulb remained sufficiently constant throughout the experiments. Electricity through rarefied Gases. 789 further and to investigate whether positive ions by striking against the cathode are really able to liberate electrons from its surface, independently of the part which such elecirons may play in the process of the discharge itself. 5. In such experiments the following method of three electrodes was adopted :—The strip s, coated with aluminium phosphate and strongly heated, was connected with the positive terminal of the battery of small cells , the cathode P being earthed through a galvanometer, and the third elec- trode 7 raised to a “positive potential higher than that of the heated strip. The apparatus was kept well exhausted throughout the experiments, so that the comparatively smal] potential difference applied to the electrodes (not exceeding 2000 vy.) could not produce a discharge through the bulb. The positive ions emitted by the strip move towards the eathode P and may produce’electrons either by collision with the gaseous molecules or by impact with the surface of the cathode. These electrons move necessarily towards the third electrode r charged to a higher positive potential tlan the strip, and the current eee by them is measured by a galyanometer. The ratio of this current to that carried by the po-itive ions to the cathode P, gives the rate of pro- duction of electrons by the positive ions in the bulb. It was evidently necessary in the first place to inquire into the validity of the method, since it may seem doubtful whether all the electrons produced in the bulb will actually reach the electrode ry. For this purpose the electrode 7 was raised to different potentials, and curves were drawn show- ing the current flowing through this electrode as a function of the potential difference beacon it and the heated strip, other conditions remaining the same. These curves appeared to be similar in shape to the well-known saturation curve in ionized gases, the current increasing at first rapidly with the potential difference and finally remaining constant. A potential difference of 200 v. appeared sufficient to assure the saturation being reached, and in most of the experiments to be described the electrode 7 was raised to a potential of +2000 y., the strip being charged to +1800 v. Further, it was found that the current received by the third electrode was strictly proportional to the positive emission from the strip, which could be varied within wide limits by changing the temperature of the latter, so that the rate of production of electrons by the positive ions in the bulb was constant at a given pressure. ‘The experiments consisted mainly in studying the variation in the rate of production of electrons with the pressure, in 790 Mr. 8. Ratner on the Nature of scharge of the bulb. If the electrons present in-the bulb originate solely in ionization by collision between the positive ions and gaseous molecules, then the rate of production of the electrons should be expected to vary inversely as the mean free path of the positive ions, 2. e. it should be strictly pro- portional to the pressure in the bulb and should become negligibly small when the bulb is highly exhausted. If, however, electrons are also produced by impact of the positive ions against the surface of the cathode, then no such proportionality would be expected, and the current through the third electrode should remain considerable even when the highest exhaustion is reached. In the estimation of the rate of production of electrons by positive ions an important correction must be introduced, the omission of which led to misleading results in the earlier experiments. When a molecule is ionized by collision a pair of ions is. produced, so that the current through the plate P is carried not only by the positive ions from the heated strip, but also by the positive ions produced by collision. Also when an electron is set free from the surface of the cathode, a positive charge is gained by the cathode. If we denote by (,_) the current through the third electrode, by C,,) the current through the cathode P, and by R the rate of pro- duction of electrons by the positive ions from the strip, then it is easy to see that R is given not by am as at first anti- } (se) cipated, but by C ae : This correction 1s small when the bulb is well deme and consequently C,_) is small compared with C,,). : Asa result of a large number of experiments carried out in this way, it was shown distinctly that the electrons present in the bulb are produced only by collision between the positive ions and the gaseous molecules, and that there is no appreciable electronic emission from the surface of the cathode. ‘Table I., in which the results of one set of these experiments are given, shows that the law of proportionality between R and the pressure p holds well within the limits of experimental error. It was noticed, however, in some experiments that when the strip was raised to a very high temperature, a dispro- portionately large current flowed through the third electrode, even at the highest obtainable exhaustion of the bulb. This was finally found to be caused by a source of error which can, however, be easily eliminated. It was found that a strongly heated platinum strip coated with aluminium phos- phate or any other salt serving as a source of positive ions, Electricity through rarefied Gases. (feel TABLE 1. Cre C Or- P x 10 aan x hee R= ae Be x 1000 mmm, 46 268 ‘21 28 36 280 15 21 eh 255 ‘I4 18 24 245 11 15 26 295 10 14 18 235 083 12 15 280 056 9 13 275 050 8 12 285 ‘O44 6 6 260 ‘O24 3 3 295 - 010 1?) | 240 ‘004 ae emits also a large number of negative ions when being used as a cathode. ‘This is probably due to the fact that the salt usually does not cover the whole surface of the strip, so that the negative ions are emitted by the platinum surface itself. In the experiments described above the strip was used as an anode, but compared with the third electrode r it was always charged toa negative potential of 200 v., so that the third electrode received also the negative emission from the strip. This source of error is comparatively small and may be eliminated by insulating the cathode P and measuring the current through the third electrode, all other conditions remaining the same. When the temperature of the strip was not very high, this source of error could be neglected. An attempt was also made to check the results by control experiments. For this purpose the heated strip was coated with calcium oxide and used as a cathode, being raised to —1800 v. The third electrode was brought to a potential of —2000 v. and the plate C earthed. The stream of electrons or negative ions emitted by the strip produced by collision with the gaseous molecules positive ions which were directed towards the third electrode charged to a higher negative potential, and the rate of production of positive ions by the electrons was measured in the same way as before. The strip was kept at a comparatively low temperature (dull red) in order to avoid the discharge setting in. Since no positive ions could be expected to be set free by the electrons from 792 Discharge of Electricity through rarefied Gases. the surface of the anode P, it was interesting to compare the results of these experiments with the results given above. The ratio R proved to be strictly proportional to the pressure in the bulb, but at a given pressure R was much smaller (about one- third) than in the ease of positive ions producing electrons by collision. This is not at all surprising if we bear in mind that the free path of electrons moving with large velocities is much longer than that of the positive ions. It may be mentioned here that the method and apparatus described in this paper proved to be useful also for the study of some questions in connexion with ionization by collision, and for the determination of the free path of electrons and positive ions in rarified gases. — 6. In the experiments described in the previous chapter the energy of the positive ions was acquired by their fall through a potential difference not exceeding 2000 v. Attempts were also made, however, to carry out the same experiments by applying much greater electric forces be- tween the heated strip and the plate P. The difficulties in such experiments mainly arise from the fact that the strip and the third electrode have to be brought to different potentials, and the potential difference between them must be kept steady and comparatively small. All attempts to use an induction-coil for this purpose were unsuccessful. The writer hopes to continue these experiments by means of a motor-generator or other source of large and sufficiently steady electromotive force. Summary. I. It is shown that the initial discharge of electricity through vacuum-tubes is not brought about by the impact of positive ions against the surface of the cathode. II. A method is described by means of which the stream of ions or electrons emitted from a hot wire in a vacuum- tube may be isolated from the ions ver by collision within the tube. III. It is shown that positive ions impinging upon the cathode with velocities corresponding to a fall through a potential difference up to 2000 volts, are unable to liberate electrons from the surface of the cathode. In conclusion, I wish to express my thanks to Prof. W. L. Bragg for placing the necessary facilities at my disposal and for the interest he has taken in the work. The Physical Laboratory, Victoria University, Manchester. October 1920. XCII. On the use of Vector Methods in the Derivation of the Formule used in Inductance and Capacity Measurements. By H. H. Poous, Se.D.* NHE various formule used in methods, such as Anderson’s, for the comparison of Inductances and Capacities are usually obtained in the text-books by the use of algebraic methods. It does not seem to be generally recognized how easily uhese expressions can be obtained geometrically from the vector diagrams representing the currents in the various arms. This method possesses the advantage of enabling a better mental picture to be formed of the currents in the various arms and their mutual phase relationships. A few examples are given here of the derivation of the required expressions in some important cases. We assume 7 Oces that an alternating current generator of frequency aie 20 used as a source, and any form of detector, such as a tele- phone or vibration galvanometer. We further assume that the system is balanced, 7. e. that the two terminals of tlie telephone are represented by a single point on the P.D. vector diagrams, for a pure sine wave of the given frequency. We find that in order that this may be true, certain relation- ships must exist between the various arms, but that these relationships are independent of w ; and hence, if the system is balanced for one frequency, it is balanced for all. More- over, as any disturbances can be resolved into sinusoidal dis- turbances of suitable amplitudes and frequencies, the system will be balanced for any disturbance whatever. This will only be the case if the inductances and capacities used are independent of the frequency. This will be very nearly true of inductances which do not contain iron, at musical or lower frequencies, but will not be true for in- ductances containing iron cores. The capacity of a condenser with a solid dielectric generally varies slightly with the fre- quency. ‘This causes a slight variation in the balance for thie various harmonics in the rather complex wave emitted by a buzzer and transformer, so that perfect silence is rarely attained, and the predominant note of the telephone changes its pitch in passing through the point of balance. A similar effect may occur with inductances owing to the capacity between different layers of windings, which may cause an appreciable change in the effective inductance for the * Communicated by the Author. Phil. Mag. 8. 