Ve etn sem - i>ee ae - ee ee tw ee nent Sieaiiainets on takin caenaimne ate ot Gee . ao OR aN ee ee ae - or Wil Nr aS A a a re i NO ee ny oS ee ieataeniead Oe cg MO Oy fs ot, Se Ae OO epg j Ine Nee a Rt 4s, exist LONDOMN: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND CO., LD. i SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN;— aM AND YRUVE J. BOYYRAU, PARIS, “‘Meditationis est perserutari occulta; contemplationis est admirar) perspicua .... Admiratio generat queestionem, queestio investigationem, investigatio inventionem.”— Hugo de S. Vietore. “ 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 vario motu.” J. B. Pinelli ad Mazonium., ALERE \ " FLAMMAN, CONTENTS OF VOL. XXXI. (SEXTH SERIES). NUMBER CLXXXI.—JANUARY 1916. sir George Greenhill: Skating on Thin Ice .............. Mr. James Kam: A Criticism on van der Waals’ Equation and some new Equations derived therefrom ............ Dr. Allan Ferguson on the Variation of Surface-Tension with SU eT EAIAME SYS) Lian ys m6, ula Gye eon a: w enelichmy SUieal wallageys ermal Ora Messrs. C. V. Raman and 8. Appaswamaiyar on Discontinuous ree Wotton (Plate Dyed 21 gies) sie soe We elas oe 2 Prof. J. A. Pollock on the Relation between the Thermal Conductivity and the Viscosity of Gases with reference to Seem) ar WOMEN le i eights aye bye AS ey 8 Mr. E. J. Evans: Some Observations on the Absorption Spectra of the Vapours of Inorganic Salts. (Plate IT.) .. Mr. W. Morris Jones on the most Effective Primary Capacity SG DIN eR Ee oh el ia lel SVU Wya'el des GGialuas paid Specs Bb. axes Prof. I. J. Schwatt: Methods for the pied of certain {CTIDEENOW S10 [oo iN ce ea EE ar De ee eee ete : Dr. L. Vegard on the Structure of Silver Crystals ........ Notices respecting New Books :— Prof. J. Pierpont’s Functions of a Complex Variable .. Intelligence and Miscellaneous Articles :— On the Structure of the aE et of Crystals, by LE CSUR g SS by coe a eo NUMBER CLXXXII.—FEBRUARY. Lord Rayleigh on the Prepagation of Sound in narrow Tubes DIP CETTE SL SRST TATS RI REN Or ence eae Prof. J. A. Pollock on the Wave-length of the Electrical Vibration associated with a thin Straight terminated Con- DET eR Pe LH ay MR Gi Te a an AC 22. 62 75 88 89 1V CONTENTS OF VOL. XXXI.—SIXTH SERIES. Page Prof. A. Ll. Hughes on the Velocities of the Electrons emitted in the Normal and Selective Photo-Electric Effects ...... 100 Prof. I. J. Schwatt on the Partial Fraction Problem ...... 108 Prof. G. N. Watson on the Sum of a Series of Cosecants .. 111 Pte. J. Alfred Hughes on the Cooling of Cylinders in a Pureann OL VAI Le. soe eS. ae pt es wo tle err 118 Prof. W. B. Morton and Miss Eleanor J. Harvey: An Appli- cation of Nomography to a case of Discontinuous Motion On a iq Set 258k Oa Deep ea 130 Mr. C. D. Child on the Production of Light by the Recom- bination of Tons -....00 0.6. ee ee oe 139 Prof. A. Anderson and Mr. J. E. Bowen on a Method of measuring Surface-Tension and Angles of Contact ...... 143 Prot. VU. W. Richardson on the Complete Photoelectric BINMSS1ON! shee. AG ae een eee ee 149 Dr. William Wilson on the Quantum of Action .......... 156 Mr. E. G. Bilham on a Comparison of the Are and see Spectra of Nickel produced under Pressure ............ 163 Prot. W. M. Hicks on a Notation for Zeeman Patterns .... 171 Intelligence and Miscellaneous Articles :— Henry Gwyn Jeffreys Moseley ........ Pee Se eo 173 NUMBER CLXXXIIT—MARCH. Lord Rayleigh on the Electrical Capacity of Approximate Spheresiand: Cylinders... sw. wwe ocho eee Lae Prof. E. M. Wellisch on Free Electrons in Gases.......... 186 Dr. J. G. Leathem on Two-Dimensional Fields of Flow, with Logarithmic Singularities and Free Boundaries ........ 190 Dr. Horace H. Lester: The Determination of the Work Function when an Electron escapes from the Surface of a lot-Bod y's. S5.. vei aie ss Sel nee eke ee 197 Prof. OC. G. Barkla and Miss Janette G. Dunlop on the Scattering of X-rays and Atomic Structure ............ 229 Mr. 8. 8. Richardson; Magnetic Rotary Dispersion in Relation to the Electron Theory.—Part I. The Deter- mination of Dispersional Periodsmigeye. «=... eee bo ey) bo NUMBER CLXXXIV.—APRIL. Prof. C. G. Barkla on Experiments to detect Refraction of DR ABAVIS)) «oon lait ous ayo level ob « 2.22 o) SER ee PAaY Dr. 8S. A. Shorter on the Constitution of the Surface Layers of Lagpids:—Part 2... 7... sane eae 260 CONTENTS OF VOL. XXXI.——SIXTH SERIES. Mr. 8S. Butterworth on a Method for deriving Mutual- and PSN GeHANCE. DCEICS is -fa cur BS Ht ae vole eee NN oo Prof. A. Anderson and Mr. J. HE. Bowen on an Optical Test PeEN CS OL COMLICE Ash hy. oP USI ee eee be Eee Messrs. Herbert E. Ives and HE. F. Kingsbury on the Theory of the Flicker Photometer. Hale Unsymmetrical Con- Rapeeenie re NE Ok oe ee Bs, Prof. Louis Vessot King: Theory and Experiments relating to the Establishment of Turbulent Flow in Pipes and MEMES Ri he rais sees Aoiee week Se PR es Mr. W. B. Haines on Ionic Mobilites in Hydrogen—Il..... & Mr. B. C. Laws on the Strength of the Thin-plate Beam, held at its Ends and subject to a uniformly distributed Mercer Case), ib. ca giiok bo Wipes Sie MA aloe te Ss Mr. F. C. Thompson on the Electrical and Magnetic Pro- perties of Pure Iron in relation to the Crystal Size ... Mr. Alpheus W. Smith on the Hall Effect and Allied Pies “LT VETIGL & oR EIS Se OA e127 Pe GR, Sa aE Prot. Karl Pearson on a Brief Proof of the Fundamental Formula for Testing the Goodness of Fit of Frequency Distributions, and on the Probable Error of ““P” ...... Sir Ernest Rutherford and Mr. A. B. Wood on Long-range Paihia Particles from) Thortim ‘... 66 6.28 ree eee. Prot. Horace Lamb on Waves due to a Travelling Disturb- ance, with an application to Waves in Superposed Fluids. . Mr. R. M. Deeley on the Theory of the Winds .......... Prof. A. Steichen on the Variation of the Radioactivity of te vores ab Lirwal is ee oe eee eae: Dr. Manne Siegbahn and Mr. Einar Friman on the High Frequency Spectra of the Elements Gold-Uranium Notices respecting New Books :— Dr. W. Briggs and Dr. G. H. Bryan’s The Tutorial PRE AEM Pe ANON I | Mista ea slay Le ahs cele oc eg A. W. H. Thompsen’s A New Analysis of Plane Geo- metry. “inite and bitteremtiale. fo. vl. Be L. Couturat’s The Algebra of Logic Dr. A. Findlay’s Practical Physical Chemistry ........ G. W. Parker’s Elements of Optics for the use of Schools IDG VEINS Sie Ae ha RR, UW = or, putting £ m = on the surface of the clear water, and oh = W?, so that W is the velocity of ee waves a ice, gi 2 a 2 V?4 a5 2) & V?4 fe Ur= SV =e Skating on Thin Ice. 3 Neither E nor W is given numerically in the list of Hverett’s ‘ Units and Physical Constants,’ or in the ‘ Smith- sonian Tables of Physical Constants’ either, although it is curious as mathematical history that Ice was the first substance of which the Modulus of Elasticity was defined and measured by Bevan and Young (Phil. Trans. 1826; Todhunter-Pearson, ‘ History of Elasticity,’ p. 189). 3. Thomas Young measures the elasticity of a substance by > a length, & suppose, called the elastic length, such that for asmall fraction 7% of this length, of uniform cross section, hanging vertically, or standing up supported, the material would be stretched or compressed, at the top or bottom end, by an extension or compression /. Thus a length 7 hanging vertically would be stretched one per cent at the top. Or otherwise, » times the length producing extension : will give k, the elastic length. So, too, the tenacity or breaking tension may be defined by the breaking length, as of an icicle; and the stress in the material must be designed in a structure so as to be kept well below the limit of tenacity, for the material to recover itself, and obey the laws of Mathematical Elasticity. In the ‘ History of Elasticity’ (H. of BE.) p. 189, & is taken in feet as 2,100,000—sav twenty-five million inches, or two million feet, one-tenth of the value given in Thomson and Tait’s ‘ Natural Philosophy’ § 686. This makes W= V¥(gk)=8000 f/s, nearly eight times sound-velovity in air; and so explains the story quoted in Ganot’s ‘ Physics’ from Parry’s ‘Arctic Voyages,’ where at some distance the sound of the gun fired on the ice was heard before the word of command to fire; the report of the gun being transmitted through the ice and so overtaking the sound through the air and arriving ahead. So also in an experiment on long-range fire, the report is heard on the telephone long before the sound is trans- mitted by air. And the crack of a rifle is carried along by a high-velocity bullet, moving faster than ordinary sound, and will be heard up the range a little to one side when the resolved part of the bullet velocity in the direction of the ear is equal to the ordinary sound velocity ; the sound through the air from the muzzle is a appreciably later. 2 4 Dir G. Greenhill on 4. The proof of the formula in (1) §2 is given in a combination of the theory of the elastic lateral vibration of the sheet of ice, propagated in the direction of the waves, with a varying ‘upward pressure op (gravitation), derived by hydrodynamical theory from the wave-motion in the water below. Then, if 7 is the elevation of the ice at a distance xv, o the superficial density in g/cm”, and EI the flexure rigidity, i in gravitation g/cm, d*n oF = —ghI Ina t OP? Sp _ ad Hp hy ae ee d denoting the velocity function of the wave-motion. wy 2ar 2rU n Writing m for ae and n for saa = U, assume 2 ad? Shed ya ak CT Gn a On Soe = bsin (mz—nt), ji eh ee and in water of depth h, take db = Behm(y +h) cos (mx—nt), so that just under the ice, y=0, = = mBshmh cos (me—nt) =— | =nbcos(mxz—nt), mBshmh = nb,. 2 Ze = nBehmh sin (ma«—nt) = oe N) gt = = (= coth mh—g) n Substituting in (1), dividing out 7, —n'o =—gkIm+ ee p coth mh—gp, , 24 Inte coth mh+m2 p (1) (2) (3) (4) (8) (9) Skating on Thin Ice, 5 as before in (1) § 2, with et Bn ete \ P and with ed Aye nie Bd v= —thinh, Wea mae ve sW"2 ro, 1 (ge)? 4 3 V2+—(me)*W?thmh V +35( % W?th*mh 12 ee eo Lak (10) 1+, th?mh in which thmh may be replaced by unity, when the depth h of the water is more than the wave-length A, making, as in § 2, Bes = 1+methmh (11) 5. The determination of the exact minimum value of VJ? in (10) § 4 will lead to an intractable quintic equation, even with thmh replaced by unity in (11) ; but as the term ge/V? in the denominator is small, we may consider the variation of U2 as due to the numerator alone, with thmh omitted as unity, GON. Ul Ve (ea. (1) and this is a minimum by variation of V when oe lige “Wwe Cave Td v:) 3 ° ° e ° e 6 e (2) making 4 15) a Wieser 2 yt) Sagaey Be v= av > Vv teb5, that is, U about 15 per cent greater than V. This is in accordance with the simple theorem that the varia- tion of av” +ba-”" is zero when its d.c. maz”~!—nbe-"-1=0, or maa™=nba—~*, and here m=1, n=3. -As another example, the cost per mile of a steamer at K knots being given by the two terms 2 + BR’, it is a minimum when 2 =2BR?, or with the running expenses and wages twice the coal bill. 6 Sir G. Greenhill on A skater, then, who can progress up to this minimum value of U and beyond, is able to place himself at will at any point of the ice-wave he forms, saya little beyond the crest, so as to have the advantage of the downhill; and if the ice should crack he will be able to escape. Then V has the special value given in (2), W2\4... Ws (e200? ii ee qo) "Te ie m= = (gh) =e(G)- - (3) With the value of £ above in § 3, this works out for ice: One inch thick. Two inches thick. 1 e — 2 foot 12 6’ A=27 G ke) 26 AA, feet. ny: 11°5 15, f/s U 13°2 17°25, f/s (9) (12), m/h 6. But when the depth f is small compared with A, say h/A <4, the variation in thmh becomes sensible. For a value of h still smaller, replacing thmh by mA, Qme\* W? 1 /27re\t k Ue ele De a. yz 14+ meh om Qare\- ee (1) Gy Xr e and U is large when A is a fraction of e; but as > becomes a large multiple of e, U* tends to a limit 2re\? | s ur=(1-() =| gh |< of ea su that a long flat wave in ice of thickness e on water of uniform depth h will have velocity 1-3 Cry] ven, 7. The value of E and & was determined experimentally by B. Bevan (Phil. Trans. 1826, H. of HE. p. 189) with a cantilever of ice, cut out as a tongue except for one end ; of Skating on Thin Ice. T length J (100 inches), breadth 6 (10 inches), thickness e (4 inches). Working with the inch as unit of length, the bending moment equation is BAK"! = pAG—a)-4",. 1. - () from which the cross section A=le divides out, and where Kk? = 12 ; so that with E/p=k, inches, “04 = 5(P—2") EAT Se nn ae (4) a her = 5 (ve 503), ah Via eae bailey) = rece fal C Pats zt) ; (4) and if d denotes the droop at the end, where «=1, id= teat aa . . O Thus with J=100, e=4, £=25 million, inches, we find aA 0'625, say 2 inet.) 8" fs (6) And conversely, this value of d measured would lead to the value of & above. 8. The skin stress on the top or bottom of the ice, given by an equivalent length h as ph, is such that the bending moment is due to an average tension or pressure $ph, acting over be, the upper or lower half of the cross section, at a distance 4e from the neutral mean line, and is therefore 1 ip 1 5 phx x be x xe z= gphbe = = pha = 5 pA(2—a%), (1) ON yp 2 jae — a maximum = at Orsi. wey) (2) so that in the experiment h, = ee = 7500 inches, or 625 feet, . . (3) as of an icicle of uniform cross section ; and it is not likely the ice could stand it without breaking. 8 Sir G. Greenhill on But in Bevan’s experiment the tongue of ice was left supported on the water, and deflected by a weight at the free end ; and then, if the ice was deflected d inches by a weight of P lb at a distance a from the root, in a tongue 6 inches broad and e thick, Dues _ 4Pa5 xPa 9 hic perbd’ ee (4) in accordance with the usual formula. In an experiment, P=25, a=98, b6=10, e=4, and by measurement d=0°206 inch; so that with a specific volume 1 Eld = 12 pke®bd = of ice or water Pi 27°73 inch®/lb, p — EX 29x 27°73 x 98° ~ 48x 10x0:206 Inches, se.) 2a and this works out to about 20 million. Then for the skin stress, ph, LE ON SY a pke*bd = h in ded Be a gear he, Daehn (6) and this works out to i= 000012, % = 3000 inches, 250 feet, . . (7) still rather long for an icicle. 9. Pack ice is produced by thrust in the surface. LEsti- mated at a pressure P=pe, the D.H. (1) § 4 is modified into d? d‘n d? op = GEL ghee oer,» . (A) dn and for ice at rest, We 9 dp = — pn, ad: d? EL +Pes item = 0, e e e e e (2) EA d? pte tee te SEQ.) ee Here the sign of P and ¢ would be changed for a super- ficial tension, as of the capillary film in $1, ora cloth on the surface. Skating on Thin Ice. 9 With a solution »=e”, the auxiliary equation (A.E.) of (2) becomes = hemi come a (A) OE NON EE 1 a (i i 4° (*-3%), Onto (5) requiring i 1 eW? 2>- = — o> se ke, P>pa/ (she )>pa/ “5 Lt hital (0) 6 (6) for the ice to become unstable, and the surface to buckle up into waves. The vibration equation in (8) § 4 can be adjusted so as to introduce the term due to P: it makes 2 —n’o= —gEIm* + gPem?+—p coth mh—gp, . . (7) me ers ae n? INR ads m = OG i ule. * coth mh +m 1 iE ga cem+—kem? 1—cem?+ = ke*m* 12 ne 12 : 8) coth mh+me ~~ meothmh+em?’ * ( but the extra complication does not repay investigation. 10. For the deflexion of ice at rest, bent into a cylindrical surface of straight waves, by a single line-load over O, such as a log, equation (1) §4 becomes da’ Els 4 == 5p — — Pp, e . e e ° (1) 1 d y) 1 i HY 1 2 : Z aS = ee = —4q*, 12 ke? = 4g” g aT (5 ke?) one (2) suppose; so that g is the reciprocal of a length, and for positive w and 7 finite the solution is n=e-f(Acosga+Bsingz).. . . . (3) To make = =0 at «=0, A=B; and measuring 7 upward, n = —ae-“(cosga+singr), . . . . (4) <7 2age—* sin gx i = 2ag’e-*(cos ga—sin gz), dz ; dz? d’n i a a 35— GZ ‘ mai ~get Brae he A 5 Ta dag®e-% cos ga, \ nde ay COS Gx (5) 10 Sir G. Greenhill on The area displaced between the water-line and profile of the ice is then {ade = % (1-e-¥ cos 4), Se N and this area up to infinity is zs The ice is deflected into two cylindrical surfaces, symme- trical on each side of O; and if the line-load over O is W per unit length, it is buoyed up by the displacement se of water; and 2ap Loe fe W Ee ) (7) > &— Ve q ze 2p(5%") a0 The bending moment (B.M.) of the ice is a maximum under QO, and is . 2 BIS = pk Pag? = «5 keg W = ic* i(5 ke) W, (8) The maximum skin stress of the ice has then a tension or pressure length h, such that itt ae == max, olga g pketag’, © 28t ee (9) oul 3W/k 2 tahoe =(F) =, b=52(g,)-- 20) The ae force F at a ashe « is then given by 24 = Ww" a—ee COs ga) =a COS Qa es! (eo (il sane aye Joenetos. o> 6 ne ) as a verification. 11. A central line-load W at the middle O of a plank or beam, of limited length 2/, floating on water, will give a downward deflexion 7 such that, from 2 to J, 1 EIS +p), (2'—2z)y'dz’=0, ..9 ee 3 l oe I= nde! = 0, (2) Skating on Thin Ice. ial as hefore in (1) $10, with a solution n = B,chqrcosqr+Bashgersingr, . . (4) to satisfy the condition of symmetry on each side of O. The bending moment at the free end of the beam is zero, and so at c= +1, 2, ) at = —2B,q’ sh q/ sin gl+ 2B.q? ch gl cosgli=0, (5) and we put | _ a(chgacos ga , sh ga sin i”) F 1 =O (Sore ehigiicosigl ji oe The load W is supported by the extra buoyancy due to the downward displacement 7 ; so that l NPI OTe se PPL is untelt (yal alone Gdy) = in which | » { gchq¢cosqvrde = }( shqlcosgl+chgqlsingl), (8) 0 {a sh gx sin gedx = 3(—sh ql cosgl+ch ql singl), (9) W= 5 —sh gi cos ql-+ch qi sin a!) q sh gi sin qi ch gl cos q/ Be i if DOCH or: LN Be nga g tacd) = | hap male ay) so that if a was the extra draft of the plank due to the load W distributed uniformly over the upper surface, W = 2pla, sh 2q/ sin 2ql Smee sh 2gl+sin 2ql’ * (11) ag Dal ch gi cos gi ch ga cos ga+sh g/l sin gl sh ga sin qa a sh 2g/+sin 2g/ . (12) 12. With a series of parallel line-loads, W, at equal interval 2/, the surface of the ice will be bent into equal waves joined together, given between «= +1 by the general solution of (4) § 11, with the origin O midway between two consecutive loads ; and with symmetry in the wave on each 12 Sir G. Greenhill on side of O, and with a change to measuring 7 downward, d‘n a LN ole a EI, +en nue 19 Me aati 0, aie n= Achgecosgz+Bshqrsingr; . . (2) and to satisfy the condition = =0 when c= +l, n = alchq(l+2) sing(l—wx)+ch g(l—2) sing(l+2) +sh g(l+.2) cos g(l—a) +sh g(l—2z) cos g(1+<2)], . (3) = = 2qa|sh g(l+ 2) sin gl—x)—sh g(l—z) sin g(1 +2) ], (4) with an area for buoyancy, WW ana | —= ] ndx = 2- (ch2gl—cos2ql). . . (5) p ba); a When 2=0, m= 2a(chglsingl+ shgql cos ql), e2=+l, ,= 2a(chqlsh gl+cosqi sin ql), and putting W=2lw, so that w is the average superficial loading, which we may reckon in lb/inch?, _ 2w gi(ch gl sin gl+sh qi cos ql) (6) a p ch 2gl—cos 2ql oT ena __ 2w gl(ch qi sh gl + cos gl sin ql) (7) or p ch 2gi—cos 2g 7 reducing in each case when gl=0 to No = 4, = w/p = 0,0) eee the uniform depression of the ice when the load is distributed uniformly. Equation (8) (10) § 10 of the B.M. at O, and skin stress, is changed to a maximum so op eae syay CH gu sini@l— sh gi cosghi ame Pe ee ig 7 haRIe ane Sof 1 Be @) ! = 2keg? (ch ql singl—shglcosql).. . . (10) In this way the relation can be calculated between the thickness of the ice and the loading permissible, distributed as infantry or artillery. 13. Next, roll the logs in a regular procession at velocity U and equal interval 2/, to represent the passage for instance Skating on Thin Ice. 13 of a train of artillery, with the ice in waves advancing underneath at the same rate. Or else reduce to a steady motion by giving the ice and water underneath a reversed velocity U, with the logs now dn dn —, replaced by wT stationary, and dt? Then, with no pack-thrust P, | oo Lea a MM a aie Goa LOE, ats AI hI Tato Fe 14 ontp— 5A tna Ge Mi Ga) ond with E—pk, [= 1 ie Un a2i=S — — —— —— 0. e e e in eae Dg ey a ws Ty i= pe, Measuring y and 7 upward, the velocity function co) =F s= Aid, == Aods, Sere ers er nS (5) oo theca Aes eae d 9 aw to the first sag where ¢j, ¢2 are composed of terms of the form et” ; and taking the water deep, so that ¢1, d2 are zero for y= — ©, we put gi=e™* cos (ay—ba+hy), bs=e"@*™ cos (ay+ba+he), (7) for each stretch of water between two logs, with some slight discontinuity at a vertical plane of junction, of vertical slipping without cavitation, which may be ignored; and é re VY? (dis $e) = \); satisfying the Hquation of Continuity. With the origin midway between two logs, take 7 = Byn + Bon, ° . ° . e 4 - - = (8) ai enon CoOs0es ap — sham sib. «is (9) tbs 1,469) iain ee ROY and put "3= shaxcosba, y,=chazsinbr. . . (10) 14 Sir G. Greenhill on Then, with a=ccos a, b=c sina, ge cos Si = = a— In @ Cc dz 13 4 y) uf dn eth a : — += = 7 COS 2—7Np SIN a, ce ax dns dn, . cn, = — cos —— sin 1 dx esi da me Venda = 9; cos a+7, SiN a, tl dn 2 de <1 = m, cos 2a — yo sin 2a, Can 1 d’n, 9 9 = sin Za COS <&, yoo + 1 d’n, ; FORE => 3 COs 2a—n4 sin 2a, 1 dn, e be a qgi = Ms sin 2a-+7,C0S 2a, toe sin a+”, cos 1 dy, ‘ Cdn SI ae d dns CH. = a sin a +—" 00s a, fengde = —73 SiN a+, COS &. IL of f ct — = 7, cos 4a—n, sin 4a, ds se = 7, sin 4a+ 7; cos 4a, 1 dn, : (2 fae ees 4a—n,sin 4e, 1 d'n, : DA Te da+tn,cos4a. . Just under the ice, where y=0, db _ 1740 dda | ay U dx’ as d Ag 1 dd, Ae = e cos (— e sin (ba—h, +a) a UB v dn, dno dx +UB, 52 b) ba +h,) sina—e* sin (—bx+h,) cos a (ch az+sh az) [sin bz cos (hy —«) —cos be sin (hy—«) | (n+ m4) cos (hy—a)— (41 +73) sin (hy —«), 1 dds = e-“ cos (bx + hg) sin a—e~™ sin (ba + he) cos & e~* sin (—be—hy+ a) = (chaz—sh az)[—sin bx cos (hg —«)— cos bx sin (hg—«) | = (m.— 1) Cos (h— a) —(m,—73) sin(lg—a). (11) (12) (13) (14) (15) (16) (17) (18) (19) (21) Skating on. Thin Ice. 15 Le But , and 7, are absent in ae a and so must be absent in AiG a +A, a so that Ay Cos (hy —a) +A, Cos (hy —a) and Ay sin (hi a) = A, sin (hy—@) — 0, (22) A,;_ _cos(hp—«) _ sin (hg—#) A. ie cos (hy—a) sin sin (hy—«)’ iy Ss ee ates Oy meted he a BPO 7-00) tan (hj —a) = tan (ho—2), A, Pi. Ae = —2Acn; sin (h—x) + 2Acn, cos (h—«) = i els UB, ee = U B,e(3 cos a—n,sin a)+UB,c(n,sina+yycose), . . (24) UB, cosa+UB,sina = —2A sin (h—2), UB, sina—UB, cosa = —2A cos (h—z), \ UB, = —2A sin, HAS 2 COS IE ihe oe ay aa ED) ¢ = Ur+A(gi—¢z) = Ux+ Ae’(ch ax+sh ax)| cos bx cos (ay +h) +sin ba sin (ay +h)] — Ae’”(ch az—sh az)| cos ba cos (ay+h)—sin bz sin (ay +h)] = Ur+ Ce; cos (ayt+h)+Ce%msin(ayth), . 2. . we. (27) with 2A=C, UB,=—Csinh, UB,=Ccosh, .. (28) Un = C(—m sinh+7, cos h), : ea 1) U dn : Gel gat = (— 1 008 2a + np sin 2a) sin h+ (m sin 2a-+ 92 cos 2a) cos h = sin (2a—h)+n,cos (2a—h), 2. . 6 « « « (30) WU dy CA sa = msin (4a—h) +n, cos es WA eee Ng ig 3 (31) From (27) when y=0, ee = Ce[m, cos(h—a)+n,sin(h—a)]. . . . 2 (32) 16 Sir G. Greenhill on Substitute in the dynamical equation (4), and divide out C, using (6), skeet: sin (4a—h) +, cos (4a—h) | 2 +e = c*[ m sin (2a—h) +72 cos (2a—h) | —n, sinh +7, cosh 2 i =? [7 con (a—h)—m sin (@—h) |= 0, a ee and equating to zero the coefficients of 2 and 7, Lah ae U2 ae 15" c* cos Geen 7 c? cos (2a—h) + cos ams csin(a—h) = 0, (84) Pe We eu U? {9 hee sin (4a—h) Mee sin (2a—h)—sin ah ecos(#—h) = 0, (35) 2 2 ap ketc Cos ese cos 2 Ae csina+1 tanh =e ee) 2 2 P — ke®ct sin 4a+ Z c? sin Deane CCOS & 12 g g Z 2 ~ ke®c* sin 1a aye c? sin et CCOS a tan h = te” Ss MG a) 2 — ke®c* cos 4a +e U c? Cos pea csina+1 12 g 7. x 2 2 2 ke®c4 cos 4a + € Wie cos pint sin «+ 1) 12 g g LZA Po 2 Ty2 2 + (qpietetsin dated sin 2a 000s a) = 05 0: (38) 2 2 J feet cos ate -—c* cos 2a——esin a+ 1 = 0, a) Wisner nou) 2 2 pobettsindatew fin 2at—c cosa = 0, PN Ci el) 2 2 : philatbiiteS (atbir+i— (a+b) +1 = 0. Be Sire 12 Then, with a+b:=im, 2 2 ltt ies mt m4 1 = 0, Mme G tf (42) i ee MUM g 1+em as before in (9) § 4, in deep water, with coth mh=1. Skating on Thin Ice. i Introducing the condition dn dy, . dn v= +1, Fim oe sin h+ + cosh = 0, — (3 COS a—n, Sin «) sin h+ (m3 sina +n, cos «) cosh = 0, n3sin (a—h)+n,cos(a—h)=0, . . « (44) _m_ chalsin bl _ tan bl tan (h—a@) ae shalcos bl thal’ 5 6 . (45) sin(h—«) _ cos (h—a@) ae 1 chalsin ol” shalcosbl” V(2.ch 2al—cos 201), ° (46) 3 = VW (4.ch 2al—cos 2b1) cos (h—«), ns = /(4.ch 2al —cos 261) sin(h—«), . ~ (47) Sabie =) c. The extra buoyancy over the length 2/, required for the load W, is given by ‘ “i W =| —pndxe = el (cn, sin h—cn, cos h) dx au SIL 2 ; vi a [ (nz cos a +4 sin a) sin h—(—7, sina +7, c03 a) cos h] 2C : x Us [ns sin (h+a)—m cos (h+a) ] BP: 9 9 211) = Gf, sin 2av 2.ch 2al—cos 2b1), PA ne RT ae (C0) which determines C. 14. If the extra buoyancy of the water is taken into account in Bevan’s experiment of § 8, where the tongue of ice is depressed into the water by the weight applied at the free end, we take 7 = By + Bono +Byy3+ Buggy . . . . (1) dy > dx By= 0, B, + By=0, | n= Bonot+Ba(ys—m,). - - - - + (2) Phi!. Mag. 8. 6. Vol. 31. No. 181. Jan. 1916. C and the condition n=0 = 0, at e=0, requires, since Tse 18 Sir G. Greenhill on Here a=b=gq, if dn i dn» see J dns es 6 1 d?n @? a —2n2, G? da? = 2m, Gg Vie =—in, @ = = 2n3, (3) I , 2 —! = 2Byn, + By(—2,—2;) = 0, when e=1,. . . . (4) d? da B, ch gl cos gl —B;(sh g/ cos gi+ch g/sin ql) = 0, pe = thgl-+tan ql. ‘ : 3 At «=0, and measuring 7 downward, 1 Cape ey aN sma) 92 9 Pke = jo ehers- 2q°Be | = bending moment at O= Pl—pb { anda, . (6) . 0 , and integrating by parts, | anda = «\ndx—S\ nde, er m=chgqlcosql, n.=shqglsin gl, n3;=shglcosql, ns=chgisingl, (14 in which Sanda = 4(ns+74), \WWonde = tye ee § onde =H—m+n), Sfamde =-im - - @) fgnsde = 3(m +72), \\enmde = 4 Sy eee Sande = =4—mtn), [ands = =-km . - CD J Sa(ns—nade =, ((P(s—n.) = 3+), - G2) FJ f2q?andz = Blg(—n3+s) + Ba (m,—1) + 2Bslgm1—B3(n; +-94)> (18) ) 12°) : + 2lq ch gl cos gl—sh gil cos gl —ch gl sin ql ch gl cos gl cos gl + lq - anal == a —thgl—tan ql, .\... 3 See (15) Ba ( Se a 1 3 ee 1 es. shyt = 1, (16), Skating on Thin Ice. L9 and if d is the deflexion at the free end, a = Bi +73—n, = (th g/+tan gl) sh gl sin gl 3 +sh gl cos gl—ch gl sin ql _ shgl _singl _ sh 2gi—sin 2q/ (17) Mee Teh! 2chglcosgl 9 7 __ Pq sh 2ql—sin 2q/ “Si pb ‘ch? gl + cos? gl © - + (18) This reduces to the preceding value of din (4) § 8 when gl is small, and the buoyancy is neglected. Velocity of Sound in Air. i. The question of sound velocity may receive recon- sideration here. For the velocity W of longitudinal vibration through ice or a solid substance, we have taken E Mie oe Ne eek coon ety loteg CL) so that the velocity is that acquired in falling freely under gravity through half the elastic length. But in air the cubical elasticity, pe, which on Boyle’s isothermal law is equal to the pressure, becomes y times the pressure in the adiabatic compression and expansion of rapid vibration ; so that the velocity of sound in air is taken as the velocity acquired in falling through vy times half the height of the homogeneous atmosphere ; and this y is the ratio of the specific heat (S.H.) at constant pressure and constant volume, taken on the average at y=1°4. Suppose, however, a shower of rain is falling : how does this aftect the value of y, and the velocity of sound ? The thermodynamical influence is the same as that in- vestigated by Sir Andrew Noble in his “ Research on the Pressure and Work of fired Gunpowder,” Phil. Trans. 1875-94. There he found by his experiments on the pressure in a closed vessel, that the law connecting pressure with volume and density was something intermediate to the isothermal law, with y=1, and the adiabatic law, y=1°4; and he accounted for the difference by the influence of the heat stored up in the solid particles of powder in the gas. C2 20 Sir G. Greenhill on Take a pound of gunpowder, and explode it in a closed steel vessel of volume C inches*; and suppose the fraction 8 lb was solid and non-gaseous, leaving 1—£8 lb in a state of as. : In a rise of temperature of dT degrees Fahrenheit (I.) the heat dH, in British Thermal Units (B.T.U.), given out by the non-gaseous part, of 8.H. X, is given by dH = —BradT. so ee Denoting the volume of the non-gaseous part by «a, tlie gaseous part, 1— lb, obeys the gas-equation ples) Pale), ie where T, T,, denote absulute temperature F.; so that, taking logarithmic differentials, TAN — 0; ° ° ° ° . (4) a and then with (2), oe —pr(2+ +=) Ri U4 Supposing p and v to vary one at a time, _ oH, OH 4, dH = aii dp (v constant) +0 dv (p constant), (6) where, when p varies while v is constant, oH pel v—a ONE Sete LS A) bei ate ie - (7) and when p is constant and v varies, for 1—£ lb of gas, H H ofa (8) vu— Cy, C, denoting the S.H. of the one lb of gaseous product at constant” volume and constant pressure ; so that (6) becomes a od pc, (9) Skating on Thin Ice. ; 21 Equating these values in (5) and (9), | [(1—A)C, +A] P +[(—-A)C, +0] = = Oc CLO) a differential relation, leading on integration to [(1—8)C,+ Br] log p +[(1—8)C,+ Br] log (v—a) =a constants: +) 751 n( bh) ew ae) Son) Cert Oh er (e.) ST ie jeje (12) reducing to the ordinary adiabatic equation when «, B=0. In the experiments of Noble and Abel, they found By ia: BOE 0 pa I Os EN ae igs 0°57, ne 0°43, Poe Tete 13256, (13) taking the solid and gaseous products of the same density when fired ; - to CO, = '0°2324) 0, = O62.) 05) (14) making fe m=1:074, against y ae a L) Thus 57 per cent of this gunpowder would issue as smoke from the muzzle of agun. But with modern powder, such as | cordite, the solid part of explosion or smoke is insensible, and the index m regains the full value of v. The raindrops in the air are the equivalent of the non- gaseous part of the powder gas, and their effect is to reduce the value of y, and diminish the velocity of sound. 16. Next treat the air in an unsaturated state as a mechanical mixture of dry air and aqueous vapour, each obeying the laws of the Gas Equation. The question has been considered by Professor C. Niven for a mixture of any number of gases, and he gives the formula (Solutions of the Senate House Problems, 1878, p- 66) me , _ mk = ran l aliases Dhak iaihy oath ns aa for U the sound-velocity in the mixture, where m denotes the density of a gas, c the 8.H. at constant volume, & at constant pressure, and w the velocity of sound in that gas. U? 22 Mr. J. Kam on van der Waals’ Equation and Take a cubic centimetre of the mixture of m, grams of dry air and m, of aqueous vapour ; and let heat be supplied to raise the temperature 60 degrees, without change of volume ; denoting by C the S.H. of the mixture at constant volume, MC, + MC (m+ mg) C80 = my,¢,¢0 + MC, 60, C = ee 5 (2) and similarly, if K denotes the 8.H. at constant pressure, _ mk + mek, = Sie ou (3) Also for each constituent gas respectively, with gravitation measure of the pressure, ky py _ Myc, Uy? Moly Ug” uy? = gyi = oh 7 len Pr = ko 9? (4) Cy my and for the mixture, : OWE P= eer 9 = PitPos Meee nn Oh yc (5) , MyCy Uy” | Mylo Uy” Uke Pitpe _ mky+mk, ky g ke g = = 6 g C My, “+ My MC, == MoCo my, + Mo 4 ( ) the special case for two gases of Niven’s general formula, derived in the same way. The Smithsonian Tables will provide the numerical data from which U can be given in a tabular form from p, m, c, k of dry air, and aqueous vapour of given saturation. 1 Staple Inn, W.C. Oct. 1915. Il. A Criticism on van der Waals’ Equation and some New Equations derived therefrom. By JAMES Kau*. Preface. |e the following deductions I take it for granted that the factors a and } of van der Waals’ equation have the effect : The factor a of diminishing the gas-pressure towards the exterior with a value P, = — the “ Inward Pressure ”’ ; The factor b of increasing that pressure at the rate renee V Bara b ; * Communicated by the Author. some New Equations derived therefrom. 23 Thus the pressure P from the exterior on the gas plus the ** Inward Pressure” P, is the total pressure II, 2. e., P+P,=II. It will be seen that I introduce two separate factors 5 and £, the latter of which is used in place of van der Waals’ factor 0. b is the volume of N molecules filling a volume 6=(V +5), R.T at a pressure IJ= yo 8 is the volume of N, molecules filling the volume V at the same temperature and pressure. Consequently 6 is a constant, 8 a variable ; tor the smaller V is (7. e. the greater the compression) the less the number of moiecules required Lie to exert a pressure I[=—_. Vv Hence the inverse value of the volume V at any pressure i= eis not the density y,, but the density yp which as b is a constant increases less rapidly than the volume V decreases, For large values of V (z.e. small value of II), V will be almost equal to 6=(V+b), N, almost equal to N, and §& almost equal to 8. For smal! values of V these factors deviate considerably. Introduction. Le According to van der Waals, the “ Inward Pressure” P, supports the pressure P from the exterior, and the total pressure a gas thus bears is ee ls W=P+P,;= V= 5? or considering P;= = a. eee waik ee Evidently II is made equal to the pressure of a “ perfect” gas of the same temperature at the volume (V—b). 24 Mr. J. Kam on van der Waals’ Equation and In the above equation Cw NS ogee whereas it will be remembered Po=5N.m.c?; N being the number of molecules per unit of volume, m the mass of a molecule, ¢ the velocity of mean square. The molecules themselves are considered to be mathematical points, and b (the volume of N molecules) here consequently is zero. In reality } has a positive and definite value and increases the number of impacts, or the pressure, at the rate 5 was shown by van der Waals. in An actual gas consequently exercises the same pressure Py at a greater volume ; 2.e. the same volume V, contains less molecules, say only Nj, having a volume #, smaller than 0. Per unit of volume we thus get a decrease of impacts, caused by the decrease of the number of molecules at the as rate a and an increase of impacts at the rate iss B? caused by the influence of 8, and the effect of the Foam is com- pensated by the latter. It is evident that the greater the compression, the smaller the number of molecules per unit of volume of an actual gas as compared with a perfect gas of the same pressure and temperature. oul We may call 4 the specific density of a gas; its value is greatest when the density is smallest, but always smaller than 1 ; it decreases in value as the compression proceeds. This circumstance not being fully considered by van der Waals’ equation, causes large deviations. II. If at the volume V and temperature T, N molecules of a ““nerfect ” gas exert a pressure Rw a then N molecules of an actual gas would exert this pressure at a volume 6=(V +45) and the same temperature, 6 being the volume of these N molecules. T= some New quations derived therefrom. 25 The volume V would contain of the latter Ne Nemolecutes, Yes 00) 0) Vv V+6 having a volume | Vv c= are ° wah ple ° ee Sane (2) The specific density referred to in the introduction is N, Vv v= N = Wag’ we aA alate PRY SNe gore at (3) and expresses the density of an actual gas with regard to a perfect gas at the same temperature and pressure II. For the perfect gas, Ral I= ai and for the actual gas at the same volume V, the aor L I= eae e e ° e ° ° e (4) As stated above, we must of course find Nh OEE ul RET ents ICT CORY KAD aE NY, Vv— er: b On the other hand we had Il=P+P;,. P,, the inward pressure, being inversely proportional to 1 a : : the density, 2.e¢. to Taare is equal to (Vb? if a is the constant at V=1; and instead of van der Waals’ equation, we find Ge hates oe aN oe Cig In this equation 6, the volume of N molecules, is naturally ! ] constant. The equation expresses that at the density 77, and the volume V, an actual gas ‘exerts the same pressure II as a perfect gas at the density aa and the same temperature. v 26 Mr. J. Kam on van der Waals’ Equation and Making c=v’a, we could write equation (A) in the form Cs Dey eee ee a \ Jae yi Voe? (B in which equation vy and 8 have the values given by (1) and (2) and thus are variables. Equation (B) naturally leads to the equation Ch ete a quadratic equation to be discussed later on. Equations (A) and (B) are cubic equations in V, having three roots. With regard to the volume they thus are of the third degree. With regard to the density, however, they are of the (C) second degree, the density at the pressures being 757 : Thus in equation (A), V on the left only applies to the volume, on the right to volume and inverse density, and the meaning is not entirely identical, and we could not write it in the form (making ¢=(V +0)) ‘rome 3 tet g? p—d’ as here we would have an equation of the third degree of the density, instead of the quadratic which (A) is. Neither could we write (A) in the form ea egal Eye a ae as the volume of the molecules contained by V is not b but only 8, which is a variable (vide equation 2). Still we can consider (A) of the third degree of V, the volume, and its curve thus as cutting a horizontal pressure- line in three points, denoting the three values of V. The dp dv towards each other as the temperature rises, till at the critical point they coincide. For this point we then have pee two points of the curve for which—~ =0 approach more dp 4 Balle Akl, PT ANT AUBES |: > GRATE? d*p —/() 6a i Seles Bee SPOT apian Tn mies whence we obtain some New Equations derived therefrom. 27 b here is the volume of N molecules exerting the same pressure at the volume (V,+0) as N molecules of a perfect gas at the volume V, (which of the actual gas only contains N, molecules). In accordance with (1) we find for N, V. Vida. i i) N we ep ay | and t (6) Np 2 | | SNE” ) and the volume of these N; molecules is Ve b 2 b 7 B= Wasep e — BI ° ° ° ° ° ° ( ) (5) and (7) give Wi 2 B= SSG PIS Varden Ae Substituting v and § in equation (4) we find Tye el sae ned ais el Wiad apr be V.—B-> 26—2b 626 7 COV, From (6) we deduce that the density at the critical state is for all substances 2 of the theoretical one, or 2 of the density of a perfect gas exerting at the same volume a pressure equal to the critical pressure at the critical temperature. This in combination with (8) may well lead us to suppose that the gaseous laws again hold good for the critical state. We shall see presently that this actually is the case. In an entirely similar manner we deduce from (B) i dp 0 Ze). SVs tuk bo. Vis (Ves By’ dp 0 6c 2p R Te aa? Vi (V-e—B)?? whence ieee aye Ao) wc jto a ee CE) a value in accordance with (8). We have already seen that equation (A) is not identical with van der Waals’ equation. It will also be seen that the usual solution of the latter by multiplying and arranging according to powers of V, cannot atford correct values of P, and P,.. For doing so, we con- sider = the density (or the pressure) to be the inverse value of V (the volume) which for an actual gas can never be, so long as } has a positive and definite value. 28 Mr. J. Kam on van der Waals’ Equation and Where the values i . aVZ= P, ° ° ei iia: some New Equations derived therefrom. 29 and thus Ty) Me leat ero btyrs rast hintani altaya) and a a P= on = (V,+ 5) Mee bn we can write (9) 2P,... V—h. T.. 1 er a) Le Ra a ata Mis ene SL which is the “‘ gaseous law ”’ for the critical state. Hquation (10) shows that critical “ Inward Pressure” and critical pressure are equal, a deduction obtainable in an entirely different and independent manner, as will be shown later on. We will, however, first introduce yet another correc- tion which, together with the preceding ones, leads up to interesting results. tenes ° f ° ° ° e e (10) III. Another factor influencing the number of molecules per unit of volume is the temperature. In equations (A), (B) and (C) we consider V as a fraction of the original volume at the temperature T’. If, however, V is given as a fraction of Vo, 1. e. of the volume reduced to 0° C. and 760 mm. pressure, it is necessary to introduce yet one other correction. At the temperature T and pressure Po= ls the number of molecules contained by Vo is — mi 273 (molecules of a perfect gas). Hence the factor of specific density v has to be corrected in the same manner, and we obtain for equation (A) PE yee One te Ee (Vo Woe or a 1 ECTS Ey and instead of (C) we obtain Aa ih P+ V2 = wv Nia eRe he Na Mt. Metts fis) Aettine (C1) 30 Mr. J. Kam on van der Waals’ Equation and At the critical state V-=26 and (A) becomes a il < 068 aE a quadratic equation, really a special form of (C,j. Writing it in the form | ft a ay ee S55 D 'gp egg we find . Lev bei une Eas ae) 9P, and as naturally has only one value, 2 b= 2 and { E he zs =; | 4P, 4P, Or or yh = OP, We thus get the two values . | ih . . 3 Po= oy or, 2P,V.=l=constang, =o and ; | | e== SSP 907 (V, +b)? The accuracy of equation (12) is demonstrated by the figures of the table on the next page, taken from “ Landolt- Bornstein,” Physikalische Tabellen, Berlin, 1905, pp. 181-186 The values marked * are borrowed from Handbuch der Physik, A. Winkelmann, vol. iii. Leipzig 1906, pp. 859-868. Equation (12), 2P,. V-=1 (constant) or . II,. V.=1 (constant), Il being the total pressure a gas bears, shows the interesting fact that the law of Boyle-Mariotte is true again for the critica] state. Naturally, if during the eourse of the experiment mole- cules have either dissociated or condensed, we must find deviation over and above such as caused by the errors of observation made in determining P, and V.. At the critical state, small changes of pressure cause large some New Equations derived therefrom. 31 changes of volume. I therefore selected all data of experi- ments affording corresponding values of V, and P,, as will be noticed. Values of V, and P, determined independently cannot be trusted to give reliable proof. Ve. Pe (exp.). P (calc.). RM PEACE occisl hae setecius vie qeinin= 0066 oT 11 79°77 CGN eeanc-ianwtanvsddcececsesete 00713 62°76 70 AVA TOPMAte ....0...s.000sese00- ‘01710 34°12 29:2 [D720 2: ES a ae Se ea 00981 47°9 50°9* Benzene chloride ................++ 01175 44°62 42°5* MOE EM inci cicaice vias sewn ditcan dbniccie's 01334 375 37°8 yl Acetate 2.2... deaesecsvewnes 01222 39°65 41 Bithyl butyrate ..................088 ‘01744 30°24 28°80 Ethylene dichloride ............ 00982 03 51 Hthyl isobutyrate...............06 01749 30°13 28°6* Ethyl propionate .................. 01482 33°86 33°75 Hsobutyl acetate ...:2c.5:..2...+. 01717 314 29°41 HeGbubiy) Or Mate! ...2../..000-s +2 01472 38°29 37°86 Methyl! acetate ...........c0-secees 0096 47-54 52 Methyl butyrate ...............06. 01455 36°02 34°4 Methyl ethyl ether ............... ‘00873 46°27 57 Methyl propionate ............... 01224 39°88 40°8 Methyl valerate ................6 ‘OL728 31°75 29 REGO WUACCEALE ooo. cnsec.en sence 01464 348 345 210, (LF clo) 60) ne 00968 50°16 51°3 Beep CMOLide © ..........0%, <0. ‘00982 49 50°9* PEOpyl formate...........120.-r6 01203 42-7 415 Pa sss siecigcisk pyitan sss» 003864 194-61 130 HME ae hee taic ven cnevecheo an 00436 775 114 BU oe one oobi: tactean ds yen nese 00587 78°9 85 PEM ee eae ss sche cetdereeccces cos ‘0066 77 76 PINE Os ache oc 5% ddidweweenciac'eun'e + 009011 72°868 60 We can, of course, also derive the same result? from equation (C;). For the critical state we have, multiplied and arranged : 1 Cc Dye Bik NG; P, Vet p, pe! a De ae pa.e Nees eee 7,F, At the critical state V, adopts only one value. Hence (0), 1 : Cc V.= Pie Wie — PY or if C SS and ~ /Pe==)i= == Pic. 32 Mr. J. Kam on van der Waals’ Equation and We here find the same values as found in (12) and (13) [a (c= y2, a=(v75) a. See equations A & B, pp. 25, 26.) Apart from being a point of the critical isothermal, the _ ¢ritical point thus appears to belong as well to two other curves of the equations : 2P..V.=1 (constant), and | Caine vite | The latter curve cuts a horizontal equal pressure-line twice, denoting the two values of V satistying the equation. P+ At the vertex of the curve for which a =( and the two values of V become equal, we have the critical point. Thus he orauldsendl for the erie ey: = ai 2¢ 1 Vena? or V. pose a a original equation gives us which value substituted in the ik or and in accordance with (12) and (13). It will be seen that the curve satisfying equation (C,) is formed by the line drawn through the points where the isotherimals below the critical temperature change into and from the horizontal straight line, and which is known as the “‘border-curve.” As the temperature approaches the critical temperature, the horizontal straight line gets shorter and shorter and the two volumes converge more and more towards the same value, ze. the critical volume, which js reached at the vertex of the border line. some New Hyuations derived therefrom. 33 The diagram roughly shows that the critical point is the point of intersection of three curves, 7. e. the critical isothermal, the border-curve, and the curve 2P..Ve=1 (constant). Fig. 1. I) quPz su0) = A” Critical point CO 30} Atm. We found equation (14) c= x Considering c=v?.a ° ; Lig (p. 26, equation B) and Ye ere 3) We find, as V,=28, t=), and Oo eiges Ge 0 3 Ve wh eine Via nti ern ties litte (15) We can deduce the same value from equation (A), p. 25, for the critical state. As we found a P.=P,= (V.+6)?’ we get ZOE (Ve+6)? EiiwiMeg S00. Vol.ol. No L8t. Jan. 1916. D 34 Mr. J. Kam on van der Waals’ Equation and and as V,= 28, 2a 1. 9b? 2b’ whence 9 9 = sage a Ver as found in (15). Substituting V, in equation (A), V.=2, we obtain 2a _R.T, Oo ion whence ee: T= GRR e ° ° e . ° (16) As b= 8 (equation (8), p. 27), Or NE c= 97 ° B.R 9 ° ° ° e e ° (17) which is the identical form for the critical temperature as found by van der Waals. Equation (9) p. 29, gives for T, the value Debye UNG ; = POEs a ° e oy, ls . (18) and as V,=3 8, we find in combination with (17) Oleic is a Bit Qe Bea: Al or P.= SAS a a leo oe a (Vitp?t in accordance with the prece ling. It may be of interest to compare the above with an experiment. In the case of carbonic acid, Andrews found for V, and P;: V.='0066, P,=77 atmospheres. We have, equation (15), p. 33, 9 a=. V.='007425, and equation (8), p. 27, b= 5 V.="0033. some New Equations derived therefrom. 35 Hence according to equation (10), p. 29, a 007425 (V.+0)2 9. (0033)? which is the same value calculated according to equation (12), p. 30, as shown by table, p. 31, and very nearly the pressure found by Andrews. For methyl acetate, V.=:0096, P,=47'54, we find in the same manner P,=52 08. For propyl alcohol, V, =-0098, P.= 50-16, we find Ee o1-65, and so on- For T., according to equation (16) or (17), we find for carbonic acid, T.=273°, as should be, considering the value of V.='0066 is the volume relative to the original volume re luced to 0° and 760 mm. The constant a for carbonic acid calculated above has a somewhat smaller value than the one van der Waals finds (a=°00874). The value van der Waals calculates for P, is (for CO.) 61 atmospheres. fe = 76 atmospheres, IV. On a future’ occasion we will show that the equality of P, and P,. can be proved in an entirely different and ‘independent manner. A natural consequence of this equality would be the disappearance of the phenomena of surface-tension and of ‘the latent heat of vaporization at the critical point. For ‘the cohesive forces would be compensated by the thermic pressure, and the transference of a molecule from the in- ‘terior of the fluid to the space over its surface would not require any work against a force. But we cannot imagine P,=P,,, and still measure an actual critical pressure P, towards the exterior, unless we assume the law of equality of action and reaction to apply to the cohesive forces, as to all other forces. Thus the surface molecules, attracted by the interior molecules, must exert the same attraction on the latter, equally strung but opposite in direction. Between the surface and the interior the thermic pressure would be supported by the cohesive forces, the density would be greatest at the surface and gradually diminish ‘towards the interior. If II were the pressure without any molecular attraction throughout the fluid, then the pressure from the interior D2 36 On van der Waals’ Equation. would be supported towards the surface and diminished in the opposite direction, with the value of the attraction P,, if there were any such attraction,—z. e. the pressure in the surface-layer would become IJ,=II+P,, and the pressure at the surface and in the interior would become P=II—P,, as illustrated by our diagram (fig. 2). In particular, if Pe=P, we should thus have = P.+ = 2P = CA ae or P= P,,=4ll. Fig. 2. P=L-P, PNP No matter how great P,, the pressure in the interior must. be transmitted undiminished towards the surface. But the pressure between surface and interior is increased and attains at the critical state (thus for P,=Pj,) a value her + Ee ole The force opposing II, is identical with the “ Intrinsic pressure ” of Laplace in the case of liquids. Instead of causing an increase of pressure in the interior, the cohesive forces are the means of such increase between surface and interior equal to their numerical value, and of a decrease of pressure in the interior of that equivalent. This, however, does not affect our equations (A) and the following. For the gas-pressure P is equal to the pressure II the gas. would exert if there were no molecular attraction, minus the value of that attraction P,, 2. e., P=II—P,, or COM alates b WED Additional evidence in support of this view will be: orought forward on a future occasion. P+ bem ol III. On the Variation of Surface-Tension with Temperature. By AtuLan Fereuson, D.Se.(Lond.), A ssistant-Lecturer in Physicsgin the University College of North Wales, Bangor*. ot ranges of, say, twenty or thirty degrees the effect of temperature on surface-tension can be expressed with considerable accuracy by a linear formula of the type Ge ean) NAVA aye Wee) For wider ranges such a formula cannot be used with even approximate accuracy, and formule of the type Dm eee BBR Y eat ena have been used to represent the experimental facts. Such formulee, however, will not bear extrapolation, over even a mederate range of temperature, and if, in particular, they be applied to estimate the critical temperature by calculating the value of @ for which T vanishes, they invariably give wildly discordant values for 6,. The directly determined values for the critical tempera- tures of liquids are comparatively few in number, and it is a matter of some importance to be able to estimate the critical temperature of a liquid to within a degree or two; for, amongst other things, the problem of tracing relations between surface-tension and chemical constitution has been considerably obscured by a habit of making comparisons at the same temperatures. Schifff recognized that better results were to be obtained by making comparisons at “corresponding temperatures,” and chose the boiling-point at atmospheric pressure for the purpose of comparison. This, of course, implies that the ratio of the boiling-point to the critical temperature is a constant for all liquids when temperatures are measured on the absolute scale, and, as I have shown in a recent paper {, this is by no means exactly true, as the ratio is markedly affected by constitutive influences. It is very necessary, therefore, to have some means of estimating the critical temperature of a liquid with fair accuracy, and, as will be shown later, this estimate may be * Communicated by Prof. E. Taylor Jones. + Liebig’s Ann. cexliii. p. 47 (1884). t Phil. Mag. April 1915, p. 599. 38 Dr. A. Ferguson on the Variation of made (for unassociated liquids) by observations of the tem- perature-variation of surtface-tension over a comparatively limited range. Some little time ago I pointed out that the equation T=T,(1—b6)* . . (ices represented the surface-tension of benzene with considerable accuracy between 0° and the critical temperature*. Such a formula, if of wide application, would be very valuable, as, in addition to its use in obtaining the surface-tension of a liquid at any temperature over a wide range, it gives the critical temperature at once from the equation a Ke a!) Se Wile us ite ttnine anaes (iv.) and so enables us to make a comparison of surface-tensions at any desired corresponding temperatures. There are very few formule which represent the effect of temperature on surface-tension over any wide range, but I have recently noticed that Van der Waalsft has put forward a binomial expression T,= AG? p? (1—m)®, .. i.e) where m is the “reduced” temperature, and A and B are constants for all bodies following Hotvés’ law. This equation represents the experimental facts very accurately, but, demanding as it does a knowledge of the critical data, it is hardly so suitable for our purposes as (iii.), which is put in a form which enables one to calculate the critical tempera- ture from observations of the surface-tension alone. More- over, as will be seen later, {111.) fits the experimental data more accurately than (v.). The manner in which it was shown that (i11.) represents the experimental results may be of interest. Differentiating (ili.) with respect to 0, we have 66 dT | d@ == bnT (1 has Op) ame e e e e (vi.) and therefore dé 1—06 it Ti int * Science Progress, Jan. 1915, p. 445. Tt Zeit. fiir Phys. Chem, xiii. p. 716 (1894). Surface- Tension with Temperature. 39 So that if — be denoted by y, the relation between y and @ is linear. was calculated over ranges of about 20° C. dé dT by sunply taking the differences of 6 and dividing by the corresponding differences of T. ie was then plotted against 8, and in all the unassociated liquids examined the eraph between these quantities was very accurately linear ; the values of 6 and of n were read off directly from the graph. The following table * shows the results for the fourteen liquids examined. TaBeE I. (Showing values of b and n in the formula T=T,)(1—08)"). Substance. n, b. 6¢-= : ‘ Qc obs. Diff. Lucas 2 ee 1:248 | -:005155 1949 193°°8 | +0°:2 [PT7 Gh ae 1:218 | 003472 288 288°5 —0°5 | Chloro-benzene ............ 1:203 |:002793 358 359°2 —1:2 Carbon tetrachloride ...| 1:206 |°003553 281°5 283°1 —1°6 Methyl formate ......... 1-210 | ‘004695 213 2140 —1:0 Methyl acetate ............ 1°200 | -004274 234 233°7 +0°3 Methyl propionate ...... 1:202 |:003891 257 257 °4 — 0-4 Methyl butyrate ......... 1:195 |:003559 281 281°3 —0°3 Methyl iso-butyrate ...... 1:228 |:003731 268 267°6 +0°4 Hinyl formate .<.......... 1-187 |:004255 235 235°3 —0°3 Hthyl acetate ............ 1:217 | :003984 Daa a 250°1 +0°9 Ethyl propionate ......... 1192 | -003663 273 272°9 +01 Eropy! formate ...........- IQ | 003774 265 264°9 +01 Propyl acetate ............ 1:204 | :003623 276 276°2 —0-2 Meau value of »=1°210. It will be seen from the above table that the critical temperatures are given from the surface-tension data alone with a surprising degree of accuracy. It will also be noticed that n varies very little from liquid to liquid. Its mean value is 1:210 and, while it is of course preferable to determine n for each liquid separately, this value may be used with some confidence in calculations referring to any other liquid—always provided that it is unassociated—as it happens that, as a simple calculation will show, a small * The experimental values of T were taken from Ramsay & Shields, Phil. Trans. 1898, p. 647 and Ramsay & Aston, Proc. Roy. Soc. lvi. p- 162 (1894). 40 Dr. A. Ferguson on the Variation of variation in n, at moderate temperatures, has not. much gitect on the computed values of T or of 6. The figure, which is drawn to scale, shows a few of the curves obtained by the process outlined above, and their close approach to parallelism serves further to emphasise the smallness of the variations in n. iY ToT ll METHYL FORMATE il METHYL ACETATE sob L | IV ETHYL ACETATE S V BENZENE ¢ {00 The accuracy with which equation (iii.) fits the observed values for the surface-tension is sufficiently shown by Tables II. and II. below. For convenience of comparison the results obtained by the use of Van der Waals’ equation (v.) are also given. TaBiE IT. Benzene {T=T,(1—-003472 0)!218}, ALE, | V.d. W. 6 (Cent.). T obs. T cale. Diff. T cale. Diff. OF ne 30°28 my sah Ne 80 20:28 20°36 + 0:08 20-39 +0°11 100 18-02 18:01 —0-01 18:01 -—001 120 15°71 15°71 +0-00 15°69 —0-02 140 13°45 13°46 +0°01 13.43 —0:02 160 11-29 11°28 —0:01 11-24 —0°05 180 9°15 9°17 +0-02 9:13 —0:02 200 (aly 715 — 0:02 711 — 0-06 220 5°25 5:22 — 0:03 5-19 —0:06 240 3°41 3°41 +0:00 3°39 —0°02 260 1°75 1-77 +0:02 1-76 +0°01 280 0:29 0°39 +0:10 0°40 +0°11 —— ’ ’ Surface- Tension with Temperature. 41 TasueE III. Carbon tetrachloride {T=Ty(1—-003553 0)#208{. A. FB, V. d. W. 6 (Cent.). | T (obs.). | T calc. Diff. T calc. Diff, 0° ve 28-02 ne Suh be 20 2568 25-63 —0-05 25°80 +0:12 80 18-71 1872 | +001 18°77 +0:06 100 16:48 16:50 +0:02 16°53 +0:05 120 14:32 14-33 +001 14-34 +0-02 140 12:22 12-22 +0-00 12:21 —0-01 160 10:22 10°17 — 0:05 10°15 —-0:07 180 8:26 8:19 —0:07 8:16 —0°10 200 6°34 628 | —0-U6 6:26 ~ 0-0£ 220 4-47 4-47 +0:00 4-46 —0-01 240) 274 2-78 +0:04 279 | 40:05 260 1-20 1:26 +.0-06 1:20 +0:10 ‘The above tables show that, whilst equations (111.) and (v.) both fit the experimental values very closely, equation (ii1.) is distinctly superior in point of accuracy. On the average, the differences between the observed and calculated values given by equation (iii.) are about half those given by equation (v.). | Several attempts have been made to express the critical temperature of a liquid in terms of the temperature coefficient of its surface-tension, perhaps the most widely known being that given by Walden*. Assuming a linear law for both T and a? (¢= = and writing them in the form T=T,)(1—2@) and a? =a,"°(1— K@), he finds empirically, from a consideration of the observed values of «, K, and @,, that aCe Pe. ANG ING — O04) ha) 3) SACyrid) As a matter of fact 1°16 and 0 94 are the means of rather variable numbers, and the values of @, given by these equa- tions are not very close to the true values, although the mean of the two values given by equations (viii.) is usually better than either value taken separately. Table IV. below, which gives the data for some of the substances discussed in this paper and is extracted from a long table given by Walden, serves to show the kind of agreement that may be expected. * Zeit. Phys. Chem. \xv. p. 129 (1909). 42 Dr. A. Ferguson on the Variation of TABLE [V. Subsea. K, | vas eotnne He. —- Mean.| Ether...... Pe OM OSG ‘00603 |:00624; 182° 192° | 187° | 194° Methyl formate ...... 00435 |'00553 |'00526| 216 210 2138 214 Hthyl] acetate...:........ ‘00392 |:00463 |00474| 240 250 245 250 Carbon tetrachloride .|‘00337 |'00418 |-00408| 279 278 278'5| 283 IB ONZENE 52 <0b ch oteeemeee 00329 |:00416 |00398| 286 279 283 288°5 Chloro-benzene......... 00266 |00311 |-00322, 352 373 | 363 4) 3a9 Ethyl propionate ...... 00331} ... |:00400} 284 .. | 284 | 273 Bromine eee eee °00298 |:00381 |:00361 | 315 305 310 302 The meaning of the column headed 1:21 K—which is an addition to Walden’s table—will be explained later. It is evident that the agreement between the numbers in the last two columns is only approximate, and equation (ill.) 1s so much more exact in use, that it removes any further necessity for the employment of the very approximate equations (viil.). The reason for the variability of the “constants” of equations (viil.) is not far to seek. The coefficient @ is not really a constant, but is a function of the temperature @. If we expand equation (iii.) we see that, very approximately, and for low temperatures Ci N— Ls where 0 is accurately the reciprocal of the critical tempera- ture. We thus obtain aO,=1°21, .. Soe in fair agreement with Walden’s empirically found relation aOc=:1:16. Neither of these expressions, however, can be considered as anything but a rough approximation on account of the neglect of squares and higher powers in the expansion oe (OUT Ramsay and Shields have remarked that for many sub- stances capillary-rise is approximately a linear function of the temperature right up to the critical point. In which case we have heh(t=60), 0.) ee and therefore, remembering that at ordinary temperatures (and assuming a zero contact-angle) we may write a?=rh, we have Surface-Tension with Temperature. 43 Hence it is evident that Ko. or. wey, 1 again in fair agreement with Walden’s equation K@,=0°94. Hence, very roughly, the relation between the coefficients a and K is a=nK=1°21 K. The values of 1:21 K have been calculated and placed in the fourth column of Table [V., and it will be seen that the agreement is as close as one could expect. The reason, therefore, for the variability of Walden’s results seems to lie in the fact that the coefficient « is a function of the temperature, while the assumption of proportionality between a and the reciprocal of the critical temperature demands that « shall be independent of the temperature. In fact « is not rigorously equal to nb but is given by and the remaining terms of the expansion are quite appre- ciable even at moderate temperatures. It may then be consicered that for unassociated liquids equation (iii.) holds with very considerable accuracy, and may safely be used to estimate critical temperatures, even if the observations of surface-tension have only been made over a limited range of temperature. But it must be remembered that in all the cases considered the surface-tension measured is that of the liquid in contact with its own vapour, and it remains to be seen whether the formula holds for liquids in contact with air. In most cases the surface-tension of a liquid in contact with air differs very slightly from that of a liquid in contact with its own vapour, and this difference would hardiy be expected to be sufficient to invalidate the formula. Unfortunately the data at my disposal for capil- lary-rise in presence of air are somewhat scanty and scattered, and I have therefore restricted myself to a discussion of the numbers furnished by the careful experiments of Mr. J. L. R. Morgan* on the drop-weights of various liquids. As- suming proportionality between drop-weights and surface- ¢ension for drops falling from the same tip, we have Dt Ug (CEO) A Bien has aes EX) * See, in particular, Morgan & Higgins, Journ. Amer. Chem. Soc. xxx. p. 1055 (1908), and Morgan & Schwartz, xxxili. p. 1041] (1911) in the same journal. 44 Dr. A. Ferguson on the Variation of as the equation to be tested. In comparing experiment with theory I have, for simplicity in calculation, assumed in every case that n has its mean value 1°21, while for the value of 6 I have simply taken the reciprocal of the known critical temperature. Values of w (the weight of a single drop at 6°) were then calculated from (xi.) and compared with tho observed values. Better results would probably be found | by calculating values of 6 and of nm for each substance separately, but the agreement as shown in Tables V., VI., and VII. is sufficiently good to prove the validity of the formula. TABLE V. TasBLe VI. Benzene. Methyl formate. b=-003466, w,=-03699 gm. _ b=:004673, w)='02991 gm. 0. wobs. | weale. | Diff. é. w obs. | weale. Diff. 11°-4| 03524 | 03523 |—-00001 6°°7 | 02886 | 02878 |—-00008 30°2 | :03235 | :03235 |+:0000 10:0 | (02824 | -02822 |-—--00002 53:0 | °02887 | 02894 |+-00007 16-4 | -027386 | :02716 |--00020 | 68:5 | 02653 | 02664 |+-00011 27°8 | 02561*| 02527 |—-00034 Taste VII. Chloro-benzene. b='002784, wy='04226 om. 0. w obs. w cale. Diff. Cow "04108 04110 + ‘00002 39:2 "03663 03676 +:00013 50°8 08499 03514 +:°00015 63:9 03320 03334 +:00014 72:2 "03205 03221 | +-00016 It remains to discuss the effect of composition and consti- tution on the values of 6 and of n. The variation in n is * Misprinted in the original paper (Morgan & Schwartz, /. c. p. 1047) as ‘02661 gm. The formula served to detect the error, as the difference seemed to be too great to be accounted for by errors in either formula or experiment. Recalenlating from the fundamental observations given in the paper, the mistake was found to be due to a slip in the arithmetical work, Surface- Tension with Temperature. 45 very small, and as far as can be seen from the figures in Table I. is quite irregular. As errors in setting the ruler have a marked effect on the value of n, it is quite possible that, had the experimental results been treated by some more exact method, the variations in n would show regu- larities not here apparent. But the labour of such calcula- tions would have been out of all proportion to the slight gain in accuracy, and for our purposes—tlie accurate repre- sentation of the temperature-variation of surface-tension, and the calculation of the critical temperature from surface- tension observations carried out at ordinary temperatures— it is sufficient to note that the variation in n is very slight, and that the mean value of n 1s about 1°21. As far as 6 is concerned, the discussion practically resolves itself into a consideration of the effect of constitution on the critical temperature. Table I. shows clearly that in the case of the esters R.COO R, / diminishes—~. ¢., the critical tempe- rature increases—with increase of K or R, (the other radicle meanwhile being supposed constant). Also the value of b for an iso-compound is greater than the value for the normal ester—1. e., the critical temperature is lower. Agam, R and R, are not mutually interchangeable. For example, we have CUE 2COOCEH, herve 1b ="003891 CE eCOOC IEE «20st 341) b= 0038984 and | Ola OOOO Ela) 2) es) b=" 003099 ie OOOO sm Ms cat) b= O0ab2o In each case the transference of the more complex radicle to the carboxyl group increases the value of b; the effect is, however, much less pronounced in the second case than the first, and probably decreases with increasing complexity of Rand R,. If we denote by y the ratio of the absolute critical tem- perature to the absolute boiling-point at normal pressure, I have elsewhere shown that for the normal paraffins * yn =h, where g and hf are constants, and n the number of carbon atoms in the molecule. The experimentally observed values * Phil. Mag. April 1915, p. 602. 46 Variation of Surface- Tension with Temperature. for y for the esters here examined are given in Table VIIL., and it will be seen that as R, increases (R being constant) ry steadily decreases, and that a similar result holds good for increase of R (R, being constant). Tape VIII. i 9c Substance. Pile 9c obs. Tg ee | Methyl formate ...... 304-9 487-0 1597 Methyl acetate ......... 330°2 506°7 1535 Methyl! propionate ... 352°7 530°4 1-504 Methyl butyrate ...... 375 8 5543 1°475 Ethyl formate ......... 327°3 5083 1°553 Ethyl acetate ......... 39071 523°1 1-494 Ethyl propionate ...... 371°8 545°9 1-469 Propy] formate......... 353'9 537°9 1°520 Propyl acetate ......... 370°6 6492 1-467 | The data are somewhat too few in number to establish quantitative laws with much confidence, but it may be noticed that for the series RCOOCH, the relation between ry and n (the number of carbon atoms in R) is very accurately Jinear, leading to the result ey=d—n, ).) 4). .) a eee where ¢ and d are constants. In this case d=52°17 and c=33'°33; the agreement between the caleulited and ob- served values of y is shown in Table [X. From this result TasuE IX. n Y Y ae ay i observed. calculated. | (from xiii.). | (from iii.). | 0 1:597 [1565] [004897] ‘004695 i 1°535 1-535 004276 004274 2 1:504 1°505 ‘003879 003891 | 3 1-475 1°475 0033555 ‘003559 | it easily follows that the temperature coefficient of surface- tension—b in equation (iii.)—is connected with the absolute boiling-point of the substance considered by the relation Pete Be ee On Discontinuous Wave- Motion. 47 Columns 4 and 5 of Table LX. show the agreement between the values of 6 calculated frum (xiil.) and those given directly from the surface-tension observations. It is very probable that with more extended data the form ot this relation between @ and 6 will be altered, but the fact remains that there is a definite relation between the absolute boiling- point of an unassociated liquid and the temperature co- efficient of its surface-tension, so that this temperature coefficient may be calculated from observations of the boiling-point alone. It may be remarked in conclusion that associated liquids so far as I have tested them do not agree with equation (ii1.) —in fact, agreement with (ii1.) may form a very convenient test of non-association. I hope to return to this point later. University College of North Wales, Bangor. J uly 1915. IV. On Discontinuous Wave-Motion. By ©. V. Raman, M.A., and 8S. APPASWAMALYAR™. (Plate I.] N analytical discussion of the principal mode of vibration of a bowed string as ascertained from the form of its vibration-curves, leads to the result that at two epochs in each period of vibration the string should pass as a whole through its position of statical equilibrium, alternately in opposite directions. At the first epoch, according to the analysis, the veiocity at every point on the string is propor- tional to its distance from one end, there being a discontinuous fall of the velocty to zero at the other end. At the second epoch the state of matters is reversed, the motion being such that the velocities are proportional to the distances from the farther end, the discontinuous fall being at the nearer. ‘These results are of importance as defining the essentially discontinuous nature of the motion involved, but it is noticed from the literature of the subject that they are obtained by an elaborate and indirect analytical process from the observed form of the vibration-curves, and have not so far received direct experimental confirmation. Some time ago it occurred to one of us that an experimental test was * Communicated by the Authors. + See Riemann and Weber’s ‘The Partial Differential Equations of Mathematical Physics,’ pp. 216-223. 48 Messrs. C. V. Raman and 8. Appaswamaiyar on not impracticable and would be of considerable interest, as affording a direct demonstration of the discontinuous distri- bution of velocity indicated by the analysis. The present paper deals briefly with the method adopted and the results. The mode of vibration of the string is evidently determined by its configuration and its velocities at either of the two epochs referred to above. If, therefore, it is possible directly to impose at every point of a finite string in its position of equilibrium the initial distribution of velocities with a dis- continuity at one end indicated by the analysis, the resulting free oscillations should have the same characteristic vibration- curves as a bowed string. We have found it possible successfully to realize these conditions in experiment by arranging that a stretched string has initially a uniform angular velocity about one end, and that in the course of this motion one point on it impinges upon and is suddenly brought to rest by a fixed stop or bridge provided for the purpose. The length of the string between the fixed extremity and the bridge is thus isolated, and photographs of the vibration-curves of the resulting motion are secured by the following device. By setting a narrow slit across and immediately behind the string in any desired position, and illuminating the slit with the light from an electric are, any given point on the string can be caused to record its motion photographically on sensitive paper contained in a dark slide which is caused to move in a direction parallel to the string, 2. e. vertically downwards, with uniform velocity behind the illuminated slit. The necessary movement of the string itself is secured by drawing it to one side together with the weight attached to its free end, and allowing it to swing down in the manner of a pendulum before one point on it comes up against a fixed stop, which ig placed about three- fourths of the way down between the upper fixed extremity and the free end. The shadow of the string across the slit records itself on the photographic paper as a white curve on a dark ground. Six records obtained in this manner are reproduced in Plate I. One of the points of observation chosen was the centre of the string and the others were on either side of it. It will be noticed that the records show the motion at each of the points of observation, both before and after the first impulse set up by the impact reaches it. The velocity at every point in the initial motion is exactly the same as the velocity of the upward motion in the vibration-curves. The form of the vibration as shown by the records evidently Discontinuous Wave-Motion. 49 reproduces that of a bowed string in a very perfect manner, the discontinuous changes of velocity being clearly shown. Though the motion is a free oscillation, it remains practically unaltered in form for a considerable number of periods, and the experiment can thus be readily projected on the screen, and is suitable for lecture demonstration. — We shall now brietly consider the theory of the experiment described above from two distinct points of view. First, by application of the Fourier Analysis: Taking the originally fixed end as the origin (e=0) and the position of the string at time t=0 as the axis of x, we obtain the expansion y =a, sin 7 sin = + dy sin i sin = + &e., i in which the cosine terms are entirely absent, y being equal axe J js can) to zero when £=0. ‘The values of the coefficients a,, a, Ke. have to be found from the initial condition (<4) — wu. #=0 By expanding @x in a series of sines, differentiating the axpression for y, and putting t=0, we obtain () eet + &e. ) ae pa i _ 2lo —2 (si sin Shoe - sul &e.). The values of the coefficients may now be written down : nile T 1 loT ae A amt? and soon. Finally we have l ey (a8 an on sin = sin a m= To find the character of the motion expressed by the series, we have to effect its summation. This is best done by _ the series and writing it in the form =3 (— iy lo > | sin nn (T aE 7) +sin na (7+ ia 7) |: a Mag. 8. 6. Vol. 31. No. 181. Jan. 1916. 90 = Messrs. C. V. Raman and 8. Appaswamaiyar on The two series into which the expansion has thus been split up may be summed independently and then added together. s di es e It will suffice to trace the value of “” over times ranging dt from t=0 up to t=T, as the values subsequently repeat themselves. At any given point a» on the string, the velocity es aan at its initial value wx, from t=0 up to Rha ae the time t= Ba value —w(l—z), at which it remains up to the time Tl+2, 2 value w,, which it retains up to the end of the complete period T. From these values the configuration of the string may be constructed, and is seen to consist of two straight lines meeting sharply at a potnt which travels with uniform It then suddenly changes to the = The velocity then suddenly regains its original velocity = along two paravolic ares situated one on either side of the string. As the edge or angle in the configuration of the string passes over any point on it, the velocity at that point suddenly alters by the quantity Im. This sudden alteration of velocity is evidently due to the resultant of the forces acting on the element of the string over which the edge passes being infinite in proportion to its mass, and in this we may trace the propagation of the impulse originally sent out by the impact on the bridge. Secondly, by the geometrical method: Since the impulse set up by the sudden stoppage of the motion at the bhiidge should evidently travel with the ordinary velocity of wave- propagation, the character of the motion can be found from purely geometrical considerations without the aid of the Fourier Analysis, and this is really the more instructive method of considering the problem. The solution of the equation of wave-propagation on an infinite string not subject to damping is y=f («—at)+F(«+at). Differentiating with respect to time, we have = = —af'(e«—at)+aF'(a#+at). If the two terms on the right-hand side of this equation are periodic functions with wave-length equal to 2/, and ure so Discontinuous Wave-Motion. RE related that at the two points =0 and w=J, has always zero value, the velocity at any point on the finite string in the actual case can be found for any instant during the vibration by summation of the values of the two functions. At time t=0, in the experiment described, the initial dis- placements of the string from the position of equilibrium are everywhere zero, and we may therefore take half the initial velocity at each point for the positive velocity- wave -and the other half for the negative velocity-wave, and the ‘two waves my be constructed in the manner shown in the figure below. og Pees. By superposing the waves after shifting them through -equal distances in opposite directions, the character of the motion at every point on the string can be found by inspec- tion, and the configuration of the string can be found from the known velocities and the times during which they subsist. ‘The geometrical construction shown in the figure emphasizes the fact that the case is essentially one of the propagation of -a discontinuous wave. The experiments described in this note were first made at the Presidency College, Madras. The Indian Association for the Cultivation of Science, Calcutta, 27th August, 1915. EK 2 [ 52 ] V. A Note on the Relation between the Thermal Conductivity and the Viscosity of Gases with reference to Molecular Complexity. By J. A. Pottock, D.Sc., Professor of Physics in the University of Sydney *. [* the equation k=fncv, expressing the thermal conduc- tivity of a gas in térms of the viscosity and specific heat, the coefficient 7 is a numerical factor which is approx- imately constant for gases of the some atomicity. Such a fact suggests the probability of a relationship between 7 and y, the ratio of the specific heats. But long before the result, just mentioned, was fully established, the probability of 7 being a function of y was recognized, though it was not generally appreciated. As early as 1576 Bolizmann f, from theoretical considerations, obtained the expression f=3f'(y—1)/2, where 7’ is the constant for monatomic gases. It has been known for some time that the equation is physically inaccurate, but the matter does not seem to have been followed further. Recently new results for the thermal conductivities of a number of gases have been published by Hucken f. In con- nexion with these measures, Hucken discusses the dependence of f, not only on the properties of the molecule, but also on the temperature. As possibly lying outside the main lines of his investigation, he does not consider the relationsip of 7 to y, but, from the zero temperature values of the thermal con- ductivities and viscosities given by him, a relation appears to exist between the two factors which can be expressed by an equation of the form f= aly— 1) ry” where a and n are constants. The precise arithmetical adjustment of these constants may well await further measures; in the meantime, with numerical simplicity as well as physical accuracy in view, the equation may be written 7°32 (y—1 f= =o. If, in the original expression, n, the power of y, is put * Communicated by the Author. Read before the Royal Society of N.S. Wales. + Boltzmann, Pogg. Ann. clvii. p. 457 (1876); see also Schleiermacher, Wied. Ann. xxxvi. p. 346 (1889); and Chapman, Trans. Roy. Soc. cecxi. A. p. 433 (1912). t Eucken, Phys. Zeitschr. xiv. p. 324 (1913). Thermal Conductivity and Viscosity of Gases. 53 equal to unity, the equation, with an appropriate value of the constant, quite well represents the experimental results with the exception of those for the monatomic gases. This leads, in the case of perfect gases, to the simple relation mil = constant, 20 where m is the molecular mass. In the following table, with the zero temperature measures of kand y, taken from Eucken’s paper, I give, for the cal- culation of f, the experimental results for ¢ and y instead TABLE I. 1 2 3 4 5 6 Molecule. m. Te AOE |S mse LOS: | Cp. Y: 2 ea 4 336 1876 | 1:260* 1-63 J 40 39:0 210-2 | || 0-128 1-667 a 2 397 85 3°422+ 1-402+ me st. 28 566 167-6 | 0:2429t | 1419+ [ae 32 57-0 1922) O20 78 | "14024 BT cinta. 29 566 VL) | 028764) in 1-405 ss... 71 18-29 193-7 || O-1l5 1-323} Oa 28 5425 1672 | O-2502* | 1401 Brees. 30 555 1794 | 0-232 1-394, Be... 34 30°45 184 | 0-245 1340 See... gees 33°7 138 | 02010 1-300 ON seeds... | 44 35°15 186-20) 02K8 1-324 BW oe eeese.s-. 64 19°5 1183 | 0-1544¢ | 1-256 Pe a )a oe. | %6 16°15 92:4 | 0160 1-239 eee 17 51°35 92-6 0°520 1336 oe 26 44-() 943 h 1:26 GA es) e.3 +0. 16 71-45 1029 | 0591 iit ee Laas 40°7 99°66 | 0-404 1-264 as P30 42-6 SbSiAL Me tke: 1-22 * Scheel and Heuse, Ann. d. Physik, x1. 3, p. 473 (1913). + Escher, Ann. d. Physik, xiii. 4, p. 761 (1913). t Landolt-Bornstein Tabellen, All other values in columns 5 and 6, from Kaye and Laby’s Tables. of the values of c,. The figures given for f in column 2 of Table II., deduced from the equation k=fye,/y, are thus wholly dependent on the results of experiment. In this second table, the values of 7, derived from the expression f=7°32(y—1)/y'*, are entered in column 3. An idea of the physical accuracy of the calculated results may, therefore, be obtained from a comparison of these figures 54 Thermal Conductivity and Viscosity of Gases. TaBue IT. 1 2 3 4 J obs. F eal: “4 k Molecule. > (0.02 en | ee ky 782 (y—1) No No&p yi Eke or ees ae 2°32 2°45 11-6 eae alee) 2°51 O51 12:4 STAB SEL 1:91 1:90 S28 ING decade aa 1:96 1:93 13-4 OR PALES iat: 1:91 1:90 13:3 DSO Mesa 1:96 ch 135 Ole dg eas Ni eLeZ0 1-63 13°8 COM aI): 1°82 1:89 12:7 WOME io unain | 1:90 1:87 12-9 ET Sith Pate) 1-41 1:70 11-7 COMMUN ea 1°58 1°56 14-0 NOM ine 1:60 1-65 15-0 SOM a ann ses: 1:35 1:39 13:3 CREA aine: 1:35 1:32 165 Ose oN ea 1-79 1:69 126 CUE nee a ee 15°3 GEES HORE Sv, 1:52 1-61 146 CEI pete ee vs: | 1:40 1:43 159 Ghee ue in 18°3 with those in column 2. There are certainly large differences: between the calculated and observed values for some gases, but the experimental determinations cannot be considered in all cases as final. The last column of Table II. contains the values} of meyko/n. As previously mentioned, with the present experi- mental results, constancy of the value of the ratio is only to be expected in the case of perfect gases with molecules of an atomicity greater than 2. The approximate similarity of the figures in some number of instances is, therefore, perhaps more remarkable than the divergencies in the other cases. It is interesting to note the rise that has taken place in the values of the thermal conductivities. For many years the determinations of f for diatomic gases were cited in support of Meyer’s well-known theoretical deduction, f=1:6027. Now, from KEucken’s measures of the thermal conductivities, the value of f for these gases is 1°9. The University of Sydney, September 20th, 1915. a1 Ou Lt jE] VI. Some Observations on the Absorption Spectra of the Vapours of Inorganic Salis. By H. J. Hvans, B.Sc., A.R.C.S., Lecturer in Physics, Victoria University, AMlanchester*. [Plate IT.] INTRODUCTION. OME time ago the present authort published investiga- tions on the absorption spectra of the vapours of iodine, bromine, selenium, and tellurium at various temperatures, and later, as a continuation of the above investigations, the absorption spectra of the vapours of certain simple salts were examined. ‘This research was not completed, as the author’s attention was diverted to other problems, and the object of the present paper is to give a brief account of the results then obtained. It is well known that the absorption spectra of the above- mentioned elementary substances show the presence of well- defined absorption bands, and it was considered of some interest to determine whether the absorption spectra of the vapours of a few inorganic salts show the presence of similar bands. For this purpose the absorption spectra of the chlorides of ammonium and mercury, and the chloride, bromide, and iodide of cadmium were examined. It was found that all the vapours examined, with the possibie exception of ammonium chloride, showed evidence of a general selective absorption in the ultra-violet, and in no case could it be definitely proved that the vapours of the salts gave any well-defined absorption bands similar to those of Cl, Br, and I. These conclusions refer to the particular region of the spectrum (Xr 2500-A 6700) which was investigated. No measurements of the variation of the general absorption with wave-lenoth were made. EXPERIMENTAL ARRANGEMENT. A weighed quantity of the salt under investigation was placed in a quartz tube, which was evacuated to a low pressure through a side-tube connected to a mercury pump. The side-tube was then sealed off in the oxy-bydrogen flame. The quartz tube was afterwards placed at the centre of an * Communicated by Sir EK. Rutherford. This paper formed a portion of a thesis approved for the D.Sc. degree by the University of London. + Astrophys. Journal, xxxii. pp. 1-16 (1910); xxxii. pp. 291-299 (1910) ; xxxiv. pp. 277-287 (1911); xxxvi. pp. 228-238 (1912). 36 Mr. E. J. Evans on the Absorption Spectra electric furnace wound with nichrome wire, and the tempera- ture could be adjusted to any value between that of the room and about 1200° C. The temperature was measured by a Pt, Pt—Rh thermocouple, which was connected to a direct reading instrument. When the required temperature had been attained, light from the positive pole of the carbon arc was passed through the vapour inside the quartz tube, and focussed by quartz lenses on the slit of a concave grating, having a radius of 1 metre, and ruled with 15,000 lines to the inch. The spectrum was then examined both visually and photographically. EXPERIMENTAL RESULTS. Ammonium Chloride. It is well known that the vapour of this substance is dis- sociated into NH; and HCl at 350° C., and consequently the pressure of the vapour inside the quartz tube is double the amount calculated from the ordinary gas equation M fo = 18 mo where) Ri= 8:2 x 107, = absolute temperature, m = molecular weight of vapour, M = mass in grams of substance in tube, v = volume of tube in c.c. From the dimensions of the tube it could readily be deduced that if 017 gram of NH,Cl was placed in the tube, the pressure of the vapour at a temperature of 700° ©. was approximately 2 atmospheres, and in the experiments under discussion this quantity of NH,Cl was used. Photographs of the absorption spectrum were obtained at temperatures varying from 250° C. to 550° C., and visual observations were also made. As the result of a large number of experi- ments, it was concluded that the vapour of ammonium chloride does not give any well-defined absorption bands in th- region extending from A 2500-A 6700, and, furthermore, there is no definite evidence that the vapour shows any general absorption in the same region. It is, however, possible that the vapour absorbs in the extreme red and infra-red regions of the spectrum, which were not examined in this research. of the Vapours of Inorganic Salts. 57 Mercuric and Mercurous Chloride. Mereuric chloride is volatile even at ordinary tempera- tures, and its vapour pressure increases rapidly with tem- perature from 20°7 mm. at 200° C. to 370°7 mm. at 2.0° C. In the present experiments *10 gram of mercuric ciuloride was placed in an evacuated quartz tube, and by means of the gas equation it was calculated that the pressure of the vapour was approximately 1°6 atmospheres at 1000° C. Visual observations and the examination of numerous photographs taken at temperatures ranging from 20° C. to 1000° ©. showed that there were no well-defined bands in the region %2500-A 6700. On the other hand, there was distinet evidence of a general absorption in the ultra-violet which increased in intensity and spread towards the red end of the spectrum with rise of temperature, At 900° C. no light of shorter wave-length than » 3400 passed through the vapour. Mereurous chloride sublimes at about 400° C., and its vapour consists of a mixture of mercury and mercuric chloride unless special care is taken to tree the salt trom traces of moisture. About :09 gram of the salt was heated to temperatures varying from 400° C. to 800° C., and the absorption spectrum was photographed and also visually examined. The photograph taken at 400° C. showed general absorp- tion in the region > 2500-A 2800, and this eel with rise of temperature, so that at 800° C. the continuous spec- trum of the carbon are could not be photographed below r 3200. As in the case of mercuric chloride, the observations carried out both visually and photographically indicated that the vapour at temperatures varying from 400° C.-800° C. gave no well-defined absorption bands. In the experiments described, no special care was taken to absolutely remove all traces of moisture, and consequently the vapour probably consisted of a mixture of mercury and mercuric chloride. According to the experiments of Wood *, mercury shows an absorption band at 42536 which widens unsymmetrically towards the red over a range of 400 A.U. as the pressure increases to several atmospheres. If a foreign gas is present, the line widens symmetrically at first, ‘and after- wards unsymmetrically towards the red. In the experiments on mercurous chloride the presence of the X 2536 line would be difficult to detect, since the mercury * Astrophys. Journal, xxvi. pp. 41-45 (1907). 58 Mr. H. J. Evans on the Absorption Spectra would be mixed with mercuric-chloride vapour, and the continuous spectrum from the carbon are is very faint below AX 2800. The Bromide, Iodide, and Chloride of Cadmium. In the experiments with cadmium bromide, which melts at 571° C. and boils at 809° C., 07 gram of the salt was placed in an evacuated quartz tube, and it was calculated that the pressure of the vapour was slightly in excess of 1 atmosphere at 1000° C. ‘I'wo sets of experiments were carried out at temperatures varying from 600° C. to 900° C., and several photographs of the absorption spectrum were taken. ‘The first series of photographs showed no trace of any cadmium absorption line, but the second series even at 600° showed the presence of the cadmium absorption line at 13261. The line was, however, not strong on any of the films, showing that the cadmium bromide had only undergone slight decomposition at the higher temperatures. Apart from this line there were no other absorption lines or bands on the photographs. At 600° C. there was evidence of general absorption below 3100, and at 900° C. all wave-lengths between A 2500 and » 3800 were completely absorbed by the cadmium-bromide vapour. In later experiments °065 gram of metallic cadmium were placed in the quartz tube, and a photograph of the absorption spectrum was taken at 1000° C. The quantity of free cadmium present in this experiment was much greater than in the experiments with the cadmium bromide, but even then there was complete absorption only below A 3000. These experiments, therefore, strongly point to the conclu- sion that cadmium-bromide vapour shows general selective absorption in the ultra-violet. The absorption due to the cadmium-bromide vapour is shown in Pl. II. photograph 1 (a) and (6); (a) giving the spectrum of the positive pole of the carbon are, and (6) the same spectrum after the light has passed through the vapour at 900° C. The latter photo- graph shows a general absorption of light of shorter wave- length than 3800 A.U. Experiments were also carried out on the absorption of light by the vapour of cadmium iodide, which melts at 404° C. and boils at 714° ©. The iodide was produced by the combination of metallic cadmium with iodine contained in a quartz tube, which was heated to a temperature of about of the Vapours of Inorgane Salts. 59 400° C. by means of an electric furnace. The quantity of each substance used was ‘005 gram, and it is therefore seen that the cadmium was in excess. The experiments with pure metallic cadmium previously referred to show that at 1000° C. only light of shorter wave-length than > 3000 was completely absorbed, and in the cadminm-iodide experiments the amount of free cadmium was only =5 that employed in the experiments on the absorption spectrum of cadmium vapour. As the temperature of the furnace was raised, the light after passing through the quartz tube was medaen in colour, and when examined with the spectroscope showed the ordinary absorption spectrum of iodine vapour. This. colour, however, completely disappeared when the tempera- fure of the furnace was approximately 400° C., showing that the iodine had combined with the cadmium. A large number of photographs were then taken at temperatures ranging from 400° C. to 1000° C., and it was found that cadmium-iodide vapour showed a decided general absorption in the ultra-violet. At 400° ©. there was scarcely any evidence of absorption, as the light from the positive pole of the electric are could be photographed in the ultra-violet as far as 42500. When, however, the temperature of the furnace had reached 650° C. all light of shorter wave-length than X 3500 had been absorbed, and this absorption spread towards the red end of the spectrum as the temperature was raised to 1000° C., so that absorption could then be traced as far as 73800. As cadmium vapour was present in these experiments, it would be expected that the > 3261 cadmium absorption line would be visible on the photographs, but owing to the great absorption of the iodide, this line could only be seen faintly on one of the films taken, when the temperature of the furnace was 580° C. Even at this com- paratively low temperature the wave-length of the line (\ 3261) was near the limit of transmission. Finally, the absorption of light by cadmium-chloride vapour was studied. ‘This substance melts at 590° C. and boils at 900° C., and the absorption spectrum of its vapour was photographed at temperatures ranging from 600° C. to 1000° C. In these experiments the quartz tube was evacuated to a pressure of ‘01 mm. of mena and the quantities of the chloride used in two sets of experiments were ‘07 and ‘045 gram respectively. It was evident from the results of the first series of experiments that the cadmium-chloride vapour had been partially decomposed at high temperatures, for the 3261 line of cadmium appeared with increasing 60 Mr. HE. J. Evans on the Absorption Spectra intensity on photographs taken when the temperatures of the vapour were 650° C., 840° C., and 980° ©. In addition to the 3261 absorption line, photographs taken at tempera- tures above 800° C. showed the presence of more diffu-e absorption lines of shorter wave-length, but later experiments indicated that the lines could not possibly be attributed to the chloride. These lines became more distinct as the temperature increased, but their investigation was rendered difficult by the general absorption in the same region pro- bably due to the chloride vapour. In the second series of experiments, greater care was taken to remove moisture from the chloride, and two photo- graphs taken at 700° C. and 830° C. showed no trace of the cadmium absorption line. There was, however, distinct evidence of general absorption in the ultra-violet, but it was less intense than the absorption by CdBr, and Cdl, Even in these experiments a slight decomposition of the vapour was obtained at a higher temperature (940° C.), and the 3261 absorption line together with the other lines of shorter wave- length appeared on the photographic film. Atthis tempera- ture all wave-lengths between 23100 and 22500 were almost completely absorbed. The absorption of cadmium-chloride vapour is illustrated by photograph (2) (a) and (4), where (a) is the continuous spectrum due to the positive pole of the carbon arc, and (6) the same spectrum after passing through CdCl, vapour at 940°C. It shows a general absorption of light of shorter wave-length than 3100 A.U., and also the presence of absorption lines at 3261, 3171, 3162, and 3152. The 3261 line is the well-known absorption line of cadmium vapour, and the origin of the other lines will be discussed later. EXPERIMENTS ON THE ABSORPTION OF CADMIUM VAPOUR. The two specimens of cadmium used in these experiments were analysed spectroscopically to test their purity. The cadmium was placed in the positive pole of the carbon are, and two photographs were taken with the concave grating. A careful examination of the photographs showed that the two specimens did not contain any impurities. Wood * found that the absorption spectrum of cadmium vapour consisted of two lines at >) 2288-1l and 2 3261-2, which can also be obtained as emission lines. The former line, * Wood, Astrophys. Journal, xxix. pp. 211-223 (1909). of the Vapours of Inorganic Salts. 61 which is much the more prominent, was found to broaden perfectly symmetricaily when pure Cd was examined, but unsymmetrically when mercury was added. The absorption line at X 2288-1 attained a width of about 200 A.U. at the highest temperatures employed, but the X 3261 line was never very broad under these conditions. In the present experiments, only the absorption line at 13261 could be obtained, as the carbon are which was the source of the continuous spectrum is not suitable for the examination of absorption lines in the neighbourhood of A 2288-1. These experiments on the absorption of the vapour were carried out at different temperatures with each specimen of cadmiu m, and the amounts of the metal used were ‘015 gram and ‘065 gram respectively. The vapour obtained from ‘015 gram of one specimen gave an absorption line at 13261, which broadened symmetrically with increase of temperature until its width was about 100 A.U. at 1000° C. At the above temperature the vapour also gave additional absorption lines in the region 73000-23200, and a general absorption in the region A 2500-A 3000, which is possibly due to the broadening of the > 2288 line. These absorption lines were the same as those previously obtained when the absorption spectrum of CdCl, was examined, with the exception that cadmium vapour gave an extra line at 3178. The vapour obtained from ‘065 gram of the other speci- men gave practically the same absorption spectrum as the above, with the difference that the absorption lines in the region A 3000—\ 3200 were very faint and difficult to detect with certainty. It was found, however, that these lines could be more readily obtained if cadmium vapour were mixed with hydrogen. For this purpose hydrogen was allowed to enter an evacuated quartz tube containing °03 gram of cadmium until the pressure was 25 cm.of mercury. Photo- ‘raphs taken when the temperatures of the vapour were 880° and 970° showed the presence of the absorption lines in the region \ 3000-A 3200. The wave-lengths of some of these rather diffuse lines were measured, and the approximate values obtained were 3142, 3152, 3162, and 3171 A.U. It therefore follows that these lines cannot be attributed to the vapour of CdCl,, and the experiments suggest that they may possibly be due to an unstable compound of Cd and H, the lines being produced when the compound dissociates. 62 Mr. W. Morris Jones on the most Effective SUMMARY. The absorption spectra of the vapours of ammonium chloride, mercuric and mercurous chloride, cadmium chloride, bromide, and iodide were examined, and it was found :— (a) That the vapours did not give any well-defined absorption lines or bands like bromine and iodine in the region A 2500-A 6700. (5) That they gave a general selective absorption in the ultra-violet (except possibly NH,Cl) which in the case of the cadmium salts was greater for the iodide and bromide than for the chloride. The author wishes to thank Sir Ernest Rutherford for placing the necessary facilities at his disposal, and for the kind interest he has taken in the work. Manchester University, Dec. 1915. VIL. On the most Effective Primary Capacity for Tesla Cotls, By W. Morris Jongs, B.Sc., Research Student of the University of Wales*. T is generally assumed that a Tesla coil gives the best effect (2. e. the highest secondary potential for a given primary discharge potential) when the two circuits are so adjusted that, separated, their periods of oscillations are equal. This condition is expressed by the relation L,C)=L,.C,, and is commonly called the condition for ‘resonance’? or ‘“‘ synchronism,” though it does not mean that the two periods of the system are equal when the circuits are closely coupled. The above condition appears to have been first arrived at by Drudef in a well known memoir on the Tesla coil, and has received general acceptance. In a recent paper Professor Taylor Jones{ has shown that the above condition does not apply if the adjustment is made by varying the primary pH alone, but that in this case the “optimum” value of C; is considerably greater than the vesonance value. Taking the secondary potential, when the * Communicated by Prof. E. Taylor Jones, D.Sc. + Ann. de Physik, xiii. p. 512 (1904), } Phil. Mag. xxx. p. 224 (1915). Primary Capacity for Tesla Coils. 63 resistances are neglected, as given by the expression V2= He et eae m™nyt—cos 2mn,t)*, (1) it is shown in the paper referred to that if k?=0°265 the optimum primary capacity is 2°128 times the ‘“‘ resonance”’ value; and that if /?=0 36 (a value specially recommended by Drude along with the “‘ resonance’’ condition) the optimum capacity is at least twice as great as the resonance value. It was suggested to me by Professor Taylor Jones that I might test the conclusions by experiment, and also determine experimentally the optimum primary capacity and relative values of the maximum secondary potential for various values of k?, with a view to obtaining data for a complete test of the theory. In the experiments about to be described the coupling coefficient was varied by removing turns from the secondary of the Tesla coil. In this process the primary self-inductance L, remains constant, and the variation of L,, can be easily determined and allowed for. ‘The value of k? and the optimum value of m were determined at each stage. The same spark-length was maintained throughout the experi- ments, so that V, could also be regarded as constant. The Apparatus. At the beginning of the series of experiments the secondary of the Tesla coil consisted of 192 turns of bare wire (No. 16 S.W.G.) wound on a wooden octagonal frame smeared with paraffin- wax, the immediate supports of the coil being covered with indiarubber tape. Initially the coil was 80 cm. long and the turns were 25 cm. in diameter. The primary coil was of 5 turns of bare wire (No. 12 8.W.G.). The turns, about 30 em. diameter, were 1 cm. apart and wound on an ebonite frame of the same shape as that supporting the secondary. The primary coil could be removed altogether from, or put into any position over, the secondary, which was always used with its axis vertical. The primary circuit of the Tesla cuil contained the primary coil in series with an oil condenser of variable capacity charged by an induction-coil and dis- charged across a spark-gap. ‘The primary current of the induction-coil was interrupted by a mercury break driven * In this expression V, is the induction-coil discharge potential, L,, is the induction coefficient of the primary on the secondary, & is the coefficient of coupling, m is the ratio L,C,/L,C,, and n,, 2, are the frequencies of oscillation of the system. 64 Mr. W. Morris Jones on the most Effective by a motor, and the discharge terminals of the coil were made of zinc. To diminish damping due to brush discharge from the terminals of the secondary of the Tesla coil, and to prevent sparking between secondary and primary, the spark-length used was only 1 mm., and this was kept constant throughout the whole series of experiments. Brush discharge was not altogether prevented even by the use of such a short spark- length. A few preliminarv experiments were made to ascertain whether the primary discharge potential was independent of tne primary capacity and remained constant provided the spark-length-was constant. [or this purpose two condensers of widely ditterent capacities were respectively put in series with the spark-gap. Measurements of the discharge potential were taken by means of an electrostatic oscillograph*. From the results obtained for the two capacities used, the dis- charge potential appeared to be independent of the primary capacity. Measurement of Frequencies. All frequencies were measured by a wavemeter. These were usually too high to be registered by the wavemeters in common use in wireless telegraphy, and one suitable for the purposes of the experiments was constructed as follows. A standardized variable condenser was connected in series with a single circular turn (58°16 cm. diameter) of copper wire (6376 cm. diameter) the self-inductance of which could be calculated from the formula L=4nR(log. =~ —2). By means of a suitable switch two other coils of two and three turns respectively could be substituted either separately or in series for the single-turn coil. The range of the wave- meter was thus widened, and all the frequencies required to be measured came within this range. The extra coils were standardized by comparison with the single-turn coil. A carborundum detector with telephone connected across the terminals of the condenser was used in all the experiments. The frequencies of the oscillations in these experiments were within the range *710 x 10° to 3:28 x 10°. Measurement of Maaivmum Potential. In these experiments comparative measurements were required of the maximum secondary potential that could be developed for various degrees of coupling of the Tesla coil * HK. T. Jones, Phil. Mag. xiv. p. 238 (1907). Primary Capacity for Tesla Coils. 65 circuits, the adjustment being made by varying the primary capacity. A number of methods were tried for ascertaining when the potential in the secondary was a2 maximum. One of these consisted in sparking from a secondary terminal to an insu- lated electrode. The distances at which the sparks just failed to pass to the electrode were taken as indicating the maximum terminal potential for each adjustment of the capacity in the primary circuit. This method, though giving comparative values of the terminal potential, did not appear sensitive enough for the determination of an optimum capacity ™. The ‘method finally adopted consisted in measuring the distance from the terminal at which a neon tube just failed to glow when held horizontally opposite the upper terminal of the coil, the end of the tube remote from the terminal being well ene This method, although the most sensitive of those tried, was not capable of oreat exactness; and as the potential of the secondary varied slowly in the region of the maximum, the adjustment of the primary capacity to its most effective value was a difficult matter, and could not be made very accurately. The method, however, is one in which practically no disturbance of the conditions is intro- duced, owing to the fact that the tube glows when at a considerable distance from the terminal of the coil. To ensure that the distance of the tube from the coil depended upon the terminal potential alone and not upon the length of the coil, a wire about 2 metres long was attached to the upper secondary terminal. This wire pro- jected horizontally from the coil and was suspended by silk cords from the ceiling. The neon tube was held opposite this wire at right angles to its length. Distances were always measured when the tube was in this position. The glow of the tube was affected by the presence of earthed bodies. It was important tnat the disposition of these bodies near the wire should remain constant during the experiments. This condition was secured by the projecting wire being at a fixed height from the floor, all other conducting bodies (including that of the experimenter) being kept in a fixed position when readings were being taken. * It may be mentioned that an aluminium-leaf electroscope connected to one of the secondary terminals also shows a distinct maximum as the primary capacity is varied, such that if the capacity is increased or diminished from this value, the deflexion of the leaf falls. There is thus an optimum capacity for “mean square,” as well as for “maximum ” secondary potential. Phil. Mag. 8. 6. Vol. 31. No. 181. Jan. 1916. EF 66 Mr. W. Morris Jones on the most Effective The method of taking a reading was as follows :—The tube was first placed at a moderate distance from the secondary terminal, and it was found that it glowed continuously over a considerable range of values of the primary capacity. The distance of the tube was then increased, the range of C, diminishing, until the tube failed to glow continuously but flashed out only at the good primary discharges. By this — time the range of C, was reduced to fairly narrow limits, and the middle point of the range was taken as the optimum primary capacity. The tube was further removed until these isolated flashes just ceased to appear. The distance of the tube from the terminal was then measured, and this distance was taken as a measure of the maximum secondary potential for the given degree of coupling and the given length of spark-gap. The frequency-ratio of the electrieal oscillations of the system, when so adjusted, was found by means of the wave- meter. Period of the Primary Circuit. After the capacity had been adjusted to its most effective value, the period of the primary circuit was determined. This was done by means of a “dummy” primary. This coil was similar in all respects to the primary of the Tesla coil. It had the same number of turns and was made of the same wire. The turns were also of the same diameter as those on the primary, and both coils gave the same reading on the wavemeter when connected up respectively to a constant capacity and spark-gap. The reading obtained with the wavemeter when the ““dummy” had been substituted for the primary of the Tesla coil, gave L,C;. Determination of k? and L2C3. The coupling coefficient of the primary and secondary circuits was next found. The “dummy” primary coil having been substituted for the primary of the Tesla coil, and the latter removed to a distance of two or three feet, the primary capacity was varied until the circuit came into tune with the secondary, resonance being indicated by the glow of a neon tube attached for the purpose to the lower secon- dary terminal. By removing the Tesla coil to a sufficient distance, the range of the capacity in the “dummy ” circuit over which the tube glowed could be cut down to very narrow limits, and the “dummy” circuit adjusted fairly accurately to resonance with the secondary. This tuning Primary Capacity for Tesla Coils. 67 adjustment was made when the primary circuit of the Tesla coil was open and also when it was closed by a short wire of sufficient length to allow for the self-inductance of the leads used when the primary was connected up in series with the condenser and spark-gap. The resonance capacity in the dummy circuit was noted in each case. The ratio of the periods when these capacities were respectively in the circuit were obtained from the wavemeter readings. If T,? and T,’, to which the wavemeter readings are directly pro- portional, represent the periods of the secondary when the primary was respectively open and closed, then T,?/T,?=1—? from which i? can be immediately obtained. The value of k? was also verified in some instances by determining the minimum ratio of the oscillation-frequencies of the Tesla coil System. The reading of the wavemeter when the “dummy ” circuit had been tuned to resonance with the secondary of the Tesla coil, the primary being open, gave the period of the secondary, and therefore the value of L,C,. The capacity in the “dummy ” circuit for this case was the “ resonance ”’ capacity satisfying the condition L,Cj=L,C, The ‘“ optimum ” primary capacity was found to be always greater than the resonance value. The above completed the set of determinations required for one value of £?._ The coupling was next increased a little by removing turns from both ends of the secondary, and the primary capacity again adjusted to the “optimum.” The whole set of observations was then repeated. These adjust- ments and observations were made both with the primary coil at the middle of the secondary and the secondary terminals insulated, and with the primary at the lower end of the secondary and the lower terminal of the latter well earthed. In the earlier part of the series of experiments all the five turns of the primary coil were used; but when about 100 turns had been removed from the secondary, it was found that the “‘ resonance” value of C, became inconveniently small for the range of the oil condenser. Consequently, in the latter part of the series only four turns of the primary cou were used. This reduces L, and proportionately increases the ‘‘ resonance” value of C,, thus making it more suitable for the range of the oil condenser employed. The removal of a turn from the primary has, however, another effect. It diminishes L,, and L,, but L, is reduced in a greater proportion than L,,. Thus the ratio L,,/L, is increased; and according to the expression (1) the maximum H2 68 Mr. W. Morris Jones on the most Effective secondary potential (for given values of m and k?) should be increased by the removal of one of the primary turns. This was found to be the case, the greatest distance at which the tube glowed being considerably greater with the 4-turn than with the 5-turn primary for the same value of k?. It was thought desirable to reduce the results of the later experiments (those made with the 4-turn primary) so as to show what the tube-distance d would have been if the primary coil had remained at 5 turns. In making this correction it was assumed that the maximum secondary potential was proportional to L,,/L;. The correction was therefore made by multiplying the later values of d by the ratio of the old to the new value of Lp,/Ly. This was determined as follows:—The change in L, was found with the help of the wavemeter, the ratio of the periods of a condenser being found when its terminals were connected to 4 turns and 5 turns respectively of the “dummy ” coil. The change in Ly, was found by ballistic galvanometer experiments, the secondary being connected to the galvano- meter and the deflexion observed on the reversal of a given current in the primary, this being done both for 4 turns and for 5 turns. In this way it was found that the ratio of the old to the new value of L,,/L, was 3%: consequently, in the results given below the values of d obtained with the 4-turn primary have all been reduced by 10 per cent. At the beginning of the series of experiments the number of turns on the secondary was 192, and this number was reduced by small stages until there remained only 54. During this process the induction coefficient Ly, gradually diminished, very slowly at first, more rapidly as the number of turns became smaller. Since it was part of the object of the present experiments to determine how the maximum secondary potential varies with k?, it was necessary to make allowance for the variation of L.,, which is one of the factors to which V, is proportional. The value of the co- efficient L,,; was therefore determined (in arbitrary units) at each stage by ballistic galvanometer experiments, in which the primary was connected to the galvanometer, and the reversed current sent through the secondary. This was done with the primary at the middle, and also at the lower end of the secondary. Curves were drawn showing how the deflexions varied with the number of turns on the secondary. They showed that the mutual induction between primary and secondary was not sensibly altered until the number of turns had been reduced to 100 when the primary Primary Capacity for Tesla Coils. 69 was at the middle, and 65 with the primary at the lower end of the secondary. By means of these curves corrections could be applied to the values of d due to change in Ly caused by the removal of turns from the secondary coil. TABLE I. Primary coil over middle of secondary, Secondary terminals insulated. Primary spark-length=1 mm. 2 | | ax. s ke, _ m=L,C,/L,C,). IN| - a Fe es Neate LN cia aN 0-147 | 0:740 1-538 65:0 ? 0-166 0-725 1°52 69:0 0-182 0694 L583 67:0 0-206 | 0-588 65. 4 63:7 0-222 0°608 1°67 58:0 0259 0-553 1:80 48°5 0-276 0-551 1°83 48:0 meOs07 | 0-420 2-08 40-0 Gono |. | 0-447 DORE 29-0 | Gsle 0-437 2°10 40°3 0368. || 0-310 2:43 28°8 0374 0-294 | 2°50 230 | TasuE II. Primary at lower end of secondary. Lower terminal of secondary earthed. Primary spark-length =1 mm. B: m( = L,C,,/L,C,). | Ny. cyan: i O15 0-830 1-48 146 0-138 0-793 1:50 137 ie, C154 0-619 161 126 0-170 0°580 ova 115 | 0-181 0'560 173 110 Hy 9 0:200 0'585 173 99 0-226 0-470 1:89 94 0°247 0°553 1:87 88 0-237 0-480 1-92 8d | 0°266 0:373 2718 vig ) 0°325 0-250 2°56 63 The results obtained are given in Tables I. and IJ. The first column gives the values of k?, the second the values of m required to produce the maximum secondary potential, and the third column the values of the frequency-ratio n./n, when the primary capacity was at the optimum value. The 70 Mr. W. Morris Jones on the most Lffective fourth column gives the maximum distances at which the neon tube glowed. ‘The last three values of d in each table have been reduced, as explained above, by 10 per cent. The distances are measures of the maximum potential that can be developed in the secondary for the corresponding values of k? and m, and a primary spark-length of 1 mm. The results in column 2 of the tables are plotted as curves in figs. 1 and 2, in which the abscisse represent 1/k? and the ordinates the values of m. Some of the points do not lie well on the curves. This is due to the difficulty of accurately adjusting the primary capacity to its optimum value. ag sale 0-8 d 6 a setts ste et aati E uiene TE egeee' i Paths 206 325 Witg . 375 700. Ve However, the curves show that, for all the degrees of coupling obtained, L,C,/L,C, was less than unity. It is clear from these results that, if the adjustment of the Tesla coil is made by varying the primary capacity alone, then the optimum value of C, is greater than the resonance value, the difference between them diminishing with k? in accordance with the theory. The tables also show that the frequency- ratio n/n, when the primary capacity is adjusted to the optimum value, passes through 2 in the neighbourhood of the value k?=0°265. This is also in accordance with the theory. Primary Capacity for Tesla Coils. wo Fig. 2. RAL: Pee Pie G5 paseaneesee 7 ) 60 | ast h 5 : i) : 5 a2 : ; 7 10-00 325° 4-50 575 ~ 5.00 ye In figs. 3 and 4 the full-line curves show the tube- distances d (Tables I. and II., fourth column) plotted against 1/k?. When the correction due to the variation 72 Mr. W. Morris Jones on the most Effective of L,,, caused by removing turns from the secondary coil, is applied, the curves take the form shown by the broken line. The curves show that if Vy, L.,, and Ly, are constant, the maximum secondary potential increases as k? diminishes. Fig. 4. 170 az = = ( eee This result also might be anticipated from the theory. By (1) the amplitude of the potential waves in the secondary 1 Yh (1—m)?+4km} any value of & this expression has, when m=1—2h*, a Within the range of the coil is in this case proportional to For maximum value of : 2hy/ (1—k?) | present experiments this quantity increases as & diminishes. In fig. 5 the curve shows the variation of the maximum tube-distance d@ with the primary capacity C,. In this experiment the coils and their coupling coeificient were kept constant, k? being 0°348—nearly equal to the value recommended by Drude. The curve shows a maximum tube-distance near the value m=0°3, 7. e. at a primary capacity much greater than the resonance value given bya Primary Capacity for Tesla Coils. 73 The theoretical curve, obtained by calculation from the expression (1), is shown in fig. 6, also for the case 4?=0°348. In this curve the ordinate represents the maximum value of the quantity 2 VO oy s/\(1—m)?+4k?m} for each value of m, and is therefore proportional to the (cos 2arnyt —cos 277N5t) 74 Most Kffective Primary Capacity for Tesla Coils. maximum secondary potential, since Vo, Le, and L, are constant during the experiment. The frequency-ratio n/n, is calculated from the relation Ne __ m+ 1+./{(m—1)?+4km} ny? m+1—/{(m—1)?+4k'm} There are important differences between the experimental and theoretical curves, figs.5and 6. The theoretical greatest maximum occurs at about m=0-l, and corresponds approxi- mately to the frequency-ratio n/n, =4. Thisdoes not appear at all in the experimental curve, in which also the maximum shown occurs at asmaller value of m than does the secondary maximum in the theoretical curve. It should be remembered, however, that in the expression (1) for the secondary potential, damping due to all causes is neglected, and that the theoretical curve of fig. 6 represents only those maxima which occur near the time 1/2n,. | It is possible that in the experiment the damping might be so great in some cases as to diminish these maxima, so that they would not appear as such in the experimental curve. Possibly also the continuation of the experimental curve (fig. 5) to smaller values of m would have revealed some trace of another maximum. Time did not allow me to determine and allow for the damping factors of the oscillations. With these taken into account better agreement would probably have been found between the curves of figs. 5 and 6. Similar curves were obtained for the case k?=0°255 and the primary at the lower end of the secondary, and the experimental and calculated curves showed differences similar to those of figs. 5 and 6. The above experiments were carried out in the Physical Laboratory of the University College of North Wales; and in conclusion, I desire to record my obligation to Professor EK. Taylor Jones for bringing the subject of the above investigation to my notice and for much valuable help and advice during its progress. uli aad VIII. Methods for the Summation of certain Types of Series. By I. J. ScuHwartt*. I. a find ae 1 1 1 2 S es Beas a) Reig eet Y oe ST es oS epee aaa al a a(at1)* aatl(at2) iy Pea Thies a). Kk=0 wherein a is a positive integer. Gi.) We may write or ae (a+n)! we: (a—1) ! 2 patn Bre oo Caaany! i Cea : 7 Y hed n=0 7 ! Lit (a—1) ! r S yn _ S “| lim n=0 7e | n—0 7 ! we ! a-—1 pn — Le ial : = > | ° rv n=0 tl: (ii.) A second method is as follows : Let wun_1; denote the nth term of the given series, then Un / Uni @en’ and : y (atn)u,=r s nts a" R=! or oO (4+2)U,=P S Une: n=0 2=0 Hence r—+(a—r)S=1, d and foe fisra] = Fa eSemtne-1 dr+C]. * Communicated by the Author. 76 Dr. I. J. Schwatt on Methods for the Now Nee tre. dp = — [e-"r*-! 4 e-"(a—1)r*-? +e"(a— eae he “34... $e-"(a— 1)! = pS Is =e"(a—1)!% ben Therefore py etn r Se Mee Oe T n=0l: r Multiplying by r* and letting r=0, we find C=(a—1)! We finally obtain =a [er < 1 yn n= 2 ae II. To find go ert ert TEPC S SS 5.6.0. yy Qn-+1)2n+2)(2n#3) (.) Separating the general term into partial ~~ L al al i 1 (2n+ 1) Qn+2)\(Qn+3) nigel a 555 therefore i oO 1 foo) yr rea) 7 a 2 aoe is “ses n+1 +2 o243 a =. gt ptt =5[1+r3 n= 0 2n eo +2 5 Soe i =5/1+040)35 5 44+ ~ log (1 —r)], ye 1 1+~7 0 y2(2n+3) ik me [1+ arate 2n+3 ie ae (7) | ; But co ye2nt3) 1, 7 yz ee , ve in n—02n+3 NO cana Ua 4 ees ae r ce MAGS a a |- Be Oy. ey aera = 5 [log (1+4)— — log (l—7 )| — 72, Therefore 1 1l+~,r l+r = oS ] Bag hd a lei aaa wt a= “log (1 r) Je at 1 log +"; + 2log at a Ae ee Summation of certain Types of Series. (ii.) Another method is the following: If in form (A) of the preceding method we let f 1s yerts S T=wH an a ee ig 1> th en dS) _ 3 yenten ou dx n=0 — x Therefore 7ede 1, l+e ; as Ame eee One ae and ei i . l+r, l+r I14r sofa Togc On rds er ern => a ae 42log (1 —r)—2| : Gii.) Still another method is the following : Let u,_1 designate the nth term of the given series, then uty n(2n—1) ih (n+1)(2n+ Bre and ii Ms & (n +1)(2n+3)u,=r S n(2n—1)un_i1, ue =e (n+1)(2n4+1)unr, n= and since 1 (n+ 1)(2n+ 3)un | — n=0 2 therefore 3 (n+1)Qnt3)m=r 3 (n+ 1(Ont lint 5 n=0 n=0 a (r5+1)[20— 243 -r]8=5. Let now i (5, +1)8=y. then ee 1 ie +(3—rjy= 5 and 3-97 r 3—r ae a , \4 a it Ja a ae aie | o r1—7r) é wee 11—r({" # dr an 4 0 (1—r)? ; a 78 Dr. I. J. Schwatt on Methods for the Letting r=’, and integrating by parts, we have Be a [-—- 1 147 DTA gti ln ie Ta Therefore l-r) pen 4+ 72 s= 7" E iy, oe ci | ae l+r ae =. | = 7a Ooh ae e+ 2log (1—r)—2]. (iv.) A fourth method : If in the given summation we let r=w?, then 3 a nao(2n+1)(2n+2)(2n+ 3) We then have d oii 1 (« oe ¢ +1)( 2s Le +2\(a5 o +3)8= Sa ae Therefore | d ad RS canal l+z arial 1s? Gop eee and Sine : x pe i Ie =55 logit? = + log ia 2n Hence S= 953 ae [wlog ss ate log (1—a") | dx. ee by parts, fee. t eda ~ 978 [Flog al l-# pt ele. S|. eles ie Beeaar ane (1-2) a], i eerie ee sie Ay [ STA (v.) We might also arrive at the result in the following way: Let eee then 73 +2 log (1 —r)-2| : yen +3 RIN ale Nears fo a o (2n+1)(2n4+ 2)(2n+3)° i Ms Summation of eertain Types of Series. ‘ 79 N 3 20 ow PS, _ is dx* n=0 oe i “igo ae | Tet = 3 [081 +a)— log (1-2) ], = LST! 5[ (+2) log (1+) +(1—2) log (1-2) |, and ae (+2)—7(L+ay ENS — 5 (1-2)? log (l—2) + se 1l+2?, l+ez ce ae 5 logs +5 “log (1— x?) — 3° Therefore 1l+7 1+72 1 7 oe dye De ag 2p Sa) oa, i (ple +7 l+r ah Api yt log o1— = Oe ar 2]. If r is negative, that is, if we are to find co. 1 1 1 Beis (—r)" metry 3.4.5"' 5.6.7" = 3 2n-+ 1)(2n + 2)\Qn+ 3)’ we must substitute in the results obtained above or S= ir? for r?, and Joga + = Jian tr for log 75 neve We then obtain =e ayo r+ 2log(L+7)—2 ], — 1—log (1 ae —" tan-} 73 =3,| og (L147 = |: III. To find n—1 II (4+) = 4.5 4.5.6 al aa epee oe as + oa qo Heer a 7 BS Pri n—1 : a tH, ¥2 1B = Gite (i.) First method : n—1 Eee en Gray OP n—l Pall. lene a ? rane a i Gan) 80 Dr. I. J. Schwatt on Methods for the therefore a = yr? “— Il (n+«) «4 10 But ANS Ze — ¥ Ax SAO Sat 6 ee Hence 9 10 6 ws 5 1006) Sa ay ne 3 c=4 ( ) ce n=0 lt + K 9\ 10 6 1 ®% yt = 10(5) Co he Coa 3 heat \e— 4] pt Saae L [ a yrteK R ‘ ea n+ K Mere: then eye 2 apm K-1 ym oe == =e —— aia mys mMm=K mv ma=l mM mal 7” mee st =) o Therefore $ = 10(3)log —n) & (—He(, o>, + (2 2, Sine aye pe 7 5 m=1 ™M Zea O / K+3 ,m—K+4 = — 810257) tog (1—7) +10(3) = (-(") sah m=1 me b K+3 »6—K+m LAN (—1)"*1 ( > ee 840 a log (1-7), m=1 i SBT OR = "ar [—C—r)* log Av) at eG At AS Mian b . + — 7 — 7 Summation of certain Types of Series. 8] (ii.) Let w,_, designate the nth term of the given series, then pas iy 11 Un ently tl a f) ( 5 ¥ u+o a : = ie at n= 2 . 10 un—1 i (44«) ii (14 co nt k=0 k=0 and > (n+ 10)u, = 7 > (2 +3)Un-1, n=1 1 re (1 +4)tUn, and since (n+10)u, |,-) = 10, wo being 1, eS nt 10)u, =r Se Hu, £10; n=0 n=0 and r( 1S +(10—4r)S =10, or , d$ 10—4,r aie 10 ) r1l=r)> r(l—r) Hence "10 Bie 0) Ome ap 3 S —e ae! [c-+10f r(L—r) Aes dr | ne =| ( 7? . ] =r Sa hoe dy |. To obtain y? aan” let l—r=x, then (1—-2)° ‘Geely a" z ae Ne Bee 7 dz = (22 i = 5 -(j e+ ) -(3 )loge 7 (Js ON 9\ 1 ON al 9\ Il 1 ale ain (slant (7 )ax— (8) see + oe ee eel =r) = fat mo +(3)( en? ONS -~({)a ye + ace = (252 + (52 -()'5"+4] Page Vag... 6, Vol) dl, No. 16). Jan. 1916. G 3,f 82 Methods for the Summation of certain Types of Series. Multiplying by 7!° and letting r=0, we have : ON ty 9X1 (£9 Oya /9\1 C=10[5-(G)5 +4 (3)-(2) +G)3-(s)5 tes \8/5 or C=518. Therefore 4 18 peso ee I s “(1— rs + 36(1—r)! —84(1—r)* log (1—r) —126(1—r) + 630. —71)4 Os 7)t + 90 =(1—7) + Al _ 840 ll’, 37). aes ov | —(—*)*log 1-1) -9— 3" — + Po 59” 0 1a 20° 7° 56 eae IV. To find 1G SOI a aes e 2) Pah: S$ oe — >, 9 co. er he a 9 «k=0 wherein a and 6 are positive integers and a =§ n= yp nt+atkK oD ye ea) ae a+kK-1 ow then Sh aes rat n=atk!b n=1 70 n=1 7 +e-1 pw Sate ae ie here m=1 and ele Le So I b—a Z ee a+Kk-1 pu—K eae ae ee baa ante Hee ry n=1 But — pe a NN pryeee- a (r—1 b= a=1 ae ( K eas aS : H fe “ne S = (b—a) (oy) ce (1—r)’-e} AMT) E SY University of Pennsylvania, Philadelphia, U.S.A. IX. The Structure of Silver Crystals. By lL. Vecarn, Dr. plul., University of Christiania*. A’ a contribution to the study of the inner structure of crystals by the Rontgen ray analysis, lam going to give an account of the determination of the crystalline structure of silver. Silver crystals belong to the cubic system, and are found in nature in very fine specimens of considerable size and with well-developed regular crystal-faces. For the analysis I used the reflexion method of W. H. Bragg and W. L. Bragg f, which is based on the determination of the reflexion angle of a homogeneous beam of X-rays. The Roéntgen-ray tube was of the sort recommended by Bragg, with a rhodium anticathode. The Rontgen-ray spectrometer was in principle the same as that constructed * Communicated by the Author. t oe H. Bragg and W. L. Bragg, Proc. Roy. Soc. A. vol. 1xxxviii. P: . 84. Dr. L. Vegard on the by Bragg, although the particular arrangements were some- what different. The slits and the ionization-chamber were mounted on the top of an ordinary Fuess goniometer, in such a way that the incident beam of X-rays and the axis of the ionization-chamber were parallel to the axes of the collimator and telescope respectively. The optical arrangements, how- ever, were not removed, so the instrument could at the same time be used as a goniometer. The crystal was mounted in the usual way on the crystal table with some wax. By means of a screw underneath the instrument the table could be moved ‘up and down without any appreciable rotation of the crystal. This arrangement proved very convenient, as it was possible to set the crystal-face very accurately by means of the optical arrangement, and then move it upwards until its central part came into level with the axis of the ionization- chamber. The crystals used were two fine specimens which were kindly lent me by Professor W.C. Brogger, of the Minera- logical Laboratory. The one had the cube faces (100) and the other the tetrahedron faces (111) well developed. The crystals were good reflectors. In order that no maximum might escape notice the entire ionization curve for varying angles was determined first with fairly wide slits (1 mm.). Then a more accurate measurement was made of the plate given by the strongest Rh line of wave- length 7>=0°607 107° cm., using a slit about 0-4 mm. broad. The condition for reflexion is given by the following formula of Bragg: nv = 2d sin 6; d is the distance from one point plane to the next identical plane ; @ is the glancing angle of the incident beam, d the wave-length, and n the order-number. The zero position of the chamber, as determined from the direct beam, is perhaps less accurate than the position corre- sponding to maximum ionization from the reflected beam. From the position of the first and second maxima, however, we can easily calculate the true zero position and the true glancing angle 6. Let the observed angles for the Ist, 2nd, &c. order be 1, %,..., and the true zero position a, then a,—-a Apa ee : 0. pga emrts cot 20= 2 cosec (a,—a,) — Cot (a — a). » and Structure of Stlver Crystals. 85 For the reflexion planes (100) and (111), for which the crystal had corresponding crystal-faces, the maxima of Ist, 2nd, and 3rd order were found. The calculation gave for a the values + 1' and —1’ for the two faces respectively. Thus the fault in the direct determination of the zero position is practically negligible. The crystal, however, had no marked (110) face, and to get the reflexion from these planes the cube-faced crystal was putup with the (110) plane vertical and the edge (100) horizontal. Butin this way the image of the reflected beam will not be a vertical slit, but will be (>) shaped, and the determination of the glancing angle less accurate. Further, the reflexion from the (110) plane was so weak that only the first maximum could be measured. But as the zero position is known, the first maximum will be sufficient for the determination of the glancing angle. In fig. Lis given the maxima corresponding to the strongest Biowl: (100) 7 PS 2m OEE LE gate aD MUSE ee eo ; I6 7 18 34° 35 53° 54° (11) l Il Bef ee ehiiti ey 14° 15° 16° 29 30° si 45 46° Rh line observed with the narrow slit, and in the table is given the glancing angle @ for the first order spectrum, the grating constant d, and the relative intensities of spectra of different orders. 86 Dr. L. Vegard on the Relative Intensity. Crystal 0 d plane. ts ; Is 2. | 3 order. Manav WW) a cm. (100) 8° 36’ | 2:030107%| 1:00 0:25 0:07 (110) 12001 1438 _,, | Not observed. (111) 2h 27341 ,, 1:00 0:49 | 0°10 The faces (100) and (111) show a normal-variation of inten- sities, with the intensity rapidly decreasing with increasing order. For the ratio of the grating constants we get “au = 11533 } ses 100 Salt Se a ey j Ze ve /3 dro =0-7085 | d 100 Ayo _ 1 if A109 / 2 —-=0 7071 )- - / 2 These are the well-known ratios, which belong to the face- centred cubic lattice * (fig. 2). For if the side of the smallest cube with one atom in each corner is 2a, then we shall have a 2a dyo=a, adyyo= We d= V3, Thus the ratios of the grating constants are explained from the face-centred lattice. This arrangement would further explain the normal distribution of intensities, as all point planes parallel to one of the faces examined should be identical and equidistant. To put the lattice to a final test we shall calculate from observations the number (n) of atoms which are associated with a cube of side dyg9. In the volume of the whole cube lattice with side 2a (fig. 2) there are 4 atoms, and in the cube with a side a=digq there should be 4 atom on an average. For the number n we get w= i Ndivo- * W. H. Bragg and W. L. Bragg, Proc. Roy. Soe. vol. 1xxxix. p. 281. Structure of Silver Crystals. 87 Fig. 2. p is the density of silver (10°50) and A the atomic weight (107:93). Nis the number of atoms in a gramme equivalent (61°5 x 10”), and dyo) is given in the table. Inserting these values, n=(0°50002, or 4. Thus the face-centred cubic lattice explains the whole series of experiments, and we come to the conclusion that the atoms im the silver crystals are arranged in the simple face-centred lattice. The arrangement is the same as that previously found by W. L. Bragg for copper crystals *. In conclusion, I wish to thank Professor W. H. Bragg and Mr. W. L. Brage for valuable information with regard to the methods of the crystal analysis, Professor W. C. Brigger for his kindness in lending me the crystals and the goniometer on which the Roéntgen-ray spectrometer was mounted, and Professor V. Goldschmidt for helpful advice with regard to erystal work in general. Finally, my thanks are due to Mr. Harald K. Schjelderup for his assistance in making the readings. Christiania, Oct. 20, 1915. * W. L. Bragg, “ On the Crystalline Structure of Copper,” Phil, Mag. vol xxyili. p. 355 (1914), 886 X. Notices respecting New Books. Functions of « Complex Variable. Professor JAMES PIERPONT. Ginn & Co. 20s. net. HIS volume deals with the elementary parts of the theory of functions of a complex variable, and is the outcome of lectures given over a considerable number of years to students of Yale University. The opening chapters give an account of the exponential, circular, hyberbolic, and other functions, followed by the consi- deration of differentiation and integration, the general properties of analytic functions, infinite products, and asymptotic series, in the latter case with reference to Gamma and Bessel functions. To illustrate the general principles of the theory, a brief treatment is given of the elliptic functions, more particularly from the standpoint of their most characteristic property of double periodicity. The final chapters, devoted to linear differ- ential equations, the functions of Legendre, Laplace, Bessel, and Lamé, with their more important properties, will appeal especially to the student of mathematical physics. Professor Pierpont’s style is clear and simple and his treatment sound and thorough. The book can be heartily recommended to students who need a work in which the principles of the function theory are carefully presented. It will serve as a useful intro- duction to the subject, and will, no doubt, arouse the interest of students and induce them to read the more advanced treatises and original memoirs mentioned in the text. Even the private student, who has not the advantage of a teacher’s assistance in overcoming his difficulties, should find the book particularly helpful. XI. Intelligence and Miscellaneous Articles. ON THE STRUCTURE OF THE SPINEL GROUP OF CRYSTALS. To the Editors of the Philosophical Magazine. GENTLEMEN, — HAVE recently received from Mr. 8S. Nishikawa a copy of a paper which he contributed last December to the Proceedings of the ** Tokyo Mathematico-Physical Society.” It deals with the structure of the spinel group of crystals. In August last you were good enough to publish an account of some experiments of my own on the same subject. I need hardly say that if I had known of Mr. Nishikawa’s work I should not have written my paper without referring to it. 1 am very glad to find that though our methods were entirely different there is no disagreement between our results. Yours &c. Dec. 15, 1915, W,. A, Braae, Raman & APPASWAMAIYAR. Phil. Mag. Ser. 6, Vol. 31, Pl. ie POOL LOAC OOOO OOP CG g “SERRE ROOD. Photographs showing the initial movement and the subsequent vibration with discontinuous changes of velocity. CE LONDON, EDINBURGH, any DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] RE BIU A TAY 1916. XII. On the Propagation of Sound in narrow} Tubes of variable section. By Lord RayvuxicH, OW, F.R.S.* Ti DER this head there are two opposite extreme cases fairly amenable to analytical treatment, (i.) when the changes of section are so slow that but little alteration occurs within a wave-length of the sound propagated and (ii.) when any change that may occur is complete within a - distance small in comparison with a wave-length. In the first case we suppose the tube to be of revolution. A very similar analysis would apply to the corresponding problem in two dimensions, but this is of less interest. If the velocity-potential ¢ of the simple sound be pro- portional to e”, the equation governing ¢ is Ga. Ldd : da On, Oe ete ea ae a) where 2 is measured along the axis of symmetry and y perpendicular to it. Since there are no sources of sound along the axis, the appropriate solution is b = Jo{rs/(d?[da?+kh)}F (a), . . . (2) in which F, a function of # only, is the value of @ when oO. * Communicated by the Author. + Compare Proc, Lond. Math. Soc. vol. vii. p. 70 (1876) ; Scientific Papers, vol. i. p. 275. Phil. Mag. 8: 6. Vol. 31. No. 182. Feb. 1916. H 90 Lord Rayleigh on the Propagation of At the wall of the tube r=y, a known function of a ; and the boundary condition, that the motion shall there be tangential, is expressed by Hd ae or mie ee ermine Oe ne (3) in which (r=y) do dk yd \dF yt (a. ,.\iak da dx molest eee k ) Ae oe (4) d a ; - 3 (sath) P +E GG atk) F—.... Ee (5) Using these in (3), we obtain an equation which may be put into the form 2 A a a sf dy # dE SaVE)+eyF) =o e So aoe Og CO Ng OlnaAN Ane) ea ah oP ide 4 As a first approximation we may neglect all the terms on the right of (6), so that the solution is Aer the ou Be F(@) = >, 5) ere («) i (7) where A and B are constants. To the same approximation, Ge rors ial Meena ae Te +i?R = jade (8) For a second approximation we retain on the right of (6) all terms of the order d’y/dx?, or (dy/dx)?. By means of (8) we find sufficiently for our purpose d? 2 2 dy dF (ep i da a aa dacry aM ie dy 1 d’y Get me aan al dx; y dx? La? qd? “6 d? a)* uf i = 0, (ae F=0. Sound in narrow Tubes of variable section. 91 Our equation thus becomes Brie at sey way dew (Sat )QF) et ee et =(14+3ey) SY Fa), 2. (9) ak in which on the right the first approximation (7) suffices. Thus (ae) = me Y(B+ Ae-2) da —ete | Y(A + Betts) dab, LOG where ee ae tone sy Sey Nic SO Ae) In (10) the lower limit of the integrals is undetermined ; if we introduce arbitrary constants, we may take the inte- gration from — to #. In order to attack a more definite problem, let us suppose that d?y/dx?, and therefore Y, vanishes everywhere except over the finite range from «=0 to «=, bd being positive. When 2 is negative the integrals disappear, only the arbitrary constants remaining; and when 2 is positive the integrals may be taken from 0 to zw. As regards the values of the constants of integration, (10) may be supposed to identify itself with (7) on the negative side. Thus oe a “Y(A + Be) dx b 0 ¥ 2ik J a | “VB+Aet™yda |, (12) Dik \, The integrals disappear when @ is negative, and when z exceeds b they assume constant values. Let us now further suppose that when a exceeds } there is no negative wave, 2. €. no wave travelling in the negative direction. The negative wave on the negative side may then be regarded as the reflexion of the there travelling positive wave. The condition is Beeb Vas | ten ¥ -2ikz de = 0, (13) { Pe Mer giving the reflected wave (B) in terms of the incident H ee bo 92 Lord Rayleigh on the Propagation of wave (A). ‘There is no reflexion if 'b } ’ ( *ye-2 de = 0; ap a a e/( and then the transmitted wave (#>D) is given by Fe) = a {ase | Yao b. eagle Even when there is reflexion, it is at most of the second order of smallness, since Y is of that order. For the transinitted wave our equations give (@>b) Ae th# j IL %b coe li VY 24] F(a) = =. a : 18) 1+ rat Ydz but if we stop at the second order of smallness the last part is to be omitted, and (16) reduces to (12). It appears that to this order of approximation the intensity of the trans- mitted sound is equal to that of the incident sound, at least if the tube recovers its original diameter. If the final value of y diifers from the initial value, the intensity is changed so as to secure an equal propagation of energy. The effect of Y in (15) is upon the phase of the trans- mitted wave. It appears, rather unexpectedly, that there is a linear acceleration amounting to 1 sp |, Vee ae Jo or, since the ends of the disturbed region at 0 and 6 are cylindrical, Lh Pal la aah, (da) (L-3eyae, LS Tas from which the term in ky? may be dropped. That the reflected i ay should be very smal! when the changes are sufficiently gradual is what might have been expected. We may take (13) in the form B —Qikex all Dd 7 — Nk Fy P= 3/0 Ye de — | ge? oe Sound in narrow Tubes of variable section. 93 As an example let us suppose that from =0 to r=b pat MU COMA eet oo sel (20) where y is the constant value of y outside the region of disturbance, and m=27/b. If we suppose further that m is small, we may remove l/y from under the sign of integration, so that Seno D = — pa cos mxe7** dx 0V0 i mgy ie 4k? y Independently of the last factor (which may vanish in certain cases) B is very small in virtue of the tactors m?/k? and 7/Y. Gh cos 2ko se sim’ QED 3.)) ein at) (C2N) In the second problem proposed we consider the passage of waves proceeding in the positive direction through a tube (not necessarily of revolution) of uniform section o, and impinging on a region of irregularity, whose length is small compared with the wave-length (X). Beyond this region the tube again becomes regular of section a, (fig. 1). Big ol. It is convenient to imagine the axes of the initial and final portions to be coincident, but our principal results will remain valid even when the irregularity includes a bend. We seek to determine the transmitted and reflected waves as proportional to the given incident wave. The velocity-potentials of the incident and reflected waves on the left of the irregularity and of the transmitted wave on the right are represented respectively by hem be bone Weg enn) Oo) (22) so that at 2, and x, we have d; = Aiea ee Rae Or = Certs, ‘ y (23) dd,/da = ik(—Ae~#1 + Bet), doy/da = —ikC e-, (24) 94 Lord Rayleigh on the Propagation of When 2 is sufficiently great we may ignore altogether the space between 2, and &,, that is we may suppose that the pressures are the same at these two places and that the total flow is also the same, as if the fluid were incompressible. As there is now no need to distinguish between 2, and a’, we may as well suppose both to be zero. The condition di= > gives Bo BSC yy oa Ce ena an ea and the condition o,d¢;/de=o,dd¢./dx gives oi(-A+B) = —o,0. 2) 2 ee Thus B \ cope: kee ee These are Poisson’s formule *. If o, and a, are equal, we have of course B=0, C=A. Our task is now to proceed to a closer approximation, still supposing that the region of irregularity is small. For this purpose both of the conditions just now employed need correction. Since the volume V of the irregular region is to be regarded as sensible and the fluid is really susceptible of condensation (s), we have ds Oty dd dds Ve da, 8 die and since in general s= —a~*dd/dt, we may take AS a, _» Ub, Tee ge On oe net the distinction being negligible in this approximation in virtue of the smallness of V. Thus 7 — da, Aix, Q dte In like manner, assimilating the flow to that of an incompressible fluid, we have for the second condition O71 (oy = kVgbo. . . (28) i$, ee Pm) C2) where R may be defined in electrical language as the resistance between x, and 2, when the material supposed to be bounded by non-conducting walls coincident with the walls of the tube is of unit specific resistance. * Compare ‘ Theory of Sound, § 264. Sound in narrow Tubes oj variable section. 95 In substituting the values of ¢ and dd/dz from (23), (24) it will shorten our expressions if for the time we merge the exponentials in the constants, writing ei At by eee nC eae Cle mh ie 30) o;(—A’'+B')+o,07°=—ikVO'’, . . . (81) Al eB! Chastho el os 609 32) We may check these equations by applying them to the case where there is really no break in the regularity of the tube, so that 07; = 02, V = (%2—2,) 0, R = (a—2,)/o. Then (31), (32) give B’=0, or B=0, and Thus Cl = cee Ss = e 7 k(z2-21) Al 1+2k(v— 24) ‘ with sufficient approximation. Thus ER ese Ia ais Wn i ee 8 The undisturbed propagation of the waves is thus verified. In general, B’ 01—d+1k(a,0,R—V) A’ oy+69+ik(oyogR+ VY’ ee oF 2 Al ayto,+ik(ojo,R+V) ° When o,—<; Is finite, the effect of the new terms is only upon the phases of the reflected and transmitted waves. In order to investigate changes of intensity we should need to consider terms of still higher order. When o,=o,, we have C= Arey oa (c *R+V) re Ale~HO@R+V)/20, Nee AEA N0) IN es oh Gi vad as BBD making, as before, C=A, if there be no interruption, Also, when o,= calculated,” are given the wave-lengths deduced from this equation, and under ““k calculated,” the ratio of these wave-lengths to the lengths of the oscillators. A consideration of the evidence shows that Abraham’s expression gives a result for the wave-length which agrees with the measured value within the present limits of experi- mental error. This was Ives’ conclusion in 1910, and the results now published add to his statement but the weight attached to confirmation from independent work. The pliysical accuracy of Abraham’s deduction is now sutticiently well established for linear oscillators of known dimensions to be used as standards in connexion with the measurement of short electric waves. These results completely support Lord Rayleigh’s t view of the value of the wave-length of the vibration on a thin straight terminated rod, and at least imply the experimental verification of his contention “that the difference between the half wave-length of the gravest vibration and the length (/) of the rod (of uniform section) tends to vanish relatively when the section is reduced without limit.” The experiments lend no support to Macdonald’s caleu- lation, which requires that the numbers in the table under the heading “é& observed” should be 2°5. It would appear, then, that Sarasin and De la Rive’s well-known experiments which, hitherto, have only been quantitatively described in terms of Macdonald’s theory, still await their explanation. The University of Sydney, September 24th, 1915. * Abraham, Joe. cit. t+ Rayleigh, loc. cit. POON XIV. On the Velocities of the Electrons emitted in the Normal and Selective Photo-Hlectric Lifects. By A. Lu. HueHEs, DSc., B.A., Assistant Professor of Physics at the Rice Institute, Houston, Texas”. | erie is a marked difference between the photo-electric effect in the alkali metals and that in other metals. When polarized light falls upon a polished metal, such as platinum or copper, the number of electrons emitted per unit energy of the incident light increases rapidly as the wave- length decreases. The emission of electrons is somewhat greater when the light is polarized in the Hii plane than when it is polarized in the EL planet. ‘This, however, is fully accounted for by the fact that when light is polarized in the Ell plane, more light is absorbed than when the light is polarized in the EL plane f. So, if we measure the number of electrons emitted per unit energy of absorbed light, the photo-electric effect does not depend upon the state of polarization of the light. With the alkali metals, however, the case is very different. Many more electrons are emitted when the light is polarized in the Hil plane than when it is polarized in the EH plane. The typical relation between the photo-electric current from an alkali metal illuminated by polarized monochromatic light and the state of polarization is shown in fig. 1. Pohl and Pringsheim § distinguish between two kinds of photo-electric effect, the ‘normal effect’ and the “selective effect.” The selective effect is shown only by the alkali metals, and then only when the light is polarized in the Ell plane and limited to a certain range of wave-lengths. The selective effect is the abnormall great photo-electric effect obtained with light (limited to a certain range of wave-lengths) polarized in the Ell plane. The number of electrons emitted with light of the wave- length at which the selective effect is a maximum may be from 20 to 300 times as great as the normal effect for the same wave-length. With unpolarized light of this wave- length, the photo-electric current is mainly due to the electrons emitted in the selective effect. Pohl and Pringsheim * Communicated by the Author. t+ We shall refer to the plane of polarization parallel to the plane of incidence as the Ell plane, and to the plane of polarization at right angles to this asthe El plane. In,terms of the Electromagnetic Theory of Light, KE denotes the electric force and the sign after it denotes the. relation of the force to the plane of incidence. t Pohl, Verh. d. Deutsch. Phys. Ges. x. pp. 389, 609, 715 (1909). § Pohl and Pringsheim,' Verh. d. Deutsch. Phys. Ges. xii. p. 215 (1910). Velocities of Electrons emitted in Photo-Electric Effects. 101 believe that the selective and normal effects have their origin in two independent processes. If this were so, one would expect that the electrons emitted in the selective effect and in the normal effect would have quite different velocities and | Photo -E, lectric Current., A2000 , "15000 distributions of velocities. The indirect evidence bearing on this point, however, suggests that there is practically no difference between the velocities of emission in the two effects. Certain definite laws have been established as to the relation between the maximum emission velocity and distribution otf velocities on the one hand, and the wave- length on the other. These laws have been found from experiments on metals outside the alkali group. But Richardson and Compton *, Kadesch +, and Millikan ¢ have included in their work experiments on sodium, and it is significant that there is nothing abnormal in the results for sodium. It has already been mentioned that, with unpolarized light, the majority of the electrons emitted from an alkali metal are those associated with the selective effect. The evidence from these experiments, therefore, does not point to any appreciable difference between the two effects so far as we can judge from velocities. It should be mentioned that Elster and Geitel § record an observation in the course * Richardson and Compton, Phil. Mag. xxiv. p. 576 (1912). + Kadesch, Phys. Rev. iii. p. 367 (1914). ¢ Millikan, Phys. Rev. iv. p. 78 (1914). § Elster and Geitel, Phys. Zeits. . 457 (1909). 102 Prof. A. Ll. Hughes on the Velocties of of a research on the selective effect which would seem to show that the electrons emitted in the selective effect were on the whole somewhat slower than those emitted in the normal effect. It was therefore thought desirable to investigate the velocities of the electrons emitted in the selective and normal effects. As the evidence available tends to show that the distribution of velocities of the electrons is the same in both cases, the apparatus was designed to show whether this was true or not. To find out exactly the nature of the difference Gif any) in the distribution of velocities would require a more complicated apparatus. It is highly desirable when working with sodium-potassium alloy, which is so sensitive to traces of reacting gases, to keep the apparatus as simple as possible. The apparatus consisted of a bent glass tube about 3 cm. wide (fig. 2). The inside was plated with a semi-transparent Fig. 2. blue screen. 1 ¥ariable averature. Ahhli---[tleareh film of platinuin except for the rounded ends which acted as windows. A guard-ring G served to prevent any electri- fication from leaking over the glass trom the platinized surface to the electrode H. Sealing-wax around the outside of the platinum seal making connexion with the electrode E gave satisfactory insulation. The sodium-potassium alloy was strongly heated before being put into the apparatus to drive off occluded gases. It was then placed in the first bulb connected with the apparatus (fig. 3), which was exhausted by means of a Gaede mercury-pump and charcoal cooled by liquid air. The apparatus was strongly heated for several hours while the exhaustion proceeded, and then, by suitably tilting the apparatus, the alloy was filtered from the first bulb into the second. ‘The alloy was heated for several hours again and then transferred to the third bulb. The apparatus was now sealed off from the pump and allowed to Electrons emitted in Photo-Electric Effects. 103 stand for two days, after which the alloy was finally trans- ferred to the bottom of the wide tube where it was to be illuminated. Fie, 3, 2nd. bulb ‘st, bulb 7 i ry H WG Sra. bulb’. The light was given by a mercury lamp. A Wratten blue filter was used to limit the hght to the wave-lengths 4360 and 4050, where the selective effect is near its maximum for sodium-potassium alloy. ‘The image of a small aperture in a sheet of metal close to the mercury lamp was focussed on the surface of the alloy just under the electrode H. ‘lhe image formed a spot of light about 4 mm. square, which did not shift more than 1 mm. on rotating the Nicoi prism. The amount of light received by the alloy could be altered by varying the aperture of the jens without altering the dimensions of the patch of light on the surface of the alloy. The charge received by the electrode EH was measured by a null method. Tne plate of a condenser of capacity 250 cm. was connected to the same quadrant of an electrometer as H. The potential on the other plate was altered at such a rate as to keep the needle of the electrometer undeflected. If the potential on the condenser has to be altered at the rate of V volts in a time ¢, the photo-electric current received by H is 2 OE — ) t where C is the capacity of the condenser. 104 Prof. A. Ll. Hughes on the Velocities of The method of experiment was as follows:—When there is no field between the alloy and KH, the electrons leave the illuminated surface in straight lines in all directions. When a field is established, the paths of the electrons are curved more or less towards the electrode E which now receives a greater current. A sufficiently large field will pull all the electrons into H. The number reaching E is a function (1) of the potential difference between it and the alloy, and (2) of the velocity and direction of emission of the electrons. There are two distributions of electrons to consider, a direction distribution and a velocity distribution. If the photo-electrons liberated in the selective effect and those liberated in the normal effect had identical velocity distributions and direction distributions, then one would obtain exactly similar curves connecting the charge received by the electrode H and the potential applied to the alloy. The current obtained with an accelerating potential of 404 volts is arbitrarily taken to be equal to unity, for both effects. The resultsare givenin Tablel. Some experiments TABLE I. Photo-electrie currents. Accelerating Selective effect Notmatiodier sae potential. (S). (N). = 404 volts. | 1:00 (=295x107? 7 amp.) 1:00 (=11-4x 10 —12 amip.)||) 1600 302 *962 910 1-06 87 627 "528 119 We 307 212 1-45 12'8 261 173 hee ey 209 133 37 3°9 ‘11d ‘0712 1:61 ae 0522 0291 79 100 |} -0116 ie Bie were carried out with the same aperture at the lens for both planes of polarization. In these cases, the selective effect with 404 volts accelerating potential was 25:9 times larger than the normal effect. In other cases, the lens was stopped down to one twenty-sixth part of its area when the selective effect was being measured, so as to make the selective and normal effects of the same order of magnitude. The Electrons emitted in Photo-Electric Effects. 105 results obtained were consistent, as was to be expected. It is clear that the ratio of the selective effect to the normal increases slowly as the accelerating potential decreases. In other words, the electrons in the selective effect tend to follow the electric field somewhat more readily than those in the normal effect. The results are plotted in fig. 4. The : cane YA Oe ee fete Photo-Electric Ci urrents. volts curve for the selective effect has paseticall become flat with 404 volts, showing that all the electrons emitted from the alloy, whatever be their direction of emission, have been received by the electrode E. It is unfortunate that acceler- ating potentials greater than 404 volts could not be used in order to find where the normaicurve became flat. We may, however, estimate that the normal curve becomes flat at about 500 volts. It is worthy of note that by multiplying the coordinates of the normal curve by suitable factors, the normal curve can be brought into close coincidence with the selective curve. These experiments show that there is a small difference in the way in which the electrons are emitted in the two effects, although it is quite small in comparison with the difference in the order of magnitude of the two effects. Is this difference due toa difference in the velocity with which the electrons are emitted, or to a difference in the direction in Phil. Mag. S. 6. Vol. 31. No. 182. Feb. 1916. I 106 Prof. A. Ll. Hughes on the Velocities of which they are emitted? If the selective electrons were slower than the normal electrons, but had identical direction distributions, then the smaller potentials would suffice to gather the same fraction of the total number of electrons emitted from the alloy into the electrode E. On the other hand, if the velocity distributions in both effects were the same, but if the concentration of electrons along directions near the normal were greater in the selective effect than in the normal effect, then again it would require smaller potentials to gather in a given fraction of the selective electrons than of the normal electrons. Hence we cannot decide from these experiments whether the curves imply a difference between the velocity distributions in the two cases, or in the direction distributions. To find out exactly the origin of the difference between the curves in fig. 4 would require a more elaborate experiment. It will probably be necessary to investigate both the direction distributions of the electrons and also their velocity distributions along each direction. The maximum emission velocity could be measured by noting the least potential applied to the alloy which would keep the electrode E from receiving any electrons. Owing to the arrangement of the electrode E relative to the illuminated alloy, the apparatus was very insensitive for the test. A difference in the maximum energies of °15 volt could have been detected; no difference could be observed in these experiments. Richardson and Compton, Kadesch, and Millikan, using unpolarized light, found that the maximum emission energy of the photo-electrons was a linear function of the frequency. Thus there is no indirect evidence from these results of any departure in the selective effect from the law obeyed by the normal effect. Experiments such as those made by Pohl and Pringsheim on the numbers of electrons liberated from the alkali metals by polarized light point to a fundamental difference between the two effects. ‘The appearance of the curve for the selective effect suggests strongly that we have to deal with some kind of a resonance effect, and that each alkali metal contains electronic systems whose frequency is characteristic of the metal. These systems become unstable the more readily, the closer the frequency of the incident light to this character- istic frequency. The electrons associated with the normal effect are not related to any system possessing a frequency characteristic of the metal. Thus, investigations of the number of electrons emitted in the selective and in the normal effects suggest different origins for the two effects. But there is no evidence from the various experiments on the Electrons enutied in Photo-Electric Hifects. 107 velocities of the electrons for the view that the two effects are fundamentally different. If then, for a moment, we assume that the two effects are ultimately identical, we must look for some intermediate effect which can account for the great difference in the number of electrons emitted. If, over a certain range of wave-lengths, the light polarized in the Ell plane were absorbed with unusual rapidity in the surface layers, then we should expect an unusually large emission of electrons, for they would have a greater oppor- tunity of emerging than if they had been released in the deeper layers. The optical properties of the alkali metals in tbe region of the selective effect are unknown, and therefore there is no evidence to justify this conjecture. We cannot say that, in the case of those metals whose optical properties have been investigated in some detail, there is any evidence to support such a view as advanced above, but then these metals exhibit only the normal photo-electric effect. Summary. Although the number of the photo-electrons emitted in the selective photo-electric effect of the sodium-potassium alloy used in these experiments is twenty-six times as great as the number emitted in the normal photo-electric effect, yet the way in which the two sets of electrons are emitted differs only to a small extent. ‘his small difference may arise either from a somewhat smaller average velocity of the photo-electrons emitted in the selective effect, or else from a greater proportion of the selective photo-electrons being emitted in directions closer to the normal to the surface. These results on the velocities of the photo-electrons in the selective and normal effects are hardly to be expected, if one concludes that the great difference in the magnitude of the two effects, and also their relation to the frequency of the light used, indicates a fundamental difference between the two effects. This investigation was carried out in the Palmer Physical Laboratory of Princeton University. I wish to express my best thanks to Professor Magie for permission to work in the laboratory and for placing every facility at my disposal. The Rice Institute, Houston, Texas, Oct. 28, 1915. Posing XV. Note on the Partial Fraction Problem. By I. J. Scuwart *. fl as the methods I have given before + I add the following : I. To separate > mv" pice Were pe I] (x? + c4) p=1 into partial fractions. 2 Aga + Be < 1 (ee) haere . . 3 ° 4 (1) This relation may be written LEAL take ae apne (at) The) eee ey (2) where tae 4 b,(@) = IT (a7 +0?) I (a?+0?). yi) Y y=rt+l if From (2) follows % mat" *= (Act + Be) b(@) +(e + 2i(e). (8) Substituting in this identity ic, for x, we have S mct-2i"-0= ob, (60,)(Aie, +B) Se a=0 To obtain A, and B, we separate the first member of (4) into its real and imaginary parts. We have n : iL), mye. 8 t= Bm, 00 — my — Mm, oC. te ee a=0 a=0 : 2[5 | +(—1) Dy COE] C. ey tn —102— nat n—1 2 +1 TOLD Da Oh eae 5 tmp en | i or 2 | i= il n E 2 e e a x mcr *= & (—1)%m, p60 +4 & (—Llm, 9, set (9) a=0 a=0 'a=0 * Communicated by the Author. + Quarterly Journal of Mathematics, no. 174 (1918); (-lL SM Nan Sea Ce" A= a 5 Une ute stai (6) II (2 —c?) I (2 -&) Meal mM ie Y=K+1 Y ‘i and i 5] >, ( aa Dn Gee a=0 Be= 34 F SU ey bisent tek (Gab) (vy Ec —c) We =K+1 Hence ; n—1 n =! 3 Pema og C& (HL) mga) 2 + (— 1)", _ 2 a=0 > + a=0 a=0 ene ul Ce ae <—t sa: in To separate > ma" a=0 ; Ria) = (Oe where n< 2p, into Partial Fractions, a ny Pa A “+B a=0 we AT De (a Ce or mio CaS (Ace Bel(t +2) age) Equating the odd powers of both sides of this identity, and also the even powers, we obtain [= *] y Mn —I9— 10? = Se ensetee eam yn a) a=0 K=0 and [2] & mn Mime "SB BOE ie i w SKAD 110 On the Partial Fraction Problem. Differentiating (3) and (4) / times with respect to a”, and letting «?=—c’, we have = me . (= Iie mae ee (;;) y) and n ey Pcie B,= 2 —1)*-?mp—20€ ( Ak Therefore 3 n 4] a 8 [EE (2 mcue™)s i + =(- Dak G m,. 20% | —_ ITY. To separate S= ——_., where p< 2q, into partial fractions. Now : I (ag 2) +: I (de 2) tt (a, +2) > Ut (Ga) It (a, +2) = aia A Suen K=1 os 9 the first part being of even and the second of odd degree. Hence we may denote Il (a,+2)+ I (a,—2) ri 1 ka and fe ; II (a,+ 2) — II (a,—2) k=l K=1 .ven 2 Tat hgh 2 MOK) = a =20("). Therefore 2 2 Oey @(2’) iy (a) Ta? (een) K=1 K=1 On the Sum of a Series of Cosecants. 111 Now O(a?) ese a HELA bs 11 (a? ¥ by) Sand i =F By, therefore : ee ee (be —b,) ue (ee —b,) Similarly _ b(a”) =3 5 Ba I (2 Dee A ae ; (ge b and f= n—1 is te 4 U (Orban) II (be —6,) K=n+1 and therefore es a ah n=1 II (dg—Dn) I (Gb) ers K K=n+1 University of Pennsylvania, Philadelphia, Pa., U.S.A. XVI. The Sum of a Series of Cosecants. By G.N. Watson, M.A., Fellow of Trinity College, Cambridge, Assistant Professor of Pure Mathematics at University College, London * 1. JN modern theories of the structure of the atom, the usual assumption is that it consists of a ring of electrons (repelling one another according to the ordinary inverse square law) rotating, in the normal configuration of the atom, in a symmetrical manner round a small positive charge. If n be the number of electrons, each of charge —e and mass m, and the positive charge be of magnitude ve (where y may or may not be equal ton), while ‘the radius of the atom is a, it is easily shown that, in steady circular motion, the angular velocity w of the system is given by the formula Zz 2 1 oo as (v—48n), where T 2er —l1)zr S,= cosec— + cosec— +... + coseo t= IF n nr n—1 => cosec (mrr/n). m=1 * Communicated by the Author. NDB 2s Prof. G. N. Watson on the The expression v—i8, plays a very important part in the investigation of oscillations about the steady motion, as well as in Bohr’s theory of spectra; and an analogous expression obviously occurs in the gravitational problem of a number of equal satellites rotating in a circle about a planet—a simplified form of the problem of Saturn’s rings, considered by Clerk- Maxwell. Reference may also be made to Sir J. J. Thomson, Phil. Mag. (6) vii. (1904) p. 247, to'G. A. Sehetts: Electro- magnetic Radiation,’ and to various recent papers by J. W. Nicholson * in the Monthly Notices of the R.A.S. For large values of n, the labour of numerical calculation, in determining §,, directly, is obviously severe ; to avoid this labour the need arises for a suitable asymptotic formula for the sum n—1 Se ee: (m/mn) ; in this paper a eee asymptotic expan a of 5,98 obtained, and, by means of this expansion, &, is tabulated for various values of n from 6 to 100 and also for n=360, n= 1000. 2. By using Euler’s well-known formula \ eee 7 l+x2 ~ sinam’ (O=a=1) a} 0 we see at once that 10 1 S.=| ——— fal pF... tO da TSr |, Peay + Pin +2 ldax =(" (at —ax)dax » ed +2)d—air) =n" (l—-y"*)dy Co) ten) Rk Neher aD aE or Ne (1+e-™)(e’—1) NS Cc AR Ie a), (te eaw, on summing the geometrical progression and then writing wm—y=e*; the final step, of bisecting the range of inte- gration, is justified by the fact (which is readily verified) that the integrand is an even function of ¢. * To whom I am indebted for pointing out the desirability of dis- covering the results contained in this paper. Sum of a Series of Cosecants. 113 3. The first stage of the analysis is now complete; it remains to transform the integral, which has just been obtained, into a form suitable for calculation when n is large. The following method is the only one, of several which I investigated, which produces the desired result : First transform the integrand thus, Se heed 2e—nt 1 i (1+e-™) (e’—1) 7 e&—1 T+e- Ne Ge YN e~ SPACE Sink 1 \] PN We pete Bee ee Loy 9 the terms e~‘/t are inserted to secure the convergence of the integrals of the expressions contained in square brackets. It is well oy ae \{aca-F ha where y¥ is Euler’s constant ae e.g., Whittaker and Watson, ‘ Modern Analysis,’ p. 240). Now * ae ie et — l P #3 Me ‘par-1 iat Bidj meal Se ae) Be On OB t2rt+1 TE ae ek where 0< 61 when t>0,r is any positive integer, and Iay, Bs, ... are » Bernoulli’s numbers, On ‘substitution in the second part of the integrand con- nected with 7S,, we get Qe -nt TEn= ny + 2n{_ | Toes +f” —— Az Aa abs ae Gmtt (-Y OB, capt te ns Piano is oa rf ee) 9 o—nt ie oe =2n{ eat onl” 1 PReHE hy + 2ny Cate Aco Ree +403 (2m (2m)! Eau it+4n (2r +2)! res Wich ae 1 (° urdu. pA Deter ee atthe eth 1? and 0<0,<1, by the first mean-value theorem, because 0051 and the integrand involved in IJ,,4, is positive. * Beara ‘Infinite Series,’ p. 234. Tort where 114 Prof. G. N. Watson on the If we use Frullani’s formula Lie 9) ES AURA Nea ih 2 ee { (Ore aee as = om 0 and write nf=u, the formula for 7S, becomes l—e-u a Tn mee =i ra at (— io) y2m-ldy (—)"*1B p41 2 2rtldy +4 oa ou i evu+ | my 40, (2r uy 2) ; fe se || . It will be observed that n no longer appears under the integral signs, so that this result may be expected to give the required asymptotic expansion when the integrals have been evaluated. To evaluate the integrals, let «> —1; then ede td On Ou \ iN, low ea MOCO ee uduy> ney ete the change of order of differentiation and integration being easily justified since a >-—-1 (Bromwich, ‘Infinite Series,’ pp: 436, 437). Now, « being greater than —1, we have { e~ au (eu 1 tdu={ en au jetne + Bee (—)P-le-pu 0 Ss (ea tau eu+] 2 es Oa atl oat2 """ a+tp ee ° e—(atpudy ; eu+] emi de in O ee e-Gatprtludy Now =I1/(a+p na 1D), and this tends to zero as p tends to infinity. Sum of a Series 07 Cosecants. Thus we have (~ era ih if 1 evti” at+l po | a 3iWwGet1)—WGet 3)5> where w is the logarithmic derivate of the Gamma-function. It follows at once that “udu i — (__)m9-—m-1 (m) SS AGpyal Now, when m>1, : i fi OIE oe ee + omei t+: 7 —— (—)™—-hn ! Om-+1) say 3 and nN am (L)=(—)™ 1m ae + Ae tose + Gee ; (3)"r rs 1 1 = ( za: ye iL 14 ents onl 4m+l Taek i ==(—)"—entt_y) _m! Oi is Consequently udu Raa, See ! (1—2-™) Omi. i ev+] Finally, we have to evaluate athe da Gr hud 1 penne A. rail aarp ce da baw “l ‘h oa the change of order of integration being justified, since a>—1 on the range of integration. 116 Prof. G. N. Watson on the On collecting our results, we now see that TS,= 2n log, (2n)+2n{y—- log rw}. ao § ("Ba 2™ ae | 9g (—) Beale m=1 LS Cn ae (e+1).n*tt This formula obviously has the property (which is sufficient to give an asymptotic expansion) that the remainder after ry terms of the series on the right has the same sign and is numerically less than the (r+1)th term, and this (r+1)th term tends to zero as n tends to infinity, r remaining fixed. By using the known formula o2,=2?"—1q7?"B,,/(2m) ! we may, if we wish, write (—)™Bi ar?" (2?" — 2) f (—)"2B,,(1— 21-2") oom oo or oo m.(2m) | mi in the series on the right. 4, For purposes of tabulation we make use of the values given for o2» in Chrystal’s ‘ Algebra,’ ii. p. 367. The first few terms of the expansion then give the approximate result nz 2n[0°7329355992 x logy (2n) — 0°1806453871 | a x 0:087266 +n-*? x 0:01035 —n-*> x 0:004-+777 x 0°005. In the construction of the following table the term in n-* could be neglected throughout, and the term in n= could be neglected when n> 20. It is worth observing that, in the special case in which the charge of the nucleus is numerically equal to the total charge of the electrons composing the ring, the expression for w? given in § 1 yields a real value for w only when n—18,>0, i. e. when n<472. 5. The analogous series To v= > e0see (“™ -38) m=1 n Z may be investigated in a manner similar to that in which S,, was considered. This series occurs in the radial com- ponent of force on an electron rotating round a nucleus in company with n—1 other electrons, when that particular electron is displaced along the circle an angular distance 8 from its position in steady motion; it is supposed that —2nrln +2nh le ee r —)"B (—)’*1B,. +4 2 ro ne Sees epi ae 1025 parti ert where Jin { ee du, eo (0 ' ' and 0<0,<1. 118 Pte. J. A. Hughes on the On using the formula of Gauss eo —t an ey | {a3 d oe e 0 t we get, without much difficulty, aT, =2n[log {dn cot (sdr)}—IW(1—An) —by(1 +n)] , (—)"Bn Buk 4 ve (2m) ! (Qnieet {re Dis —4n) tape HE +2) ~ye=- Lp —apOm-D LE ENG, the remainder after r terms having the same sign and being numerically less than the (r+ 1)th term ; where X is equal to B/(27). The corresponding series occurring in the transverse component of force is UU = 2 cos( 7 — 6) cosec? — 38) _9 ATs LO aoa TON this may be determined from the asymptotic expansion for T,, which may legitimately be differentiated term by term with respect to 2. XVII. On the Cooling of Cylinders in a Stream of Air. By Pte. J. AtrrED Hucues, B.Sc., H Coy, R.A.M.C., for- merly Research Student, University College of North Wales, Bangor*. HE interchange of heat between a solid and a moving stream of eas 1s a subject of:considerable technical import- ance, and also not devoid of scientific interest. A considerable amount of work has been done on the cooling of thin wirest in a current of air, mainly with the object of constructing instruments for the measurements of air-velocity. These experiments have shown that the heat lost by the wire is proportional to the difference of temperature and to the square-root of the velocity. There are, however, no data available relating to the convection of heat from ‘botlies of large diameters; and the following experiments were under- taken with a view of throwing some light on the problem of convection in these cases. * Communicated by Prof. E. Taylor Jones, D.Sc. + King, Phil. Trans. 1914, p.873. Morris, ‘ Klectrician,’ Oct. 4, 1912. 5) Cooling of Cylinders in a Stream of Air. ig Apparatus. The measurements were carried out on a series of copper tubes all of the same length, but with external diameters ranging from ‘433 cm. to 5°06 cm. These tubes, of about a metre in length, were placed vertically in the wind-channel, and were heated by steam from a boiler placed on top of the channel; after 2 steady temperature was attained, the weight of the water which condensed in the cylinder in a given time measured the heat which it lost through convection and radiation. The wind-channel, fig. 2, consisted of a wooden tunnel placed horizontally, which was of square section, of side 3 feet, and was 10 feet long. This terminated in a rectangular box of greater section in which the fan rotated. The fan was made by attaching four narrow steel biades to the axle of a three-phase motor, the speed of which could be regulated (a) by a rotor rheostat, and (b) by varying the voltage of the alternator supplying the current. A good range of wind-velocity could thus be obtained. The velocity of the air currents produced by the fan was measured by the ordinary Pitot-tube and gauge. This consisted of two glass tubes of 3%; inch bore, the dynamic tube facing the wind, and the static tube being placed with the plane of its orifice parallel to the direction of the flow. These tubes were placed close to one another in the centre of the channel, imme- diately in front of the cooling cylinder. By means of rubber tubing they were connected to a water-gauge, the difference of level in which measured the difference of pressure, which was proportional to the square of the velocity as given by v=V/ 2gh, h being the pressure difference in metres of air, and v the velocity in metres per second. The difference between the two sides of the U-tube was measured by a cathetometer microscope reading to ,4, of a millim. The gauge-tubes were 2 cm. wide so as to avoid surface-tension errors. Large oscillations were produced in the liquid in the gauge on account of variation in the wind-pressure, and this caused a difficulty in measuring the wind-velocity. The oscillations were damped, and the amplitude reduced by loosely stopping the bottom of the U-tube with cotton-wool. In some of the later experiments, thick paraffin was adopted as the liquid in the gauge. With this, the ditference of level was increased on account of the smaller density of the liquid, while the oscillations were reduced by reason of its greater viscosity. 120 Pte. J. A. Hughes on the The Pitot-tube determined the wind-velocity at the middle of the channel; and since an average value of the velocity over the cylinder was required, a determination was made of how the wind-velocity varied along a vertical line through the position of the Pitot-tube. A reading was taken with the Pitot-tube in its ordinary position: then readings were taken for the same wind velocity at different points across the channel. Thus an average value was calculated for the channel, and this was done for different velocities of the wind. A graph was plotted (fig. 1) showing the relation between the Pitot-tube reading at the centre of the channel and the average reading for the section. From this an average value can be read for any value of the axial wind-velocity. Observed fh in mms. Average h in mms. Calibration of Tunnel. The heat lost was determined by finding the weight of water condensed. The arrangement of the steam generator is shown in fig. 2. The boiler A is placed on the top of the channel, and the steam passes into the top of the tube through the water-trap B placed directly over the tube. The whole was shielded from air-currents by an asbestos cover. With cylinders of the largest diameters and with the stream-line tubes, a metallic steam-trap covered with cotton-wool was used, and the tubes were soldered to the bottom of the can as shown in the diagram. Cooling of Cylinders in a Stream of Air. 121 The condensed water was collected in a beaker placed underneath the tubes, a glass funnel being attached to the bottom end of the copper tube to ensure that all the con- densed water should be caughtin the beaker ; the excess of steam which came out with the condensed water escaped to the air past the sides of the tube. The end of the copper tube and the funnel were covered with cotton-wool. A wooden box was built round this receiving apparatus beneath the wind-channel, to shield it from air currents produced by the fan. The experiment was carried on in the following manner. The water was boiled and the steam allowed to pass through the copper tube for about five minutes, to ensure that the steady temperature had been attained. The first reading of the Pitot-tube was taken, the motor started, and after a short interval, a weighed beaker was placed underneath the end of the cylindrical tube, and the stop-watch started. Readings of the water “head” in the Pitot-tube gauge were registered, and after 10 minutes the beaker was taken away, and the motor stopped. The second zero of the gauge was taken, and the beaker and water weighed. The temperature of the air and the atmospheric pressure were also taken. Phil. Mag. 8. 6. Vol. 31. No. 182. Feb. 1916. K 172 Pte. J. A. Hughes on the In the case of the large cylinder, 20 cm. length, 15°5 em. diameter, the arrangement of the apparatus is shown in fig. 3. The condensed water was allowed to collect in the cylinder Citself, the whole being suspended from a balance B placed on the top of the channel, the steam connexion being made by rubber-tubing R. When the balance-beam is down, the cylinder rests on a stand S in the channel ; the excess of steam escapes through the exit H. The water condensed, which measures the heat lost, is estimated by taking the weight of the cylinder at the beginning and end of the interval of time, viz. 10 minutes. Steam is passed through for about 5 minutes, and after the steady temperature has been attained, the weight is taken and the motor started. After 10 minutes the motor is stopped and the weight again taken. The weight of the water condensed will be the sum of three parts: (1) that condensed by convection and radiation in still air in the interval from the first weighing till the motor was started; (2) the condensation during the wind-current; (3) the con- densation in still air during the interval between the stopping of the motor and the time when the final weight is taken. The “water equivalent” of the excess of time above 10 minutes is found, and this subtracted from the total weight gives the weight of water condensed by the wind alone. Still-air condensation is small, as can be seen from Table II., and therefore no serious error is introduced by this method. Corrections. Most of the heat is lost from the cooling cylinders by convection, but some also by radiation. A correction can be applied for the heat lost by radiation, and this subtracted from the total loss of heat gives the heat lost by convection alone. The radiation correction was calculated from the equation R=k(#*—@,*), where R is the heat lost by radiation; k the coefficient of emissivity in gram-calories per sec. per sq. cm.; §—6, the excess of temperature of the hot cooling pede above the temperature of the surroundings. k* was calculated from Bottomley’s results to be ‘63 x 10-” gm.