6. Vol. 40. No. 240. Dee. 1920. ot 794 Dr. H. H. Poole on Vector Methods for higher harmonics. This is in some ways rather an advantage, as after a little practice one recognizes by the sound which side of the best balance has been reached. In what follows, one point of the system O is taken as the zero of potential. Points on the circuit diagram which are at the same potential when a balance has been attained are represented by the same letter, being distinguished by suffixes. Currents are denoted by c’s with suffixes. Capital R’s are used for the entire resistances of arms possessing self-inductance (L). In some cases I is used for the impedance of such an arm and @ for its phase-angle, p being used for the total impedance of two or more arms in series or parallel. Small r’s represent non-inductive re- sistances, M’s mutual inductances, and K’s capacities. With each circuit diagram is shown the corresponding vector diagram. The vectors in every case, except fig. 8, represent falls in pressure in the corresponding arms. The diagrams are lettered to correspond, points on the vector diagrams not corresponding with any points on the circuit diagrams being distinguished by the letters P, Q, S, ete. V represents the P.D. generated by the source which is assumed to be constant in some cases. Any phase-angles employed are marked on the vector diagrams. Comparison of Two Self-Inductances | Maxwell’s Method]. ieee Circuit Diagram Vector Diagram Here Aj, is at the same potentiai as A,, both being repre- sented by Aon the vector diagram. The current c, in OA; also flows through A,B, similarly the current c, in OA, flows through A,B. It is evident that the vector diagram repre- sents equally well the branch OA,B or the branch OA,B, every point on the diagram for one branch coinciding with the corresponding point for the other. Formule used in Inductance Measurements. 795 Hence Qi pe Cal's c, Ry = ehva oe ¢, Lye = Co Liga pene Di, Ry Ly’ Hence both these conditions must be fulfilled in order that a balance may be obtained. The method is evidently much more easily employed if one of the inductances is variable. It is further considered later. Comparison of Two Mutual Inductances [Maawell’s Method}. Fig. 2, Oo rs YOUY P0009 3 | as Mz 8 v Circuit Diagram Vector Diagram Here the points O,; and O, are at the same potential, so the fall in potential due to the impedance of the arm O,AQ, must be balanced by the E.M.F. due to the current c’ in the primary of the mutual inductance M;. The triangle OPQ represents the vector diagram ofthis arm. The same cur- rent ¢c flows in the arms O,AQO, and O,BOQ,, and the current ¢’ in the primaries of the two mutual inductances is also iden- tical. Hence the triangle OST representing the vector diagram of the arm O,BQ, is similar to OPQ. Hence cRy _ claw _ eM cR; clw c'M.o’ R,_ i, _ ™M a Eee Me Hence a perfect balance can only be obtained if the self- inductances are in the same ratio as the mutual inductances. If, however, we make the non-inductive resistances in series with the coils L, and L, large, the angle @ will become very R, M, small and a fair balance will be attained when=- = —. he ML 3 F 2 796 Dr. H. H. Poole on Vector Methods for In comparing two mutual inductances, there are evidently four possible arrangements, as we can interchange the pri- mary and secondary of either coil. ,We see that it is generally best to use the low-inductance side of each coil as secondary except in the case where some other arrange- ment would make tlie ratio of the secondary self-inductances near the desired value. Comparison of a Mutual with a Self-Inductanee [Maxwell’s Method]. Fig, 3. 2 SOs: . Oni A c,R, iP Circuit Diagram Vector Diagram Since A, and A, are at the same potential, c, and c, are : a 2 : a ak 2 evidently in phase, so the current in the primary of M is ¢;+¢,. The vector diagram is obviously as shown, being identical with fig. 1 as far as the branch OA,B is concerned. In the branch OA,B we have a driving E.M.F., (¢,+ @)Mo, represented by BQ. This must be subtracted from the fall of potential c,Lym [PQ], with which it is in phase. The difference must be equal to cli. Hence C111 = Co? | ge | 1 RK, _ L,—M C1 1Vy = Colr9 cue to Re eee ¢,Ly@= Cy Lew + (cy + ¢2)Mo@ a Re 1 Ly + (1+ “) M. In the Campbell Inductometer Bridge, as used with unequal arms for measuring large self-inductances, the inductance L to be measured is inserted in series with, the secondary L’ of the mutual coil, so that L;=L+4+WL’. Formule used in Inductance Measurements. 797 . . V9 f : ‘ A balancing inductance, Ly=— L’, is inserted in the arm 1 A,B. This evidently cancels out with L’, and we are left : , | , : pee with L=(1+ nt) M. The most convenient ratio for — is V9 Vo either 9 or 99, the appropriate balancing coil being ac- Lag by cordingly or —. When used with equal arms for small self-inductences the secondary of M is divided, half being in each of the arms A,B and A,B. The vector diagram becomes. slight ¢} +¢c,)Mo 2 alf of the mutual inductances being altered, as PB is now equal to ¢,loe+ BQ to See h in each arm. [ lhe connexions, of course, are such that when the E.M.F. induced in the lower arm is in the direction A,B, that in the upper is in the direction BA,.] The equations remain as before, and since the value of L’ is the same in each arm this cancels, so we get L=2M where L is the self-inductance to be measured, and, as before, forms part of the arm A,B. Comparison of Two Capacities | De Sauty’s Method). Fig. 4. A QO 3] = Ct is wy Nal Circuit Diagram Vv Vector Diagram The argument is identical with that for self-inductances. The upper vector diagram refers to the case of condensers which are perfectly free from either dielectric hysteresis or leakage, so that the currents through them are in exact quadrature with the P.D.’s across their terminals. 798 Dr. H. H. Poole on Vector Methods for Evidently C111 = CoP Ss io Lee eee rs Ko K.o In the case of an imperfect condenser some power is absorbed and the angle OAB is obtuse. This waste of power may be due either to actual leakage or to dielectric hysteresis, or both. In any case, the current will be nearly identical with that which would pass through a perfect con- denser K and a resistance R in series [or alternatively a condenser K' and a resistance R’ in parallel]. K and R will probably depend somewhat on the P.D. and frequency, and the current through the actual condenser will probably not be perfectly sinusoidal, so the representation is only approximate. If we obtain a balance with two such condensers, the con- ditions must be represented by the lower vector diagram. Kividently, as before, We shall almost certainly have to insert a resistance in series with one condenser in order to obtain a balance. If we have a perfect condenser K, [say, an air condenser | we can use this method to measure the effective value of Ry for another condenser Ky, and hence find its power-loss for any given frequency and voltage. We cannot hope for per- fect silence, as a sinusoidal P.D. wave would probably not give rise to a sinusoidal current through an imperfect condenser. Comparison of a Self-Inductance with a Capacity [ Anderson’s Method]. The branch OA,C is similar to OA,B in fig. 1, and is similarly represented on the vector diagram. The current c; through the condenser (assumed perfect) is 5 in advance of the P.D. (OA); hence, as AB represents the fall in pressure due to this current passing through the resistance 13, the angle OAB is right. The vector OB must represent the fall in pressure in the arm OB. The lengths of these vectors are Formule used in Inductance Measurements. 799 evidently as shown in the figure. The current in BC is the vector sum of c, and ¢s, so in finding the components of BO Fig. 5. c,lw O hate eae Aer) ep Vector Diagram Circuit Diagram along and perpendicular to OP we may treat each current separately. Resolving along OP we have C3 C71 = F— = Coo COS ee: Ko 2! 9g ) and cRy = C94 COS ) 5 et cal aes the condition for steady current balance. 1 4 Resolving perpendicular to OP, we have C313 = CoN, SIN and CyLeo = crs + csr + cory sin d, 1314 = Cg, [rst rt as] ; Ts or since C=c7"Ko and ryry=7.R, we get L=K [114 a 13(71 _ R)]. If we can obtain a balance with r;=0, i.e. the points A, and B coincident, we have the simpler formula L=K7ry74. This method is, however, less convenient unless either L or K is infinitely variable. 800 Dr. H. H. Poole on Vector Methods for Comparison of a Mutual Inductance with a Capacity — [ Carey Foster's Method]. Fig. 6. ah Circuit Diagram Vecter Diagram Here, since A, and A, are at the same potential, the triangle OPA evidently represents the P.D. vectors for the arms OA, and OA, The fall in potential due to the im- pedance of the arm A,BA;, through which the current ¢, passes, must be balanced by the E.M.F. induced in it by the mutual inductance M. Now, the current in the primary of the latter is the vector sum of c, and ¢, therefore this E.M.F. must be the vector sum of c;Mw and c,Ma, each vector being 5 in advance of the corresponding current vector. [The primary of M must obviously be so connected that the E.M.F. is in the desired direction.] Hence. the figure AQST represents this part of the circuit, the lengths of the vectors being as shown. The triangles OPA and SQA are evidently similar, so : C9 coR Wo SSS SSS SS ? Keer, ¢y¥M@’ M=Kr,R 5 Col c,(L—M)o and cos b= 2 = ) ’ CyVy ¢;Mo Peale ree M 1A We must remember in using this method for measuring a capacity by means of a variable mutual inductance, that if we make 7, infinite, co becomes zero, and we obtain a perfect balance with M=0. If we start with a very large value Formule used in Inductance Measurements. 801 of 7, the vectors AQ, QT, and TS, which are proportional to ¢g, are very small, and though a i ue balance is impossible, we shall obtain the least sound by making M verysmall. To avoid this trouble it is best to commence balancing with 7, small or even zero. It is fairly evident that we would not expect a very sensi- tive balance if the triangle SQA is very arch; sinaller than the triangle OPA. This implies that 7, should not be very much bigger than Mo. The latter is not likely to exceed a few ohms, even if we use the variable mutual inductance near its major Jimit, as is evidently the best. On the other hand, unduly reducing 7, will considerably reduce the avail- able P.D. OA across the condenser terminals, and may even eause the current c, to become so large as to cause over- heating. We must ‘remember that OA does not represent the full voltage of the generator, as the effective impedance of the mutual. inductance primary must be considered. Ligfect of a Small Deviation from an Exact Balance. It is interesting to consider the application of the geo- metrical method to the calculation of the telephone current caused by a slight variation from exact balance. This calcu- lation becomes somewhat cumbrous if treated exactly, all the factors being taken into consideration. If, however, we assume, as a first Lg Aira Se that the impedance of the telephone is so high that the current through it is negligible, we can, in certain cases, easily find the P.D. across its terminals. We thus arrive at certain conclusions as to the best conditions for accuracy which are probably near enough to the truth to be of use in practice, at least in cases where a high-resistance telephone is used. We can then consider the effect of the current passing through the tele- phone, and also the loss of voltage due to the impedance of the generator. These impose further conditions which must be approximately satisfied if the most sensitive arrangement is required. The method is best suited to measurements not involving mutual inductance. Two such cases are considered below. Comparison of Self-Inductances. Suppose we are measuring a self-inductance L; by com- parison with « standard irae We want to arrange that a given error in L, should produce the maximum ‘effect on >» 802 Dr. H. H. Poole on Vector Methods for the telephone. We must also consider the effect of a small change in one resistance, say Ry. (1) Let Ly, become L,+6Ly, all the other quantities re- maining as before. Neglecting the telephone current, the vector diagram of fig. 1 now becomes as shown in fig. 7. The points O, Ay, P., and B are unchanged from their - original positions, but A, and P, have separated as shown from A, and Py. Fig. 7. It is evident that as the angle OPB is always right, P moves on a semicircle on OB as diameter. Also, since the resistances are unchanged, the triangle OA,A, is clearly similar to OP,P,, so that A, lies on the semicircle OA,Q on OQ as diameter, where OO ea B OB Ty Je Ie = 79+ Ra" For a very small motion A,A, is a tangent to this circle, so that it is evident that the angle OA,A, is equal to 5 — a, Let oE be the P.D. (A,A, on the figure) across the telephone terminals due to the increment 6h. Then COS & =¢ 7", cosa (tana). Formule used in Inductance Measurements. 