-cals. per sec. per sq. cm. for tar- nished copper; * Phil. Trans. vol. clxxxiv. p. 591 (1898). Cooling of Cylinders in a Stream of Air. 123 and therefore R per cm. length =k 1(6!—0,'), A=area in sq. cms. 1 =length in cms. ¢ =time in secs. The values obtained are given in Table I. TABLE I. RB (cals. per cm. length). | Diameter of Cylinder. 4 calories. "43 cm, 124 |... ‘Slane. 28°7 be) 1°93 bP) ae 506, | 2343, Us are | Se Stream-line Section. It has been assumed that the temperature of the outside surface of the cooling cylinder was the same as the tempe- rature of the steam. This is practically true, as can be shown by calculating the value of 6,—0@, from Hd (92-01) = QerrkT? 6,—6, being the drop of temperature in the tube, H the largest value of the heat which passes across the tube per cm. length in time T, 7 the mean radius=15°4 cm., d the thickness =°2 cm., k the conductivity of copper. The value of 6,—0, is (03° C., so that the temperature of the outside of cylinder can be taken to be 100° C. The quantity of water condensed depends on the density and the temperature of the air, so to obtain uniform results acorrection was applied, and the quantity of water was reduced toa standard temperature and density. Results. Experiments were performed with five copper cylinders of diameters *43, °81, 1°93, 5°06, 15°5 cm. respectively, and the results obtained for the loss of heat in calories per em. length (H) and the velocity (V) in metres per sec. are given in lable IL. K 2 08-6 S6FE 86-4 9-218 960-8 1-60¢ OFL-01 8-198 801-8 6-18 991-8 OTs 9-9 8-EGL 9LE-L ¢-89F CSL-O1 oss | 899.8 8-686 8-8 1808 088-9 gel POEL L329 =| Ss PPP-G qgeg |: 9ST-8 6-063 ETL 8183 PPL-G 8.189 849-9 Sch | (96-6 Lees | (OGL 9.916 986-¢ L093 OFG-¢ G-989 O6F-¢ 6-907 801-6 6968 | Gord 6-606 806-4 029 P¥6-P L889 OP-F 9498 | 994-8 L91g | 8FL-9 9-981 966-4 1GES 9LL-F G09 PLP L-C&E 968-8 6-918 | FPG 8-161 ES 806-F 68ZG IGL-8 6-109 L-? LES 8FS-8 9818 || GGE-F 1-991 = 999-6 COLI 88-8 b-19P 6FG-E 1-40 796-1 PCIE | IIes L-F91 s 0 a9 LLL €-6L8 980-8 8-666 £60-L 8.616 0 189 = 0 8-061 888: 6-676 199-8 $906 © 0 $18 0 6-28 at ee SSS Se | | See Z "A ‘HD "A a “A ‘H "A ‘H "A ae : ‘Ud G.¢[ = wWvIg "U0 90.G= "WRIT "umd E6.[= 'Weiqg | "Ud [Q.=‘UlBIG "WO Ep. =" UBITT 0O angle of contact. Thus if we plot values of 7 as abscisse and values of R as ordinates, we get a curve the inclination of the tangent to which at the origin is tan-! (cos @). If the tangent makes an angle of 45° with the axis of «, the value of @ is 0° and the liquid may be truly said to wet the tube. The following tables give readings of r and R for glycerine and turpentine respectively :— Turpentine. Glycerine. i R. 7 | R 385 674 "385 549 247 341 “247 284 162 178 154 16 128 137 09 093 09 | 093 | ‘038 038, OBE oa) O88 | 148 Measuring Surface- Tension and Angles of Contact. The curves (fig. 2) were obtained by plotting r against R for water, glycerine, olive oil, turpentine, and mercury. It will Radius of Curvature. Radius of Tube. be noticed that all appear to approach the origin at an angle of 45° with the exception of mercury. This points to a zero angle of contact with all except mercury. Mercury gives an angle of nearly 41°, or rather, a little greater than 180°—41°=139°. University College, Galway 149 2 XXI. The Complete Photoelectric Hmission*. By Professor O. W. Ricwarpson, F.R2.S., Wheatstone Professor of Physics, University of London, King’s College. [ is well known that when light of sufficiently short wave-length is allowed to fall on metals, an emission of electrons takes place under its influence. Since all substances emit light when they are raised to a high tem- perature, an emission of electrons from hot bodies owing to the action of light will occur even when they are not illuminated from an external source. The emission of electrons which arises in this way may conveniently be termed the “complete photoelectric emission,” to indicate that it is excited by the complete (black body) radiation with which the material is in equilibrium at the temperature under consideration. It is of interest to enquire whether this emission will resemble in its behaviour the thermionic effects which have been investigated experimentally, and to seek to determine how its magnitude compares with that of the observed thermionic emission. The most striking property of the thermionic emission of electrons is that expressed by the current temperature relation Pe OMe AE Gh BI ey (¢=maximum current from unit area, T=absolute tempe- rature, A, A, and 6 constants). The writert has shown that it follows from simple thermodynamic considerations that the complete photoelectric emission is also governed by an equation of type (1)§. The thermodynamical argument does not determine the numerical value of the constants A and b (X is unimportant) which enter into the formula for the complete photoelectric emission. It does, however,. determine the meaning of b, which is closely related to the work done by an electron in escaping from the metal. The values of the minimum frequency of the light which is able to excite any photoelectric emission from metals show that b is not very different in the case of photoelectric and thermionic emissions from a given material. It is not * Part of an address delivered by the Author in opening the dis- cussion on Thermionic Emission at the Manchester Meeting of the British Association, 10th September, 1915. + Communicated by the Author. ¢ Phil. Mag. vol. xxiii. p. 619 (1912). \ § The same conclusion has been reached by W. Wilson by a direct oe of the quantum hypothesis (Ann. der Physik, vol. xlii. p. 1154 (1913)). 150 Prof. O. W. Richardson on the certain that the values of 6 for the two effects are not identical for the same substance. The value of A, on the other hand, is left entirely arbitrary by the thermodynamic considerations, which do not therefore enable us to determine the scale of magnitude of the complete photoelectric emission. Thus the theoretical argument leads us to the conclusion that the complete photoelectric emission varies in the same general manner with temperature as the observed thermionic effects: it is possible, although not certain, that the indices rand 6 are identical in the two cases contrasted, in which case the two emissions would be in the same relative pro- portion at all temperatures ; on the other hand, the absolute value of the complete photoelectric emission is left entirely undetermined, so that we are unable to determine by such calculations what proportion it bears to the observed therm- ionic emission. This information can, however, be obtained from known photoelectric data in the case of the metal platinum. The calculations are not exact, but it is improbable that the sources of uncertainty in the calculations will lead to errors in the final results which are as great as those pertaining to the experimental measurements of the absolute values of photoelectric and thermionic emissions from a given material. Data* now available give the number of electrons emitted from certain metals, including platinum, when unit light energy of the different effective frequencies falls on them at normal incidence (or at some other angle which is definitely specified). The magnitude of the complete photoelectric emission will not, however, be obtained if we simply multiply this number by the corresponding intensity of the light in the black-body spectrum and integrate the product over the whole range of frequency, on account of the different optical conditions in the two cases. In the photoelectric experiments in which a beam of light is incident normally, the intensity of the exciting illumination is greatest at the surface and falls off exponentially as the depth of penetration increases. In the natural emission, on the other hand, the electro- magnetic radiation is isotropic, and its intensity is the same at all depths. This particular difference between the two cases can be allowed for if we have a knowledge of the coefficients of absorption of the electromagnetic radiations of different wave-lengths and of the electrons which they cause to be emitted. It is usual to assume that both the light and the electrons * Richardson and Rogers, Phil. Mag. vol. xxix. p. 618 (1915). Complete Photoelectric Emission. 151 are absorbed according to an exponential law, although such an assumption can be regarded as only a very rough kind of approximation when applied to the very slowly moving electrons with which we are now dealing. In reality the problem involved here is a very complex one: the stream of electrons which travels in a given direction is depleted in number both through the electrons being stopped and through scattering, and it also suffers loss of energy. Little is known definitely either as to the relative importance or as to the precise etfect of these different actions. In addition, the electrons which escape lose most of their energy in passing through the surface ; although this fact need not prevent their absorption being approximately exponential when they are travelling in the interior. In any event, the exponential law of internal absorption is the only assumption with which it is possible to arrive at any result in the present state of the subject. | Consider first the case of a beam of light of definite frequency incident normally, and let I be the energy crossing unit area just within the surface of the metal in unit time. Of the electrons ejected from atoms ina layer of infinitesimal thickness perpendicular to the beam, let the proportion e~* reach a parallel plane at a distance x from the layer. Let @ be the coefficient of absorption of the light and N the number of electrons ejected from the atoms of the metal when unit energy is absorbed by them from the light. A caleu- lation following well-known lines, which allows for the absorption of both light and electrons in accordance with the assumptions just indicated, shows that the number N, of electrons which reach unit area of the surface bounding the metal in unit time is given by I ee NL ea We assume that a definite fraction, y, of these escape from the surface. In general y will be a function of the frequency of the light, and will be different for different materials. Using Planck’s notation, the intensity i,dv of the isotropic natural radiation inside the material, within the frequency range from pv to v+ dy, is ” hy dy adv ==) D givikD 1? eV hemittal! We\h' .e (3) where 7 is the real part of the complex refractive index of the material. A calculation similar to that leading to (2) 152 Prof. O. W. Richardson on the except for the fact that the intensity of the radiation is now constant at different depths, shows that this radiation delivers at each unit area of the surfacein unit time the number N,dy of electrons, where Nog aN ay ait a I a LAV, yh anne from (2). This method of calculating assumes that the liberation of electrons from the atom by light is a consequence solely of the absorption of the light and is not a result, for example, of the process of emission of light. The total number of electrons which escape is obtained by multiplying this by the factor y and integrating throughout the spectrum. The complete photoelectric emission current J is therefore equal to x 87re( a+ ByN hy? J ey eyN.dv= wu ( 2 ae gwlkD _ 1" e (5) Cv, For any particular temperature T, the integral on the right- hand side of (5) can be evaluated graphically if we know B/e, 7, and yN,/I for all frequencies. In the case of platinum, values of (1—p) ce for all frequencies for which the factor hy*/(e”*"—1) is appreciable have been given by Richardson and Rogers*. In this expression p is the reflexion coefficient from platinum at normal incidence for light of frequency v. This quantity has only been measured over part of the effective spectrum, but the data given by W. Meier t, which extend as far as = 2°57 x 107° em., show that we shall not go far wrong if we take p as having an average value 1/3 over the region under consideration. This region extends from X=1°9 x 107° cm. to X=2°75 x 10-° cm. approximately, the effective average value of % being in the neighbourhood of 2°46 x 107° cm. Data also given by Meiert show that the effective average value of r? can be taken to be not far from 1'3 over the same region. The only datum bearing on the relative values of « and 8 is an observation by Rubens and Ladenburgt, who found that when ultra-violet light passed through a thin gold leaf the emission of electrons from the front side was 100 times as great as from the side of emergence, whereas the intensity of the incident light was 1000 times that of the emergent * Loe. cit. + Ann. der Physik, vol. xxxi. p. 1027 (1910). t Ver. der deutsch. Phys. Ges. ix. p. 749 (1907). Complete Photoelectric Emission. 153 light. From these numbers Partzsch and MHallwachs* have calculated that for gold #=1:03x10° cm.7? and 8=0°59 x 10°cm.~!. Since gold and platinum do not differ much from one another in atomic weight and density, the value of the electron absorption coefficient « will almost certainly be much the same for both metals. Meier’s katoptric measurements with gold at wave-length X=2°57 x 10-° give numbers which lead to @=0°56 x 10° cm.~!, in satisfactory agreement with that deduced from Rubens and Ladenburg’s observation. The absorption coefficient for platinum in this part of the spectrum is larger than that for gold in the ratio 1°65 to 1:14 according to Meier; so that in dealing with platinum the value of @ should be increased to 0°85 x 10° cm. ' instead of 0°59x10°cm.-!. In this way we estimate the value of (~+)/« over the part of the spectrum which is photoelectrically active to be (eS CH a cie IoP UE MPU Me AU (55) In arriving at the value of the complete photoelectric yt emission from platinum at 2000° K. the product ee has been taken outside the integral in (5) and the foregoing average values substituted in it. This is not exact, but it is the best that can be done with the data at present available, and in any event it is accurate enough for the immediate purposes of these calculations. The remaining integral i) 3 { (1—p) ue us “— dv has then been evaluated graphically, e 0 ekT een using the values of pee € given by the measurements of Compton and Richardsonf{ and Richardson and Rogerst. It is not claimed that the results of these calculations give more than arough idea of the magnitude of the complete photoelectric emission. For one thing, the absolute value of the photoelectric emission when measured under a given illumination varies considerably with the state of the metal surface, and in a manner which cannot be said fully to be understood. In the second place, a number of approximate or doubtful assumptions are involved in the calculations. Of * Ann. der Physik, vol. xli. p. 269 (1913). + Phil. Mag. vol. xxvi. p. 549 (1913). - { Loe. cect. Phil. Mag. S. 6. Vol. 31. No. 182. Feb. 1916. M 154 Prof. O. W. Richardson on the these, the uncertainty as to the exact law of absorption of the electrons inside the metal and the approximation involved in taking certain factors outside the integral have already been alluded to. In addition, it is likely that y will be alittle different for the electrons liberated by light of a given frequency when isotropic and when incident normally. It is, however, quite improbable that any of these considerations can introduce erroneous factors large enough seriously to affect the order of magnitude of the final result. Proceeding in the way indicated above, it appears. that the value of the complete photvelectric emission current density for platinum at 2000° K. should be i= 2°1x 107" amp. per em.?) 2) ee Let us now compare this value with the observed thermionic electron emission for platinum at the same temperature. In considering these questions platinum has been used because it is the only element for which the requisite photoelectric, thermionic, and optical data are available. There is one: further point. The value (7) has been obtained on the assumption that the hot body is in an enclosure surrounded by radiation in equilibrium with it at its own temperature. This condition is not satisfied in the thermionic measurements. Most of the electromagnetic radiation emitted by the hot body never returns to it. The radiation density inside the hot body will thus be something less than that assumed in calculating the value (7). Thus (7) is to be regarded, subject to the limitations as to accuracy already referred to, as an upper limit to the photoelectric contribution to the electron emission from the hot body. The following values of the thermionic current densities at 2000° K. from a number of elements, including platinum,, have been found by Langmuir under very good conditions as to freedom from gaseous contamination:— OMEN EG: 2 4) aioe os — > W. Ta. Mo. Pt; C. Thermionic current -3 vant Ley ay, ms (amp. per cm.2). —> 3x10 7X10 13x10 6x10 10s These are enormously greater than the value (7) of the complete photoelectric emission calculated for platinum at this temperature. The smallest thermionic currents from platinum ever recorded in the neighbourhood of 2000° K. Complete Photoelectric Emission. 155 are those observed by H. A. Wilson* with well oxidized wires. He found 4x 107° amp. per cm.? at 1686° C., which corresponds to nearly 10~‘ amp. per cm.’ at 2000° K. Hven in this extreme case the observed thermionic current is about 5000 times as large as the complete photoelectric current. We are thus led to the conclusion that the complete photoelectric emission gives rise to an insignificant portion only of the observed thermionic currents. While this con- clusion cannot be held to be established with absolute certainty, on account of the doubtfulness of some of the factors entering into the calculations, it does, nevertheless, appear extremely probable on the evidence. In view of its importance, it is desirable that the question should be settled quite definitely ; but it is questionable whether any considerable advance on the present calculations can be effected without a material extension of our knowledge of the conditions involved in photoelectric action. If photoelectric action had proved successful in accounting completely for thermionic emission of electrons, one would have been tempted to speculate further, and to ask whether ordinary evaporation—the emission of uncharged molecules instead of charged electrons—might not be attributable in a similar way to the direct action of radiation. One could imagine the emission of molecules in evaporation to be the result of a kind of photo-chemical action, the effective radiations being mainly in the infra-red. Such a position would obviously be in line with current ideas as to the nature of heat-energy which are based on the quantum hypothesis. It is not improbable that certain radiations may be found to have a selective and direct effect in increasing evaporation, in addition to the quite obvious but less direct effect due to the heat generated by the absorption of the radiation ; and this effect is worth looking for. But judging from the analogy with the relation between ordinary photoelectric action and the emission of electrons from hot bodies, it seems likely that such effects, if they exist, will account for only a small fraction of the emission of molecules in ordinary evaporation. * Phil. Trans. A. vol. ccii. p. 262 (1903). M 2 Ee XXIT. The Quantum of Action. By WiuvtaM Wison, PhD., Lecturer in Physics, University of London, King’s College ™. o a recent paper on the Quantum Theory +, Jun Ishiwara starts out from an assumption which closely resembles an hypothesis published by me about the same date{. The aim of Ishiwara’s theory also appears to be the same as that of mine—namely, to furnish a common basis on which the theory of complete (black body) radiation, spectral series, and other phenomena can be established. It is therefore desirable that the two hypotheses should be compared and the differences between them clearly pointed out. Ishiwara’s assumption may be stated as follows :— “Let gig2---.qj and pipe....p; be the positional and impulse coordinates of an elementary material system in a state of steady periodic motion, or of a system consisting of a very large number of elementary systems in statistical equilibrium, and let each pair, g;, p, be represented by rectangular coordinates in a plane; then the motion of the system is such that we may divide each plane into regions of ‘equal probability,’ whose mean value, for any state of the system, 1 7) j x \ pi dq:= h is a universal constant ’’ §. The hypotheses which I have proposed as a foundation for the Quantum Theory are stated at some length in the paper mentioned above ||, and may be put rather more shortly in the following form :— (1) Hach dynamical system behaves as a conservative one during certain intervals, and between these intervals are relatively very short ones during which definite amounts of energy may be emitted or absorbed. (2) The motion of a system in the intervals between such discontinuous energy exchanges is determined by Hamil- tonian dynamics as applied to conservative systems. We may speak of a system, during such an interval, as being in one of its steady states. * Communicated by Prof. J. W. Nicholson. + J. Ishiwara, Toky6 Stgaki-Buturigakkwai Kizi, 2nd ser. vol. viii No. 4, p. 106. t W. Wilson, Phil. Mag. xxix. p. 795 (1915). § This is a free translation from the original, which is in German, || W. Wilson, Zoe. ett. The Quantum of Action. 157 (3) The discontinuous energy exchanges always occur in such a way that the steady motions satisfy the equations :— Sp: dq,=ph \ps dgg=ch \ ps dq3=th where p, o, T,....are positive integers (including zero) and the integrations are extended over the values of q;, p; corre- sponding to the ee The factor A is a universal constant. Ke Here also we may represent each pair g,, pi by rectan- gular coordinates in a plane, and it is a consequence of the third hypothesis that the state of any system, at any given time, will be represented by a point on a certain locus in each g, p plane. These loci divide all the planes into equal areas, This third hypothesis is clearly very different from Ishiwara’s, though it is obvious that it will lead to similar and, in some cases, identical consequences. The essential differences between the two assumptions are :— (a) When a large number of systems are in statistical equilibrium their representative points are, in Ishiwara’s theory, uniformly distributed through any region in aq, p plane. These regions have not necessarily equal areas, but the mean area of the regions in which the representative points of any one system are situated is equal to the universal constant A. In my theory each g, p plane is divided into regions of equal area, and the representative points of a system are always on the boundaries between these regions. (6) Another distinction between the two theories is con- nected with the limits of the integral Sp dg. Ishiwara extends the integration over the whole period of motion of the system, whereas in my theory it is extended over the period corresponding to the coordinate concerned. A short critical discussion of the various forms of Quantum Theory, especially those of Planck, Ishiwara, Bohr, and that recently put forward by me, will not be 158 Dr. W. Wilson on the out of place here. Regions of “equal probability” consti- tute a feature of Planck’s later theory and that of Ishiwara. The essentials of Planck’s theory, which appears to me to be included as a special case in that of Ishiwara, can be very shortly stated. He deals with a very simple type of system, the equation of motion of which is , d? m2 +Kg=0 when it is in a steady state, 2.e. neither emitting nor absorbing energy. We easily deduce the following relation between p and g :— 2 2 p CEN Orn ke ee where pam , A is the amplitude of the motion, and v the frequency. ‘This is the equation of an ellipse. For different amplitudes we have a number of similar and similarly situated ellipses, and it can be shown in a very simple way that the energy of the system, or oscillator, is equal to Hy, where H is the area of the corresponding ellipse. This result is obtained by the use of the principles of ordinary dynamics. If, now, we suppose the oscillators to be capable of interchanging energy with one another, and proceed to investigate, by the same principles, the law of their distribu- tion throughout the g, p plane, under statistical equilibrium, we assume the distribution to be uniform in an infinitesimal region dp dqg=dH, and write dN=Nfdp dq, where N is the number of oscillators and dN the number in the region dpdg. We readily find i LA ae ae where H is the area of the ellipse on which the element dp dq is situated and k is the “gas constant” reckoned for one molecule. This result may be called the Maxwell law of distribution. It leads to the conclusions, (a) that the * We are assuming, as Planck does, that the directions of vibration are perpendicular to a fixed plane, and that the oscillators are fixed. Quantum of Action. 159 average energy of an oscillator is AT, and (6) the Rayleigh- Jeans law when equilibrium is established between the oscillators and the ether. These conclusions not being in accord with experimental facts, Planck has recourse to the following main hypothesis. He supposes the oscillators to be uniformly distributed through a finite (not infinitely small) region H’—H=A, bounded by two of the ellipses mentioned above. The whole of the q, p plane is thus divided into areas each equal to h. These are his regions of ‘equal probability.” It is now seen that the area of any ellipse separating two such regions can be expressed in the form \pdq=(n+)h, where n is any positive integer (including zero) and the average energy of the oscillators in the region is E,=(n+34)hy. The law of distribution is now ERER an) any f=(l-e @)e *, where 7 is the fraction of the total number of oscillators situated in the region n. The average energy of all the oscillators is = af 1 K=hAv ( 5 a= =| - efT_} There are, in my opinion, objections to the hypothesis of regions of “equal probability.” It appears to involve energy exchanges of a continuous character side by side with discontinuous exchanges. Indeed, Planck assumes * that his oscillators absorb energy in accordance with ordinary dynamical laws, but emit it in a discontinuous manner. So long as no other way of accounting for the phenomena concerned can be found, such hypotheses may be justifiable ; but no theory can be held to be satisfactory which applies one system of dynamics to the process of absorption and another to that of emission. This, however, is what is done in the theories of Planck and Ishiwara f. Moreover, there is at least one consequence of the con- tinuous absorption hypothesis which is not in good accord * M. Planck, Theorie der Warmestrahlung, p. 150, 2nd Edition. + Cf. Jeans, ‘Report on Radiation and the Quantum Theory,” Phys, Soe. 1914, p. 82. 160 Dr. W. Wilson on the with experimental results. It demands a _photo-electric “accumulation period.” The experiments of Marx and Lichtenecker* seem to show, however, that this period (if it exists at all) must be much shorter than that calculated from the principles of Planck and Ishiwara. The later theory of Planck and that of Ishiwara are not, and cannot be, extended to ether vibrations, since such an extension would involve the consequence that the ether would possess energy at the absolute zero of temperature. In fact, Planck, in developing his theory, has been at some pains to leave the ether within the domain of ordinary dynamical methods. In consequence of this, he is forced to load his theory with a highly arbitrary additional hypo- thesis connecting the probability of an emission of an oscillator and the intensity of the radiation of like frequency in its neighbourhoodt. These objections cannot be advanced against the form of Quantum Theory I have suggested, while it has the advan- tage over those of Planck, [shiwara, and Bohr that only one hypothesis is used which goes beyond ordinary dynamics. This is the hypothesis (3) stated above, which lays certain restrictions on the interchange of energy between material systems and between such systems and the ether. It is not claimed that this hypothesis goes far enough, or that it is sufficiently complete to lead in all cases to unambiguous results; but it leads, I believe, to all those results of the other theories mentioned which are in agreement with experimental facts. Some of these have been dealt with in my previous paper [. It leads to HWinstein’s law without requiring an “accumulation period.” This can be shown in the following way :—The average energy of the vibrations in the illuminated solid is given by hy 3 eft —] if we only consider vibrations perpendicular to a fixed plane. Now, for the whole range of temperatures within which photo-electric measurements have been made, and for values of v in the ultra-violet or even in the visible part of the spectrum, this expression reduces to an exceedingly small fraction of hv. Most of the vibrations therefore in this range of frequencies have no energy at all. The number having the energy of Ay will be a very small fraction of the * HE. Marx & K. Lichtenecker, Ann. d. Phys. xli. p. 124 Se) + M. Planck, loc. cit. p. 159. LN, Wilson, loc. cit. Quantum of Action. 161 total number, and for systems having the energy 2hv the fraction belongs to the second order of small quantities. Tt follows therefore that in all but a very small number of cases the energy emitted must be equal to hy, and if an emission is accompanied by the ejection of an electron, its kinetic energy cannot exceed this value. A very important consequence of hypothesis (3), to which I wish to draw attention here, is that any emission of energy of frequency vy to the zther must be equal to | nhv, where n is an integer. Such considerations as those in the last paragraph lead to the conclusion that n must be unity for frequencies in the visible range of the spectrum and for higher frequencies. This result appears in the theories of Bohr and Ishiwara as a special assumption which cannot be deduced from the other hypotheses they adopt. Any form of Quantum Theory must involve assumptions intimately connected with physical quantities having the dimensions of an action, and the most important of these is angular momentum. The hypothesis (3) given above makes the angular momentum of a system equal to nh 20 (where n is an integer), even in the case of an electron the orbit of which is elliptical*. This is not inconsistent with Bohr’s hypotheses, and coincides with his views in the case where the ellipse degenerates into a circle. The assumption (3) lays certain restrictions on the possible values of the eccentricity of the electron orbit. This can be shown as follows :—Let g, represent the distance of the electron from the nucleus and gq, its angular distance from a suitably chosen fixed line. Then we have for the equation of the ellipse ee) 11 T+ cos do. and therefore r= = SIN Yo, and om i TAS ine * sin? g,.dq2 {p: Hla Sa + COS Go)?’ * That there is a real relationship between angular momentum and the constant 4 was first noticed by Nicholson some years ago (Monthly Notices of R. A. S. xxii. p. 679 (1912)). 162 The Quantum of Action. This can be transformed into (" 2 fp: dq= 16. 7 axrdx where a=l+e and B=l1—e. On evaluating the integral we get Ut 22)t Now, by hypothesis (3) \p: dqy =ph, and 21 p,= ch, where p and o are positive integers. When these values are substituted we have (p+o)(1—e?)F=o. The part which is common to all forms of the Quantum Theory can be described very simply. When systems are not exchanging energy, their equations of motion are obtained by giving a stationary value to the integral j 2L dt, where L isthe kinetic energy of the system, subject to the condition of constant energy (Principle of Least Action). The Quantum Theory lays restrictions on the interchange of energy between different systems. These restrictions require that under certain circumstances the following equation must hold \2L dt=nh, where fA is a universal constant, n is an integer, and the integration is extended over the period of motion of the system. Wheatstone Laboratory, King’s College. November 1915, bey 1639p] XXII. Ona Comparison of the Are and Spark Spectra of Nickel produced under Pressure. By EH. G. Bitnam*, wesc. A. te. C.Se. ia a recent paper T, Prof. W. G. Duffield has made a detailed comparison between the results of his investiga- tion of the effects of increase of pressure upon the arc spectrum of nickel {, and those obtained by the present writer §, in which the mode of excitation was the spark between nickel electrodes connected to the secondary coil of a resonance transformer. Many interesting points of simi- larity are to be observed, but one’s attention is immediately arrested by certain very remarkable differences. The most noteworthy feature of the results is that, whereas the mean displacements towards the red of the lines in the spark spectrum are, upon the whole, in reasonable agreement with the means derived from the are spectrum—assuming direct proportionality of displacement to increase of pressure, —the displacements in the arc under ten atmospheres pressure are, on the average, about twice as great as in the spark spectrum. An exception is to be observed in the case ot lines which broaden slightly but symmetrically under pressure, without reversal, 2. ¢., lines of Class III. in Gale and Adams’ notation. These lines appear to suffer approxi- mately equal shifts in the two cases. For other unreversed lines the results are somewhat discordant. This is not surprising when one considers that the diffuseness of the lines under pressure renders the measurements rather un- certain even under the most favourable circumstances. From the point of view of making a rigorous comparison between the results of the measurements of the displace- ments produced under the two conditions, it was somewhat unfortunate that the pressure used for my main series of experiments (namely, ten atmospheres above the normal pressure) should have happened to coincide with a region where Duffield observed an abnormally high rate of shift per atmosphere. The means at my disposal prohibited the use of pressures much higher than ten atmospheres. Apart from the fact that very radical alterations in the design of * Communicated by the Author. + Phil. Mag. Sept. 1915, p. 385. { Duffield, ‘‘ Effect of Pressure upon Are Spectra,” No. 5, Nickel A 3450 to A 5500, Phil. Trans. A 215, p. 205. § Bilham, “The Spark Spectrum of Nickel under Moderate Pres- sures,” Phil, Trans. A 214, p. 359. 164 Mr. E. G. Bilham on a Comparison of the the pressure chamber would have been necessary, the diffi- culty of maintaining a spark under pressure is by no means inconsiderable. The chief source of trouble lay in the rapid disintegration of the electrodes. Working with very small spark gaps, of the order of about a millimetre, a frequent result was that a short circuit was produced by a detached fragment of nickel lodging between the electrodes. Another frequent mishap was that the spark gap rapidly increased in width until a very violent discharge between projecting parts on the outside of the pressure chamber announced the fact that an easier path had been found. Clouding of the window was also a serious source of trouble. Turning to a different aspect of the case, an inspection of the appearance . of the photographs produced under a pressure of eleven atmospheres points very forcibly to the conclusion that, in very many instances, measurement of the spectrum “ lines” was rapidly approaching the limit of possibility on account of their diffuseness. The above considerations will perhaps serve to show that the investigation of spark spectra under high pressures is” attended with formidable difficulties, although such data would unquestionably be of great interest and value. Our knowledge of the variation of displacement with pressure in the case of spark spectra is thus limited to comparatively low pressures, and, in attempting to draw any conclusion from the published results, this must be borne in mind. There can be no doubt, other things being equal, that the most desirable course to take in attempting to estimate the comparative magnitudes of the displacements in the are and spark spectra, would be to compare measurements actually made under the same pressure in both cases. ‘This pro- cedure, as Duffield has shown, leads to the conclusion that in most cases the shifts in the arc are much higher than in the spark. The question arises as to whether such a con- clusion fairly represents the case. when we consider that Duffield’s measurements of the shifts in the are at this pressure are, in general, about double the mean shifts caleu- lated from the measurements made at pressures of from twenty to a hundred atmospheres. Although most investi- gators have reached the conclusion that the increase of wave length is directly proportional to the increase of pressure, © there is, so far as I am aware, no reason for supposing that no departure from this law is possible in a particular instance. Moreover, the possibility exists that the relation between displacement and pressure may be different for the are and spark spectra of the same element. Duffield has Arc and Spark Spectra of Nickel under Pressure. 165 shown, further, that the displacement does not depend solely on the applied pressure, but also on the density and tempe- rature gradients, so that there is no @ prior’ reason for expecting strict agreement between the results for arc and spark either in general or in any particular case. One must confess, however, that the discrepancies actually found are of a higher order of magnitude than might be anticipated. Farther work on the spark spectrum would undoubtedly be desirable before making anything like a final decision. Personally, I should hesitate to draw any important con- clusions from the fact that my values at five atmospheres increase of pressure are relatively higher than those at ten . atmospheres. The smallness of the shifts renders the order of accuracy rather low, and confers additional importance upon the personal error and upon the various causes tending to produce spurious displacements. Among these may be mentioned the effects of granularity in the plates and of lighting during the measurements. The spectrograph used for my experiments was carefully designed to eradicate, as far as possible, displacements due to changes of temperature of the grating or mounting—a fertile source of trouble in all work where the highest precision is essential. Even so, it was found impossible to completely eliminate errors due to this source, except, indeed, by increasing the number of plates for measurement. With exposures of about the same length, more uncertainty would be introduced in the cases where the displacements are smallest, that is, at the lowest pressures ; and this applies with equal force to all other sources of error. With a large grating mounted in the Rowland manner, as was employed by Duffield, the tempe- rature effect would undoubtedly be very much larger. Without labouring the point unduly, it will be realized that we have here a cause which might easily account for the whole of the observed discrepancies between Prof. Duffield’s displacements at ten atmospheres pressure and my own. Some further light is thrown upon the question by referring to the results obtained by Humphreys and Mobler* at pres- sures of 9?, 125, and 143 atmospheres, and by Humphrey-T at the higher pressures of 42,69, and 101 atmospheres. ‘he measurements at the lower pressures are somewhat meagre, but show, on the whole, no large differences from the results obtained by taking the means of the measurements at all pressures. Thus the mean shift per atmosphere of eight lines under pressures up to 14} atmospheres is 2°0 * Humphreys and Mohler, Astrophysical Journal, iii. p. 114. + Humphreys, Astrophysical Journal, xxvi. p. 36. 166 Mr. KE. G. Bilham on a Comparison of the thousandths of an Angstrém unit, compared with 1:8 from the 42 atmospheres results and 1:9 from all measurements. These lines belong to the types for which Duffield found the greatest discrepancies between the arc and spark measure- ments at ten atmospheres pressure—namely, those which are easily reversed. There is, in general, good agreement amongst the results of the two investigations of the are. It seems difficult to avoid the conclusion that for some reason Duffield’s measurements at ten atmospheres pressure are, on the whole, too high, and, from what has been said with regard to sources of uncertainty, the effect of temperature changes suggests itself as the most likely cause. Comparison of Arc and Spark Displacements. In Table I. are given the mean displacements per atmo- sphere in the arc found by Humphreys and Duffield respectively, and the averages of the two sets of data, for comparison with the spark displacements. Where only one set of results is available, it is given in brackets under the heading “‘ Mean Displacement in Arc.” Duffield’s values at ten atmospheres and Humphreys and Mohler’s low-pressure values are also given for reference. The class number for each line according to its behaviour in the spark spectrum under pressure is given in column 2. (Note :— Class __T. includes lines which reverse symmetrically. Class II. sp i se » unsymmetrically. Class IIT. is is » Yemain bright and fairly narrow. Class IV. i a », remain bright, but are very much broadened sym- metrically. i zi 55 are very much broadened unsymmetrically towards. the red. Class V. The following classification exhibits a few alterations from that given in my original paper. After a careful re- examination of all my photographs, the lines 3510-47, 3561°91, 3624°87, 3739°38, 3783°67, and 3831°87 have been transferred from Class I. to Class I]. The line 4359-76 has been transferred from Class III. to Class V. Lines enhanced in the spark are indicated by the letter E.), Are and Spark Spectra of Nickel under Pressure. TABLE I. 167 Mean Displacements per Atmosphere in thousandths A.U. Wave-length | «), «5 Xr. ‘ t. 2: 3391-21 Ra: 93°10 oe 3413°64 as 14:96 ae 23°80 a 33°71 8 37°45 +s 46:34 12 53°04 f, 54°29 E | IV. 58°59 I, 61°78 T 67°63 ne: 69°64 EE. 72°68 RE, 83°95 BE. 93°10 19 3501:00 a. 10°47 ph its pW ik 19°90 RE: 24°65 I. 28°10 EE, 48°34 II. 61:91 IBD: 66°50 iM 71-99 EH: 76-91 E | IV. 88°08 TL, 3597-84 ‘KE 3602°41 uit 09°44 I, 10-60 ii 12°86 II. 19°52 yo 24°87 EL 35:07 1g, 62°10 PEE 64°24 in 69°39 Lk 70°57 ‘LE 74:26 igis 88°58 II. 94:07 ET ARC. Humphreys. Duffield. Low All 20-100 Pressures. | Pressures. 10 Atm. Atm. 34) 4. 5. 6. it 1:4 se Gee ae es i me IS ane Wee is 1°5 shel pu: + ae 270 i Wee 2:2 1:9 nae ate ee 1% Ae su nee 15) une se at 2 oe #2 dif 1 yi ht uae a 1:5 bah we 2:0 ie na ney 2:0 cts ey. 2:2 1°8 ot (1:5) ee 2:0 Be 2°6 2°8 25 ay aes ae 18 hs 1-4 2°4 ise. wee Lae 2:0 Aa 1-2 15 Re OF 2-7; a, || 2°4 ony 19 2°2 A tat 2°4 4:9 ) 271 2:0 ol 1°8 1:8 29 1:0 2°4 6°4 2:2 2:0 4°4 1-4 16 4°7 2°6 15 3°4 0:8 Pee 4:7 125) is of poe iG 4°8 Ly wes ond 1:3 hey 3°9 21 1:0 4°2 1 1:0 37 18 es 2°6 LS Mean Mean displace-| displace- ment ment per Atm./per Atm. in Are. |in Spark. ri 8 (1°4) (1°5) (1°5) (1°7) (2°0) (2°3) (1-9) (1:7) 17 (1°5) 1:3 a 15 (2/1) 1:8 (1-7) 2:5 (1:5) (2°3) (2:0) 1-4 sy 1:5 (2:0) 15 1:7 1:5 2:3 2:2 (2'5) 2-7 1-6 1:9 (2:3) 3-0 i) 1:8 1-6 21 11 i 21 20 21 1:8 hy 8-2 16 14 2-2 21 1:9 16 1-4 16 2:3 2'6 17 1°6 2:1 1:8 1-1 0-7 (1°5) 51 1-2 3°5 1-7 2-2 (1:3) 1-2 1:8 25 1-4 1-9 1-4 20 (1°3) 2:5 168 Mr. H. G. Bilham on a Comparison of the Arc. Mean | Mean ieee ones Wave-length Humphreys. Duffield. ment. ment. he i Class. —| per Atm.| per Atm. i Low All 10 At 20-100 | in Arc. jin Spark. Pressures. | Pressures. ai Atm 1. 2. 3. 4. 5. 6. We 8. 3722°63 ial Bs 16 au 2-9 1:9 2-9 36°94 IL. 463 U2 ie 16 i4 1-0 39°38 -TI. . be 1:2 (2) 1:0 69:62 E | IV. 4°0 1:4 (1°4) (ore Rou ssh a 13 1-0 15 1°4 2:2 83°67 i Me 0-9 1:3 1:3 Livalb 16 3807°30 UE, es 11 30 15 Loe 18 31°87 Il. 32 18 (1°8) 36 4970 E | LY. oe rise cae 14:9 58°40 if Ne tei 39 2A 129 21 3972°31 III. ee 11 1:2 0°8 10 28 73°70 We as 20 4°4 Ib 2°0 58 4121-48 | TIL. 39 17 (1:7) 2°6 42-47 III. 10°5 RE A 32 4288 20 V. 16:2 10°9 (10:9) 12:1 Seuss 21 63 5:0 26 | 44 59°76 We By 115 12:2 (12:2) 10°6 4401-70 Vv. 4°8 116 12:0 8: 10°2 4459-25 Wy 6°2 10:2 109 86 9°3 62-65 Wi. ae 12°3 9°5 (9°5) 9-9 70°61 V. 8:9 12°4 11:0 SH 89 4520:20 ud 18 31 1-7 18 92°69 V. 8:5 14-1 10°3 9-4 11:8 4600-51 IV. 11:3 13°8 9°3 10°3 05°15 V. 7:8 12:0 9'5 8°7 48°82 Wi 81 14:0 11°4 9°8 86°39 WOE 8:0 184 IUeyy 9-4 4714-59 Vv. 6°7 17-1 111 8:9 56°70 Ne Wea 15°7 11:3 9°3 The mean shifts for the lines in each class have been com- puted for comparison with the values found by Duffield for the arc. Only lines figuring in both investigations have been included. TABLE II. Number | Mean displacement per Atmosphere. Bene Class. of g aa ys ms Lines. Arc (Duffield). Spark. jee It . oe) 4 1:95 1°88 0:96 Thy 24 1:60 1°85 1:16 III. 6 1:57 3°48 2°22 IV. 1 14 Ta 550 V. 8 9°84 9°83 1:00 — —$<—$—$<——$ $$ Are and Spark Spectra of Nickel under Pressure. 169 If Humphreys’ data are included in obtaining the means for the arc, the results are only slightly modified as in Table ITI. TaBueE ITT. Mean Displacement per Atmosphere. | Number Batic oe I oe Are (Humphreys Rae Spark : Arc. ee and Duffield). oar i 11 1-94 2-00 1-03 TL P04 -| 1-64 1-84 1-12 III. Suqea 1:55 3-48 Wie: IV. ieee 14 | 77 S30 v. | uaa 8:84 | 9-83 paar From an inspection of Tables II. and ILI. we arrive at the interesting conclusion that the are and spark shifts are, on the whole, very nearly equal for lines in Classes I., IL, and V., but for lines in Class III. the spark displacement is rather more than twice that in the arc. The values for Class IV. rest upon the results for a single line, so that the probability of error is too great to permit of any important conclusions being drawn. It is a significant fact that certain lines in Class V. show signs of reversal in the spark under a pressure of eleven atmospheres. For example, the reversal of the line 4401°77, a typical specimen of this class, can be distinctly seen in Plate 1. of my original paper. In the are this line is given by Duffield as suffering immense unsymmetrical broadening, without reversal even at the highest pressures. If we regard the continuous spectrum, which is so prominent in the more refrangible region of the spark spectrum, as being due to immense broadenings of Jines of Classes I. and II. it is evident that Class V. lines have something in common with those of Classes 1. and IJ., although at first sight nothing could appear more dissimilar. The view I have taken—namely, that Duffield’s displace- ments at ten atmospheres pressure are uniformly too high— thus leads us to the conclusion that the behaviour of lines easily reversed, or tending to reverse, is approximately the same whether developed in the are or spark under pressure. Phil. Mag. 8. 6. Vol. 31. No. 182. Feb. 1916. N 170 = Are and Spark Spectra of Nickel under Pressure. Alnormalities or discrepancies are exhibited by unreversed lines. It will be observed that this result is exactly the opposite of that reached by Duffield’s analysis of the data, and supported by the theory he gives concerning the part played by the density and temperature gradients. Ttremains for further investigation to decide whether this view is correct. The whole question turns very largely upon the reality of an abnormally large rate of displacement with pressure in the nickel are at ‘relatively low pressures ; and confirmation of Prof. Duffield’s conclusion may be regarded as one of the most pressing problems in this branch of spectroscopy. The part played by the temperature and density gradients in determining the position of the maximum of emission or absorption in a spectrum line isa problem which is becoming increasingly more prominent with every fresh advance in this line of investigation. It must be confessed that we are still far from a complete interpretation of all the manifold phenomena which present themselves. All that we can really be sure about at present is that the measured displace- ment is made up of a component directly due to the increase of pressure and probably proportional to it, together with a component of uncertain magnitude which has its origin in the changes in the structure of the line due to the alterations of the conditions prevailing in the source of emission. To what extent it may be possible eventually to disentangle the two components one cannot at present say. Meanwhile, as Dr. Royds* has recently remarked, the most valuable method of attack appears to be that of the electric furnace, in which the vapour density is likely to be influenced by pressure to a much less extent than in the are or spark. In any case, the recognition of the existence of these pbenomena is of the first importance, not only in interpreting existing results, but also in determining the most profitable directions for future research. Kew Observatory, November 1915. * Kodaikanal Observatory Bulletin, No, xliii. eee | XXIV. A Notation for Zeeman Patterns. Oy WM. iene foie." “a convenience of a generally recognized and concise notation to represent the configuration of the com- ponents when a spectral line is decomposed in a magnetic field is evident. I have found the following very useful for entering on handlists of spectral lines, and offer it as a suggestion for general use. Runge has shown that the displacements of the various components in a given field are multiples or submultiples of a constant unit. The wave- number displacement choseu by him as the unit is that of what is commonly called the normal triplet—the theoretical value for an electron revolving round a centre of attraction. It is convenient to take one third of this, in order to avoid the frequent appearance of fractions. Using the very careful determination from Cd and Zn by Fraulein Stellenheimer + working in Prof. Paschen’s laboratory, this constant is given by 107-°dd/\W7H=1'584. As the displacements take place in general symmetr cally on both sides of the undisplaced line, it is only necessary to tabulate those on each side and to indicate the distances as multiples of the unit. From analogy with the usual diagrammatic method of indicating a Zeeman pattern, the displacement of lines vibrating parallel to the field may be written as a numerator, whilst those perpendicular are indicated in the denominator. Thus the normal triplet would be written : or printed 0 | 3. In cases where it is desirable to indicate relative intensities subscript figures may be attached, or—for printed matter where such subscripts are inconvenient—by figures in brackets, as is done below. With this notation the patterns for the various kinds of series are then represented as follows :— Doublets. Si, letps Sz, Py. 2 | 6.10, 4| 8. Dio. Di. Do. TW GENCR 1 Oo QO} 5. The satellite pattern (D,,) does not seem to be the same * Communicated by the Author. + Ann, der Phys. xxiy. p. 284. ~—— 172 A Notation for Zeeman Patterns. for all metals. Thus for Tl Dj. itis 5 | 34.6.10, for Ca and Spoil, tor Ba 5 | 3.6.10: amd mone appear that the typical form is 5 | 3.6.10, but that some of the perpendicular vibrations (3.6.10) may be too faint to be observed. In Meg—which does not exhibit satellites—the line keeps to the Dy, type. In others—e. g. in the Cu line 5700—the pattern is 5 | 3.64.10. It belongs to a doublet of the D type which shows no satellites, and the pattern seems to be that of a combined Dj, and Dy. The corresponding doublets in Ag and Au show satellites, andin these each component exhibits the normal pattern for each. Triplets. Sa. See S3 Oe Ome .lle2. SM Cale 0) 12 The remark above as to the absence of a component being apparent only is well illustrated by the case of HyS, and HyS,. Thus indicating intensities by numbers in brackets after the fioure giving the displacement, we have for the first three triplets:— Judging from the rate of decay of intensity with in- creasing order, however, we might have expected the com- ponent 12 to be the last to go in §,(3). D,3. Dio. Dy. Do». De. Ds). ? PN pale ene’ 0 | 6, .0nes The satellite patterns, as in the case of the doublet series, are not uniform. Thus for D,,, Ca has 0 | 64, Sr 0] 15, Cd 4 | 10, Hg 0.6 | 3.9.15. It is interesting to note that the abnormality for Sr and Cd is shown also in their deviation from the rule that the mantissee of the denominators of D,3(2) are multiples of A,—in these elements the multiple occurring in D,. and D,, instead*. For D,,. the normal type is probably 6 | 3.9. Ca and)Znesive 6 | 6, Sr Oi dean Hg 6|3.9. Hg D(2) has very complicated and abnormal patterns, and as is known the satellites themselves have * Phil. Trans, A. ecxiii. p. 37-4 (1918). Si(1). S.(2). Si(3). .3(3) | 6(4).9(5).12(3), 0(2).8(1) | 6(2).9(1).12(8), 04) | 61). S,(1). S2(2). S.(3). 3(4) | 9(4).12(5), 3(1) | 9(2).12(2), AO: Intelligence and Miscellaneous Articles. 173 separations quite out of step with the typical values. The measurements of Runge and Paschen for Hg D(2) give D,3. Dye. Dit 0.6| 3.9.15, 2.43.74 | 4.64.9.11, 2] 7 instead of 0 | 7, Dre». Das. Ds). 6 | 3.9.12, CIS. 7; 0.3 | 3. For Hg D(3) the strong lines appear normal, whilst that for Hg D,,(4) is 0 | 6—possibly observation error for 0 | 7, as the components are very faint. The University, Sheffield. XXV. Intelligence and Miscellaneous Articles. Henry Gwyn JEFFREYS MOSELEY. rus brilliant young physicist who was killed in action on August 10th at the Dardanelles was the only son of my close triend and fellow student the late Professor Henry Nottidge Moseley, F.R.S., of Oxford. He was educated at Eton, where he entered as a scholar, and at Trinity College, Oxford, where he gained a Millard scholarship. He obtained a First Class in Mathe- matical Moderations and Honours in Natural Science. Harry Moseley was a most loveable boy and early showed sreat enthusiasm for science and marked originality. On leaving Oxford in 1910, he was appointed by Professor Rutherford ot Manchester as lecturer and demonstrator in the Physics depart- ment of the University. After two years he resigned his lectureship and was awarded the John Harling Fellowship, which enabled him to devote his energies entirely to research. In 1918 he returned to Oxford to live with his mother, and con- tinued his experiments in the laboratory of Professor Townsend. He went in the summer of 1914 to the meeting of the British Association in Australia; but on the outbreak of war made as speedily as possible for England, and resigning all thought of con tinuing the researches in which he was so successtully engaged, applied for and obtained a Commission in the Royal Engineers. He was, I am assured, offered work suited to his scientific capa- cities at home, but deliberately chose to share with others of his age the dangers of active service. He was made signalling officer to the 38th Brigade of the First Army, and left for the Dardanelles on June 13th, 1915. There, in the beginning of August, he was instantaneously killed by a bullet through the head as he was in the act of telephoning an order to his division. He was only 27 years of age. 174 intelligence and Miscellaneous Articles. Sir Ernest Rutherford writes that Moseley was “ one of those rare exainples of a mau who was a born investigator.” This quality he inherited from his father ; and it is noteworthy that his grand- father Canou Moseley, F.R.S., was a distinguished and original mathematical physicist, whilst his grandfather on his mother’s side, Mr. Gwyn Jeffreys, was also a Fellow of the Royal Society and for many years a leader in the study of Oceanography and Marine Zoology. Professor Rutherford tells us that Harry Moseley’s ‘“‘undoubted originality and marked capacity as an investigator were very soou ungrudgingly recognized by his co-workers in the laboratory, while his cheerfulness and willingness to help im all possible ways endeared him to all his colleagues.” Mr. C.G. Darwin, a grandson of Charles Darwin—who worked with him at Manchester and produced with him a joint paper in which they mapped out accurately, for the first time, the spectrum of the characteristic X-radiation from an X-ray tube with a platinum anticathode,— writes that he was without exception or exaggeration the most brilliant man whom he had ever come across. Others who came to know him well in Manchester write of his charm of manner and personality, of his kindliness and unselfishness and care for the interests of others. Asa boy (when I knew him best) he was a keen and observant naturalist and knew every bird and bird’s nest in the neighbourhood of his home. In this and in the collection of flint implements he was enthusiastically aided by his sister. His last letters home from the East were full of observations on the plant-life, the birds, the beasts, and the flint implements of all ages which he found in a day’s ramble on the hills where he was encamped. A most happy lite of experimental research, with natural history and his garden as relaxations, was assured to one so greatly gifted and beloved, for he had private means and an ideal home. Whilst abroad on active service, in view of possibilities now alas! realized, Harry Moseley expressed the wish (which will be eventually carried out) to bequeath any property at his disposal to the Royal Society of London “ for the furtherance of scientific research.” I am enabled by the kindness of Dr. Bohr of the University of Manchester, who has been aided by Dr. Makower, to add to this short personal sketch a notice of Moseley’s scientific work which will, I am sure, be highly valued by the readers of the Phil. Mag. KK. Ray Lanxester, Dec. 24, 1915. Mosrney came to Manchester in the spring of 1910 as lecturer and demonstrator in the Physical Laboratories of the University. At that time a great number of scientists from all parts of the world were workiag in the laboratories under the direction and inspiration of Sir Ernest Rutherford. Moseley at once caught the spirit of the laboratories and applied himself with charac- teristic energy and enthusiasm to the difficult and important problem of determining the number of /-particles emitted by a Intelligence and Miscellaneous Articles. 175 radioactive atom on disintegration. This problem had been in- vestigated only for the active deposit of radium, and even in this case no high degree of accuracy could be claimed for the result of previous investigators. Moseley succeeded in im- proving the method and in obtaining much more accurate results. He also extended the investigation so as to include most of the radioactive products emitting 6-rays*. His results have siuce gained additional importance on account of their bearing on theories of the origin of Z- and y-rays. The knowledge and experience which he gained in these experiments Moseley subsequently used to obtain high potentials in vacuo by means of the charge acquired by a radioactive substance during the emission of G-rays+. He was thus able to obtain higher steady potentials than had previously been reached in this or any other way. Whilst engaged in these difficult researches Moseley still found time to devote attention to other problems. In collaboration with Fajans he developed a most interesting method of deter- mining the life of very rapidly decaving radioactive products +; with Makower he discovered that radium B emitted y-rays which were so easily absorbed that they had not been previously detected §; and with Robinson he measured the total ionization produced by the 6 and y radiation from radium B and radium C ||. In 1912 Moseley resigned his lectureship, and having obtained the John Harling Fellowship at Manchester University, he was able to devote all his time to scientific investigation. In the same year a new field of Physical research was created by Laue’s dis- covery of the interference of X-rays in crystals; and interest was soon intensified by the brilliant work of W. H. and W. L. Bragg on the constitution of X-rays and the structure of crystals. Very soon after Laue’s discovery Moseley, working with Darwin, started a thorough investigation of the properties of X-rays by means of the new method. Their results were published in June 19134]; and although many of the results were discovered and published earlier by the Braggs, this paper contained a great number of most interesting experimental details and theoretical considerations, and constituted an important step in the rapidly increasing knewledge - about the nature of X-rays. Immediately after the completion of this work Moseley undertook a systematic investigation of the characteristic X-radiation from as many different elements as possible. This investigation involved great experimental difficulties, partly on account of the fact that the chemical nature of many Moseley, Proc. Roy. Soc. A. lxxxvii. p. 230 (1912). Moseley, Proc. Roy. 80c. A. Ixxxvii. p. 471 (1913). Moseley and Fajans, Phil. Mag. xxii. p. 629 (1911). Moseley and Makower, Phil. Mag. xxiii. p. 812 (1912). | Moseley and Robinson, Phil. Mag. xxviil. p. 327 (1914). {| Moseley and Darwin, Phil. Mag. xxvi. p. 210 (1913). r+t+—+ 176 Intelligence and Miscellaneous Articles. elements makes them rather unsuitable as anticathodes in X-ray tubes, and partly on account of the extreme absorbability of the radiation from many of the elements. Nevertheless, by his wonderful energy and by an ingeniously simple experimental arrangement for photographing the X-ray spectra, Moseley in less than half a year obtained measurements of the wave-lengths of the most intense lines in the high-frequency spectra of the greater part of the known elements and discovered the fundamental laws, which will always bear his name*. This work, begun in Manchester, was completed in Oxford in the early spring of 1914. As is well known, Moseley found that the frequencies of the principal lines in the high-frequency spectra are simple functions of the whole number which represents the position of the elements in the periodic table of Mendelejeff. The extreme importance of this result is that it reveals a relation between properties of different elements far simpler than any which could be expected from pro- perties previously investigated, all of which, including the ordinary visible spectra, vary in an intricate manner from element to element. Moseley’s discovery therefore gives a most important clue to the question of the internal structure of the atom which has received so much attention in recent years. While it is hardly the place here to enter into this problem in any detail, the general importance of Moseley’s results is perhaps best illustrated by the fact that they enabled him to predict with certainty the number of possible elements hitherto unknown and their position in the periodic series. In this way he was, for instance, able to fix the number of possible elements in the group of the rare earths ; and just before he went to Australia he was occupied in collaboration with Prof. Urbain on an investigation of the high-frequency spectra of the elements of this group, which no doubt will throw very much light upon this field of investigation which hitherto has given chemists so much trouble. A full account of this investigation has not yet been published, but Moseley gave a paper on the general question before the Meeting of the British Association in Sydney. Every reader of Moseley’s papers will be strongly impressed by his penetrating theoretical understanding and his great experimental skill which, together with his unique capacity for work, have secured him a place among the foremost workers in science of his time, although he was not able to devote more than four short years to scientific investigations. * Moseley, Phil. Mag. xxvi. p. 1024 (1918), and xxvii. p. 708 (1914). THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. ;— [SIXTH SERIES.) /“" © MARCH 1916. .\.. 9 me, a Ser XXVI. On the Electrical Capacity of Approaimate Spheres and Cylinders. By Lord Rayveien, 0.1, F.RS.* N ANY years ago I had occasion to calculate these capacities | so far as to include the squares of small quantities, but only the results were recorded. Recently, in endeavouring to extend them, I had a little difficulty in retracing the steps, especially in the case of the cylinder. The present communication gives the argument from the beginning. It may be well to remark at the outset that there is an important difference between the two cases. The capacity of a sphere situated in the open is finite, being equal to the radius. But when we come to the cylinder, supposed to be entirely isolated, we have to recognise that the capacity reckoned per unit length is infinitely small. If a be the radius of the cylinder and 0 that of a coaxal enveloping case at potential zero, the capacity of a length / is t 4] log (b/a)’ which diminishes without limit as b is increased. For clear- ness it may be well to retain the enveloping case in the first instance. * Communicated by the Author. + “On the Equilibrium of Liquid Conducting Masses charged with Electricity,” Phil. Mag. vol. xiv. p. 184 (1882); ‘Scientific Papers,’ vol. ii. p. 130. ¢ Maxwell’s ‘ Electricity,’ § 126. Phil. Mag. 8. 6. Vol. 31. No. 183. March 1916. O 178 Lord Rayleigh on the Electrical Capacity of In the intervening space we may take for the potential in terms of the usual polar coordinates | b= Hy, log (r/b) + Hyr-! cos (0—e) + Kyr cos (@—«e') +. . + Har-" cos (nO —e,) + K,7r" cos (n? — en’). Since 6=0 when r=, an = Eng K, = —H,0-*, and §=Holog (r/d) +H, (= — 75 ) eos (041) rap fellas 20 L ne a( ja — jx) 008 ( eh a ic (4) At this stage we may suppose 0 infinite in connexion with 1, He, &ec., so that the positive powers of r disappear. For brevity we write cos (nO—e,)=F,, and we replace r~? by u Thus @= —Holog (ub) + HywF,+How?Fo+... . . (2) We have now to make @=d, at the surface of the approximate cylinder, where q, is constant and U=Up+ Ou= Up (1 + CLG, + Co.Ge+ ee Ne Herein Gni= cos (nO—e,), and the C’s are small constants. So far as has been proved, e, might differ from en, but the approximate identity may be anticipated, and at any rate we may assume for trial that it exists and consider G, to be the same as I’,, making u=upt du=u(1+C,F,4+C,F.+ ee Dh e e ° (3) On the cylinder we have d; = —H, log (29) + Byugk + sug hs =I os eile _ ou + a, —H)+ Hyuk + 2H ou,’ F, + 3H zug? F3 + ine .t 0 Ou ‘ 1 D) 3 + a) {5 Hy + Hem, F,+3H3u PY, + Once 0 +4p(p—1)H,w?Fp},. . . (4) and in this du/u= CF, +C,Fo+C3F3+... Mm Ni iif hc Approximate Spheres and Cylinders. 179 The electric charge Q, reckoned per unit length of the cylinder, is readily found from (2). We have, integrating round an enveloping cylinder of radius r, Wie db wae Hy eae, ip’? — =D 3 ° ° e ° (6) 2 ? and Q/¢, is the capacity. e now introduce the value of du/w from (5) into (4) and make successive Lianae ee The value of H, is found by multiplication of (4) by F,, where n=1, 2, 3, &e., and integration with respect to @ between 0 and Dar, when products “such as F iF,, F,F3, &c., disappear. For the first step, where ©” is neglected, we have 0 = Fw” F2d0 ey H,0,,f F,7d@, - < bY, ° . (7) or Be hae Lee cen Wy ail feat sbkhies tes hi COE) Direct integration of (4) gives also $, = — H, log (ub) +(e ae {Hyuk +2 Hem? Fs Stig Pst o. +o le ney (9) cubes of C being neglected at this stage. On introduction of the value of H,, from (8) and of du from (5), $1 = — Hy log (ub) +1 Ho {8C)7+5C,7+7C;?+ ...}. (10) Thus $;/Q=2 log (ub) —4{8C2 +502 4+70;7+ ...$. . C1) In the application to an electrified liquid considered in my former paper, it must be remembered that wo is not constant during the deformation. If the lquid is incompressible, it is the volume, or in the present case the sectional area (c), which remains constant. Now Qe ={" ae any 2, (494 1- 2H 4.3(=) b = <7 (14+ 4(Cr+ Cr+ C+ “yy Y. so that if a denote the radius of the circle whose area is o, ue =a"{14+3(C724+C/7+C,?+ ...)}. . Pasta 9 180 Lord Rayleigh on the Electrical Capacity of Accordingly, log u?= —2 loga+3(C?+C%+ C+ ...), and (11) becomes $,/Q =2 log (b/a) —C,2—2C,?— ... —(p—1)C,”, . (13) the term in ©, disappearing, as was to be expected. The potential energy of the charge is $¢,Q?. If the change of potential energy due to the deformation be called P’, we have Pl = —1Q*{ 0742024... 4+(p=DCZ),. in agreement with my former results. There are so few forms of surface for which the electric capacity can be calculated that it seems worth while to pursue the approximation beyond that attained in (11), supposing, however, that all the e’s vanish, everything being symmetrical about the line 6=0. Thus from (4), as an extension of (7) with inclusion of C?, dé 0=(H,u,"— HC.) | = ie ah oF (OF, +CoFo+ ...)(HiuoFi + 2Hau? Fs + BW RHO ihe + oe , + (EGR + Colt Colt oe) 4 at) et eee or with use of (8) 27 ug? H,,/ Ho= o,f oe P(CyF, a CLF. + C.F + ee :} 0 T (3C,F 4. 5C,F y+ 7CsEy4: Gee ae by which H, is determined by means of definite integrals of the form 27° a ) F.E,F 0, ... . 7 0 n, p,q being positive integers. It will be convenient to donote the integral on the right of (16) by L,, In being of the second order in the (’s. Approximate Spheres and Cylinders. 181 Again, by direct integration of (4) with retention of C%, dO o\= —H, log (ugb) +(2 (Cy Fy + ORO + C3P,+ oe 3) (Hyuk, a 2H.u,?F. + JE aug ft eis ) 41H, ( OF +CE: + O:Fa+ ...) dé + (cont Gl a+ C.F + ge)? (Hau F 2+ 33k, : -/- ioe +43p(p—1)H,u?F;,). In the last integral we may substitute the first approxi- mate value of Hy from (8). Thus in extension of (11) $)/Q=2 log (ub) —3{ 3024+ 50.2+702+ ...} +C,I,+ 2C,[,+3C31; + eee 2m 16 | =| ia (C,F,+C,F,+ CF; + ee .)?4 CaF +3C3F3 + eae +4n(p—1)C,F,} . ° e (18) The additional integrals required in (18) are of the same form (17) as those needed for I,. As regards the integral (17), it may be written 27 \ dé cos n@ cos pé cos gf. 0 Now four times the latter integral is equal to the sum of integrals of cosines of (n—p—q)@, (n—p+q)\6, (n+ p—Q)8, and (n+p+q)0, of which the last vanishes in all cases. We infer that (17) vanishes unless one of the three quantities n, p, g is equal to the sum of the other two. In the excepted cases GR dor Me hen sy a ee LOD If pand g are equal, (17) vanishes unless n=2p ; also whenever n, p, g are all odd. } We may consider especially the case in which only C, occurs, so that u= uo(1 + C, cos p@) - 2 ~ ° e ° ° (20) In (16) ome ro: V2 atin 1,= (2p i 1)Cp \ Oar ple so that I, vanishes unless n=2p. But Ie, disappears in (18), presenting itself only in association with Cs), which. 182 Lord Rayleigh on the Electrical Capacity of we are supposing not to occur. Also the last integral in (18) makes no contribution, reducing to ip(p—1)Cr |” SF oos pe, which vanishes. Thus : $:[Q=2 log (ub) —(p+3)Cy, - - . . QI) the same as in the former approximation, as indeed might have been anticipated, since a change in the sign of C, amounts only to a shift in the direction from which @ is measured. The corresponding problem for the approximate sphere, to which we now proceed, is simpler in some respects, though not in others. In the general case u, or r~', is a function of the two angular polar coordinates @, m,and the expansion of éu is in Laplace’s functions. When there is symmetry about the axis, w disappears and the expansion involves merely the Legendre functions Pa(y), in which w=cos@. Then U=Up + du=uU{ L+C,P)(~)+C.P2(u) + oe “> Site (22) where C,, Cy,...are to be regarded as small. We will assume ou to be of this form, though the restriction to symmetry makes no practical difference in the solution so far as the second order of small quantities. For the form of the potential (@) outside the surface, we have @=Hyuw+HywP (4) + Hw Po(s)+.-.5 - «eee and on the surface by = Houy + Hyp? Py + Hew? Pot... + 6u{ Hy + 2u)H,P,4 3u,?7H,P,+...} + (6u)?{ HP) 4 3uH,P.+...+4p(pt+ 1) uo? HePp}, (24) in which we are to substitute the values of du, (du)? from (22). In this equation ¢; is constant, and H,, H.,... are smnall in comparison with Ho. The procedure corresponds closely with that already adopted for the cylinder. We multiply (24) by P,, where n is a positive intever, and integrate with respect to u over angular space, 2.e. between —1 and +1. Thus, omitting the terms of the second order, we get ughH, ee Ei pCe) s.r as a first approximation to the value of H,. Approximate Spheres and Cylinders. 183 Direct integration of (24) gives pifdu= Hyuofdu + ul {CPi +C,P2+ ...} { 2upH, Py + 3u?HePo+.. }dp = Houp\dy ~ up {2uoH \C,P,? + 3uo? HCP.” +4uH3C;P2+ ...bdp, or on substitution for H, from (25) b= Hoy4 1-30-80 —. i — de . (26) inasmuch as “Pp? d z ae Pi) dua es RM) As appears from (23), Hy is identical with the electric charge upon the sphere, which we may denote by Q, and Q/¢; is the electrostatic capacity, so that to this order of approximation a ae Capacity=uy? 1 +2C7+.. (28) Here, again, we must remember that ne differs from the radius of the true sphere whose volume is equal to that of the approximate sphere under consideration. If that radius be called a is 20730. | 20,3 } Uo tea 41-7 j Tee Bt apce I sears (29) op anh. C,? n—1 > Capacity=a 1+ tee. + oa 4 fe COO) and in which C, does not appear. The potential energy of the charge is 3Q?+Capacity. Reckoned from the initial configuration (C=0), it is ae ie + i sae Bie Jt has already been remarked that to this order of approximation the restriction to symmetry makes little 184 Lord Rayleigh on the Electrical Capacity oy difference. If we take éu/u,=F + F.e.). + Fay 2 eee where the F’s are Laplace’s Se ons, 1 2 . Ce" = \\F dudw corresponds to Sa cci This substitution suffices to generalize (30), (31), and the result is in harmony with that formerly given. The ‘expression for the capacity (30) may be tested on the case of the planetary ellipsoid of revolution for which the solution is known *. Here ©O,=4¢?, e being the eccen- tricity. It must be remembered that a in (80) is not the semi-axis major, but the spherical radius of equal volume. In terms of the semi-axis major (a), the accurate value of the capacity is we/sin~* e. We may now proceed to include the terms of the next order in C. ‘The extension of (25) is 14+] EPH Cea On an NG duP{CP, 2.) eet f20,P,+...+ (¢-+ Pa ees where in the small term the approximate value of H, from (25) has been substituted. We set +1 { dp (all Oh eae eet CpPpb a {20,P, +....+ (¢+ DC,P.) =) ee where J, is of order C? and depends upon definite integrals of the form “Pll ie Paleo kee dp, . ° o: | (0, A ienn itis aamite (35) n, p, ¥ being positive integers. In like manner the extension of (26) is d;/Qup=1—202—2C0"—..- - 2 + (90,0, 4 3Csie bailed, fa ot -{" du(C,P,+C,P.+ ...)?{C:Pi+3CoPet ... 5 Lindh C,P>).” es Here, again, the definite integrals required are of the form (35). * Maxwell’s ‘ Electricity,’ § 152. Approximate Spheres and Cylinders. 185 These definite integrals have been evaluated by Ferrers * and Adamst. In Adams’ notation n+p+q= 2s, and (35) = 5. : ied ASP abe?) where ss (37) ee liye pice 1) 9 ale eh (38) yas 3. aie In order that the integral may be finite, no one of the quantities n, p, g must be greater than the sum of the other two, and n+y+gq must be an even integer. The condition in order that the integral may be finite is less severe than we found before in the two dimensional problem, and this, in general, entails a greater complication. But the case of a single term in 6u, say CpP,(w), remains simple. In (36) J, occurs only when multiplied by Cn, so that only J, appears, and Jp ( IOS | Peed) idl oa eo) Thus (36) becomes pert) (tp ea oe $;/Qu)= Rip tt CP + 1 ce, ae eas. When p is odd, the integral vanishes, and we fall back upon the former result ; when p is even, by ( 37), (38), ae Fn zZ {A (3p) }? it By ay “Beil AG at ot syv(4d) — For example, if p=2, aig 4 3 —— Sam i eae Ea Ne bea $1/Quo= 1— § Ca? + 35 Ca’. Sebrewans Mil (Ca Again, if two terms with coefficients C,, C, occur in du, we have to deal only with Jz, Jo. The integrals to be evaluated are limited to ame NP AP dp, (eae de {Pe dp. If p be odd, the first and third of these vanish, and if g and * ‘Spherical Harmonics,’ London, 1877, p. 156. + Proc. Roy. Soc. vol. xxvil. p. 63 (1878). 186 Prof. E. M. Wellisch on be odd the second and fourth. If p and gq are both odd, the terms of the third order in C disappear altogether. As appears at once from (34), (36), the last statement may be generalized. However numerous the components may be, if only odd suffixes occur, the terms of the third order disappear and (36) reduces to (26). January 20, 1916. XXVIII. Free Electrons in Gases. To the Editors of the Philosophical Magazine. GENTLEMEN,— a the May number of the ‘American Journal of Science’ T published a brief account of the results of certain experiments dealing with the motion of ions and electrons through dry air: it was shown that the electrons remained free during their motion through the gas and that their relative number increased with diminution of pressure, the electrons coming into evidence in my experiments at about 8 cm. pressure. These results were found to contain the explanation of the apparently anomalous increase of the mobility of the gas ion at reduced pressures; when the ions were considered apart from electrons, the mobility remained normal throughout. The existence of free electrons in gases at relatively high pressures had previously been demonstrated by Franck for argon, helium, and nitrogen at atmospheric pressure; it was found by him that when these gases were carefully freed from impurities (especially oxygen) the negative carriers were practically all electrons ; the slightest trace of oxygen, however, was sufficient to convert the carriers into ions. This ability to contain electrons in the free state was regarded as being a peculiarity exhibited by the inert gases ; the demonstration of their existence in air, which was indeed so a priori improbable on account of the presence of the electro- negative constituent oxygen, rendered it at once evident that Franck’s result was a particular case of a more general principle, and that the free electrons along with ions would be found for all the ordinary gases, the inert gases being conspicuous by reason of their reluctance to form negative ions. Immediately after the preliminary publication I resumed the experimental work ; this was completed last July, but, Free Electrons in Gases. 187 owing mainly to the time involved in travelling, considerable delay has arisen in publishing the new results. I think it therefore advisable to make this brief communication, dwelling especially on certain features of the subject which it appears to me should be given prominence if satisfactory progress is to be made. The free electrons were investigated for the gases carbon monoxide, carbon dioxide, and hydrogen ; for the two latter gases the electrons were considerably more numerous than in air at the corresponding pressure, a fact which was con- sistent with the result that the abnormal increase in the ionic mobility set in for these gases at higher pressures than for air. In hydrogen, the electrons appeared in considerable numbers at atmospheric pressure. A number of vapours were tried, but for the most part they showed no evidence of free electrons ; however, in these cases there was always a pressure of several mm., and it is quite possible that lower pressures would bring them into evidence. An interesting and remarkable exception is the vapour of petroleum ether (a mixture of pentane and hexane); for this vapour, whose molecules contain only atoms of carbon and hydrogen, the negative carriers appeared to consist entirely of electrons, the negative ions being absent: the vapour was, however, particularly sensitive to impurities, and, after standing for a few hours ina closed metallic vessel, would no longer contain any considerable number of free electrons. -I desire here to make some remarks with regard to the effect of impurities on the relative number of free electrons and negative ions in a gas; this is a point on which there exists considerable misapprehension. Franck found that the free electrons in the inert gases were especially sensitive to oxygen asan impurity. In the Octcber number of the Philo- sophical Magazine Mr. Haines has stated the existence of free electrons in nitrogen and hydrogen at atmospheric pressure, and has mentioned oxygen as one of the impurities which if present in small amount would rob the former gas of its free electrons. Mr. Haines does not enter into detail with regard to the impurities in hydrogen, but I have myself found that a small quantity of oxygen in hydrogen at atmo- spheric pressure would deprive this gas of its electrons. It has probably appeared to many to be extremely strange that a trace of oxygen could have this effect and yet that free electrons should exist in air at several cm. pressure. Asa matter of fact, the difficulty is more apparent than real. Although the results to which reference has just been made are perfectly currect, still, when the subject is viewed more 188 Prof. E. M. Wellisch on generally, the electrons are not sensitive to oxygen as an impurity. To consider only the case of hydrogen, as I did not experiment with the inert gases: I found that as the pressure was reduced the sensitivity to oxygen was markedly decreased ; é. g., to take actual figures, although a small amount of air (less than 1 mm. pressure) would rob hydrogen at 1 atmosphere of its free electrons, the electrons existed in considerable numbers in a mixture of hydrogen at 825 mm. and air at 25mm. At still lower pressures it was found possible to increase the number of electrons by adding small amounts. of air ; this arose from the extra number of electrons supplied by the air more than compensating for the diminution in the number supplied by the hydrogen. For the explanation of these results, and indeed of many diverse results that have arisen during the course of the experiments, I have been constantly guided by the following idea, which has proved of the greatest assistance even should it subsequently be found not to correspond with reality: viz., that an electron cannot effect a permanent union with an uncharged molecule to form a negative ion unless the relative velocity at collision exceed a critical value characteristic of the molecule concerned ; in other words, there is a definite potential of ion formation justas there is a definite ionization potential, the former of course being smaller than the latter, probably of the order of one half. To revert to the case of hydrogen with oxygen as the impurity. Jf an electron be expelled from a hydrogen molecule, we can imagine a sphere A drawn round the parent molecule of such a radius that the electron will be effective in forming a negative ion for any collision with a hydrogen molecule within this sphere; also a sphere B of larger radius corresponding to the formation of a negative oxygen ion. Now in pure hydrogen the free electrons that are observed are those which pass the boundary of the sphere A; if a trace of oxygen be present, an electron may form a negative oxygen ion in the region between A and B. In hydrogen the electrons must have at collision a fairly high degree of elasticity, so that we may regard the velocity of the electron between A and B as being approximately a function of its distance from the parent molecule. Now the chance of meeting an oxygen molecule between A and B is much greater if the hydrogen be at a high than at a low pressure, because in the former case the time taken for the electron to cross the region is considerably greater owing to the larger number of collisions. The sensitivity of the Free Electrons in Gases. 189 electronic effect to traces of oxygen increases, therefore, over a wide range with the pressure of the hydrogen. An interesting question arises as to whether in the case of certain vapours the electron has sufficient energy at ordinary temperatures to form with a molecule a negative ion: in such cases permanently free electrons would not be possible except with field strengths high enough to effect ionization by collision. I was not able to make a systematic study of vapours at low pressures to ascertain whether free electrons did really exist, but an examination of the decay of the free electrons in petroleum ether vapour as ageing occurred, indicated the appearance in the vapour of another constituent capable of absorbing electrons at ordinary temperatures. This arose probably from the walls of the containing vessel, although the possibility of a formation of aggregates by the vapour molecules themselves has to be considered. In any event these “electron sinks’? undoubtedly exist, and should be carefully distinguished as regards their effect upon electrons from impurities such as oxygen. The study of tbe free electrons in various gases is one of great importance and should repay careful investigation. They are really observable in practice, especially if a high- frequency commutator be employed ; elaborate precautions are not necessary if it be merely required to bring the electrons into evidence—e. g., in my preliminary experiments I was able to observe the electrons in considerable numbers at atmospheric pressure, in hydrogen which was prepared from commercial acid and zinc in a Kipp’s generator and dried in the usual way before passing into the apparatus. What is urgently needed is a determination of the pro- portion of ions and electrons in a pure gas at a specified pressure ; to say that the electrons occur in air at 8 cm. or in hydrogen at 1 atmosphere provides little information, because the sensitivity of the measuring apparatus is involved. A ease of particular interest would be to ascertain whether any negative ions are present in pure inert gases, as a small proportion of ions might not unreasonably be expected. I hope within a few months to be able to publish the full account of my work, Yours very truly, The University, K. M. WE LtIscxH. Sydney, 7 ae 22nd, 1915. Bey) XXVIUI. On Two-Dimensional Fields of Flow, with Loga- rithmic Singularities and Free Boundaries. By J. G. Leatuem, M.A., D.Sc.* 1, [NTRODUCTION.—In a recently published paper + the writer has defined conformal curve-factors, discussed some of their properties, and employed them in the solution of problems of two-dimensional liquid flow. In all the configurations there discussed the fixed boundary has been an open polygon, rectilineal or partly curvilinear, whose continuity is not broken at any point, and the field has been free from singularities. It 1s now proposed to show how the same calculus may be applied to regions whose fixed boundary is broken and to fields which contain logarithmic singularities, as, for example, to cases of liquid flow between two boundary stream-lines and cases of flow due to sources or vortices. Methods of solving problems of this kind, with a special view to the determination of the forms of free stream-lines, have been given by Mr. J. W. Michell], Prof. A. HE. H. Love §, and Prof. B. Hopkinson ||, who all deal with cases in which the fixed parts of the boundary are rectilineal polygons. It will be shown that the method of curve-factors deals with such problems in a different and more concise manner, and is further applicable to some cases in which part of the fixed boundary is curvilinear. The analysis is, of course, capable of interpretation in terms of fields of electric flow or force, and it is believed that the method of curve-factors may be regarded as a comprehensive mode of approaching all those classes of physical problems which can be formulated in terms of conformal transformation. : 2. Double transformation.—The general problem is that of determining a relation between two fundamental complex variables, namely, z=2+7y the variable of the geometrical configuration, and w=@+t¥ the variable specifying the field of flow or force (the velocity being the vectorial rate of decrease of @). When the field is free from singularities * Communicated by the Author. + “ Applications of Conformal Transformation to Problems in Hydro- dynamics,” R. 8. Phil. Trans., sec. A, vol. cexv. (1915). “The Theory of Free Stream-Lines,” R.S. Phil. Trans., sec. A, vol. elxxxi. (1890). § “The Theory of Discontinuous Fluid Motions,’ Proc. Camb. Phil. Soe. vol. vii. (1891). \| “ Discontinuous Fluid Motions involving Sources and Vortices,” Proc. Lond. Math. Soe. vol. xxix. (1898). On Two-Dimensional Fields of Flow. 191 what is wanted is a conformal representation of the relevant region in the z plane on the half-plane of w for which is positive ; but when there are singularities the (z, w) relation is not conformal, and an intermediate variable €=£+7” is introduced, and relations are established between z and 6, and between wand ¢. The (z, €) relation is a conformal representation of the relevant region in the z plane upon that half-plane of € for which 7 is positive, and is generally a differential relation formulated in terms of Schwarzian facters and curve-factors by the methods, and subject to the limitations, explained in the previous paper. The (w, €) relation is an explicit formula for win terms of ¢, constructed by taking account of the specified singularities, with such images as are required to make y=0 when n=0. The difficulty of integration, inevitable in problems of this kind, arises only in connexion with the (z, €) relation. 3. The geometrical relation.—In constructing the (z, £) relation it is to be noted that a gap in the boundary at infinity, corresponding to the parallelism or divergence of two parts of the boundary, is to be regarded as a corner and dealt with by means of a suitable Schwarzian factor. Curves in the fixed part of the boundary are represented by curve- factors selected from known types of suitable angular range, and these curve-factors (unfertunately not usually the curves themselves) are among the data of the problem. Curves which are free stream-lines are represented by curve-factors which are among the quesita of the problem. 4. The jield relation.—In constructing the (w, €) relation a distinction must be drawn between singularities in the boundary and singularities at other points of the field. The only singularities contemplated are logarithmic, namely, in cases of liquid flow, sources and vortices, doublets being got by differentiation if desired. A source in the boundary at the point €=c, whose rate of output into the field is m, is represented in w by a term —(m/sr) log (€—c). A vortex in the boundary is inadmissible since it involves an infinity of vy. if the point €=c corresponds to a point at infinity in the z plane, the singularity represented by the above term is a source at infinity, that is an influx of liquid at the rate m through an infinitely distant gap in the boundary. A source whose rate of output is m, situated in the field at a point €=a2+78, must be counterbalanced by an image source at the point €=a—18. Hence it is represented in w by a term —(m/2q) log {(f—a—i8)(f—a+ip)}. . . (1) 192 Dr. Leathem on Two-Dimensional Fields of Flow, Similarly a vortex at the point €=y +23, round which the circulation is mw, is represented in w by a term (tp/2m) log {(€—y—160)/(C—y+10)}. . . (2) 5. Motion of a ship in a canal.—As a first example one may consider the case of a doubly pointed ship moving along the central line of a uniform straight canal. On super- position of a motion equal and opposite to that of the ship, the ship becomes part of the fixed boundary. Only half the configuration need be studied, and the z diagrain is as shown in fig. 1, wherein the values assigned to € at the important Fig. 1. SIT EEL E ESEES BE IE SE ETE PEE I STS yee ie Lo Wee git pir points are indicated. The angles at bow and stern are taken to be 2pm and 2qr. The velocity at infinite distance being V, there is a source of output JV at €=a, and an equal sink at =o. The geometrical and field relations are mitainnn GCS )ab Pk nl es = (Fe oeaate) 7 8 Oe where E is a positive constant and @ is a curve-factor in €, having linear range from —c toc, and angular range (p+q)7. The curve-factor may be any one of several known types, and the shape of the ship depends on the form of @ and the values of ail parameters in the geometrical relation. One condition which the parameters must satisfy corre- sponds to the seometrical relation giving the proper breadth to the canal at C=a and =x. The abrupt changes in the value of y indicated by this relation lead to d 7THG (a) ME i CO (a+c)(a—e) a eR (4) Thus, for example, if the selected curve-factor be 6 oe = 10 — hk) 4) (Ce) it is necessary that {r(a—k) + (a—e)*(ate)' “PP (ate) (ac) P= (a +1)P™ with Logarithmic Singularities and Free Boundaries. 193 6. Effective inertia of a ship in a canal.—A compact formula for the longitudinal inertia-coefficient of the ship (assumed to have the same mass as the liquid which it displaces) can be deduced. If X is the impulse required to set up a motion of the ship with velocity V in still water of density p, it can be shown that >] p"X=E4) (—adg/da)dy, . Maas, the summation referring to two straight lines of integration across the canal, one (with sign +) ahead of the ship, the other (with sign —) astern. To right and to left w can be expanded in the forms Vet ey+ 2c, exp(—nmzl™), Ve+co’+2tn exp(nmzl), (7) where the n’s are integers. Only the terms ¢ and ¢' con- tribute to the limit of the right-hand side of (6) when the transverse lines tend to infinite remoteness. Hence p2X = 2(q—a')=—21( (dw—Vdz), . . (8) wherein the subject of integration is supposed to be expressed in terms of € by means of formule (3). 7. Electrical flow in a flat conducting strip pierced by a symmetrical hole-—The analysis of the previous article may be interpreted in terms of two-dimensional electric flow, g being taken to be the product of the electric potential and the constant specific conductivity, and the “ ship” being represented by a hole cut in the strip. In this state of flow let dz, be the element of geometrical displacement corresponding to df when w=0 ; and let dz, be the corresponding element in what would be the flow if there were no hole in the strip, in which case dw would equal Vdz. Then i). mee it c i ; zg See Bae ain tes nie) and the latter integral represents the limit of the difference between the lengths of the strip, with and without the hole, which correspond to the same difference of potential. In fact it represents the increase of the ohmic resistance of the strip, due to the presence of the hole, in terms of the resistance per unit length. Phil. Mag. 8. 6. Vol. 31. No. 183. Mareh 1916. P 194 Dr. Leathem on Two-Dimensional Fields of Flow, Any distribution of electrodes, symmetrical about the central line of the strip, can be dealt with by introducing into w terms of the type of formula (1). 8. Liquid flow with free stream-lines, with or without sources and vortices.—When the fixed part of the boundary is a rectilineal open polygon the (z, €) relation is dz = E(C)FC)dé, where F(€) is a product of Schwarzian factors determined in the usual manner, and &(€) is an as yet unknown curve- factor representing the free boundary. The problem is reduced to quadrature when the form of @ has been ascer- tained, and it will now be shown how @ 1s determined from the condition that | dz/dw | is constantin the free boundary. The field relation is of the type m a —y—16 waAt-¥ Mogi (faa) +a +5H2 log {FIV a0) so that dw/df is an algebraic fraction in € of known form ; and dz/dw=@¥dé/dw. The modulus of this can be made constant in the linear range of @ by building up @ by factors, each factor being itself a curve-factor of such a character that, when it is associated with a particular factor in nume- rator or denominator of Fdé/dw, the combination has (for values of € corresponding to the free boundary) a modulus independent of € Every factor of Fd&/dw beiny dealt with in this way, the complete product of the curve-factors is @. If the algebraic sum of the outputs of the given sources, including a possible source at infinity of output 7A, is not zero, the free stream-line will extend to infinity. In this case the linear range may be taken from €=—a to €=0, and the synthesis of @ is as follows :—(i.) A factor (¢—a)”, where a is real and positive, occurring in Fdé/dw,is to be associated with &jx", where 43;=a'?+¢'7, which is known to be acurve-factor of the assigned linear range with modulus (a—f)'” in that range. (ii.) A product of conjugate com- plex factors {(€¢—a)?+ 8?} may occur in the denominator of d¢/dw, and will certainly occur in the numerator if there are sources or vortices clear of the boundary. Now Got (a? + 62)? + 42a? + 87)? +20h7?6? = (11) is a curve-factor having the assigned linear range, with modulus {(—«)?+ 6?}’” in that range. Hence Gx {(C—2)? +8} with Logarithmic Singularities and Free, Boundaries. 195 has constant modulus. (iii.) There may occur in the denomi- nator of d&/dw a factor €+06, where 6 is positive. ‘This vanishes at a point in the linear range, and it is impossible for a proper curve-factor to have a modulus which vanishes in its linear range. Hence the only way of maintaining constancy of modulus of dz/dw, so far as this element is con- cerned, is to introduce into Ga similar factor €+6. This is a corner-factor, and introduces a corner of internal angle 27 into the free stream-line ; in fact it represents a cusp in the free boundary, pointing inwards to the field. If 5 were zero one branch of the cusp would belong to the fixed boundary. Thus Gis completely determined as a product of powers of terms of the type of 43, G1, and (€+0). If the free boundary does not extend to infinity the linear range of & must be finite, and may be taken from €= —c to ¢=c; and the synthesis of @ is as follows :—(Q.) If a factor (€—a)", where a’?>c’, occurs in Fdé/dw, put a=c cosh y, and note that Gual—ce 4+ (C—e)"?. we (12) is a curve-factor whose modulus in its own range is {2ce~V(e cosh y—£)}”. Thus ;°"(€—a)” has constant modulus, and 5” is the proper factor to be introduced into @ A particular case of G is A=t4+(C—c?)'", whose modulus in its own linear range is constant. The apparent possibility of introducing into @ any arbitrary power of 4 might seem to indicate indeterminateness of Z but the power of @ is not arbitrary, being determined by consideration of the angular range of the (z, €) transformation. (ii.) A pair of conjugate complex factors {(€—2)?+ 87} occurring in df&/dw is counterbalanced by introducing into @a suitable power of a factor of the type G=(6—k) cosh y+ (“’—c?)”’ sinhy . . . (13) The parameters are determined by the relations cosh y=[{ (ate)? +f? +4 (a—c) + B?}" 1/20, k = a/cosh*y, and the modulus of &; in its own linear range is {(¢— a)? -+- 82}, (iii.) If the denominator of dédw contains a factor £—), where c? >6’, the same factor (a corner factor) must be intro- duced into & This indicates an inward-pointing cusp in the free boundary. Pi 2 196 On Two-Dimensional Fields of Flow. The angular range which the (z, €) transformation must have is known from inspection of the fixed boundary ; it is generally —7z. As the angular range is a times the order at infinity of dz/dé, there can be no uncertainty about the power of 4 which must be introduced into in order to give the desired angular range. Thus the synthesis of Zin terms of &, &, &, and corner factors, is determined. When there is flow to infinity between two free stream- lines which tend to parallelism, one factor is employed to represent the double curve. ‘The linear range is taken from €=c through €=+0 and €=—x tof=—c. Two forms of curve-factor, namely, 65.3 (€+4) sinhy—i(C—e)"", G,=(k—f) sinhy—i(0?—c*), (14) where k>c, can be adapted to counterbalance the variability of modulus of real or pairs of imaginary factors in Fd€/dw ; it is unnecessary to go into details of the algebra. As an example consider the case in which the fixed boundary consists of a semi-infinite and a finite straight line meeting at right angles, and the motion is due to a single source. The transformations are _ —HGdg RMMNLn pUauy ya ido ae (€—c) w= on log { (f o.) +B tae ° (15) whence dw/dg= — (m/2n) (E—a){(E—a)? +}. The rules of synthesis indicate that G= Gis gt?) (a? + Boyan Or (oO? + ee (G= a) Ga (16) according as @ is positive or negative. 9. Free stream-line problems in which part of the fixed boundary is curved.—The possibility of specifying solvable problems of this type may be illustrated by the case of flow through a semi-infinite straight pipe with a symmetrical curved nozzle. Fig. 2 shows half the configuration (which Fig. 2. 8 9 fa Pe eee <1 es o~ AS) 3 is all that need be considered), and the values assigned to ¢ at the various points. The transformations are of the form —Hédé oo Gary l w=— 5 log (E—}), is eee Work Function of Electron escaping from Hot Body. 197 so that dz/dwx @(€—a)-”. Itis required of & that it be a curve-factor having linear range from €=—o to =a, angular range pr, and such that | &(€—a)-? | is constant for negative values of €. Forms of & known to satisfy these conditions are (Cr + a'?\PG3, and &#, where Ga t(Cr ra yer —a)ie—alee, . . (18) Bate? 4a) hr PO(C— aa exp(—t a ab (19) Other forms of curve-factor which can be used in this way may be got by assigning convenient forms to / in the formula 6 Exp x| @log {(6?+8")/(0—6%)}00, - (20) or by using another formula given in article 40 of the previous paper on the subject. In each case the form of the nozzle depends upon the selected form of & The difficulty of integrating the (<, ¢) relation may be formidable. eee SSS XXIX. The Determination of the Work Function when an Electron escapes from the Surface of a Hot Body. By Horack H. Luster *. HE equation for the thermionic emission from a hot metal in a vacuum has been deduced by Richardson f and others from theoretical considerations involving the assumption that the potential energies of an electron inside and outside of the surface are different, so that an escaping electron must do work against an equivalent adverse potential difference. This difference in potential is repre- sented in equivalent volts by the symbol ¢. The work done by an escaping electron is represented in the well- known Richardson equation , == Oe by the constant b. In order to justify completely the assumption involved in b, such a work function should be found to exist, and its magnitude should be identical with that of 0. In 1903 Richardson + showed that the escape of electrons * Communicated by Prof. H. L. Cooke, M.A. 7 Proc. Camb. Phil. Soc. 1901, p. 286. } Phil. Trans. A. vol. cci. p. 497. 198 Mr. H. H. Lester: Determination of Work Function should involve a loss of thermal energy, due to the fact that energy is rendered latent by the potential difference at the surface, and that this loss ought to increase rapidly with the temperature. Wehnelt and Jentzsch* announced the discovery of such a cooling effect, and attempted its mea- surement. However, they seem to have neglected to compensate for the disturbing action of the thermionic current in their bridge system, so that the effect they meastred was considerably obscured by the thermionic current disturbances. Cooke and Richardson f found such an effect in 1913. They described a method of measure- ment, and published values on the cooling effect, for osmium and tungsten. Before this, however, viz. in 1910, the same experimenters had announced the discovery of a converse heating effect t and had published values for a series of metals. The value of $ was calculated from both the heating and the cooling effects, and was found to be of the expected order of magnitude. The present paper contains a discussion of their work on the heating effect, and presents an extension of their measurements of the cooling effect. Some related features are discussed. The paper is divided as follows :-— I, Experiments of Richardson and Cooke. II. Measurement of the cooling effect for carbon, molyb- denum, tantalum, and tungsten. III. Related features : . Identification of 6 with ¢. . Relation of ¢ to contact potentials. . Effect of gases on d . Nature of surface films. . A new method for determining the temperatures of hot filaments. Ou Oo bo I, EXPERIMENTS OF RICHARDSON AND COOKE. These experiments carried out in this laboratory showed the existence of the heating effect. It is not claimed for them that they show more than the order of magnitude. In fact, Richardson was inclined to the view that, owing to the way in which the contact potential entered, the effect found rather measured the cooling effect at the surface of the hot * Ann. der Phystk, (3) vol. xxviii. a 537 (1909). + Phil. Mag. April and Sept. 1913 ¢ Phil. Mag. July 1910 and April 1911. when an Electron escapes from Surface of a Hot Body. 199 electrode than the heating effect at the surface of the cold one*. The rate of production of heat by a current 7 flowing across the boundary of the cold electrode is given as i(g+V+2@,-4)), where ¢ represents the equivalent potential at the surface, V the potential between the electrodes, a twice the gas constant calculated for a single molecule, e the charge on an electron, 8, and @) the absolute temperatures of the hot and cold electrodes respectively. 2 is the saturation current and is independent of V. The @’s are constant; hence the expression inside the parentheses is a linear function of V. The heating effect was measured by comparing the change in resistance produced by the influx of electrons with that produced by a known increment of current through the metal of the electrode. The effects thus found for a series of voltages were plotted against V,and the intercept of the straight line thus found with the heating-effect axis gave the value per unit current of $+—(A1—%) = H ; from which ee H—2(,— é,). The materials under investigation were mounted in thin strips forming a grid between two osmium filaments which served as the hot cathode. The electrodes were mounted in a brass vessel that was made airtight by means of waxed joints. It was impossible to observe in such an arrangemnt the precautions that Langmuir has since shown to be necessary to insure consistent behaviour of thermionic phenomena. Their results showed that small changes in the gas-pressure did not affect the value of ¢. Evidence was found to support a theory put forth by Richardson that the heating effect is a function of the cooling effect through the relationship Wi = W. + ev. dy = d2o+ Ve, where V, represents the contact potential between the hot * O. W. Richardson, “ Rapports du Congrés Int. de Radiologie,” Brussels, 1910. or in terms of @, 200 Mr. H. H. Lester: Determination of Work Function and cold electrodes, ¢, the cooling effect at the hot, and ¢, the heating effect at the cold electrode. Langmuir has shown that the big effects due to the presence of residual gases disappear only with the last traces of the gases* ; hence, in view of the lack of ideal conditions, it is not surprising that Richardson and Cooke did not find that changes in gas-pressure affect the heating effect. The dependence of the heating effect of the anode upon the cooling effect of the cathode was shown by some peculiar results obtained in experiments with osmium as the source of the thermions and iron as the receiving surface. It was found that the osmium developed two ranges of thermionic emission ; 2. ¢., for a given tempe- rature two values of the saturation current were possible. Apparently two values of ¢ were involved. The larger saturation current gave larger values of the heating effect, and the lower saturation current gave lower values ef the heating effect. The theory of the dependence of the heating upon the cooling effect regards the work done when an electron is taken completely around the circuit. If W, and W, represent the work-functions at the surfaces of the hot and cold electrodes respectively, and if V- is the contact- potential between the two metals, then this work may be expressed as W,—eVe— Wet = 0, where includes work done against external resistance, gas pressure, and Peltier potential. The terms in ¥ are negligible in comparison with the first three; hence we have Wa = Wet eV. ic 1 et According to this relation, (W.+eV,) varies directly with W,. Now the thermionic currents at a given temperature vary inversely with 6. Hence, since ¢ and 0 are supposed to be equivalent, it is evident that the observed change in the heating effect should have been inverse to the change in current ; that is, W.+eV¢, the effect measured, should have been smaller for the larger current. This was contrary to their observation. It is possible that a contrary change in eV, more than compensated for the change in W. The same experimenters later measured the cooling effect * Phys, Rev. Dec. 1913. when an Electron escapes from Surface of a Hot Body. 201 for osmium *. According to the theory given above, this should have been equal to the heating effects already mea- sured. They found for the cooling effect at the osmium surface, 4°7 equivalent volts. The values in eq. volts for the series of metals whose heating effects were deter- mined (omitting iron) were 7:26. 5°3, 7:1, 5°82, 5°6, 5°19, and 7-4, which gives a mean of 6°23. No one of the measurements gave a value as low as that obtained for the _ cooling effect of osmium, and the difference from the mean was 1°5 volts. It seems, therefore, that the experimental evidence does not verify the relation expressed in (1). Further reference to this discussion will be made when contact potentials are considered. It will be shown that the sign of V, in equation (1) probably should be changed. II. MeasurEMENTS oF ¢ FROM THE CooLine EFFEct. In 1913 Cooke and Richardson published experiments that showed conclusively the existence of the cooling etfect, and described a method of measurement that is capable of giving quite accurate results. This method has been employed in the present investigation and will be described in some detail. Their rather lengthy mathematical deduc- tions are omitted. For a complete discussion of the method the reader is reterred to the original paper. The expression for the loss of thermal energy due to the escape of electrons is the same as that for the gain of thermal energy in the heating effect, viz. i(¢+%(6—4,)). The symbols have the same significance as before. The cooling effect, as in the former case, was measured by noting the change of resistance caused by the change in temperature (in the cathode in this case), and by comparing this change of resistance with that produced by a known change of electrical energy. The experimental arrangement is shown in fig. 1. Battery B, supplied energy to a bridge network in one arm of which the filament was inserted. The other arms consisted of a 10-ohm Wolff standard, a 1000-ohm ratio-coil of a Leeds & Northrup bridge, and the variable resistance from the same bridge. The bridge current could be reversed by the commutator K,, and could be increased by a very * Phil. Mag. April 1918. 202. Mr. H. H. Lester: Determination of Work Function small fraction of itself by means of a 100-ohm shunt (y) around a 1-ohm cell (#) in series with the battery. These resistances were Wolff standards. The l-ohm and the 10-ohm coils were maintained at constant temperature Figs. by means of an oil-bath. A Weston ammeter (A), reading to 1500 milliamperes, served to measure the bridge current. It was compared with another ammeter for which the calibration was known. The battery potentials were read on a Weston voltmeter (V) reading to 75 volts. It was compared with a Weston standard voltmeter and found to agree. ‘The standard was later calibrated by means of a Wolff potentiometer and a Weston normal cell. The thermionic current was introduced into the bridge when an Electron escapes from Surface of a Hot Body. 203 system in either of two ways. In the first method it was introduced at the middle point of the arm CF by means of the high-resistance shunt PQ. In the second, the shunt was placed around the arm FE by means of the key K3. With this arrangement the thermionic potentials were applied to the middle point of the filament L. ‘The thermionic potentials were measured by means of the micro- ammeter, I. This instrument was a Paul testing set, giving 1 scale-division per 2 microamperes and reading up to 375x10-* ampere by means of suitable shunts. It was calibrated by means of a Wolff potentiometer and a Wolff standard resistance. The thermionic potentials were supplied by a 120-volt storage battery, across which was shunted a 1500-ohm slide-wire resistance from which suitable potentials were tapped off. The potential used was ordinarily about 40 volts. The thermionic currents were started or stopped by reversing the commutator K,. The galvanometer used was of the d’Arsonval type, with a resistance of 225 ohms, and gave a deflexion of 1 scale-division for a current of 9x 10-° ampere. Method of Observation. Observations were carried out as follows. The key K; was closed, so as to shunt the 10-ohm standard in the arm CF. With the thermionic current off, K, was opened and closed and the resulting galvanometer deflexions were recorded. Usually seven deflexions constituted a set. Then, with K, open and Kg, alternately reversed, the resulting deflexions due to turning the thermionic current on and off were recorded. From seven to fifteen deflexions were taken in this case, depending on the magnitudes of the deflexions and the steadiness of the galvanometer.. The mean de- flexion was determined in most cases by the method described in Richardson and Cooke’s first paper*. The ammeters A and T were read, the voltage of B, determined, and the resistance of L was read from the bridge system. K, was now reversed, and the whole system of measurements repeated. The two series of observations constituted one complete set for the “Standard Shunted” arrangement. An exactly similar complete set was taken with K; closed, so as to shunt the filament instead of the 10-ohm standard. This second set with the ‘‘ Filament Shunted ” arrangement gave the data for calculating an independent value of @. The two methods gave consistent results. * Phil. Mag. July 1910. 204 Mr. H.H. Lester: Determination of Work Function Theory of Measurements. The methods consisted chiefly in comparing the loss of heat due to turning on the thermionic current with the gain in heat when a known increment of current was supplied to the filament. ‘The comparison was effected by assuming that OH, the change in heat supplied to the filament, was in each case proportional to OR, the change in resistance of the filament. The increment of resistance due to an increase in the current was accomplished by shunting z with y. The final expression for the change in the supply of energy is given by the expression Alt 2° 2 ye L OH=(1+9)"i°(2R.2E + OR), . : | ee where : 1+4( ie G ) 3 as G+DE(1+9+ a) De, (3) where OE = increment of the rate of supply of energy ; g = fractional part of bridge-current through the high-resistance arms ; 2; = current through the bridge; R= resistance of the filament ; G = resistance of the galvanometer—225 ohms; « = shunted portion of external resistance—1 ohm. y = shunt about «—100 ohms; OR, = change in resistance of the filament due to shunting 2; D. = galvanometer deflexion due to shunting w ; DE and CD = resistances in the arms DE and CD respectively. The change in the resistance due to starting the ther- mionic current is not so easily determined. The presence of the thermionic current in the bridge system disturbs the galvanometer balance, and there is a change in the resistance of the filament due to the Joule heating effect of the ther- mionic current. The first effect is minimized for the “standard shunted”’ method by introducing the thermionic current at the middle points of the two lower arms. With this arrangement the current should distribute itself sym- metrically around the network and should not influence the galvanometer. For the “tilament shunted” arrangement the current theoretically departs from and returns to the same pointin the filament, and is not present as a disturbing factor in the bridge network. As a matter of fact, perfect compensation is not achieved in either case. In the first when un Electron escapes from Surface of a Hot Body. 205. case, the thermionic potentials are placed on one end of the filament. ‘The superposition of the bridge current causes an unsymmetrical emission of electrons, so that the ther- mionic current is not quite symmetrically distributed about. the bridge network. In the second case, the superposition of the bridge current causes a shift in the centre of emission from the centre of resistance to which point the current was returned, so that there is a resultant potential difference along the wire. These effects are supposed to reverse with the bridge current. With z first in one direction and then in the other, alternately, high and low values of ¢ may be obtained. The mean of these values is supposed to be the correct one. In practice these two values are not worked out. The readings for the two values are grouped into a single equation, but the validity of the method depends upon the truth of the above assumption. Inasmuch as Cooke and Richardson did not test the above assumption experimentally, it was deemed advisable to perform such a test in this in- vestigation. The two values of ¢, found with z first in one direction and then in the other, differ by a considerable amount. If, however, an alternating current be used to heat the wire and a small superimposed direct current be used to operate the bridge, two values of ¢ should be found that differ by a much smaller amount. This experiment was performed, and the prediction was fulfilled. The two values found for the cooling effect were 4°742 and 4:°797, the mean for the two being 4°766. Unfortunately, an acci- dent prevented the securing of direct-current measurements. on this particular filament. However, characteristic values from a similar filament by the direct-current method were 6°210 and 3°315, the mean being 4°763. The results of the experiment were regarded as satisfactory proof of the validity of the assumption that the mean of the high and the low values in the direct-current method really eliminated the uncompensated portion of the disturbance produced by the thermionic current in the bridge system. The Joule heating effect of the thermionic current presents more difficulty. It cannot be eliminated and can be calcu- lated only approximately. Cooke and Richardson performed this calculation and incorporated the correction term found in the general expression for the total diminution of energy due to the escape of electrons. we found for ae total rate of loss the expression oR oR, oH=T($+ = (@:—6)) + Bin" OR _ 9 oR E -2f" in(1+ or) (1 + 21 [ (“ide Ja, @ 206 Mr. H. H. Lester: Determination of Work Function where OH = rate of loss of heat ; OR; = change in resistance of the filament due to turning on the thermionic current; R = the resistance of the filament ; Ry = the resistance of the circuit of battery B ; j = the thermionic current flowing into unit length of the filament ; a, £ = coordinates along the filament measured from the centre of distribution of the thermionic current ; p = the resistance per unit length of the filament with T off ; -'T = the thermionic current. The other symbols have the same significance as in (2) and (3). Expression (4) divided by expression (2), the change in the rate of supply of energy due to shunting w with y, ‘gives the value of the ratio OH/dE, which is equal to OR:/OR., or, OH te OR; ues Dr—pw Sh TOR De? 1 ieee (5) where py is the deflexion of the galvanometer due to the uncompensated part of the disturbance produced by the thermionic current in the bridge circuit. Expressions similar to (4) and (5) are obtained for the bridge current reversed, except that —2 is substituted for +7, and —py for +p. Substituting (4) in (5) for the two cases, adding, and simplifying, the expression for the cooling effect is v obtained in the form : a cok 2(1 —2q)i2 (Do+ Dz) Re? x +y co) + e (0,—0))= (T+T)D.V(e+y) (1+ “tar, (6) where the primed letters refer to the values with current reversed, and OR, is given by (3). Mounting and Preparation of Filaments. The filaments were mounted axially in glass tubes as illus- trated in fig. 2. The tungsten and tantalum filaments were welded to copper leads, which were in turn welded to stout platinum leads that were sealed into the ends of the tubes. The molybdenum was silver-soldered to the copper leads, and the carbon was attached by means of a paste which was supplied for the purpose by the Edison Electric Lamp Com- pany of Harrison, New Jersey. The anode was made from clean copper gauze bent into a cylinder to 4t the glass tube when an Electron escapes from Surface of a Hot Body. 207 and crimped over at the ends so as almost completely to enclose the filament. Outside connexion was made through a platinum wire sealed into the glass. Fig. 2. L=filament ; H = copper gauze anode. After a filament was mounted as described the tube was attached to a rotary mercury-pump and enclosed in a vacuum furnace. During exhaustion the furnace was maintained at a temperature of about 600 degrees centigrade. When the McLeod gauge showed a pressure of about 1 x 10-* mm. of mercury, the furnace was turned off, connexion with the 208 Mr. H. H. Lester: Determination of Work Function pump was broken and exhaustion was completed with charcoal and liquid air. Liquid air was also placed on a trap near the furnace so as to prevent mercury vapour and other eon- densible gases from diffusing back into the tube. As soon as it had cooled sufficiently the furnace was opened and the filament was glowed out at a temperature near its melting- point. This process was accompanied by a considerable evolution of gas. This evolution did not continue long, so that after about fifteen or twenty minutes the pressure decreased to about 2x 10~-° mm.,in most cases with the wire hot. The tubes were now sealed off and were ready for use. In one case readings were taken on a tungsten filament before sealing off. The above procedure was not followed with carbon except in one case where the tube was sealed off when the pressure was down to 1x1074 mm. with the wire hot. It was found that the carbon gave up a very great quantity of occluded gas. It seemed impossible to glow it out to a point where no more gas was given up. Tor reasons that will be dis- cussed later it is thought that no serious error arose from the lack of ideal conditions in the case of carbon. The temperature of the filaments, except in the case of tungsten, were estimated by comparing the colour of the filament with that of a tungsten filament for which a tempe- rature-current curve was available. The tungsten tempe- — ratures were given by the curve referred to above, for which thanks are due to Dr. Irving Langmuir, of the General Hlectric Company Research Laboratory. Dr. Langmuir was kind enough to furnish most of the filaments used in this research. The tungsten and tantalum filaments were fine drawn wires similar to those used in the construction of metallic filament lamps. The molybdenum was in the form of a fine wire about ‘1 mm.in diameter. The carbon filaments were taken from burned-out carbon lamps of the “ metalized ” filament type. The filaments of these lamps are baked out in the process of manufacture at a temperature of 3000° C., thus changing most of the carbon to graphite and greatly purifying the material. The other materials were regarded as pure, though no investigation was made except that a test on tungsten showed no trace of thorium. The values of ¢ are given below in the form adopted by Cooke and Richardson. In their work they neglected the factor 1/i in the expression for OR,. This omission did not materially affect their values for tungsten, but it is very important for many of the values found here. when an Electron escapes from Surface of a Hot Body. 209 CARBON.—Filament #1. Fil. sh. St. sh. 0. 1. 1 Dee lealOs Ve a a a e iH d+—(A8).*) b+ — (AB). | 7(A8).| > ‘7'70| 38:8 8:8 | 7912 iG: | 24-7 4:°948 O81 | 472 approx. }270).39°3 | 1:55) 7915 T49 | 24-7 1650 roe eto ics | S000) 122-5 | 30-4 heal. Oso On P8004 | 122 30°5 5 UE HI ea Gerke 22739 (a ar) | 800) 40°4 Sado fw Le2 24-7 RCS on TES Ate Ree AS -240 | 4°88 1700 SOU} 41°3 TSuivog Jae 132 24-7 800} 39:7 Py Oro | Lob DAT el ease 5084 -240 | 4°79 S00) 46:8") 138-9.) 7-970 | 136 es OFF 835} 5:1 1:7 | 8249} 289 30°4 1730 *835| 4:5 DEVE tt ALS 3074 eit te ieee we Rl tes 248 | 4:140 8383p) 5:3 te OR aOIL,. 280) SO ase ca. 4-6683 948 | 4420 835} 4:9 | S200) Qs 30°4 8901 7-0 | 36 | 8211! 700 | 24-7 1800 890) 66 53 | 8-206! 705 24:7 ASASOW ital wasnt 256 | 4°50 890| 71 89 | 8290! 780 2A Tee aoe 4:‘676 256 | 4°42 ‘890! 6:7 Fa 8 2Ol| 8t0 24:7 900} 50°0 DO | S289) “Gd 248 E55) LC Gy: POT Ey ER aaling a w 264 | 4:90 900} 52-1 3°39 | 8:196| 649 I24°8 500) 52-3 5 1) 82784 ‘63k A tlhe Aes ae 4:637 264 | 4:38 900) 51°8 1-1 | 8-284] 666 24:8 900) 51:0 | 23-7 | 8:194!] 610 24:8) 4°647 ate Sean 264 | 4°37 1850 900) 52°7 | 34° 8189) 611 24°83 900) 5:5 50 | 8426!) 882°6 | 30°5 CAO D A Ae Flin gene eZ 264 | 4:64 900} 49 58 | 8-420} 835 30°5 900} 59 9-8 | 8429! 815 3025) ane sa cc 4:799 264 | 4°54 900) 53 1-4-3430 1825 30°5 4 Mieane cacssee 4-500 Filament #2. 9 | 1-49| 21-34] 11-65 4-037] 718 | 29-7 Hstimated.|1-49| 21-46] 493] 4034; 685 |297| 4878 | ...... 256 | 4-61 1800 | 1-49] 20:78] 7-90| 3997] 658 [297] _...... 5214 256 | 4-96 1-49) 20:51] 848) 4002] 670 |297 1900 _ | 162) 28:2 | 36-0 | 4-095] 3085 | 30-2 1-62| 21-2 | 149 | 4095| 2063 [202] ou... 4-43 279, | 4:16 igo9 | 130) 22:02] 50 | 4:0338| 738 | 29-7 = | 4-50] 21-66) 13-08| 4-:035| 775 |297/ 5018 | ...... 26 | 476 Mirena (2, .ca ee 4°62 * AO represents (@,—9O,). Phil. Mag. S. 6. Vol. 31. No. 183. Alarch 1916. () 210 Mr. H. H. Lester: Determination of Work Funetion CARBON (continued).—Filament £3. (Alternating current-method.) a Fil. sh. St.sh. |. Q, A aD ah Des AR. AE >< 10 -SainI: 2 n9 hora lars 4 6+ 2 (a8). | o+ Scag. 2°} -698| 34 | 338 | 45211 1605 | 36 Prey Mischa? s| 338, | 4510), T6Sa Mee lai ce 5-231 301 | 4:93 % -700| 365 | 59°83 | 4510] 1613 | 36 -700| 346 | 10 | 45391 1600 | 36 4-449 | ‘301 | 4:14 Mean) 202508 4:53 Filament ~4.—After exhaustion in vacuum furnace. 889 68 | 548! 8441| 660 |24/ ...... 52072 | 240 | 4:97 : 939) 6-7 | 275 | 8527) 692 | 24 4936 9. i ee -240 | 4:70 882| 68 | 697 | 8527] 700 | 24 895) 68 | 56 | 8398) 565. | 24) 4-621 24 | 4:38 i749 | 89| 6S | 16 | 8898) 555 | 24 : 895] 68 | 3:03 18309] 521 | 24 4667001) aoe 24 | 4:43 ‘8981 65 | 3:56 | 8317| 510 | 24 Tonge | oot) 028) F251 BSI Ole: et) a 5-104 27 | 4838 934, 7-25} 6:19 | 8518| 1022 | 24 Meanie sine 4°66 Mean of all values ........ 4°55 Value of “6b” corresponding to 4°55 = 52800. MOLYBDENUM. | Fil. sh. St ish sat A =Gnn C pas (a Q. Daley oes rene aug Divs p+ (20), 64% (ad) “(a0).) @ MM aio 4a. ie oes! 959 las ae i759 | 1410) 844 | 17-3 | 1-259) 2785 | 24 ASTON eee 248 | 4-622 te 1:410| 34:8 | 20% | 1:259] 279-5 | 24 1-410| 34-2 | 11-9 | 1-268] 2792 |24 | 9 ...... 4-246 248 | 4-598 | 4-560! 38-2 | 33:5 | 1-290] 5635 | 24 Mi ice! sang | 15CO) 88:0 | 25°95| 1-201) 5625 [24 | ...... 4-943 264 | 4-679 “0 | 1-500! 383 | 369 | 1-285) 557-5 | 24 | 1:500| 37-9 | 19-9 | 1-290] 557-5 | 24 #862) 264 | 4598 | 4.5451 899 | 33:2 | 1-381| 10145 | 24 4 848 O74 | 4574 1999 | 17545] 401 | 659 | 1831) 1070 | 24 : 1:550| 40-4 | 580 | 1°331/ 1050 ‘| 24 1550 40-6 | 42:0 | 1-336! 10725 |24 | ...... 4-863 274 | 4-580 | yrs] 42-0 | 69-9 | 1326) 1150 1935 GG jong | LT) 428 | 364 | 1328) 1150 256] 4861 | 281 | 4-580 (9 | 1-575 | 41:3 | 43°75| 1:329| 1160 | 23-5 1°575| 43:1 | 645 | 1:329| 1188. |935] ...... 4-745 981 | 4-464 Meanie ae 4-588 “6” = 53200 > when an Electron escapes from Surface of a Hot Body. 211 TANTALUM. | p 0. Dee Daa Re VPs 1075.| Vi p+ (1 —%) 7A). gi Fil. sh St. sh eZ ‘605! 4:05 | 2°03 | 4-438! 69-07 | 24:5 17909 | 605) 4°28 | 1-43] 4434) 65-7 [245 4974 fo. 240 | 4-734. -605| 4:00 | 2:01 | 4-436] 64 | 245] 605} 418] 1:12 | 4443] 64-4 245) ae 4-579 240 | 4:519 639] 5:57 | 5:29 | 4365! 3135 | 24 v59 | 689] 5:05 |11°30 | 4356] 316 = | 24 ATOR A Riess ee 248 | 4-455 ; 6391 5:04 | 8-45 | 4:351/ 321 [24] ...... 4-732 248 | 4-534 639] 504 | 832 | 4372) 325 | 24 1775 | 648) S12 |1897 | 4-401) 545 | 24 648] 5°37 | 825 | 4-411] 5355 | 24 ALSO WINi Mae vane. 250 | 4:340 foo soe te EE ee ne aaremay e ) O eea a | ee ee Be ee ee | Ee ee eee 660! 5°13 110-33 | 4514! 402-5 | 24-1 1800 | 660) 5-00 | 842 | 4520] 3925 243) 4984 | oo... 256 | 4:72 ‘660! 5-01 | 11-29 | 4522! 3935 |243) 0... 4-644. 238 | 4383 ‘660| 4:97 | 5-495] 4592} 378:5 | 24-3 662] 5°55 | 13-02 | 4-480] 495-5 | 24 _ |662) 5°33.) 11-44 | 4-440) 5045 |24 ] ou... 4-791 257 | 4534 688] 5:42 | 24-84 | 45891 742-35 | 24:3 1959 | 688) 561 | 7:10 | 4579] 7441 [243) 4758 | o..., 364 | 4-494 688) 5°36 114-36 | 4572! 7506 |24:3) ...... 4-796 264 | 4532 688} 5:41 117-73 | 4581] 766-5 | 24:3 ee US MS OSES Aaa se OS 7131 663 |25°36 | 4597/1715 12438) 4845 | 1... 274. | 4-571 1900 (| 718) 5-74 |49-42 | 4620] 1712 | 24-2 713] 52213818] 46011 1660 |242) 0... 47582 274 | 4-308 ‘715| 7°97 |35°51 | 4600] 1912 | 24-2 | i | | Mearn(an le 4-511 T=) 212 Mr. H. H. Lester: Determination of Work Function TUNGSTEN. Wire #1. (After thoroughly ageing out.) Fil. sh. St. sh. A 6. 2p. Dp. | Ro sOaae at ‘ 2 Z “ o-+-(A0). o+-(A9). a ? 321) 176 112389 19815 588 |245) 48499 ae 301 | 4-547 oggo | B21| 175 | 5539/1915, 615 | 245 2 3901 17-7 1142-621 20-2191 615 [245] ...... 47635 | “801 | 4-462 -391| 18:1 | 49-05| 20-208 690 | 24-5 -390| 16-9 | 116-6 120-020) 570 | 24-2 3901 16-9 | 563 |20:063' 582 [242] 44797 ion 301 | 4479 Wire ¢ 2. -294| 29-8 | 27-4 |19-180/ 90 | 10-3 soo | 294| 2e4 | 256419180] 91 103] ...... 4-550 -290 | 4:260 - 295] 29:94| 21-2 119-00 | 94:5 | 10:3 -295| 30:2 | 29-82/19:00| 94:5 |103| 44890 | ...... -290 | 4-190 295] 28-921 28:3 19-070, 925 |103| 49493 | ...... 290 | 4-552 -295| 29:08] 21°38] 19-0701 88 |103 |-293| 28-94] 25-90 19-425] 92:5 | 10-3 | 993] 28-94] 26-10119-425| 1025 1103] ...... 46887 | -290 | 4-399 Wire +3. 2050,, |203| 11:50] Bre7|1z534) 61 | 24 - -303| 11°64| 15-72/12531/ 61 | 24 4511) 300 | 4-211 3331 13-7 | 51-08/13-559| 291 | 94 o1so |333| 188 | 3949135501 201 [24 |... 4-858 316 | 4:542 - 3331 13-98| 39:26113:555| 286 | 24 3331 14-2 | 50°33/13555| 286 | 24 47865 | 1 316 | 4-470 -352| 15-61 | 103-561 14-1671 663 | 24 Met gop | 301] 15°62} 90:59| 14-163} 663 | 24 4-9994 nay -394 | 4:598 2 351] 15°75| 21-67/14-400| 721 | 24 S51) 153848598 14-400| 715 eto) oo... 49575 | -324 | 4633 3001 19:35} 579112681 65 | 12-22 soo |200| 20:19] 1554)12-684| 66 |1222) 47840 | oun... 290 | 4-494 - -300| 20-01} 10°54112-497, €1 | 12:22 300 20-1 | 10-17/12-496| 61 [12-221 ..... 4-869 290 | 4-579 BAGMOS80) 47-7 419-672) 369 Gene. | 316 oiso. || 236) 248 | 489 | 13673] 360 1242) 48607 | 2 ae 4-545 2 936] 24-2 | 70-6 |13:884 386 | 12°12 336] 94-5 | 323 |13:883| 385 [12491 ...... 49292 | -316 | 4-606 -935| 94-42| 32-74| 13-872) 3871 {12-11 pis «, | S30 oe| m24) 13'872) 875 a 5040. | 4a 316 | 4-724 : 336) 95-20| 49-491 13-661| 350 |1211| ...... 4-670 316 | 4:354 336) 25-79| 48°62 13-6621 352 | 12:11 ( when an Electron escapes from Surface of a Hot Body. 213 It is difficult to make any definite statement as to the exact accuracy of the above measurements because of some apparently unavoidable errors. In the cases of tungsten and tantalum the galvanometer was subject to slow drifts, Sometimes in one direction and sometimes in the other. It is thought that these drifts were caused by a lack of homo- geneity in the filaments. A tungsten filament when passed between steel rolls broke up into five or six finer filaments. It may be that these wires are never quite homogeneous. It is thought that errors of this sort are largely eliminated from the mean. For tungsten the mean variation from the mean of the eighteen values found was ‘11 volt or 2 per cent. It is thought that this represents about the accuracy for the tungsten and tantalum measurements. Owing to the lack of ideal conditions for carbon and to the fact that the small temperature-resistance coefficient caused the galvanometer deflexions to be small, the carbon values are probably not accurate within less than 4 per cent. Only one specimen of molybdenum was examined. The readings were good, however, and may be regarded as accurate within the same limits as the values for carbon. It is interesting to note in this connexion that values of & found for one wire that was known to be not homogeneous were more nearly normal than were corresponding values of “b” found by observing the temperatures and currents. From this observation it was concluded that measurement of the value of ¢ is less subject to error than are corre- sponding measurements of the value of “6.” It may be that the value of can be more accurately determined from the cooling effect than it can be from the current-temperature relationship. III. Certain FEATURES RELATED TO THE PROBLEM. 1. Identification of b with ¢. It will be remembered that in the theoretical derivation of the Richardson equation, b is supposed to represent the work done when an electron escapes from tlie surface. Since this work is measured by the potential difference represented by @¢, it follows that 6 and ¢ should be equivalent if the assumption in regard to 6 is correct. The relationship between ¢ and 6 should be given by _ pe ger +1 1) CN Re a) where e¢ is the charge on an electron and R is the gas constant calculated for a single molecule. Zi4 Mr. H. H. Lester: Determination of Work Function Jt is important that the equivalence of ¢ and 6 be-estak- lished in order that the validity of the Richardson equation should be more completely demonstrated, and also for another reason. The assumption involved in } rests upon the further assumption that the translational kinetic energy of an elec- tron in a hot metal is equal to the energy of a gas molecule in a gas at the same temperature as the hot metal. This latter supposition has been proved true by the experiments of Richardson and Brown *. ‘Their work receives additional support if it is demonstrated that ¢ and b measured inde- pendently are shown to be equivalent. The term - (6,—6,) in the expression for ¢ is based upon the results of Richard- son and Brown’s work. However, as pointed out elsewhere, this term contributes but little to the value of @, so that its presence does not seriously affect the validity of the check. The experimental verification of equation (7) was attempted by Richardson and Cooke, but, as explained above, their values of @ obtained from the heating effect were of insuffi- cient accuracy to give more than an approximate test. Their work on tungsten, however, was carried out under ideal con- ditions and by a method that is capable of giving accurate results. They found for tungsten db = 4:24 eq. volts, from which, by eq. (7) 6 = 49400. Langmuir ¢ has determined b very accurately for the same metal, and has found b = 52500. Recent work in this laboratory by Dr. K. K. Smith} gives b = 54700. Dr. Smith’s value was obtained as the mean of a large number of determinations in which high and low values are given equal weight. It 1s probable that the low values are more nearly correct. If Langmuir’s value is given weight 2 and Smith’s weight 1, then the weighted mean of the two determinations gives 6b = 53600. The Cooke and Richardson value as calculated from ¢ differs * Phil. Mag. vol. xvi. p. 853 (1908). t Phys. Rev. Dec. 1918; Phys. Zezt. vol, xv. p. 516 (1914). { Phil. Mag. June 19165, when an Electron escapes from Surface of a Hot Body. 215 from this by nearly 7 per cent. It seemed to the writer that this discrepancy was great enough to leave a reasonable doubt as to the exact equivalence of } and ¢. On account of the importance of the relationship it seemed advisable to obtain more specific information on this point. The present measurements on tungsten were taken with this endinview. ‘Tungsten was selected for the purpose because it is the only metal for which extensive measurements have been made under ideal conditions to determine the value of 6. Cooke and Richardson investigated only one speci- men in their measurement of the cooling effect. It was thought possible that an extended investigation might yield results more in harmony with the published values of 0b. The mean of the eighteen values found for tungsten gives h@ = 4°478, from which be o213503 This value is within ‘7 per cent. of Langmuir’s, and differs from the weighted mean of Langmuir’s and Smith’s value by 1°6 per cent. Hach of these differences is well within the limits of accuracy of the present experiments. The above result leaves little doubt as to the exact equi- valence of ¢ and b within the limits of accuracy of the above determinations, and consequently justifies the assumption in regard to b involved in the Richardson equation, and lends additional support to the findings of Richardson and Brown concerning the kinetic energy of electrons. 2. Effect of Gases on the Value of ¢. Langmuir has shown that residual gases strongly influence ‘the value of 6. It seemed worth while to test for corre- sponding effects in the value of ¢. Osmium when heated gives out gas copiously. An osmium filament mounted and treated in the usual manner but not completely glowed out, would give up gases that are probably free from water- vapour. These gases react with the hot cathode to form compounds which condense on the walls of the tube, so that there is a gradual improvement in the vacuum, a gas-free state being finally arrived at. A tube so treated gave initially for the cooling effect 5°93, after a few hours heating the value was 5°16. The filament burned out before the gases were all consumed, so that the final value was not obtained. A tungsten filament similarly treated but not at first sealed off from the charcoal tube gave values as 216 Mr. H. H. Lester: Determination of Work Function follows :—Before sealing off, 5-799 ; immediately after seal- ing off, 5-372. Succeeding values taken at various intervals for two or three hours were 5°44, 5°399, 5°270, 5°116, 4°79, 4°85. The subsequent values oscillated around 4°8. Carbon behaved similarly to the osmium and tungsten in that the initial values were high and the later values were lower and constant. A series of readings to determine the variation of @ with time was not taken ; it was noticed, however, that it took carbon much longer to become ‘“ normal” than it did tungsten. Usually the carbon filaments were heated from tweaty-four to forty-eight hours before readings were recorded. Langmuir investigated the reduction of the residual gases in the presence of a hot wire, and termed the phenomenon the “clean-up effect””*. Itis evident that in the progress of the clean up the vacuum was cleared of all active gases and successively lower pressures were produced. We have seen that successively lower values of @ were also obtained. Tne value found for tungsten in the unsealed tube is what was expected as the initial value immediately after sealing off. It is noticed that the latter value was greater than the former. This discrepancy was undoubtedly due to the fact that certain gases were liberated from the portion of the glass that was heated during the sealing-off process. It was concluded from the behaviour of osmium and tungsten that residual gases affect the value of @ in precisely the same way that they affect the value of 6. This result was expected. The behaviour of carbon is interesting. It may be re- membered that except in one case the carbon filaments were not subjected to the vacuum-furnace treatment. There were no waxed or ground-glass joints, except one that con- nected the glass tubing to the pump, and the pump was kept running continuously. Since there could be no addition of gas to the experimental chamber, it was thought that after sufficient time a condition would be reached in which the active gases would be consumed and the carbon would be surrounded only by residual gases inert to carbon. If the variation in d@ is due to the influence of active gases, then the elimination of these ought to produce the same con- stancy of behaviour in regard to the cooling effect as the elimination of gases by the clean-up effect. In fact the clean-up effect removes only the active gases. The results of the carbon showed practically the same behaviour as was * Journ. Amer, Chem. Soc, vol, xxxiyv. p, 1310 (1912). when an Electron escapes from Surface of a Hot Body. 217 obtained with tungsten and osmium. The pressure in the tube during the carbon experiments varied from 1 x 107*mm. with ihe wire hot to 3x 10~° mm. with the wire cold, ap- proximately. This behaviour of carbon verifies for the value of ¢, found when inert gases are known to be present, an observation that Langmuir made on tungsten. He found that the presence of argon in the experimental tube did not affect the value of b. Argon is inert to tungsten. It would be interesting to find out if any pressure of inert gas would affect the value of d. This was not attempted in the present investigation. 3. Nature of Surface Films. It is probable that active gases affect the value of ¢ through the formation of surface films. It is interesting and important to consider the nature of these films. The work of Richardson and Cooke shows that the heating effect measured under conditions that are not ideal is greater than the cooling effect measured under ideal conditions. It is probable that the heating effect, like the cooling effect, is increased in the presence of active residual gases. If the film acts merely as a high resistance through which the escaping electron must go, then the work that an electron does in escaping would be exactly represented by an equivalent Joule heating effect. Since we may regard the film as at all times in thermal equilibrium with the hot surface, this Joule heating would act to decrease the cooling due to the escape of electrons, and we should expect the cooling effect to be decreased bythe film. ‘This is, however, contrary to observation. The alternative view is to regard the film as charged. A charged film might be produced by the ionization of surface molecules of a surface compound and the loss of electrons to the interior of the metal, or the arrangement of ions into a double layer with positive ions on the outside. The field within such a film wouid oppose the passage of escaping electrons and accelerate the passage of entering electrons. Since both the heating and cooling effects are increased, there must be some such field acting at each surface. There is reason to expect such an arrange- ment of donble layers at a metallic surface. The same force which retains electrons within a metallic surface would also tend to retain negative ions within that indefinite region called the “surface.” Positive ions would gravitate to the outer portion of the region occupied by the ions and molecules that make up the so-called film. 218 Mr. H. H. Lester: Determination of Work Function 4. Contact Potentials. Richardson has developed a relation for contact potentials in the form eV. = W, a W, “+ ab, where V¢ represents contact potential, e the charge on an electron, W, and W, work done on an electron in leaving and entering the metallic surfaces, and w a term of tle order of the Peltier effect. This expression may be written in terms of ¢ as Ve = @1-— 0) s+ where the Peltier term is neglected. This expression gives the contact potential as the difference of the ¢’s for the two metals ; so that to test the much discussed question as to the existence of intrinsic potentials, it is only necessary to measure @ for a series of metals and substitute in eyuation (8). Measurements of @ for metals far enough apart in the voltaic series to test the above equation have not been made with sufficient accuracy for the purpose. However, Richardson noticed that the values of the heating effect tor the metals examined by Richardson and Cooke lay close together. In explaining this seeming coincidence, he suggested, as stated above, that the heating effect is connected with the cooling effect through the relation db; = dot Ve. © ee is ieee aaa (9) He recognized that the non-existence of contact potentials would explain the results as well. It has been shown that the observations on iron cannot be taken to support the relationship as he expressed it. Moreover, it was shown that the heating effects measured by Cooke and Richardson did not equal the cooling effect of the hot emitter as measured by the same experimenters, although the theory demanded that they should, since the effect measured as the heating effect was supposed to be (do+V.). The ¢$’s so far determined from the cooling effect have been identical within their limits of accuracy, so that between the metals for which this effect has been measured, no contact potential exists as great as two-tenths of a volt. It was observed that quite different values of @ were obtained when active residual gases were present. Tor this ease there would ordinarily be differences in the @¢’s for different metals, and the differences would be of an order of magnitude sufhcient to account for contact potentials. when an Electron escapes from Surface of a Hot Body. 219 This condition prevailed in the Richardson and Cooke experiments. The question immediately arises, Why did equation (9) not hold for their values? ," The same expression holds if the primary and scattered beams make equal angles with the surface of scattering substance, as was the case in these experiments. I) and & were the same for different radiators, X for the emerging scattered X-radiation was determined from absorption experiments, consequently relative values of s, the scattering * Barkla & Ayres, Phil. Mag. Feb. 1911; Crowther, Camb. Phil. Soc. Proc, 1911, and Roy. Soc. Proc. 1911; E. A. Owen, Camb. Phil. Soc. Proc. 1911. Phil. Mag. 8. 6. Vol. 31. No. 183. March 1916. R 226 Prof. Barkla and Miss J. G. Dunlop on the coefficient, were obtained from the various scattering sub- stances. Owing to the different qualities of the fluorescent (characteristic) X-radiations from these substances, the details of the methods adopted depended upon the particular substance. Copper.—The characteristic radiation (series K) has a wave-length about 15x 10-8 cm. This was excited by all the primary radiations used, these being of shorter wave-length, consequently the complex secondary radiation was passed through a sufficient thickness of aluminium to absorb all the characteristic radiation and leave only the scattered radiation to traverse the measuring electroscope. The intensity of this, as shown by the ionization produced in the electroscope, was compared with that from aluminium when this also had been transmitted through an equal thickness of aluminium. It was seen that the two radiations measured—one from copper and the other from aluminium—after transmission through the particular thickness of aluminium were of the same penetrating power. When the radiation was soft, however, perfect similarity of the two secondary radiations was never obtained, though it was known that all the “K” characteristic radiation from copper was cut off. The difference in penetrating powers was, however, very small and was accounted for, as seen later, by the fact that the primary beam was not quite homogeneous, and the softer constituents were scattered from copper in somewhat greater intensity than the harder. When the primary radiation was of a penetrating type, the two secondary radiations were as nearly as measurable identical. This indicated that they were not characteristic radiations but purely scattered radiations. The thickness of aluminium used varied from about °08 cm. to 2°2 cm. according to the penetrating power of the X-radiation experimented upon. A series of comparisons was made between the intensities of scattered radiations, through a long series of penetrating powers of the primary radiation, by both methods mentioned above. Silver.—The radiation of series K characteristic of silver has a wave-length about -56x10-'cm. When a very soft primary radiation was used—one not containing an appreci- able constituent of shorter wave-length—this characteristic radiation was not excited, consequently the secondary radiation from silver, after transmission through a very thin absorbing sheet in order to absorb the “ L”’ radiation, was entirely scattered radiation. When the primary radiation became a little harder (with constituents of shorter wave- Scattering of X-rays and Atomic Structure. 227 leneth) the “ K” radiation appeared. At this stage this had a penetrating power differing little from that of the primary and thus from that of the scattered radiation, so it became impossible to separate them by the absorption method. Consequently no values were obtained of the scattering from silver of radiation of wave-length near to that of the characteristic radiation. When, however, radiation of very much shorter wave- length was used, the fluorescent radiation of series K was still excited, but this was removed by transmission through a sheet of aluminium, leaving only the very penetrating scattered radiation to be compared with that from aluminium. As an example of the results showing the similarity of the two penetrating radiations scattered from silver and from aluminium, the ratios of the ionizations produced by these in the two electroscopes, after transmission through various thicknesses of aluminium, are ae. below. | | | mnsetricds a Al | Ratio of Ionizations produced by | the scattered radiations from Ag traversed. | and Al in arbitrary units. USS A Se 2°2 em. -223 222 2-8 “219 | “219 The constancy of the numbers in column 2 shows that the two radiations were transmitted through aluminium to exactly the same extent. Tin and Lead.—The methods applied to copper and silver were used in examining the scattering from lead and tin respectively, the Cu “‘ K ” radiation being not unlike the Pb “TL” radiation, and Ag and Sn “K” radiations being of neighbouring wave-len oth. Considering the lack of perfect homogeneity of the beams employed, the agreement between the results obtained by the two methods was quite close. All the general features which are indicated below were shown by both methods ; consequently it was unnecessary to take anything like a complete series of observations by both these methods. The second method (6) was subsequently adopted, so that the results given in Table II. are strictly comparable, and any irregularities occurring are not such as might be expected R 2 228 Prof. Barkla and Miss J. G. Dunlop on the to appear in the results obtained by two different methods, each giving only approximate values. The difference between the results of the two methods was usually less than 10 per cent., although in experiments from copper a variation as large as 40 per cent. was obtained. Subsequent exami- nation of the observations taken suggests that the discrepancy was possibly due to the fluorescent radiation from the copper not having been completely absorbed in the experiments the results of which are shown by an asterisk. However that may be, the conclusions stated below hold equally well, consequently the experiments were not repeated. Relative Scattering. Absorbability of X-radiation. (S/e)ou | (S/e)gn (S/e)pp (=) Sa | Soy =| | Seou o/ Al. (2) @) | © © | ee 33 172 179 “345 *1-44 1:06 35 16 9 1:47 : | 21 23 56 *1-79 1-25° 75 185 135° | 1-41 *2-44 2:0 2:18 235 29 | The only results obtained by the first method (a) are givem above along with the corresponding results by method (0), to show the order of agreement. The final results of the measurements are given in Table IT. The numbers given in columns 3, 4, 5 and 6 indicate the intensity of radiation scattered from copper, silver. tin, and lead relative to that scattered from aluminium; the wave- length and absorbability of the radiation are given approxi- mately in columns 1 and 2. The absorbability of each radiation was measured by finding the thickness of aluminium necessary to diminish the ionization in the measuring electro- scope by about half and applying the equation for homo-- geneous radiation I=Ije~”. This of course gives only an approximate measure of the absorbability of a radiation not strictly homogeneous. The approximate wave-lengths were: then obtained by plotting Barkla’s absorbabilities against. Scattering of X-rays and Atomic Structure. 229 Moseley’s wave-lengths for the more homogeneous character- istic X-radiations, and finding the wave-lengths corresponding to the determined values of X/p * As the scattering intensities given in Table II. are relative to the scattering from aluminium, it remained to determine if the amount of radiation scattered from aluminium varied with the wave-length of the radiation. A similar deter- mination had been made by one of us when paper and air were the scattering substances ; in these cases little or no variation with wave-length had been observed. It was TABLE II. Showing relative intensities of scattered X-radiation of various wave-lengths from several metals. f Scattering from Cu, Ag, Sn, and Pb compared Mradinition. with that from equal masses of Al. Approximate | Spproximate Blo en | (S/P) ag | (S/e)sn | (S/e)pp, wave-length. tc | Sea | Cla | Sela | Sodas P/Al- fo erat =e Uy | 9610 “cm. 13 ca sei EG des ‘91 11 Ba ierob. 2) aioe He "63 | 33 25 a6 | iets i-2 "59 28 uae? LP LSA ad Ge CS IR is) 585 27 he M9639 sae oe | t3 52 18 RaW gk pin arene “ 9:0 ‘47 1:3 |! igs) bs see 58 43 ‘95 | 1:5 Be a3 4°4 38 65 1:05? ee whe 2°85 316 37 | 1°12 fo we 2°69 314 36 1:07 key eae oo 2°1 31) 39 a bee | 1-47 ss ‘306 33 | i TSO oi ou 1s8 "305 "32 1:05 | =e 1a thought possible that aluminium, an element of higher atomic weight, might show an appreciable variation in scattering through along range of wave-length. The relative intensities of the primary and scattered beams were therefore compared by means of the ionization they produced in two electroscopes as the wave-length of the primary radiation was varied. The penetrating powers of both primary and scattered radiation were also examined throughout. * The wave-length of the most prominent line, the a line, was taken as representing the wave-length of the K radiation sufficiently accurately. The short wave-lengths were obtained by extrapolation. 230 Prof. Barkla and Miss J. G. Dunlop on the It is not necessary to give the detailed results obtained in this investigation as some points need further examination. We have, however, established the approximate constancy of the scattering coefficient in aluminium for radiation within the range of wave-length used in these experiments. Not only these experiments but a number of independent investigations indicate a slight increase of scattering from aluminium with an increase in the wave-length of the radiation. Within the range of wave-length of these experi- ments it is, however, certainly small; the error will be of no moment if we neglect it. Conclusions. The conclusions to be drawn from the results of Table IT. may be summarized as follows :-— The scattering of X-rays of very short wave-length by equal masses of various substances varies only little with the atomic weight of the scattering element. There isa slight increase of the mass scattering coefficient with an Increase in the atomic weight of the scatterer. (Hlements of low atomic weight have previously been shown to scatter to approximately the same extent mass for mass not only X-radiation of short wave-length, but X-radiation of any wave-length between widely separated limits.) The increase of the scattering with the atomic weight of the scattering substance becomes more marked with X-radiation of greater wave-length, until for long waves it becomes very considerable; that is, the heavier elements scatter much more mass for mass than light elements, when the radiation is of long wave-length (“‘soft’’). Thus the intensity of X-radiation of given wave-length scattered by equal masses of different elements increases with the atomic weight of the element, and the intensity of X-radiation scattered from a given material increases with the wave-length of the radiation of given intensity. As previously stated, these variations are inappreciable among elements of low atomic weight for radiations of the wave-length which may be regarded as ordinary Rontgen radiation. In attempting to interpret these results, it should be pointed out that the scattering is proportional to the number of scattering electrons only when these act independently of one another. The condition is apparently satisfied in light atoms when traversed by Roéntgen radiation of any wave-. length within wide limits. Each electron is then influenced Scattering of X-rays and Atomic Structure. 231 by the electric field in the primary wave for only a very short interval of time; its actual displacement is small and the forces called into play within the atom are small; it emits energy at a rate proportional to that at which primary radiation energy passes over it*. Even in the heavier atoms. in which the electrons are more closely packed, and in which they are held by stronger restraining forces as shown by their higher natural frequency of vibration, it appears that this independence in scattering action is maintained provided the primary waves are sufficiently short. Only under such conditions is the scattering coefficient in a given substance independent of the wave-length, and only then does the intensity of the scattered radiation give a direct indication of the relative numbers of electrons in the atom. It appears from the scattering of very short waves that there is little deviation even among the heavier elements from the law of approximate proportionality of the number of electrons per atom and the atomic weight. There is indication that with waves still shorter than those employed, the mass scattering coefficients would have become Pees constant for all elements, s showing the number of electrons per atom in the heavier elements to be of the order of half the atomic weight,—as in the case of the lighter elements. The independent action of tne electrons might be expected to disappear as the wave-length of the radiation increased. Interference with this independent scattering action might be expected to take place in the heavier atoms for shorter waves than in the ease of the lighter atoms. In such a case the scattering would approximate to a scattering by groups of electrons rather than by the individual electrons. If we considered the extreme case in which n electrons might be considered as a scattering particle of charge ne, we should expect scattered radiation n times as intense as that from n independently scattering electrons , the rate of radiation : 2 pe? /? from a charged particle e being given by a and the acceleration f being identical in the two cases. Though the actual phenomenon may only approximate to this it seems probable that the variation in size (including charge and mass) of what may be called the scattering unit is sufficient to account qualitatively for the above experimental results. * This, of course, does not apply to the few electrons actually ejected: from atoms. + Assuming of course the joint mass of the particle to be m times the mass of a “single electron, a condition almost certainly holding with the distribution of electrons in the atom. 232 Mr. 8. 8. Richardson on Magnetic Rotary The magnitude of the variation, too, is as nearly as one can estimate of the right order; and it is worth noticing that the rapid change in intensity of scattering with wave-length takes place over a range of wave-lengths approaching the diameter of the atom. The possibility of a degradation of type—that is, a lengthening of the wave-length—of radiation in the process of scattering of long waves, and of an influence of a change of intensity of the incident radiation on the process of scattering, are points for further investigation. We wish to express our thanks to Mr. J. HE. H. Hagger— now Inspector of Munitions—for valuable help in these experiments. XXXI. Magnetic Rotary Dispersion in Relation to the Elec- tron Theory. Part I. The Determination of Dispersional Periods. By 8. S. Ricuarpson, B.Se., AlkeCaSe: Lecturer in Physics, Central Technical School, Liverpool *. | Oa amare developments of the electromagnetic theory of dispersion, particularly at the hands of P. Drude, H. A. Lorentz, and M. Planck, whilst differing with respect to the mechanism to which the absorption is attributed, lead to equations which are essentially of the same form, but with constants of different meaning. The natural dispersion in regions of the spectrum not bordering on absorption bands is in each case represented by an equation of the Ketteler- Helmholtz type,— a,” Ag” oa? tore t a Suro aes “Ail (1) or, separating the constant terms from those of a purely dispersional character— nv7#—l= b b (fen amen sw Oo 2e = 5 (2) In the visible and ultraviolet the effect of infra-red resonators is almost negligible and the dispersion is found to be controlled from bands lying in the ultraviolet usually, and mostly in the Schumann region. As this portion of the spectrum lies beyond the range of ordinary spectrographs, and as moreover the work of Lyman f has brought to light the prevalence of general absorption in this region, which * Communicated by Prof. L. R. Wilberforce, M.A. + ‘Spectroscopy in the Extreme Ultraviolet,’ ch. ii. Dispersion in Relation to the Electron Theory. 233 would thus add to the difficulty of detecting selective effects, in such cases the values of A,, A», K&e., can only be obtained from the dispersion curve itself. In cases where the dis- persional bands lie in the near ultraviolet (‘4 w to ‘2 w), and are therefore amenable to direct experimental investigation, it is still desirable to determine the dispersional frequencies from the dispersion curve itself. For according to the theory of Lorentz-Planck the periods Ay, A», .... do not coincide with the free periods of the resonators responsible for the absorption but are sensibly shorter than the latter ; and the constants a,, a, .... acquire a meaning different from that ascribed to them in the original elastic-solid and earlier electromagnetic theories. In the latter aj, a, ... are proportional respectively to the number of resonators of each period in the unit volume. In the Lorentz-Planck formula we may write 3 We Beg 2 eS where g is proportional to the number of resonators in unit volume and X is the wave-length corresponding to the free period. Thus r2 n?—l1= ae 2 e x ( Af ) treet The well-known labours of Rubens, Nichols, Paschen, Martens and others have provided ample experimental con- firmation of the application of formule of the type (1) or (2) above so far as natural dispersion is concerned. The observations can, however, seldom be pushed far enough into the ultraviolet to determine the differential effect of two bands in the Schumann region, and the dispersional period deduced is in most cases an effective mean value only. Denoting this by X, we may write 2 n?—1l=a)+ wo SOND) ihe fe iva: (3) where ais the contribution made by electrons whose frequency is so high that their effect on the dispersion is too small to be detected experimentally, and —cnd? represents the small residual effect of infra-red resonators in the region of the spectrum to which the formula applies. As, however, there are here four adjustable constants, the formula acquires a somewhat empirical character, and it is therefore desirable 234 Mr. 8.8. Richardson on Magnetic Rotary to look for confirmation of the value of 2, from: other sources. In a paper on “ Light Absorption and Fluorescence ”* Prof. H. C. C. Baly has shown that the dispersion of water- vapour, hydrogen chloride, chlorine, ammonia, and nitrous oxide in the visible spectrum can be accurately calculated from a simplified Sellmeier formula by taking a dispersional frequency which is some integral multiple of the frequency of some one of the infra-red bands. In the case of water- vapour for example, taking the infra-red period at 2°95 pw, if we put A, =2°95/31 the calculated dispersion curve is steeper than that observed ; with \,;=2°95/33 it is less steep than the observation curve; but with A;=2°95/32 the agreement. is remarkably good, and it does not appear that a better result would be obtained by taking an intermediate value. Prof. Baly’s argument rests upon the theory of energy quanta and the supposition that the absorbed energy is emitted at an infra-red frequency. A radiation theory of absorption appears to be quantitatively inadequate in the case of solids, but it is of course possible that some other mechanism may account for the interdependence of ultra- violet and infra-red frequencies. In Part II. of the same paper Prof. Baly points out that the central frequencies of the absorption systems in the near ultraviolet of the vapours: of benzene, toluene, and the xylenes are in each case almost exact multiples of the frequencies of certain infra-red bands. If similar relations can be proved to exist in general for solids and liquids, they will provide a valuable aid in the development of optical theory. There are some indications: that this may be possible. The absorption band of carbon- bisulphide at *321 ~ corresponds to the infra-red band at. 3°2 w observed by Coblentz,and Martens has shown that this: band (-321) gives rise to a small amount of fe disper-: sion. (Cf. also the result for benzene, infra.) | An eae method of determining Ay, Neen 1s: provided by the Faraday effect, and since the application of an intense magnetic field induces a slight variation in the periods of the resonators, it 1s to be expected that the phenomenon of magnetic rotary dispersion may furnish in- formation not to be obtained from natural dispersion alone. - The general efiect of the application of a magnetic field to a material medium is to superpose on the existing motions: of the electrons or electrified particles a uniform angular velocity eH/2mC ; the effect of which, in turn, is mainly a * Phil. Mag, April 1914. Dispersion in Relation to the Electron Theory. 935 diamagnetic polarization. If in addition the molecules contain rotating masses of finite magnetic moment, a pre- cessional motion will be set up in these, the extent and duration of which will depend upon the disturbing effect of collisions and therefore upon the temperature, but will be capable of developing a mean paramagnetic polarity. It may also be expected that the applied field may in many cases produce a molecular distortion, which, reacting on the vibrators, will influence their free-periods. Sir J. Larmor’s treatment of the problem of magnetic rotation * from the point of view of the first of the above effects is based on general dynamical principles and gives a result independent of the actual mechanism of dispersion. The formula obtained, namely (in symmetrical units Tf) Lae dn ee Wet 2 Te a Meier PAY Net is equivalent to the empirical formula of Becquerel, involves no adjustable constants, and agrees with experiment in giving rotations of the same sign on the two sides of an absorption band. It involves the assumption that the constant e/m is the same for all the resonators concerned. If we take formula (2) for the dispersion we obtain Bie sity BUX. bor Hence Bes. ih r” 2 2 2 7 Ce ome = eee, ele aoe ik a summational formula of the same type as that obtained by Drude. If we assume that the summation holds good in this form with different values of e/m, we may write ak 1 aa es 7 byes x? a ~ 3 ee —-) g See ae Hadi } ) If the infra-red bands are attributed to vibrators of atomic mass, the value of e/m for these is so small that the corre- sponding term vanishes from the summation, and this very completely in the visible and ultraviolet where the A factor is very small also. If, however, any such band is due to a comparatively small number of electrons, the term must be retained. Also referring to equation (3) we see that ndn/dr * ¢ Aither and Matter,’ App. F, p. 352. Tt See note infra. : 236 Mr. 8. 8. Richardson on Magnetic Rotary will not contain ay, and therefore the electrons of extra-high frequency have no influence on magnetic rotation. The effect of molecular distortion on the periods of the vibrating particles may be represented by a small change in the impressed field, as it must be proportional to the field in order that the direct proportion between 6 and H observed experimentally may be retained. Representing the effective field by (1+¢)H, we have BE els if sores r? ) bre> Xe ) } b= seat | ce(E+ Sarna) aE ge) We may note in passing that the effect denoted by € is similar to the effect of solvents or reagents in opening up the electromagnetic force-fields of a molecule, which has been shown by Prof. Baly to account for the development of new bands in the absorption spectrum. Reference must here be made to Drude’s earlier or mole- cular-current theory of magnetic rotation*. The phenomenon . is there attributed to the displacement of the molecular orbits (or orbits of revolving electrons) about positions of equilibrium by the passing of the light-wave, such displace- ment being always in the wave-front. If, for simplicity, we confine ourselves to the case where the magnetic field is directed along Ox, and to plane waves propagated in this direction, an examination of Drude’s analysis shows that the term responsible for the magnetic rotation arises from the periodic variation of the mean molecular field through the element of area dydz. ‘This variation, as Drude states, must be due to a rotation of the mean molecular field about the axis Oy, and this rotation is then represented by dj dé dy dz An dt ), where y; is the density of the molecular field and (&, », ¢) the displacement of the orbit. Such a term, however, represents merely a shearing of the molecular orbits parallel to the plane wz, and would not produce a variation in the flux through the element dydz unless accompanied by rota- tion of the orbits themselves about axes parallelto Oy. Such rotation is, however, not assumed by Drude, for as he points out the magnetic field in the light-wave is too small in com- parison with the field impressed in actual experimental measurements. Hence we may conclude that no effect arises in ordinary cases from orbital displacements. More- over, as is well known, the theory requires opposite rotations on the two sides of an absorption band, and the only known * ‘Theory of Optics’; or Optik, 2te Aufl. S. 417. Dispersion in Relation to the Electron Theory. 237 instances of this are exhibited by certain absorption bands of one or two rare earths examined by G. J. Elias *, and in those of crystals (tysonite, xenotime) at the temperature of liquid air brought to light by the remarkable experiments of J. Becquerel t. These isolated instances (the substances are all paramagnetic} doubtless arise from some other cause. Becquerel, in fact, explains them as due to positively charged vibrators combined with differential absorption. In the case of very strongly magnetic substances cohesion between the components of a molecular chain might induce an appreciable variation of flux through orbital displacements, and it is possible that the rotation observed by Kundt in transparent films of iron and other magnetic metals may be due to this cause. In view of the fact that the wave-length of light is some 10° molecular diameters, no effect is to be expected from such cohesion, except in the case of ferro- magnetic metals. In general, therefore, the orbital dis- placement will be without effect. A modification of equation (4) has been introduced by Ulfilas Meyer f, namely, : Gy ae te 2nC? n * dn * In this expression n’ is such that n'?—1 is the value of the right-hand member of (1) when the terms corresponding to the infra-red periods are omitted. Itis deduced from the Hall-effect formula of Drude by omitting the infra-red and high-frequency terms and assuming that e/m has the same value for all the terms retained. Meyer obtains a good agreement between theory and experiment in the case of sylvin, rock-salt, and fluor-spar. In what follows we shall retain the summational formula. Putting e i\¢@ a=(1+6), a=(1+6)2, we have 1 we \2 2 \2 6= Innr202 Vb,(<5-53) +b2( 55) + ... } 5 (7) b or setting @=ndd” and k= ae &C., r2 2 rn 2 b=h(g ae) tHe) to * Ann. d. Phys. (35) p. 299 (1911). + Phil. Mag. xvi. p. 153 (1908). + Ann, d. Phys. (80), p. 607 (1909). 238 Mr. 8. 8S. Richardson on Magnetic Rotary. For substances which are transparent in the visible and near infra-red, and for which the infra-red values of z are negligible, X,, A»... will represent ultraviolet periods only. If 2», is considerably less than 2,, then since b,=a,A,2, )y = Ago”, the ratio b./b,; and therefore f/k, will be much smaller than a,/a,. Further, the ratio ee Niaok eS ey Cee (if ~>2,) will be less than its square. The effects of two free-pertods are therefore more sharply differentiated in magnetic rotary dispersion than in ordinary dispersion. Further, the resonators which give rise to the constant term ad in (3) have no appreciable effect on magnetic rotation, and these therefore do not give rise to the term £/A? in the expression for 6 as is sometimes stated *. Hence the values of Ay, Ay -.- in (8) are those of the dispersional periods only in the ultraviolet. If there is only one such period we have rn2 2 =H 3) ° e ° e e ° (9) If now this equation is used to determine ), from a series of values of 2, say approximately equally spaced along the spectrum, the values so obtained should agree within the limits of experimental error with the value of 2, obtained from natural dispersion if there is only one free-period. Want of such agreement, particularly if accompanied by a progression in the calculated values of Ay, indicates the existence of at least two free-periods. In the latter case, and where sufficient progression exists, a closer approxi- mation to A, may be obtained by neglecting the dispersional part of | whilst retaining its constant part. Hquation (8) may then be written r 2 b=k, 57-33) + ko ove" (ieee ny and Ay, ky, k2 deduced from three values of ¢. In the experiments recorded below I have carried the measurements of magnetic rotation as far into the ultra- violet as the nature of the liquids used and the apparatus at my disposal would allow, as it seemed probable that an extended dispersion curve would give in most cases more information concerning the dispersional-periods than one confined to the visible spectrum. * Cf. Wood’s ‘ Physical Optics,’ p. 530. Dispersion in Relation to the Electron Theory. 289 Heperimental. The method adopted is the same in general principle as that already used by several experimenters *, and consists of the use of a half-shade polarimeter in conjunction with a spectroscope, which may be placed so as to disperse the light either before or after the transmission through the polarizing prisms. In the former case the spectroscope takes the form of a monochromatic illuminator. As the theory of the half- shade method has been discussed elsewhere +, remarks here may be confined to a description of the disposition of tho apparatus which has been found most suitable for measure- ments in the visible and ultraviolet. The light from the source S (fig. 1) passes first through a quartz-condensing lens of 20 cm. focal-length, and is con- centrated by this on the aperture O in the electromagnet, the ultraviolet rays coming to foci between this aperture and the polarizing prisms P. After traversing the substance in T and the analyser A, the rays are collected by a quartz- fluorite lens QF which forms a sharp image of the dividing- line of the polarizer on the slit V of a wave-length spectro- meter of the constant-deviation type. The rays are deflected to this slit by a totally-reflecting glass prism which is mounted so that it can be racked down out of the path of Fig. 1. A L SSS Q—--°_.----- Fy o thee aa}. * -- ff ------ [Z-------4 ° a QF $ P the light when required. In this case the light falls directly on the slit of a quartz-spectrograph SP. With this arrangement the condition of the illumination could be tested at any time during a series of photographic exposures, a great convenience when using sources of variable intensity. As the prism of the constant-deviation spectrometer absorbs somewhat strongly in the violet, the visual observations were not continued beyond X="4958y. Observations on * Landau, Phys. Zeits. p. 417, 1908; and Lowry, Proc. Roy. Soc. 1908, A, p. 472. + Landau, loc. cit., and G. J. Elias, Ann. der Phys. (35) p. 299 (1911). 240 Mr. 8. 8. Richardson on Magnetic Rotary this wave-length were made both visually and _photo- graphically. The magnet available gave only a comparatively small potential (5000 cm.-gauss with a 1 cm. gap), the polar faces being 3 cm. square. Reduction of this by additional cones was found not to increase the magnetic potential ; the set- back of the main pole-pieces which their introduction entailed counteracted the effect of concentration of the field. The pole-pieces were bored with holes which, in order to preserve the uniformity of the field, were only -25 inch in diameter. Preliminary experiments both with an ex- ploring coil and by magnetic rotation showed that the magnetic potential increased with increasing interpolar distance up to 4 cm. gap and then diminished. The tube used for most of the liquids tested had a length 3°62 cm. A current of 7°5 amperes was used which carried the magnetization a little beyond the “ knee” of the curve. No advantage can be gained by the use of stronger currents, since the small increase of field is obtained at the expense of a rapid increase in the heating of the magnet coils. With the above current the coils were only just warm to the touch after a 2 hours’ run. The current was obtained from a battery of 50 accumulators, and as the resistance of the magnet coils and battery leads was only 1°63 ohm, by the use of a large ballasting resistance the current could be maintained constant to 1 part in 750 for several hours at a time. The ballast consisted of four open coils of bare eureka 16 S.W.G., each 2 metres long and about 5 ecm. diameter. These were arranged in series with two parallel wires of the same material und gauge, which, with a spring brass connector, formed an exceedingly convenient and reliable fine adjustment. ‘The current was measured with a millivolt-amperemeter by Siemens. A change in the field-strength may be produced through the heating of the magnet cores. A careful ballistic test showed, however, that with a current of 7°5 amperes the error due to this cause would be quite negligible. The polarizing prisms were of the square-ended Foucault type permitting complete polarization of a beam of small angular diameter to about % °220 uw, and were supplied by Hilger. A two-part polarizer was employed, the angle of half-shade being adjustable. For most of the spectrum lines selected an angle of 1°°5 could be used, but owing to the narrow aperture in the pole-pieces and the consequent reduction of the intensity of the light, it was not found to be advisable to reduce the angle below this value. Dispersion in Relation to the Electron Theory. 