803 Now cosa= pick where p; is the total impedance of a branch OA,B, Lie wdL and tana= an so O(tana)= aR. Hence 6E= 2 el om et P1 py (2) Let R,; become R,—6R,, the inductances and all the other resistances remaining at the balancing values. P again moves in the same direction along the semicircle OP.B to some point P, (fig. 7), Let A,’ be the corresponding posi- tion of A so that A,’A, represents the P.D. 6H’ across the telephone. Since 7, and L, are unchanged, OA,’ mt BP, (lia (y+ Ocy OA, ) BE: ef Cy ; Also angle P,OP,=angle P,BP., so the triangles OA;A’y and BP,P, are evidently similar, so that when 6a is small the angle OA,A,/=7—a. Hence we see that A,A, is perpendicular to A,Aj’, 2. e. the P.D. across the telephone due to a small difference between L, tr Sa : - = and — is in quadrature with that caused by a small i) L, “ difference between = and =. This fact greatly helps the 2 2 operation of balancing, especially if our standard inductance ’ pak ss | . is variable, as, even if =" has not exactly the right value, 2 the inductance ratio which gives the least sound is very nearly = Similarly, if this adjustment is imperfect, the 2 R ; : salt value of —* which gives the least sound is ~* Hence we R, Ue) can make the final adjustments independently and succes- sively, repeating the steps until the best balance is reached. We have SE/— C704 cr, cosa. 6 (tan a) sina tan « : 804 Dr. H. H. Poole on Vector Methods for a LodR : Hence, as 6 (tan a)= Gon: , OR; being a decrement in R, apes a aroR VréR. oH’ 5 PA P1 If there are errors 6L, and 6R, in L, and R, respectively, the resultant P.D. across’ the terminals of the telephone is evidently Vin, /abLT+ Re Pr In order to measure a given inductance as accurately as possible, we must arrange that a given error dL, in it shall produce the maximum value of 6H}. R, should. evidently be as small as possible ' since (7, +R,)?+ L,?o? occurs in the denominator. Unless our standard inductance L, is variable, it will probably be neces- sary to introduce a non-inductive resistance in series with the coil L, in order to effect a balance, but this should be as small as possible. e e 6 e en e e It is obvious that 51, vanishes if 7, is either zero or 1 infinity, so there must be some value of it for which this quantity is a maximum. Since ee oe a Bhi pr Ge By + Lie! 8E,\_ V | Oe e ill +R)? + L?o?— 24(r, +R, )] 5 e oH, so for a maximum value of —_ we must make Se r2=R?+ Lo? =L2, where I, is the impedance of the arm A,B. This implies that on the vector diagram (fig. 1) OA=AB. The same / condition evidently ensures a maximum value of oR,’ Formule used in Inductance Jleasurements. 805 We see that for a given value of @ it is best to make 7,;=1,, so that but pr=r; +12+27,1,cos@ (see fig. 1), =21,7(1+cos@) when 1=], and J, sind=L,o ; ep On arse Sat eae a eh mond -fcuse) eh 2 Hence we should use the highest available frequency in order to make @ as large as possible. In the case considered above the available voltage V i assumed to be fixed and the telephone impedance infinitely hich. In this case it is evident that the points O, Ay, and B on the vector diagram being fixed independent of Ly, the sensitivity is independent of the magnitudes of 7, Ry,and Lo, and all we need do is to make the ratios of these quantities correct. In practice, a current flows through the telephone and reduces the P.D. 6H, the amount of this reduction depending on the resistances and reactances of the bridge-arms as com- pared with the telephone. It is not necessary to consider the question very exactly, as it is obvious that the smaller 7%, Ry, and L, are, the lower will be the impedance to this transverse current and the smaller the reduction in 6H. We do not, however, gain much by making 7r,, Ry, and L, much car ile than 7,, R,, and L, respectiv ely. The latter quantities may be regarded as already fixed, since it has been shown above that 7, should be approximately eqnal to I). On the other hand, it is evident that the smaller 7, Ro, and L, are, the larger will be the current c., and, if the generator resistance is appreciable, the smaller the available voltage V. There is not much use, however, in attempting to reduce the total generator current by making the impe- dance py very much greater than the parallel impedance Pi- Hence we see, in a general way, that it is best to have ry, Bo, aad L, of about the same size as r,, R,, and L, respectively. 806 Dr. H. H. Poole on Vector Methods for Summarizing these points, we see that in measuring a given inductance L, by comparison witha standard Ly we should endeavour to satisfy the following conditions :— (1) The frequency used should be as high as possible. _ (2) The resistances R, and R, should be as~small as possible compared with the reactances L,w and Low. (3) The resistances 7, and 72 should be about equal to the impedances of the corresponding inductive arms. (4) The inductances L,; and Ly, (and hence the corre- sponding resistances) should not differ greatly in magnitude. The first three conditions may be further summarized by the statement that the triangle OAB should be approxi- mately isosceles and as nearly as possible right angled. Comparison of a Self-Inductance with a Capacity. Suppose we are measuring a given self-inductance by comparison with a standard condenser, it is evident that if we neglect the current through the telephone and assume V constant, the points O, As, B, and C (fig. 5) are unaffected by the changes of L and R, while A, moves exactly as in the last case, and we arrive at the same conclusions. As before, the current that actually flows through the telephone renders it expedient to make the transverse im- pedance as low as possible. In this case it is perhaps worth while considering the magnitude of this impedance more fully. Strictly speaking, we should also consider the angle between this impedance vector and that for the telephone, but this would involve a knowledge of the angle of lag of the telephone. As, however, this angle of lag seems unlikely to be very large, and hence the angle between the two impe- dance vectors will probably be small compared with a right angle, it will suffice to make the network impedance as low as possible and disregard its phase-angle. Let us neglect the generator impedance, so that as far as the small transverse current, which may be regarded as super- imposed on the main currents, is concerned, we may treat the network as two impedances p’ and p"’ in series, p’ being the impedance of the branches A,BO and A,BC and the arm A,O in parallel, and p” that of OA, and CA, in parallel. | Formule used in Inductance Measurements. 807 Yo"4 The resistance of A,BO and A,BC is r;+ Chane but since "gh 4 saga ae a ae gp iaee a ils oe ee Pgh 4 'R’ ; L this reduces to Ko, +R) ; Dee The reactance of A,O is ra K?’p*» 2p 2 A oe ta 1 hae i where p is the impedance of the branch OA,C. Sop = ze and if « is the angle of lead of the joint current tana= (jig. 5). p' is most easily obtained from the admittance vector diagram for the arms OA, and CA, in parallel. rare so a is equal to the angle POC Fig. 8, °T Here | is the impedance of A,O and @ its angle of lag. We have 1 Phe. COs ot pe gees PLE 07 Ee COUT Ibe a p ; ate also sin B=" sin 0= sin 6; p J so 8, the angle of lag of the total current in these arms, is 808 Vector Methods in Inductance Measurements. evidently equal to the angle OUA in fig. 5. The triangles QST (fig. 8) and CAO (fig. 5) are clearly similar, so angle QTS (fig. 8) is equal to angle « above, and a+ B=80. Hence, to find the total impedance to the transverse current we must compound the two vectors p’ and p'’, the angle be- tween them being @. Now, this angle is always acute, and the vector p'’ may be regarded as fixed by the condition that 7, should be equal to I, so to obtain a minimum transverse impedance we should make p’ as small as possible, 7.e. K should be large. Two rather interesting conclusions have here been reached. (1) Given 7,, R, and L, the transverse impedance de- pends solely on the value of K, and not on the distribution of the resistance necessary for balan- cing between the arms 79, 13, and 74. (2) Owing te the angle @ being acute, no reduction in the transverse impedance by ‘‘ tuning ” is possible, i.e. there is no value of K giving a minimum impedance for a given frequency. Under the best conditions of working, 7,=I[= Le nearly and cos @ is small, so = ie p=Le V2, ‘| z pen and =—=; Yo Rowo en so the total transverse impedance is not much greater than gly eek /3 (iczg2 +1"). drs In most cases —=— will be greater than Low. For example, Ko if w=3300, corresponding to the Note C in the treble clef, and our largest available condenser is 1 microfarad, <= 300 ohms. If L=0:05 henry [which is a compara- @ tively large value for a coil without an iron core], Low—=165 ohms. Hence in most cases we should use the largest available condenser, though little would be gained by — ss ee! the use of one whose capacity greatly exceeded Toe even if a it were available. . It must be remembered that a large value for K entails a small value for 7,744 73(7;+R). As before, it is a disad- vantage to have 7, and 7, small, as this causes excessive Elements in the Sun. 809 current and reduces the available voltage ; hence 73 should be as small as possible, in order that for a given condenser v2 and 7, may be as large as possible. If we have to measure a condenser, it is easily seen, as before, that we should choose a self-inductance coil with a L yp . . large value of R? °° if we employ a variable self-inductance, it should be used near its upper limit. ‘The larger the value of L, the higher will be the transverse impedance, but the less the “lost volts”? due to generator impedance, It is probable that the Jargest available self-inductance will be . gh . considerably smaller than Ko 8° the transverse impedance @ due to it will not be very important, and its use will be advantageous. We may conclude by summarizing these points as follows :— (i) The frequency should be as high as possible. (2) R should be as small as possible. (3) r, should be about equal to I, the impedance of the inductive arm. (4) 73 should be small. (5) Since K is in most cases considerably smaller than 5 Soe it is best to use the largest available condenser for Lo measuring a given self-inductance and wice versd. Physical Laboratory,. Trinity College, Dublin. September 8, 1920. XCIII. Elements in the Sun*. (Paper B.) By Mrcu Nap Sawa, D.Sc., Lecturer on Physics and Applied Mathe- matics, University College of Science, Calcutta +. JT is a matter of common knowledge that the continuous spectrum from the photosphere of the sun is crossed by a number of dark lines, which are called Fraunhofer lines in honour of their eminent discoverer. The correct inter- * Much of the introduction is taken, mutatis mutandis, from Fowler’s Report on the subject; vide ‘Journal of the British Astronomical Society, May 1918. + Communicated by the Author. Phil, Mag.'S. 6. Vol. 40. No. 240. Dec. 1920. 