241 The condensing lens C was mounted upon an electrically- driven tuning-fork of frequency 30. ‘The lens was thus maintained in vibration in a vertical plane, and the light distributed more uniformly over the field of the polarizer. The use of a lens vibrating harmonically, though possibly open to theoretical objections, was in actual practice found to render the results much more consistent and reliable. The projection lens was a quartz-fluorite combination of 32 cm. focal-length, giving exceedingly good achromatism through the ultraviolet spectrum. It was corrected for spherical aberration at °300 yp. A suitable source of light of the required intensity for this type of experiment is difficult to find. A quartz-mercury lamp possesses many advantages, but lines of the requisite intensity are not very evenly spaced along the spectrum. Probably the best source is a quartz-tube amalgam lamp, but in the absence of the latter and after trials with a simple mercury lamp, it was decided to use the iron are as the source. ‘The difficulty here is to obtain sufficient steadiness. Owing to the relatively large current which must be used, the negative pole melts, a circumstance which, it was found, prevents the use of a rotating arc, excellent as this is for ordinary spectrographic work. Iron-magnetite, and iron- cassiterite arcs were also tried, but it was ultimately found that the best work could be done with a simple iron are arranged as in fig. 2, The negative electrode is vertical and Fig. 2, adjustable while the positive is horizontal and connected to the circuit at both ends through equal manganin resistances, R,, R.. A current of 7 to 8 amperes was used with poles 1:5 cm. in diameter. With this arrangement, once the poles become thoroughly hot, the molten iron from the negative Phil. Mag. 8. 6. Vol. 31. No. 183. March 1916, S 242 Mr. 8. 8. Richardson on Magnetic Rotary falls and forms a very symmetrical button on the positive and the arc remains in consequence very steady over long periods. The tube to contain the liquids under test was about 1:5 cm. diameter, the ends being ground square and smooth. To avoid contamination no cement was used, the end plates being held on by placing between the tube-ends and the poles two small rubber rings of the same diameter as the tube. For the ultraviolet, plates of fused silica were at first employed, but were found after a time to develop strains introducing a small amount of elliptic polarization which would reduce the sensitiveness of the apparatus. For the experiments recorded below microscope cover-glasses ‘01 cm. thick have been employed. These transmit quite well to 275 p. | The temperature was observed with a copper-constantan thermocouple in series with a suspended-coil galvanometer, preliminary experiments showing that the &.M.F. of this couple was not affected by the magnetic field employed. The cold junction was kept in ice. The tube containing the liquid was enclosed in a cubical box at the bottom of which was a small heater consisting of 50 cm. of No. 24 constantan wound on a rectangular frame. This coil was joined in series with an accumulator, amperemeter, and sliding rheo- stat. It was thus found possible to maintain the tube at a temperature of 20° C. within a few tenths of a degree. As the heating effect in the magnet coils was scarcely per- ceptible, the above method of maintaining the liquid at a constant temperature was as a rule practicable for a 2 hours’ run. ‘The small temperature variations allowed could be neglected on account of the small effect of temperature on magnetic rotation. The scale attached to the analyser mounting could be read to 1’, but most of the readings were taken on an accessory ‘‘ quartz-scale ” (100 divisions= 21° 40°) with vernier read- ing to 1'*3. The disposition of the apparatus shown in fig. 1 allows the settings to be made hy the observer at the spectrometer. In the visual region the value of the rotation for each line was deduced by taking the mean from 72 settings of the analyser, the current being reversed after each group of 6 settings. The extreme variation in the readings was seldom more than 4! and with the brighter lines much less than this. The values of 6 quoted in the tables below are probably correct to the third significant figure where the magnet was used, and to 1 part in 1000 with the solenoid. Dispersion in Relation to the Hlectron Theory. 243 In the ultraviolet, when using a spectrum rich in lines, a _ choice of two methods may be made. (1) A particular line may be selected and a photograph taken for each one of a number of settings of the analyser. Provided the latter are not too far apart (3' or 4') the particular reading which gives equality of illumination in the upper and lower spectra ean be ascertained with the same degree of accuracy as in visual observations. (2) The analyser may be set to a given position, and the particular line with equal density in the two spectra mav then be found by inspection. The latitude of error in the former method is clearly independent of the rotary dispersion and depends only upon the light-intensity and the degree of half-shade. But in the latter method, although the accuracy is increased by reducing the angle of half-shade, it also depends to a large extent upon the luminosity-gradient along the spectrum, and therefore upon the rotary dispersion. If this is small, as it is with small magnetic potential, there is difficulty in locating the position of exact equality of illumination. For obtaining the rotation corresponding to an accurately known wave-length, therefore, the former procedure is to be preferred. When, however, the dispersion-curve is viewed as a whole, the limits between which it may vary through experimental errors are not very different in the two cases, and for the same reason no advan- tage accrues from a steepening of the luminosity gradient with an auxiliary plate of quartz or other optically active substance. In the present investigation, where the electro- magnet has been employed, the first method has been used to obtain the final results, hie second being applied in most cases only to the preliminary or “ pilot” plates. The follow- ing lines were selected as being (a) sufficiently monochro- matic, (b) of considerable intensity, and (c) nearly equally spaced along the spectrum :— 6708 (Li), 5893 (Na), 4958 (Fe), 4529 (Fe), 4046 (Fe), 3631 (Fe), 3306(Fe), 3100 (Fe). Wratten Double-Instantaneous plates were used throughout, the delicacy and fine grain of the film being well suited to this class of work. The exposures necessary varied from 15 seconds to 2 minutes. The correction for the end-plates, being only just outside the limits of experimental error, was obtained with sufficient accuracy by calculation from the law of the inverse square (80272), the mean rotation of glass for the D line being taken as ‘016’. S 2 244. Mr. 8. 8. Richardson on Magnetic Rotary As very satisfactory extinction was obtained with crossed nicols, it was concluded that no appreciable quantity of doubly-reflected light entered the spectroscope and no correc- tion was deemed necessary. In introducing such corrections on theoretical grounds only there is danger of over-correc- tion, as absorption within the substance under test during the double transit or a very slight want of parallelism between the end-plates would prevent the reflected light from entering the spectroscope-slit. _ From the observations on pure carbon bisulphide for the D line, and at 20° C., using the value for Verdet’s constant obtained by Rodger and Watson * : §=-04347 —-0,737 t, the change of magnetic potential on reversal of the field (C=7°5 amperes) was found to be 13,322 cm.-gauss. Experiments with a Solenoid.—For liquids which can be obtained in considerable quantities and which are sufficiently transparent in long columns, a solenoid possesses several advantages over an electromagnet. A much larger mag- netic potential is easily obtained, with consequent increase in rotation and dispersion ; and if the ends of the experi- mental tube are allowed to project beyond the ends of the coil, the method lends itself particularly to the determination of absolute rotations +. For measurements of the rotary dispersion the chief disadvantage is that the field is strictly proportional to the current-strength, and it is therefore necessary to keep the latter more nearly constant than when using a magnet with iron cores brought nearly to saturation; and the large resistance of a solenoid leaves little room for ballasting resistance unless a battery of an exceptionally large number of cells is available. The heating of the coils causes a rapid decrease in the current, but I find that with a water-cooled coil the variation of current is very small after the initial stage of the heating provided the flow of water is steady, and with the aid of a small rheostat can be kept con- stant to within 1 part in 1000 by occasional adjustment. A diagram of the coil designed for use with a battery of 50 accumulators and for a current of 7° amperes is given in fig. 3. A brass tube B, 7°7 cm. in diameter, is mounted on two ebonite uprights EH, 1:6 em. thick and 21 cm. high. The tube is attached to the ebonite by means of two brass flanges,. the inner faces of which are lined with fibre. The coil, * Phil. Trans, A, 1895. + Lbid. Dispersion in Relation to the Electron Theory. 245 consisting of 3750 turns of No. 18 double cotton-covered copper wire in 12 layers, is wound in the space C between the fibre cheeks, each layer being carefully insulated with shellac (which was then dried by passing a current). The ends of alternate layers are brought out to separate terminals Fig. 3. on the ebonite uprights. The finished coil is 11 em. external diameter and 50 cm. long. The tube B carries an inner one of 6 cm. internal diameter, the space between the two being closed by rings at the ends, so forming a space for circula- tion of the cooling-water. A thin brass rod bent into a helix S is fitted between the tubes to give a spiral motion to the flow of water. The tube to contain the liquid under examination, A, is mounted in a wider tube T forming a water- jacket, the latter being held in position in the inner brass tube by means of brass collars provided with spring clips. The water-jacket carries also two thermometers, and a spiral of No. 24 constantan. By means of the latter coil, which was connected to an accumulator, amperemeter, and rheostat, it was possible to maintain the temperature constant within one or two tenths of a degree. A temperature of 20° was chosen for all the experiments. The disposition of the remainder of the apparatus was the same as in fig. 1, the coil taking the place of the magnet. By choosing a suitable position for the projecting lens OF, it was possible to keep the light reflected from the inner surface of the tube quite clear of the upper and lower extremities of the slit of the spectrometer. The insulation of the coil was carefully tested, both between successive layers and between these and the brass tube, under a pressure of 100 volts. No leakage could be detected except in the case of the bottom layer, which was not used in the experiments. 246 Mr. 8. §. Richardson on Magnetic Rotary Water. Pure conductivity water only was used and the rotation was measured with the solenoid. For the ultraviolet the values were deduced from the photographs by the second method described above, and these were converted into absolute rotations by the use of Rodger and Watson’s value for the D line. The refractive indices n given in the table are those of Gifford * corrected for the temperature difference 20°-15° C. The change of magnetic potential on reversal was 42,238 cm.-gauss, giving a doubled rotation of over 9° for )=="5893 and nearly 43° for the line *3034. For convenience of reference in this and the following tables the observations for different wave-lengths are distin- guished by the letters a, 5, ¢, ete. Obs. Ax 104. 0 x 106 Re, X10" (obs.). | ¢ x 1014 (calc.). | Ag Geet 5893 (Na) | 3808 | 1:5330 1763 1764 +1 b......| “4958 (Fe) | 5529 | 13366 1°816 1-816 0 Ceri | 4678 (Cd) | 6276 | 13381 1-838 1:839 aioe Eases ie “4341 (HH) 7431 | 1:3403 1-877 1-875 —2 Cis a: 3962 (Al) | 9°159 | 1:3435 1°93 1-929 — eee 38611 (Cd) |11:°380 | 13475 2-002 2-001 —1 CORNER 3303 (Zn; | 14218 | 1°35238 2098 209i = Wee es 3034 (Sn) |17°620 | 1:3580 2-203 2°206 +3 The values of d, calculated from equation (9) are G0 = 109 bd Tt aad) = EO df = -112 dh = “114 ja The values of X chosen for the calculation must be as widely separated as possible, as the effect of a small experimental error becomes greatly magnified when A, is calculated from two points on the dispersion curve which he too close together. Lowryf has determined the dispersion ratio of the magnetic rotations for the two mereury lines ‘5461 and °4359 for water and a number of aliphatic compounds, using a very powertul magnet. In the case of water, (4859) 1.645. (5461) * Proc. Roy. Soc. Ser. A, lxxviil. + Jour. Chem. Soc., Jan. 1914. Dispersion in Relation to the Electron Theory. 247 Taking 76461) = 1°3345 and 743.5. =1°3402, formula (9) gives A, ='113 pw, a value agreeing well with the above. A small but decided progression is apparent in the values of 2, which points to the existence of at least two free- periods responsible for the dispersion in water. The smallness of the progression is probably due to the value of d, being small in comparison with the shortest » of the experiments, and for the same reason the longer free-period cannot be calculated with accuracy from equation (10). The mean value obtained from observations acf and beh is approxi- mately -1375, which gives k,=°9499 and k,='7012. Thus 2 2 $x 10% =-9499 —) +:7012, r2=01891. TAIL The differences between the values of # calculated from this formula and the observed values are given in the last column. Flatow * finds that the natural dispersion in water is very accurately represented by an equation of the form (3) from "089 to -214 4; the constants for 20° C. being : | ay = °37512 ce = 013414 a, = *38850 A, = 712604. As id, is here an effective mean value, this result agrees well with the deduction made above, namely, that the dispersion in liquid water is controlled by a period near X='1375 and at least one other of smaller value. The following table gives the values of the magnetic rotation (6) relative to the value for 4958 for a number of lines in the ultraviolet spectrum of iron. The only previously obtained values—those of Landaut+—obtained with an electro- magnet of small power, and recorded to three significant figures, are worked out as ratios for comparison. AX 104. 6 (Landau). 6 (Author). | | = x | i | "4958 4-000 1-000 “4529 1:226 1-221 “4405 1:301 1°299 ‘4308 1:366 1:365 -4199 1446 1°449 4046 1°575 1:576 “3886 dere) Piss *3609 2°065 2-060 *3100 3005 3017 * Ann. d. Phys. (12) p. 85 (1903). +t Phys. Zeits. (13) p. 417 (1908). 248 Mr. S. S. Richardson on Magnetic Rotary Benzene. In the case of benzene and the liquids following, the electromagnet was employed in order to reduce absorption by the use of a short column of liquid. The values of n are deduced from the observations of Simon. The values of 6 are here all obtained from the photographs, the analyser settings being varied from each line until equality of illumi- nation was obtained. t=20° C. The absolute rotations are obtained by using Perkin’s water-ratio for the D line (2°3053). Obs. Ax 104. 0X 10°, eee o x 1014, Ref....| °5893 8-778 15004 4-574 aay, 4529 16°49 1°5187 51387 (Merete 4046 22°21 15311 5567 ei "363 1 30-32 1°5475 6186 dehy 3306 40:95 1:5670 7014 Os -3100 51°51 1°5850 7846 Calculating the values of A, from equation (9) we find progressive values, namely :— go acv= N68 cd = *1734 NS ae de = “1702. As the amount of increase is here considerable, at least two free-periods probably widely separated may be expected. Using equation (10) to obtain a closer approximation to the longer period, we obtain :— from bce, 2A, =°1907 ace, X, =°1903 > Mean = 1902. A? =Osois: Dae. eh | loot ky = 2°468, C == 1°498. 14 ae e ee Was | ea 2468 (5 aa The differences between the values of ¢ calculated from this and the observed values are given below. y 41-498, Dispersion in Relation to the Electron Theory. 249 Ax 104. @ (obs.). (calc.). Diff. "5893 ort. 4°574 (ref.) "4529 5°187 5137 0 “4046 5°567 5°566 —l "3631 67186 6°187 se) "3306 7014 7013 —l 3100 7846 7847 sell The natural dispersion in benzene has been measured by Simon * in the visible and ultraviolet, and by Rubens in the infra-red. I find the results well represented by formula (3), taking the mean effective dispersional period at ‘1815 p, Mid; "02511, ag="55695, c="0017. Thus: 2 1155695 + 0017 0. 1 The differences between the observed and calculated values are given in the table below. Diff. Diff. } Obs. A x 103, a 7 (obs.) s M (cale.). (Author). (Martens). | +2837 1-6190 16190 | 0 0 2881 1°6120 1°6122 +9 +5 2981 15983 15986 | +3 +190 3404 1:5603 15605 i +2 ede, ‘3467 1:5564 P5565. 1. el i 3613 1:5485 1°5484 = +8 e "4341 1°5229 1°5227 ary 0 g -4678 1:5158 1°5155 = 3 B -4800 1°5137 1:5136 =) —5 4861 1°5125 1-5123 =} 3 "5086 1°5094 1:5090 —4 == 5849 1°5060 | 1°5056 = 8 ‘5893 15005 | 1:5003 = aS 6563 1:4959 1:4957 29 =ailul 7682 14907 1:4905 ait} —I12 810 1-4890 | 1:4892 Eo 8 864 1-4874 | 1-4873 uy] ee ‘926 1:4859 1:4864 +5 Bet | = ‘997 1:4848 14851 +3 =s B 1-080 1°4834 1:4839 +5 =6 5 1-178 14822 1:4826 +4 —5 pS 1:297 1:4813 1°4817 +4 —6 1-439 1:4801 1:4806 +5 —2 1°621 1:4792 1:4796 +4 —] 1°850 | 14784 1:4785 a 0 * Ann. d. Phys. (53) p. 542 (1894). 250 Mr. 8.8. Richardson on Magnetic Rotary It will be seen that since the mean dispersional period "1815 w is considerably greater than the mean of the values obtained above from formula (9), Meyer’s equation cannot apply strictly in the case of benzene. A similar remark applies to the liquids water, metaxylene, naphthalene bromide, and carbon bisulphide. Martens * deduces a formula for benzene in which dy = 14528, 4,="1264, c=0, andiA, = "1745." The dimerenes= for this formula are given in the last column above. Prof. Baly associates the central line of the absorption spectrum of benzene vapour (-2466) with the infra-red band observed by Coblentz at 2°49 uw, the ratio being nearly 1: 10. In this connexion it is worth notice that the principal disper- sional period deduced above from the magnetic rotation 1902 w, is almost an exact snbmultiple of 2°49 yw, the ratio being 1:13. The absorption bands of benzene in the near ultraviolet with a centre at ‘2466 have very little effect on the dispersion or magnetic rotation. Introducing a term for this band, I find that the constant 4 is only 1/740 of the sum of the remaining constants. The absorption is here clearly due to 2 comparatively small number of electrons. A probable explanation being that the period -2466 is only acquired by one or more electrons in the molecule at com- paratively rare intervals, due to an accidental distortion of the molecule through collision. M. Henri ft, from photo- metrical measurements, finds that in acetone only one molecule in 36 is effective in producing the characteristic bands of this substance, and Baly and Tryhorn have recently obtained a similar result for aniline. m. Xylene. The observations were conducted in the same manner as Obs. Ax 104. 0X 10°. Wace Sie Ota Gina ‘6708 5416 «14914 3°634 Dress 5893 7210 1:4965 3747 ENS "4958 10°84 15061 4014 NEON 4529 13°51 15131 4194 ioe. "4046 18:19 15242 4°538 Hp boat ‘3631 24°72 15381 5:014 * Ann. d. Phys. (8) p. 603 (1901). Tt Phys. Zeits. p. 515 (1918). Dispersion in Relation to the Electron Theory. 251 those for benzene. The values of n are deduced from Rubens’ measurements. t=20° C. The absolute values are obtained from Perkins’ water- ratio for the D line (1°8957). Using equation (9) the values of 24, from sufficiently separated values of A, are progressive. a ues He A. ieee be = "1613 ce = 1614 cf = *1633 df = +1653 The existence of at least two dispersional frequencies is thus indicated. Proceeding to apply equation (10) we obtain from ace, A= °1819 bdf, Ay = "18386. The second value obtained from the larger values of 6 is more probably free from the effect of experimental error. Using this value, k, = 1°9916, ms 1:2790. Thus 104¥xd=1 9916 (5 a557) 1:279. nae ee — Bah Ax10%, |g (obs). @ (calc.). Dif. | 6708 | 3634 3-627 —-007 | 5893 3747 3751 +004 : -4958 4-014 4-002 Lees ABOO led Via Ae1 04 | 4-194 000 | ‘4046 | 4588 | 4580 — 008 3631 5-014 5-014 000 From Simen’s* values for the natural dispersion of m. xylene in the ultraviolet I find Ole a, = “SOU an ae —GO00a4 the mean effective value of A, being 1830. Thus he 7 NS (co FA : Da F, 2 n*? = 1°61444+4°56073 \703349 00054 v2. * Loe. cit. 252 Mr. 8. 8. Richardson on Magnetic Rotary In the following table the infra-red values are those of Rubens, whose values are reduced by ‘0011 in order to bring them into line with those of Simon. Ax 104, av (obs.). a (cale.). Diff. 2981 1:5861 1°5856 5 ‘3081 15751 1°5750 2a ‘3133 1 5701 1°5702 ai ‘3261 1:5596 1:5598 49 3404 1°5501 1:5502 42) 2467 1:5464 1:5465 | 2 ‘3613 1°53905 1°53905 0 S 4340 1:5154 1°5154 0 wo ‘A678 1:5089 1:5087 266 -4800 1:5069 1°5068 a | ‘A861 1:5059 1°5059 0 5086 1:5030 1:5028 9 5849 1:4999 1:4998 ey | "5893 1:4950 1:4949 a 6563 1:4908 1:4907 ay ‘768 1:4860 1-4860 0 ue "893 1:4857 1-4846 0(—11) i "878 1°4845 1-4833 | | 940 14834 1:4822 a8) ee 1-012 1:4822 14811 0 2 1:096 1:4808 14801 +4 Sy 1:195 1-4795 1:4791 ty a 1:316 1:4784 1:4783 +10 B 1-461 1:4775 1:4775 4-11 = 1-645 1:4768 1:4784 £97 1-881 1:4760 1:4760 +11 From Rubens’ values Martens deduces \;='1366. But this value is clearly much too small, the differences for :3403, *3261, °3133, °3081 being respectively —16, —33, —56,and —69. Martens assumes a>=0, c=0, and takes aj=1°1724. The sudden bend in the experimental curve in the region of X=1°'645 is doubtless due to selective dispersion produced by the band observed by Coblentz at 1°74. Similar but smalier fluctuations appear in the experimental curve for benzene, for which substance Coblentz has recorded weak bands at °8 w, 1:0, 1°4, and 1°7. Naphthalene Bromide. The liquid used was a specimen of e-monobromnaphthalin, supplied by Kahlbaum as pure, and carefully redistilled. Dispersion in Relation to the Electron Theory. 253 The values of n are deduced from the observations of Simon *. t= 20°. Obs. Ax104. éx 10°. May: @x10". ieee. 6708 17°64 16475 7858 Sa 5893 23°83 1:6576 8-283 ae ‘4958 38-21 16783 9-370 7a 4529 50°19 16942 10:296 nae 4046 74-36 17231 12°17 Fest ‘3735 105-4 17531 14-70 The values of X, deduced from equation (9) are progressive. OTE elie y wd == “2087 — -919 a be = °2166 — 3 ce 198 of = +2268 The exceptionally high value of A, combined with the high densjty (1°487) of the liquid accounts for the very large magnetic rotation produced. Proceeding to estimate the value of the larger dispersional period from equation (10) we obtain :— fromnace, . Ay =" Zar, bdjf,, Ay =.°2563, The latter value gives k, = 3°1286, ke = 3°5245 ; and the former gives ky = 3°2151, k, = 3°4858. Taking the value obtained from the larger rotations as the more accurate, we have 2 2 $ x 10! = 31286 (<—055e9) + 35945. AX 104. (obs.). ¢ (calc.). Diff. [| ‘6708 7-858 7814 — ‘044 ‘5893 8'283 8°283 000 "4958 9°370 9°351 —-019 "4529 | 10-296 10°295 —001 "4046 12°17 12:25 +08 "3735 | 14°70 14:70 00 * Loc. cit. 254 Mr. 8. 8. Richardson on Magnetic Rotary For the natural dispersion in this liquid Walter’s * values have been taken (to four decimal places) and appear to be fairly represented by a mean period X1=°2370, the formula I obtain being : rn2 2_] = -954924- a n?—1 = °95422 + °66659 x? —-OB617" ] | Ae Ax 104. a (obs.). n” (calc.). Diff. ‘7621 1°6405 16408 +3 "6884 1-6464 16463 —|I ‘6563 16495 1°6495 O "5893 1:6582 16581 —1 ‘5270 1°6705 16793 2 4861 16819 16819 0 4341 17041 17041 0 "4308 1-7059 17059 i) “4102 1-7185 1-7190 +5 | "3970 1:7289 | 17291 tie Carbon Bisulphide. The liquid used was purified by allowing it to stand in contact with mercuric chloride for several weeks, with sub- sequent repeated distillation over mercury. ‘The product was almost odourless. The values of » were deduced from those of van der Willigen. In this case the observations were confined to the visible spectrum. t=20°. Obs. | A104, | 18% 10°. wie: $x 10", LOND 6708 1468 | 16182 | 6-604 Bian 5893 | 19:90 16291 6-912 Bee 4958 30°71 1-6508 7-550 ail 4326 | 44:59 1-675 8-344 Bi tein ‘4144 | 50°50 16885 8-653 fivwen| 4046 54-14 16952 8-863 The first three observations were obtained by the visual method, the latter three photographically. * Ann. d. Phys. 1891. Dnspersion in Relation to the Electron Theory. 255 The values of A; calculated from equation (9) are ac = 1803 bd = 18h6 Mean A, = 1804. ce = 1805 dj 193 No progression is here shown, formula (9) thus repre- senting the magnetic dispersion with sufficient accuracy within the visible spectrum. Verdet’s values for the D and F lines of the solar spectrum give a value quite near to the above. Pion Ube, 2 == Lo2gm 8, [dq = 11234, n= 16538 bn cata The value so obtained does not, however, represent an actual dispersion period or position of metallic reflexion as it is too small to account for the natural dispersion. The mean dispersional period for the latter required by formula (3) is placed by Martens at °2175, and by Flatow at °2255. The formule obtained, however, do not fit at all well in the extreme ultraviolet, e. g., Xr. 2(obs.) (Flatow). Diff. (Martens). Diff. (Flatow). 274 2°0053 —°0073 +°0320 As the errors are of opposite sign it is probable that the mean band lies between the two values here given. The low value obtained from the magnetic rotation shows that the constant /, must be introduced, as in equation (10), but this equation cannot be applied to calculate the longer dispersional period without a more extended series of obser- vations or measurements carried to a higher degree of accuracy in the visible. I hope to repeat the determination shortly with a more powerful electromagnet. Symmetrical Units —The system of units adopted in the formule above is that in which all electrical quantities, including current and resistance, are expressed in electro- static units, all magnetic quantities in electromagnetic units, the connexion between the two being established by the two circuital relations : os = = ‘eprli bn i = = curl X. 256 Magnetic Dispersion in Relation to Electron Theory. This system, which is that adopted by Lorentz in his ‘Theory of Electrons’ (in which, however, Heaviside’s rational values are used), appears to be the best adapted to the interpretation of experimental results. Summary. (1) The calculation of ultraviolet free-periods from ordinary dispersion leads in general to a mean effective value only. New formule are obtained for benzene, m. xylene, and a-monobromnaphthalene. (2) When a similar calculation is made from magnetic rotary dispersion using the formula r2 2 b= 52) the value obtained is usually much shorter and often progressive. As the latter formula contains no adjustable constants, this result shows that in the case of the liquids here examined Meyer’s formula does not apply. (3) When a progressive result is obtained in (2) a close approximation to the longest dispersion period in the ultra- violet can usually be obtained from r2 2 b=hi(sgoya) +h This appears to be due to a much sharper differentiation in the effects of bands of decreasing wave-length in the mag- netic case. (4) In the case of benzene, and possibly also in m. xylene, the longest period is almost an exact submultiple of the period of one of the infra-red bands. 7 I desire to express my cordial thanks to Prof. Wilberforce, who has kindly placed the necessary apparatus at my disposal, for the interest he has taken in the experiments ; to my friend Mr. G. Calderbank, B.Sc., for much valuable help in the evaluation of the numerical data; and to Mr. F. W. Pye for the care and attention bestowed on the construction of the solenoid. The George-Holt Physics Laboratory, University of Liverpool. ; THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE [SIXTH SE ss 5.1 Aan) HN a a y "L9p = AP RED, Mateus a / XXXIT. Note on Experiments to detect Refraction of X-rays. By Cuartes G. Barxua, f.R.S* Q* the simple theory of dispersion the refractive index (w) of a substance is given by an expression of the watt Ne? ee mm(no?—n?)’ where N is the number, per unit volume, of particles of mass m, charge e, and vibration frequency no, and n is the frequency of the incident radiation. As the frequency of incident radiation changes from n, that for luminous radiation, to n', for penetrating X-radiation, each “infra-red term ” is diminished appr oximately in the ratio n?/n'?, and each “ ultra- violet term” (including “M,” “L,” and “ K” characteristic X-radiation terms) in the ratio 1 age? But if this fre- quency n’ be higher than that of the natural vibration of any of the constituent electrons, all these terms Ne? mm(ny?—n'*) become negative, that is, all the vibrating systems affect the refractive index in the same way. If then the theory were applicable even in a moditied form to radiation of such short wave-length as that in Roéntgen radiation, it seemed just possible that by careful experiment refraction might in certain cases be detected. * Communicated by the Author. Phil. Mag. 8. 6. Vol. 31. No. 184. April 1916. f 258 Prof. C. G. Barkla : Experiments to Experiments had previously been made on the same subject by a number of investigators, but they had not been of such a kind as with our present state of knowledge to justify any hope of success. | The refracting surfaces were usually ordinary optical sur- faces, in which the irregularities were in general large compared with the wave-length of Réntgen radiation. This being the case, any refraction occurring would be irregular refraction which would produce only a certain amount of diffusion in a transmitted pencil and consequent blurring of the “image.” The methods of detection, too, were not the most sensitive. | In the following experiments a prism of large angle and with crystalline refracting surface was used; the pencil of radiation transmitted was an exceedingly narrow one; the experimental arrangement was such as to make a small deviation measurable; and the material of the refracting prism and the transmitted radiation in certain cases were such that the frequency of the latter was not far removed from the natural frequency of vibration of one set of electrons in the other. The refracting substances used in these experiments were rock-salt and potassium bromide, the two refracting surfaces being crystalline planes at right angles. A narrow pencil of X-radiation was passed through the refracting right-angled prism and was received on a photographic plate on the other side. In the initial experiments the X-radiation was sent through two parallel slits in parallel lead screens. In the path of this pencil beyond the second slit was placed the prism with its refracting edge parallel to the slits but intercepting only one-half of the beam. Thus, of the very narrow pencil, one- half * passed through the air just outside the prism, while the other passed through the prism. At first curious effects were produced on the photographic plate beyond, some of which were suggestive of refraction : the two portions of the beam were apparently separated, for a clear line appeared between the two dark “images” on the negative. It was, how- ever, found that such effects, and fringe effects observed at the edge of a shadow, were due to (1) the finite magnitude of the source, and (2) the lack of proportionality between the density of a photographic image and the intensity of the radiation producing it. These results led to nothing fundamentally new; they are * Of half the width. detect Refraction of X-rays. 259 only worth mentioning as of a kind likely to lead to wrong interpretation. The final experiments are, however, probably worth record- ing. The radiation from an X-ray tube passed successively through two narrow vertical slits in parallel lead screens about 50 centimetres apart, on to a photographic plate at a distance of about 150 centimetres from the second screen. The photographic impression produced by the pencil of radiation was then a thin vertical line 3 centimetres long and _ about ‘01 centimetre in width. A few centimetres away from the second lead screen were placed two crystals of potassium bromide arranged as two refracting prisms end to end—that is, one above the other—with refracting edges vertical but with their bases on opposite sides of the transmitted radiation {see figure). Thus any refraction-of the radiation in trans- mission through the prisms would result in displacements in opposite directions of the upper and lower halves of the thin pencil of radiation. An exposure of a rapid plate was made for a period of from five to eight hours. It was found that the photographic impression was still a straight line from top to bottom. There was no evidence of the slightest break or lateral separation between the two halves of the image. When looking at the line image in a direction making a sinall angle with its length, it was estimated that a relative. lateral displacement of *025 millimetre could have been detected. Y2 260 Dr. S. A. Shorter on the Constitution Thus the transmission through the crystal had not pro- duced a deflexion of 012 millimetre at a distance of 140 (about) centimetres from the refracting prism ; that is, the deviation was less than 2 seconds of are. We conclude from this that the refractive index of potassium bromide for — radiation of wave-length in the neighbourhood of *5 x 10~* cm. is between 0°999995 and 1:000005 *. It is of course possible that if only radiation of frequency within certain narrow limits were refracted appreciably, the intensity of this would be toc small to produce a distinguish- able feature on the photographic plate. Experiments with more nearly homogeneous beams of radiation might indicate such refraction for a particular wave-length. Although this is a possibility, there is little reason for expecting that such experiments would lead to results differing from those already obtained. For the radiations used in these experiments covered a great portion ~ of what may be regarded as the K absorption band asseciated with the bromine “ K”’ radiation. I wish to express my thanks to Mr. J. H. Hagger,— now Inspector of Munitions—for his assistance in these experiments. XXXII. On the Constitution of the Surface Layers of Liquids. ByS. A. Suorter, D.Sc., Assistant Lecturer in Physics in the University of Leeds.—Part L. + CONTENTS. § 1. Introduction. § 2. The General Case of a Binary System. §3. ‘The Principle of the Relativity of the Surface Magnitudes. § 4, Special Cases of a Binary System. § 1. Introduction. sie principles of thermodynamics were first applied to . the question of the constitution of the heterogeneous layer separating two phases, by Gibbs in his well-known work “ On the Equilibrium of Heterogeneous Substances ” f. Asin the case of many other extremely interesting problems, Gibbs’s mode of treatment of the subject presents mathe- matical difficulties, which are insuperable to practically all of those to whom the subject is of interest. The object of * Bromine K radiation has a wave-length about 107° em. + Communicated by the Author. t Scientific Papers, vol. i. p. 235. of the Surface Layers of Liquids. 261 the present work is to develop the thermodynamical theory of the subject in an elementary manner, preserving at the same time perfect mathematical rigour and the utmost generality of treatment. The first attempt at an elementary exposition of the sub- ject was made by Milner *, who cbtained a formula for the “‘surface excess”’ of a solute, applicable to the case of a dilute solution in contact with a third substance. The theory developed in the present work is applicable to solutions of any degree of concentration—and therefore to the case of a mixture of two perfectly miscible liquids. Since in this latter case the distinction between solvent and solute is a purely arbitrary one, the theory must take account of the “surface excess” of both solvent and solute. This consideration of the “surface excess” of both components of a solution leads to a curious and interesting principle, which I have termed the “ Principle of the Relativity of the Surface Magnitudes.” This principle underlies all Gibbs’s theory, though it is never explicitly referred to in it. § 2. Zhe General Case of a Binary System. We will first consider the case of the interface separating two phases of a system containing two components. This general case includes a number of special cases, which we will consider in § 4 of the present paper. The most elementary method of treating the subject is to consider the working of an ideal “engine” in which the contractile tendency of the surface and the expansive tendency of the volume are both used as a means of pro- ducing an external mechanical effect. Such an engine is represented in fig. 1. Q is a vessel of rectangular hori- zontal cross-section, containing masses My, and M, of two components Cy and C, respectively, in two phases @ and q’ separated by a plane surface of area o and tensiont. The area of the surface may be varied by means of a movable late P, which fits accurately to the sides of the vessel parallel to the plane of the paper. The horizontal portion of this plate passes through a slot in one of the vertical sides perpendicular to the plane of the paper. ‘This side is sufli- ciently thick in the neighbourhood of the slot for the plate to be incapable of any motion but a horizontal motion of translation. The volume v of the system may be varied by means of the piston R. * Phil. Mag. Jan. 1907. 262 Dr. 8. A. Shorter on the Constitution Suppose now that the engine goes through a reversible isothermal cycle. According to, the first law of thermo- dynamics, the total work done by the system is equal to the Fig. 1. total heat absorbed by it. According to the second law the total heat absorbed is zero. Hence the total work done is Zero. Let the cycle consist of the following four stages :— (1) The surface-tension being maintained at the con- stant value 7, let the volume v of the system be changed till the surface increases from o too+6ce. During this change the pressure p will remain constant. (2) The area of the surface being maintained con- stant, let the pressure p be changed till the surface- tension changes from 7 to 7+ 67. (3) The surface tension being maintained constant, let the volume be changed till the area of the surface changes from o+6éce to a. 4) The area of the surface being maintained con- stant, let the pressure be changed till the surface- tension changes from 7+ 67 to 7. of the Surface Layers of Liquids. 263 If we take values of o as abscissee and values of 7 as ordinates, the cycle is represented by the elementary rect- angle PQRS of area de67, described in an anti-clockwise Fig. 2 Pa iti. P Q O ‘Cue Sci hy aoe G direction (see fig. 2). Hence the system does an amount of work d¢6r7 against the force controlling the surface*. The force controlling the volume must therefore do an equal amount of work on the system. Suppose now that we take Co Fig. 3. vi values of v as abscissee and values of pas ordinates. The eycle will be represented by a parallelogram P’Q'R’S’ (see fig. 3) with two sides (P’Q' and R’S’) parallel to the v axis, '* A certain force will be necessary to counteract the effect of the difference in the hydrostatic pressure on the two sides of the vertical portion of the plate P. Since the work done by this force in any dis- placement is equal to the product of the total weight of the components and the increase in the height of their centre of gravity, the work done in a cycle will be zero, so that this force may be left out of accouns in considering a cycle of operations. 264 Dr. S. A. Shorter on the Constitution of length (dv/dco), 60, a distance (dp/dr) 67 apart. Now the perimeter must be described in an anti-clockwise direction. Hence the above two differential coefficients must have the same sign ; so that equating the areas of the parallelograms PQRS and P’Q'R’S’, we obtain the relation dv dt (=), = a ° e ° e ° ° (1) Since in a binary system the constancy of 7 and 6 is equivalent to the constancy of p and @, this equation may be written * dv az cou = (a). 5 . S . ° . (2) We see from this equation that if the surface-tension depends upon the pressure, it is necessary to change the volume, when the area of the surface is changed, in order to keep unchanged the nature of the different parts of the system. Such a change would obviously be unnecessary if the two phases were homogeneous right up to the surface of separation. Let py and p; denote the densities in the @ phase, and p)' and p;' those in the @¢' phase, of Cy and C, respectively. Let V and V' denote the respective volumes of the phases, and My, and M, the respective total masses of the components. We see that in general the equations My = poV +p V’, M, = piV+p,'V, will not be verified. Let us write aly == Mo— (poV + po V"), - . ¢ (3) ol’; — M,-— (p:V +p1'V') staan : : = (4) We will call (following Milner f) each of the quantities IT, and TI, (defined by the above equations) the “ surface excess’? of the corresponding component. It is evident that Ty and T', will depend only on the nature of the system, and not on the dimensions of its parts. The differential coefficient (dv/do)», is easily evaluated in * This relation may be deduced more concisely, though in a less elementary manner, from the equation dU=@dS—pdv-+rdo (U=energy, S=entropy), by writing it in the form d(U —0S+pv)= —Sdé+v¢dp+rdo. + Loc. cit. Gibbs uses the term “surface density.” of the Surface Layers of Liquids. 265 terms of I), [, and the densities. Since the constancy of p and @ involves the constancy of pp, pos P1, Pi, Vo and 14, we have from equations (3) and (4) Todo + pod V + py dV' = 0 e 5 4 A (5) and Rida +tpidV +p, dv 0.) 2 ats. .. -(6) Since a s= *V AGN a er. (7) we have dab VN eal 20h strana Co) From equations (5), (6), and (8) we readily obtain the desired relation (Z) — Viee—po')—Po(pi— pr) do Qp Po P1— Pop’ Substituting this value in equation (2) we obtain the equation (9) Pipi Po— Po (Po—Po ) (pi—p1 ) dp oe Ri I Po P1— PoP. =) (10) This is the most general relation possible (in the case of a binary system) between the surface magnitudes and quantities capable of experimental determination. §3. The Principle of the Relativity of the Surface Magnitudes. It will be seen that in the case ofa binary system our theory gives only one equation connecting the two quantities IT) and I’, with quantities capable of experimental determina- tion. We cannot obtain separate equations for Ty and T;. At first sight, this seems to imply some incompleteness or imperfection in the thermodynamical theory. A closer ex- amination of what is meant by the term “‘ surface excess ” shows, however, that this idea is wrong, and that the theory yields all that could be expected of it. Suppose that we take as abscissee the distances of points from some reference plane parallel to the surface, and as ordinates the values of the density of a component. The graph will consist of two straight lines parallel to the axis of abscissze (corresponding to the interiors of the two phases) joined by a curved line (corresponding to the region of 266 Dr. 8. A. Shorter on the Constitution heterogeneity). A simple type of such a curve is shown in fig. 4, It is evident that the position of the surface of Fig. 4. Position. separation may be assigned quite arbitrarily within the limits of the region of heterogeneity. If we assign a position AB, the surface excess is given by T'= area C’N’R—area CNR. The value of I. may vary within wide limits with the arbitrarily assigned position of the surface of separation. If we choose the position of AB so that area CNR = area O'N’R, the surface excess is zero. We will call the plane, relative to which the surface excess of a component is zero, the “ zero-plane”’ of that component. If the density assumes in the surface-layer values greater than in either phase, the position of the “zero-plane” is fixed (see fig. 5) by the relation area CED-+area C'N’R= area RDN. If the increase in density in the surface-layer is very great the ‘“‘ zero-plane”’ may lie in the interior of one of the phases (as in fig. 6). 26% of the Surface Layers of Liquids. “QUP ld ele Ay Isuad Pile 010d Ties een me re ee ee eee eee Position. 268 Dr. 8. A. Shorter on the Constttution The value of the surface excess relative to a plane a distance < from the ‘*‘zero-plane” in the direction of the o phase is evidently given by T= (p—p')z. Consider now a system of two components Cy and ©;. The “zero-plane”’ of each component will have a definite position, and the distance between the two will have a definite value, characteristic of the system. Suppose now that [, and [, are the respective values of the surface excesses, relative to a plane whose distances from the “zero-planes” of OC, and C,, measured in the direction of the ¢ phase, are z) and z respectively. We then have Ly = (Py—Po )Zo and T= (pi—py')é1- Hence, if denote the distance of the “ zero-plane” of C, from that of Cy measured in the direction of the @¢’ phase, we have foes Piz eu) Poem h =e, —2 Referring back to equation (10)—the general equation for a binary system—we see at once the reason for the apparent incompleteness of our theory and the particular form of the equation. The quantities y and T, are purely arbitrary magnitudes which could not possibly appear singly in any thermodynamical relation. They may, however, be combined in such a way as to yield a quantity which is not arbitrary, and this quantity appears on the left-hand side of equation (10), which may be written in the form ‘01 — Pop’ dr jy Ree eae ( eee (Po — po’) (Po— pr) Kap) ue We may, however, interpret equation (10) in another way. The value of the surface excess of C, relative to the “ zero- plane” of Cy is evidently (p;—p,')h or ae ! pas P1— Pi 1: ! Pomeeo Let us adopt the notation of Gibbs and use the symbol of the Surface Layers of Liquids. Zod Is) to denote the surface excess of any component C, in a system relative to the “zero-plane” of another com- ponent C,. Hquation (10) may then be written Po P1 —Popr ( 7) Tee Bo OS Poa en 3) 1(0) Po— Po’ dp Q ( ) The value of the surface excess of Cy relative to the “* zero- plane” of Q, is evidently equal to —(py9—po')h or whe / pe oo Ty Pi P1 so that equation (10) may be written ‘oy ' (aT Bh eee 0 PUTO EN IL ON Mn De Gales 0(1) oS 5 €a) (13) The above general considerations apply, of course, to a system containing any number of components. Thus in the case of a system of n components, there are n ‘ zero- planes,’ and the theory should yield n—1 equations for the n—1 lengths which fix the relative positions of these planes. In recent years the surface excess of a solute in a binary liquid mixture has been determined experimentally by ob- serving the change of concentration caused by a large extension of surface. We have seen that the surface excess of a component is a purely arbitrary magnitude, so that the interesting question arises as to what is given by the above experiment. The only case which has been investigated practically is that of a solution in contact with a third substance. This case will be considered in Part II. of the present work. We will consider here an imaginary experiment with a two-phase binary system, which could not be realized in practice because of the smallness of the changes involved, but which will serve to illustrate the point in question. Suppose that a closed vessel contains a binary liquid mixture in contact with the vapour phase. Let the vessel be tilted so that the area of the surface separating the phases is increased from o tog+Azs. This will in general cause the vapour pressure to alter, Suppose that by the addition of an amount AM, of the solute C, the initial pressure (and therefore the initial concentration of the solution) is restored. One would at once say that the surface excess of the solute is AM,/Ac. Let us examine IO Dr. S. A. Shorter on the Constitution the question in the light of the above theory. Initially we have My=poV + po V' +T oo, M;=piV +p; V'+Tye, and, finally, Mo=pol VY +AV) +p) (V'+AV')+To(o+Ac), M, + AMy=p,(V+AV)+—)'(V'+AV')4+Ti(e+ Ac) ; and since the total volume is constant AV 7 AW 0. From these equations we readily find that AIG oh Pim Pi. op \ ANG = ie Po=i 1(0)9 e e . (14)} i.e. that the value of the surface excess of C, deduced from the experimental data is the value relative to the “ zero- plane ” of Cp. § 4. Special Cases of a Binary System. Case I. Two Immiscible Liquids. The simpiest case of a binary system is that in which each phase consists of a single substance, z. e. that formed by two immiscible liquids. Let us suppose that the @ phase contains only the component Cy, and the @¢’ phase only the com- ponent C,. Hquation (10) then takes the form a es dt / == = ) . ° ° e e 1 Pi. Po G U ce ‘This equation may also be written in the forms Dio =P, iB). Pram ec (16) at e Die ae ae This result is of importance in relation to the question of the composition of the surface-layers of pure liquids. It is and of the Surface Layers of Liquids. 271 often asserted that the surface-layers of a liquid are ina state of compression*. According to Lewis+, who has developed a quantitative theory of this supposed surface compression, the variation of the densities of two im- miscible liquids in the neighbourhood of the surface Fig. 7. Position. separating them is as shown in fig. 7. In the case of water it is calculated that the mean density in the surface- layer is about twice that in the body of the liquid. An effect of the same order of magnitude is calculated for a number of other liquids. In the exceptional case of mercury the mean density in the surface-layer is only °84 of the bulk density. Let us examine this theory of surface compression in the * See, for example, ‘The Chemistry of Colloids,’ by W. W. T (Arnold, 1915), pp. 229 & 250. i io i 5 oe Mag. Sept. 1910, p. 502; Kolloid- Zeitschrift, vol. vii. p. 197 272 Dr. S. A. Shorter on the Constitution light of the thermodynamical theory given above. Hquation (15) may be written in the form dt (2). o) (18) where fA, it will be remembered, is the distance of the ‘‘ zero-plane”’ of C, from that of Cy measured in the direction of the ¢! phase. Now in the case of a system having a density diagram of the kind shown in fig. 7, A is evidently negative—as is also the case if only one of the liquids exhibits surface compression. Hence the theory of surface compression leads to the conclusion that the tension of the interface separating two immiscible liquids is diminished by increase of pressure. ‘This result may, of course, be deduced from quite elementary considerations. The change of pres- sure produced by an extension of the interface at constant volnme must be such as to increase the tension. If the liquids are compressed in the surface-layers this exten- sion produces a diminution of pressure so that dt/dp is negative. The effect of pressure on the interfacial tension has been investigated by Lynde* in the case of a number of pairs of immiscible or partially miscible liquids. In the case of water and mercury it is found that the interfacial tension is increased ‘74 per cent. by an increase of pressure of 5000 lb. per sq. inch. This gives (assuming 7=370 dyne/cm.) for dr/dp the value 8:2 x 10~° cm. or ‘082 wy. This result is, of course, inconsistent with the view that either the water or the mercury are compressed in the surface-laver. It is interesting to calculate the order of magnitude of dt/dp required by Lewis’s theory. Since the mercury does not suffer any superficial increase of density its “‘zero-plane” will be situated somewhere in the surface- layer. Since the mean density of the water in the surface- layer is double the bulk density, the ‘*zero-plane” of the water will lie in the interior of the mercury, a distance from the nearer boundary of the surface-layer equal to the thick- ness of the surface-layer. We therefore have Kee dp If we assume the value 10-° cm. for the thickness of the > thickness of surface-layer. * Phys. Rev. vol. xxii. p. 181 (1906). of the Surface Layers of Liquids. 273 surface-layer, we obtain a value of dt/dp opposite in sign to, and arithmetically more than a hundred times as large as, the value obtained experimentally. A similar discrepancy between theory and experiment exists in the case of the ether-mercury interface, Lynde’s experiments giving for dr/dp a small positive value, and Lewis’s theory a relatively large negative value. Our theory shows, therefore, that the theory of surface compression is inconsistent with experimental data relating to the effect of pressure on the interfacial tension. Case II. Two Partially Miscible Liquids. Suppose that the two phases are the liquid layers formed by two partially miscible liquids, the ¢ phase being the Cy layer. If we write co=py /po and c,=p,/p,', we have Pigs — 2) (27) Pprcner i ais 6 1—¢p dp or 1—coc at we Sal ee (1—e@)(1—«¢,) \dp : ae Lynde’s results for three pairs of partially miscible liquids, together with the calculated values of h, Vo), and T'\¢; are shown in the following table. dr ©, Oe ¥ dp h 1 XEN) Py (0) in dyne/em. inpy. inpyp. n10-*gm./em.2 in 10~-® gm./em.” Water. Ether. 97 —'060 —-068 +°6 +°4 Water. Chloroform. 27 — 006 —-006 +:°06 +09 Water. Carbon bi- 42 +037 +041 —4 —'d sulphide. It will be seen that in two cases drflp is negative. The distance between the “zero-planes” is, however, much too small to justify any assumption of surface compression. Case IIT. Liquid Phase in contact with Vapour Phase (both components volatile). In this case we may in general neglect the densities in the vapour phase in comparison with those in the liquid phase. Phil. Mag. 8. 6. Vol. 31. No. 184. April 1916. 3) 274 Dr. 8. A. Shorter on the Constitution Equation (10) then becomes 1 20 (BL) (a7) ie) ik P: Po \po pil’ \dple ! We BS (eee de eels 10) Gs p1 dp), (22) If we suppose the vapour phase to consist of an ideal mixture of gases of molecular weights m) and m, respectively, the above equation reduces to hae d ig (oa ty Bi ) es or . Po P1 dp where R is the “ gas constant,’ and po and p, the partial pressures of the vapours of Cy and C, respectively. Lf we suppose the state of the system to be specified by the con- centration c of the solution of OC, in Cy we may write (x) Me dp & 6 Pi ee ol mes) Tyo) — RO Po pi (24) The concentration ¢ may be defined in any of the three usual ways vated ial fs Po and | Pa Pot pi Let us now consider the case of an ideal dilute solution of C, in Cy. In such a solution the partial pressure of the solute is proportional to its concentration, and the relative lowering of the pressure of the solvent is equal to the “molar fraction” of the solute. We have therefore p= Kei, «0 oe where K is some constant * about the magnitude of which * Or more precisely, some function of the temperature. of the Surface Layers of Liquids. 275 the Theory of the Ideal Dilute Solution makes no state- ment; and BoiriPo! Mtoe (26) 1a4 ny Myo. ° where P,) is the vapour-pressure of the pure solvent. Substituting the values of po, p;, and dp given by equations (25) and (26) in equation (23) we obtain the result m dt [hae — Te ip), ince MEMRAM O37) which differs from the well-known result deduced by Milner * only in respect to the more precise definition of the ‘surface excess.” We thus see that the equation applies to the case of a dilute solution of a volatile substance in contact with the vapour phase. We will consider the question of liquid mixtures in Part I]., when we deal with the case of a solution in contact with a third substance. Practically all the experimental data available for illustrating the theory relate to this latter case, which is a special case of a ternary system. Case IV. Liquid Phase in contact with Vapour Phase (only one component volatile). If ©, is involatile equation (22) becomes as. r et) eeu Net: Ogre ee) (28) If the solvent vapour behaves as an ideal gas of molecular weight mm, we have __ MoP1P aT 5) To= ae Py LI gh In the case of an ideal dilute solution equation (27) is obviously valid. The University of Leeds. Feb. 3rd, 1916. * Loe. ett. ps6 XXXIV. On a Method for deriving Mutual- and Self-In- ductance Series. By 8S. ButteRrwortH, M.Sc., Lecturer in Physics, School of Technology, Manchester*. 1. UMEROUS series formule have been obtained for the mutual induction between coaxial circles f. In general, these formule have been completely determined (i.e. the range of convergency and general term are known) only in the case of circles far apart. It is proposed in the present paper to develop a method which will yield the general term for formulee which hold when the circles are close together. 2. Let the radii of the circles be A and a, and let 6 be the distance of their centres. Then the mutual induction between them is given by the Elliptic Integral formulat Pr) open M=47 VKa{ (,-#)K-ZEt, afer (1) in which 2— 4AAa ~~ (A+a)?+0? and K and E are complete Hlliptic Integrals of the first and second kind respectively with & as modulus. By differentiating (1) twice with respect to &, remem- bering that di) H—K A eee dk ony dit. k(L= i?) es ! M ?M a Rea to obtain expressions for ae and Aye? and eliminating E and K between these expressions and (1), we find that a » &M ef ke(1— 8?) 5 — kL +k’) ae —3M=0., 29) By the substitutions M=/%y, k?=.«, (2) transforms into S(34+2y=¢0+8)y, ..1. eee in which Bee dx * Communicated by the Author. + For a collection of such formule, see Rosa, Bull. Bureau of Standards, vill. p. 1 (1912). ty +t Maxwell, ‘ Electricity and Magnetism,’ il, art. 701. On Mutual- and Self-Inductance Series. 277 This equation is of the type SS+y—l)yy=a(St+a)(S+B)jy, . . - (4) the equation of the hypergeometric series, so that, employing the usual transformations for this type, we obtain as equiva- lent forms of (3) a i 0 (ote Sel eis at's) COE) 0, (3,> Dee ee 33(93+2)ys=x3(S3t+8)?yp - - + + + Ge) SPys=A(Sy+5)(Ss—3)yy - - (8a) in which — mee =r aaa = "=Y; Y=ey, ys=(l—e)iy, y= ary, v a—l o—t— eto 1— 2s, 2 aR Baral marie and ad J, = Ly =— Ax, 3. In forming the primitives of equation (3), it must be noted that y is, in all cases, a positive integer, so that the usual second particular solution of (4), viz., go B(aty—1, B+y-1, 2-y, L), is no longer valid. However, when y¥ is a negative integer, the complete primitive of (4) may be obtained as follows :-— Denote by G(a, 8, y, 6, xz) the series 14 Pn MOTB +1) A ee S(O + 2) oe reer and apply the operator =3($+y—1)—a(S$+2)(8+8) to y=rAaG(atra, Ba, ytra, L+A, 2). Then p(y)=M(A+y—1) x and w( SL) =r r+ 2y—2).A+N ty — 12 loge so that It, —09 @ (At BS)y} —1)h 278 Mr. 8. Butterworth on a Method jor or a possible solution of (4) is y=lhoo(A+BS) AoGtetr, B+, ytd, 142, a)}. a OH ae When y is anything but a negative integer or zero (0) reduces to y=BE (a, B, Y> ye the usual first solution of (4), but when y is a negative integer (say —7), all the terms in G(ata, BHA, yr, 14+A, wz) beyond the (n+1)st contain % as a factor in the denomi- nator, so that both the particular solutions implied in (9) will remain finite when A vanishes. Hence, in this case (5) yields the complete primitive. Moreover, if y is a positive integer greater than unity, the Paeteatian y=ua'—7 applied to (4) will convert that equation into one of the same type in which the new y is a negative integer or zero. Finally, if y is unity, the complete primitive of (4) is nee -0(A+BS 5) {aN (a, B+r, be alee 4. Using (5) and (5a) to obtain the eee of (3), we find that Tie a inaonl(A+ BS) QP GA—B AH ACL AGL A}. © =It 04 ee ie MER ALL AL eee =lt,.-97-¥ As+B,S){eAGA—3, N—h ALL, A+L, w)}. (68) =k =otl( A, +B. Onan 2G(A—3, A—} A—1, AFI, #)} (6) =Hh.=0( A+ BS ){edG A+, NEOUS Ea in which the constants An, B, have still to be determined. deriving Mutual- and Self-Inductance Series. 279 Since M, x, x; vanish with k, B=B,;=0. Also when £ is small M/43,/ Aa =7h?/16=7e!/16 = —rra32/16 by virtue of (1). Under the same conditions the right hand members of (6) and (6 c) become Aw3/32 and Ag3x53/32 respectively. Hence A= — As 2am Using these values in (6) and (6 ¢) we find an 3? 32 . 5? Sn aera le \ a VKa i8(1+ we ONE G6 ae ero foe T oe 3? . 5? . 7 4 VAap(I— 3 gH + 5 1G Ree AR Oro ) 2 in which E £Aa ee) oe SO mie 7 (Arabs (A) is valid for all distances of the circles, but converges most rapidly when the circles are far apart. It could readily have been obtained by substituting the usual series for K and E in (1). (B) converges only if k°<4. When the circles are of equal radius = 2a/b, and the series is then identical with one obtained by Havelock *. Ly, &2, &, vanish when & is unity, and when either of the independent variables is small (6a), (66), and (6d) assume the form A,+Bnzlog a. Under the same conditions with k'?=1 —k?, aa 4 M/47,/Aa= log : ~2= log ore 2, iy Therefore * Havelock, Phil. Mag. xv. p. 832 (1908). (A) (B) 280 Mr. 8. Butterworth on a Method for Using these values in (6 a), (6 6), and (6 d) we find :— From (6 a) 52 amt (o> S45 rakgt.-), ©) in which 4 ) go= logs, —2 | ey am Ve ace Lege r PT Bo 1 2M tii 1 : tr $ra= 5 mal aOaFD | From (6 6) pie ia s are = 5 ($0 + hte! + _ prgek'bsl + in which A go = logy —2 | dy a2? | / (fpr sees = dy — qo aan | a 1 | unl: ane be le od \ | Hilal: cecal Ae | a ari: | > oma bn! — dn “he oes) ees From (6d) a oR ene |) Me Gomes Lol Bee ka ek, ee ete oe to) ate" + +...(H) deriving Mutual- and Selj-Lnductance Series. 281 in which | dy’ =log 4u—2 | tt! T-3tG gr" 3-5-3 | wml =5-7-3 | | be teas ar 8 | ($) and (D) are valid for all distances of the circles, but converge most rapidly when the circles are close together. (H) converges only if 4?>4. The first five terms of this formula have been given by Rosa (loc. cit. p. 16). 5, Self-induction of a Solenoid. Take the diameter of the solenoid as unit of length, let the axial length of the solenoid be z, and the number of turns be nz. Then, if M represents the mutual induction between two equal coaxial circles at distance z, the mutual induction between the solenoid and an equal circle in the plane of its end is af Mdz, and the self-induction required is 0 < al z\"M dz. 0 0 Now from (3 d), since yz=M, 4=—2’, d os CA ad < }=320 dz er?) dz ee so that by integration dM 4 (e+e) =32) Mdz—3) dej Mdz, (2+2)M=(1+32*) | Mdz—32\ dz \ Mdz, 282 Mr. 8. Butterworth on a Method for from which \M dz=(z+ 2°) (1-25"), | IM Pe Ae fdz{Md2=2(2+2) { 3-M—(1+ 327)" | | Using the value of M given by (1), and remembering that I k? = ——. Tha we obtain from (7) ‘hee: .(K=B), fae fata: = 27 (a—-k)K— 21m}. | Hence the self-induction sought is = ant “de M dz, 0 0 _ Amn? BEF {(1—k7)K—(1—2 * )E—}. This holds when the diameter is unity. ‘Tf the diameter e © = is 2a and the length 0, put z=0/2a, so that a and (by dimensions) muitiply by (2a)?. Then 32 an? tay LS He This expression was first given by Lorenz*. Proofs have also been supplied by Coffin + and by Russell t. The author believes the above proof to be new. 6. By treating equation (8) in the same way as (1) was treated, various series can be obtained for L. {d—-2) K-24) n—ey a \ Thus, hee =F Bhs tae satisfies the equation 2 2 LY we Re: dy ee (1) SH + 5k(1— i) F + 8y=0, * Wied. Ann. vii. p. 161 (1879). + Bull. Bureau of Standards, i li. p. 123 (1906). t Phil. Mag. xiii. p. 445 (1907). = deriving Mutual- and Self-Inductance Series. 283 or putting h?=2 s=x a ; dé 3 1 S(S+ Lu=o( $+ 5) (s- 5)% ary co), an equation of the type (4). The solutions of (9) with the various possible independent variables yield the Ne serles :— ca 15 ( a ai Be) = mab (F) ive ae Te 2 2, B)— = fe 7 . (G) = Anna bd ¥(; G. as eg [ =e) 3 5) >) = 4r’n*a*b ? F 3° 9? 2, — =) See CB) in which F(a, H Y> ‘* denotes the hypergeometric series a8, #(at+)BB+1) » ea 1.2.y(y+1) © The above series are suitable for long coils. (I) and (G) are convergent for all lengths. (H) and (1) converge only if b>2a. U= ay, ~ 5, l+7~ ee {5a B+ Ry + 5 sak Abs ae Tee LSC OST oO cw age 51g 4608" Ue a (J) in which a i} Wi= logs, —— | pape ee) eal | a A gags AY pee Wane a hig ae a | i eae Te Win shams ra | J 284 On Mutual- and cia Series. ff 2 pe ae ts ey a eae Sak, Ie per ES I! oo NM —) 4.406° \3 0 oun ce a (K) in which 3 7 Wi = logs — 5 OR jer ye a a We = 5 ae 3 ; pcg Lg A alee eo Wa go eee reer Dn a 8 ia Heke Gapeeliaee Al Vrs soos TG 3 4 J 2 1D ILA ING DL lee 3 ee 243 er Min ees ee I! L= 16mntatd (5-) va +54(5,) Ws oe =(5.) ¥ eae Sisdly coun Toe Lee a taj caer ate ie oe (L) in which oa) licens Frusace V a ee Wri lo i 3 | BPs cage) CAPs i! Pde Ss I. Sant d beard i.e TAM PO poate zat pata Vs 2 ane as pea 3 | La a (SEN PBA ys ia aheet ay ca te Wy ie oy (her 2. sala) Ye pads Oe re ee 2) i Y 2 2S 4) > RG (5, Vi ane by QD On an Optical Test for Angles oj Contact. 285 in which ie toa ae Wowie EA 5S Ww = 74 6-5-7 | Wow tah J ‘I'he above series are suitable for short coils. (J) and (K) are convergent for all lengths. (1) and (M) converge only if b < 2a. XXXV. Onan Optical Test for Angles of Contact. By Prof. A. ANDERSON, I/.A., and J. E. Bowsn, M.A.” lL the February Number of the ‘ Philosophical Magazine ’ we gave a method of determining surface-tensions from observations on the heights to which liquids rise in tubes, which is free from corrections of any kind, and the accuracy of which depends entirely on the capabilities of a measuring microscope. We showed how, by plotting curves giving the relation between the radius of the tube and the radius of curvature at the centre of the meniscus, the angle of contact could be deduced. The curves obtained for water and some other liquids showed that—at least, very approximatel y—this angle was zero. The following experiment was designed to test directly whether any angle of contact differing from zero really exists in the case of water and some other liquids which were examined. If a rectangular piece of very thin glass be dipped in a liquid and then held in a vertical plane with two of its edges horizontal, there will be a long cylindrical drop attached to it. Its cross-section or the appearance it presents when looked at endwise is one of the forms represented in the figure (fig. 1). * Communicated by the Authors. 286 Prof. Anderson and Mr. Bowen on an Fig. 1 (a) is drawn on the supposition that the angle of contact is zero. It will be seen that the drop forms two cylindrical lenses, one convex with centre at the point O', the other (only the lower half of which is present) being igi P ! / Nv hi s\n ae [ee A fi os 28 \ | hei concave with centre at O, the point where the glass plate is tangential to the contour of the drop. The upper edge of the drop may be along the line through O perpendicular to the plane of the paper, or may be higher up at some point N. Ifa Optical Test for Angles of Contact. poi a collimator with its axis horizontal and with a horizontal slit be placed to the left of the drop, parallel light will fall per- pendicular to the glass plate as shown in the figure, and the concave lens will form an image of the slit, which will be a straight line through O perpendicular to the plane of the paper, and in the same horizontal plane as the corresponding line through O. If now a low-power microscope be placed to the right of the drop with its axis along OA, the image of the slit formed at A can be seen and arranged to coincide with the horizontal cross-wire of the microscope. If the microscope be now moved back through a distance AO it will be focussed on the glass plate, and the image in the microscope of the upper edge of the drop should either coincide with the horizontal cross-wire of the microscope or it should be necessary to raise the microscope vertically to make it do so. Tf an angle of contact existed, the drop would present the appearance of fig. 1 (0). The continuation of one side of the drop has been dotted in to the point P, where it would become vertical ; and it will be observed that in this case the centre of the lens O which is in the same horizontal plane as P is above the upper edge of the drop. Thus after focussing the microscope on A, getting the image to coincide with the horizontal cross-wire, and then moving it back through the distance AO so as to focus on the glass plate, the microscope must be lowered a distance ON to get the image of the upper edge of the drop to coincide with the cross-wire. The microscope used was the Bailey & Smith microscope used in the previous experiment (loc. cet.). This was capable of horizontal and vertical motions, and was arranged with axis along the line of parallel light and its horizontal motion along this line. A clean microscope slide-cover was used as the glass plate and the liquids examined were those of the former experiment—water, glycerine, olive-oil, and turpentine. In all of these when the drop was freshly formed a perfectly straight image of the slit was seen, and to get the image of the upper edge of the drop (sometimes marked by a line of extremely minute air-bubbles) to coincide with the cross- wire when the microscope was brought back the microscope had to be raised. The image of this edge was not straight. As evaporation went on the upper edge came down to the same level as A (which of course also moved, both vertically and in a horizontal direction perpendicular to the plate); and as evaporation proceeded further the image became curved and took the shape of the upper edge of the drop. No indi- cation of the state of things represented in fig. 1 (b) could be 288 Prof. Anderson and Mr. Bowen on an observed. Hence it was concluded that the angles of contact in the cases examined were zero. As the liquid on the two sides does not evaporate at quite the same rate, the upper edges on the two sides do not always coincide : thus a better examination can be made if only one side of the plate be wet and only half the drop shown in the figure be formed. If the argument used above is correct, it is possible to use the experiment as a rough method of measuring surface-tensions. To do this we consider the convex lens also, which has its centre at O', and which brings the parallel light to a focus at B. Let a = OA. b==0'B. h = vertical distance between OA and O'B. 7, = radius of curvature of each face of concave lens (assuming curvature the same). 7, = radius of curvature of face of convex lens. pw = refractive index of liquid. p = density of liquid. Pi= pressure inside liquid at O. p= 9 9 9 O’. a = atmospheric pressure. T = surface-tension of liquid. Treating the lens as thin, 1 2 L 2 pe ie ar nee al pe roe 5 age Oe teh iy 1 ;) P—=1 (= +) ~ 2(u¢—1) Gta ; But Do2—pi=gph ; T= 2g9p(u—1)hab]/(a+0). If only one side of the plate is wet with the liquid, ,the formula becomes: T =gp(u—1)hab/(a tb). Optical Test for Angles of Contact. 289 As the microscope was fitted with verniers for both hori- zontal and vertical movements, the readings were easily taken. The method is, of course, not capable of great accuracy, owing to the changes taking place in the drop due to evaporation. In the case of a liquid which evaporates quickly (e. g. turpentine) the readings must be taken very rapidly. The following are some measurements taken in this way, which appear to be sufficiently close to show that the argument is correct : Water. a= ‘644 6='544 h=:392 T= 56 “(64 673 *350 Ton "932 ‘66 "294. 73°d Turpentine. a= ‘674 b= "381 A='l51 T=26°9 -076 *396 ‘15 28:2 ‘661 "304 "148 O74. Glycerine. : a='443 b= -287 h=°370 T=74:9 "435 344 corel for *358 ‘400 -944 75°9 °367 *3388 "392 (44 *308 °369 "382 74:5 a, 6b, h areinem., Tin dynes per cm. It will be noticed that though the method gives values for water and turpentine in the neighbourhood of those generally accepted, the values for glycerine, though consistent with each other, are too high. This may be due to the fact that the glycerine drop is much thicker than either of the others, and the tormula for thin lenses may not apply; or it may be due to the rapid absorption of aqueous vapour from the air. University College, Galway. Phil. Mag. 8. 6. Vol. 31. No. 184. April 1916. x XXXVI. Tie Theory of the Flicker Photometer.—lI. Un- symmetrical Conditions. By Herspert Ki. Ives and Hi. F. KINGSBURY *. SYNOPSIS. . Introduction. . The Sensibility of the Flicker Photometer. . Critical Frequency-[lumination Relations with Superposed Steady and Flickering Illuminations. 4, Critical Frequency-Hlumination Relations with Unequal Light and Dark Intervals. (a) Theoretical. (6) Experimental. -(e) Discussion. 5, The Flicker Photometer with Unequal Exposures of the Compared Colours. 6. Some Effects of Accidental Dissymmetry on the Critical Frequency and Flicker Photometer Phenomena. 7. Summary. Co bo 1. Introduction. N a previous paper f we have developed a theory of the action of the flicker photometer. ‘The essential part of that theory consists in the treatment of the visual appa- ratus as a conducting layer of matter receiving and trans- mitting intermittent radiation according to the general physical laws of conduction. The principal assumptions are as follows :— 1. That the stimulus is transmitted by a medium having a certain ‘diffusivity,’ by reason of which the amplitude of a fluctuating stimulus is reduced. 2. That the disappearance of flicker occurs when the ratio of maximum to minimum of the transmitted impression has a certain value. . By the application of the Fourier conduction equation, on the basis of these assumptions, it was shown that the behaviour of the flicker photometer toward different colours and at different intensities may be derived from the beha- viour of the colours separately with respect to the critical speed of disappearance of flicker. The previous paper dealt only with what may be termed symmetrical conditions. Critical frequency relations were studied only for equal dark and light exposures, the flicker photometer was considered only for that case where the two * Communicated by the Authors. + “The Theory of the Flicker Photometer,” Ives & Kingsbury, Phil. Mag. Nov, 1914, p. 708. The Theory of the Flicker Photometer. 291 compared colours are exposed for equal times, and the only setting of the flicker photometer which was the subject of numerical study was the equality point. In the present paper a study is made, on the basis of the same assumptions, of certain unsymmetrical conditions. These conditions are much more difficult to treat. They necessitate, for instance, definite knowledge of the value, under various unusual conditions, of the Fechner fraction, and of the relative effect of impressions of different wave- form on the ultimate receiving apparatus. Their complete treatment is In some cases practically impossible without certain simplifying assumptions, which are not in entire accord with known facts. Partly—perhaps, principally—on this account the further development of the theory presented here is not entirely satisfactory. But, while it fails to fit the experimental facts quantitatively as closely as we would hope, it does handle the principal phenomena qualitatively in a very striking minner, and, moreover, has prompted experimental work whose results must receive consideration in the development of any theory of flicker photometry. 2. The Sensibility of the Flicker Photometer. Of the two methods of utilizing flicker for the measure- ment of light, one—the frequency of disappearance of flicker—is quite insensitive, the other—the point at which two alternated illuminations show a minimum of flicker— is quite sensitive. Itisfor this reason that the flicker photo- meter proper is deserving of consideration as an instrument of precision, while the critical frequency phenomena are of little value in measurement. The low sensitiveness of the critical frequency method is, as far as the mode of treatment here followed goes, simply a matter of experimental fact. The error of setting for critical speed amounts to one or two cycles per second, which means for ordinary intensities 25 to 50 per cent. in illumination. Add to this the fact that variations in con- ditions of adaptation, &c., cause uncertainties fully as large, and the unsatisfactory character of the method for measure- ment is clear. How, then, can the flicker photometer, whose behaviour we have endeavoured to predict from the critical frequency phenomena, possess the sensibility it does? This greater sensibility we find is shown by the straightforward appli- cation of the theory as presented in the former paper, as follows :— X 2 292 Messrs. H. H. Ives and E. F. Kingsbury on the Let us consider the case of no colour difference. At the condition of equality, with an ideal flicker photometer, no flicker results, no matter what the speed of alternation. The criticai speed is therefore zero. As the equality point is deviated from in either direction a flickering condition is produced due to the alternation of the two unequal illuminations. The speed necessary to make this flicker disappear may be determined by the ordinary experimental method for various known differences of illumination. When sufficient points have thus been fixed. the lines joining them enclose a space within which no flicker oceurs. At any given speed the limits of the no-flicker space indicate the limits within which the flicker photometer setting must lie. The sensibility of the flicker photometer may thus be indi- cated graphically by the narrowness of this region. Let I, be the illumination of one side of the flicker photo- meter. I, that of the other side. (We shall in this paper distinguish between the actual illumination and the apparent illumination, which latter refers to the appearance of the disk when running, and is always Ix angular opening. This usage causes the difference of the constants in certain of the equations common to the two papers.) Let us, as in the previous paper, deal with stimuli repre- sented by simple sine curves. Then, as before, the ranges of the two transmitted 1 impre essions at a depth X are live 2K | K = diffusivity | ey ae, I,e ja ok w =speed | The total resultant range is the difference between these or € 2K PRp— EB]: eo. eo ae The fractional part this is of the whole, or the fractional range 1s ka ye (ee 3) Ie ai IG | e . e e e . ¢ / Calling this fractional range a constant, 6, according to our previous assumption, and solving for w, we obtain I,-I, 2[logs + log ye = neh) Me * [loge]? Theory of the Flicker Photometer. 293 In fig. 1 are shown calculated values of w, for the simplest possible photometric condition, namely that where the total illumination is maintained constant. This means that we are here free from any questions as to the rate of change of diffusivity or of the Fechner fraction with the illumination. Pigs Critical Frequency , Cycles per Second. The sensibility characteristics of the Flicker Photometer as exhibited by the critical frequency of disappearance of flicker for various ratios of the two compared lights. Calculated curves for the case of total illumination maintained constant. Various values of the Fechner fraction (8). Curves are shown for three values of 6, namely, -01, -001, and ‘0001. An experimental value for w has been taken for the limiting conditions, and the value used for the diffusivity solved for from the equation (4). - The figure shows with what extraordinary rapidity the critical speed rises as the two illuminations become unequal. Thus a variation of only about one per cent. causes as great a change in the critical frequency as did fifty per cent. in 294 Messrs. H. E. Ives and EH. F. Kingsbury on the the ordinary critical frequency relation. The sensibility of the flicker photometer is measured by the width or the no-flicker region, as already noted. This is extremely small ——a matter of a few per cent.—for low speeds, even for the largest value of 6. The figure clearly shows the desirability of using as low a speed as possible, which means the careful elimination of all mechanical flicker. It is obvious why the sensibility of the flicker photometer decreases with increasing colour difference, since this necessitates a higher speed and consequent wider zone of possible setting. The case just considered was not studied by us experi- mentally, as our arrangement of apparatus did not make this convenient. We have, however, obtained data for the common case where one illumination is maintained constant while the other varies. The calculation of this case is like that of the preceding, except that it becomes necessary to introduce a changing value of the diffusivity (and possibly of the Fechner fraction) as the mean illumination changes. In accordance with the results of the previous paper we have assumed the diffusivity to vary according to the relation b= X*(alog], + 6)) 21. (ea where I, is the average value of the illumination, the con- stants being derived from experimental critical frequency relations and the formulz previously developed. Fig. 2 shows the caleulated curves for various values of 6, and a set of experimental points. (The apparatus used is described under a later section.) It is to be noted, first of all, that both theoretical and experimental curves show a lack of symmetry, the median lying for low speeds toward the fixed light, at high speeds toward the movable one. ‘This is a point of very great practical importance. The obvious moral is that the photo- metric arrangement should be such that the mean illu- mination remains constant, that is, as the illumination on one side is increased that on the other should be decreased, as was done in the first case considered and shown in fig. 1. The effect of this lack of symmetry is, at the speeds which would ordinarily be used, to displace the setting toward the fixed illumination, which is therefore nnder-rated. In most practical cases, with a good design of flicker photometer, the effect of this lack of symmetry would be largely eliminated where a strict substitution method is employed. If, however, colours are measured which call for greatly different speeds, the displacement will be different for each. This difference could amount to as much as two per cent., judging from Theory of the Flicker Photometer. 295 speed-data recorded in an earlier paper *, for red and green light, using the experimental values of fig. 2. This is, there- fore, a point which should receive careful attention in the future design of apparatus to be used in large colour- difference work, such as the determination of spectral luminosity curves. Second Cycles per Sensibility characteristics of the flicker photometer. Calculated curves for various values of 6 for the case where one illumination is constant, the other variable. Experimental values are shown by dots. The second point to be noted is that the experimental values agree most nearly with those calculated trom a very small value of the Fechner fraction, namely ‘0001. The actual value of this fraction for simultaneously presented juxtaposed surfaces is between ‘0025 and -01. Why is this small value apparently called for ? * “Spectral Luminosity Curves obtained by the Equality of Bright- ness Photometer and the Flicker Photometer under Similar Conditions,” Ives, Phil. Mag. July 1912, p. 149. 296 Messrs. H. E. Ives and H. F. Kingsbury on the This point illustrates the remark made in the introduction, that the cases to be treated in this paper demand definite numerical values which are not in all cases known, and call for simplifying assumptions which are not completely justi- fied. We may state in regard to this quantitative discrepancy between the calculated and observed facts that : 1. We do not actually know what the value of the Fechner fraction is for successively presented impressions; it may be much lower than for the cases so far studied experimentally. 2. Our theoretical treatment is for a simple sine curve stimulus, while the experimental work was done with disks having an abrupt dividing edge. 3d. Any unevenness of “the surface of disks such as we used will tend greatly to increase the speed called for at all points except at what is the left-hand end of the diagram shown. This point will be treated in detail under another section. : Whether these considerations are entirely sufficient to account for the apparent exaggeration of the experimental over the calculated conditions we are unable to determine. However, it is evident that the theory does account extra- ordinarily well for the principal facis regarding the sensi- bility of the flicker photometer. Before leaving the question of sensibility we may call attention to an apparent consequence of the theory of some practical importance. It has to do with the character of the transition from one stimulus to the other. Should this be abrupt or gradual? Referring to fig. 1, it is to be noted that if the transition, which is there considered as gradual, is made abrupt, the limiting critical speeds, where either light is reduced to zero value, will, for fairly obvious reasons, be increased. Since the lower limit of speed is fixed (zero), this means that the no-flicker region will be narrower with the abrupt transition. In short, the sensibility for a given speed should be greater for an abrupt transition than for a gradual ; for instance, it should be greater for a sharp tran- sition flicker-photometer field in focus than for the same field out of focus. We are informed that some recent experimental work confirms this. Where, however, the speed is adjusted to the minimum set by the disappearance of colour flicker, it is probable that the minimum will be so much lower with the gradual transition as to leave the uitimately attained sensibility about the same with either type of transition. Theory of the Flicker Photometer. 297 3. Critical Frequency-Lllumination Relations with Super- posed Steady and Flickering Illuminations. Before leaving the equations developed to indicate the sensibility of the flicker photometer, it is of interest te note that they show as well the behaviour, on our theory, of a flickering illumination superposed on a steady one. Thus equation (4), while developed to represent the case of ¢wo dovetailed flickering illuminations of unequal intensity, represents as well the case of a steady illumination, of the value of the lesser illumination, upon which is superposed a fluctuating illumination of the amplitude of the difference of the two illuminations. What will be the behaviour of a fluctuating illumination of this character as the illumination is varied? So far as the authors are aware this case has not been studied experi- mentally. We obtain an answer by combining equations (4) and (5), which give 9 are 2 2| logs + logy aT [log e]? 0) S= apoio Tete Bilei' sria)! ait 1CO)) The most striking characteristic of this expression is evident upon inspection, namely that each different ratio of steady to flickering light calls for a different slope (coefficient of log I,) in the critical frequency-log I plot. We may calculate the critical frequency-log I lines as follows :— Let experimental values of » be taken at two illuminations for the ordinary experimental case of I,=0. Assume a value for 6. From this a and 6 may be obtained, which constitute all the data necessary. In fig. 3 are shown, first, an experimental critical fre- quency-log I line, for the case of I,=0, obtained with an abrupt transition disk, then the corresponding line as calcu- lated for a ratio of = of *8, for three values of 6. Finally 1 are shown lines obtained by experiment for ratios of 2 of | 1 ‘83. and “56. It will be noted that in their most novel characteristic—the variation of slope—the experimental lines verify the prophesy of theory. As in the matter of sensibility, the correspondence between theory and experiment is not quantitatively all that could ‘ 298 Messrs. H. B. Ives and EB. F. Kingsbury on the be desired. The best correspondence would call for a value of 6 of about ‘001, but even with this small value the slope of the theoretical line is too great. We would again point out that while the theoretical work is done for the case of Fig. 3. Cycles per Second. 10 1G 2-0 Phenomena of superposed steady and flickering illuminations. Dashed lines calculated from theory. Full lines, experimental values, showing varying inclinations as suggested by theory, but actual slope and magni- tudes not as calculated from simplest assumptions. gradual transitions, the experimental is done with abrupt transitions. In addition, our assumption that the diffusivity varies with the logarithm of the mean illumination may be wrong. It is, for instance, possible that the diffusivity is a function of an illumination intermediate between the maximum and the mean. Mechanical defects of the photo- metric field may as well be active as disturbing causes. These points are, however, secondary to the fact that the theory accounts for and, in fact, has prophesied the pre- dominating characteristics of the cases considered. In both it is evident that information of some value would be furnished by performing the experimental work with appa- ratus giving the exact kind of transition which is treated by the theory. The experimental difficulties of obtaining a sine-curve transition from one illumination to another are, however, rather formidable, and we have not felt the slight additional support which might be furnished to the theory to warrant undertaking the great additional labour. Theory of the Flicker Photometer. 299 4, Critical Frequency-J llumination Relations with Unequal Light and Dark Intervals. (a) Theoretical. Our theory of the flicker photometer is essentially a theory of the action of the eye toward intermittent excitation by light. The action of the flicker photometer follows on this theory directly from the critical-frequency-illumination laws. In undertaking an examination of the flicker photometer for the case where the two lights are exposed for unequal intervals—one of the principal objects of the study—it is therefore necessary to test the sufficiency of our theory to explain the critical frequency relations for unequal light and dark intervals. The most extensive study of this case is that of T. C. Porter*. He experimented with a series of disks, part white, part black, and obtained results which he found well ‘represented by the equation n=a+(b+clogI)logw(360—w), . . (7) where w is the size of the white sector in degrees, n=cycles per second, [=illumination, a, 6, and c constants. This equation, which has no theoretical basis, indicates that the critical frequeney should be a maximum for w=180 degrees; that the curves connecting critical fre- quency with angle should be symmetrical about 180 degrees ; that the log I critical frequency straight lines should have a different inclination for each angle of opening (except complementary angles, which have the same). His actual numerical values call for a Jarge negative frequency at 0 and 360 degrees, instead of zero, and the equation is not satis- factory for small angles at illuminations below the change in slope of the straight critical speed-log I lines. While it is not necessary that our theory should yield the same formula as that developed by Porter, we must be able to explain: the dominating phenomenon, namely the reduc- tion of speed as the white opening is either increased or decreased from 180 degrees. Let us consider two complementary disks, of white angles less than and greater than 180 degrees respectively. Let these be rotated at some finite speed. Suppose the diffusivity of the receiving and transmitting medium of the eye to be the same as either disk is observed. It is obvious that (under this restricting condition) the two damped fluctuating impressions which are transmitted to any given depth must * “Studies in Flicker,” Porter, Proc. Roy. Soe. vol. Ixili. p. 347 ; vol, Ixx. p. 315; vol. Ixxxyi. p. 445. 300 Messrs. H. E. Ives and H. F. Kingsbury on the be exactly complementary ; in other words, their range of fluctuation of value must be equal. But while the actual range is alike, the fractional range, in terms of the mean value, which on our theory determines the disappearance of flicker, will be small for the large white opening, and large for the small white opening. It is easy to see that (the diffusivity being fixed) the critical speed must increase pro- gressively from zero to 360 degrees white opening to large values for small openings. ‘But, according to our theory, the diffusivity varies with the illumination and in the same direction. It is to be expected that it is the average illu- mination (or, strictly speaking, the average brightness), the product of illumination and white opening, that determines the diffusivity. Accordingly we may expect the diffusivity to decrease from a maximum at 360 degrees to very low values for small angles. Lowered diffusivity calls for lower speed of alternation to produce the same range. We therefore have, according to our theory, two opposing effects : one, which we may call a contour effect, calling for increasing speed as the white angle decreases ; the other, the diffusivity effect, which calls for decreasing speed as the white opening becomes small. If our theory is correct, the resultant of of the two should have a maximum at 180°. The adequacy of these effects to explain the experimental facts can, of course, be determined only by quantitative calculations, which we shall now attempt. Here we cannot, as in the case formerly treated, assume a simple sine curve stimulus. We have therefore appreached the problem ina different manner. We have first developed the complete Fourier series expansion for unequal light and dark intervals, with abrupt transitions, as follows: ag le i Toh cos wt + 5 in 2a cos Zot T I,=I¢+ By , = 3 sin 37h cos sot+....}. (8) We have then combined this, in the same manner as in our earlier paper, with the fundamental Fourier conduction equation. The result is the following infinite series: pee, AN sea cue a = ZK c =I —/ a ~=1l@t+ ss E sin 1h cos( wt \ aK Le kay { zs w Pe Kain 2mpeoos (2a—N4/®)+ nu} (9) fed Theory of the Flicker Photometer. 301 where I,=transmitted impression. {=intensity of stimulus. @=fraction of cycle during which stimulus is acting. ¢=time. w = frequency. X= distance from surface of medium. K = diffusivity. Were we in possession of extensive tables of values of this expression for a wide range of values of ¢, X, w, and K, it would be possible to pick out those values of » which, for any chosen combination of the other variables, would make the fractional range of fluctuation whatever we chose. Such tables are not, however, available, and this is a pro- blem in heat conduction which has not apparently been investigated experimentally. No simple method of solving for the range of fluctuation or for » has been found by us, and, as a consequence, we have had to be content with a rough and approximate solution. In default of the complete solution we have worked with the simplified expression furnished by the first two terms only. ‘he range is given by the difference between the maximum and minimum values of the cosine term, or i ON R=— le ASTID aaa Ne (10) the fractional range is this divided by the mean value, and this we assume, for the condition of no flicker, to be a certain small quantity which we have called 6, or WwW Wy ox one 5=—e ie simmmq@an cn says CET) ORG Taking logarithms, rc ee +log sint@—logd—logé }7_, = 7 = = 12 Os) log e Nes z ( ; If, now, for the diffusivity we introduce the value derived in the former paper (equation (7) of that paper), substituting the mean illumination for the illumination there used, and also using the actual values of the constants, in place of the. 302 Messrs. H. E. Ives and Kh. F. Kingsbury on the symbols (m and p) there used, we finally obtain for the complete expression, log +log sin r@—log p—log oy o= nee cae (a log Ig+6), (13) where a and 6 are the experimentally found constants for a 180-degree disk under an illumination I. (The quantity outside the brackets should, of course, reduce to unity for the case of 180 degrees opening. The outstanding error is one of the consequences of using the approximate two-term expression. The term log — has been dropped in most of the calculations below.) at In fig. 4 are plotted curves showing the values of @ as Fig, 4, 10 Z0uy 30 40 20 Cycles per Second. Lower diagram.—' ull lines, calculated critical frequencies for differen” ‘angular openings. a, high illumination; 4, medium illumination; ¢, low illumination for blue light. Circles and crosses give experimental values. Upper diagram.—Calculated critical frequency-log I lines for three angular openings. Dotted lines indicate characteristics for blue light at low intensities. All calculated data are on a basis of 6=-001. Theory of the Flicker Photometer. 303 given by this equation under various conditions of illu- mination. These are all worked out for a value of 6 of ‘001. Curves a and 6 show the speeds called for at two illumina- tions for white sectors from zero to 360 degrees. It will be seen that the maximum near 180 degrees, and the drop toward zero for both large and small angles, are predicted, in general agreement with the findings of Porter. Unlike the curves given by Porter’s equation these are somewhat unsymmetrical. Curve ¢ shows a very interesting consequence of our theory. The experimental results obtained by the senior author on blue light at Jow intensities, are interpreted on the present theory as indicating that the diffusivity of the transmitting medium is a constant for those conditions. Now, if the diffusivity becomes constant there is nothing to bring the critical speed back to zero when the opening becomes small. Consequently the speed should continually increase with decreasing angle. ‘The actual shape of the eurve for illuminations below which the critical speed is independent of the illumination is obtained by making the term within brackets in the above equation constant, choosing an experimental 180-degree value. No relationship of this character between critical speed and angular opening has been observed before, so that its existence or non-existence forms a severe test of the new theory. The straight lines d are the critical frequency-log I lines as predicted for various openings. It will be noted that they are of different slopes, the slope varying continuously with the angle. As a consequence, the curves showing the relationship between angular opening and speed are unsym- metrical to different degrees at different illuminations, as shown also by curves a and b. The experimental study of these points is described in the next section. (b) Heperimental. The apparatus used in all the experimental work described in this paper 1s shown diagrammatically in fig. 5. Ly, and L, are point source “ 100 candle-power” carbon lamps, L, being fixed and L, movable along a photometer track T. F is a white surface of magnesium oxide on which falls the light from the lamp L;. D is a sector disk of the form shown in the separate sketch at the side whose purpose is to interrupt the view of the surface F when the apparatus is used for critical frequency studies, and to receive the light from the lamp L, when the apparatus is used as a flicker } 304 Messrs. H. HE. Ives and H. F. Kingsbury on the photometer. In the former case the disk need be merely opaque, in the latter case it must have carefully bevelled edges and be evenly covered with magnesium oxide. A series of disks having various openings were provided, several Fig. 5. Plan of Experimental Apparatus. L,, L,, light sources. M, motor to drive disks. T, photometer track. S, ‘electric speed-counter. F, white surface. E, observing aperture. D, sector disk, N, No Nicol prisms. being arranged for use with the flicker-photometer arrange- ment, the majority, however, being simple thin opaque disks of metal not whitened. These latter were used exclusively in the critical frequency experiments. M is the electric motor used to drive the disk. Its speed is regulated by a series resistance and read upon an electric speed-counter 8. The latter consists of a small toy motor (provided with dis- proportionately large and close-fitting brushes) used as a generator, its H.M.F. being measured on a millivoltmeter. This speed-counter was calibrated by stop-watch and revo- lution-counter and found to give an accurately rectilinear speed-voltage relation. EH is the observing aperture, of one square millimetre area. N, and N, are Nicol prisms, by means of which the brightness of the observed field can be varied over a wide range. The effective area of the field * of view has a diameter of about 94 degrees. sd ca a Theory of the Flicker Photometer. 305 Details of the method of observation were quite similar to those employed in an earlier investigation to which reference may be made™*. The first point to be studied was the shape of the curve connecting critical frequency and angular opening. ‘This was investigated by obtaining the critical speeds for a series of opaque disks whose openings were 30, 120, 180, 240, 300, 330, and 345 degrees. The experimental values are shown in fig. 4 along with the computed ones. Two sets are Fig. 6. Cycles per Second 20 ie) 20 log L Critical frequency-log I data for 80° and 380° degree white disks, showing absence of symmetry at all illuminations. shown, one set, for a comparatively high illumination, exactly as obtained, a second set for a lower illumination, derived from the others from critical frequency-log I data to be discussed presently. It will be seen that the computed and observed data are, in general, similar. The chief difference between the ex- pression used by Porter to represent this case and the one * «Spectral Luminosity Curves obtained by the Method of Critical Frequency,” Ives, Phil. Mag. Sept. 1912, p. 862, Phil. Mag. 8.6. Vol. 31. No. 184. April 1916. ¥ 306 Messrs. H. H. Ives and EH. F. Kingsbury on the used by us is that ours calls for a lack of symmetry about the 180-degree axis. This lack of symmetry appears in the experimental results, and, being a point of some importance, we have established it with some care by a series of obser- vations for the complementary openings 30 degrees and 330 degrees over a long range of illumination, as * shown in fig. 6. These observations were made alternately with each angle at each illumination with complementary opaque rotating disks. The next point to occupy our attention was the slope of the critical frequency-log I lines for various angular openings. Data obtained by us on several occasions are shown both in the figure just referred to and in fig. 7. While there are 20 \ (o} Cycles per Second. ~ 345, ; ; ge ° é 5 Zi q ' o ta FA6 N fo) Critical frequency-log I data for various white angular openings showing no systematic relation between angular opening and slope. (The different sets, because obtained under ditferent conditions, are to be com- pared only for slope, and not for relative magnitude of frequencies.) differences in the slope of certain of these lines, and while as well the region covered by experiment is too short to establish the presence or absence of some systematic difference of slope, it is clear that there are no such variations of slope Theory of the Flicker Photometer. 307 as are called for by our theory as it has been worked out here. The various straight lines are very nearly parallel. Discussion of this discrepancy between our theory and experiment will be reserved till the next section. The third point studied was the behaviour of the eye toward blue light at low intensities, for different sector openings. The blue light was obtained by the use of a monochromatic blue filter placed over the eyepiece, and the _ illumination was lowered until the field appeared no longer coloured, but grey. Observations were then made for three angles, 30, 180, and 330 degrees, for four illuminations. The resultant horizontal straight line plots of the data are shown in fig. 8. The approximate symmetry which holds Fig. 8. Cycles per Second Re Seana set og : ) rick Q 7 Critical frequency-log [ data for blue light at low intensities, for exposures of 30°, 180°, and 330°, at high illuminations has entirely disappeared. These data are plotted as well in fig. 4 along with the points prophesied from the theory. It is evident that the new phenomenon indicated is a reality, for the critical speed increases con- tinuously as the opening is decreased, instead of returning toward zero as it does for higher illuminations. (v) Discussion. The conflict between our theory and the data on the slope of the critical frequency-log I lines for various angles is perhaps due to the simplifying assumptions which we found necessary in order to handle the case at all. -We nave, in fact, used practically the case of a simple sine function superposed on a steady stimulus, and this, as we have already seen (Section 3), gives both from the theory and by experi- ment, lines of slope varying with the relative magnitude of ¥ 2 308 Messrs. H. E. Ives and H. F. Kingsbury on the the two illuminations. it is to be noted, also, that besides using only the first two terms of the complete expansion, we have made the simplest possible assumption as to the relation between the diffusivity and the amount of light entering the eye. We have as well assumed the Fechner fraction to be the same, no matter what the shape of the transmitted impression. It may very well be that in place of the constant 6 some function of the angular opening should be used. Whether these various possible explana- tions of this discrepancy are sufficient we have not been able to establish. It appears probable to us that, in the simplified form of the theory as worked out, differences actually existing in the slopes of these lines have been much exaggerated. This is a point which can be settled only by the complete solution of the equation from which @ is obtained. We suggest that this might be obtained experi- mentally by studying the conduction of heat through slabs of various thicknesses, the heat being applied periodically in a manner to imitate the intermittent exposures here studied. In spite of the apparent disagreement between theory and experiment on this one point, we think it may be claimed that the theory does give the principal facts in a sufficiently striking manner to warrant its being considered as sub- stantially correct. We have felt justified in using it in the remainder of this paper, with the sole caution of confining ourselves to illumination values at which its indications are close to the experimental data. With regard to the newly discovered behaviour of the eye at low illuminations toward various angular openings, we suggest that it would be an interesting experiment to make measurements with various openings on a totally colour-blind observer. If the behaviour of the eye toward blue light which has become colourless at low iiluminations is similar to that of the totally colour-blind eye at normal illumina- tions, as appears established from various researches, then it is to be expected that with such eyes critical frequency will (1) be independent of the illumination ; (2) depend upon the opening of the sector, and will decrease as the sector opening is increased. Von Kries and Uhthoff * have made critical frequency * “Ueber die Wahrnehmung des Flimmerns durch normale und dureh total farbenblinde Personen,” Von Kries, Zevt. fur Psych. und Phys. der Sinnesorgane, vol. xxxii. p. 118 (19053). Theory of the Flicker Photometer. 309 measurements on a totally colour-blind observer (Uhthoff), and found that the critical speed for him was far below that for a normal observer and seemed little affected by varying the illumination. Unfortunately their account is very meagre of exact data, and they appear to have discontinued the experiments after merely establishing that there is a difference between the two classes of observers. 5. Lhe Flicker Photometer with Unequal Exposures of the Compared Colours. In the design of the flicker photometers a question apt to require answer is, What effect, if any, is tobe expected from an unequal exposure of the two fields? An illustrative case is furnished by those flicker photometers which alternate the two fields by the oscillation of a lens or mirror. If the amplitude of the oscillation is small, one side of the field is exposed for a majority of the time to one colour, the other side for a majority of the time to the other colour. Again. if the mechanical centre of the alternating system does not coincide with the optical dividing line unequal exposures result. What will be the effect on the readings ? A qualitative answer to this question is furnished on the basis of our fundamental assumptions by consideration of fig.9. Here are represented two complementary dovetailed Fig. 9. Diagrammatic representation of behaviour of Flicker Photometer with unequal exposures of the compared colours. A and B represent the transmitted stimuli as given by two complementary openings. These have approximately the same fractional range. Their dovetailed sum- mation S, shows outstanding flicker. On increasing B to B’, the amplitude of fluctuation is increased not only in the proportion of B’ to B, but still more due to the increased diffusivity. The new summation S, is constant. 310 Messrs. H. E. Ives and EH. FY. Kingsbury on the impressions as transmitted to the receiving apparatus. Hach is due to the same illumination (by different colours) acting on complementary disks. The critical speed is nearly the same for each, consequently the fractional range of each is nearly the same. ‘This, however, means that the actual range is much less for the disk of small opening than for the disk of large opening. The summation of the two impressions is not therefore constant, but partakes more of the nature of the impression due to the larger opening. In order to make the summation constant it is obvious that the illumination of the smaller disk must be increased, whereby the range of fluctuation of the impression due to it will be made Jarger, and at the same time the diffusivity proper to its colour will be increased, which will assist to the same end. It follows from this reasoning that that colour of two under comparison which is exposed for the shorter time will be under- rated. An important point to notice is that this condition holds only if each colour under comparison preserves its indi- viduality to some extent in the process of transmission, as it would do if its transmission is due toa separate mechanism (e.g. set of nerves). It is only by this means that the two complementary impressions can be assigned different diftu- sivities and consequently different ranges. In the case of no colour difference the two impressions must be transmitted by the same channel and the diffusivity must be the same for both—that due to the mean illumination. When the illumination is the same on both sides of such an unsyim- metrical flicker photometer the two impressions will exactly dovetail, with a constant summation, which is obviously the case. This point has bearing on the question of the exact com- putation of the flicker photometer’ s behaviour with different colours. We cannot assume that eacn separately distinguish- able colour has its own channel. Three-colour vision theory would point to three such channels as probable. We must therefore expect that the diffusivity governing the trans- mission of each colour is, to a Greater or less extent, modified by the other colour under comparison. This means that such computation as can be made on the basis of each colour having its own diffusivity can be expected to be only qualitative. In the previous paper we have confined our- selves to such qualitative study of the effects of different colours, and a similar restriction holds with the following discussion of the effects of unequal exposures with different coloured lights. iy Theory of the Flicker Photometer. 311 As in the earlier paper, we may calculate the position of setting of the flicker photometer by equating the two ranges, as given by equation (10). Doing this we obtain the equation EF it) ee — 100 Va, log lot V ag log 1g(1—d) + by (assuming the substantial correctness of the relation used to connect diffusivity and illumination), where F, a, as, b,, 62 are constants. Inspection of this equation shows that, as was the case with different colours equally exposed, the first term on each side becomes relatively more important the higher the illu- mination, that is, the position of setting becomes more and more nearly the equality of brightness point. But in the present case the practical coincidence with the equality of brightness setting is pushed to very much higher illu- minations. We have taken numerical values for red and blue light from the second of the series of papers on the experimental work on the flicker photometer * and inserted them in the equation (14), with the result shown in fig. 10. It is evident Fic, 10; log I, — The Flicker Photometer with unequal exposures of the comparec lights. Relative “slit widths” as calculated for different relative ex- posures of red and blue light, for different illuminations, showing under- exposed colour to be under-rated. from these curves that, if each colour acted on entirely separate channels, unsymmetrical flicker photometers would read very differently from the symmetrical one, the difference amounting to from 20 to 50 per cent., even at high illu- minations. * “Spectral Luminosity Curves obtained by the Method of Critical Frequency,” Ives, Phil. Mag. Sept. 1912, p. 352. i (14 » , 312 Messrs. H. HE. Ives and H. F. Kingsbury on the We have confined our verification of the theory to a careful series of unsymmetrical flicker-photometer measure- ments on green versus “‘ white” light, the green (furnished by a pot-green glass over the “ white’ light) being exposed for 330 degrees, the “white” for 30 degrees. We first made a measurement with two 180-degree exposures with no colour difference, then with the green glass in place, thus determining the transmission of the green glass. The working illumination was maintained constant, and the measurement with no colour difference was made in order to eliminate any errors due to lack of symmetry in the surface of the rotating disk—a matter which will be dwelt on at length in the next section. A 330-degree white disk was then put in place of the 180-degree disk which formed the rotating part of the symmetrical arrangement and exactly the same procedure gone through, giving a new value for the transmission of the green glass. Our results were as follows : | HE Transmission with 330 degrees green exposure _ ‘~~ Transmission with 180 degrees green exposure oR do. = 1°034 1:053 These figures, showing an over-rating of the longest exposed light, are in agreement with the theory. The amount of this over-rating is less than would be calculated on the assumption of separate channels, but this, as we have shown, is to be expected. | There remains to be considered the question of the sensi- bility of the unsymmetrical flicker photometer. This we may study in exactly the same manner that we have studied the symmetrical case—by finding and plotting the critical speeds for various relative intensities of illumination from the two compared colours. For this purpose we use the equation es 9 5 1 2 it Dog ee naa XCogep obtained by combining the range equations for the two exposures. In fig. 11 are shown curves giving the calcu- lated values for the critical speed where one illumination is held constant, the other varied, the constant illumination being exposed for 180 degrees, 30 degrees, and 330 degrees, { 2 +log sin rf —log 3 | » (15) Theory of the Flicker Photometer. oils as indicated on the respective curves. All values are calcu- lated from the experimental value of 35 cycles per second for the 180-degree case, using the value — 3 for log 6. These curves show clearly one point which appears at once on experiment, namely, the lower sensibility of the unsym- metrical instrument. There is, too, a somewhat greater lack of symmetry about the equality point than with the equal exposure arrangement. Fig. 11 exhibits as well in convenient form the phenomena Fig. 11. 40 s c S Al Vv e € a | w 2 Vv ° < | > QL 4 = DAE dé “| Zo S|: We i i | {7 ees § =|:00/ 42 4 16 tS 2- ie) Be) 1:0 to 10 Calculated sensibility characteristics of Flicker Photometer with unequal exposures of the compared lights. One fixed, one variable illumination. Variable illumination exposed 380°, 180°, 330°. visible in a flicker photometer in which the alternation of the two lights is performed by an oscillating lens or mirror whose excursions are too short. While the centre of the field of such an instrument is equally exposed to the two lights, one side is exposed longer to one light than the other, 314 Messrs. H. EH. Ives and E. F. Kingsbury on the and vice versa. The three curves of the figure then represent the centre and two sides of such a field. At any given speed flicker disappears, as the illumination from the varying light is increased, first on the large-angle side, then on the small-angle side, then in the centre. Effects of this sort have been noticed by experimenters. 6. Some Liffects of Accidental Dissymmetry on the Critical Frequency and flicker Photometer Phenomena. Insistence has been made in previous papers on the im- portance in the flicker photometer of avoiding all mechanical flicker, such as that from black dividing lines between the alternated fields. One of the most important experimental results of the present work, and one largely unexpected, has been to show in definite quantitative form the extreme seriousness of any sort of lack of symmetry in the flicker photometer. In fact, we believe that many of the contra- dictory results obtained by experimenters are to be traced back to negligence in the important precautions of securing the greatest possible freedom from mechanical imper- fections and in using a strictly substitution method of observation. The manner in which these effects were discovered offers the best way of presenting them. It was our original intention to test the symmetry of the curve connecting angular opening with critical frequency by alternating observations, first on the illuminated rotating disk and then on the illuminated surface F (fig. 5) as interrupted by the same disk. By first equating the brightness of the disk and white surface through the use of the disk edge to form an equality photometer, it was thought that ‘this alternate measurement with one ight and then ‘the other would furnish the best possible test of the symmetry ot the ettects of com- plementary disks. The result of our first series of measurements so made, with a disk of 30 degrees opening, is shown in fig. 12. While the points obtained using the disk merely as an opaque interrupter plot in the normal straight line form, those obtained with the whitened disk fall on a well-marked curve, partly above and partly below the straight line. The physical cause of this peculiar difference was at once found to le in the imperfect smoking of the disk with mag- nesium oxide. ” varying with the relative exposure. 4, When a flickering illumination is superposed on a steady one, the critical frequency is connected with the mean illumination by the same logarithmic relation as that just given, but the constant ‘‘a” varies with the ratio of flicker- ing to steady illumination. 3 (6) Behaviour of the flicker photometer. 5. The high sensibility of the flicker photometer is shown to be due to the very rapid increase in the critical frequency of disappearance of flicker on each side of the equality setting. 6. In a flicker photometer which exposes the compared colours for unequal periods, the less exposed colour will be Theory of the Flicker Photometer. 321 under-rated. This form of flicker photometer is less sensitive than the ordinary equal exposure arrangement. 7. Mechanical imperfections in the flicker-photometer field are shown to seriously shift the equality point even with no colour difference, emphasizing the necessity for strictly substitution methods in flicker photometry. Theoretical. The behaviour of the visual apparatus toward intermittent light is closely parallel to the action of a layer of matter, obeying the Fourier conduction law, in which the diffusivity varies as the logarithm of the intensity of the illumination, this layer being exposed to the intermittent light on one side, while its condition on the other side is measured b an instrument whose sensibility is governed by the Weber- Fechner law. Or, given the experimental data on the behaviour of the eye in perceiving flicker with equal dark and light sectors for different colours at various intensities, it is possible from the most general law of conduction, on the basis of the simple assumptions here made, to predict with con- siderable accuracy the phenomena occurring with disks of varying openings, at both high and low illuminations, the phenomena with superposed flickering and steady illumina- tions, the sensibility characteristics of the flicker photo- meter, the behaviour of the flicker photometer toward different colours at various illuminations, the occurrence of colour flicker and brightness flicker, the effect of unequal exposures of the coloured lights under comparison, and the disturbing effect of mechanical defects in the photometric field. The phenomena explained or predicted by the theory constitute, in fact, all the known experimental facts in con- nexion with the flicker photometer. Where the corre- spondence between theory and experimental fact is more qualitative than quantitative, we have advanced reasons for believing these differences to be due to the approximate nature of the solutions given of the mathematical work, or the insufficiency of our experimental knowledge of certain factors. Physical Laboratory, The United Gas Improvement Company, Philadelphia, Pa., Oct. 1915. Phil. Mag. 8. 6. Vol. 31. No. 184. April 1916. Z ee! XXXVII. Theory and Experiments relating to the Establish- ment of Turbulent Flow in Pipes and Channels. By Louis Vessot Kine, Jf. A.( Cantab.), D.Sc.( McGill), As- sociate Professor of Physics, McGill University, Montreal*. INDEX TO SECTIONS. Part I.—THr MATHEMATICAL THEORY OF VISCOSITY AND ITS EXPERIMENTAL VERIFICATION. (1) Historical Survey of the Theoretical Development. (2) Steady Motion between Parallel Planes. (3) Steady Motion in a Pipe of Circular Cross-Section. (4) Shearing Motion between Parallel! Planes. Part Il.—On THE STABILITY OF LAMINAR FLOW IN PIPES AND CHANNELS. (5) Historical and Critical Survey of the Theoretical Development. (6) Discussion of Experimental Results in the Flow of Fluids. (i.) Liquids. (ii.) Gases. Part J. THe MATHEMATICAL THEORY OF VISCOSITY AND ITS EXPERIMENTAL VERIFICATION. Section 1. Historical Survey of the Theoretical Development. S the fundamental problem of fluid-resistance in both laminar and turbulent flow is intimately connected with the mechanism of shearing stresses across adjacent layers of a fluid in relative motion, it may not be out of place in the present paper to undertake a critical survey of the theory of viscosity and of its experimental verification. The fundamental hypothesis of viscosity asserts that when layers of a viscous fluid are in motion relatively to each other, the mutual tangential stress per unit area is proportional to their relative velocity divided by the distance between them. To be more precise, suppose the fluid to be flowing in layers parallel to the plane wy with velocity U in the direction of the x-axis. If the gradient of velocity in the direction of the z-axis be dU/dz, the shearing stress per unit area of the plane xy in the direction of flow is given by F=p—, Ee uw being a constant depending only on the physical properties of the fluid and called the “ coefficient of viscosity.” The insertion of the fundamental hypothesis (1) in the * Communicated by the Author. Turbulent Flow in Pipes and Channels. 323 general scheme of hydrodynamical equations seems to have first been effected by Navier? and Poisson’, the justification for this procedure resting on theoretical considerations as to the mutual interactions of the ultimate molecules of fluid. Since dU/dz represents the rate at which shearing strain is produced by the shearing stress, the hypothesis (1) may be expressed in the language of elastic solid theory by saying that the ratio of the shearing stress to the rate at which shearing strain is produced is equal to yp, the coefficient of viscosity. From this point of view the familiar equations of viscous-fluid theory were developed by Saint-Venant ® and Stokes*; from this mode of presentation it appears that (1) is the simplest hypothesis consistent with the linearity of the general equations. The experimental justification of the accuracy with which the fundamental hypothesis is able to give an interpretation of reality rests on the comparison of a comparatively small number of simple experimental conditions of viscous flow with the theoretical solutions. Some of these will now be considered in the following sections ?. Section 2. Steady Motion between Parallel Planes. The origin is taken half way between the planes and the velocity U is measured along the axis of « parallel to the planes, while the axis of y is taken perpendicular to the planes. If Op/dw is the pressure-gradient along the axis of a, the equation of steady motion is © Oop 9 ane San ee (2) As long as the flow is laminar there is no component of velocity parallel to the y-axis, and Q@p/dw is an absolute constant given by Op/dw= (Po—py)/l, - 6 . . e (3) where p.—j}; 1s the pressure-difference between two points at a distance / apart. Assuming that the coefficient uw is an absolute constant depending only on the properties of the ' Navier, MWém. de Acad. des Sccences, t. vi. p. 389 (1822). > Poisson, Journ. de Ecole Polytechn. t. xiii. p. 1 (1829), 3 Saint-Venant, Comptes Rendus, t. xvii. p. 1240 (1848). + Stokes, Trans. Camb. Phil. Soc. vol. viii. p. 287 (1845); Math. and Phys. Papers, vol. 1. p..75. * A valuable historical and critical account of viscosity is given by Brillouin (Marcel), ‘ Legons sar la viscosité des liquides et des 2072,’ Gauthier- Villars, Paris, 1907. ® Lamb, ‘ Hydrodynamies,’ p. 542 (1906). Zi2 324 Prof. L. Vessot King on fluid and not on the relative velocity of adjacent layers, the general solution of (2) is U=A + By-+ (l/2p). Op/02).y > aes A and B being constants of integration. Experimental evidence’ is consistent with the condition U=O at the boundary y=-+0; under these conditions U=—(1/2n) @—y?). (Op/d2), - . . (5) so that the velocity at the centre Up is given by Uo=- (Ce). @:—poft a The total flow per unit breadth is given by the expressions wh eo yu 2 OF OP tne 0: aU Rb sacs ; 40) a Udz= 2 ude) 3 ual) (7) and the mean velocity is thus given by U=—10%/4).@p/de)=2U,. This simple case of steady motion has been made the starting point of some of the various theoretical treatments of the stability of laminar motion briefly reviewed in Part II. As far as the writer is aware, no experiments have been carried out with a disposition of apparatus corresponding to this solution. As the linear hot-wire anemometer developed by the writer and described in detail in a previous paper ® is especially suited to the study of this distribution of velocities, the present solution was made the basis of a detailed experimental investigation of gaseous viscosity by studying not only the variation of the total flow with pressure-gradient, lengths and breadths of channels, &e., but by an actual examination of the gradients of velocities themselves. The high resolving-power and sensitiveness to low velocities of the linear hot-wire anemometer make this instrument exceptionally useful for this type of work. The results obtained by this means (which will be fully illustrated 7 See Lamb, ‘ Hydrodynamics,’ p. 544. In the case of gases, recent experimental work by Knudsen (‘ La Theorie du Rayonnement et les Quanta,’ Gauthier-Villars, Paris, 1912, p. 183 e¢ seg.) and by Dunoyer (‘Les Idées Modernes sur la Constitution de la Matiere,’ Gauthier- Villars, Paris, 1913, p. 215 e¢ seq.) on ultra-rarefied gases indicates that molecules actually embed themselves in the material of the boundary, to return into the gas at some later time with a velocity whose direction is entirely independent of the previous collision with the walls. 8 King, L. V., “On the Precision Measurement of Air Velocity by means of the Linear Hot-Wire Anemometer,” Phil. Mag. vol. xxix. April 1915, pp. 556-572. Turbulent Flow in Pipes and Channels. 325 and discussed in Part III.) indicate the somewhat surprising result that the parabolic distribution calculated from the usual viscous fluid theory is only a rough approximation to a much more complicated state of affairs. Section 3. Steady Motion in a Pipe of Circular Cross-Section. Taking the axis of < to coincide with that of the circular cylinder, the pressure-gradient in this direction is constant and is given by OP/O2 = (ame Pi by. ar ee) hee P2—pi being the pressure-difference measured at points a distance / apart. The equation for steady motion is ® WwW 2 (ur) == r( po) ee of which the general solution is W=—??(po—p,)/(4ul) +Alogr+B. . (11) Under conditions of finite velocity along the axis r=0 and no slipping at the boundary r=a, we have W = (a*—1") (po—71)/(4e1). . . ~ C2) The maximum velocity at r= 0 is given by W g=a7(ps— 1) /pdyy se 2 aS) and the total flow is o= | W . 2arr dr=tra*( p.—p;)/(8ul)=s7a? Wo, (14) we 0 and the mean velocity over the cross-section is Ui Sone Ahlen th, ened nck) This solution forms the basis of most of the practical methods of determining the coefficients of viscosity of a liquid or of a gas; it is also the starting point of some of the theoretical treatments on the stability of stream-line flow, and, until the writer’s own experiments on the subject, all observations on the laws of flow were confined to the case of pipes of circular cross-section. It is generally assumed that the theory of viscosity obtains its application to reality in that equation (14) contains all the laws found experi- mentally by Poiseuille’®; although these laws have been ° Lamb, ‘ Hydrodynamics,’ 1906, p. 543. 1° Poiseuille, Comptes Rendus, vol. xv. p. 1167 (1842), 326 Prof. L. Vessot King on amply verified in the case of tubes of capillary diameters, a marked deviation is easily noticeable as soon as the diameters exceed a few millimetres, even at velocities considerably below what is taken to represent the “ critical velocity.” Section 4. Shearing Motion between Parallel Planes. Measuring y perpendicular to the moving planes at a distance d apart, the equation for steady motion is d?U/dy’=0, of which the solution appropriate to U=0 at y=0, and U=U, at y=d is U = Uoy/d, «| 9th) Gari eae (16) which, it will be noticed, does not involve the viscosity. The shearing stress between the planes per unit area is F=p(dU/dy)=pUiid. .. eee In this case experiment measures the shearing stress F ; it may be remarked here that in these circumstances the fundamental law of viscosity relates to the traction between the fluid and a solid boundary and not to the traction between layers of fluid, as would be revealed by an experi- mental analysis of the velocity-gradients themselves. The simple case just considered is not realizable experi- mentally, although it is made the starting point of one of the important cases of laminar flow examined theoreticaily from the standpoint of stability. The appropriate experi- mental arrangements involve the measurement of the tractions between circular disks, as in Maxwell’s classical experiments, or between concentric spheres or cylinders; observations are generally carried out by oscillation methods, although an apparatus has recently been constructed by Gilchrist 2 which enables the torque due to viscous shear between relatively rotating coaxial cylinders to be measured. Taking the case of a cylinder of radius a, rotating with angular velocity wy inside a coaxial cylinder of radius b, the angular velocity in the fluid at radius 7 is given by ” or=(a/r). (b?—71?)/(P?—a?) .@a, . . (18) while the couple exerted between the cylinders is, per unit length, L= —4rrpa’a, . b?/(b?—a*). . . . (19) Gilchrist points out, in a review of the best experimental data available at the date of his paper, that small discrepancies - ne Gilchrist, Phys. Rev. vol. vii. p. 124 (1918). 22 Lean, Hy drodynamics,’ 1906, p. 546. Turbulent Flow in Pipes and Channels. 327 of about 0°5 of one per cent. exist in determinations of the coetficient of viscosity by the various methods employed, and that these cannot be attributed to errors of experiment. As has already been pointed out, measurements of the viscosity of gases carried out by these methods depend on the shear between a solid surface and a gas, the laws for which may not be identical in all circumstances with those relating to tractions between successive layers of gas. Moreover, all these methods entail very low relative velocities and rates of shear, while any slight departures from the theoretically specified motion have generally been attributed to the breakdown of viscous stream-line flow to “turbulent flow” without further examination. Any direct measure- ments of velocity-gradients do not appear to have been carried out, although this is now possible owing to the development of the linear hot-wire anemometer described in a previous paper by the writer. Any factor other than the viscosity resulting from the free-path transfer of momentum might easily have escaped observation in the type of experi- ments referred to, as the inertia of the moving surfaces employed would tend to smooth out any extraneous irregu- larities in viscous tractions which might exist. The writer hopes, at some future date, to undertake the analysis of velocity-gradients in these cases by means of the hot-wire anemometer; it may well happen, as in the case of flow between parallel planes, that factors other than the viscosity due to free-path phenomena may play a part in determining the velocity-gradients and tractions referred to. Part II. ON THE STABILITY OF LAMINAR FLOW IN PIPES AND CHANNELS. Section 5. Historical and Critical Survey of the Theoretical Development. The origin of the theoretical work on the subject of the present chapter dates back to Helmholtz’s remark that surfaces of discontinuity in perfect fluids are unstable”. The subject was treated mathematically at an early date by Rayleigh, especially in connexion with the stability of jets and the explanation of phenomena relating to sensitive 13 Helmholtz, Phil. Mag. Nov. 1868; Gesammelte Abhandlungen, i, p. 146 (1882-3). 328 Prof. L. Vessot King on flames. Osborne Reynolds’s experimental work on the subject in 1883 marks an important point in the history of the subject !®. By studying the nature of the flow by means of colour-bands, it was shown that ata certain “ critical ” mean velocity the stream-like flow broke up into violently eddying motion; it was also shown from the theory of dimensions that the criterion of breakdown should depend on the ratio DUp/u only, D being the diameter of the tube, U the mean velocity of the fluid, p its density, and yp its coefticient of viscosity, all these quantities being measured in C.G.S. units. The passage from laminar to turbulent flow was indicated both by the colour-band method, as well as by a change in the law of resistance as observed from the discontinuity in the curve connecting the total flow and the pressure-gradient, and was shown to take place for a value of U given by the relation DUp/a=K,, 542 oe which will be referred to as “‘Reynolds’s Criterion,” while the number K will be referred to as “‘Reynolds’s Constant.” In the case of water Reynolds found, both from his colour- band method as well as from a study of his own experiments and those of Poiseuille’® and Darcey!’ on the laws of fluid resistance in pipes, that K had a value in the neighbourhood of K=1900 to 2000. According to theory K should have the same value for all viscous incompressible fluids. Several series of observations, carried out since that date by various observers on liquids and gases, when examined by the same methods as those employed by Reynolds, agree in assigning fairly consistent values to the constant K; these are discussed in further detail in the next section. Revnolds’s observations gave rise to a series of theoretical contributions, among the first of which were papers due to Kelvin’® and Rayleigh**. Kelvin examined the types of flow discussed in Sections 3 and 4 from the point of view of stability, and came to the conclusion “ that the steady motion is wholly stable for infinitesimal disturbances, whatever may be the value of the viscosity (w) ; but that when the disturbances are finite the limits of stability 14 Rayleigh, Various papers between 1879 and 1914; Collected Works, Arts. 58, 66, 144, 194, 216, 217; also Phil. Mag. vol. xxvui. p. 609, Oct. 1914. 15 Reynolds, O., Phil. Trans. vol. elxxiv. p. 9535 (1833); Collected Works, vol. ii. p. 51. 16 Poiseuille, Comptes Rendus, vols. xi. & xii. (1840-1). 1T Darcy, Comptes Rendus, vol. xxxvili. 18 Kelvin, Phil. Mag. vol. xxiv. pp. 188, 469, 529 (1887). Turbulent Flow in Pipes and Channels. 329 become narrower as p diminishes.”’ These views have been criticized by both Rayleigh 79 and Orr”. Rayleigh’s earliest papers on the subject deal with the laminar steady motion of a perfect fluid between parallel planes in which the distribution of velocity is continuous, while the rotation or vorticity changes suddenly on passing from one layer of finite thickness to the next. Rayleigh's general conclusion is thus stated: “The steady motion of a non-viscous fluid in two dimensions between fixed parallel walls is stable provided that the velocity U everywhere parallel te the walls is such that d?U/dy? is of one size throughout, y being the ordinate measured perpendicularly to the walls. It is here assumed that the disturbance is infinitesimal.” The extension of this ideal problem to include the case of an actual fluid possessing viscosity brings us at once to the main (theoretical) difficulty of the subject, and one which cannot. yet be regarded as (theoretically) settled. The subject was proposed for discussion in the Adams Prize Essay of 1887, and may be considered to constitute the simplest case of fluid resistance. Many writers, including Reynolds himself ?!, have since that date contributed to the theoretical aspect of the subject, and have succeeded by various methods in establishing theoretically determined values of Reynolds’s Constant K, for cases of flow which not only include that between parallel planes but also the more difficult one of the circular tube. These are reviewed in an important memoir by Orr”, who himself gives an original treatment of the subject. An authoritative account of the present state of the problem, from which the writer has quoted the above passages, has recently been published by Rayleigh”. It is therefore unnecessary to deal at greater length with this aspect of the subject in the present paper, beyond stating that the most recent theoretical treat- ment of the subject “by Hopf”, following a method due to 19 Rayleigh, Phil. Mag. vol. xxiv. pp. 59-70 (1892); Collected Works, vol. iil. p. 575. Ore Proc. Roy. Irish Acad. vol. xxvii. pp. 9-188 (1907). This important memoir gives a critical account of the theoretical work on the sudject to the year 1907 in great detail. 21 Reynolds, Phil. Trans. vol. clxxxvi. a, p. 123 (1894): Scientific Papers, 11. p. 535. *2 Rayleigh, “On the Stability of Viscous Fluid Motion,” Phil. Mag. Oct. 1914; see also Phil. Mag. Sept. 1915. a8 Hopf, Ann. ad. Phys. xliv. no. 9, pp. 1-60 (1914). See also a recent discussion of the subject by Taylor, R.1., Phil. Trans. Roy. Soe. vol. ecxy. A. pp. 28-26 (1915). 330 Prof. L. Vessot King on Sommerfeld *4, tends to confirm the original conclusions of Kelvin and Rayleigh as to the stability of a state of steady motion. The unsatisfactory state of knowledge at the present time en the subject is clearly shown from the Table given at the end of this paper, from which it is seen that the theoretical estimates of Reynolds’s Constant K differ very markedly from each other, and very greatly from the generally accepted values obtained for liquids and gases. It may be noted that all experimental data refer to tubes of circular cross-section, whereas the simplest theoretical case refers to the flow of fluid between parallel planes ; in the latter case observations have not hitherto been availabie. The deficiency is now made up by the writer’s own experl- ments on the flow of air to be described in detail in Part III.; the results indicate the existence of hitherto unsuspected conditions of laminar flow, both at extremely low velocities as well as at velocities considerably higher than those which are taken to represent, according to the usual interpretation, critical velocities. By means of the linear hot-wire anemometer developed by the writer’, the velocity-distribution over the cross- section of a two-dimensional channel (section 0°45 em. x 5°08 cm.) was measured for various pressure-differences. Owing to the high resolving power of the instrument, it was found possible to measure velocities at intervals of 0°05 mm. over the cross-section, and in this way a velocity-distribution curve representing as many as 100 observations could be obtained, the pressure-difference under which the flow took place being maintained constant to 1/10 of one per cent. Even at velocities considerably below the “ critical velocity,” as usually defined, the distribution curves showed consistent and interesting deviations from the parabolic form demanded by theory. (i.) The experimental curves developed weil - marked “humps” in the neighbourhood of planes midway between those dividing the distance between the walls of the channel in equidistant parts; these appeared at very low velocities (maximum velocity 20 em. per sec.), and, with increasing velocities, the number of “humps” increased, as many as nine being easily discernible in the curve corresponding to a maximum 1 velocity of 650 cm. per second. (ii.) Although the general shape of these distribution curves is parabolic, the maximum falls very much below that demanded by the theoretical viscous-flow theory, as 21 Sommerfeld, Proc. Int. Congress of Mathematicians, Rome, 1908. Turbulent Flow in Pipes and Channels. Sai though some factor contributed to increase very materially the ordinary kinetic-theory coefficient of viscosity. As the peculiarity (i.) would seem to associate the pheno- menon with transverse stationary sound-wayes (in the particular instance quoted the pitch of the gravest mode would be much above audition), it would seem not unnatural to attribute (1i.) to the same cause. Making use of a conception which appears to have been first put forward by Brillouin * in another connexion, the “rationale” of the phenomenon would appear to be as follows :—When flow of a compressible fluid takes place under conditions which permit of the establishment of stationary modes of sound- vibrations (as in flow in non-capillary tubes, channels, &c.), the wave-front is refracted by the velocity-gradients which exist in the medium. As rate of change of momentum is propagated along a sound-ray, it follows that, under these conditions, the existence of compressional vibrations will result in a component of rate of change of momentum pro- portional to the velocity-gradient being transferred across an element of surface parallel to the direction of flow. Hence we must write for the viscosity, w=po+ms, where fo is the ordinary viscosity of the liactio. theory, and p, is a “quasi-viscosity” due to the existence of compressional waves and is proportional to the intensity of the wave-motion. The theoretical velocity-gradients caleulated on this theory give a very satisfactory socscee of the experimental results : the full details of the theory and the comparison with experimental results will be given in Part III. of the present paper. It thus appears that the gradual modification with increasing velocities of the laminar flow in channels through the interaction of transverse sound-vibrations with viscous tractions is an essential factor in ultimately leading to the breakdown of stream-line motion. It appears to the writer that in the combination of compressional vibrations and shearing-motion lies the origin of a vortex rotation which, as the amplitudes of the sound-vibrations and their frequencies increase, may ultimately result in the formation of jinite vortex filaments, resulting finally in “turbulent motion” in the generally accepted sense of the word. As the factor of compressibility, hitherto neglected in theoretical treatments of the stability of laminar flow by Reynolds, Rayleigh, Kelvin, Sharpe, Orr, and_ others, appears to be of considerable importance in the light of the °° Brillouin, L., “ Conductibilité calorifique et viscosité des liquides monoatomiques,” Comptes Rendus, vol. clix. pp. 27-30, July 6th, 1914. 332 Prof. L. Vessot King on experimental results briefly described above, it is hardly necessary to consider in further detail the various theoretical treatments referred to. While the condition of incom- pressibility assumed in these discussions might possibly be justified in the case of liquids (and some evidence seems to point to compressibility as a determining factor affecting. stability in this case as well) **, it is hardly to be supposed that this factor could be entirely ignored in the case of gases. In the light of the preceding remarks it appears that still another factor may playa part in determining the instability of flow in tubes and channels, that is, ihe determinateness of the various transverse modes of sound-vibrations which may be set up. In a two-dimensional channel (or channel of elongated cross-section) the nodes and loops are extremely determinate in position, and we should expect laminar flow to persist to very high velocities. In the case of a tube of circular cross-section, however, some of the normal trans- verse modes (e. g. those having diameters as nodes) are andeterminate in position, and in an actual case would be determined by accidental inequalities on the interior of the tube or by slight departures from circular form. It would thus appear that in such cases (which include practically all experimental data) Reynolds’s Criterion of ‘ sinuous ” motion would be decided, not by instability of steady flow in the usual sense, but by the appearance of the first transverse mode having a diametral node. Relatively to the fluid, these indeterminate modes would tend to twist around, resulting in a state of affairs which, if continued long enough would, in spite of the stabilizing effect of viscosity, result in a complete destruction of conditions of steady flow. Many phenomena relating to the “critical velocity’? are thus capable of explanation; for instance, the “flashing” first observed by Reynolds?’, making use of the colour-band method, 2. ¢. a state of “sinuous” motion over a short length of tube followed by a region of stream-line flow. The effect of the material of the tubes on the critical velocity observed by several experimenters is probably due, for the most part, to the degree of mechanical finish of the interior surface obtainable with the particular substance employed’. In the ° See footnote (36). Reynolds, footnote (15), fig. 16. According to the ideas developed in the preceding paragraph, it would seem that the acoustic properties of the walls of the tube or channel may not be without influence on the velocity-gradients in the immediate neighbourhood of the boundary. wr —! a Turbulent Flow in Pipes and Channels. Se" flow in tubes of large diameter, the velocity-distribution over a particular cross-section is nearly always unsymmetrical with respect to the axis. Then again, if the curves” connecting the total flow with pressure-gradient be carefully examined, it will be noticed that slight discontinuous changes of slope can be detected some time before the “critical flow” is reached; the effect in this case is probably due to. sudden changes of velocity-distributions resulting from a number of symmetrical transverse modes being brought into. existence before the first indeterminate mode leading to a complete breakdown of stream-line flow. If this is the. correct interpretation of Reynolds’s condition of “ sinuous flow,” it would seem capable of mathematical formulation leading toa more satisfactory explanation of the experimental! constant K. Such an investigation would hardly be profit- able in the present state of experimental knowledge until! the much simpler and more determinate cases of two-dimen- sional flow have been thoroughly worked out, both experi- mentally and theoretically. Section 6. Discussion of Haperimental Results on the Flow of Viscous Fluids. (i.) Liguids—Reynolds’s original experiments on_ the. subject have already been reviewed in the preceding section, and a possible explanation of several features of the pheno-. menon discussed. Further experiments along the same lines by Barnes and Coker *° give additional support to the view that the passage from stream-line to turbulent flow depends. on the determinateness of the transverse modes of the compressional vibrations rather than on the instability of steady viscous flow, as generally understood in theoretical investigations on the subject. The observers just mentioned, by exercising extreme care in preventing the formation of eddies in the tank previous to the entrance of the liquid into. the flow-tube, were able to obtain stream-line motion for velocities as much as four times as high as that given by Reynolds’s Criterion. In some cases, after passing through a state of turbulent flow, stream-lines reformed at higher velocities ; in such cases a sharp rap on the tube sufficed to. break down the motion, which immediately became stable °° Such, for instance, as the curves for air obtained by Kohlrausch, K. W. F., Annalen der Physik, xliv. p. 297 (1914); and more especially those obtained by Sorkau tor water and various organic liquids. (Foot. note 36.) © Barnes, H. T., & Coker, F. G., Roy. Soe, Proc. vol. Ixxiv. p. 841 (1904). oa4 Prof. L. Vessot King on when the rapping ceased; the phenomenon of reformation of stream-lines at higher velocities was observed both by the colour-band method and by a special thermal method. It was also noted, in passing, that the amount of colour employed had a marked effect on the velocity at which the colour-band broke up. According to the views expressed in the preceding section, the reformation of stream-lines at the higher velocities would correspond to the excitation of symmetrical transverse modes of higher frequency than the indeterminate mode giving rise to the first appearance of turbulent flow **. Among the more accurate observations carried out for the distribution of velocity over the cross-section of a circular pipe through which water was flowing, may be mentioned the experiments of Morrow*. Measurements were carried out by means of a Pitot tube, and it was found that the distribution-curves changed gradually from approximate paraboloidal distributions to those typical of velocities above the ‘critical velocity.” The scope of experiments of this type was extended by Stanton **, and recently in an exhaustive memoir by Stanton and Pannell**. The distribution of velocities over the cross- sections of a number of circular pipes was accurately determined for air- and water-flow above the “critical velocity’? in order to obtain evidence over as wide a range of density and viscosity as possible as to the accuracy of Rayleigh’s Principle of Dynamical Similarity to cases involving surface-friction between solid and liquid surfaces. Tf R be the resistance per unit area, v the kinematic viscositv of the fluid v=yp/p), and pits density, Rayleigh’s Principle * gives in its application to the particular case considered, R=pU?F(UD/)),. «0.0. where D is the diameter of the pipe, U the mean velocity 31 A close examination of Reynolds's curves (Scientific Papers, vol. i. diagram i, p. 90) connecting velocity and pressure-gradient indicates the existence of discontinuities at higher velocities than that taken as the critical velocity. 32 Morrow, Roy. Soc. Proce. vol. Ixxvi. a, p. 205 (1905). 33 Stauton, T. E., Roy. Soc. Proc. vol. Ixxxv. a, p. 366 (1911). 34 Stanton & Pannell, “ Similarity of Motion in Relation to the Sur‘ace Friction of Fluids,” Roy. Soc. Trans. vol. cexiv. a, pp. 199-224 (1914). See also a discussion of these results by Lees (Roy. Soc. Proce. vol. xci. a, p. 46, Nov. 1914), who derives from these observations accurate empirical formule based on the Principle of Dynamical Similarity, expressing the relations between mean velocity and pressure-gradients for the flow of water and air in pipes of circular cross-section. 35 Rayleigh, see footnote (19). Turbulent Flow in Pipes and Channels. 339 over the cross-section, and F some function of the variable UD/v. This principle was fully confirmed in its application to pipes of widely different diameters for fluids differing in density and viscosity to the extent represented by air and water. The value of UD/y corresponding to “critical” changes in the character of the motion is not very sharply defined, but leads to a value of Reynolds’s Constant in the neighbourhood of _ K=2500. It does not appear to the writer that the theory of the mode of breakdown from stream-line to turbulent motion, suggested in the previous section as depending on transverse modes of compressional vibrations, is in contradiction with the principles just discussed, except that the form of the function F(UDJ»), and especially the value of UD/v corresponding to a change in the type of motion, would depend on the form of the cross- section of the tube, but would probably remain the same for geometrically similar shapes as in the particular case covered by existing experiments on circular cross-sections. For this reason it would seem highly desirable to obtain the corresponding data on two-dimensional channels or cross- sections having an elongated elliptical form ; it would be expected that the appearance of the ‘critical’ flow would occur at a higher velocity than that given by the usual Reynolds’s Criterion, owing to the high degree of determin- ateness of the transverse modes. Mention should be made of numerous experiments by Sorkau * on the flow of water and of various organic liquids through a short capillary tube (diameter 0-423 mm.: length about 5cm.). Observations were made of total flow against pressure ; although, as has already been pointed out, such observations are not suitable for revealing discontinuities of flow unless they are well marked, the experiments of Sorkau indicate the existence of three distinct régimes of turbulent flow occurring at velocities considerably less than that given by Reynolds’s Criterion. Quoting from the last-mentioned reference given below, the value of K for water varies from tt a 4 CC. to 410°5 at 25° CL "Above 25° 6.’ the three turbulent régimes are well-marked. Similar results were obtained in the case of various pure organic liquids. (ii.) Gases.—Accurate observations on the flow of gases with reference to a study of the turbulent régime have only comparatively recently been carried out; in such cases the 36 Sorkau, W., Phys. Zeit. xii. pp. 582-595 (1911); xiii. p. 805 (1912); xiv. pp. 147 et seg., 759-766, 828-831 (1918); xv. pp. 582-587, 768-772 (1914) ; xvi. pp. 97-102 (1915). 336 Prof. L. Vessot King on curve connecting the pressure-gradient and the total flow is one of nearly continuous curvature, so that a “ critical velocity”? in the usual sense is not sharply defined unless the velocities and pressure-gradients are large and the diameters of the tubes small. An elaborate series of experi- ments under these conditions was carried out by Ruckes *7, who employed capillary tubes of internal diameters between O-L and 2°2 mm. at pressures from a fraction of an atmo- sphere to about 50 atmospheres; Reynolds’s Constant, as determined from the curves connecting total flow and pressure-gradient, had values in the neighbourhood of K=2000 ; in several cases stability continued until velocities corresponding to values of K between 3000 and 8000 were reached, resembling in this respect the observations of Barnes and Coker*®® already discussed. For metal capillaries K had values as low as 400 and 500, a somewhat surprising result, probably due, as has already been mentioned, to the degree of mechanical finish of the interior of the capillaries. Experiments by Grindley and Gibson ** on the flow of air through a long lead pipe (108°2 feet long and diameter 0°125 inch) gave a value K=2200. Among the earliest observations on the distribution of velocity over the cross-section of tubes may be mentioned those of Becker**, the velocity-measuring device involving the aerodynamic resistance of small aluminium spheres. Experiments by Fry and Tyndall“ were carried out by the use of Pitot tubes, and very sensitive manometers specially designed for the purpose ; the curves of velocity- distribution differed markedly from the paraboloidal distri- bution required by the viscous theory, and, as in Morrow’s observations *2, the transition to distributions above the critical velocity was gradual; the change took place most rapidly in the neighbourhood of a velocity corresponding to a value of Reynolds’s Constant K =2500. Very elaborate observations along these lines have recently been published by Kohlrausch *! and are worth careful study. Five glass tubes were employed of diameters between 0°76 and 3°6 cm. Several series of curves connecting the total flow and the pressure-gradient are given ; a careful inspection of these curves shows that, while they exhibit a decided 37 Ruckes, W., Ann. d. Phys. vol. xxv. p. 983 (1908). 38 Grindley & Gihson, Proc. Roy. Soc. vol. xxx. p. 114 (1908). 39 Becker, Ann. d. Phys. vol. xxiv. p. 863 (1907). 40 Fry & Tyndall, Phil. Mag. March 1911, p, 348. ) 41 Kohlrausch, K. W. F., Ann. d. Phys. vol. xliv. p. 297 (1914). Turbulent Flow in Pipes and Channels. 337 curvature over the range of velocities below the “critical velocity,” there exists an indication of a tendency for the series of points to show more than one interval of discontinuous slope ; the “critical” flow is not very sharply marked, but a value is assigned corresponding to a value of the ‘critical constant” in the neighbourhood of K=2080. Distributions of velocities over the cross-section are studied in detail for one of the pipes by means of a Pitot tube; the results indicate a gradual transition of shape from the “stream- line” to the ‘turbulent ”? distribution, and a close inspection of these curves, even at the lower velocities, shows a tendency for the curves to assume forms similar to those obtained by the writer for two-dimensional flow of air. The observations of Kohlrausch are not sufficiently numerous, nor the resolving power of the Pitot tube method of measuring velocities sufficiently high, to reveal in their full detail the type of distributions observed by the use of the linear hot-wire anemometer. Perhaps the most remarkable observations on the subject are those of Dowling*”, which show an indication of an approach to the results of the writer. The criterion of a change in the flow distribution was obtained by an ionization method. The most marked change was obtained for a mean velocity. giving values of K averaging 2480 for air and 2500 for carbon dioxide ; in the case of long thin tubes the values of K were somewhat increased to 3120 and 3080 respectively, and the anomalous effect of metallic tubes was also noted. The most significant result of these observations was the discovery of a second well-marked critical velocity con- siderably below that from which Reynolds’s Criterion is established ; in fact, many of the curves obtained by Dowling resemble in their general appearance those obtained by the writer, connecting the velocity at a certain point and the pressure-gradient in presenting a succession of well- marked points of discontinuous slope considerably earlier than that which is generally supposed to mark the commence- ment of the “ turbulent” régime. ® Dowling, J. J., “Steady and Turbulent Motion in Gases,” Roy. Dub. Soc. Proce. vol xiii. p. 375 (1912). In these experiments, air ionized over a short portion of the tube by a radium source, was examined for electrical conductivity by the usual electrometer method at a distant point. The rate of recombination depends on the nature of the velocity-distribution curve in the tube between the ionizing source and the exploring electrode; the electrometer leak is plotted against the mean velocity and the resulting curve gives a remarkably sensitive indication of sudden changes in the nature of the flow. Phil. Mag. S. 6. Vol. 31. No. 184. April 1916. 2A 338 Turbulent Flow in Pipes and Channels. Table of Theoretical and Experimental values of Reynolds’s Constant relating to the Flow of Fluids in Channels and Pipes: K=DUp]p. Flow in Circular Pipes. a b Value of K and Authority. Reference. Method. Roe Pa (ISHALDC se. ana. Trans. Am. Math. Soc. Theoretical. 470 2 vol. vi. p. 496, 1905. Oe IShanpeisasis ay, ew oe pees Theoretical. 363 (corrected by Orr). 8 | Orr ............ See footnote (20), p. 184, Theoretical. 180 AL 1907. (Reynolds ... See footnote (21), 1883. peste and 1900-2000 pressure-gradient. Couette ...... Ann. Chim. et Phys. (6) 2150 Volfexxi vip. too) (quoted sqm lalallala: by Orr), 1890. Barnes and See footnote (30), 1904. (Colonr-band, pressure- (1950 and with pre- Coker. < gradient, and special< cautions as high as (thermal method. | 4860. “= | Morrow ...... See footnote (32), 1905. 1 Distribution of velocity 1980 by Pitot tube. 3 Stanton and _ See footnote (34), 1914. (Extensive observations ( Principle of dynamical | Pannell. on air and water by similarity estab- ! Observations on Liquids. | 4 pressure- gradient | lished. MK=2500. and velocity-distri- | butions. Reutekces ttt See footnote (37), 1908. aan in capillaries: (2000 and in some cases pressure-gradient. < 38000-5000. Metal | tubes 450. Sorkau......... See footnote (36), 1915. ( Extensive observations { K=413°4 to K=410°5 on water and pure for water. organic fluids: pres- U sure-gradient. 2 (Grindley and See footnote (38), 1908. { Pressure - gradient in 2200 Gibson. long lead tube. Fry and See footnote (40), 1911. (¢ Velocity-dissribution 2500 Tyndall. by Pitot tube. | Dowling ...... See footnote (42), 1912. Tonization method. 2500 © | Kohlrausch. . See footnote (41), 1914. { Pressure-gradient and 2080 < _ velocity-distribution | by Pitot tube. Stanton and See footnote (34), 1914. See above. 2500 i Pannell: Observations on Gases Flow in Two-dimensional Channels. 3 (Reynolds ... See footnote (21), 1894. Theoretical. 517 ee Sharpe.. ...... See above. Theoretical. 167 > | S (Orr ............. See footnote (20), p. 130. Theoretical. 7 Note.—The only experiments available on the flow of air in two-dimensional channels are those obtained by the writer by means of the linear hot-wire anemometer; it appears from these observations that there exist a series of critical velocities, of which the more prominent seem to correspond to those calculated by Reynolds, Sharpe, and Orr. [| 339s XXXVIII. Jonie Mobilities in Hydrogen —Il. By W.B. Hates, B.Se., DI.C., Beit Scientific Research Fellow™*. 1” a recent paper? the author gave the preliminary results of experiments showing the presence of several kinds of negative carriers in pure hydrogen ionized by a-rays. Experiments have now been made at different pressures, so that a revised and extended series of measurements can be given. Heperimental. As described in the former paper, the apparatus is so arranged that the ions move under the influence of a simple harmonic alternating field between two metal plates. From readings of the current carried by the ions for different values of the amplitude of the field, the critical value of the amplitude can be found at which the ions just travel the distance between the plates and back in one cycle of the field. A mercury manometer tube was attached to indicate the pressure of the gas in the experimental chamber. Measure- ments could be conveniently taken between atmospheric pressure and about 8 cm., so that a filter-pump sufficed for lowering the pressure. In order to ensure that the hydrogen should retain a high state of purity at lowered pressures tlie chamber had to be rendered thoroughly air-tight, which is not an easy matter with different parts of glass, ebonite, and brass held together with sealing-wax. After one or two attempts this was satisfactorily accomplished by covering all the joints, when carefully made with sealing-wax, w ith a soft wax requiring only a ‘moderate degree of ‘heat to render it quite fluid, so effectively filling all minute cracks and pores. The method adopted as giving the most consistent results was first to fill the chamber at the pressure desired with pure hydrogen as it comes from the cooled charcoal bulb. The gas was then left standing and at intervals a complete series of readings of current against voltage was taken, so that the changes in the ions which go onas the contamination increases could be observed. In most cases the hydrogen was freshly prepared from zinc and dilute sulphuric acid, but the same results are noticed when the gas has stood for some days in a holder, provided that it is freshly purified * Communicated by the Author. + Phil. Mag. October 1915. 2A 2 340 Mr. W. B. Haines on with charcoal before the measurements. The changes which take place in the ions are very large, as will be seen by a reference to figs. 1-3, which show some typical results taken Fig. 1. Cugrent D Cc B A Volts. Current in the manner described. In its initially pure state the gas. gives an ionic current for very low voltages, and (in those cases where the readings are completed betore the conditions change appreciably) the curve is quite smooth over a large Lone Mobilities in Hydrogen. 341 range of voltage. From this it is to be inferred that the carriers in the gas are all electrons, moving so freely that very few become permanently attached to hydrogen mole- cules. This state of things does not last long. During the first few hours the electrons diminish in number and Fig. 3. c heating 5 ap ioe ve | i ge. 3hrs| P*8-6cr. | | | | | | 4 | | Current — n Oo ye ee = UN) Sy > ell ; j ; i lo] 320" B A DB A Volts. freedom, while the appearance of heavier ions of definite ~ mobility is marked by changes in slope of the curve occur- ring at definite voltages. Fig. 1 shows a typical case ata pressure of 68cm. The initial curve has a slight upward curvature, showing that some material ions are already formed, the smoothness of the curve indicating that the electrons forming these ions traverse part of the journey between the plates free, and the remainder attached to a molecule. For an abrupt change of slope in the curve it is requisite that the ions be in definite allignment at one plate when beginning their journey, and that they preserve their character throughout the journey. The second curve was taken after an interval of three hours. It shows a diminished electronic freedom (D), while the two ions C and B appear at about 100 and 230 volts respectively. The readings have not been extended far enough in this case to indicate the normal ion A, but, judging from other curves, it is likely that there would only be a small proportion present at this stage. After one day the ion a is shown largely pre- dominant, B and C being indicated in smaller proportions, while D is wholly absent. If at this stage the collected impurities from the charcoal bulb are admitted to the 342 Wr. W. B. Haines on chamber, the mobility of A is much reduced (20 per cent., say), and at most only traces of lighter ions can be detected. The values deduced from a large number of curves are here tabulated. Notation, as in previous paper :— Distance between plates, d=(1) 6°4 cm. (2) 4:2 cm. H=reading of voltmeter at critical voltage =amplitude /,/2. n=50 alternations per sec. Mean temp. =15° C. TABERG Positive Ion. | Press. E. ul. 1/u. PX. | cm. volts. | em./sec. | 21 36 12% ‘0079 266 50 ol 63°71 ‘0158 316 9°8 57 34:3 | 0292 337 | 14°8 81 2471 0415 356 18°5 236 19-4 0516 359 24-6 303 salt ‘0663 371 29:4 304 aS, O774 380 35'0 416 11:0 0910 385 39°3 197 9°88 1012 388 44-9 517 8°85 1130 397 O27 255 Geers) 1307 404 60-0 290 6°74 "1486 404 63-2 303 6°45 "1552 406 70°9 357 )'80 1725 415 orga S 365 5°36 "1867 : 415 Pap iE ie Negative Jon A. Press. | 4H. i a. elas Press. | E. | 1/u. em, | volts. | cin./sec. || em. | volts. | cm./sec. 86 COM Oz 0142 || 583 440 10-4 | °0961 13°8 LOT AZ 0234 || *58-4 413 11-1) .|26902 184 | 144 31-7 70315 || 606 | 462 9°9) || get 23°39 182 25°] “0399 63°35 480 95 | ‘1050 26°3 200 22°8 ‘O438 Gi onan 468 9°8 "{021 35°77 | 270 16-9'))) O59 1S 168 4) sole 8-9 “1120 3772 | 282 NG:2 3] SOG LT, Hig muahiods a te220 8:5 ‘hia *45°8 300 138 0722 aT Ope 506 on "1104 *50°6 | 1365 12:3” | 0798, emo Zot 78 "1285 old 388 11:8 0848 | Mean value P x ~=604'7 Lonie Mobilities in Hydrogen. 343 Tasxe III. Negative Ion B. ness: |) |) ae | Vu. || Press. |) Eh u. 1/u. cm. | volts. | em./sec. | || em. | volts. | em./sec. Taek | as saad [eda Aaa 86 Sayin’ glad ‘0072 *56°0 195 23°4 ‘0426 18°4 Wp ose | Olas 583 220 20°8 0481 23°5 Sie (owl 0192 || ~=60°6 200 e199 0503 24-4 94 48-7 | 0206 | 63:5 244 | 187 0534 26°3 97 47-1 | °0212 || *68-4 255 180 0557 36°3 138 3a°1 | -03802 capid Ul 232 19°7 0507 39°3 63 aleO |, “03823 || *f5°5 230 19°9 0503 41-3 | 159 288 | *0348 76:4 137 15°7 0536 *49°8 176 26°0 | -0385 775 126 15-1 0645 513 194 236 | “0424 || Mean value Px w=1206 * The asterisks in the Tables indicate those values which are plotted but were excluded from the calculation of the mean value of Pw. TABLE LV. Negative Ion ©. | | | | Press. | E. | Ul. 1/w. | Pxu. | cm volts. |” em./see. | 56-0 85 53°8 Osh 2 2010 | 67°7 100 45°7 ‘0218 | 3105 | : To 113 | 40°5 0247 | 3054 | Mean value PX w=3084 Discussion. In fig. 4 the reciprocal of the mobility is shown plotted against the pressure for all the ions. The values from the tables are marked by darker rings. For the sake of com- parison, the values of Chattock + and of Franck ¢ and Pohl are marked by crosses with initials (atmospheric pressure). The differences in my values indicate that purifying the gas has the effect of making the positive ion less mobile and the negative ion more mobile. A similar result is noted by Wellisch § for the ions in air when the drying is very thoroughly carried out. + Chattock, Phil. Mag. 1899, ser. 5, vol. xlvili. p. 401, and 1901, ser. 6, vol. i. p. 79. { Franck & Pohl, Verh. d. Deut. Phys. Ges. 1907, p. 69. § Wellisch, Amer. Journ. Sc. May 1915. 344 Mr. W. B. Haines on There was no consistent evidence of more than one kind of positive ion. The values indicate a departure from the law Pu=const., especially at lower pressures, when the mobility is smaller than would be predicted. Both the negative ions A and B follow this law very closely, the constant in the one case being just half that in the other. At pressures higher than half an atmosphere some readings give a mobility higher than is indicated by the formula, the dotted curves showing approximately the limits Recriprocal of Mobility Pressure within which the values lie. This would be the case if the limit were exceeded at which the proportionality of velocity to field breaks down, 7. e. the velocity impressed on the ion being sufficient to cause it to break up and become lighter. Lone Mobilities in Hydrogen. 345 To test this, the distance d was lessened so that the voltages concerned in calculating the mobility were smaller. These values are distinguished in fig. 4 by squares instead of circles. The same tendency to higher values was noticeable, though perhaps not so marked. It is concluded that in some speci- mens of gas at higher pressures there is a genuine tendency for the normal value of the mobility of both A and B to be exceeded. At atmospheric pressure the velocity can be taken as proportional to the field up to at least a velocity of 700 cm./sec. With regard to ion C* the results are not so definite or complete. In order that the interpretation of a curve shall be above question the ions D and B should be shown on either side, and in order that the measurement shall be accurate there must be a large number of the ions concerned present. These conditions are rarely fulfilled together. Also with the lighter ions the various portions of the curve are more crowded together, and a single inaccurate point may obscure its meaning. Three reliable values are recorded in Table IV., and several others are plotted. This represents the best that could be done with the present disposition of the apparatus. There is not sufficient evidence to decide whether or not more than one kind of ion is included in this group. The required changes are being made in the experimental arrangements in order that a further inves- tigation may be made in this direction. The more mobile ions are always the first to disappear as the contamination grows, and they are much more sensitive to impurities at higher pressures than at lower. The relative proportions of the various ions can be roughly estimated from the change in slope of the curve at the point of inflexion con- cerned. Thus in two cases where the gas had stood for a day the ratios of B/A were :—2 at 76 cm. and 2°0 at 24 em., showing clearly the greater stability of ion B at the lower pressure in spite of the fact that the contamination of the gas must be much larger in this case. Assuming that the effective impurity is the oxygen content of the air which leaks into the apparatus, rough measurements give :— Press., 9cm., 2°0 per cent. oxygen, ratio B/A=3:4 18 6 5 *6 ae 2 Hs “2 * Note.—It should here be remarked that the values given in the first paper as referring to ion C are spurious. The ions really present at the time of measurement were A and B with a small number of a lighter ion, the latter just sufficing to obscure the measurement for B.. The curves were not traced out in sufficient detail to make this distinction clear, 346 Mr. W. B. Haines on At higher pressures the contamination is largely that from the walls of the apparatus. These facts are in the direction that would be expected, since at lower pressures there are fewer encounters per second between the molecules of the gas and therefore less opportunity for the ionie aggregates to increase their size. In other words, there is greater stability of light ions at lower pressures. [Tor similar reasons the foreign molecules present in the hydrogen have greater influence at higher pressures. Since the electrons do not readily attach themselves to hydrogen molecules, the first step in the formation of most of the hydrogen ions is, no james the capturing of an electron by a foreign molecule having the necessary affinity. And the greater the frequency of collision the greater the chance of the electrons being quickly taken captiv e. The formula for the mobility of an ion as ee ived from the theory of gases is given by Townsend * a _ Xet UO Panic. ¢/V being the mean time between collisions. For a hydrogen molecule with a single electronic charge e/m=4°78 x 10°, we 10. The values + for ¢/V vary between 1-052 x 10-2 (N.7. P.) and 1:186x 10° (15° C., Kundt amd Wanaenaye The former value gives u=41°0 em./sec., which corresponds very nearly to the measurements for ion C (w=40°6 at 76 cm.). The second value gives w=46°2, and this bears a close integral relationship to the measurements for the other ions. ‘his is made plain in the accompanying Table, in which the theoretical mobilities for various molecular aggregates are shown, based on the assumption that the time between collisions is the same for aggregates as for single molecules, and the measured values placed beside them. Ton. Neg. C.") Neg. "Nes: ALY Pos: Experimental values wv... 40°6 15:9 79 54 Theoretical values w ...... 462 15-4 Cath 5:1 | 76 cm. press. No. of mols. per electronic and 15° ©. CREO Pees wiarebincine cee « 1 3 6 9 neg. A, and We conclude that the ions neg. C, neg. B, * ‘Townsend, ‘ Electricity in Gases,’ 1915, p. 84. + See Meyer, ‘ Theory of Gases,’ p. 192. Lonie Mobilities in Hydrogen. 347 positive are composed of aggregates whose simplest elements contain 1, 3, 6, and 9 molecules of hydrogen, respectively, er electronic charge. These integers are marked out as a scale in fig. 4, along the ordinate at one atmosphere, com- paring theoretical values of w with actual. These experiments explain clearly the results recorded by other workers on the subject, using methods which failed to distinguish between the different kinds of negative ions. Lattey and Tizard *, working at low pressures, remark on the need to use freshly prepared gas to obtain the high mobilities they record. They state that inconsistent results were obtained from hydrogen after it had stood for some time in the apparatus, which was, no doubt, due to the variable proportions in which ions A and B appeared. Also the large rise in mobility as the voltage increased was no doubt due—at least in part—to the failure to distinguish between ions of various mobilities. Chattock and Ty ndall f, working at atmospheric pressure, found an exceptionally high mobility in pure hydrogen, and record quantitative measure- ments of the effect of the addition of small proportions of oxygen. Summary. Measurements are recorded of the mobilities of the positive ion and three different negative ions in hydrogen over a range of pressures between 8 cm. and-76 cm. The results for two of the negative ions are expressed by the equations Puw=604°7 and Pu=1206 respectively. The work for the third and lightest ion is not yet so complete. Theoretical considerations indicate that the three negative ions are built up from elements containing one, three, and six molecules of hydrogen per electronic charge respectively, and the element in the positive ion has nine molecules per electronic charge. Imperial College of Science and Technology. * Lattey and Tizard, Pree. Roy. Soc. 1912, p. 349. + Chattock and Tyndall, Phil. Mag. 1910, ser. 6, vol. xix. p. 449. Baa XXXIX. Strength of the Thin-plate Beam, held at its Ends and subject to a untformly distributed Load (Special Case). By. CU, Laws, BiSey Aa Se." N steel floating structures, as e. g. floating docks and ships, the shell plating supported at and rivetted to the frames or girders of the vessel is subject to hydrostatic pressure ; and, if we consider a section of the plate in a direction perpendicular to the frames, forms a case partaking of the nature of a beam continuous over its supports and encastré—subject to a uniformly distributed load. The plating undergoes or tends to undergo distortion, and its appearance between consecutive frames is somewhat as indicated—but exaggerated for clearness—in fig. 1. In this Big. 1. flate Beam With o Uniformly Distributed Load wf ler Square inch. figure vr and 7, denote the lines of rivets connecting the plate to the frames F', F,, which latter are assumed to form rigid supports. The plating is deflected outwards between + and s, and 7; and s,;; and inwards between the edges of the supports s and s,, retaining the normal condition tan- genital to the supports at r and r,, where it is obviously subject to a pull or tensile force. It is the unknown value of this force which increases the difficulty of the problem. In determining the thickness to be assigned to the plating, it is customary to consider the problem as a simple encastré beam of span something intermediate to ss; and rr. The assumption is erroneous, and the only explanation for it is that by its means the work of calculation is thereby very much reduced, while the results obtained are considered to be sufficiently near the truth for practical purposes. From the mathematical standpoint, the problem should be approached as it is actually found to exist. The solution, by a direct process of rigorous mathematics, leads to com- plications due to the unknown conditions—of slope, force, * Communicated by the Author. Strength of the Thin-plate Beam. 349 &c.—existent at the ends of the beam. It may, however, be determined indirectly in a fairly simple manner by the aid of nothing more than the ditferential equation of the second order, and the object of this thesis is to indicate the method of attack. Consider a strip of plate, of unit width and thickness ¢, constituting a beam of length r7,=2a, and subject to a uniformly distributed load w per unit length due to a head of water h. . Take the axis of X at the mid-surface of the plate, 2. ¢., in a plane distant ¢/2 from the supporting surfaces, o—the centre of the unsupported span—as origin, and the axis of Y downwards. The appearance of the beam is shown diagrammatically in fig. 2. Fie. 2. Duagram -shewtrg Disposition of Forces Q Q f acting on beam Subzeot Lo une gormly dewsCribntat t | load w The solution is obtained by dividing the problem into two parts, viz. :— First. Assume the beam, under the action of a uniformly distributed load w per unit length (Q being removed), subject to a horizontal pull P and retaining its horizontality at the ends d and d, (fig. 3). Load w only 3 ( Q removed) Second. Take the beam under the action of a concentrated 350 Mr. B. C. Laws on the Strength load Q only at ¢ and ¢, with no pull at the ends and free to slide horizontally (fig. 4). Fig. 4 Lead Q only is — Obtain the deflexion at ¢ (or c,) in each case and impose the condition that these deflexions cancel each other—by reversing the direction of Q in fig. 4; we then bane the beam as it actually is, and as indicated in fio. 2.) "ihe solution is given below :— First. Consider w only (fig. 3) :— ne 1¢4 =—M"—P , Les = 2 tea Put P=m?E.I and let @ be the inclination at any point of the curve, then = tan 0 = 8, since 8 must be small. av dey a) ME, ede ee ne ee dé ip akin ataes Wie And aa a. hos ee The solution cf this equation 1s :— (ayo ae eu Se By ove UO. Ww er mE? and eb) ah a =m. ETA. jh. a / where ¢, is the base of the natural or Naperian logarithms. dy w.a When «= 4, Ae a G=0. ..A.e bem nl =i) of the Thin-plate Beam. dol When «= 0, ty Bean) Aes OAror A= —-B, Ax WwW. 1. Ex ea —B= 5 Meh meee n’?EK. Ife, e ) De ifn : 12, w i Miia Oe é. pet g ea lee es ¥ Sime a ee m.