3G - 810 Dr, Megh Nad Saha on pretation of these lines was given by Kirchhoff in 1859, wh showed that most of the principal lines can be attributed to the absorption of light of proper frequency by the cooler layers of the vapour lying above the photosphere. Since this epoch-making discovery, it has become a part of the routine work of astrophysicists to catalogue and properly measure the wave-length of these lines, and identify them with the emission lines of elements obtained in the laboratory. The most extensive mapping of the Fraunhofer spectrum is due to Rowland, who counted and catalogued about 20,000 lines, but of these only 6000 have been identified with the lines of known elements. By this means, the pre- sence of thirty-six elements has been definitely established in the sun, with doubtful indications of eight or more. The following are the elements of the existence of which in the sun no evidence has yet been obtained * :— (A). Rubidium, Cesium; Nitrogen, Phosphorus, Boron, Antimony, Bismuth, Arsenic; Sulphur, Selenium; Thallium, Praseodymium. Doubtful indications have been obtained of the existence of the following elements :— (B). Radium ft; elements of the inert group with the exception of Helium, which is obtained in the flash spectrum, Osmium, Iridium, Platinum, Ruthenium, Tantalum, Thorium, Tungsten, Uranium. The following elements are represented by very faint lines in the Fraunhofer spectrum :— (C). Potassium, Copper and Silver; Cadmium and Zinc ; Tin and Lead; and Germanium. (D). Chlorine, Bromine, Iodine, Fluorine, Tellurium, and many other elements have not been investigated at all. No satisfactory explanation has yet been offered of the complete non-existence of the lines of elements mentioned in group (A) or (B), or of the faint occurrence of the lines of elements mentioned in group (C). Similarly, it has not yet been made clear why certain elements like Ca, Fe, V, Ti are so unusually prominent in the solar spectrum. They are represented not only by the absorption lines of the neutral atom, but also by the absorption lines of the ionized atom (enhanced lines). It is sometimes assumed that these phenomena are due to the chemical composition of the sun—in other words, the * Pringsheim, Physk der Sonne, p. 116. + For the controversy regarding the existence of Radium, and the members of the inert group in the flash-spectrum, see ‘ Observatory,’ vol, xxxv. pp. 297, 357, and 402. Elements in the Sun. Sit elements of which no lines are found either in the Fraunhofer or the flash spectrum are totally absent from the sun. But this view is most unsatisfactory, and can only be regarded as a stop-gap. ‘There is, @ prior, no reason why, in the sun, certain elements should be preferred to the exclusion of others. On the contrary, it seems natural to infer that the sun is composed of the same elements as the earth, and contains all the 92 elements known to the chemists on the earth, ; It therefore becomes increasingly necessary to investigate why certain elements should entirely fail to be recorded on the Fraunhofer or the flash spectrum. It may be supposed that certain elements fail to be recorded because, on account of their heavy atomic weight, they are practically confined to the photosphere. But it is not merely a question of atomic weight, for in the list of missing elements we find light elements like boron and nitrogen side by side with a heavy element like thallium. The view which is urged in the present paper is that the varying records of different elements in the Fraunhofer spectrum may be regarded as arising from the varying response of these elements with regard to the stimulus existing in the sun. The stimulus existing in the sunis the same for all elements, viz., that arising from a temperature of about 7500° K., but owing to different internal structure, elements will respond in a varying degree to this stimulus. The manner in which we can quanti- tatively estimate the effect of the stimulus has been sketched in papers A and C*. In paper A, the effect of the stimulus on the alkaline earths Ca, Sr, and Ba was estimated. It was shown that while on the photosphere 30, 40, and 57 per cent. of the atoms are respectively ionized, the percentage of ionization increases with height and becomes practically complete for Ca at a pressure of 10-° atm., for Sr at a pressure of 10~*, and for Ba at a pressure of 10-2 atm. In this connexion it may prove interesting to compare the results with the following remarks of Fowler :— “We find further, that while many of ‘the metals are represented by both are and enhanced lines, there are some which are identified only, or mainly, by their enhanced lines alone. ‘Thus, although Ca shows both classes of lines strongly, Sr and Ba practically show enhanced lines alone.” Fowler ascribes the different behaviour of Ca on the one * Paper A—“ Ionization in the Solar Chromosphere,” Phil. Mag. Oct. 1920; Paper C—‘‘On the Temperature Radiation of Gases” (to appear shortly). 3G2 : 812 Dr. Megh Nad Saha on hand, and Sr and Ba on the other hand, to their differences in atomic weight, but according to the view presented in paper A, this is mainly due to the varying values of the ionization potential. The author’s belief is that in the sun and the stars, the attraction due to gravity is largely com- pensated by selective radiation pressure, and atomic weight is of much less consequence than can be supposed. The method sketched in paper A has been extended to the alkali metals, and a few other elements. It will be seen. that the theory accounts in a most gratifying manner for the varying behaviour of sodium lines in the Fraunhofer spec- trum, and its intensification in the sun-spot spectrum, for the faint occurrence of the potassium lines, and for the complete absence of the lines of Cs and Rb, and for the varying behaviour of the lines of Mg and Me™, thongh, on this last point, the results are not so satisfactory. There is very little doubt that if proper data be available, the method can be extended to the explanation of all the details of the Fraunhofer spectrum. For the explanation of the method, the reader is referred to Sections 2 and 3 of paper A. The temperature of the photosphere has been taken to be 7500° K., the pressure 1 to 107! atm., while for the high-level chromosphere a temperature of 6000° K. has been used. 1. Tae ALKALI ELEMENTS IN THE Sun. (a) Sodium. The following table shows the ionization of sodium in per cents. under varying conditions of temperature and pressure :— Tonization Potential= 5°12 volts=1°:17 x 10° calories. Pressure ......... Bef LOT 10S 10 ae one Temp. [(0) 6 O ener Mati 6 19 53 89 98°5 GOO0) Ree: 21 56 90 98-5 | OOO bees easels 46 &5 98 Ges anee O00. hi eteay. 60 72 99 | Complete BROOD! Meeks ke ues 72 96 lr ; outer onization. Hlements in the Sun. 813 The table shows that under the solar conditions, 60 per cent. of sodium atoms are ionized in the photosphere, and ioniza- tion is practically complete at a level where the pressure falls to 10-* atm. The result is in very good agreement with observational facts, for according to Mitchell the D, ‘and D, lines reach a level of only 1200 kms. Over this height, only ionized Na atoms are present, the chief emission lines of which lie, according to Goldstein *, in the remote ultra-violet, and so escape detection. Taking the temperature of the spot =5000° K.f, we see from the tables that only 6 to 19 per cent. of the atoms are ionized. So over the spot, there is a great increase in the proportion of unionized Na-atoms, and we should expect a much stronger absorption of the D, and D, lines. The following table (taken from Kayser’s Handbuch der Spektro- skopie, vol. vi. p. 114) shows that this is actually the case :— Intensity Intensity in Line. Series-Description. in the Sun, Sun-spot Spectrum, D,—589615 ......... (1, s)—(2, po) 20 60 D,—589019 .. ...... (1, s)—(2, p,) 30 90 BGS2 00) ooo... (2, p,)—(8, @) 5 12 5688°26 ......... (2, p,)—(8, @) 6 12 Olot4t ........ (2, p,)—(3, $) 2 8 BIGLS5. *!. ....:. 2, p)—G, 5) 3 9 (b) Potassium. The identification of Potassium is rather doubtful. Rowland has identified only two faint Fraunhofer lines X= 4047-36, 4044°29, with the emission lines (1, s) —(4, 7), (1, s) —(3, po) of potassium. The following table shows that owing to the low value of the ionization potential, potassium is highly ionized throughout the whole of the solar atmosphere. * Goldstein, Ann. d. Physik, vol. xxvii. pp. 773-796; Schillinger, Wien. Ber. p. 608 (1919); Nelthorpe, Astrophysical Journal, Jan. 1915, + Emden, Gas-kugeln, p. 443. See aiso numerous papers on the spectra of sun-spots by Fowler, Hale, and others, 814 Dr. Megh Nad Saha on Ionization of Potassium. Tonization Potential=4°318 volts=1:00 x 10° calories. Pressure .. ...... ly 100 Ly One. 0 Ome nee Temp OOON Mie 95 9 98 66 AOD eee 3 ful 32 72 97 OOO ete he. 15 44 83 98 is SS BOO Vee va 41 81 0. eee WOOO See 70 OR ae re Sate te 81 98 | Complete S000: Ae ees 87 99 Ionization. SOOO ees Oh = 1O0OOR ere 98 | The identification of potassium is to be carried by the lines (1, s) —(8, pi), (1, s)--(8, .) which under all cireum- stunces are likely to be much less intense than the leading pair (1, s)—(2, p,), (1, s) —(2, pe) which lie in the infra red, X= 7665°3, 7699°3. This fact, combined with the high proportion of ionization, tends to make the identitication rather difficult. The lines of the ionized atom lie in the ultra-violet, and so escape detection (Goldstein, Astro. Journal, xxvii. pp.. 29-34 (1908) and Ann: dene OOS Gitte, )n | Over the spots, the potassium lines ought to be strengthened, but no observation seems to have been made on this point. (c) Rubidium. As has been mentioned in the introduction, no lines of Rubidium have been identified in the Fraunhofer spectrum of the sun, though in the usual flame and are spectrum there are many strong lines within the available range. The following table shows the ionization of rubidium. It will be apparent from the tables that in the sun rubi- dium is completely ionized. Consequently, even if it be present, we shall get the lines due to the ionized atoms only, which, aceording to Goldstein *, lie in the ultra-violet. The spot spectrum should show some faint rubidium lines. But no observation seems to have been made on this point. * Goldstein, loc. cit. Elements in the Sun. 815 Ionization Potential U=°96 x 10° calories==4°16 volts. Pressure ......... ie lon eeelOre ee Ores) 10-4). ST0-°, Temp. BOON er oS: 1 5 ek eRe 2 6 20 2 eee 1 4 13 37 78 21 ee 4 1 39 80 OP sero 7 Se 19 51 Siete cee aT 50 Co a 46 85 98 | ty f An Bie. | Complete 2) ere 90 lTonization. A) ets ts. 3. 96 | TOOOG: >. os... 99 (d) Cesium. No Ceesium lines have been identified in the Fraunhofer spectrum. The case is identical with that of rubidium, only in a more marked degree. A table of the ionization of ceesium is appended below :— U=3°'88 volts=:90 x 10° calories. Pressure ......... els eo hee NO ra On Temp 2) 3 8 oS) 1 4 11 34 ee eek 4 12 35 15 i ee 7 20 53 89 O85 a ae A 25 62 QR eee | Le 56 90 eee A Oe Sea 81 97°5 Complete 3) | | Si oes ae 88 99 Tonization. ae 5 gg een ee a 98 | Oxygen. The presence of oxygen in the sun was a matter of great controversy until a few years ago. The well-known bands A, B, « were shown by Jansen ™* to be of telluric origin— i. ¢., caused by the absorption of the solar light by the mole- cular oxygen of our own atmosphere.. But Runge and Paschen identified the weak Fraunhofer triplet, \=7772°20, * Pringsheim, Physik der Sonne, p. 119 et seg. 816 Dr. Megh Nad Saha on 7774'43, and 7775°62, lying in the extreme red, with the three emission lines of oxygen, having the series formula (2, s)—(m, Pi), (2, s)—(, Po), Q, s)—(M, ps). In the sun, therefore, oxygen exists in the atomic state. The heat of decomposition of the oxygen molecule is not yet known with certainty. From Langmuir’s * observation that at 2400° K. and 107! atmospheric pressure oxygen is completely dissociated, [ have calculated provisionally the heat of decomposition to be less than 50,000 calories, which is less than the corresponding value for hydrogen (84,000 cal.). Since the equation of chemical equilibrium is almost the same in both cases +t,it follows without any calculation that oxygen should be completely decomposed into atoms in the sun. The ionization potential of oxygen is probably large, and no lines due to ionized oxygen seem to occur in the sun. The lines of O* and OTT have been qualitatively studied by Lunt, Fowler ft, and Brookshank, but no series-formulee have yet been obtained for these lines. But Fowler’s identification of certain of these lines in the Bo-class of stars and in Wolf-Rayet stars respectively seem to indicate that the first step ionization of oxygen is reached at an approximate temperature of 20,000° K. and second step jonization at probably not less than 30,000° K. . Magnesium. The wave-lengths of the strongest lines of the emission- spectrum of Mg, viz. lines (1,8)—(2, P), (1,8)— (2, py), lie beyond the range of atmospheric absorption, so that we have to fall upon the next strongest lines, the triplet (2, ) (3, di), i=1,2,3, 2%=3838°34, 3832-46, 3825°51, which are very prominent in the flash spectrum, and reach, according to Mitchell, a level of about 7000° km. The height reached by the line (1, 8)—(2, P), X=2852, if it were available for observation, would probably be somewhat higher, say about 9000° km. The brief is strengthened from an examination of the table of ionization given below. x Langmuir, Journ. Chem. Soc. vol. xxxiv. pp. 864, 1080 (1912). + Vide Hydrogen in the Sun, paper A, p. 488. t Fowler and Brooksbank, Month. Not. Roy. Astr. Soc. April 1917. Elements in the Sun. 817 Ionization of Magnesium.’ Ionization Potential=7°'65 Volts=1°76 x 10° calories. Wo ee ae tO, 105%, Pressure <2: i... 5.55: ke Temp. BE ee eee mee XW ok Baie! 2 9g | AS et 5.10-2 1 4 At 32 TEE tele Ae One ae 2 6 18 50 87 WO Faecal ace Saks 6 20 54 89 98 BME Baka. Eisisb he 11 32 73 96 epee ROT AL Se hao 17 47 85 98 | oC gga nee! Bien eumena | OT geo | MIO poe Sarge Nathan 56 90 98 | WA, eG cece aks io 96 aoees | PT tl oS 86 98 | BON cuca nt sia: 93 99 | Gampletc LE (a 96 en UT oe wal cow's ais 98 Ionization. BON. ec katieccs was 99 As the ionization potential of magnesium is rather high, it is ionized to a lesser extent than the other alkaline earths. Total ionization is reached at a pressure of about 107° to 10-7 atm. in the chromosphere. This is in excellent accord with observational results. The lines of ionized magnesium have been studied by Fowler. The lines which should theoretically * turn out to be the strongest, all lie in the ultra-violet, as the following table shows -— (1, s)—(2, pi)......2795°5 ALU. —(2, po)...... 2802-7 (2, Pi) — (3, d;) saberecee AT ISL (2, po) —(3, dg)...... 2790 (3, d,)—(4, f)...... 4481, so that the only line available for observation is the leading member of the Bergmann series X—4481 A.U. According to observation, this reaches a level of 450 km. only. According to the considerations presented in paper C, lines of this description under all circumstances require not only a higher stimulus, but also a high density of the * Fowler, Phil. Trans. vol. cexiv. The intensity given for 4481 is relatively much greater than the intensity of the (1, s)—(2, p) lines. 818 Dr. Megh Nad Saha on radiant particles. Though the upper layers of the chromo- sphere favour relatively stronger ionization, yet the density is so small that such lines fail to be recorded. They are recorded only from the lower regions where the density is sufficiently great, and the stimulus is not much weakened. Nitrogen in the Sun. The emission spectrum of Nitrogen is a rather difficult matter to deal with. Under the stimulus in which most of the other elements can be made to emit their characteristic line spectrum or even their enhanced lines, nitrogen gives only several classes of band or molecular spectra. ‘The line spectra seem to require a very great stimulus for their production. But the existing knowledge on the subject is too meagre to allow the discussion of the line emission of atomic nitrogen from the present standpoint According to a recent paper, the ionization potential of N is 17-18 volts, but this must be made up of the energy of dissociation of the molecule plus the energy of ionization of the atom. {f the so-called cyanogen band having its head at V=3885 * be really due to molecular nitrogen, the con- clusion follows that in the sun nitrogen occurs in the molecular state. There may be a fair proportion of atomic nitrogen, but the stimulus is not sufficiently strong to make it emit the lines we are familiar with. All these charac- teristics are in very good accord with the chemical inertness of nitrogen, which is, again due to the high value of the heat of dissociation of the N, molecule. The fact is only qualitatively known. According to Langmuir +, less than 5 per cent. of nitrogen is dissociated at 3500° K. and at atmospheric pressure. From this we may calculate the heat of dissociation in the following manner. The reaction takes place according to the scheme N, == N+N-U, and if x be the fraction dissociated, P the partical pressure, we have eee Geet ei Sets Now >C=2Cy—Cy,,. * This is the opinion of Runge and Grotian (Phys. Zerts. vol. xv. 1914). Fowler, on the contrary, believes that the band is due to CN (cyanogen). See also Barratt, Proc. Roy. Soc. Lond. vol. xevii. + Langmuir, Journal of the American Chem. Soe. vol. xxxiv. p. 860 (1919). Elements in the Sun. 819 According to the Tetrode-Sackur formala, Ox=—16+5 log M="119, and IN, = —'05 * Taking «=*03 in Langmuir’s experiment, we can calculate U from the above formula ; we obtain U =1°50 x 10° calories, in round numbers. 1s) log oe =2e. ‘Uhis becomes 1°390.. If P=1 atm., almost 98 per cent. of the nitrogen ought to be completely broken up into atoms. For P=10 atms., the proportion is 83 per cent. Thus the calculation does not seem to favour the sugges- tion that a certain percentage of nitrogen occurs in the sun in the molecular state. But the data used for calculation are of the roughest nature, and nothing definitely can be said until better data are available. Helium. Tt is well known that none of the Helium lines occur in the Fraunhofer spectrum, though occasional reversals of the D3 have been observed in the spectra of the penumbra of sun- spots +. But in the flash spectrum, helium lines are very prominent, the D; reaching a height of 7500 km. in the chromosphere. ‘These facts may be explained in the fol- lowing manner :-— The lines by means of which helium is detected belong to the series-combination (2, p)—(m,d), or (2, s)—(m, p). These lines cannot be absorbed by the ordinary He-atoms, which possess (1, s) orbits, but by such atoms as. possess the (2, p) or (2, s) orbits. These orbits can only be produced under very great stimulus. It is known from the ‘ Harvard Annals’? (vol. xxviii. 91) that the absorption lines of helium disappear below stars of the Ao class. Hence a temperature of about 11,000° K. (temperature of: the Ao class) is required for producing a sufficient number of He-atoms with (2, p) orbits, when the pressure is one atmosphere. These conditions are not attained on the photospheric level; but at great heights, owing to * Laski, Phys. Zeits. xx. p. 269 (1919). + Pringsheim, Physik der Sonne, p. 123. one. T—7V000° (.. we can now ecaleulate the value of 820 Dr. Megh Nad Saha on diminished concentration, uot only (m, p), but also (m, d orbits are produced. As these absorbing atoms occur only at large heights in regions of low concentration, they are not in sufficient number to produce a weakening of the corresponding regions of the continuous spectrum by ab- sorption. The occasional reversals of He-lines in the penumbra of sun-spots seem to be an effect of diminished concentration without a corresponding loss in temperature. The temper- ature of the penumbra is intermediate between those of the spot and the undisturbed photosphere, and may be supposed to lie between 6500° K. and 7000° K. We have no direct observational result which can give us some idea of the pressure, but Hvershed* has found that gases are flowing through the penumbra radially outwards, parallel to the surface of the sun, with velocities ranging from rather small values to about three or four kilometres per second. As the mass-motion of a quantity of gas is always attended with a fall in pressure, the penumbral regions certainly possess lower pressure than the undisturbed photosphere. The physical conditions in the penumbra are therefore favourable to the production of a larger percentage of He-atoms with the (2, p) orbits. Probably this accounts for the occasional weak reversal of He-lines over the penumbra. The Flash Spectrum. From paper A and the foregoing part of the present paper, it will be seen that the Fraunhofer spectrum can mainly be regarded as function of a single phy sical condition, viz., the temperature. The flash spectrum is, on the other hand, a function of temperature and concentration, a low concentration favouring a greater percentage of lonization, and consequently a relative intensification of the enhanced lines. Special attention is called to the word “ relative ’ when terrestrial sources of limited extent are considered, for though the percentage of ionized atoms becomes greater, the absolute number of particles becomes less, and hence all lines, as a rule, become fainter. But the lines of neutral atoms would become much fainter than the enhanced lines. All this was substantially stated in paper A, but at the time of writing this paper | was acquainted with no experi- mental work which could be cited in support of the view. I have since come across some works which support my case. - * Evershed, Astrophysical Journal, vol. xxv. (1909). Hale, Joe. evt. vol. xxviii. (1910). Elements in the Sun. 821 The following is an experimental work by Loving on the comparative intensities of Ca and Meg lines in ordinary are and vacuum are. (Astrophysical Journal, vol. xxii. p. 300, 1905.) Fowler also noted that the enhanced lines of Mg, Te and Cd are greatly intensified in a vacuum are (Fowler and Payn, Proc. | Roy. Soc. Ixxii. 1903). isenent 2-1 Open | Vacuum | Relative Spark Chromo- are are. weakening. | sphere. ees ee | BA ne ——— K—3934 ..... 500 | 25 - | 10001 | 275 | H—3u68 ...... 300 | 20 a 500 | 60 g-4997 ...... 1000 8 oN Ci aa Magnesium. | : | | 9352 500 8 ee ae NE “8 ke 235 | 60 | | 4481 (+ Mg) 0 20°. | — fe OP =I 1 | | | Thus the tables show that in the vacuum arc the ratio Cat/Ca is 6-8 times its value in the ordinary arc. Unfor- tunately the pressure, which is a vital point, is not stated. In the case of Mg, the stimulus at the open arc is not even sufficient to excite the ionized line 4481. But a diminution in concentration brings about ionization. This case is rather remarkable because it affords an experimental basis for the view that temperature remaining the same, a diminished concentration can bring about a higher stimulus. Conclusion, The foregoing work will probably make it clear that the theory of temperature ionization developed by the author in paper A, and more fully applied in the present paper, is eapable of throwing much light on many obscure and puzzling questions of solar physics. We are precluded from making further applications owing to lack of necessary data, viz., the value of the ionization potential of elements or the value of the term or terms (1, s). If this and the spectral properties of all elements were known with as much certainty and exactness as in the case of the alkaline earths, details of the Fraunhofer spectrum could be very satis- factorily explained. We can, however, make the following suggestions with >. 822 Dr. Megh Nad Saha on regard to future work on the subject :—Hlements like Fe, Ti, V, Ni have probably ionization potentials varying from 6 to 9 volts, so that their behaviour is almost parallel to that of calcium and magnesium. Hlements which are missing from the sun can be broadly subdivided into two groups: 1st, those which are completely ionized, e.g. Rb, Cs, and pro- bably thallium ; 2nd, elements of which the ionization and radiation potentials are so high that they are not in a state capable of absorbing those of their characteristic lines which occur in the continuous photospheric spectrum (vide paper ©). Helium and most of the inert gases fall within this group. The case of helium has already been considered. Neon, Argon*, and other inert gases have very high ioni- zation potentials, the value being 16 volts for argon. Their principal emission lines (1, s)—(2, p) lie in the ultra-violet, and identification is to be carried on by lines (2, s)—(m, p), or (2,p)—(m, d) or (2, p)—(m, s). As the intensity of the photospheric radiation in the region (1, s)—(2, p) is very small, the compensating value of the radiation- pressure is also small, and the elements fail to reach great heights. In the lower regions, the stimulus is not sufficient enough to convert the atoms to the states (2, s) or (2, p) and hence these lines do not occur in the Fraunhofer spectrum T._ THE CASE OF THE UNIDENTIFIED FRAUNHOFER LINES. As has been mentioned in the introduction, about 60 per cent. of the Fraunhofer lines catalogued by Rowland still remain to be identified. According to Fowler, a large proportion of these are due to molecules and molecular com- pounds. The best known example is the so-called cyanogen band with its head at X=3883, now attributed by Grotian and Runge to molecular nitrogen. Other examples are Newall’s identification of the G-group of the Fraunhofer spectrum with the band-lines of hydrocarbons, and Fowler’s detection of ammonia and water-vapour bands in the ultra-violet part of the solar spectrum. These identifications raise a very important point, viz., Is it possible, under the conditions prevailing in the sun, for * For the spectral grouping of the lines Ne and A, see Paschen, Ann. d. Physik, vol. lx. + Probably the metalloids As, Sb, Bi, etc., fall in this category, for it is well known that they can be made to emit their line spectrum at a comparatively low temperature, vzde paper C, Introduction. Elements in the Sun. 823 any molecule or molecular compound to exist in the undis- sociated state? The problems are essentially of the same nature as those which are treated by the physical chemist in the laboratory, only repeated on a scale which is not available on the earth. But thanks to the recent developments in thermodynamics by Nernst, Planck, Sackur, and others, we can now handle these problems from the theoretical side in a very satisfactory manner, if only proper data are available. Examples have already been given; it has been shown that in the sun, hydrogen and oxygen are in wholly dissociated con- ditions, while some proportion of nitrogen may remain in the molecular state. The chief data which are required are heats of dissociation, the specific heat of the components, and the value of the chemical constant of the components. Thus to take a con- erete case, let us consider the chemical equilibrium of NHg. The reaction is of the type NH,*N+3H-U,, and the chemical equilibrium is given by the law 3 pt Pe aes =-sayn+ poe Pes where w is the fraction dissociated. Now vCp=(Cp)w + 3(Cp) ue — (Cp) we, and 2vC=Cy + 3Cu— Oyu, ‘ All of these quantities can be calculated theoretically ex- cepting Cyu,. (The theory of the chemical constant for polyatomic gases is yet to be developed—see paper men- tioned below *.) It should be remembered that Uj, is gee from the energy evolved in the reaction, 2NH;=N.+ 3H,— le if N,=>N+N—Uy, and H,=H+H—Ug, it is easy to see from the energy principle that we Unt aE asl? — and cannot be estimated before all these quantities are known. * “On the Chemical Constant of Diatomic Gases.” Leon Schames, Phys. Zeits. = xxl. p. 41 (1920). 824 Elements in the Sun. For a chemical reaction of the type Ti = O,= TiO, -- WP the equation of chemical equilibrium is given by the formula eae SN: U Spue “8 p2apd—a) 7 BORE? og ee which reduces to P2z3 iar U Ge 2Ar= 7) a | S30Re if, in accordance with the kinetic theory of specific heat, we put (Cp)noo= This class of chemical. reaction is likely to have wide application in the treatment of the spectra of sun-spots, for a- large number of spot-lines were identified by Fowler, Hale, and Olmsted with the band-lines of MgH,, TiO., and CaH, (Tri-atomic compounds) *. It appears that no attention has yet been paid to the study of the spectra of the faculee, which are believed, on the basis of very sound physical arguments, to be regions of higher temperature than the photosphere. Supposing the temper- ature of the facule to range between 8000° K. and 9000° K., their spectra are likely to show very important differences from the spectra of the ordinary photosphere, and to be similar to the spectra of the Foto F;G classes of stars, just as the spectra of sun-spots are similar to those of the Antarian stars (K-class), At any rate, the subject seems to offer a very rich field for investigation. If the necessary data be available, these questions will be taken up in a future communication. log +3 log T+ One +2Co— Crio,, I have much pleasure in recording my best thanks to my students of the Post-graduate classes in Physics for much useful help in the preparation of this paper. University College of Science, Calcutta. May 22, 1920. * Pringsheim, Physik der Sonne, pp. 211 to 217. XCIV. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 284. ] January 21st, 1920.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. Mr. Ricuarp Dixon OxLpHaM, F.R.S., V.P.G.8., gave a de- monstration on a Model to Illustrate the Hypothesis of a somewhat Rigid Crust resting on a somewhat Yielding Substratum, as applied to the Problem of the Origin of Mountain- Ranges. He remarked that geodetic measurements in the Himalayas, the Pamirs, and the Andes show that in each case there are systematic departures from equilibrium, in the form of parallel zones in which the surface-level stands alternately above and below the level of equilibrium, the differences being very considerable, and amounting to the equivalent of somewhat over 2000 feet thickness of rock of average density. These zones run parallel to the direction of the axis of greatest elevation of the range, and are explicable by an hypothesis that the elevation of the ranges was due to direct uplift produced by changes in volume of the material underlying the erust, if this material be supposed to possess a certain limited amount of compressibility or plasticity and the crust to have a certain amount of rigidity, which would offer resistance to an exact adjustment of the uplift of the surface to the varying amount of uplifting force developed in the material below the crust. The model is designed to visualize the consequences of such an hypothesis. It consists of two strips of spring steel, supported at regular intervals by connecting links to a series of blocks capable of vertical movement. For one strip these links are of fixed length, representing a condition in which surface-elevation will be exactly equivalent to the magnitude of the uplifting force. For the other the links have a limited possibility of variation in length, representing a condition where the rigidity of the crust is given a certain possibility of influence on the resulting elevation of the surface. On giving differences in height to the elevating blocks, to represent the varying amount of uplifting force supposed to exist under the mountain-range, it is found that the two strips do not run at the same level, but the second runs alternately higher and lower than the first, just as geodetic measurements have shown is the case in the great ranges of mountains. The model is of no value as evidence in favour of the hypothesis whieh it was designed to illustrate, but is regarded as of some interest in visualizing the consequences of an hypothesis which seems worthy of closer in- vestigation than it has yet received. Plul. Mag. 8. 6. Vol. 40. No. 240. Dec. 1920. 3H - 826 Geological Socvety. February 4th.—Mr. G. W. Lamplagh, F.R.S., President, in the Chair. The following communication was read :— ‘Geological Sections through the Andes of Peru and Bolivia: le Pyonr the Port of Mollende to the Inambari River. By James Archibald Douglas, M.A., B.Se., F.G.S. The paper gives a description of a geological section across the Andes of Southern Peru, from the port of Mollendo tv the Inam- bari River, a tributary of the Madre de Dios. The deflection of the Pacific coast-line of South America north of Arica towards the north-west brings to light a zone of ancient granite and gneiss comparable with the rocks of the coastal Cordillera of Chile. These rocks are shown to be of ‘alkaline’ type, and are contrasted with the ‘calcic’ granodiorites forming ‘the batholitic core of the Western Cordillera. It is suggested that their formation preceded the uplift of the folded chains. The Jurassic zone of Northern Chile has been almost entirely stripped from the underlying plutonic core, but its continuation has been proved at more than one locality, and in the inter-Andean region strongly-folded fossiliferous beds of Bajocian age are found beneath an unconformable Cretaceous series. The batholitic core is shown to comprise at least three distinct phases of plutonic intrusion, represented by granodiorites, diorites, and adamellites. The volcanic cones of the Western Cordillera have given rise to an extensive series of lavas and tuffs comparable with the Mauri- River Series of Bolivia. Cretaceous limestones here take the place of the red gypsiferous sandstones farther south, and are transgressive on to Devonian rocks. The latter contain abundant fossils of Lower Hamilton age. The post-Cretaceous line of dioritic intrusion, formerly described as running through Coro Coro and Comanche, once more appears on the line of section. . The Permo-Carboniferous fauna of Bolivia has not been dis- covered in the district here described, but beds of similar lithological character are found overlying fossiliferous limestones assigned to the highest part of the Avonian sequence. The eastern flanks of the Cordillera are composed of a great thickness of barren shales, slates, phyllites, and mica-schists, the only fossils discovered being eraptolites of Llanvirn age. This area is further characterized by a well-marked ‘alkaline’ province of igneous rocks, comprising el:olite-syenite-porphyry and rocks closely related to laurvikite, ditroite, and durbachite. A comparison is made with a section drawn through Northern Chile and Bolivia (from Arica to the Bolivian Yungas), and an attempt is made to reconstruct the history of the Cordillera. Jouy. Phil. Mag. Ser. 6, Vol. 40, Pl. XVI. Fig. 1. Basalt. (1.) Granite. (II.) Fig. 4. ira 5 Ly ae ; ap ame, i 4h { % : he —" +e © 4 3a 8) - 4 ms ¢ Toy & Guosn. Phil. Mag, Ser. 6, Vol. 40. Pl. XVII. Fig. 2. Ratio of Intensities=23: 1. Fig. 4. PeeMer ci, cao ? ae erm ny ane INDEX: tro. VOL. XL. .—~ 828 Carbon monoxide, on the ignition of, by electric sparls, 355. Cavities in rocks, on the closure of, 681. Chadwick (J.) on the charge on the atomic nucleus, 734. — Chapman (D. L.) on the equation of state, 197. Chapman (Prof. 8.) on magnetic storms, 6665. Chatley (Dr. H.) on cohesion, 213. Chemical action, on the rate of, 569, O71. : — elements, on the mass-spectra of, 628. Chromosphere, cn ionization in the solar, 472. Clarke (J. R.} on the thermal con- ductivity of some insulators, 502. Coal-gas and air, on heat-loss by conduction in explosions of, 318; on the ignition of, by electric sparks, 356. Cohesion, on, 213. Colloids, on the precipitation of, 578. Compton (Prof. IX. T.) on ionization and production of radiation by electron impacts in helium, 553. Convection, on, 692. Copper, on the electrical conduc- tivity of, fused with mica, 281; on the hght radiations emitted by the vapour of, 296; on the nuclear charge of, 743. Crystils, on the’ arrangement of atoms in, 169; on the rate of chemical action in, 569. Curve, on a method of finding a parabolic equation for a, 513. Cylindrical wall, on radiation from a, Datta (B.) on the stability of two rectilinear vortices of compressible fluid moving in an incompressible liquid, 158. David (Dr. W.T.) on heat-loss by conduction in explosions of coal- gas and air, 318. Davis (A. I.) on convection of heat and similitude, 692, Douglas (J. D.) on the geology of the Andes, 826. Duffield (Prof. W. G.) on the alter- nating-current carbon arc, 781. Edgeworth (Prof. F. Y.) on the application of probabilities to the movement of gas-molecules, 249, DN Dx Electric are, on the pressure on the poles of an, 511, 780; on the alternating-current carbon, 781. charges, on a method of finding the scalar and vector potentials due to the motion of, 228. - conductivity of copper fused with mica, on the, 28]. -—— discharge, on the disappear- ance of gas in the, 585. disturbances due to tides and waves, on, 149. resistance, on the measure- ment of changes of, by a valve method, 291. sparks, on the ignition of gases by, 345. Jilectricity, on the discharge of, through rarefied gases, 785. Klectrocapillarity, on the theory of, 363, 375. Electromagnetic waves, on the pro- pagation of, round the earth, 163. Electron theory of the metallic state, on the, 746. Electrons, on ionization and reson- ance potentials for, in vapours of lead and calcium, 738; on the ionization velocity for, in helium, 440; on the internal energy of Lorentz, 494; on the collisions of, with molecules of a gas, 505; on optical effects due to scattering of light by, 718, Elements in the sun, on, 809. HKnergy, on the kinetic, of molecules, 19%. Entropy-temperature diagrams, on, 211, 501. Equation of state, on the, 197. Everett (Miss A.) on a projective theorem in optics, 113. wing (Sir J. A.) on the specific heat of saturated vapour and the entropy-temperature diagrams of certain fluids, 501. Kixplosions of coal-gas and air, on heat-loss by conduction in, 318. Films, on the permeability of, to hydrogen and helium, 272. Fluorescence, on light absorption and, 1, 15; onthe, of- i0dme vapour, 189, I luorine, on the mass-spectrum of, 628. Tucussing the image of a light source, on a method of, 316, INDEX. Foote (Dr. P. D.) on ionization and resonance potentials for electrons in vapours of lead and calcium, 73; on atomic theory and low voltage arcs in cesium vapour, 80. Friction, on static, 201. cones, on an