@ at i TE ae dx? m ee — 0) €5 is the equation for bending-moment at any point of the beam. When d*y i i mM. ale Mm. ee i ) L=Aa4, pee ge MS ae ey & nr MS Dea oe er § 0 0 d?y i w 2a.m pe I MS ee 0 0 M.C —mM.c, a7 w we im. ale, Bie pan) ee | aw =! 0 See Ax c mM } Cae —M.a 0 0 Now taking equation (1) we have :— w w 9 Ww Wo Gel When v=, M, = aaa, =i ee I.y, + Siglo Bie where yi = deflexion at o. . . . e U WwW Jombining this with the values for M, and Me we have:— M.a —71.4 ‘ MAA me, ee w. a? — a + ny a gre haa anre rf y) 0 0 Similarly, if 7" be the deflexion at c, nH. .y = m.a —m.a Tahal on™ ¢ 2H T (1 WO ( (€ ane, Gas ) 4 w(a? —_— CF =) m ° oy. vas ear m a Page g firs 4 Second. Consider Q only (fig. 4). There are two divisions of this problem: (a) For values of w ranging from 0 to c, and (b) for values of w ranging from c to a. 302 Mr. B. C. Laws on the Strength (a) For the portion oc of the beam :— meet 14 = —M, +Q(a—c). _ dy ME =Qle~e) Lieu iB. é There is no constant of integration, since when # = 0. dy dx At c we have dy _ My e—Qla~cie aay E.I f (6) For the portion ed of the beam :— —B.194=—Mi +Q(a— Q di MQ ONG “Geo BOL 2 one where f is a constant. Q M, vat Gage Dc ae a When w=a, as ma(0 eam ee EcT + omoT- aU ica OMG is) See “de 1OVRME 7) DIL M: (a-—c) _Q(a—c) . eG mm : ? da Equating the two values for = atc, we have Qe 2,2 i ea), 2a 2 72) When «=a, —E. Te Y yee ee dav 2a Q Q When v=, = 1h eo M, =—M, + Q(a—e) = Q Q@ QO(a—c)? eon ae ie 5 = MM; = a eG When 2=c, — of the Thin-plate Beam. 353 Again, referring to the portion cd :— Q dy __ Ma (a—2) | Qla—2)’ ant JD 7 a 2 m3) 9 0 ae a a a aE 3(« v—av?+ 5) +0: Now when r=a, y=0. pe M: ny. Q a am wor 2 Ses _Mi(a—2)? Qa-2) ey... eeene Q(a—a)? fa—e a— a} 2H .I 2a 5) and deflexion at ¢ = iene HOS ACT) ; r Ts Os yee en Also referring to the portion o¢ :— Q dy _ M, Rat Q(a— he, dx Kt ; Q = eee _ ate Cie Now when «=c, pe M,— Q—e). 2 _ Q(a—¢)*(a+3e) I~Je = Ho One ss 12a Des _ Q(a—e)f(a+2e) | Q(a—c)” hs 12K .1 dak .I ° _ @ Q(a—cP(at 20) and deflexion ato=y, = 128 v1 Now, in the general case fig. 2, ¢ is level with d. Therefore, putting y, = ye we get :— Q(a—c)?(a+3c)_ —w.a (cy er daeteeaen ) is w(a?—c?) = oo 12a a m? ' aaa ; } Im? BK 0 0 an equation from which Q may be obtained when m is known. Phil. Mag. 8. 6. Vol. 31. No. 184. Apri 1916. 2B 354 Mr. B. C. Laws on the Strength The resultant bending-moments at 0, c, and d, are :— u 2 e ee Ate Se m2 Mia an ee 2a 2U.C . —-mM.C pa eke Ate: MW’ Met 2 ft x ¢ m= pe ae Cae ; 2a ° 0 0 Qe mere + eta Fees w Q Ww 0 ; Q(a bea ) . SQ ay Oc sO ia Oni: O's eae Atd: My—MP}=— 2) — 20-1} 4 ue wae It will be found that the resultant bending-moment at ¢ is the greatest, and must be used for calculating the stress in the material of the plate due to bending. Fie. 5. i be t 1 @——— eo 2c. oo ee t pc 8 ao Pe | i Now consider the diagram fig. 5, we have :— 242 Stress due to P = = _E, “ 242 Ora. and elongation of the plate (beam) = 2a x aah al 12 eo L Aly? ength of ‘the curve cc; =S = 2e + a 8d? ei 3 16 of dici—s = (a-—¢ ; de or dyey= s= (a—e) + CE Therefore elongation of the beam = § + 2s — 2a, pe De 16d? m*ta Be" 3 (a0) ane 5 C Bnet a). \C or 1S ae 06 8 P) an equation from which to determine t when D and d are known. Now M.G —MAa , D= y- an y> = : J Laur si (a mao =) Jo 9 iD ; T t m? M.A —m.a on io of the Thin-plate Beam. 355 And d may be taken—without appreciable error—as the : sii: a+e ha ace deflexion at the point z= Bak: therefore, by considering the deflexion over the portion cd of the beam we have :— oe a mla+e) mM. oh —m.a 4 as ) gu ie ue Ce on: mH ~ yi M.A —M.a ClO w(sat+e){a-c) Q(a—c)*(2a+3e) 8m7K . 1 A8akK .I It is noteworthy that the magnitude of d, being relatively very small, is only of theoretical significance, and, in the application of the above principles to practical problems, d may be neglected in comparison with D. Summarising, the equations to be used in the calculation for strength of the beam are :— (1.) To find Q from :-— mM. Tae —m.a NL.€ —m.c Q/a—c)?(a+3c) Vs eae ee el eae ree ee 12a ae eG a 2m? t 0 (IIL.) To find ¢ from :— 2 a Oe Aye ei D = my / 4, Cc where Py _ wa @ +e 1 ee a Q(a—c)?(a+ 2¢) ‘oe cc I EN |" men Z 2m? i ‘ (III.) To find the bending-moment from :— Bending-moment at c we os) i Q(a—c)? ra ARE SPE KT sion Da hae MG —M.a é — 0) In any specific case we have in (I.) and (II.) three equations from which to determine four unknown quantities Oa, Ds. If we assume a value of m then Q, D, and é may be determined, thence the bending-moment from (III.), and, finally, the stresses due to bending and stretching. As an example take the case w eee = a=11 inches, c=10 inches. w= ‘007 ton per inch run (equal to 36 ft. head of fresh water). 5B? aad 396 Strength of the Thin-plate Beam. Assume for the purpose of our calculation HE =13,500 tons, and take m= °'1. From (1.) we get O= op. : From (II.) we get by the aid of (1.) :— eel) ee De aes Mes & De a oe 50 aT ee and ¢ = ‘287 in. Fig. 6. Maximuim stress in tons per square: inch, | | “2 3 4 5 “6 Thickness of plate in inches. From (III.) we get :— Bending-moment atc — +2071. Therefore :— se Stress intensity due to stretching = sla ~via 924 tons/in.” sec’. area t a e ;,. bending ("= aS = 15°216 tons/in.? Or Maximum tensile stress = 16°14 tons/in.? » compressive ,,. = 14°29 tons/in Hlectrical and Magnetic Properties of Pure Iron. 357 Similarly, by giving to m other values, we may determine corresponding stresses and plate thicknesses, from which data to construct a curve of “stress with relation to plate thickness,” as shown in diagram fig. 6. If we have previously decided upon the stress to be allowed in the material, we are able by the aid of this eurve to determine the necessary thickness of plate re-— quired. Thus for a stress of 10 tons per square inch (corresponding to a factor of safety of about 3 with mild steel) the thickness of plate required would be °376 inch. XL. The Electrical and Magnetic Properties of Pure Iron in relation to the Crystal Size. By EF. C. THompson, M.Met., B.Sc., Demonstrator in Metallurgy inthe University of Sheffield ™. mes late years very considerable attention has been paid to the crystalline boundaries in metals, and the theory has been advanced, and received consi- derable support, that the metal at the crystal junctions is amorphous, corresponding in many respects to the liquid metal in a drastically undercooled condition. The mecha- nical effects resulting from such a structure are now fairly well known. The electrical and magnetic properties have not yet, however, received full examination from this point of view. : So long ago as 1902, Professor W. M. Hicks t, commenting on the high permeability of iron alloyed with aluminium, pointed out that the elements aluminium, silicon, and phosphorus, which increased the permeability of iron, were the elements which had been shown by Arnold f to increase the crystalline size of that metal. “ It is probable, therefore, that the increase of permeability due to these substances is a secondary effect due to the size of the crystals.” This idea would imply that the magneto-motive force required to send the magnetic flux through the body of the crystal itself is less than that necessary to send it across the glassy material at the crystal boundaries. In an investigation on the relationship of the elastic limit of a metal to the size of the crystals §, some electrical and * Communicated by the Author. + Hicks, ‘Nature,’ xv. p. 558 (1902). t Arnold, Journ. Iron & Steel Instit.i. p. 107 (1894). § Thompson, Trans. Faraday Society, i. p. 104 (1915). 399 Mr. fF, ©. Thompson on the Electrical and magnetic measurements were made on samples of very pure iron in which, as a result of appropriate heat-treatment, the size of the crystals varied to a marked extent, the material remaining otherwise unchanged. [ron was the metal chosen, since a greater variation of grain-size can be induced in it than is the case with most other metals. An ingot weighing 28 ib. was cast from a specially pure Swedish bar-iron, melted by the coke-crucible process. Before casting it 0:04 per cent. of aluminium was added to render the metal sound and free from blow-holes. Theiron gave the following analysis :— Carbonic iin LRN 0-049 per cent. Silo) He aes 0-04 ot Manmamese 0s) vi Gia 003 be Re CUO ORT SNM OIRO gh 0-02 if hosp mone) WM eiaee 07016 ine A trace only of aluminium remained in the ingot, whence the iron percentage by difference was 99°87 per cent. It is therefore one of the purest, if not the purest, ingot of iron ever made by this process. The specific resistance and the magnetic properties were determined in the following five conditions, in each of which the number of crystals per cm. leneth was carefully determined microscopically :— (1) As received from the rolls. (2) Normalized, 2. e. re-heated to 900° C. in a muffle and cooled in air. (3) Re-heated to 900° C. and cooled in a muffle during twelve hours. (4) Re-heated to 900° C, maintained at this temperature for fifteen hours, and cooled in the furnace over another forty hours. . (5) Re-heated to 900° C., quenched in cold water, and tempered at 650° C. (Tbis temperature was chosen since it is well above that—520° C.—at which iron deformed by cold work loses its strain™. Quenching strains which might conceivably vitiate the results arethus removed. Also the small percentage of carbon present will practically revert to the condition in which it is present in the other samples.) By these treatments the number of crystals per cm. was varied from 690 in the quenched and tempered sample to about 10 in the drastically annealed one. In the latter case, however, some considerable irregularity in the crystal size— which though not readily explicable, is probably dependent * Goerens, Journ. [ron & Steel Instit. Carnegie, vol, 11. (1911). Magnetic Properties of Pure Iron. ye on the crystallization which occurs at the critical temperature Ac;—rendered it unsuitable for the purpose of directly cor- relating the crystal size with the magnetic and electrical properties. The five specimens were cut consecutively from a single length of the rolled bar, between the two ends of which no difference of chemical composition could be detected. After treatment each of the small bars was turned down from the original diameter of one inch to a centimetre. Absolute freedom from alterations in composition due to surface decarburization was thus ensured. (a) Llectrical Lesistance. During the passage of an electric current through a normal metal the opposed resistance may be considered to be the sum of two factors, one for the actual crystalline material, the second depending on the resistance of the amorphous metal at the crystal boundaries. ‘This amorphous metal, which, as already mentioned, is of the nature of an undercooled liquid, may be expected to possess a higher resistance than the crystals themselves, since a substance at its melting-point possesses a higher conductivity in the solid ‘crystalline condition than in a liquid. The thickness of the amorphous films is exceedingly small, probably of the order 10-°>cm.* Thus the lenguh of the crystalline metal to be traversed by the current is essentially unaltered, whatever be the number of crystals per unit length. That part of the observed resistance due to the crystals themselves will, therefore, be a constant. On the other hand, however, the resistance of the boundaries will be pr oportional to the number of these regions of decreased conductivity which have to be traversed. Thus, if p is the observed specific resistance, Pz 1s the specific resistance of the crystalline portion, n the number of crystals per cm., and R the boundary resistance for each grain, then Pi Bo aie lu Dae Say ree a I) In order to check this relationship experimentally, bars 20 cm. long and 1 cm. in diameter were turned from each of the heat-treated samples. The specific resistance of each was determined by comparing the fall of potential over a definite length (15 cm.) with that in a standard low resistance through which the same current was flowing. The deter- minations were carried out at 16° to 17°C. In Table I. the * Thompson, Trans. Faraday Soe. i. (1915). 360 Mr. F. C. Thompson on the Electrical and results are collected, together with those calculated by means of equation (1). In every case an increase in the crystal size is accompanied by a corresponding decrease of the specific resistance. ‘'o about 1 per cent., which is quite as near as the mean crystal size can be determined micro- pee een, the relationship given above is found to hold good. | TABLE I. Specific Resistances of Pure Iron. No. of Crystals) Specific Resistance in microhms Treatment, ae oe per cm. Observed. | Calculated. | Difference. Aisreceiviediy ites ona 426 7563 7°564 —-001 Normalized: eee cso 343 7513 — 7-422 +:091 Ammealedinnes Mir ass. ok Be eee 276 ap ageooll 7307 — ‘056 Drastically annealed ...... ? T1522. [So Quenched and tempered a €90 7986. |. Ong — 033 The values experimentally obtained render it possible to calculate the true specific resistance of the crystalline material free from junctions, and substituting the mean values thus found in equation (1), the specific resistance of any sample of pure iron becomes p=—6'33 + I-72 x 10>* microhm per c:cma een The value 6°83, the only rational one to adopt for the constant for pure iron, is a value much less than that usually adopted. In view of this result, no determination of the specific resistance of any metal is complete which does not give the crystal size and also sufficient data for the caleu- lation of pg. The wide divergence of the specific resistances (7°15 to 7°99) seems to explain the great variation in the values obtained for such “ constants’? by different investi- gators, even when pure materials have been employed. ‘The variation observed in the case of iron is probably greater than in that of most other metals, ror instance copper, since in these latter cases the crystalline structure is always coarse compared with the normal crystallization of iron. Hence the factor n is subject to smaller variations, and different determinations show less discordance. One case in which Magnetic Properties of Pure Iron. 361 equation (1) is extended to metallic alloys is worthy of brief consideration here. Such alloys as consist entirely of a single solid solution should obey the rule equally with pure metals. It was found, however, in the course of an investi- gation on the effect of annealing on the electrical resistance of the copper-zinc-nickel alloys which belong to this class, that in many cases with a high nickel content the size of a erystal exercised no appreciable influence on the resistance. The values for an alloy containing 61°2 per cent. of copper, 9°81 per cent. zinc, and 28°6 per cent. nickel (a) with the very fine structure obtained immediately that re-crystalli- zation had been induced, and (b) after subsequent prolonged annealing at 696° C.to coarsen the size of the crystals, were found to be 41°2 and 41:0 michroms per c.c. respectively. An explanation of this lack of dependence of the conductivity on the crystal size in these alloys, however, is not difficult, and the validity of the relationship (1) is not called:in question. There is now lhttle doubt that in solid solutions such as this alloy an internal strain of considerable magnitude must be assumed, which will re-act on the electrical conductivity, lowering it until, in certain cases, the conductivity of the erystalline and of the amorphous parts is practically the same. Returning to the values of the specific resistance of iron, that obtained for the sample as received from the rolls (7°56) is in excellent agreement with the value (7°6) arrived at by Benedicks* by extrapolation of his results on the specific resistance of steels in the same condition. It is evident, however, that his equation for the specific resistance of steels p=7164+26°83C, where >C is the sum of the carbon and the equivalents in carbon of the other elements present in solution, can only be be an approximation to the truth, since heat-treatment, e. g. annealing which varies the grain-size of the material without any chemical change, would result in a change of specific resistance. An observation made during the present work on pure iron would seem to throw light on the conductivity of carbon steels. The drastically annealed sample of iron showed the curious irregularity already referred to. About nine-tenths of the mass consisted of crystals so coarse as to be readily * Benedicks, ‘ Thesis for Doctorate,’ Upsala, 1904. 362 Mr. F. C. Thompson on the Llectrical and visible to the naked eye after etching. The rest was composed of very much finer grains. The conductivity in this case was the greatest of any of the samples measured, but it was distinctly less than that corresponding so a specimen con- sisting of big crystalsalone. The conductivity was, however, greater than would be expected from the relative proportions of the large and small erystals. These regions of fine and coarse cr ‘ystallization are essentially so many circuits of higher and lower resistance, and it is to be expected that the current will pass prefer ential ly through the coarser crystalline parts of lower resistance. ~ Now it has been shown by Benedicks * that the electrical resistance of steels containing less than 0°5 per cent. carbon varies with the carbon per centage in a different manner from that of steels with a higher carbon content. As a result of much careful work, it was concluded that about 0°27 per cent. carbon is present in solution in these latter steels giving a con- stituent “ferronite.” It would seem, however, possible to explain the altered behaviour of the electrical properties of steel above and below 0°5 per cent. carbon ina simple manner. A low carbon steel consists of a matrix ofiron crystals (ferrite) with isolated areas of pearlite in which the earbon present as H'e,0 is segregated. The electrical resistance of these latter areas is considerably greater than that of the ferrite (the specific resistance of pearlite being about 20 microhms per c.c.). Hence in such a steel the current will preferentially take a path through the ferrite. As the carbon content of the steel rises the pearlite increases at the expense of the ferrite, and at 0-45 per cent. the volumes of each are the same. Beyond this percentage, however, the relative order is reversed, and it is now the ferrite which occurs isolated in a background of pearlite. In this structure the continuous paths of the less resistant con- stituent are wanting; and it is not difficult to realize that the specific resistance now bears a somewhat different relationship to the carbon content from that which holds for those steels in which ferrite is the predominating constituent. Such an explanation removes the necessliy for assuming the new constituent ‘“‘ferronite.’ The fact, too, that, sofar asthe author has been able to determine, the pearlite areas in pure carbon steels free from silicon and manganese (which some of Benedicks’ steels certainly were not) are directly proportional to the carbon content, is also * Benedicks, loc. cit. Magnetic Properties of Pure Iron. 363 opposed to any theory involving any large solubility of that element. The mathematical expression here given for the relationship between the electrical resistance of a metal and the size of the component crystals is very similar to that given by Von Weimarn*. Regarding the question as being ana- logous with the electrical resistances of colloidal solutions, it was argued that the observed resistance of a metal is the sum of two values, the first of which is proportional to the volume of the sample, the second being dependent on the surface energy of the intercrystalline amorphous films. This latter value is obviously a function of the number of erystals per unit volume, while the firstis a constant for each metal. The relationship, therefore, is almost identical with the author’s, though it has been arrived at from a totally different point of view. Its validityis dependent on certain assumptions from which equation (1) is free, and no attempt was made to supply experimental evidence in support of the expression. The speculation of Lord Rayleigh} concerning the in- fluence of the Peltier effect on the electrical resistance of binary alloys offers a still more interesting subject for comparison. ‘This was shown to result—in the case of ummiscible components—in a back electromotive force (which simulates an added resistance) arising from the couples set up at the surtaces of contact of the two constituents. In a pure metal a similar result should flow from the presence of the amorphous films between the crystals. In the case of an alloy of fixed composition the increased resistance was shown to be independent of the number of the couples,7. e. of the number of crystals of each present. In the case under consideration, however, the conditions are somewhat different, since, as is probably the case, the films are of the same thickness with both big and little er ystals. The quantity of the cement present in the latter case is thus increased, and considering the mass to be composed of disks of crystalline and amorphous material alternately, the amount of metal present in the latter state is nm times the thickness of each film where nis the number of films, i. e. crystals per unit of length. If ¢ is the thermo-electric power of the couple formed by the components of the alloy, kand k’ the thermal-conductivity * Von Weimarn, ‘ International Journal of Metallurgy,’ ili. p. 65. + Rayleigh, ‘ Nature,’ liv. p. 154 (1896). 364 Mr. F. C. Thompson on the Electrical and of the components, and pand q the fractional volume of each, then it was shown that the added resistance r J ees =, a Poe a result which does not containn. Now ina pure metal if p is the fraction of the volume composed of the crystalline material, and q that of the remaining amorphous metal, and if & and k’ are of the same order of magnitude, then a Kj / is negligible compared with 2 since the amount of the amorphous constituent is so small. Hence, approximately, Since g is directly proportional to 2, the increased resistance in a pure metal due to the intercrystalline films is also - dependent on the number of crystals perem. ‘The total resistance of the metal is, therefore, expressed by an equation of exactly the same type as equation (1). Unfortunately, the data required for the comparison of this result with that (2) already determined experimentally are wanting, but the coincidence of the general type of the equation with that found experimentally is of the greatest interest, and affords further confirmation of the truth of the, at first sight, unlikely theory of the amorphous intercrystalline cement. (B) Magnetic Properties. The magnetic properties of the same samples of heat- treated iron were determined ballistically with the aid of Hicks’ yoke. No correction has been applied for the reluctance of the latter, since relative values are sufficient for the present purpose of correlating crystalline size with magnetic properties. The results are summarized in Table IL, and the hysteresis loops are reproduced in fig. 1. These results indicate that the maximum induction for a field of 95 gauss and the remanent magnetism are both, to all intents and purposes, independent of the size of the constituent crystals. They depend, therefore, apparently on the nature of the crystalline material rather than on the nature and number of the crystalline boundaries. The co- ercive force, however, increases regularly with the number Magnetic Properties of Pure Iron. 365 TABLE II. Magnetic Properties of Pure Iron. No. of | Bamax.) | |Hysteresis| Brinell Treatment. Crystals, for H | Barem.).| He. loss. | Hardness. | per em. | =95. As received ...... 426 | 18250 | 10400 | 25 922900 | 86 Normalized ...... 343 | 18550 | 10000 | 2-0 21400 | 83 Annealed ......... 276 | 18200 | 10000 | 20 | 192900 | 74 Drastically | | ; | = ae } 2 | 18400 | 10000 | 40 | 24400 b cnee Quenched and |) agg_| ; rp eet 660 | 18409 | 9600 | 50 34600 105 | 1 division =20 of crystals per unit length (the anomalous case of the dras- tically annealed sample will be considered later}, and at the same time a corresponding increase in the hysteresis loss per eycle is found. In each case the relationship is an approxi- mately linear one. Along with the magnetic results the hardness values of the samples determined by Brinell’s method are recorded. The load used was 500 kilograms, and the ball 10 mm. in diameter.. The results are of interest in confirming the general im- pression that magnetic and mechanical hardness go side by side. With the exception, again, of the first sample, if the hysteresis loss per cycle is e and the Brinell hardness H, then pei. 2) Nae 2) 366 = =Electrical and Magnetic Properties of Pure Iron. When dealing with the electrical resistance of the dras- tically annealed sample, it was mentioned that the micro- structure was remarkably uneven. The large grains, however, were the ones which seemed to exert most influence on the conductivity, which was the greatest of any of the specimens tested. The magnetic results, however, point in quite the opposite direction. With the exception of the quenched and tempered iron, the magnetic “ hardness” of the drastically annealed sample was the greatest of all ; and the conclusion is inevitable that the minute crystals (despite their relatively small mass) had impressed their own properties on the whole sample. The results are of practical importance in pointing to the possibility of producing by suitable annealing iron for trans- formers, &c., with far lower hysteresis losses than any yet obtained. The constant values obtained for the maximum induction with varying crystal size hardly appears to be concordant with the view that the high permeability of iron alloyed with aluminium, silicon, or phosphorus is due to the large crystals which these elements induce. As an alternative suggestion, it may be noted that each of these three elements is used to remove the oxygen taken up by metals during melting. In the unalloyed iron it is probable that the oxygen is present partly as oxide in solid solution, a con- dition which in general decreases the permeability and increases the coercive force. The presence of silicon or aluminium would tend to eliminate the oxide from solution ; and it may be that to this fact the high permeability of the alloys of these elements with iron is to be referred. The results here recorded were obtained in the Physical Laboratories of the Department of Applied Science of the University of Sheffeld. The author wishes to express his sincere thanks to Prof. J. O. Arnold, D.Met., F.R.S., for his eae and advice, and to Mr. EH. H. Crapper, B.Eng., M.1.0.E., Senior Lecturer in Electrical Engineering, for the Ae often ed in carrying out the piece and magnetic tests. To Professor W. M. Hicks, D.Sc., F.R.S., the author is extremely indebted for his very kind criticism and help. ee i XULI. On the Hall Hjffect and Allied Phenomena. To the Editors of the Philosophical Magazine. GENTLEMEN,— N the Gctober number of the Philosophical Magazine, in the paper by Mr. Livens on “ The Electron Theor y of the Hall Biffect and Allied Phenomena,” there are two points to which I would like to call arent In discussing the reversal of sign which often occurs in the Hall effect, in the Nernst effect, in the Leduc effect, and in the von Httings- hausen effect, it 1s stated that, ‘* It 1S, bowever, of importance to notice in this connexion that the signs of Al the effects are found to be reversed if one of them is, or in other words, the effects always have the same relative direction. This obser- vation is important as implying that the cause of the reversal is common to them all.” An examination of the accompany- ing table, in which the direction of these effects has been indicated for a few metals, will show that the statement quoted above is inaccurate. ‘There are some metals in which the reversal of signs occurs in accordance with the statement of Mr. Livens, but in the majority of cases it is otherwise. Metal. | Hall BiyBse ae ieee | | Effect. | Effect. liffect. Daan | | | RAN IEL DN. oishisesc. ses) _ | = | x | te Melle 2.22... > — als at Aniammory); 532.2.) | + = 8 we PONE 9 ds one eck =a = af a BSUUVED se Gadie enc ccnees | — - = OEE oie sc aces: | - a0 Ls Ss ia 1 ze ba ‘) i et. | | In view of this discrepancy it seems impossible to account completely for this reversal of signs by a local magnetic field inside of the metal. In some cases the presence ‘of this local magnetic field may afford a satisfactory explanation of the reversal o£ signs in the effects, but it does not afford a general explanation. 368 On the Hall Effect and Allied Phenomena. In discussing the influence of temperature on the Hall effect in the ferromagnetic metals, Mr. Livens states that, “In these cases as the temperature is increased the Hall effect coefficient increases, exactly parallel with the magnetic permeability, until the critical temperature is reached, when it decreases more rapidly to a value more akin to that found in the simpler metals. This would appear to be almost con- clusive evidence of the appropriateness of the explanation of these irregularities suggested above.” That there is a sudden decrease both in the Hall effect and in the permeability at the critical temperature cannot be questioned, but from this fact it is not to be inferred that the coefficient of the Hall effect and the permeability depend on the temperature in the same way. The Hall effect seems to be a more complicated function of the magnetic permeability than the statement quoted above seems to indicate. Kundt has shown that the Hall electromotive force is proportional to the intensity of magnetization for a particular temperature. If E is the Hall electromotive force ; I, the intensity of magnetization; f, the susceptibility; H, the inagnetizing force ; and A, a constant of proportionality, H=AxI=Axkx H. For a given value of H the relation between the Hall electro- motive force and the susceptibility is given by the equation H/k=A< xX constant. If the Hal] effect and the permeability increase parallel to each other, the ratio of the Hall electromotive force to the susceptibility should be independent of the temperature, that is A should be independent of the temperature. Hxperiments have shown that A isa function of the temperature and that it increases rather rapidly with the temperature. Hence the Hall effect and the permeability depend in different ways on the temperature, and the variation of the Hall effect with the temperature cannot be entirely accounted for by the variation of the permeability with the temperature. Very truly yours, ALPHEUS W. SMITH. Ohio State University, Columbus, O. [ 369 ] XULIT. On a Brief Proof of the Fundamental Formula for Testing the Goodness of Fit of Frequency Distributions, and on the Probable Error of “P.” By Kart Pearson, a.” 1. FN a contribution to the ‘ Philosophical Magazine’ in 1900 (vol. 1. pp. 157-175) I gave the first proof of the fundamental formula for testing goodness of fit of a theoretical frequency curve or surface to observational data. The observational data were supposed to constitute a sample of definite size drawn from an indefinitely large population obeying the theoretical curve or surface of frequency. The problem proposed was: To measure the probability that a sample of the proposed size drawn from the indefinitely large theoretical population would present as great or ereater divergence than the observed material does from the sampled population. Was, in fact, the observed popu- lation so improbable a sample of the theoretical distri- bution that it must be rejected as a sample by the prudent statistician 7 Let the theoretical population consist of M individuais failing into g classes with the frequencies mj, mg, ... mq, all these classes being indefinitely large, but having’ finite ratios. Then, if a sample of size N give the classes M4, N2,--- 2g, the mean value of n, in a long series of samples wiil be ms=Nm,/M, and the distribution of n, will follow the binomial { Ts Bi aye. and accordingly, if no class be extremely small as compared with N, the distribution of any frequency like n, will be approximately normal and the combined system of fre- quencies will follow the generalized Gaussian law ad } S, (= a) asl (eel a) ( z= fe A og 2. A asas' i 1 ee G:) where a sees Ns Li—he—, Of = Ne, 1-x f the squared standard deviation of the binomial, and the correlation of #, and zy or 75s, 18 given by * Communicated by the Author. Phil. Mag. 8. 6. Vol. 31. No. 184. April 1919. 20 370 Prof. Karl Pearson on Testing the Further, A stands for the determinant { 1 ier TASSELS ee 944, 9 g=ll To 1 To Mal ks Tobey, z) Toad “ool e\ von e - | = and A,,, is the minor of the constituent 7.” In the paper already referred to, I found ithe value of the exponential power by actually reducing the above determinant and its minors. I have since deduced other proofs, of which the present seems for lecture purposes the briefest. In the expression (i.) above we must include only gq—1 variate frequencies, as the gth frequency follows at once from these by the relation N= n+nrot... +7, the size of the sample N being fixed. Accordingly, 8, is a sum for s from 1 to g—1, and S2 a sum of products for all unlike pairs sand s' out-of 1, 2,3. ...3.-./s/eumyg oe Now consider what happens when we fix the variates Ly, #2 .+.Xq-1, or all but 2,5 then the equation to the frequency of 2s will be given by 3 2 ne (ts—hs)* Bm ange Qe as all the other terms being constant and falling into 2’. 3? =; A/A,, will give the standard deviation of this distribution. We can to determine >, give the other 2’s their mean values, in which case we shall have samples of constant size 7;+n, to be taken out of the population. The standard deviation of such samples will then be given by $2. Gilet) Ms fie Ms ) Ms + Mg Ms + Mg ee Ce Xs = (th) Pets) s q 5 qg Accordingly : Ag i Ay Ss] ae (il.) Thus the coefficient of x? in our exponential is determined. Goodness of Fit of Frequency Distributions. d71 Now consider the partial correlation coefficient of 2, and 2, for constant 2), 2... %-1 for all the g—1 variates except x, and 2. We may put these constant variates zero, and it is clear that we have to find the correlation of z, and x, when we take samples of size ns+n +7, from the ms, ms, and mz, classes only. But in this, if &/,, =’. be the standard deviations, we must have “ us hs Ms Ms Se) | 8 M+ Mg + Mg Ms + Met Mg —_— Ns Ts (Net +19) eee = Ns | —-——_——— =) Se aS 8 . a} Vie (iii. ) Ng + Ng + 709 Ng tt Ng t Ng and, similarly, | Ng (Ne +72,) Ys -bRae My Further, if pss be the required correlation, Ms Mg! Ms + Ms! + My Me+ Ms +My Pss' ye Sy = (Ns =e Ns! == Ny) ns + Ng + Nq [ Thus Van. ig) OS ee eS Vis tig VN + Ng But by the general theory of multiple correlation, vat ee Ass! nya oP hy BN ee oso yh/ o7A ony or MB V Ng Ns! ‘e eke wt GOsN Vig +g Ving + iq ns alte n) using the values found for (ii.). Thus we deduce imental OROeAN Te (v.) Accordingly we have found the coefficient of the term 2.2, in the power of the exponential, which may now be written —— sqen - = . 2 - = ° (vi.) a2 Prof. Karl Pearson on Testing the where we have AG Vs 8s Ugh 5! aie 8:(F #2) +28, (4¢ OC 5! we | =, {2 (+ = \ +28, { tt b xe = 8, a tz (Ua to oh ° Bosnia Now (egy = IN eases) NG and therefore Py kya) ade ate =O and (tat... +ay-1) = ae. Thus finally we have for x? 2 Bey lei sy WG = (=). where the summation is for all values of 2, from 1 to gq. There are, however, only g—1 independent variables, and accordingly the chance that a sample of N will occur with a value of the deviation-complex of magnitude x” or greater is | a ae Oey | KEN eK This is the value of P calculated in the tables of the ‘“‘ goodness of fit”? and accordingly all turns on the deter- mination of (ng— 7) ae Jee 2. On the Probable Error of the Determination of Goodness of Fit. CER 2 a a sf {oer then in a recent memoir * the probable error of x? has been found, or rather o wee * Young and Pearson: Lrometika, vol. xi. part ii. Goodness of Fit of Frequency Distributions. 373 Now * P, = ~%, fs —2x? bal dy — p—? Xn Q—-2 pln = ae x aX 9 i 72X92 dy or 3X" ,9-3 OPo = Sx eX ae x { e72X 2-2 dy a 22 x WAX)s where e-3X n9-3 W(X) ae ea , Now suppose g is even: then i eX yi-2 dy = (¢g—3) (q—5).. ill a 3X" dy. Thus, since { eX dy = V/V 77/2, 9 ae i q—3 was spe? 3 Nun, SAN a YalX) Vier arama = Py (x!) —Pa-a(x’) + Next suppose g is odd: then { e-2X'yI-2dy = (q—3)(q—5) ... 2( e-3X'y dy. Thus, since { eX y dy = 1, 0 wh? (x) = Po (x?) —Po-2(x’) T; the same value as before. * Phil. Mag. loc. cit. p. 158, Ha. (iii.), remembering that n=q—1 =number of independent variables. + Phil. Mag. loc. cit. p. 159, Equations (v.) and (vi.). 374 Prof. Karl Pearson on Testing the Hence generally, — Cr. a 3 y2 (Po (x7) — Pq—-2(x”))- The values of P for the value of y? and for q and g~2 can be taken directly from the Tables for Goodness of Fit *. The value of o,2 is given by : .. g g(g—2))\? where g=number of cells as before and H is the harmonic mean of their contents+. It is thus fairly easy to determine the standard deviation of P, and so ascertain whether the deyree of ‘ goodness of fit” is sufficiently stable to provide in any case an adequate measure of fit. — Illustration I—We may illustrate the method on the data provided in the introduction to my ‘ Tables for Statisticians,’ p. xxxll, for the goodness of fit of the cephalic index of 900 Bavarian crania to a Gaussian distribution. The following table gives the observed and Gaussian frequencies :— Cephalic Index. Observed. Gaussian. Winders 7 DOs s: eate | SS 12°4 CO O-MOO | aateceas: 12°5 12:7 MOOT EO) th aeeeeee 17 221 TOMO) aistnaccee 37 35'3 MOG) Lc watade | 55 51:9 WIO=OUD .aesees | 715 7071 SOS=815D" se ..te | 82 87-0 coil EH eo) ES 116 99-4 S25=83D oss se-ue| 98 | 104-2 (ste ao ee ence 107 | 100°5 84°5-85°5 ..... 2. 82 Tt Soa SOOO Dm uate ns: 74 | 726 | S6°5-Si0) Greet 58 54:3 O19-88'O.\ ee seaices 345 37°4 SOGOU secten-cs 19 | 23°7 895-90 Seteleee 10 | 13°8 SO;o=0085 (ye eee 8 | 74 | Over Ol ar eo n<.e 9 | 6°3 These 18 groups give y?=10-27, leading to P=-8909. The x’s are, of course, given by the Gaussian column, and from this H can be caiculated by aid of tables of reciprocals. Thus q — i — Oo 5 SS) a = si = "1 8b.21,2,00- * ‘Tables for Statisticians,’ p. 26. + Young and Pearson, Joc. eit., where we must note that y2=N¢, Fruit. — Goodness of Fit of Frequency Distributions. 375 Further, 2(q—1) = 34, g(q—2)/N = 32. Accordingly, a9 2(q—1)+ a +2 3 Me 35°106,272,53, and Gye = 135°106,272,53 55 = 5°9256. Since y? is only 10°27, this indicates that the material might quite easily be a sample from the Gaussian given. Now EaCyaes 0009 and! Poo@/) = "8016, L1P,(y2)—Py-o(y?)} = 04465. This leads to op» = 59256 x 04465 = *2646, and probable error of P=:1785. The result may therefore be given as It will be seen that the test of goodness of fit is subject to a larger probable error than has hitherto been realized. It will not in this case affect our judgment, for P is so large that the large probable error will not reduce it to a really small probability, but it indicates that, especially in considering relative goodness of fit, we must proceed with caution in this matter. Illustration IT.—I will consider the application of the present theory to fourfold tables, and take as examples two tables cited on p.xxxiv of my ‘Tables for Statisticians’ for characters of fruit and flower in Datura and for Eye-colour in father and son. The tables are :— thus (i.) Datura. (ii.) Hye-colour in Father and Son. Colour of Flower. Father. Violet. White. Totals Light. | No ‘totals gee Totals: ‘| Light Prickly .... 47 | 21 | 68 _ | Light ...... 471 | 148 | 619 | - | Smooth ....| 12 | 3 | 15 “| Not-Light.| 151 | 230 | 381 Totals...) 59 24 83 Totals...) 622 | 378 | 1000 376 Prof. Karl Pearson on Testing the The ‘theoretical’? tables in this case are those with independent variates, 7 e é., 483373 | 19-6627 | 68 Gi) s8o018 | 233982 | 6 ll. 10-6627 | 43373 | 15 236-982 | 144-018 | 381 59 | o4 | 83 622 | 378 | 1000 The y? are Gi.) 2 = +7080, Gi.) 1? = 133°3265. The reciprocals of 7’s are to be found from the “ theo- retical” tables. a Cc a+c¢ But if we have a fourfold table : a little consideration shows that pone | ae | d | e+d | b+d | N N3 (3) (a+b)(c+d)(a+c)(b+d)’ if the condition of zero contingency be satisfied. Hence we tind aca H=°* Hos 2) *395,8866 for (i.), °018,0344 for (ii.) ; 2(¢q—1) = 6 for both; q(q—2) _ N Accordingly, ye = 36°492,2 ‘096,3855 for (i.), = 008 Tor (il) 721}# = 2-5480 for (i.), = {6:026,0344}? = 24548 for (ii.). It is clear that c,2 depends chiefly on the term 2(g—1)=6, and will be generally for fourfold tables very near V6 Thus we have oS = 2°4495. ‘7080+ 1°7186 for (i.), y? = 133°3265+41-6557 for (11.) ; Goodness of Fit of Frequency Distributions. ati and record that the total frequency in (i.), or 83, was inadequate to determine y’, while that in (ii.) was adequate to give y? with about 3 per cent. possible error. We now consider the standard deviation of P. This for a fourfold table is Tp = 30,2 - 1 Pay?) — Po(x’) }- The value of Po{y?) is not tabled in the Tables for Goodness of Fit, for the simple reason that it is provided in the Tables of the Probability Integral*. In fact, ror n= 4. Tf. 7° 7 = a => 2 = —3X? Vee Gas Vere a Tayo ini Nine eo = fe (—e-8 —y2e-2X + e- 2X ) a Q ‘ =— = —3X" : ae ve 2X (y dy) 5 1 pa or Gar 5) By ary: oy2 : ee eo Wdy. _ _ 2 pee zs mak 5 =e ( Pal) =e dx } or Pay) =4/ 2 [e-Bay = 2x 41-2) =2x 1-4 +4)}, vX where $(1+) is the probability integral ¢. Now y?=-7080 and therefore y=°84143 for (.), and x’ =133°3265 and therefore y=11°54671 for (ii.). We can find P,(y) for (i.) at once by interpolating between °84 and ‘85 in the tables for the probability integral. There results 4(1+«) = °799945, and there- fore }(1—a) = 200055 and P,(yv) = °40011 for (i.), but Pa(y) =°8713. Hence = : 2-°5480(°8713—°4001) = °6008. op Thus we deduce 4049 for the probable error of P in (1.). * This fact has, of course, been known since the first “goodness of fit” table was calculated, but was overlooked by certain American writers. I therefore gave a proof in a joint paper by Dr. Heron and myself, Biometrika, vol. ix. p. 312. + ‘Tables for Statisticians,’ p. xxxi. f Loecat.p. 2: 378 Testing the Goodness of Fit of Frequency Distributions. In the Eye-colour case the determination of P, (11°54671) must be a little less accurate, for the ordinary tables do not go to such high values of the argument. Interpolating between the values of —log F tor 11 and 12 in Table IV. of the ‘ Tables for Statisticians,’ we find —log F(11°54671) = 30°46966, or log F = 31°53034. Hence 4(1—a) = 3°3911/10* and Psy) = 677822/10" = 007/102: But* ~~ Pig) HAvsa Aes thus op = : 2+45.48( 1035/1028 —-007/1025) = 1-2618/10%. Thus the probable error of P is 851/10. The numerical results for the two cases are: Gi.) P=-87134-4049, (ii.) = P = 1:035/10%++851/102. Now the whole range of P is from 0 to 1, and the probable error of P shows that within the limits of random variation P for the Matura table may be anything from 0 to 1. The data are accordingly absolutely inadequate for the purpose of indicating whether or no colour of flower and prickliness of fruit are associated. In the second case the probable error is such that P might really be zero, but it is not such that P could ever be replaced by anything but another indefinitely small probability. In other words, the data are amply sufficient to demonstrate that eye-colours in parent and child are definitely associated. On the other hand, the large value of the probable error of P, relative to the size of P itself, justifies a further emphasis of the caution given above to compare indefinitely small magnitudes of P of different orders with considerable reservation. * © Tables for Statisticians,’ p. xxxv. [ 379 XLII. Long-range Alpha Particles from Thorium. By Sir Ernest RutHerrorD, F.R.S., and A. B. Woop, M.Sc., Lecturer in Physics, University of Liverpool *. N the course of an examination of a strong source of the active deposit of thorium by the scintillation method, one of us observed the presence of a small number of bright scintillations which were able to penetrate through a thick- ness of matter corresponding to 11°3 em. of air at 760 mm. and 15° C. These scintillations were undoubtedly due to alpha particles and of greater velocity than any previously observed; for the swiftest alpha particles hitherto known, viz. those from thorium C, have a range in air of 8°6 cm. The number of these long-range alpha particles is only a small fraction of the total number emitted by the source. The actual number of long-range particles decreased ex- ponentially with time, falling to half value in 10°6 hours— the normal period of decay of the active deposit of thorium. Owing to the pressure of other work, the experiments were kindly repeated and extended by Mr. A. B. Wood, who examined in detail the variation of the number of scintil- lations with thickness of matter traversed. There are still a number of points that require further examination, but as neither of the authors is likely to have time to continue the experiments in the near future, it has been thought desirable to give a brief account of the preliminary results. Heperimental arrangements. The end of a brass rod, 1 mm. in diameter, was exposed as negative electrode in a small vessel containing a strongly emanating preparation, either of radio-thorium or meso- thorium. By suitable adjustment of the electrodes, the active deposit was concentrated almost entirely on the end of the rod—a condition essential to the accurate determination of the ranges. After two days’ exposure to the electric field, the wire was removed and placed end-on at 4 mm. distance from a small screen of zine sulphide viewed with a low- power microscope. Care was taken that the axis of the microscope passed through the centre of the rod. The screen was permanently covered with a mica plate whose thickness corresponded to 8°6 cm. of air—the maximum range of the alpha particle from thorium C. All scintillations then observed were due to alpha particles which had passed through the mica plate and 4 mm. of air, 7. e. a distance * Communicated by the Authors. 380 Sir E. Rutherford and Mr. A. B. Wood on corresponding to 9 cm. of air. With the most intense source available, about 20 scintillations per minute were counted on the microscope. The number fell off rapidly with increase of distance of the source, but an occasional « particle was still observed at a distance of 2 cm. from the source. In order to determine the variation of number of these particles with distance of matter traversed, thin screens of aluminium, each corresponding in thickness to 1:25 mm. of air, were successively interposed between the source and screen. It was found that the number of long-range particles remained constant between 8°6 and 9°3 cm. of air, but decreased in number from 9°3 cm., vanishing at 11°3cem. The grouped average of all the observations in a large number of experiments 1s shown in fig. 1. Fig. 1. 8-6 Bh noes 10:1 10-6 a eS Range ii CMs. It is seen that the curve shows evidence of two fairly definite slopes AB, BC, as if there were two sets of alpha particles present of different ranges. This important point Long-Range Alpha Particles from Thorium. 381 was very carefully examined, and the results of a special series of observations are shown in fig. 2. It will be seen that there appears to be a fairly definite change in the slope when the number of scintillations is reduced to about two-thirds of the total. Fig. 2. ) $ o @ So" © eo) Relative number of a particles 1-6 =] Y—|_9—-—| 00I 9:6 Of [- ‘sua ul asuey 9-01 Seif Sl Tt will be seen from the curves that the alpha particles start decreasing in number from about 2 cm. of the maximum range. The variation of number with distance is much slower than that to be expected for a single group of alpha particles of corresponding range. This is brought out in fie. 3, which shows the results obtained when the scintillation- distance curve in air was obtained for the two well-known groups of alpha particles emitted from thorium C of ranges 50 and 8-6 em. respectively. In these cases, the scintillations fall off rapidly, beginning at about 1 cm. from the end of the corresponding range. The difference between the slopes of the scintillation-curve for the long-range alpha particles and those from thorium © cannot be explained by the oblique Doe Sir EH. Rutherford and Mr. A. B. Wood on path taken by some of the rays through the mica on account of the nearness of the source and screen. Calculations showed that the influence of obliquity could only account for a small fraction of the difference actually observed. Fig. 3. 0 2 4 Gor nie te 10 Range in’ cms. The results we have so far obtained certainly seem to indicate either that (1) the long-range alpha particles are expelled with variable velocities over a comparatively narrow range, and in this respect differ markedly from alpha particles from ordinary radioactive products which are known to be expelled with identical velocity, or (2) that two homogeneous, groups of alpha rays of characteristic ranges are present. In order to distinguish definitely between these two hypotheses, it would be necessary to count many thousands of alpha particles, but other evidence suggests that (2) is the more probable explanation. The slope AB ends at about a range 10°2 cm. and when the number of alpha particles is reduced to about two-thirds Long-Range Alpha Particles from Thorium. 383 of the total. This suggests that two groups of homogeneous rays are present, one-third of maximum range 10°2 cm. and two-thirds of range 11°3cm. ‘The slope of the scintillation curve to be expected on this hypothesis agrees within the experimental error with the observed curve. This division of the alpha particles into two homogeneous groups may be compared with the two well-known groups of alpha particles emitted from thorium C, for it is known that one-third have a range 5°0 cm. and two-thirds a range 86 cm. This suggests that the new groups of alpha particles have their origin in thorium C, and that one-third of range 10-2 cm. accompany the alpha particles of range 5:0 cm., and the remainder of range 11°3 cm. accompany the alpha particles of range 86 cm. While it is very difficult to prove the correctness of such a deduction, the numerical agreement in the divisions of alpha particles of different ranges is certainly striking. We have not so far examined experimentally whether the new alpha particles are expelled from the alpha ray product thorium C, but this seems very probable. To settle this point, it will be necessary to prepare a strong preparation of thorium © and to determine whether the period of transformation, measured by the new alpha particles, is in agreement with the accepted value for thorium O, viz. half value in 60 minutes. Number of Long-range Alpha Particles. Since the number of long-range particles decreases at the same rate as the alpha-ray activity of the active deposit of thorium, it is convenient to express their number as a fraction of the total number of alpha particles emitted per second from thorium C. For this purpose, the number of long-range particles per minute was measured with the source fixed at a known distance from the screen. ‘lhe active deposit was then allowed to decay in setu for 32 hours. The absorbing screen was then removed and the number of alpha particles from thorium C measured at distances from 6 to 7 cm.,so as to include only the longer range aipha particles (8°6 cm.) from thorium C. In this way it was found that the fraction of long-range alpha particles was 1/6700. Taking inte consideration that the alpha particles of range 8°6 cm. from thorium © are two-thirds of the total, the fraction becomes 1/10000. Preliminary observations by different methods gave a somewhat lower value, but the above number cannot be much in error. We thus see that the long-range alpha particles 384 Sir E. Rutherford and Mr. A. B. Wood on are expelled in a very small proportion (1/10000) compared with the ordinary alpha particles. Unless a very intense source be employed, it will not be easy to detect the presence of the long-range alpha particles when the ordinary alpha particles are first absorbed by a layer of 8°6 em. of air. In the Bragg ionization curves from thorium © given by Marsden and Perkins*, a small residual activity is to be noticed beyond the distance 8°6 cm. which is relatively more marked than for the corresponding curve for radium C. This no doubt is to be ascribed to the effect of the very long- range alpha particles. Discussion of Results. It is now well established that thorium C is anomalous in breaking up in two distinct ways. One-third of the atoms are transformed with the emission of alpha particles of range 5°0 cm., and the remainder gives alpha particles of range 8°6 cm. These modes of transformation of thorium C have been examined in detail by Marsden and Darwin J, and an ingenious scheme of changes has been suggested to account for the facts observed. It is known that the products corre- sponding to thorium C in the radium and actinium series, viz. radium © and actinium C, also have two distinct modes of transformation. F ajans t showed that 1/6000 of the atoms of radium C give rise to a new product of half period 1°38 minutes, which emits beta rays in its transformation. In a similar way actinium C has been found to emit two sets of alpha particles of range 5-4 and 6'4cem.§ This is ascribed to a double mode of transformation, 1°5/1000 of the atoms breaking up with the emission of alpha. particles of range 64 cm. Assuming that the new alpha particles of thorium can be divided into two homogeneous groups of range 10:2 and 11°3 em., it is seen that thorimm © must break up in four distinct ways with the expulsion of alpha particles of ranges 5-0, 8:6, 10°2, and 11:3 em, at a7 C. The possible modes of transformation of thorium C are thus more complicated than was at first supposed, and it is obvious that the suggestions given by Marsden and Darwin as to the modes of transformation of this substance can be * Marsden & Perkins, Phil. Mag. xxvii. p. 691 (April 1914). ' caer & Darwin, Proc. Roy. Soc. eke: rill. p. 17 (1912); see also Marsden & Barratt, Proc. Phys. Soc. xxiv. 1, p. 50 (1911); Marsden & Wilson, Phil. Mag. Xxvl. p. 354 (19138). i Fajans, Phys. Zeit. xii. P. 369 (1911); xiil. p. 699 (1912). § Marsden & Perkins, Phil. Mag. xxvii. p. 604 (1914). Long-Range Alpha Particles from Thorium. 385 only a partial explanation. From the close analogy of the ** ©” products of radium, thorium, and actinium, it is probable that further examination will show an analogous complexity in the modes of breaking up of radium © and actinium C. The loss of energy in the form of expelled alpha particles is very different in the four modes of transformation of thorium C, and in consequence it does not seem likely that the resulting products can be the same in all cases. The differ- ences in the energies emitted by the two branch products of thorium C formed a serious difficulty in the original expla- nation given by Marsden and Darwin of the two main modes of transformation of thorium OC, and this difficulty is now further increased. A more detailed discussion on these interesting points will be reserved until further experimental information is available. The relation found by Geiger between the range of the expelled alpha particles and the life of the radioactive product, suggests that the average life of the atoms which expel the long-range alpha particles must be exceedingly short, and of the order of 107 and 1071 sec. for the products emitting alpha particles of range 10-2 and 11-3 cm. respectively. The following table gives the velocity of the four groups of alpha particles from thorium ©, taking as the basis of calculation the measurements of Rutherford and Robinson that the velocity V of the alpha particles from radium C of range 6°94 em. at 15° C. is 1:°922 x 10° cm. per second, and assuming Geiger’s relation V?=kR where RB is the range. Ratio of Calculated Range at 15° C. velocities. velocity. Miporimm Co. ..c....05.-| 4°95 cm. il 1-71 10° em. Borin ©, cece. on. SiO 4; 1:205 ZG eck ee New product (C,)...... 102° :. L275 ph ie New product (C,)...... TAs ae 1°32 NO ea ee Recently one of us* showed by direct measurement that the velocities in two main groups of alpha particles from thorium C were in good agreement with the calculated values if the range of the alpha particles from thorium C, is 4°95 cm. instead of 4°&0 cm.—the value usually taken. From our measurements, there appears to be no doubt that the higher value is more correct. * A.B. Wood, Phil. Mag. xxx. p. 702 (1915). Phil. Mag. 8. 6. Vol. 31. No. 184. April 1916. 2.) 386 Prof. H. Lamb on Waves due Summary. Evidence is given that the active deposit of thorium emits a smill number of alpha particles of greater velocity than any previously observed. These alpha particles are believed to have their origin in the transformation of thorium OC, and appear to be divided into two homogeneous groups of maximum range 10°2 and 11°3 cm. The number of these alpha particles is about 1/10000 of the total number emitted from thorium C, two-thirds of the number having a range 11°3 cm. The results indicate that the atoms of thorium © can break up in three and probably four distinct ways with the emission of four characteristic groups of alpha particles of ranges 50, $76, 1072, and’ 11-3 cm: University of Manchester, February 1916. XLIV. On Waves due to a Travelling Disturbance , with an application to Waves in Superposed Fluids. By Horace Lams, F.R.S.* iL. | be any case of wave-propagation in one dimension a distribution of impulse of amount cos kx per unit length gives rise to an oscillation of the type n=O(k) coskxe™, . a where o is a function of £ determined by the theory of free waves. ‘he effect of a concentrated unit impulse at the origin is accordingly given by a Fourier expression 1= : ( b(k) cos kv e''dk T No sy ae ge 1 »\ pt(ot— kz) i) bi hye dk+ 5 \ Qo = b(kjetot dk. 2) 0 To avoid possible indeterminateness in the sequel we may introduce a factor e-"’ to represent the effect of slight dissipative action. If it is required in order to make the integrals convergent, we may suppose yp to be a suitable function of &, but it will serve our purpose if we take pu to be constant, and in the end infinitesimal. To find the wave-system due to a concentrated disturbing force of unit magnitude travelling with the velocity c in the direction of, w-negative, we may take the origin at the instantaneous position of the disturbing influence. The result of an impulse 6¢ delivered at an antecedent time ¢ is * Communicated by the Author. toa Travelling Disturbance. 387 given by (2), if we replace « by ct—w, and multiply by é¢. apm from t=0 to t=, we obtain a b(het- Ke erwtdit [ p( (i: On Aiad| dt. NM 23) i the absence of the factor e- the result would in some cases be indeterminate, the physical reason being that we can superpose a system of free waves having the prescribed velocity c, if such are possible. The as with respect to ¢ gives thr ee thr meee je ak x 1 ob lkyedk te = 55 9 b-U(o—ke) * 203, p—iwotke) The coeificient ~ being supposed small, the most important part of the result will be due to values of & in the first integral which make Ge Cie Ure ee rahay approximately, if such exist. Writing, eae k=K+k', where « is a root of {5) and &' is assumed to be small, we have o—he= (Se) = (UAT rags nearly, where U is the group-velocity corresponding to the wave-velocity c. The important part of (4) is therefore THE m ead eae Nga ae oper (7) since the extension of the range of integration to k= +2 makes no serious difference. The integral comes under a known form. IE a be positive, we have ‘7 Seam (ane =, (xe 0) asa! ote. Or 3 Lea 1a eZ dm nS O . [e>0] ; 2 a—im ={ Der ee Opa tt (9) Hence if U0, 2 GO) ig, he Bet Lin ae rd) dt). a ae) TUK ML) and, for «<0, a p (x) Ba i(k+ B)z+ Lie lems Gad ldbye) 1 ts (16) The wave-system now extends on both sides the origin. For a moderate range of « we may put @=0 in the exponentials; thus Hi Pl), Uy eee) I+ J QudUjdeye ay) If dU/dk is negative we must take the lower sign in (13). The result is, with the simplification referred to, a (x) i(exr— 17) pire C= dWidee oo Here, as in (17), the upper or lower sign is to be taken = according as #0. 3. The formula (12) may be used to reproduce a number of known results. Thus, in the case of an impulsive pressure cos kx on the surface of deep water, the consequent surface- elevation is (18) y= — 4, sin ot cos he se. pias pete ite) en (elt) To conform to (1) we must put | ik aj ee ue 2 p(k) po” se e es ee es ( 0) and take the real part. Hence, for the train of waves following a unit concentrated pressure, we find 9 n= — Fee KX, - ° ° co + (21) which is the known result. We have here made use of tlie relations c=xc, U=te. Again, in the case of water of finite depth h, the result of a surface-impulse cos ka is easily found to be n=— * tanh khsinotcoskz. . . . (22) We therefore write | gh oe po 1227 N10 eh SR INET) 390 Prof. H. Lamb on Waves due In this case we have og’ =gk tambith, |...) ay tenance if Dieh i> Aa aT Ta 2 ° e e 2 an e(1 sinh Eh) (25) Hence for the effect of a travelling disturbance, we find and 2 sinh? ch 7 7 ~~ oe?" sinh eh cosh kh— Kh ee This, again, is in accor dance with a known result *. The critical case of U=c occurs when we have water- waves subject to gravity and capillarity combined, if the velocity ¢ of the travelling disturbance is equal to Kelvin’s minimum wave-velocity. Since P=gk-TR, . 3. ee where T! is the ratio of the surface-tension to the density, we find, in the critical case, dU c ak = im, ee e e ° e e ° (28) Taking the value of (4) from (20), and substituting in ou we have Lae U n= Fy gsin («e+ a7). oe The critical values of « and ¢ are given by =V/ (9/1), C=2”7 (gl)... 0 eee 4. The formula (12) supplies all that is essential for calculations of wave-resistance. If we denote the potential energy of the waves, per unit length of the axis of «, by it D) On’, where C is a coefficient depending on the nature of the medinm, the mean energy, both potential and kinetic, in the wave-train will be |? pe ble ot 3 GU CY) absolute value being indicated in case ¢(«) is imaginary. Supposing, for detiniteness, that Uc, the fixed point must be taken in advance of the disturbing body, and c—U is replaced by U—c. 5. The application to waves in superposed fluids has an interest, owing to its bearing on the phenomenon of “ dead- water” *, Suppose we have a stratum of depth A and density p resting on a liquid of greater density p’ whose depth is prac- tically infinite. The problem of free waves in this case was solved long ago by Stokes+. If the origin be taken in the upper surface, with the axis of y drawn vertically upwards, we may write @=(Acoshky+Bsinhky)coskxcosat, . (33) op —=CeVeos kx cos ot, 0. a (BAY the two formule relating to the upper and lower fluids respectively. If 7, ’ denote vertical displacements of a particle, we have Ops Op, dy OG! = mr ce aero fF (35) whence 7=— : (A sinh y+ B cosh ky) coskxsinat, . (36) f= — * (elveos ASI Gila een se The variation of pressure about a particle is given by P=Pp (3 —9n) 5 PP (% —s') =) (38) In order that the pressure at the upper surface may be constant we must have, therefore, 2 Ah BO ot. (B89) 3 gk * Ekman, l. c. enfra. “On the Theory of Oscillatory Waves,’ Camb. Trans. vol. viii. (1847) ; Math. and Phys, Papers, vol. i, p. 212. 392 Prof. H. Lamb on Waves due The continuity of pressure at the interface between the two liquids requires 2 (= cosh kh+ sinh ih) pA— ( o” a sinh kh+ cosh kh) pB o2 ue 2 ze 1) eth... (40) The condition that the vertical displacement at the inter- face must be the same for the two liquids gives Asinhkh—Bcoshkh=—Ce-* . . . (41) This system of equations has two solutions, which it is convenient to distinguish by suffixes. In the first of these we have Cy" =e. e ° ° ° ° e « (42) as in the case of a single fluid, with BC) = Ne e ° e . ° ° (43) In the second solution we have ane, gk(p'—p) sinh kh — “~~ p' cosh kh+ p sinh kh’ (44) with Bower: Os per o,? A, gk? A, (pot The wave-velocities corresponding to these two modes of oscillation are given by O71 Oo Cle Tie ier ak ° ° ° e ° (46) As the wave-length (27/k) increases from 0 to w, ¢, increases from 0 to an upper limit co, given by = (1-4 he gee 0 0 )o (47) If U,, Uz be the corresponding group-velocities, we have iL oy Ui 5 ot : ° e ° . ° (48) a p' kh p' cosh kh+ psinh kh”* sinh kh J Hah ~(shm) 2 2) eae ay 0 ae ke As the wave-length increases from 0 to «©, Us, increases from 4c, to Co. toa Travelling Disturbance. 393 From this point we may without confusion use the symbols n, n to denote the elevations at the upper surface and at the interface, respectively. Hence, putting y=0 in (36), and y= —h in (37), and combining the solutions, we have n=— = (oA; sin ojf+o,A,sinogt)coskx, . . . . (50) g =— (axes sin ot — ie o,e**A. sin ot) cos kx. (51) Use has here been made of the relations (39), (43), (45). These formule correspond to the following initial distribution of d, ¢’, V1Z. 2 o= { Ase + As (cosh ky + ok sinh ky ; coskx, (52) 2kh 2 g= {Aa Fe Aa} eb 003 kn BPP Shs re sede) Pg The coefficients A,, A, are to be regarded as functions of k, depending on the particular manner in which the oscillation (of prescribed wave-length) is started. When it is necessary to call attention to this, they may be written as Ai(k), As(h). The wave-trains due to a travelling disturbance are accordingly given, on the analogy of (21), by == ou) Aj(«1) sin «2 — ren Ag(x2) sin ket, (94) o.eT kh E Co eka : a 1! = — meu, U,) Aj(k;) sin Kyu— p! = ge = Up) A»(2) sin x2, (59) where ae (56) and «, is the positive root of the equation 2_ 9 (p'—p) sinh kh Ee : kh’ p’coshkh+psinhkh ~ Oe It is also understood that in the preceding formulze 01=— KC, Oo=Kol. . e ° ° ° (58) If c>co, x2 is imaginary, and the second terms in (54) and (55) disappear. The periods 27r/o,, 27r/c. being different, the mean energies, and the mean rates of transmission of energy, in the two systems of waves may be calculated independently. 394 Prof. H. Lamb on Waves due The potential energy, per unit length of the axis of a, being if 1 . “9 9 9pt + s9le py, sy GR) the mean energies are 2 | | B= = {e+ (p —plem*h } A2(«,), el), 2 o2koh E,= pee aia [e+ (p pent Az*(k2). (61) The total wave-resistance is h= kh, +R,, . ° . e . A (62) where R= =n, R= ny 5 it 6B) Hence Ri= “{p +(e’ ple} AXe), © - +. . G4) p o2e2koh pannel apt Bota ug —2koh aap Bye Uy) Pt PO ae 6. As regards the determination of Aj,(k), A,(k), two special cases may be considered. | If an impulsive pressure coskx be applied to the upper surface at the instant t=0, we have, initially, PPHCOs ka) oa for y=0, and y PO=DO a2 os) ae (67) for y=—h, the latter condition securing the continuity of the initial distribution of impulsive pressure. Hence from (52) and (53) par PA ai. aie 1 kh 2 2 (p’—p)e-* A, — p (cosh kh— psn kh + i ae “5 )As=0. (69) With the help of (44), we find 1 (p' —p)e72h* ti 1) = : A Saat CTT ig 1 PRS ia ESN : Ai(k) o(k) pip+(p'—p)e 2} (70). toa Travelling Disturbance. 3995 Hence, in the case of a concentrated pressure travelling over the upper surface, guia Aip+(p'—p)e =}? 5 2 45 —2Kolt ne lie “saab ive iby aiie- ae 11 2G Ui) glen @ pe mae 0? Next, suppose that an extraneous impulsive force cos kz is applied downwards to an infinitely thin stratum at the interface. The initial conditions are now i= OW gS yas Gtr ate) pi —po=eoshe oi. Js ID) seaGial.) R= for y=0, and fory=—h. Thus A, +A,.=0, and (p'—p)e-**A, —p(cosh kh — ar =1. (75) Hence Ah A,(h Ae (4) = —A,(4) = aati Rigneamns 1 ve (76) The comparison with (70) shows, as we should expect, that the second type of waves is now favoured relatively to the first. In the case of a concentrated vertical force travelling along the interface the two components of the resistance are accordingly —2k ih ge ~ pt (pi peat} 2 p C K2 eZee Us) gp tip en, It is to be remembered in these problems that R, and R,’ vanish when c>¢p. The formule as they stand are open to the objection that they make some part of the resistance increase indefinitely as c is diminished. Thus (71) makes Ry=a for c=0, whilst (78) makes R,’=« under the same condition. This paradoxical result is easily understood if we remember that we have imagined a finite force to be concentrated on an (77) (78) 396 Prof. H. Lamb on Waves due infinitely narrow area. ‘To get anything like a representation of real conditions we must suppose the force to be diffused to someextent. Mathematically, this is most easily attained by introducing a factor e~ under the integral signs in (2), and making the consequent changes in the subsequent formule. The effect is that the pressure is now distributed on each side of a central line according to the law b ” a0 at)’ (79). the intensity falling to half the maximum when «= +b. In this way we find, for a disturbance advancing along the upper face, gota 80) 7 Hee pe 7 ! — 2k h- 2kod Ree ee : 81 1p 2€e—Wa)igo-+ (oS pn is whilst for a disturbance travellin g along the interface . Gamat a 2K,6 89 7 oe Cee (82) a 2 = 2kob R= : cate (83) p'—p 2(¢— U2) ap + (p!— pe" f 7. The name ‘‘dead-water” is given to a phenomenon occasionally observed in Norwegian fiords, and other similar localities, where the sea is covered by a stratum of fresh water brought down by rivers. It consists in an abnormal resistance experienced by slowly sailing vessels, which disappears if the speed can be raised above a certain limit. The vessel appears to be gripped by some mysterious power, and cannot be properly handled. The matter was brought by Nansen to the notice of V. Bjerknes, who attributed the phenomenon to the inter- facial waves generated at the common boundary of the fresh and salt water. At his suggestion the question was taken up by Ekman, who has devoted a long memoir™ to the subject. ‘This includes descriptive accounts from various * “Scientific Results of the Norwegian North Polar Expedition,’ 1893-96, pt. 15. (Christiania, 1904.) toa Travelling Disturbance. 397 sources, and an account of tank experiments made in verification of the above explanation. There are also some indications of mathematical theory. A fuller publication was promised: but I do not know whether this has taken place. From the theoretical point of view the matter is not altogether a simple one. The difference of densities is actually very small, and although interfacial waves of con- siderable amplitude may be readily generated, their energy for given amplitude is relatively slight. It is obvious, in fact, that if the ditference be small enough no appreciable effect of the kind contemplated could be expected, since the case would be practically that of a homogeneous fluid. It is difficult to devise a case which shall admit of mathe- mitical treatment, and at the same time be comparable with actual conditions; but in default of this we may examine whether any indications can be gathered from the preceding formule. In the phenomena referred to the ratio (p'—p)/p has some such value as ;5. The expressions Boe ee 8 Pip ple 7%, which occur in our formule, may therefore be replaced without sensible error by p. The values of R,; and Rj’ in (80) and (82) are therefore practically the same as if the fluid had been homogeneous. Since 9?/c*=«,?, we have from (80) and (81) ae 2 C Ceres Be ae Uy cred One approximately. A glance at the numerical table at the end of this paper, which is calculated on the basis of (o' —p)/p=.45, shows that for values of c in the neighbourhood of ey this. ratio may be enormous, in spite of the smallness of the first factor, if 6 be at all comparable with h. That is, even when the disturbing pressure acts on the upper surface, the resistance due to the interfacial waves may greatly exceed that (in itself very small) due to the waves on the upper surface. When the disturbing forces act at the interface this resistance is again greatly increased; thus from (81) and (83) R,’ Pp ) Dk 6 — ; ee anes bee siya nee, ig 85), Ry (; —Pp Sie 398 On Waves due to a Travelling Disturbance. Ekman’s experiments showed, in fact, that the resistance was greatly increased when the bottom of his model boat approached or penetrated the surface of separation. As regards the actual magnitude of the resistance, an estimate, necessarily somewhat vague, can be made in a certain case as follows: It has been computed ™* that the wave-resistance to a horizontal cylinder of radius a towed at right angles to its length with velocity c, at a depth A in homogeneous fluid, is R=47’qpatefe", . RSM woth (cic) where k,=g/c? as before. If we assume that when the cylinder is towed along the interface the wave-resistance at the interface bears to that at the upper surface the ratio of R,' to Ry’, as found from (82) and (83), we get R=) | dapat ee cp 2a eee aN) To make the formule more comparable, we have put b=0, which involves an under-estimate, and leads to no difficulty for values of ¢ in the neighbourhood of $c). Thus if we put Kyh=2, and h=250 em., which was the estimated thickness of the freshwater layer in one of Nansen’s experiences, we find Reta.) ae in grammes per centimetre of length of the cylinder. | This corresponds to a speed c=='498c)=39 cm./sec., or about 1:4 kilometres per hour. Considering the somewhat precarious nature of the above comparison, no importance can be attached to the precise numerical value obtained for R.; but the order of magnitude of the result (which almost certainly errs in defect) seems decidedly to support the adequacy of the theory advanced by Bjerknes and Ekman. * Annali di matematica (3) vol. xxi, p. 827 (1918). mh Oo OO Or Hn Bo erie 2a) | 910 1. 9538 = LO HK od GO oe Os ot) The Theory of the Winds. 399 ie: Ay /h. | X,/h. 790 628 | ‘008 395 B26 win 316 157 | 020 224 209. |. 088 161 old "039 125 | 3:53 | -050 916 | 628 | -069 19°4 7:86 079 681 | 105 092 ‘Say GI) i a 109 48°3 | 31-4 150 44:0 | 62:8 145 1:00 1:00 1-00 1:02 1:08 119 1°46 1:69 2°09 3°08 5°45 10:57 | Lel(c—U,). | (wahe~*2*)2, 213 10-7 | OT 10-8 53910" DieicLO ae hor x 10-4 1:1210-1 1:35 «107! £29 x 107! Lesx 10? LASS AO =2 2°68 x 10-? 83-19x10-2 The symbols 1, A. denote the lengths of the waves at the upper surface and at the interface, respectively. XLV. The Theory of the Winds. To the Editors of the Philosophical Magazine. GENTLEMEN,— N a recent paper™ I suggested that the reason why the winds of middle latitudes blow as a general rule towards the poles, in opposition to the surface temperature gradient, is that the stratosphere is much warmer over high latitudes than it is over low latitudes. If the winds blew in accordance with the surface-temperature gradients, there would be north- easterly and south-easterly winds moving from the poles towards the equator, and the equatorial regions would be areas of low pressure whilst the polar regions would be areas of high pressure—the high-pressure belts which are now at latitudes 30° north and 30° south of the equator would move Such a distribution of the winds would result in a very considerable increase in the size of the frigid polar to the poles. areas. Dr. Walker has recently suggested that the effect of solar radiation on the upper atmosphere may be an important consideration. The comparatively high temperature of the stratosphere over the polar areas is not accounted for veryeasily. In the summer, no doubt, the atmosphere in these areas enjoys continuous sunshine, and for some time receives more heat * Phil. Mag. vol. xxx. July 1915, pp. 138-33, 400 | The Theory of the Winds. than does the equator ; but the reverse holds good. in the winter, yet the winter as well as the summer temperature of the stratosphere above the poles is higher than it is above the equator. One suggestion * made was that the high temperature of the stratosphere may be due to the presence in it of ions and meteoric dust. But it is necessary to suppose that these are most numerous in high latitudes. That ions exist in the upper atmosphere is now considered to be very probable. Prof. H. Nagaoka suggests that they may result from two causes. The first cause is the ultra- violet light of the sun, which he believes is capable of ionizing the atmosphere down to about 40 kilometres from the earth’s surface. The second is the stream of electrons emitted by the sun. These electrons according to Arrhenius would be caught by the earth’s magnetic lines of force and would be deflected in long spirals along the lines to the poles. Some heating of the atmosphere may result from their loss of velocity on reaching the atmosphere. They may also arrest some of the sun’s heat for the following reasons. At ordinary temperatures ions load themselves by attracting surrounding molecules of gas. They thus form complexes consisting of as many as thirty molecules. Such groups of molecules may possibly arrest rays other than the ultra-violet ones, and raise the temperature of the gas in which they move. Tam not aware, however, that any experiments have been made to ascertain the effects, if any, ions have on radiant heat. The variations which take place in the direction of the flow of the winds from time to time seem to demand some variable cause. Both the quantity of meteoric dust, and the number of ions in the stratosphere, no doubt vary from time to time. If ions can intercept the heat of the sun or warm the atmosphere by impact as they reach it, and owing to the magnetic field of the earth are deflected towards the poles, the « comparatively high temperature of the stratosphere over high latitudes might be accounted for. Yours very truly, Abbeyfield, R. M. DeEcey. Salisbury Avenue, Harpenden. Feb. 15, 1916. * Tbid. p. 31, fi aaa XLVI. The Variation of the Radioactivity of the Hot Springs ae lua. By A. Sreicuen, 8.J., Ph.D., Professor of Physics, St. Xavier's College, Bombay * N his interesting paper contributed to this Journal f, Professor R. R. Ramsey gives two instances of the variation of the emanation-content of springs. He finds the greater radioactivity at the time when the springs are yielding the greater amount of water. My experience with a highly radioactive spring here in India is just the opposite. Tuwa is a village in the Kaira District, Bombay Presi- dency, 319 miles north of Bombay. The geographical coordinates of the place are lat, 22° 43’, long. 73° 30’. Not far from the railway-station at Tuwa there is a small area, about 20 by 17 metres, where numerous hot and cold springs issue from the ground. The highest temperature of the springs which I measured is 67° C., ‘the lowest 28°C. On the 13th December, 1910, the Rev. H. Sierp and ee ao the radioactiv ity of the hottest and of the coldest springs tf. The method of measurement was the Schmidt shaking method ; the instrument used was a Wulf quartz- fibre electroscope with an ionization-chamber. We applied Duane’s correction for the absorption of rays by the walls of the cylindrical condenser §. The radioactivities are calculated by means of Laborde’s empirical values ||. All the measure- ments were made at the springs. We obtained the following results :— Hot spring : Saturation current...... 82-1 Mache units (M.U.). Radioactivity :......... 33°00 x 10-9 curie (C.). Cold spring : Saturation current...... 84°25 M.U. Radioactivity............ 33°88 x 1079 C. On the 11th April, 1911, we visited the springs once more, in order to ascertain whether the radioactivity of the * Communicated by the Author. + Phil. Mag. vol. xxx. pp. 815-818 (1915). t Indian Medical Gazette, December 1911. § Kohlrausch, Lehrbuch der Prakt. Physik, 11th ed. p. 657. || Laborde in "Handbuch der Radiuwm-Briologve und Therapie, p. 41: Phil. Mag. 8. 6. Vol. 31. No. 184. April 1916. 2H 402 Variation of Radioactivity of Hot Springs at Tuwa. water had undergone achange. We found that the supply of water from the springs was considerably less than in December 1910. There bad been no rain during December, January, February, March, and April. The actual strength of the flow, however, was not measured. The radioactivity of the hot spring was examined in the same way and with the same instrument as in December 1910. We obtained the following results :— Saturation current............ 154:37 M.U. Radioactivity «0. 62 00x 10s ae: Thus the radioactivity of the hot spring was almost double of what it had beenin December. For want of time the cold spring was not examined on this occasion. Here, then, we find the greater radioactivity when the spring yields less water. This result can be accounted for by considering the local conditions. The cold spring showed practically the same radioactivity as the hot spring. This seems to indicate that at Tuwa the radium-bearing rock is near the surface, as the cold spring is not likely to come from a great depth. Moreover, the radioactivity of the neighbouring wells decreases with their distance from the springs. ‘three wells were examined for radioactivity in April 1911”. Ist well. Distance from the hot spring about 200 metres. Saturation current............ 19°85 M.U. IA ClOACEiVaLy 7c is ocean 98x TOm ae 2nd well. Distance from the hot spring about a quarter of a mile. Saturation current............ 31 MOU: Radioactivity: 0. .0\.ax....) 3°20 | x. 57 9918 | 3427 | 160 ee 6-2 625 | 6835p 9908 | 2489 | 148 | 1475 | 679 | 6-75 oa92 | 3458 | 199 | 1995 | 7-72 | 7-75 2879 3473°D| 11385 se 8°81 8°78 8:80 2872 3482 105 La: 9:52 9°35 9°50 2869 3486 101 ses 9°90 9-73 977 2866 3489 98 98:75 | 10713 10°08 10°15 2861 3495 92 92 10°87 10°85 2860 3497 90 A 11-11 11:28 2855 3503 84 84 11°90 11:90 2789 3987 0 2730 3663 76 wag 13:16 13°3 2726 3668 81 12°35 12-4 2724 3671 84 84 |-11:90 11:90 2718 3679 92 92 2712°5| 3686°5 99°5 98°75 | 10°13 10°08 10°15 2696 3709 122 se: 10°32 10°53 2690 311% 130 L293oy heed 775 2685 3724 137 one 7°30 2681 3730 143 ne 6:97 6:95 6°95 2678 3734 147 IAT Dy | iOFO 6°75 \ eos © (eo) =] — on) QO Or It may be seen that the agreement between the calculated and observed values is exceedingly good. It 1s an inter- esting fact that the absorption-band group as observed for pyridine vapour is not the same as for that observed with a solution of pyridine, for the centre of the former is at 1/X=3587, while the centre of the latter in dilute solution, as shown in Table I., is about 1/A=3910. This fact has already been noted by Purvis, and is of some value in the present connexion because it enables us to determine whether the relation between the wave-numbers of the centres of two absorption-band groups holds good. Now the difference between the above values of 1/X for the two absorption bands is 323, and therefore this must be very nearly the fundamental infra-red wave-number of pyridine. Since the value of 1/A=3587 may be taken as an * Publications of the Carnegie Institution, Washington, No. 35 (1905). + Astrophys. Journ. xxxix. p. 243 (1914). 2 2 420 Prof. Baly and Mr. Tryhorn on Light accurate measurement of the central wave-number of the less refrangible ultra-violet band, and since, further, this must be an exact multiple of the true fundamental wave- number, it follows that the true fundamental wave-number 3587 Li divisor which gives a value a little larger than 323. It follows, therefore, that the wave-number of the fundamental infra-red band of pyridine vapour should be 1/326-°09= 3°066 pw, and also that the central wave-number of the more refrangible pyridine vapour band should be 12 x 326:09= 3913°08. As was pointed out above, the values given in Table I. show that the central wave-number of the absorption band at infinite dilution is a little greater than 3910, and it is therefore evident that with increasing dilution the centre of the absorption band approaches the true value for the vapour. The measurements in Table I. show that the maximum shift occurs in 10 N solution, which almost exactly corresponds to the molecular ratio of C;H;N to H,O of 1:1; and therefore it may be concluded that the maximum effect of the water, as far as altering the free vibration period of pyridine is concerned, occurs when the solvent and solute are in the proportions agreeing with the monohydrate C;H;N,H,O, and that the influence of the water decreases with dilution until, finally, the free vibration period becomes the same as that of pyridine vapour. It is also evident that we now have an explanation of the variation in position of an absorption band when a substance is examined in solution, namely, that it is due to a combina- tion or a tendency to a combination of the substance with the solvent. Moreover, it also follows that the fundamental infra-red band must also shift with solution; and in the fourth column of Table I. are given the wave-numbers of this infra-red band at the various dilutions, 7. e. 1/A +12. The wave-lengths of the infra-red bands are given in the fifth column of Table I. For obvious reasons it is impos- sible to measure the infra-red band for dilute solutions, but from some observations made in these laboratories it is clear that this band shifts ina manner comparable with that shown in Table I. The wave-numbers of the infra-red band have been plotted against the logarithms of the corresponding volumes of solution, and the curve obtained is shown in fig. 1, curve I. From the value given in Table I. it is clear that the fundamental infra-red band of pure homogeneous liquid pyridine should lie at 3°096. The absorption at this = 326:09, because 11 is the only integral Absorption and Fluorescence. 421 region of the spectrum of pyridine is somewhat complex owing to the bands at 2°95 w and 3°25 pw, due to the nitrogen atom and the hydrocarbon chain respectively. The first and second harmonics of the fundamental band for a thin film of liquid pyridine have, however, clearly been observed by Coblentz at 6°25 and 9°35 and by Spence at 6°35 w and 9°50 pw. We have also examined the less refrangible band of salicylaldehyde, and the values obtained are shown in Table ITI., the solvent in this case being alcohol. Tasue III. Vv. | 1/r. Factors. Infra-red Band. | 01042 | 3084 | 19x16235 | 616, 0208 | 3074 19x1618 | 618 | 1 | 8059 Lox eb Nii) 62 10 | 38045 | 19x160-27 | 6-24 16 eso Per oNeteon mills igas 20 | 3039 19x1599 | 6254 |. 820 8049 1 NGOS | Gs | 1280 8066 19 XD6I-4 | 7620 50000 | 3101 | 19 x 163°2 6:13 | It was not found possible to examine solutions of less con- centration than V=50,000 owing to the limitations set by the spectrophotometer, but it is obvious that the limiting value of 1/X is greater than 3101. Now Purvis* has measured the wave-lengths of the component lines of the more refrangible band of salicylaldehyde vapour, and tnere is no doubt that the central line in this case is at \=2514 or 1/A=3978. On the other hand, the infra-red spectrum of salicylaldehyde has not been observed, and therefore we can gain no direct information as regards the value of the fundamental wave-number in this region. It is, however, clear from analogy with other compounds, since 1/A=3978 is the central line of one absorption band of salicylaldehyde vapour, that, if v, be the fundamental infra-red wave-number, xv, must equal 3978 and yvz must equal some number which is rather larger than 3101, « and y -being two integers. We have observed both absorption bands of the aldehyde in alcoholic solution when V=1280, * Trans. Chem. Soc, cv. p. 2482 (1914). 422 Prof. Baly and Mr. Tryhorn on Light and the centres of the two bands are then at 1/~=3066 and 3874 respectively. In this case, since the concentration is the same for both bands, it is obvious that S08. SOME the wave-number of the infra-red band y a at the concentration V =1280. Since 2 and y are two integers the only possible solution is 3066 _ 3874 Ne Comes. 161-4. It follows that the wave-number of the less refrangible band in alcoholic solution is 19xv,, and that the wave-number of the more refrangible band is 24xv, The true value of v, for salicylaldehyde vapour must therefore be eet 6s Ta Z4 In the third and fourth columns of Table III. are given the wave-numbers of the infra-red band and the corre- sponding wave-lengths calculated on the basis y=19. In fig. 1, curve II. is shown the relation between v, and the Fig. 1. Log V. concentration from the values given in Table III. The dotted portion of the curve is extrapolated to show that by analogy with the corresponding pyridine curve, the caleu- lated value of 165-75 is very probably correct. The general conclusions, therefore, drawn from pyridine are thus fully confirmed by salicylaldehyde, namely, that on the addition of a solvent the absorption band first shifts towards the red, Absorption and Fluorescence. 