improved design for, 386. Frumkin (A.) on the theory of elec- trocapillarity, 368, 875. Gas molecules, on the application of probabilities to the movement of, 249, Gases, on the ignition of, at reduced pressures by electric sparks, 345 ; on the yariation of the specitic heat of, with temperature, 357 ; on the reiative ionization poten- tials of, 413; on the ignition of, at reduced pressures by transient ares, 450; on the disappearance of, in the electric discharge, 585 ; on the discharge of ‘electricity through rarefied, 785. Gels, on the cohesion of, 213. General Klectric Company’s Re- search Staff on the disappearance of gas in the electric discharge, 585. Geological Society, proceedings of the, 247, 825. Gerrard (H.) on electrical disturb- ances due to tides and waves, 149. Ghosh (Dr. J. C.) on the absorption of light by the Goldberg wedge, 775. Giles (Miss I.) on fused copper-mica mixtures, 287. Gilmour (A.) on the measurement of changes in resistance by a valve method, 291. Glass, on the thermal conductivity of, 502. Goldberg wedge, on the absorption of light by the, 775. Gossling (B. 8.) on the ionization potentials of gases as observed in ‘thermionic valves, 413. Gravitational fields, on space-time manifolds and. 703. Hardy (Prof. W. B.) on static fric- tion, 201. Heat, on convection of, 692. Helium, on the permeability of thin fabrics and films to, 272; on the ionization velocity for electrons in, 440; on the ionization of, in 829 the sun, 486; on the series con-. stants of, 489: on ionization by electron impacts in, 553; on, in the sun, 819. Hemsalech (G. A.) on the light radiatious emitted by the vapours of magnesium, copper, and man- eanese, 296; on a method of focussing the image of a labora- tory light source, 316. Hinshelwood (C. N.) on the rate of chemical action in the crystalline state, 569, Horton (Prof. F.) on the ionization velocity for electrons in helium, 440, Hot-wire anemometer, on the, 640, Hydrogen, on the secondary spec- trun of, 159; on the spectra of the positive rays of, 240; on the permeability of thin fabrics and films to, 272; on the ignition of, by electric sparks, 350; on the ionization of, in the sun, 483; on the series constants of, 489. Jonition of gases at reduced pres- sures by electric sparks, on the, 345; on the, by transient arcs, 450. Inductance formule, on the use of vector methods in, 793. Insulators, on the thermal conduc- tivity of some, 502, Todine vapour, on the dissociation of, and its fluorescence, 189. Tonization by collision, on, 129, 505 ; in the solar chromosphere, on, 472; by electron impacts inhelium, on, 553. potentials for electrons in lead and calcium, on, 738; of gases, on the, 413. velocity for electrons in helium, on the, 440. Jackson (L. C.) on variably coupled vibrations, 329. James (Rk. W.) on the crystalline structure of antimony, 233, Jeffery (G. B.) on the path of a ray of light in the gravitation field of the sun, 327. Jevons (W.) on electrical disturb- ances due to tides and waves, 149, Johnstone (J. H. L.) on the relative activity of radium and uranium, 50, Joly (Prof. J.) on the closure of small cavities in rocks, 681, 830 Kinetic theory of gases, on the application of probabilities to the, 248. Konno (S.) on the variation of thermal conductivity during the fusion of metals, 542. Landau (St.) on the dissociation of iodine vapour and its fluores- cence, 189. Lead, on ionization and resonance potentials for electrons in the vapour of, 73. Light, on the path of aray of, in the gravitation field of the sun, 327 ; - on the scattering of, by unsym- metrical atoms and molecules, 393; on the scattering of, by electrons, 718; on the absurption of, by the Goldberg wedge, 775. absorption and fluorescence, on, 1, 15. —— radiations emitted by the vapours of magnesium, copper, and manganese, on the, 296, source, on a method of focussing the image of a, 316. Lindemann (Prof. F. A.) on the velocity of chemical reaction, 671.. Loetschberg tunnel, on the radium content of the rocks of the, 466. Lubrication, on the, of bismuth, 201. McAulay (A. L.) on an electrical method for the measurement of recoil radiations, 763. Mackey (Miss I.) on the electrical conductivity of metals fused with mica, 283. Mclennan (Prof. J. C.) on the permeability of thin fabrics and films to hydrogen and_ helium, 272. Magnesium, ou the light radiations emitted by the vapour of, 206, on, in the sun, 816. Magnetic fields, on a double solenoid for the production of uniform, 519. —— storms, on, 665. Manganese, on the light radiations emitted by the vapour of, 296. Mass-spectra of chemical elements, on the, 628. Megeers (Dr. W. F.) on atomic theory and low voltage arcs in ceesiuin vapour, 80. Metallic state, on ‘the electron theory of the, 746. {NDEX. Metals, on the variation of thermal conductivity during the fusion of, 542. Methane, on the ignition of, by electric sparks, 352, Meyers (C. H) on a vapour pressure equation, 362. Mica, on the electrical conductivity of copper fused with, 28]. Micrometer, on the ultra-, 634. Milner (Dr. 8. R.) on the energy of the Lorentz electron, 494. Mohler (Dr. fF. L.) on ionization and resonance potentials for electrons in vapours of lead and calcium, 73. Molecules, on the scattering of light by unsymmetrical, 393. Morton (Prof. W. B.) on the adie cation of the parabolic trajectory on the theory of relativity, 674 ; on Einstein’s law for addition of velccities, 771. Murray (H. D.) on the prec pitation of colloids, 578. Nitrogen in the sup, on, 818. Nucleus, on the charge on the atomic, 734. Observations, on the adjustment of, 217, 680. Oldham (R. D.) on a model to illus- trate the origin of mountain ranges, - 825. Optical effects due to the scattering of light by electrons, on, 713. rotation, optical isomerism, and the ring-electron, on, 426. Optics, on a projective theorem in geometrical, 113. Orbit of a planet, on the, 499. Oxygen in the sun, on, 815. Parabolic equation for a graphically faired curve, on a method of finding a, 515. trajectory, on the modification of the, on the theory of relativity, 674, Pearson (Ii. 8.) on the advance of perihelion of a planet, 342. Pederson (Prof. P.O.) on the theory of ionization by collision, 129. Pendulums, on triple, with mutual interaction, 611. Perihelion of a planet, on the ad- vance of the, 327, 342. Phosphorus, on the mass-spectrum of, 682. Planet, on the orbit of a, 499, EN DE X. Platinum, on the nuclear charge of, 742. van der Pol (Dr. B., jr.) on the propagation of electromagnetic waves round the earth, 163. Polarization, on rotation of the plane of, due to the scattering of light by. electrons, 713. Poles of an arc, on the pressure on the, 511, 7380. Poole (Dr. H. H.) on the use of vector methods in the derivation of formule used in inductance and capacity measurements, 793. Poole (J. H. J.j on the radium con- tent of the rocks of the Loetsch- berg tunnel, 466. Porter (Prof. A. W.) on the specific heat of saturated vapours and entropy-temperature diagrams of fluids, 211. Potassium in the sun, on, 815. Potentials due to electric charges, on the scalar and vector, 228. Precipitation of colloids, on the, 578. Prescott (Dr. J.) on the torsion of closed and open tubes, 521. Probabilities, on the application of, to the movement of gas- -molecules, 249. Projective theorem in optics, on a, 113. Quantum theory, on Rydberg’s law and the, 619. Radiation from a cylindrical wall, on, lll. Radiations, on an electrical method for the measurement of recoil, 763, Radium, on the relative activity of, and actinium, 50. content of the rocks of the Loetschberg tunnel, on the, 465, Raakine (Prof. A. O.) on the dimen- sions of atoms, 516. Ratuer {S.) on the pressure on the poles of an electric arc, 511; on the discharge of electricity through rarefied gases. 785. Reactions, on the velocity of uni- molecular, 461. Recoil radiations, on an electrical method for the measurement of, 763. Refraction, on, due to the scattering of light by electrons, 713, 831 Relativity, on a new reading of, 31; on the bearing of rotation on, 67 ; on the parabolic trajectory and the theory of, 674. Resonance potentials for electrons in lead and calcium, on, 738. Richardson (Prof. A. R.) on sta- tionary waves in water, 97. Rideal (Prof, HK. K.) on the velocity of uuimolecular reactions, 461. Ring-electron, on optical rotation and the, 426. Rocks, on the radiuin 466; on the closure of cavities in, 681. Rotation, on ‘the bearing of, on rela- tivity, 67. content of, small be hlaunts in the sun, on, 814. Rydberg’s law, on, 619. Ryde (J. W. H.) on the disappear- ance of gas in the electric dis- charge, 585. Saha (Dr. M. N.) on the secondary spectrum of hydrogen, 159; on ionization in the solar chromo- sphere, 472; on elements in the sun, 809. Sampson (Prof. R. A.) on the bearing of rotation on relativity, 67. Shand (Prof. 5. J.) on a rift-valley in W. Persia, 247. Shaver (W. W.) on the permeability of thin fabrics and films to hydro- gen and helium, 272. Silicon, on the mass-spectrum of, 628. Silver, on the nuclear charge of, 743. Similitude, on convection of heat and, 692. Slate (Prof. I’.) on a new reading of relativity, 31. Sodium in the sun, on, 812. Solar chromosphere, on ionization in the, 472. Solenoid, on a double, for the pro- duction of uniform magnetic fields, 519. Space-time manifolds tional fields, on, 703. Specific heat of a gas, on the varia- tion of the, with temperature, 397. Spectra of the vapours of magnesium, copper, and manganese, on the, 296 ; on the series constants of the, of hydrogen and helium, 489; on the mass-, of chemical elements, 628, and eravita- - 832 Spectral emission, on Rydberg’s law and the quantum theory of, 619. Spectrum of caesiuin vapour, on the, 80; on the secondary, of hydro- gen, 159; on the, of hydrogen positive rays, 240. Static friction, on, 201. Stead (G.) on the ionization po- tentials of gases as observed in thermionic valves, 413. Stenz (H.) on the dissociation of iodine vapour and its fluorescence, 189. Stewart (R. M.) on the adjustment of observations, 217. Stimson (Dr. H. F.) on ionization and resonance potentials fer elec- trons in vapours of lead and calcium, 73. Sulphur, en the mass-spectrum of, 631. Sun, on the path of a ray of light in the gravitation field of the, 327 ; on elements in the, 809. Thermal conductivity of some solid insulators, or the, 502; on the variation of, during the fusion of metals, 542. Thermionic valve, on the ionization potentials of gases as observed in the, 413; on an application of the, to the measurement of small dis- tances, 634. Thomas (J. 8. G.) on the directional hot-wire anemometer, 640. Thomson (G. P.) on the spectrum of hydrogen positive rays, 240. Thomson (Sir J. J.) on the scattering of light by unsymmetrical atoms and molecules, 893; on optical effects due to scattering of ight by electrons, 713. ; Thornton (Prof. W. M.) on the ie- nition of gases at reduced pressures, 345, 450. Tides, on electrical disturbances due to, 149. Time, on the measurement of, 161. Tobin (Tf. C.) on a method of finding a parabolic equation of the rth degree for any graphically faired curve, 5138, INDEX. Todd (Prof. G. W.) on the variation of the specific heat of a gas with temperature, 357, Torsion of closed and open tubes, on the, 521. Townsend (Prof. J. 8.) on the col- lisions of electrons with molecules of a gas, 505. Toy (FE. C.) on the absorption of light by the Goldberg wedge, 775. Tubes, on the torsion of closed and open, 521. Tunstall (J.) on the crystalline structure of antimony, 283. Tyndall (Dr. A. M.) on the pressure on the poles of an electric are, 780. Ultra-micrometer, on the, 634. Unimolecular reactions, on the ve- locity of, 461. Uranium, on the relative activity of radium and, 50. Vapour pressure equation, on a, 362. Vapours, on the specific heat of saturated, 211, 501. Vector methods, on the use of, in inductance and capacity formule, 793. Velocities, on Einstein’s law for addition of, 771. Vibrations, on variably coupled, 329. Vortices, on the stability of, 138. Waller (Miss M. D.) on the alter- nating-current are, 781. : Waran (H.P.) on an improved design for friction cones, 386. Waves, on stationary, in water, 97; on electrical disturbances due to, 149. Whiddington (Prof. R.) on the ultra- micrometer, 634. Williams (A, L.) on the electrical conductivity of copper fused with mica, 281. W1.son( W.) on space-time manifolds and gravitational fields, 703. Wireless telegraphy, on the current generated in a receiving antenna for, 168. Young (Dr. IF. B.) on electrical dis- turbances due to tides and waves, 149. END OF THE FORTIETH VOLUME. Printed by Taytor and Francis, Red Lion Court, Fleet Street. Foore & MEGGERS, Phil. Mag. Ser. 6, Vol. 40, Pl. I. « D se em in Y mn eoOrmM wie © iD ONT Se © © Neh 6 AND tt 4) 09 © yNOD be 7-0 Ci 5990-96 Va 23 Applied = Be wlerative Low voltage arc spectra Les potential of caesium Vapor accelerating potential in volts Bese ty — - " ; ) . , 7 ha Na “y ae ' Ea r fe : : ta he ‘ ee 2 < 5 . New . SN See, ee Soe : aia rar iar sire NOTED ampere oe gst iS oe 25-0) i, AC ee 7 i en ri Pyar! 7 : ; twuncena oe > See) ety ney ee , ; ae teat he * ‘ 2 c roe te r 2 tye . te i ‘ i ee see TS pirelae aah . f te Sa res 2d Se ? ee ae ee Pie sa Bags moi nro beeping 3S ee apa i tay go Sesh gees ce APR CCU 2 ce : 4 . y . 1 : ; = x ain yg 3 nee pee et Fee daa Wr - ar ere | re sy ns : i t bean : % 1 Dies ome we Gt ay ‘ = : " j : { { i A ; ( : n \ . \ . 3 \ * ) os aN 4 < 2 7 ; * ‘ *~ Phil. Mag. Ser. 6, Vol. 40. Pl. IT. ELECTR 9 ICAL BASE LINE 50 FT. LONG. yA Resistance of Circuit 50 ohms. SEA SLIGHT. a A pene A tener ei in ea ate cnet inhale Noel __ 12 3 6 9 12 Noow 5 -9-/8. 9 Ai fe SEA SLIGHT. g Set A ee Bie OBSERVED TAFE (12 KNOT) ESTIMATED QGIRECTION OF T70E AW ' WATER eS — as TO B Start Bay,ECTRODES. Observigg Yards. rance, 29th July, 1918. knot W. by S. gy: WBYS. > Vira ag MV. ey b> Ace — dq brarboard, —— Youne, Gernann, & Jevons. Fie. 1, Ty ENTRANCE TO DARTMOUTH HARBOUR. SHOWING POSITION OF MOORED ELECTRODES. Depths shown in Feet. Mik Cove_ Mcmrielierevarey! [MD Yonesrone Gv0¥- ELecreooas on 60 Vewricae Base Live Yan 1098 Yano $. MOORED ELECTRODES M, AND C, 2000 YARDS APART. Miervoe TS. SEA ROUGH min, fa MEIVILTS, 8 SEA MODERATE 5x0 GSS ~ Se _pletiel Vie ag ERS TEN car SEA_ ROUGH. Mw AW, ike tin. | e} 0 4 6 9 q a [eb mu. 14:6 rv. 3|-20 43-6 77. IGS 17.v. pea PN Nee ee Cauor. By SEA MODERATE 13-9 -/8 5; LW ‘Ww 72 win, oor io DEEL (ecules §L f QA y [D4 ro. FSET 3-20 417m wv. u 3#:8rrv- PAA q p a MMAR) aac ee (Os Nay! $]- 49 aa SEA CALA SEA MODERATE. SEA SLIGHT 13-9-18 Li. L4-9-/8 yee “in, CALA HOON. Apt sh i £ V2 a § ia ie \y 17-5 7.0 Simu. AS ¢e8 Om 5 mesa Acai Sina SEA_CALM, rn) Wher 3|-401 SEA_ROUGH. z= SEA MODERATE, Resistance of Circuit 530 ohms. Deflexions i indicate M,+ve with respect to C,, i Fie. 4. MOORED ELECTRODES M, AND M, 200 YARDS APART, 74 SEA_CALM, £0: SEA VERY HOUGH. Ares iH 3 eM [een Ay) 3 SEEN BO : 3° Seem Et AEE ii = oo iw Ww 6 O ms L Noon. 25-9-15 oe Sea MOgERATE Onis | 207 eee eet a eA oA 5 t i 9| 10: : N of HIN iy Hin rae mn, aN, aN Yl 9G a ine 5 GF ae SiuuGen oo ime Tiae Sope aiG Nes SEA CALM SEA CALM Sea Cau 4) iy 4 3 ome FFD YI OW a ; $ 9 2 b 6 9 ‘ Noor 26-978. SEA SLIGHT ' i tin nin “in zim wine 3 tec 3 n 1 ' 1 1 D Hi 1 NE 6 9 G I a $s 6 69 Resistance of Circuit 235 ohms. Deflexions * inaicate M,,+ve with respect to M,. Tia. 5, TOWED ELECTRODES. Base Line 100 Yards. Start Bay, Beli Buoy, 25th July, 1918, Observed Tide 1 knot S.W. by W. SE ay S—> {furning} ve te pepe HW ay NH Yeni ag| oS Ab ay (Across Tide) Bt (age acash tide) peed Potl\ (Across Tide) eee | ie : i ‘f ai HAAN * 3 oT ssh= 30 hop 3] ~Ysuie cores $-7" (Max 101) aac TICALLY NO ROLE Tiyan > 2 Veay £17726 ROLLING, Wo rote. HaAav’d Boe Fic. 9, Phil. Mag, Ser. 6, Vol. 40. Pl. IT. MOORED ELECTRODES ON VERTICAL BASE LIN E 60 FT. LONG. Upper Electrode 4 feet below Surface. Res: stance of Circuit 50 ohms. ¢) y SEA Rover. we, 5 3 es ere A SerGnr s se N SA ea eS lta i i H 3 a Aoou 5-9-8 A_ SLiscwr 7 SEA Stignr 3] sig = a ne NTS Ney metal = q 3 6 9 a 3 6 ° AD =SEA BECOMING CALNER SEA CALA 5 GAL AL fs “ a] 0 FN AN ee ete = 3 ¢ 3 n 3 6 4 ig aoow Panis oO ST Sta ROUGH Sea Serenr. N 5] Ae ences NY a ee hminari es wie 311] woo f 4 7 SEA SLIGHT SEA _SLIGNrs 3 g Ooty a ted fe Arn ANA te a nn eA Nee 3 6 a a 5 6 9 N egos, Te, 6, TOWED ELECTRODES, Base Line 100 Yards, East of Portland, 10th Oct, 1918. 5 (A) ‘ A 3 aN 2) SW—> SE ld, NE NW 4 i ¥ 4 q 8 R| q $2 y “3 ~s (8) E bh KE ts » > Conronenrs CASE AVEO EMA 3 \Y — eesuir Ae (Oz) == Oaseaver Tiak (1 Nor cs 98 ESr/NATED D/NECTION O* s-+:---- COURSE RELATIVE TO Sea WATER ———Acruat Course. T1a.7. ‘ TOWED ELECTRODES. Base Line 100 Yards. Dartmouth Harbour Entrance, 29th July, 1918 Obscrved Dido 4 knot W. by 5: oy N—> Vurmag| Say > (Turning) MPS ~ fire A Mar Nr 4] t (a ! ‘| 2 forboond |e Boon Eo.5 y— {, : SPN pa Se 3? is 3 Ni 2 7 Ta, 8. Phil. Mag. Ser. 6, Vol. 40. PJ. III. Zine sulphide (Zinc-blende). THOMSON. Phil. Mag. Ser. 6, Vol. 40, Pl. LV. He Hy NVo./ Comparison Spectrum, Ho He mixed No.2 Hg CO band Hg Hy~4267 4835 4358 ° C All lines in No. 2 not otherwise marked belong to the second spectrum of hydrogen. SISIOHCE DHOED SaSee eee a COON Ht SHR i UBER 2 as | se Ser. 6, Vol. 40. Pl. V, 24 CVD MICA Ce or HCE eH So085 S085 GSioag uy Samus cussaaeteee REC H BO GU! ASAD BADE DORs BENes ae GSU (MSGR CWeUs SARA ORees eee 7empevol Cenligrade 1 22 eR 0 VRBO 2BeSe Ree aweeSe UAE GHES0 PAHS CSASs PAOsa Secs Eee epee Pett tt | tC n 60 Graph N21. COPPER — MICA RESISTANCE - TEMPERATURE CURVE Ht H “ zi tt 70 EY 0 100 10 1720 130 Temperature — Degrees Censigrade Resistance - Ohms Groph N22 COPPER - MICA RESISTANCE - TEMPERATURE CURVE R00 HEH wana HH eesee Eee FH H+! iat ry 1 este : re EEE 1600 H +H rH H z 4 eaearasaiae Hee 140 HEE : 1200 4 CHEE 3 nee Hf 100. H soeen ease sage seegeseae | Bi te rot rH 4 i 600} rH EPR EEE gage eaae! + 1 900 — i 4 : H shi 200) Ht — t Lt + EHH t iH fail H Here 125 «(150 175 200 225 250 275 300 Temperature - Degrees Ceniigpaie, Groph N23 COPPER AND MICA 4001 = yaa bau: greets HH f ; | 30 + HEH HH HHH FH i El. 2 ities & a are H ty S HHH He H | 2001 - 0 FRaEHE Hite HtttttE Hitt SS maa i t jugsusgueaue & : at : ge HEE 1000 a a + rt a t | ie oad need audi esiionsceate | Ht HH Et : rasta Hi HEH : tt Ht EH t : HEE : Q 100 200 300 40. Temperature - Degrees Cenligrade —Thousond Obs Aesislarce Phil, Mag, Ser. 6, Vol, 40. Pl. V. Graph N° 4 COPPER AND MICA + + Temperoluve - Degrees Centigrade in HME ETE tH EEE HH pagedagsegtne ess Hitt + Ha # rH ay HH Ht + ea i t a i Ht Ht | H THEE EH +H yer Ht i i sped cened ieeeaigs 200 ‘Joo 900 ee - ee Sie Hoy a ee Ares/slance —- O18 ‘hil. Mag. Ser. 6, Vol. 40, PI. VI. 28 IVD SELENW/UM Eta: fst HSEEE ate eet TACT aaa eae eee eee ceces reas CHOSE TaSTSanTas GSES GOTaRETaET SEaETSRETS saat BuGL CoEReouned Saas sus saceret Feet serettPeeaehrerreereeerererers ses eetes canes ueeaeees Suseeseateeesttes teeasteest easttente settcttens subesteoad cntastoste tentecsaisoseeatoct scitaitoci usttateed otbottedt Soaeeaees Coste ateou asitostas feassonses esos eoeeeaeetoaeeaeaeares subbsaiessoseestectenstt caeoztecszontesdtest est otttitas testtantes Sunes Bobs Guzed Feens Gueeacnved fates fears uses ceres saves resaeraanactscy = ere ee eet at ee eee aE seceesiae Sea ine ern nat ead eal aus ESE Soe aes aa aaa Seung babes SaNEs EsSESERTas ortacnacessesarcesescasreeasraeereseoree SuSzs Fated SEaeTGrars asasastas seaasaeer soveuovanefoverpaens soravarare 7 /;| Gomes Sonus SESE GEESE SURES SEtEe GOSSESSnG GOBER SERGE SEESE GEESE SSESSESEES Esaaaseed egy ase Pe Hatt PH HEE Ee eettne eet Sous autos teaabestee titaatieesatecdteed focatesteevittoittes fenitastte i Sag sees Sunes Sones Snces raice erect Etat Sues EEE EEE EEE EEE sansa snees suaey Sones Gases fuics ceseaseeea seene canes sever tear creer eres suuaz tascecuseuaeeed fates actaavaraanaes sever aerat raves ieratiaia? oH Suess cease seassceeus SoueC Se! G0 g0nueGuUGE GSES HERS SUUU” GEESE ZEEEECTEE Sasa esiet catbeteeaateocdtnieeasttontee aeteaetacteteetiteeteasias sunsa been esos susee creer sures eee sees sraseeaeessenaetee cea Sccaittecs eosstfest etesihos iterate ssineoostifteoctit PCH ae SA augue Sosy ee feassstirectamntntneoee CoE eet : Pane aaa is a Jas ae Se GSuBegueus Geeus CoSEEGEUGECUSEESGEREGEE fd caPEan oz tetas tetas teeesaneee annECDE HEE O 10 O00 300 Tee Bit Centigrade Groph N2 SF /RON AND MICA eueaee PAE ai eae tH i equa H HH H — a8 ro a a tt EH HH 100 c 200. 00 Temperature — Degrees Centigrade Resislance — Thousand Ohins Phil. Mag. Ser, 6, Vol. 40, PI, VI. ee, es SS AOR ee ea hae HAUL rs ° COPPER AND SELEWUM 7 7 5 ; 9 FETE SEEUEREEDEREEUEEGGE EEEETGEUIWAZGHaW AREEEEETEDE FEET 1 PERE Patt Ht EH janet tesedeage ieatea aGug sey HHH Rais ATE Ht HH HH = aac EEE 4 f t Hy spewawaea peel : th } ttf : Ht Sea Hannatae | aH HH HH : H H f Zee eae =e Sep +} H t Hi ary iH i i bat Fesesess 30 = tf t f HH aueane 4 H t ' : rH f H HEHE HH i Esta seas cad atetei titel HH +H feet Ht tH HH H tH g H tt H H eben & He 8 Ht va at ‘a Ht a aa rf S Ly t 20) : NE HEE setieitiaadt (| ts i HAT HHERATH ae at : ; HH Hiiieee HH if + v i H mt ii EH EE HH H v EH Besapbusee tes! tH Ht N HH H fj HHT a ttt N : £ HEH +t + t S ia aaa esi icesicad s ane aoa t tt HEEL soegeaeas ea : H FEE a) = PH 4 3 aH ry 1 HH tee aii SE > HH H fF & eee FF Ht stot SS i +H 4 + LH EH Hp +H + 1 to sent Seepai Guin al ene an piuiueiie dt ee : +H t ona HA +H + Hf i EH eee ee ; scrdstaaie Ht : gn ee - tt at HH Ht HEH iH HERE Het + . 4 HE HH Bee H HLH 7 ah HELE H HHH : He sacs HEH HH HH aieeuaai Ht + stecesdi és HEH He Et FEE tH HEE HATH 150 2L0 3/0 410 100 200 wee UG 700 200 500 Temperature — Degrees Centigrade. Temperature — begrees Centigrade Temperature. - Degrees Centigrade Cr ahr ee PF 2 me caste eC ST ae ee ee oy + Sie cee mene RE aT EF Se ais, Phil. Mag. Ser. 6, Vol. 40. Pl. VII. Phil. Mag. Ser. 6, Vol. 40. Pl. VII. Wnts. 2 pa MVE OES 2 fone tii Oe) yaa ore > at Phil. Mag. Ser. 6, Vol. 40, Pl. VIII. | HEMSALECH. Red Bands of Calcium in Fringe and Luminous Vapour. : a : Q NI + em temios 8 | @ a a] t N yh & wo] © © © Mode of ava Excitation. Upper Luminous Vapour thermo - chemical. | thermo- electrical. Upper Red Fringe. Eage of Plate. thermea?/. Lower thermo-electrica/. Red Fringe Lower Luminous Vapour. thermo -chemicea/ Drop of Potential along plate: about 8 volts cn, Plate temperature : about 3000° C. Hemsaech. Thermo-chemical and Thermo-electrical Excitation of Light Radiations é nt B c ¢ s 3 3 > : Ri Se oat ae) 5 ae fos a 3 jars ra ine Jo ° Q a 7 < Q a) € Q aa) c 2 7) ° lz G [e} ae a& Ot le YS 72) Some > $ > v s 2 Nye aes Gace vo SQ b> ° SQ 5 % Q ° % we Sc OS .S % cf S % S 2 SR , a RES r La Se ———— as -___ — 49827, ~ 4425 44 36> Co. —da55 —5/06 Cu, Davis} Cu. SWISS TSE —4535 7). ~3/33hn: =52/0 73829 — 4554 Ba. 52/8 Cu. ~ 3832/ Mg. ——FIASO —4607 Sr —3483 C, oe — 5536 Ba. -g -4754 —3920 Fe. —F7E3°Mn. , —3934 Cea. —5700 Cu. —J8ed ss —5782 Cu. 73956\-7. mas SEIS ICTS) r —3968 Ca. i eee he ; ald wOSCIG : 4 ee : -—4934 Be “ eles oO Se a o¢ ae ce + a & ¢ o g mh > eS ~ > > 2s + & (Xz 3 3 3 3 3 3 2 3 “22> ie A G8, ° a 5 2 a eee ae ! 1 ~ fe) vane > % 3 2 o'|& 2 e 3 y % 5 34 3 + = & y S io Ra v x g 5° ee oes mess ~ NS Phil. Mag. Ser. 6, Vol. 40, Pl. IX. in Metal Vapours. a ee volts. em, Plate temperature : about 3000° C. Drop of Potential along plate: about 8 an AA OA ECA 4 i # fh 4 a | i ; \ \ Y wt VIF i oe oe ae | ; ‘YX ‘Td ‘Or TOA ‘9 299 ‘SPM Ud ots 72 = 4 , IXY \/ ST AVS" ff s a ‘Gh-[=" pue 19.0=4 Joy SUOTIeIGIA JO Sa0RTE ‘ILG-0-=" pue G.[=4 IO} SUOIJCIQIA JO S9DB1IL Gog YIMOT GFIIVTASIG FOF YIMOT Fog YIdd/) bs SNS ¥ dS gS ~m 2 m% S 85 Gog YAIaaS/) “NOS. Phil. Mag. Ser. 6, Vol. 40. Pl. X. Traces of Vibrations for p=1°5 and »=0-571. LOWER B08 UPPER BOB DISPLACED. DISPLACED. UPPER LOWER UPPER P BOB. BOB. LOWER BOB UPPER BOB. 79 0%, DISPLACED. UPPER B08 LOWER BOB. 83 BS Wn S38 49 UPPER BOB DISPLACED UPPER BOB. LOWER BOB UPPER BOB 72 19%, VAAN ip \ LOWER B08 DISPLACED LOWER BOB nee ls ey ee te wr pe Pk ; vhs a em Barton & BROWNING Phil. Mag. Ser. 6, Voi. 40. Pl XI. } ! i } j ! j | Bartoy & Browntva. Phil. Mag. Ser. 6, Vol. 40. Pl. XII. Ey 4 s f u Phil. Mag. Ser, 6, Vol. 40, Pl. XIII Barton &'BROWNING. is Sieg ara ett SRE So ae ok a, ate ae ae Seman) ASTON. b-H9 SHS Phil. Mag, Ser. 6, Vol. 40. Pl. XIV, | q RA HH “8 245-~%@ “s _-& - 4g -95 ?6-B- 9:5: ~ oT yp) 3 ~ 32- Se “20 = = i ee pan ere tm -/2 see no SE -235 ' ~ * -24-5 40-™ ei -7 hee /4- ™ 14S 4g. mm Wi 2g re Oe 47-™ & -8 ™-16 49-% ° 49- — Re 16 32 -8/ =o°5 * 19 19 - m -10 20° & 40 Bi) 66 - &% SI W 22 e-12 4G &H) WI gy eg eS Wurppvincrtoy. Phil. Mag. Ser. 6, Vol. 40, Pl. XV. General view of apparatus shown diagrammatically in Figure 1. Metal boxes containing circuits A and B shown open. The letters correspond to those used in Figure 1. ‘eee ie ea iL 7 nth j a, ¥ iM oe La wig 9088 01202 5060