423 and then on further dilution it shifts towards the shorter wave-lengths, until it finally reaches the true value for the vapour. One further example may be given, namely, that of aniline, which is interesting from the fact that it shows two absorp- tion-band groups in alcoholic solution. The central wave- numbers of these two bands have been measured at very great dilution (V=84,000) and were found to be 3496°7 and 4255 respectively. Now these two wave-numbers must be even multiples of the fundamental infra-red wave-number, and therefore we have 3496°7 4255 x gee where v, is the true fundamental infra-red frequency of aniline and 2 and y are whole numbers. The only solution is given by e=23 and y=28, whence the true values of v, are found to be 152°03 and 151:97 respectively. Now Parvis* has observed the less refrangible abserption band of aniline vapour, and from his measurements it is clear that the central wave-number is 1/A=3496°1. As this value is obviously more accurate than that obtained from the solu- tion, we may take the true value of v, for aniline to be 3496°1/23=152, which is a mean of the two solution values. Tt follows, therefore, that aniline vapour should show a very strong absorption band at 1/152=6°58 yw, or at the first multiple of this, 3°29. The absorption of a liquid film of aniline has been observed by Coblentz t, and in comparing the absorption spectra of vapour and liquid it must be remembered that in the latter the bands will have slightly longer wave-lengths. The value obtained by Coblentz was 3°34 uw instead ot 3°29 pw. These results are of some importance, for, in the first place, they show that the relationship between the infra-red and ultra-violet absorption bands pointed out in the previous. papers, namely that the central wave-numbers of the latter are whole multiples of a fundamental wave-number in the infra-red, holds good in solution. In the second place, they would seem to offer an explanation of the well-known fact that in general there isa shift in the position of an absorption band when the substance is dissolved in a solvent. Since the central wave-number is a whole multiple of a fundamental infra-red wave-number, the shift must be due to the fact that: the fundamental wave-number changes on * Trans. Chem. Soc. xevil. p. 1546 (1910). + Loe. cit. 424 Prof. Baly and Mr. Tryhorn on Laight solution, and it may be noted that a small change in this fundamental wave-number will produce a relatively larger shift in those wave-numbers which are its multiples. Although we have observed that the fundamental infra-red band shifts on solution by an amount and in the direction comparable to that of its multiples, yet no accurate obser- vations of the infra-red absorption spectra of substances in solution have been published. We have examined the absorption spectrum of an alcoholic solution of @-naphthol ethyl ether in the neighbourhood of 3, where both solvent and solute each show strong absorption bands. We found, however, that the two types of molecules do not separately exhibit their own absorption bands, but that the solution shows only one absorption band, which moreover is not a mean of the two due to the two components. ‘The solution absorbs as if it were a single entity with an absorption band of its own which differs from that of each component. It is possible that this will explain the interesting fact that the wave-length of the infra-red emission band of the Bunsen flame is not the same as that of either carbon monoxide or carbon dioxide. On the other hand, the two oxides are in equilibrium in the flame, and it is this equi- librium which is emitting the radiation. Just as a solution seems to exhibit an absorption band of its own and not the bands peculiar to its components, so the equilibrium of carbon monoxide and carbon dioxide present in the Bunsen flame emits a radiation of its own and not that of its com- ponents. We thus arrive at a reasonable explanation of the change in position of absorption bands when the absorbing substance is dissolved in a solvent, namely, that the solute acts as a single entity with its own fundamental frequency of vibra- tion in the infra-red. Since the frequencies of the ab- sorption bands in the visible and ultra-violet regions are multiples of this fundamental frequency, we see that these must shift in the presence of a solvent. As has been pointed out in the previous papers of this series, one of the results of the relationship between the fundamental infra-red frequency and the phosphorescent, fluorescent, and absorption bands in the visible and ultra- violet regions, is the existence of constant differences between the wave-numbers of the latter. Since the relationship can obviously only hold good when all the wave-numbers are characteristic of one molecular entity, the constant differences may be taken as evidence of the presence of one molecular entity. The existence of the constant differences enables us Absorption and Fluorescence. 425 to be independent of a knowledge of the infra-red spectrum, a matter of great importance in view of the fact that the infra-red absorption spectra have been observed of only relatively few substances. This is partly due to the fact that the experimental technique is relatively difficult, but also to the fact that no investigations have been made of the absorption of solutions in the infra-red region. The result has been that solids of high melting-point have not been examined. In view of what was said above about solutions, it is to be hoped that investigations of their infra-red absorption will be undertaken, and, indeed, we ourselves look forward, when more pressing matters have been disposed of, to carrying out some of the work in this field. In the absence of knowledge of the infra-red spectra, it is possible, as stated above, to argue from the constant wave- number differences between the bands in the visible and ultra-violet regions. The existence of these constant differ- ences has now been proved for so many compounds, that the relationship may finally be accepted as absolute. lt therefore becomes possible to use the converse argument, namely that, if a substance exhibits different absorption bands in different solvents and if the constant difference relation holds good between the various absorption bands, the change in absorp- tion in passing from one solvent to another cannot be interpreted as an indication that the substance has changed its constitution. It is, however, believed by many chemists that a change in absorption produced by a change in solvent always means a change in the primary preeennee of the compound i in the chemical sense, the belief being apparently based upon the conception that a particular absorption curve must be characteristic of a definite molecular structure, and that itis not possible to call into play any free period of vibration which previously was latent. As may readily be understood, it was not possible to prove the existence of the constant difference relation for solutions until the variation in the position of the absorption bands with concentration had been worked out. Now that this variation is understood, it is possible to apply the argument to the problems of chemical constitution. This has now been done for the phenols and nitrophenols with results that are eminently satisfactory, but a description of this work must be reserved for a further paper. The second point to be discussed is the variation in the absorptive power with concentration of the absorbing com- pound in a given solvent. 426 Prof. Baly and Mr. Tryhorn on Light According to the well-known law the absorptive power is expressed by the relation I 0 L SY ae where I, and I are the intensities of the incident and emer- gent light respectively, d the thickness of the absorbing — layer, c the concentration, and & the absorption constant. In actual experiment the absorption is best expressed in terms of that exerted by 1 cm. of a molar solution (one molecular weight dissolved in one litre of solntion). The value of k, which according to Beer’s law should be constant, is readily enough found from log pac ae In practice, however, it is rarely found that Beer’s law holds good, but up to the present no expression has been found which connects the value of & with the concentration. Some previous observations of the absorptive power of aqueous solutions of pyridine* have shown qualitatively that it increases very materially as the concentration decreases.. An interesting fact in connexion with pyridine is that its absorptive power is much greater in acid solution than in neutral solvents+. This at once affords a considerable range of solvents, and we have determined quantitatively the value of k for pyridine solutions in the following solvents—water,. alcohol, N/5000 aqueous hydrochloric acid, N/1 aqueous hydrochloric acid, and concentrated sulphuric acid. In the case of the neutral solvents, considerable care was necessary to guard against the absorption of atmospheric carbon dioxide which at once gave too high values fork. In the case of each solvent, the absorptive power of pyridine in- creases rapidly with the dilution until a constant value is. reached, and the values observed are given in Table IV. In previous papers ft the view has been put forward that the molecules of every substance are surrounded by a more or less condensed force field of electromagnetic type. These fields are opened or unlocked by the influence of a solvent,. such opening up taking place in definite stages. Hach of * Baly and Rice, Trans. Chem. Soc. ciii. p. 91 (1918). + Hartley, zb¢d. xlvii. p. 685 (1885), and Baker and Baly, zbzd, xci. p. 1122 (1907). {t Baly and Krulla, Trans. Chem. Soe. ci. p. 1469 (1912); Baly and. Rice, zbid. ci. p. 1475 (1912), ciii. pp. 91 & 2085 (1918) ; and Baly, Phil. Mag, xxvii. p. 632 (1914), xxix. p. 223 (1915). Absorption and Fluorescence. 427 these stages is characterized by its power of absorbing light. rays of definite wave-length, and consequently when only one absorption band is shown, as in the case of pyridine, only one stage in the opening up is present, and we are therefore only concerned with the equilibrium between the closed fields and the fields when opened up to one only of the possible stages. It is obvious, from the fact that the absorptive power increases with dilution, that the solvent increases the amount of the opened up stage present. The mass-action law as usually conceived does not therefore hold good, since if bis the fraction of the molecules opened up, — = constant. In other words, the fraction opened up, and therefore 4, should be independent of the concentration. On the other hand, an increase in the mass of the solvent increases the relative number of opened-up molecules in the equilibrium up to a constant value when the solvent has opened up the maximnm possible for that solvent. If /be the fraction of that maximum which exists at any given concentration, it is clear that f must lie between 0 at V=O (infinitely large concentration) and 1 at V =~ (infinitely small concen- tration), and, further, that 7 is some function of the con- centration. The simplest possible relation connecting f with the mass of the solvent under the given conditions f=l—e-, the mass-action law leads to the relation where « is a constant characteristic of the substance and the solvent used. It is a justifiable assumption to make that the absorptive power () is proportional to the concentration of the opened up stage, and we may therefore put k f=_ where K is the maximum and constant value of k in the given solvent. We thus have k x= ton ae whence K—k _ a rag and lo K = aV. 428 | Prof. Baly and Mr. Tryhorn on Light From this formula the absorptive power of pyridine at various concentrations in the five different solvents has been calculated, and the curves showing the relation between k and V are given in fig. 2. For convenience in plotting the eae 2 pe dlls oo lest Bae effi Pee eae Se ae a HES roe Gl 24 ie TOG asa a epee ae Serio Pe ee ace ee | pa FS | J} eH fie a Ps a | FD Me PA, i ae ae EcEECEE Eee . | | | J ea pt) 19424 a aoe Wt mae: Be feos — os aE 4682 | I is | amc sunanesasaeia: 446178 00020406 O8 1-0 2 14 16 18 2022242628 3:0 32 3436 38 Log V. = (eo) =| Ny a= Absorptive power of pyridine. Curve IV in N/1 HCI. Curve I in water. | V in concentrated H,SO,. II in alcohol. ITI in N/5000 HCl. ”? rh) 9) logarithms of & have been placed on the ordinates and the logar ithms of V on the abscissee. The crosses on the curves are the values experimentally found and shown in Table IV. As may be seen, these experimental values lie on the appro- priate curve in every instance. The values of @ were calcu- lated from the absorptive power of pure pyridine and the 429 Absorption and Fluorescence. k th ‘or c a = y—aV e formula K L—< Water. Alcohol. N/5000 HCl. N/1 HCl. Cone. H,SO,. Vv. Cale. Obs. Cale. Obs. Cale. Obs. Cale. Obs. Cale. Obs. 0:08065 285°6 285'6 285°6 285'6 285'6 285°6 285°6 285'6 285'6 285°6 Ol B01°5 356 352°5 3852°9 300 354. O'1¢ 549°5 550 555°9 558°5 562 566 0:2 676°6 688°5 693°2 700 706 0:3 977°2 1009 1022 1040 1059 0'4 1255 13815 1539 373 1401 0°6 1750 1884 1940 2021 2110 1:0 2536 2500 2873 2880 3020 2990 32438 3260 3499 3521 2:0 3699 4660 53138 5915 6906 4:0 4475 6462 7640 9930 13457 5:0 4587 4600 6892 8362 11425 16604 6'0 4639 7160 8865 12657 19669 10:0 4680 7534 7510 9760 9750 15765 31150 20:0 7599 10083 18040 55060 400 18416 87494 60:0 184238 106601 100°0 18424 124487 200°'0 183926 400'V 4682 4682 7600 7600 133990 4000:0 10041 10041 18424 18424. 1383990 32000°0 7600 7600 10041 10041 18424 18424 183993 133993 80000:0 183993 1839938 Taste TV.—Absorptive Power (k) of Pyridine in Various Solvents calculated from Values of #:—Water, 0°33889; Alcohol, 0:20621 ; N/5000 HOI, 0°15537 ; N/1 HCl, 0:04072 ; Concentrated H,SO,, 0°011491. 430 Mr. H. F. Biggs on Energy of Secondary Beta Rays appropriate K only, and hence the agreement: between observation and calculation is eminently satisfactory *. It would seem possible that with suitable modifications the formula faire’ might ‘find a more general application to solution phenomena, for in such cases as the dissociation of strong electrolytes the mass-action law again does not hold. In its present form the expression only deals with the simplest case of one molecular entity changing into another molecular entity. Tf, for example, the solvent convert one molecule into two entities, e.g. ions, then the effective concentration of the new entities will be proportional to “V, and we shall have f=l—e-«VV, Tf for f be put A/Ao, where % and A» are the molecular con- -ductivities of a solution of concentration V and at infinite dilution respectively, then No = ee av V, ‘This formula is found to express the molecular conductivities of weak electrolytes with absolute accuracy at concentrations smaller than V=4, and of strong electrolytes also over a fair range of concentration. A discussion of this cannot he given here, but the matter is mentioned in order to show ‘that the formula may find considerable application to solution phenomena. log The University, Liverpool. LI. On the Energy of the Secondary Beta Rays produced by Partly-absorbed Gamma Rays. By H. ¥. Biaes, B.A., Assistant Lecturer in Mathematics in the University of Manchester t+. Object and Results of the Experiment. UTHERFORD + has brought forward strong evidence to show that the 6 particles which give the lines of the magnetic spectrum of the # radiation from RaB and RaC earry energy in whole multiples of the quanta corresponding * We are indebted to Mr. James Rice of this University for sug- gesting this formula to us, and take this opportunity of expressing our cordial thanks to him for the interest he has taken in this investigation. + Communicated by Sir E. Rutherford, F.R.S. t+ Rutherford, Phil. Mag. xxviii. p. 305 (1914). produced by Partly-absorbed Gamma Rays. 431 to the frequencies that Rutherford and Andrade* found for the y rays of RaB and RaC respectively. Further, Rutherford, Robinson, and Rawlinson f, by using an eman- ation-tube wrapped in lead foil, obtained the magnetic spectrum of secondary @ rays produced from lead by the vy rays of RaB and RaC, and found that all the observable lines were nearly, if not quite, identical with the primary -ray lines of RaB. Lines corresponding to the faster primary @ rays of RaC were probably also present, but were difficult to observe with certainty owing to their relative faintness and the fogging from scattered radiation in this region of the plate. It is thus indicated that the secondary 8 rays produced from lead by the y rays have a spectrum identical, or nearly so, with that of the primary 8 rays. and therefore, like the latter, derive their energy in quanta from the y rays. The question then arises whether the number of quanta which y rays of given frequency may impart to secondary B particles is independent of the energy of the impinging y rays, or whether the y rays, when they lose energy in passing through matter, may lose also the power of producing 8 rays of great energy—whether, for instance, y rays that originally can impart energy div to an electron may be able, after passing through a centimetre of lead, to impart at most the energy 3hv. For it seems probable that in passing through matter y rays may lose energy insome other way than by the production of 8 rays. Thus the scattering of y rays investigated by Florance{ may be due to the forced vibration of electrons that remain in the atom, and therefore absorb less energy than those projected as secondary @ rays. If this is so, we should expect that the y rays, after passing through a considerable thickness of matter, would be, as it were, shorn of some of their available quanta, and that therefore their secondary @ radiation would be, on the whole, less penetrating than that of the unaffected y rays. It was to investigate this point that Sir Ernest Rutherford suggested the present research, which is a modification of a research carried out by Eve § in 1904. It was found that if this effect of matter on the y rays exists, 1t lies beyond the range of the method used. * Rutherford and Andrade, Phil. Mag. xxvii. p. 854 (1914); Phil. Mag. xxvill. p. 263 (1914). + Rutherford, Robinson, and Rawlinson, Phil. Mag. xxviii. p. 281 (1914). t Florance, Phil. Mag. xxvii. p. 225 (1914). § Eve, Phil. Mag. vii. p. 669 (1904). 432 Mr. H. F. Biggs on Knergy of Secondary Beta Rays Incidentally, the results confirm Rutherford’s theory that the observed great energy of the faster @ particles is derived from the known y rays, each 8 particle carrying several of their energy quanta, and is not due toa hypothetical y radia- tion of such high frequency that only one quantum of energy is required. Apparatus. The source, 8 (fig. 1), of y rays was 64 mg. of RaBr, with its products in a steady state. The radium was contained =) 1Ocm. ina platinum tube enclosed ina sealed glass tube. The glass tube was fixed to a lead plate so that it could be removed (for safe keeping) and exactly replaced. Pb is a lead block so placed that the shortest line from any radium to the nearest point of the electroscope lay through about 20 em. of lead. The electroscope, E, 10 cm. cube, was of block tin with a lead casing 3mm. thick. One side was closed only by aluminium foil. Opposite this side could be hung on a light frame the radiator, R, a piece of lead foil 10 cm. square produced by Partly-absorbed Gamma Rays. 433, and -12 mm. thick. Absorbing screens Af could be placed over the side of the electroscope, and other screens, Ay, directly over the source. Method. Series of observations were taken of the ionization in the electroscope both with the radiator in place and with the radiator removed. The ionization due to the secondary radiation from the radiator is then obtained by subtraction of the smoothed curves. Two methods of arranging the observations were used. In the first, a series of absorption curves for the secondary 8 rays was taken, the different curves corresponding to different thicknesses of matter traversed by the primary y rays. Each curve shows the ionization plotted against the thickness of aluminium in the screen AS which absorbs the secondary radiation. Any marked change in the energy- distribution of the secondary radiation should then be made apparent in a change of shape of the curve as we pass down the series. Fig. 2. lonization O AB :25 =i. oF /5) 1-O 1-25 5 mm. Al. Thickness of screen over electroscope. Fig. 2 shows a pair of experimental curves typical of these observations, that is, ionization (in arbitrary units), Phil. Mag. 8, 6. Vol. 31. No. 185. May 1916. 2G 434 Mr. H. F. Biggs on Energy of Secondary Beta Rays plotted against the thickness (in millimetres) of aluminium through which the secondary rays have passed. The screen Ay over the source is, in this case, °76 cm.. of aluminium. Curve R is for the radiator in place, curve X for the radiator removed. Fig. 3 gives (in curve ii.) the result of subtracting curve X from curve R in fig. 2, using the smoothed curves, with the Fie. 3, N v1 20 O AB. ‘5mm. AL other curves of the same type; that is, curves 1., 11., 111., 1v., Pb, are absorption curves in aluminium for the secondary radiation due to primary radiation that has passed through 0, ‘76, 1°52, 2:28 cm. of aluminium, and 1:71 cm. of lead respectively, produced by Partly-absorbed Gamma Rays. 435 together with the platinum and glass of the tubes enclosing the source. To facilitate comparison, the logarithms of the ordinates are also plotted. The curves thus obtained would, of course, be identical if the energy-distribution of the secondary 8 rays were independent of the thickness of matter passed through by the yrays. The observed differ- ences between the curves may be accounted for by the relative weakening of the softer primary yrays. In fact, the curves form a series very much like what would be expected if the B rays produced by y rays of given frequency had always the same velocity, or velocities, 7. e. if the energy- distribution of the secondary @ rays due to any one fre- quency of y rays were independent of the energy of those ry rays. i the second method, the screen Af over the electroscope is kept constant for each curve, while the screen Ay over the source is varied. Thus the ionization produced by 8 rays that have traversed the same matter is plotted against the thickness of matter traversed by the primary y rays that produce these 8 rays. This procedure should show up at once any change in the maximum energy of the secondary @ rays when the primary y rays pass through matter; for, if we consider the case where the electroscope-screen is so thick that only the faster 8 rays get through when the y rays are unscreened, it is obvious that a fall in the maximum energy of the secondary rays will be shown as asudden drop inthecurve. Ideally, of course, the same curves might have been obtained by comparing the ordinates of curves such as those in fig. 3, but a glance at fig. 3 will show that the irregularities of scale make it preferable to obtain consecutive readings in the way mentioned. It should be noticed that this method is only a variation of the ordinary experiment on the absorption of vy rays, for there also the y rays are probably measured only by the ionization produced by the secondary 8 rays. The essential difference is merely that in the present experiment the secondary @ rays are controlled, being made to pass through known thicknesses of matter before producing ionization. We should therefore expect the exponential law to hold as in the direct experiments ; also, since tke softer @ rays are cut out, the effect of the softer y rays that produce them should be less apparent, and therefore the exponential law would begin to hold for smaller thicknesses of screen absorbing the y rays. Such turns out to be the case. 2G 2 436 Mr. H. F. Biggs on Energy of Secondary Beta Rays Fig. 4 gives a typical pair of curves obtained by this method. Fig. 4. 10 Ae) AN " fonization FF J fe f (09) Scale of Logarithms Oo Ay “58 1-16 1-74. BSBem Pb. Thickness of screen over source. As before, curve R is for the radiator in place, curve X for the radiator removed. Curve L is obtained by plotting the logarithm of the difference of curves R and X. It is seen that the exponential law holds for thicknesses above 0:2 cm. of lead. Fig. 4 shows the results for varying Ay, the screen over the source, when AQ, the screen over the electroscope, is kept at ‘47 mm.of aluminium. Similar results were obtained for A®B=:25 mm. and A@=°86 mm. aluminium. The following table— AB, Limits of 2, Ly, mm. Al. cm. Pb. em.—L "25 “6; 2°3 64 “47 3, 1:2 66 86 “4, 12 67 produced by Partly-absorbed Gamma Rays. 437 gives the absorption coefficients, w, of the y rays in lead as measured by the falling-off of the ionization due to the secondary @ rays projected from the lead radiator, which ionization follows the law I=Ce-#*, where x cm. is the distance the y rays travel through lead. The first column gives the thickness in millimetres of the aluminium screen, AB, over the electroscope through which the secondary 8 yrays pass. The second column gives the range of x from the point where the logarithm-graph becomes straight to the point where measurements cease to be reliable. It will be noticed that the values of u agree well among themselves, and also agree roughly with that found directly by Rutherford and Richardson * and by Richardson f for the hardest y rays of RaC. The most important deduction to be made from this result is that there is no trace of y rays more penetrating than those whose absorption-coefficients and frequencies are already known, even in the production of the hardest 8 rays. Indeed, the values obtained for u may be said to amount to -a proof that a secondary § particle derives its energy, not from vy radiation whose frequency is so great that one quantum suffices to account for this energy, but from y rays of considerably lower frequency, which must there- fore impart several of their quanta to each @ particle. For instance, if we consider the @ rays that pass through °86 min. of aluminium, which are produced by y rays having an absorption-coefficient, w="67 cm.~1in lead, we find, using Varder’s f results, that these 8 rays must have energy at least equal to 6°2 x 10e, where ¢ is the charge of an electron in electromagnetic units ; again, the absorption-coefficient ‘67 cm.~' in lead corresponds to a frequency for which hy=1'2x10¥e. This number is deduced by Bragg’s law, vc py?/?, from the absorption-coefficient in lead, 1°5 em.7}, ascribed with much probability by Rutherford § to the y rays of RaB whose wave-length is 1:°37x10-%. Thus no less than 5 quanta are needed to account for the energy of these B rays. It shouid also be remarked that the results support Rutherford’s conclusion that the preponderance of the harder 8 radiation is due to a single y radiation. This is shown by the fact that the exponential law begins to hold good for smaller thicknesses of lead than when the absorption * Rutherford and H. Richardson, Phil. Mag. xxv. p. 722 (1913). + H. Richardson, Roy. Soc. Proc. A. xci. p. 398 (1915). t Varder, Phil. Mag. xxix. p. 725 (1915). § Rutherford, Phil. Mag. xxx. p. 356 (1915). 438 - Mr. A. J. Dempster on the Tonization eurve of the vy rays is obtained directly. Again, if the divergence between the value here obtained for yp (°66 em.~’) and that found by Rutherford and by Richardson (‘50 em.) is a genuine one, this is further evidence that the hard 8 rays are mainly due, not to the hardest y rays reflected at 43’ from rock-salt, but to those reflected at 1° 0’. There are two possible disturbing factors to be reckoned with, primary @ rays producing secondary or scattered 6 rays, and secondary (scattered) y rays from the radiator. The former, however, must be almost negligible, even for no extra screen Ay, since the primary radiation has already passed through the platinum and glass in which the radium is contained. In fact, the curves themselves show that the primary 8 rays have been already almost totally absorbed. The effect of the secondary y rays, on:the other hand, is by no means negligible, though that it is small compared with that of the B rays is shown by the closeness with which the curves of fig. 3 approach to zero. We may, in fact, safely assume that the values found for wu really represent the falling-off of the secondary 8 radiation. I wish to thank Sir Ernest Rutherford for suggesting this research and for the kind interest which he took in it while work was in progress. LIL. Zhe Ionization and Dissociation of Hydrogen Molecules and the Formation of H;. By A. J. Dempster, late 1851 Exhibition Scholar of the University of Toronto”. Y the analysis of positive rays, J. J. Thomson has shown that in a discharge-tube containing hydroven there are present charged atoms, charged molecules, and sometimes a constituent with a mass three times that of the atom of hydrogen. The pressure used was about :003 mm. of mer- cury, and consequently the potential necessary was of the order of 20,000 volts. In the following experiments a different method of getting the positive rays was used. Electrons from a Wehnelt cathode C are accelerated in the field CA. They ionize the gas, and the positive particles produced are given a velocity which carries them past the edge of C (2 mm. wide) and through the narrow tube T. These positive particles are then deflected by magnetic and electric fields, and fall on a screen which has a parabolic * Communicated by Prof. R. A. Millikan. and Dissociation of Hydrogen Molecules. 439 slit S. Hach constituent of the rays is drawn out into a parabolic curve, and, by increasing the magnetic field, the various parabolas may successively be brought to fall on the slit. When this occurs the charged particles pass through Sand give up their charges to the Faraday chamber F. The mee Le Fig.t. advantages obtained by using a Wehnelt cathode are, that any desired potential may be used, in particular low potentials, and that the pressure of the gas may be made as low as desired and may be varied without changing the potential. The change that occurs in the constituents of the rays as the pressure is decreased is described in this paper. With 800-volt rays and hydrogen at a pressure of about ‘O01 mm. the curve in fig. 2 was obtained. The abscissee represent the strength of the deflecting magnetic field, the zero being to the left of the origin, and the ordinates give the charge obtained in the Faraday chamber as the field was increased so as to bring the three lightest constituents of the rays over the parabolic slit. There are present hydrogen atoms, hydrogen molecules, and very much of the constituent with atomic weight 3. The curve obtained with a pressure of ‘0017 mm. is given in fig. 3 ; there is a noticeable decrease in the relative amount of both H, and H;. Fig. 4 gives the curve when a charcoal bulb with liquid air was used. Here the pressure was less than ‘0005 mm., and H, and H; have practically disappeared. That this change was due to de- creasing pressure, and not to the removal of some constituent of the gas by the charcoal, was shown by the fact that when hydrogen was admitted, while the charcoal and liquid air were on, H, and H, regained their original relative intensities. 440 .Mr. A. J. Dempster on the Tonization Since in the high vacuum the free path of the molecules -is-very great, the positives which are still formed by the dense stream of corpuscles coming from the Wehnelt cathode make very few collisions with the hydrogen molecules. H, (7 ie. 2 © ly QO H i | O @ -) j (4) q) © ee C) ® Cc Y) @ D (J , 0 o Magnetic Field Hence these positive molecules are analysed in the condition in which they are just after they have been ionized. We must conclude, then, that electrons ionize only by detaching and Dissociation of Hydrogen Molecules. 44) a single electron from the molecule, and are not able to dissociate the molecule into atoms. When the pressure is greater, some of the positive molecules collide with the molecules of the gas before the cathode, and this collision Cnaree:. 268 Magnetic Field results in a dissociation of the gas into atoms. A positive atom thus formed may attach itself to a neutral molecule and give rise to Hs. The experiments thus confirm for electrons of this speed the conclusion reached by Millikan from his experiments with oil-drops (Phil. Mag. xxi. p. 753 (1911)) that gaseous ionization produced by §-rays, or by X-rays of all hardnesses, consists in the detachment from a neutral molecule of a single elementary charge. They also verify the theory advanced by J. J. Thomson to account for the results ob- tained in his experiments on positive rays (Phil. Mag. xxiv. p. 234 (1912)), that the electrons and positive rays produce 442 Tonization and Dissociation of Hydrogen Molecules. different types of ionization. The results show also that H3 cannot be regarded asa stable gas, but that it is a temporary complex formed only when the hydrogen is in a dissociated state. In his paper in Phil. Mag. xxiv. p. 241 (1912), J.J. Thomson finds that H; “ occurs under certain conditions Magnetic Fiela of pressure and current,” but later (‘Rays of Positive Electricity’) regards it as a stable gas which, among other properties, combines with oxygen and mercury when under the influence of the electric discharge. Itis possible that the disappearance of H; in the latter case is due to a higher vacuum, for I find that the pressure is much reduced when the discharge is passed through mercury-vapour and hydrogen. ‘ Summary. Hlectrons of 800 volts speed ionize hydrogen by detaching a single elementary charge from the molecule. They are not able to dissociate the gas. Coefficients of Mutual Induction of Eccentric Coils, 443: The positive molecules so formed are able to dissociate the gas. When this occurs the complex H; is formed. H cannot be regarded as a stable gas, since it is not present when there is no dissociation of the hydrogen molecules. Ryerson Physical Laboratory, University of Chicago, January 22, 1916. LIII. On the Coefficients of Mutual Induction of Eccentric Cols. By 8S. Burrerworts, M.Sc., Lecturer in Physics, School of Technology, Manchester *. 1. PN certain types of variable inductancest one coil moves so that its plane remains parallel to, and at. a constant distance from, the plane of a fixed coil. By this. means the mutual induction between the two coils may be made to range from a considerable value to zero and then to change sign during a comparatively small motion of the coil. This use of eccentric coils lends a certain interest to the problem of determining the mutual induction between two non-coaxial circular filaments. 2. The principle to be made use of is to treat the mutual induction between tlie two circles as a potential function of the position of the centre of the moving circle. The proof of this is as follows :— Replace the moving circle by its equivalent magnetic shell and consider the variation in potential energy of this shell as it moves in the magnetic field due to the current in the fixed circle. It is clear that the variation in potential energy of the individual particles of the shell follows the law of potential, so that, since for motions of translation the displacement of every particle is the same, the potential energy of the whole shell also varies in accordance with the law of potential. Identifying the potential energy of the shell with the mutual induction between the two circles,. the proposition is proved. For parallel circles the mutual induction (regarded as. a potential function) has a symmetrical distribution about the axis of the fixed circle, so that if the mutual induction is. known in the coaxial position the mutual induction in any other position can be derived by the usual methods for determining potential. * Communicated by the Author. + Campbell, Phil. Mag. xv. p. 155 (1908). 444 Mr. 8. Butterworth on the Coefficients of 3. Equal Circles. Take the radius of either circle as the unit of length and let x be the distance of their planes. Then when w is large and the circles are coaxial, the mutual inductance is * 2a” 3 25 245 m= ( ~ a? * Qhat — 19808 * i) ) the multiplying factor to obtain the nth term from the preceding term ee 2n— sa) n—1 a+ eel Hence, using hes coordinates with the centre of the fixed circle as origin and its axis as the axis of coordinates, when the centre of the moving circle is at r, 0 the mutual inductance is Der? 3P, 25P, 245 Pz m= =| oa ee ” 7 in which P,=P,, (cos@) is the zonal harmonic of order n. When the circles are coplanar, O==, and (2) then becomes 2Q ) 9 125 8575 162 + 1924 + Bi9278 T° ) (8) the multiplying factor to obtain the nth term being ih = n—1 2y? ee Formule (1), (2), and (3) converge if r>2, but the convergence is rather slow if 7 is less than 3. When e is small and the circles are coaxial the mutual induction 1s T 15x 31 hth cab eyi sh) Sa. Oe Mys=4mr4 2o i i ti Fe (Mo =) aa = =) BT ay) ( —ta)- } als (128) ue Ne 910 Sn aa (4) aan X= log. >. 2 M,=—",(1+ To obtain the mutual induction for non-coaxial circles * Havelock, Phil. Mag. xv. p. 332 (1908). + Coffin, Bull. Bureau of Sianaanial ii. p. 118 (1906); Havelock, loc. cit. Mutual Induction of Eccentric Coils. A45, from this formula we must replace the terms w” by 7*P,, and the terms x” log, « by P,, for these clearly satisfy Laplace’s Equation, and since oP, On they reduce to 2” and x" log, x respectively when 0=0. In applying this transformation the following explicit UU, P,=! cwhen. d—0, On jee : Co =P,log dt ytyn. - - « (): in which p=cos 8, 2n—3 n= 2 an ay —P,1)- 2( a= 1) (P,- Pe) il ON 1 at jae —) AG) (P,—P») i. mae) and in particular Wo= 0, = — F(1—my(L+ 7), | Wi= Gad —w)@1+ 241p—- 11325339), + (T —e = (1—u)(185—2957 + 3728p? + 180083 ~ 3247 u4— 181075). J On making the necessary substitutions in (4) the required. mutual induction is given by fe ek 2+ gr {P 2(r =3)= — 2 — zona" 12-50) ] * aay {Oat} ae SAP | Sesiae this HORSE in which’ ooo 446 Aor end to tabulate aq) aes Vowece Mr. 8. Butterworth on the Coefficients of For purposes of computation it is convenient to write (8) 6 — log. r(1+ Bir? + Bor*+ ByrP +...) . done in Table I. When #=0 the circles are coplanar and (8) reduces to (9) as functions of w. ‘This is Me 3 2( LUO aaa 4( yee Pe ce uae MING icy ahisie ay Gt =0) 175. esl (512)? (u- aa)o do 0 ° ° (10) in which Ny = Og: long as r is less than unity. Formule (4), (8), and (10) converge fairly rapidly so approximation up to r=1°6 but fail for larger values. TABLE I. Values of coefficients in formula (9). They will give a rough 4, In order to test whether the formule are adequate to represent the mutual induction for the range which is most useful in practice, 2. e. from the coaxial position to the position where the mutual induction is zero, Table II. has been prepared. In this table, w is the distance of the planes of the circles and p the distance of their axes, the radii of the two circles being unity. The results are plotted in HB. Oy. oie ae mag | By. Bo. Bs- 0°0| 0°7726 | --0°1817 | —0:00487 | —0-00066 | —0-0988 | —0:00549 | —0:00067 0-1| 06778 | -—0:1414 | —0:00217 |—0-00009 | —0-0909 | —0-00495 | —0:00053 0-2) 0:5903 | —0:0962 |+0-00208 |+0-00071 | -—00825 ! —0:00340 | —0:00017 | 0:3; 05102 | —00473 |+0:00577 | +0:00197 | —0-0684 | —0°C0107 | +0-:00028 0:4) 0:4861 | +0°0044 | +0:00829 | +0:60098 | —0-0488 | +0:00166 | +0:00063 0-5! 03671 | +0:°0578 |+0-00921 | +0-00045 | —0°0234 | +0:00424 | +0:00069 0-6 | 0°3026 | +01123 | +0:00818 | —0:00021 | +0:0075 | +0:00597 | +0-00037 0-7 | 0:2420 | +0°1671 | +0:00507 | —0°00090 | +0:0441 | +0:00604 | —0:00027 0-8) 0:1848 | +0:2216 | —0 00006 | —0:00098 | +0-0862 | +0:00341 | —0:00084 0-9} 0°1808 | +0°2752 | —0:00701 |—0-00019 | +01341 | —0:00805 | —0-00052 1:0} 0:0795 | +0:0274 | —0:01531 |+0°0017 | +0:1875 | —0:01462 | +0:00214 Mutual Induction of Eccentric Coils. 447 fic. 1. It is seen that the induction changes sign when pis less than 1:7 in all the cases taken. This practically corresponds to the range of formula (9). — a ——— ee DAB m iE, Mutual induction between equal parallel circles. Radii of circles=unit of length. x=distance of planes; p=distance of axes. a2=01 x=0°25 2—075 o. | M/4z. pL. 0 M/47. o- | M/4z. 0: M/4z. 0 |infinity | 1:0 0 2°39 0 150 0 0°88 0:2 Zar | OO. | 0°05 2°34 | 0121; 145 | 024) 0838 0-4 coos). | O07) 2727) | OES, P88) 2) 037,076 06 reo Om || O102) 219)" | 0256) E30 ~) O51 |). 067 08 086° || 06 | 0133; 2:10 | 0384) 1°20 | O67 | O57 10 0-530) 0:5).).\ O7173)| 4 1:98) )) OS8Se FEO) .| O87 | 0-48 1-2 Oey. | 0229) 1-82.) Os7S 7 O90 | 115 | 026 1-4 O12 || 03 | 0318; 1°60 | 0°796| 0°65 1:59 | 0:03 16 | —O17 || 0°25 wei see ots bite 1:94 | —0:10 1°8 02 | 0-49 1:24 |, 1:225) 0:26 20 |—043 | 0°125/ ... sie 1:98 | —0°24 01 1:00 0°53 | 0°05 | 2:00 | —0°36 M = Sy ONO) (0, eK Pe ra 7 Pe 3) oh ee 448 Mr. 8. Butterworth on the Coefficients of 5. Unequal circles. Let the radii of the circles be a and A and Jet a be the distance of their planes. Then if 2 is large and the circles are coaxial, the mutual induction is * 77707 A? 3 A? 3 2 3. Die 2 ( To Oh ede aK. 5 7) Ket in which K,=14+, | | Kinl4 35 + ge | ease a r g=il+ eo eee | ap Seago a: K,=B(—n—-1, —n, 2, =) where I'(a, 9, y, z) denotes the hypergeometric series a(a+1) B( (8 +1) 1.2y(y+1) Hence for non-coaxial -circles the mutual induction is Rolie ie 1-4 4 OE a4 cee 5 uO 7 46 fs reducing for coplanar circles to mar A? oe ee Bee Beebe ae, WAG + eee = Stee (13) Formule (11), :(12), and (13) converge if r>A-+a, 1.e. in formula (13) if one circle is entirely outside the other. When « is small it is convenient to choose the difference in radii of the two circles as the unit of length. Then, * Havelock, Joe. cit. (11) ), (12) Mutual Induction of Eccentric Cozls. 449) when the circles are coaxial *, 3 (1427 1 My=4n/Aay rea in Aa 1(m— 3) Ins ee a parce 1024 Ae ri Oa wale ee t, (14); in which r= log, The transformation of (14) to the formula for the non- coaxial case requires the determination of a solution of Laplace’s equation, which will reduce to (1+ 27)" log, \/1+4? at all points on the axis of wz. This solution will now be: found. 2 6. In cylindrical coordinates (a, p, 6) and with =e Laplace’s equation is Oa Orv Lio’. owt dp? 7 p op Transforming to spheroidal coordinates by putting pi (te hoe ea 8. (LS ay SO al Ale (15) becomes 2 fa-wyF }+ of 14) oo t= 0, . (16) a possible solution of which is 0 = 2 (Ps(u)Pair)} Pal) QalW) ie —ZAP(m)P.(iv), . (17) in which i=/—1. When »=1, that is, when p=0, v=2, (17) reduces to OP n() On * Havelock, doc. cit.; Rosa, Bull. Bureau of Standards, viii. p. 15: (1912). Phil. Mag. 8. 6. Vol. 31. No. 185. May 1916. 2H i —Qn(tv)—SA-Pi(iv)- 2. (18): 450 Mr. 8. Butterworth on the Coefficients of Since eg) Din I 2n—5 Onig=te, logs; —_ Coen Oi Ute esans Pie 2n—9 fn and ~ 5(n—2) eo oe) Pye ae) 2n—1 2n—3 +2 — — (Pa Pea ate Jn (P,—P,_2) 2n— * 3(Gn— ee Ea) ae f the logarithmic term in (18) is Pv) log 44/1 +0". Also, if s takes the values n, n—2, n—4, 1) ema 2n—1 2n—3 2Qn—5 92 CoM SENS Ri ae ih ek Bl WME Es ie ee eA on Ban yt AC oAee \ He ane (aa ater 2n—7 (19) Bets re the value of V along the axis of symmetry becomes P,(iz) log ./ 1+ 2". Therefore to obtain the solution sought it only remains to expand (1+.?)” in a series involving P,(iz) and to apply the results (17) and (19) to each term of this series. The series in question is to. deal : PING) mom , (1L+a2?)™=(—)”2 ie ear ne eT $4 m+1)Pan(it) m 4m+1 —(4m—3) i ima Imp ten 2(12°) m(m— 4m+1)(4m—1 +(4m—7)- n(n : Dm DMD ayn i (20) From which 1+2’?=— * Pa (iar) + = Po(ia), 8 16 NS (1 - LP 35 P,(iz) = 91 Po(ix) + jp Pot). Mutual Induction of Eccentric Coils. 451 7. On applying the results of Section 6 to (14) and inserting the values of P,,(w), P,(tv), the mutual induction between non-coaxial circles is found to be = { ie. eae po). 1 M=4m,/Aa[X—2+ 15454 (23) $2-x ike L 31 ~ 10z4 A®a? a 39 esx: ae fle (22) iii A= log, 16V Aa Q (L+p) V¥1+v* 2 ; p2= a 3 Po(m) Po (ev) 1 in which 5 (+e) (1-3n4)}, ae 16 Bch 8 . d.= G 3j P.(m) Pov) + 35 P,(p) Pa(av) ee a (3+ 2p? + 3p) —2v2(1 + bu? —15 4) + v4(3—30p? 4-354) }, Xo= g(L—m){d—p)—v*(1 + 7p) 5, 1 : X= gg Cw) {8(l—p) (74 Qu t Ty?) — 6V°(5 —p— pw? +59u?) +y4(21 4+ 2414—113u?—533u7)}, and from (15 a) py”, —v* are the roots of at bp ae OL a) In (22) the difference in the radii of the two circles is unity. If A~a=c, then replace in (22) 1/Aa by c?/Aa, multiply by c, and make p?, —v? the roots of GUC —2)—7 0.4. . . aa} To test the formula, let ¢ approach zero. The limiting value of wis a/,/a?+p?=2/r, and that of cv is r. Using these in (22), we obtain formula (7). 2H 2 452 Mr. 8. Butterworth on the Coefficients of 8. When «=0 the cireles are coplanar. 2 ee. 0, 77 — a—l and the mutual induction is —— : Zee M,=4r VAal —2— = ee = \(1- =) 45 or OC pkO.e 8192 wa mgetae) _(27_Ske | 216 4 a |; (24) in which N= log LOM AG, r This formula holds good when the two circles intersect. 2 Mime. hy One aoe - , and the mutual induction is au iL if M,’ = 4m Kal d'—2 39 kA (u/— 5) +e) 50a) ea 8192 Ata? t (x'— a CT ee 1 — (Lay + ut Tae) 4 aes Co teze) 16 Vv Aa “ol +m) This formula holds when one circle is entirely inside the other. If r=c, the two circles touch internally and the mutual induction is Orme iy A 97 Wee ei aan | rn —- 2+ + 35 (a" =?) s99 peal" 3) Aa LEE RE owen ee git in which Mi = los. It is interesting to notice the similarity between formule (26) and (10). Mutual Induction of Eccentric Coils. 453 9. The ycintiles for unequal circles close together do not readily admit of simplification even with the aid of tables. The following calculations (Table III.) have been made to test the range of the formule. The ratio A/a=2 has been chosen so as to compare the results with the experimental curves of Campbell (Joc. cit.). Otherwise it is not a very favourable ratio for the formule. The curves obtained (fig. 2) show very good agreement with the experimental SEEETaEaTeera cS ) ! p/a 2 4 curves and give approximately the same zero points. The separate terms are tabulated so as to indicate their relative importance. The labour involved in obtaining the third term is altogether out of proportion with the importance of that term, especially as in the cases where it is large the fourth and succeeding terms are likely to have an appreciable effect. It will be found that for most purposes the first two terms only in the formule need be used. It will also be noticed that in the most important range of the curves (viz., from p/a=1 to M=0) the curvature is slight, so that after calculating a few points within this range from the theoretical formule, an empirical formula of the type p” M= a9 +016 tons + cae should be sufficient to determine intermediate values, 454 Mr. 8. 8. Richardson on Magnetic Rotary TaseE III. Mutual induction between unequal circles. i ae ala=0. x/a=0°2. Ist Qnd 3rd Mie. Ist nd 3rd (MI Term. | Term. Term. |4z / Aa a Term. | Term. | Term. 42 Aa | 0°426 07196 | —0-005 0-617 || 0:00 0-406 0°202 | —0:008 0-603 0°521 0167 | —0:003 0°685 || 0°45 0°453 0:187 | —0-005 0°635) 0°649 0-144 | —0-003 0-790 || 062 0:501 0-174 | —0-004 0°671 0-783 07125 | —0:002 0:906 || U°'74 0:548 0‘160 | —0:003 0°705 0-937 0-112 | —0-002 1:047 || 0°84 0°597 0:147 | —0:003 0-741 1 IY) 0:107 | —0:002 1:224 0°93 0-640 0-133 | —0-002 OTT! 0:773 0:000 —0-003 OTF 1:02 0-671 0-118 | —0-002 0-787 | 0-570 | —0:085 | —0:006 0:479 || 1:15 0:673 0 074 | —0-001 0:746 | 0-426 | —0°149 | —0-011 0-266 || 1°39 0-591 0:047 | —0:000 0-638 | 0°314 | —0°208 | —0:016 0-090 || 2°22 0-220 | —0-129 | +0-000 0-091 | 0:223 | —0°260 | —0:022 | —0:059 || 2°69 0°052 | —0°225 | +0°COl | —0°172 | rja—05:; tja=0; 0:314 0:232 | —0:0U7 0°539 0:00 0:089 | 0:330 | —0:015 0-404 0°342 0:216 | —0:007 0°551 || 0-65 0:075 | 0:313 | —0-008 0°380 | 0-366 0:202 | —0 005 0°563 || 0:96 0:061 0:297 | —0:003 0°355 0°382 0:185 | —0-003 0-564 1:24 0°032 0-280 | —0-001 Oslt | 0°385 0:165 | —0°001 0549 1°53 | —0:002 , 0°250 | +0:010 0-258 0:367 07142 | +0-001 0-510 |; 1°94 | —0°091 | 0-213 | +0°028 0:150 0:313 0-107 | +0-004 0°424 | 247 | -—0207 0-184 | +0-068 0 045 0-192 0-045 +0-012 0-249 || 3°32 | —0:392 071338 | +0:200 | —0°059 =0:053 | —0:070 | +0042 | —0-081 ae LIV. Magnetic Rotary Dispersion in Relation to the Electron Theory.—Part II. The Number of Electrons and Additive Relations. By S. S. RicHARDSON, ISAS A TeCiSe Lecturer in Physics, Central Technical School, Liverpool*. Cae of the problems arising from the theory of electrons of resonators present in the molecule of a substance. is the determination, from optical data, of the number The subject has been examined by Reiff t, Lorentz f, Drude §, and Erfle || in connexion with the dispersion constants, and by Keenigsberger and Kilchling { with respect to the § Ann. d. Phys. (14) pp. 677, 936 (1904). || Ann. d. Phys. (24) p. 698 (1907). Q Ann. d. Phys. (28) p. 889 (1909) ; * Communicated by Prof. L. R. Wilberforce. + Wied. Ann. (55) p. 83 (1895). t Versl. k. Ak. Wet. Amsterdam (6), pp. 506, 555 (1898). (32) p. 843 (1910). Dispersion in Relation to the Electron Theory. 455 constant of absorption. The present paper deals with the application of magnetic rotary dispersion to the elucidation of the same problem. If N denotes the number of resonators present in the unit of volume, p the number per molecule, m the mass and e the charge of each; d the density and M the molecular weight of the substance; and A the absolute mass of the hydrogen atom : then d Ey TL Rae ARG de oo (1) The ratio of e, to h,if we confine our attention to electrons, is equivalent to the electrolytic constant—96530 coulombs or 9653 x 3 x 10'° electrostatic units per gram. When Nye; can be obtained from optical data, (1) gives at once the value of 7. (A) Number of Electrons deduced from Natural Disper- ston.—In the case of ordinary dispersion we may write ayn? Wr? According to Drude’s theory, , is the wave-length corresponding to the free-period of the vibrator and ie Ne?” ~ arm,C? * Nie —— ifr n2—l1=> This expression for a; holds good also in the theory of Lorentz and Planck, but A, is less than the wave-length dr, corresponding to the free period, the two being connected by the equation fe AP 1 Nya? pels A WE rary cone Oot Bint ig 1—g’ Wok 3 mm,C? In either case, therefore, if %, is the wave-length of the dispersional period, determined from the course of the dispersion-curve, 2 aC Za C Nye; = - ee NERY TOR TAS th ae (2) 5 : Ay fF my i When the constants a,, a...., Ai, Ag---, have been completely determined, equations (1) and (2) can be applied to evaluate 7, ps, &c. This, however, is not in general practicable, as the value of the dispersional period is as a rule only an effective mean. This is particularly the case when Ay, As, Ke., lie in the Schumann region. In the visible and ultraviolet the effect of infra-red bands is very small, and can be represented by a term —cA*. 456 Mr. 8. 8. Richardson on Magnetic Rotary Thus 2 v7—1Ll= ane aa —chr, where the terms under the summation sign refer to ultra- violet resonators only (electrons). Drude (loc. cit.) shows that if we confine our attention to this case, we may determine a lower limit to the number of electrons affecting the refractive index. If x, x, denote the reciprocals of the squares of two wave-lengths in the visible or ultraviolet and 71, 7%, the values of n’?— 1 + chr for these wave-lengths, and if d eine. v= Ly— Xe” then the lower limit referred to is given by the inequality M 7475 pot... @ Drude has apphed equation (3) to the data for a large number of organic and several inorganic compounds, and arrives at the conclusion that ‘the number of electrons in the molecule which influence the refractive index is equal to the sum of the valencies of the several atoms in the molecule.” Similar calculations have been made by Erfle for a number of aromatic and other unsaturated compounds. The following table will suffice to indicate the general character of the numerical results. t, e 7 Pit Pet -- oe Soe) cle NO er TABLE I. (Drude.) v. p (Erfle.) v p MQMIORI HEN Gs cnpeeses ae stecs 4 (16) 9°) | Aunigleme are etsecance ase ee 30 15°3 QNuraie7 Wester vacates see 8 (16) 8:2 || Propargy] alcohol......... 18 (22) 9°9 Carbon bisulphide ...| 8(16) | 7:3 || Tribrom-ethylene......... 12 (80) | 15-7 WEAIGGTY Wiad atehe sa pwiincad 4 (8) 39 || Naphthalemeiaes......ceen 48 13°9 Siva aastearaieses cies 2 (8)7) 9 Co | @riethwlamime (0 e7.ee.c2 42 (44) | 23°3 Rock-saltwilbcos..scesee 2 (8) 5'6 || Silicon bromide ......... 8 (82) | 21°6 Methyl alcohol......... 10 (14) 84 || Hexapropyl disilicate .../186 (164)} 90°5 Hthyl alcohol ......... 16(20) | 12°5 || Tetramethy] silicate...... 40 (56) | 29-9 IBENZONG) masta. eee. 30 12:0 || Hydroxylamine ......... 8 (14) 55 Polueneriareeissed- cence ee 14° Sigil Diamiowdy ea. ..02.5 5. dee + 2°7 XM Vilene graces nn eeeaecee 176 MEbydrocents. or.) ow. scsdcat 2 1°9 Cane-sugar ............ 99 (136)) 149 INTEROP Eis pasha. sehen eaee 6 (10) 4°5 DIspersion in Relation to the Electron Theory, ADT Here v is the total number of valencies in the molecule and p is the number of electrons calculated with e/m= 4-5 x 10" e.s.u., which is somewhat smaller than the normal value. In the above no attempt is made to distinguish between the resonators which influence the refraction but have no measurable effect on the dispersion and those to which the dispersion is due. This of course can only be done when from an extended series of measurements of n the constants a have been calculated. There are a few substances for which these constants have been determined. The results for these, recalculated for the normal value of e/m(5°325 x 10"), are given below. TaBLE ITI. | | Substance. v. ‘pi | Substance. | v tS ae oP? SNE VE [ces NEON Pad ahead, aed | | PUGATDZ Geshe. .2s:- Slo east! Wiatertnecses-s...2deec ee | 4 (8). 88 Fluorite............ 4 (16) | 3:49 | Carbon bisulphide ... 8 (16) 1:93 ByLVINE .... act. 2a (Ss) Wi O™ | Benzene) .2 20. Sencseh- | 30 | 3°35 Bock-salt, ......... | 2 (8) | 354 Naphthalene bromide.) 48 (54) | 3:06 Cale spar (ord.).... 12 (24) | 4:42 | Methyl iodide veseeceee| 8 (14) | 2°90 As X, in the formule is a mean value, the above numbers give the upper limit to the number of electrons of longest ultraviolet free period, except in so far as the unknown value of e/m may affect the result. The values of pe/m used for the first seven substances are those calculated by Drude, and for the last three those of Erfle. The refraction measure- ments of Erfle were confined to the visible spectrum, but carried to a high degree of accuracy, and the constants obtained do not differ notably from those given in Part I. of this paper. Drude concludes that ‘‘ the number of electrons in the molecule which influence the dispersion is equal to or less than the sum of the valencies of the atoms.” (B) Number of Electrons deduced from Magnetic Rotation.— It has been shown in Part I. that the only electrons in- fluencing the magnetic rotation are those whose periods are sufficiently long to affect the dispersion, and the mag- netic property therefore provides an alternative method of estimating the number of electrons of this type. Before proceeding to deduce the formula we must give certain secondary effects some further consideration. 458 . Mr. 8.8. Richardson on Magnetic Rotary Voigt’ theory of magnetic rotation and birefringence, which attributes the two phenomena to essentially the same cause, fails to account quantitatively for the latter property in liquids, the observed value being some 10? times as great as that calculated. Cotton and Mouton & Langevin * have shown that the property of magnetic birefringence is satis- factorily explained by a molecular orientation set up by the magnetic field,—a theory which also accounts for the rapid decrease in the effect with rise of temperature. Now such orientation may bring into play intermolecular forces, either electrical or magnetic, and influence the periods of vibration of the resonators. As a first approximation the variation in the magnetic rotation so produced may be represented by a virtual change in the dispersion constants, which therefore become (a+aH?). The experiments of Cotton and Mouton have shown that among organic liquids the magnetic double refraction is exhibited most strongly by the unsaturated and aromatic compounds, saturated and aliphatic substances showing scarcely any trace of the effect. It is therefore reasonable to expect that the value of & will be negligible for liquids of the latter class. Voigt’s theory suffices to explain the birefringence in metallic vapours, and E€ may be taken as zero for substances in the gaseous state. We may also regard it as inoperative in solids, for the applied field is always very small in comparison with the intense intermolecular fields brought into play during crystallization. From the change in diamagnetic susceptibility during the solidification, Oxley finds the latter equivalent to magnetic fields of the order 10‘ gauss. It may be added that -the hypothesis of intermolecular fields is not essential to the theory. The efective free periods of an assemblage of seolotropic molecules when these are partially orientated by the field may be expected to differ from the effective periods of the same molecules when their indiscriminate motions render the medium as a whole isotropic. Thus, apart from the change eH/2mC directly imposed by the field, the change in the effective period will influence the rotation, which will therefore differ from that calculated in terms of the dispersion constants of the zsotropic medium, and this influence we may also attempt for the present to express by & As we have seen above, it is probably only in unsaturated compounds that it may be great enough to be measurable, and in general therefore it can be omitted. In the formule e/m has been taken to have different values for electrons of different frequencies. The velocity of the vibrating electron is so small in comparison with that * Le Radium, Sept. 1910. Dispersion in Relation to the Electron Theory. 459 of radiation, that of course no appreciable variation of electro- magnetic mass is to be expected from this cause ; but the mutual action of neighbouring electrons may increase their effective mass, and it is pr efer: able, for the sake of generality, to retain separate values for e/m. If we write nelm for the ettective ratio, where e/m is the normal value deduced from the Zeeman effect in gases (5°325 x 10" e.s.u.), and retain only this normal value within the constants a, a,..., ky, ky. + the equations for refraction and magnetic rotation may be written 2 r2 — ** prea aren 2 Cal OUR aTE Meike Maen AME NR Laure ety ae (6) os ve | mi7by(1 + £1) —= | + anatba(l + $2) n A Cetin .) } 7) The factor €, as mentioned in Part I., represents the effect of the impressed field on the intramolecular fields. It may also be taken to include the influence of the impressed field on electronic coupling, such as would give rise to abnormal Zeeman resolution. It has been shown that the mean value of the dispersional period deduced from magnetic rotation may be smaller than that obtained from natural dispersion. When €=0 we may conclude that this result indicates that 7 is smaller for longer periods than for short or 2 LOR ys Neue are, im sorder of diminishing magnitude, 7,” will have preater welght in comparison with re than N29 has in comparison with N11. The effective value of e/m is always less than the normal value, and 7 is therefore always less than unity. In proceeding to obtain an expression for p we may omit ¢, bearing in mind that the result will need modification if this factor is operative. We have 9141 = lie ry?) (2m 9 9 Ny AG N TUR es 9 Be Ny Di cae Le 2C?m 19\rCm) “1 Taking 24, Ag... in order of diminishing magnitude, we may write and he ” =a ag Ne es © RE ED BBG EE 1) rv : praia: Ay Woe BP ye 4 aN &e. 460 Mr. 8. 8. Richardson on Magnetic Rotary Unless 7 varies more rapidly than the inverse square of the dispersional period, a,, 8,... will be all less than unity, » AU ne (>) e” ? git 00 Tr pea 4 eae a)IM +N + NB +. | JL It e 2 2 2 | Pas 75 (tm) | com [N, + Noo? +N3@?+ Hence | (n?—1—ay)? _ 2e [Ni + Naz +Nj8+ .. 7 i 6 tN Ne Ne _ 46 = In applying this formula we may note that * (i.) N'>N,. Hence the result will furnish an upper limit to the number of electrons of the longest ultra- violet period. Gi.) When Ay, A,... are not widely separated, N’ approximates closely to (Ni; +N,+N3+...), and the result gives the total number of dispersional electrons very nearly. (iii.) As a is known for very few substances, it must usually be neglected and the resuit treated as an augmented value of ». When condition (ii.) is satisfied, the result will be a close approximation to the fotal number of refractive electrons f. (iv.) In the case of unsaturated compounds, the value of N! will be influenced by the small values of «, 8, etc. Using equation (1) to find the number of electrons per molecule, we have a (n?—1l—a,)? M 1 é ae ) a Eh a (9) Denoting the specific rotation and molecular rotation relative to water by p and R,, respectively, we obtain gt } P™ -01308 x -0002909 (n?—-1— ay)? M De for. A="d89s ms Diz * ae x /4109, . 4. 2) WA, and se Gane ce” 02285. . 2) anata * See Note 1. + Of. results for aliphatic compounds, Table III. Dispersion in Relation to the Electron Theory. 461 The expression for p obtained from natural dispersion contains e/m, but it will be noticed that this quantity is eliminated in the present calculation. The values of p for some typical organic substances calculated from equations (10) or (11) are given in Table IIL, a, and € being in all cases neglected. Normal values of v only are given. TasBue ITT. Substance. Ue pp: fy Bip Pp, (min.). PMI ec Rees doa cetve. cs 4 I ae a4 Carbon bisulphide ......... 5 33 oa. 12'5 Methyl alcohol ............... 10 Bs 1°64 10:0 uayealeolvol’ .. .....2....<: 16 aes 2°78 14°6 m. Propyl alechol......... A 22, yaa 3768 20°5 Pammiyl alcool ......:.. +0... ot "989 a 31:0 Acetaldehyde ............... 14 = 2°385 13°6 ropaldehiyde ............--: 20 oe 3°382 19°] RRECHOME: foc esaccet scwses ss 20 44 3°514 18°3 MIOEMIE ACIG 05... .0..00e00--- 10 sie 1671 ILC POEWIECICIO | on scc0ncecccs. +s 0 16 “ 2°525 16°7 PROPOME ACIG 20.5 5....0-.-. 22 ss 3°462 22°3 MEyEIe ACIC 2. ..5..0c0c8-. 3. 28 5 4-472 20-9 Walerianic acid .............. 34 ake 5513 33°4 Gnanthylic acid ............ 46 wei 7552 44-5 Methyl acetate ........... ... 22 she 3°3h2 24:2 Mitiwlacetabe: ......0.c.+.-- 28 Bb 4-462 27°77 Propy! acetal, ©.......-....--: ot a 5487 33°4 Petey oye es co ctlae S32 38 ay 6°670 33'5 LETC a ee 30 2°305 16°6 int Tost 42 1-989 27°3 Diphenylmethane............ 64 2°560 33°2 Triphenylmethane ......... 92 2°770 49°5 iG ae re 58 2°940 29°7 Dimethylorthotoluidine ... 52 2°316 29°6 Dimethylparatoluidine ... 52 2°869 25°9 MA oe oo vciad wo nicinn'v ss 44 2°448 278 PEGATINGEHOS: ....0556-00060--22 46 2°040 30°1 Orthotolylmethyloxide ... 44 2°208 26°2 Dimethylresorcinol ......... 46 2°095 29°4 WictapuBiAcOy ....2c-j02.-+.:- 40 2°428 22°3 a-naphthylethyloxide ...... 62 3°588 27°7 - a Ne a eee 62 3°336 285 Kithylie anisate ............... 58 egg 40 9 m. Phenylenediamine ...... 38 3°576 13°7 Dimethyl]-6-naphthylamine 64 5144 23°2 a-monobromnaphthalin ... 48 3°826 v7 fell A comparison of the columns v and p shows that probably in every case the number of electrons influencing the refraction is not greater than the total number of normal valencies of the atoms. 462 Mr. 8. 8. Richardson on Magnetic Rotary The valne of a is known fairly accurately for the few substances whose dispersions have been measured over a large range of wave-lengths. Thus for quartz, fluorite, sylvine, and rock-salt we have ay = 37, °35, °25, °19 respectively, and the values of » calculated from (10) as given below. TABLE LV. Substance . 6 M d nv i (minutes), ah i D e QA U ZA Monee en en auasaeeen ‘01664 60°4 2°65 154438 4°9 Biltroritie aueaeed. se cenaae "00897 78:0 3:18 1:4333 5:10 ROC Sait: wusnccnanseetacsen "08280 58:5 215 1:5443 412 SylwiMene hours memes: ‘02670 74:6 1°95 | 1-4904 4:87 In fluorite, reck-salt, and sylvine, Meyer™ finds that the fundamental formula e adn =F ea MA Le) agrees well with the experimental results when n does not ‘involve infra-red influences. Hence for these substances tthe dispersional period in the ultraviolet deduced from magnetic rotation should agree with that deduced from dispersion, and if more than one period is involved the effective e/m is therefore the same for both. Introducing m as above (n}=7.=7), Meyer’s results give for ONGC) ae oe eg Ps 1:21 10" (eam oekssalits tee ot so eesielhee Ib Sybvine, wae aence 55 =e bad. The magnetic dispersion in quartz in the visible spectrum thas been subjected to very accurate investigation by Lowry 7. Taking the relative values obtained for °6708 yw and °4800 w mear the ends of the visible spectrum, namely, °646 and 1-318, with n=1:5415 and 1:5501 respectively, and using equation (9) (Part I.), we obtain Y= "106 B. The period thus deduced from the magnetic effect agrees * Ann. d. Phys. (80) p. 607 (1909). + Phil. Trans. A. vol. cexii. p. 295 (1913). Dispersion in Relation to the Electron Theory. 463 very closely with that obtained by Rubens from natural dispersion, namely, Mie 108 Hb. Hence we may conclude that only one value of 7 occurs here. For this substance Rubens gives a,\,?=°010654. Substituting these values in equation (12), and taking o6=°01664' for X= ‘5893, I find n © =1:37 x 107 e.m.u. m Tf the values of the effective e/m thus determined are used in place of the normal value for calculating the value of p from the natural dispersion (Table II.), we obtain *:— Onartz ...... pol Sylvine ...... p=2A, Fluorite...... p=orl, Rock-salt ... p=4+2. For other substances the value of ais not known with the same degree of certainty, since the determinations of n have not been made with sufficient accuracy over a large range of the spectrum. ‘Taking the values at present ob- tained, the results for several liquids are given below, equation (10) being used. TABLE V. Substance. p- d. ie aq. Pp. CO Eos ol ae eee i il 1°333 ‘268 1‘4 Carbon-bisulphide ......... 3°250 1:263 | 16307 634 49 TEST UR Re a 2-305 *879 | 1:5005 Dom Heil CON SCS a 1896 878 | 1:4950 ‘614 67 Naphthalene bromide ...... 3'°826 1:487 | 1°6582 SOL 2Ort It will be noticed that the values of p in comparison with v are always much smaller in unsaturated compounds than in others. This is in all probability due to the existence of a long dispersional period widely separated from the next shorter period. The calculation of p from the formula for magnetic rotation alone could be made if the constants were known. From the formule obtained in Part I. for some of the above * See Note 2. 464 Mr. S. 8S. Richardson on Magnetic Rotary liquids * we may calculate the number of electrons of longest period in terms of 7 and (1+), since bas gt £\'N i= ( )M oO in 1€- Neglecting ¢, we obtain: — Water Si pi 22005 m. Xylene... py} Benzene... p,7,?= 1-038, Naph. brom. .. paj7?@— sem As ), enters to the fourth power,a small error in its value will considerably affect the result. The above numerical results may now be reviewed in con- nexion with the electronic theory of valency. According to the electrostatic theory of J. J. Thomson and Ramsay, the force of chemical affinity is produced by the transference of a negative electron from one atom to another ; in that of Stark the electron is attached to neither atom but exerts an attraction on both ; whilst in the hypothesis recently put forward by Arsemt,the particle is in a state of continual oscillation from one atom to the other. Without, however, formulating any hypothesis as to the mechanism by which the electron sets up the force of chemical affinity, whether electrostatic or electromagnetic, we may conclude that only one electron is concerned in the production of each “‘ bond” of affinity ; and the theory of J. J. Thomson and Ramsay may be used as the most convenient way of distinguishing between these electrons and others associated with the atom. Remembering that the molecule as a whole is neutral and com- mencing with the atoms of which it is composed assembled in their neutral state, it is clear that the number of electrons to be transferred is equal to the number of active positive valencies in the molecule. Now according to the theory of Abegg t, these will represent either normal or contra- valencies according to the group of the periodic system in which the element falls. Thus, tor example, we have: GiCOWDs elsnt 1 2c), Bea O60 a Blement:..... Na Me) Aliiesr) Po) SiC +Valency... +1 +2 +3 +4 +5 +6 47° 0 oe Be Sus Sa aye ed Normal. Contra. * In the tables given in Part I. for naphthalene bromide and carbon bisulphide, the values given under 6 are those of 70. + Jour. Amer. Chem. Soc. (86) pp. 1655-1675 (Aug. 1914). t Zeit. anorg. Chem. (89) p. 330 (1904). Dispersion in Relation to the Klectron Theory. 45: Hence the sum of such values for the atoms present in a molecule gives the maximum number of electrons capable of producing affinity; and this number should be taken for v in place of the normal valencies. The numbers in paren- thesis given under v in the tables are calculated in this way. It is to be expected that the electrons which produce the attraction between the atoms in a given compound, and which according to Thomson’s theory are transferred from one atom to the other, will be those subjected to the weakest controlling foe and will be, therefore, the long-period electrons which give rise to the dispersion and magnetic rotation. Various considerations support this view. When the number of bonds increases relatively to the number of atoms, as in unsaturated and aromatic com- pounds, the dispersive power undergoes a marked exaitation. The numerical values obtained above for the upper limit to the number of electrons of longest period are too small to be associated with any but the active bonds. In quartz, where the dispersion constants are probably more accurately known than in the case of any other substance, we have p,=4, 2. €. one electron for each bond in the molecule. In elements, however, which show high positive contravalency, particu- larly those of high atomic weight, the electrons not associated with the bonds may influence the dispersion owing to weak controlling forces. Thus the six contravalency electrons of sulphur are probably responsible for a large part of the dispersion and refraction in carbon bisulphide, and the same is probably true of nitrogen in the highly refractive amines. The point of view assumed in the present theory will perhaps be rendered more definite if we make a distinction between the various electrons which may influence the optical properties as follows. Class I. The recent application of photometrical measure- ments to absorption phenomena has shown that the character- istic absorption bands in many cases must be attributed to electrons whose number in the unit of volume is only a small fraction of the number of molecules. Henri™ finds the absorption in acetone is produced by 1 resonator in 36 mole- cules; Baly and Tryhorn ¢ find the number of molecules operating in the production of the less refrangible aniline band is 1 in 30; Keenigsberger and Kilchling { deduce for the bromine band at °412 w only 1 in 60. Hallo, from observations on the magnetic rotation in sodium vapour, * Phys. Zeit. (14) p. 515 (1918). + Trans. Chem. Soc. (107) p. 1121 (1915). t Ann. d. Phys. (28) p. 889 (1909). Phil. Mag. 8. 6. Vol. 31. No. 185. May 1916. 21 466 Mr. 8. 8. Richardson on Magnetic Rotary obtains an exceedingly small fraction for the ratio of the effective to the total number of molecules. It appears that the frequency brought into play in these cases is due to some accidental or seldom-recurring condition otf the molecule. On account of their small number, electrons of this type have little influence on the refractive index or magnetic rotation, except quite close to the absorption band to which they give rise, but are probably capable of giving rise to fluorescence and phosphorescence. Class II, Valency electrons of dispersional type :— (a) Under this heading are included the electrons through whose agency the attraction between the atoms in any given molecule is produced. very such particle represents one valency bond. (6b) The contravalency electrons of certain elements, par- ticularly N, P, 8, Cl, Br, I, may have a frequency sufficiently low to aftect the dispersion as well as the refraction. This is more marked in the case of the elements which readily part with these electrons, e.g. sulphur and iodine. Class III. Valency electrons of non-dispersional type.— These differ from class II. in possessing a frequency so high that their effect on the dispersion cannot be detected experi- mentally, whilst they still exert a measurable influence on the refraction. Class IV. Intra-atomic electrons.—This term may be used for referring to electrons present in the atom which take no art in deciding the valency. Such electrons will undergo so little displacement relative to their positive complement, that it is probable that each will exert only a very small effect on the refraction. An examination of the numerical values given in the preceding tables shows the following relations hold good :-— (a) The minimum number of electrons demanded by the refraction calculated on the normal value of e/m is never greater than the total number of positive (normal or contra-) valency electrons ; it is never greater than the sum of those of class II. and class III. (6) The minimum number allowed by the dispersion and magnetic rotation is never greater than the sum of those of classes II. (a) and IL. (0). (c) The maximum number of electrons of the longest period allowed by the dispersion and magnetic rotation is never greater than the number of electrons of class II. The empirical character of the four-constant formula Ay" 9 n?—l=ayt+ VAL —cnr Dispersion in Relation to the Electron Theory. 467 has already been referred to. When there are more dis- persional periods than one, the shorter ones are more likely to escape detection than the longer and the tendency is to obtain too great a value for 4, with the result that a, is too small and ay too great. The accuracy with which the constants can be determined depends of course upon the range of spectrum covered by the observations and upon the accuracy with which the individual values of n have been determined. Where both conditions have been ful- filled, as in the case of quartz, rock-salt, sylvine, and fluorspar, the value obtained for aj is alwaysa small fraction. In a homologous series of organic compounds each addition of a methylene group adds an equal number of electrons to the molecule, but as such addition also produces an approxi- mately constant increment in the molecular volume, the number of electrons in the unit of volume or the quantity p/Vm tends towards a constant value as we proceed to the higher members of the series. As this applies to the elec- trons of each class independently (except class I.) we ought to find that the value of ay which depends upon the number of electrons of class III. (possibly also to a slight extent of class IV.) in the unit volime approximates to a constant value. The following table shows that numerical results are quite in accordance with this view. jp, is put equal to the number of linkages (class II.a) and a is calculated from equation (11). As the compounds are aliphatic con- dition (ii.) is satisfied. TaBie VI. Substance. Py: Qo. Substance. Das Qo. EPHEAHO 02. 505 cvai con oss TG? 7206, ||| Pormice acidin.s.3. 04.30: 5 | °286 ii 12220. | MACEHEC AGIGL) +2222. eeees.. 8 | :27] BRIE fb 82% 55152 30: | 25 | ‘233 || Propionic acid ........- LE 205 Methyl alcohol......... | “5 | 293 || Butyric acid ....)..2.... 14 | -278 Ethyl alcohol ......... 8 | -221 || Valerianic acid......... Lg ear, n. Propyl alcohol ...... 11. | -246 || Ginanthylic acid ...... 23 | *287 Acetaldehyde............ 7 | 219 || Methyl acetate ......... UN 208 Propaldehyde ......... 10 | -238 || Ethyl acetate ......... 14 | :255 vel 2/1 001 ei et 10 | °222 || Propyl acetate ......... 17 | °262 ( In view of the fact that N’, equation (8), is only an approximation to N,+N,+N3;+.... the uniformity in the values of agis even greater than might have been expected*. In the following section we proceed to consider the bearing of the present theory on the additive optical properties of organic substances. * See Note 3. yoiga a 468 Mr. 8. 8. Richardson on Magnetic Rotary The Correlation of Additive Properties in Magnetic Rotation and Refraction. In seeking to obtain additive relations between the optical properties of a substance, it is necessary to bear in mind that the electrons associated with the atom of a given element will possess different periods in different substances. The polarization of the medium set up by the electric force of the light-wave which, if averaged or smoothed-out would be neutralized by the polar field of the medium as a whole, owing to the discontinuous structure of matter, is locally coucentrated in the vicinity of the individual molecules. The polarization near the molecules therefore exceeds the polar field, and the excess constitutes a force acting on the electron in opposition to the controlling force exerted by the atom. ‘The period of the electron is thus increased, and must be distinguished from the period which the electron would have if the atom were isolated. The degree of augmentation, since it depends upon the degree of polari- zation, will vary with the nature and density of the substance. Any term of the dispersion formula which represents the effect of the electrons of a given kind of atom will possess different constants in respect of different substances. This of course applies equally to magnetic rotation. In order that the dispersion formule for different substances may all contain the same term for the electrons of a given kind of atom, it is necessary to eliminate the polarization effect and to obtain a summational expression in terms of the zsolation frequencies of the resonators. Whilst the necessity for doing this has been generally recognized in the case of refraction, it is noteworthy that, with regard to the magnetic rotation, additive relations have been sought for always from the values of this quantity as directly measured. The influence of polarization on refrac- tion has been fully investigated by H. A. Lorentz, and the components of molecular refraction are usually determined from the well-known Lorenz-Lorentz formula. The results are more consistent than those obtained from magnetic rotation, and the latter quantity has therefore been regarded as more highly constitutive than refraction. Whilst this may be quite true, it is clear that polarization effects should be eliminated in the magnetic case and the contribution of each electron expressed in terms of the isolation frequency of that electron. Lorentz * deduces an expression for the rotation which, * “Theory of Electrons,’ p. 163, eqn. (246). Dispersion in Relation to the Electron Theory. 469 using 6 for this quantity, may be written a aty t 40 UlatyP te GaP where v is the frequency (angular velocity) of the incident light, and putting vy for the angular velocity of an electron vibrating with its isolation frequency, and a for the polari- zation constant, =x, x (V0? VO) la; ’ pe: ys wikk » Me? aR eOIN ex A more convenient expression has been obtained by G. H. Livens* which, in terms of the symbols used in the preceding parts of this paper, becomes 9 I gk m,(vy2>—v") ca na—l= = 5 ae” 1-= evH m(vy2 —v?) + C in which the + and — suffixes distinguish the values of n for circularly polarized light of opposite chiralities. Since Vv 2 — ——(n7—-n inc | a we have yee ah eHy* myr(vey—v?)? He Dae ae? = Le My (Vy — Vv") Taking the polatization constant a as 1/3 and writing 2 e 5p lal ni "oa Ba Rig ia va 5 ahs Ee : nN it G we obtain ‘ o= ni—l= = : nit2= an eS anes * Phil, Mag. August 1913, p. 362. Mr. 8S. 8. Richardson on Magnetic Rotary 470 Hence w @_ i es 1 oer j aed _ @,—o_ = =e (1-3) 2 y 2 — € " ) (@,—@_) Now e2 oO, = Ged coal ob H my (Yy = y”) + o e2 O>0.0.-85 oe FR oo oO orc oOo 6 m(v?—v?) — a Therefore Pov H 1 ae ae NP eit on @ oO_= ce on account of Thus eS and on= WOR ne or re m?(ve—v?)? — ( 1 “ual the relatively small value of (evH/C)?. 2e3vH - 3 nt=—("**) 2vH e nei 3 CO me ae cotaauas i )) 2vH e° Ani BP we | mr Ow rae a Gass = s e° a One? ( 3 m,7(V,2 —v?)? _- 2H (ety sf et oe wl 3 ae en But from Lorentz’s expression for the component of the refraction *, we may put for the effect of each electron n’?~—1l 1 e° ne t2~ 3m (v2—v?)" * «Theory of Electrons,’ Note 58, p. 308. Dispersion in Relation to the Electron Theory. 471 Hence 2H. pai Ba — Feu +2 B(O TT) os Converting from rational to ordinary symmetrical units and taking v in periods per second, we must write H yeh res for H, e/4m fore, and 2zyv for v. Thus 6 7 av wes (5) ° POzey Gil phim Nie 432 The summation here extends to all the electrons in the unit of volume; but as every type of electron present in one molecule is repeated in every molecule (except those of class I. which do not produce a measurable effect) we may write mv?H (n?+2)? n~—1\? Ny —1\) ? ae LN (ag) +4¢N(Te5) } ; where N is now the number of molecules in the unit of volume, and the summation extends to the p individual effective electrons in the molecule. The components of the molecular refraction as usually calculated with the same notation are expressed in n?—1 M n M nag g =(Nazaz) gt--: top terms, =P,+P.+ SOG DO + Py. The expression for the absolute molecular rotation R (= M6/d) may therefore be written _ mv (n*+2)? Sei P es Pe urea, we C22 n But since N=d/Mzh this becomes Tv" ea) 2 = are n Y {PP+ Pit... + Pe}. = Hence the quantity in which we must seek the additive relations is not the molecular rotation R simply *, but n (n? +2)?" * See Note 4. 472 Mr. 8. 8. Richardson on Magnetic Rotary The magnetic rotation is usually measured for the sodium line, and the molecular rotation generally recorded is not the absolute molecular rotation but the ratio of this quantity to the corresponding quantity for water. Calling the ratio R, and substituting the usual numerical values of the constants, we have nN Ri, (n? +2)? in which the molecular rotation is expressed in terms of the molecular refraction components. According to the theory put forward above, only the electrons of class II. (a) or IL. (b) influence the dispersion and magnetic rotation. In general, where the resonators IT. (6) are not operative, the number of terms in the sum- mation will depend on the number of bonds only, and the value of P for each bond may be calculated. Using the notation (CC), (CH), to denote the value of P for the electron con- cerned in each linkage respectively, we may deduce the numerical values as follows. Taking the two hydrocarbons pentane and hexane, for example, we have by experiment, Borthexanens: Ant. Ry = 6670) «' =a oss Kor pentane... % Ry, =0'638, 1. — ase Substituting these values in (2), 26°50 = 5(CC)?+14(CH)? (hexane), 22°665 = 4(CC)?+12(CH)? (pentane), X43 TAS SAPs = Bo? hs ee ° (2) whence (CH)2=1°881, (CC)2==-172. On account of the approximate constancy of the differences for a CH, group, only two independent equations can be obtained from a homologous series. Hence we cannot obtain evidence from equation (2) of the null effect of electrons of classes III. and IV., and for the present must rely on the considerations already given. Numerical con- firmation will be obtained when we come to the calculation of refraction for the alcohols. For the value of the CO linkage we may consider one of the ethers. Selecting ethyl ether, i, = 4300 aio, 19°315 = 2(CC)?+10(CH)?+ 2(CO), whence (CO)2 = 330m Dispersion in Relation to the Electron Theory. A473 Selecting n. propyl alcohol for the determination of the value of (OH), we have Be Ol OO, |) ey LOO. 14°867 = 2(CC)?+ 7(CH)?-+- (CO)?+ (OH )?, whence (OED)? "== Waieo. Ta the case of the carbonyl group we cannot separate the values of P for the two linkages, and can only determine the sum of the squares. In this and other cases of unsatu- ration the summation may be indicated by prefixing Deh Choosing propionic acid for this, R, = 3°462, np = 13866, 13°648 = 2(CC)?+ 5(CH)?+4+ (CO)? + (OH)? 4 &(CO)?’, which gives DCO)? 23: Using the above values we may calculate the value of R,, if the value of n, is known, tor organic compounds involving only these linkages. TABLE VII. Substance. ees | R,, (obs.). | R,, (cale.). Diff. Thain CWA Se 2 rr 13581 | 56388 | 5638 ref. WTEMABIOY oc iees sot cise ee eae 13754 -{- 6670 | 6670 ref, retamien cate cot Stee Le S963 ee 8 1220 er oF +002 Methyl alcohol ............... 1°3294 L640 Wi LT De sell Mey AICOWOL. & .s.5c0cee: 00s 13023) I DSO" | 2 for — 028 m. Propyl alcohol ............ 13854 3°768 3°768 ref. m. Butylaleohol ............ 1°3991 4760 | 4785 +°025 Giyeerimen se. 55.52).08.s ioe 1-4729 40938 | 3944 —149 BPP VVETMET bude wetiee des clone 1:3529 4-799, |, 4798 ref. Bere Bet! 5 oe.-.--2- 254050: 13714 LOM Ne Lai —170 miceric acid: 22408.5000.2 4.0% 1:3718 2525 | 2:464 —:061 PTOPIONIC ACID (6.02 cee +08 13866, |) 3:462.) 4) 3462 ref. BULTIC ACIG 2, es. .c- conan te: f- Soom Arent) A469 —°003 Walerianic acid............... 14043 5513 | 5474 — ‘039 Cfnanthylic acid ............ 1:4215 7552 =| = 17-5384 — ‘018 EMMY TIC BCA. ics. sie ln s0 1°3930 Braga | AAD SLO ACAALOCHY ES |) o5:% 500 000ce 13316 2385. | . 2429 +044 Propaldehyde ............... P3636 2° | B'dd2) lo a atoo +107 INCCEOMEID Pras ae. biti wdoekie 1°3591 3514 | (3428 — 086 Methyl acetate 2)... 0..0.03... 1°3610 Sie02 |) abdd +:193 Methyl butyrate ............ 15SBO0 i OSA 0 OTL +:197 Ethyl formate ............... de lh OOD Mal er DOe + 001 Hthyl acetate slo... .:. | 1°3726 | 4-462 4°546 + 084 Kthyl butyrate <......)....... [) 13960) /) G477) 6-580 +°103 Ethyl valeriate ............... 13970 | 7500 | 7562 +062 Hthyl oxalate ).2.45..54:32. 1-4104 6654 | 6-457 — 197 Propyl acetate t.2...5.c06.>- 1:5844 O4et |", o°b04 + °067 | A474 Mr. 8. 8. Richardson on Magnetic Rotary The values of np are those of Conrady, and R, is that of Perkin. It will be seen that although the values of the linkages are those deduced from R, for a single substance in each case, no attempt being made to work out mean values, a very fair agreement between the observed and calenlerees values 1S Spend Turning now to the refraction, we have ‘n?—1 = |e =a) ad = Pit Pi +.-.+ Fy The values of P,?, &e. oo by the magnetic;rotation for the linkages (CC), (CH), CO), (OH) enable us to calculate at once the corresponding valties ee Ege (Cll) = aeao3: (CO), = oie (CC) == lid: (OH) ==) Lrlfioe The sum of the P’s for the electrons of classes III. and IV. of the atoms C, H, O must first be calculated, for which we may use the following values for the molecular refractive power with reference to the D line. iPentanewvucsuee ee P = 25°17 ihhexane wee... sick P = 29°71 Dea 3) Pegs ga NC. yr P = 22°37 Thus denoting the sums just referred to by the symbols. C, H, and O, we have 29°71 = 5(CC) +14(CH)+6C+14H = 21:0154+ 60 + 14H, 25°17 = 4(CC) + 12(CH) +50 + 12H = 17°895+50412H, whence i= “Care Ci e245: For the oxygen atom, 22°37 = 2(CC)+10(CH)+ 2(CO) +40 +10H+0. Hence Ora Ols: The two linkages of the carbonyl group cannot be evaluated from the magnetic rotation, since the latter gives. the sum of the squares of the two P’s, and these may not be of equal value. We must obtain }(CO) therefore trom the Dispersion in Relation to the Electron Theory. 475 refraction. Taking propionic acid, for which P = 17°46, we have 17-46 = 2(CC) +5(CH) + (CO) + (OH) + (CO) +30+ 6H +20 = 829 + 6°765 + 574 +1°173 + 3(CO) +.3°735 + 522 + 2:026 whence >(CO) = E836. The additive relations between the refractive powers or the magnetic rotations deduced by the methods of Landolt, Brihl, and Perkin, rest solely on the constancy of the differ- ence introduced by the addition of the group —CH,— in different series of homologous compounds. But we must note it is perfectly arbitrary whether the summation shall be referred to the atoms alone, as in the usual theory, or to the valency linkages alone*, or to the two combined as in the present theory. For this theory it may be claimed (1) that it contains internal evidence of its truth, (2) the values obtained admit of a more rational interpretation than those of the “atomic” method, (3) it correlates the phenomena of magnetic rotation and refraction. The test referred to in (1) is provided by the alcohols. The value of the (OH) linkage may be determined from the molecular rotation, or independently from the refraction. For the latter, taking propyl alcohol, we have P = 17°46, and therefore 17°46 = 2(CC) +7(CH) + (CO) +(OH) +3C4+8H+0 whence OH = 1°141. The value of OH deduced from the magnetic rotation of propyl alcohol is 1°173, a sufficiently close agreement. The mean value of OH deduced from the refraction of six alcohols is 17136. With regard to (2) it may be remarked that the small values of the (CC) and (CO) linkages indicate higher frequencies and therefore stronger controlling forces on the electrons, than in the (CH) and (OH) linkages. This result may reasonably be associated with the more easily replaceable character of the H atoms in the latter. Further, the atomic value for H is very small (-087), which accords with its * In terms of the linkages alone I find the following values for the components of the molecular refraction for the H, line :— (CH)=1°683, (CC)=1-249, (CO) =1-493, (OH) = 1°627, (C=O) =3-404. Thus for ethyl alcohol: P,=5(CH)+(CC)+(CO)+(OH) =12°78. (Observed value 12:71.) A76 Mr. 8. 8. Richardson on Magnetic Rotary small atomic weight and low valency. The refractive effect of H is probably due entirely to electrons of class IV. ‘The values obtained by the ordinary “atomic ” method do not admit of any such interpretation, the value for H being 1:05 and that for C, an atom of 12 times the atomic weight and 4 times the valency, being only 2°5. The subjoined table gives the values of P for a number of aliphatic compounds calculated from (CC) = 415 C = 1-245 (CH) 23538 H= -087 (CO) = °574 O = 1013 (OH) = 1173 S(COY = 836 TaBLE VIII. Substance. Fy(Obs:), si) ey ealle)): Diff. : EEA SUR ste RS Hod hs laa Rem AMGL I: Sactap cee Mune eat Doel ey Ve e2oaley ref, WHO ATO! wots ne Meee, Wank I LOGE hed inne Oral ref, Ocha pr egaiaratay ss Ae ly Oa eh Bene +08 I Methyl aleohol 3%. 8 22.) OOD al eral eG BiclyWalcowol. se ss-aye Hille Teena wl OD +21 | Propyl alcohol oi... ine VAC DEAS +03 ie louiby lalcoholy nis, :s ee 22, 0G rn ei22 es — 05 I Glycerine erate) ie so) Veneto: |) (20;4om en 320:3i —'14 | Bit loin ether Way) iye 3.49. a8 Vl 2B Cog ref. | Propyl-ethyl ether ......... I \26SOy 2G: Ont +:0d NiBtommeyacid Ei, ss ocuke ae: S047) Wakes —'16 IICCLICHACICY ane se en ectecjen: be EO Ge ale — ‘04 | Propionic acid 4, | GAG) aoe ref. SUG AsIC ACI pels se tase tel 22 NOW ie 22500 —1U | Walerianie acid /))..0)/32.... 26-79 | 26°54 —°25 Capryliejaerdh Woe... 444. 31°30 | 31:08 — "22 (Enanthylie acid ............ 3o7'94 5) * 85-62 — 32 Pedal BRU ia er Cone asl Ae, eR aa 227i OBiee hy aeiOU —°08 mcetaldeliyde iaenien. assae. JMS YR ales ME SON — 02 | Propaldehydeyi. sans. PRESS 76 a AY Siler Os +°08 Duby aldelnyde yi) see seen: 20°58 20°59 +01 ) EAI A UNG RA FRc A We he i 2 SO mo i puede — 25 CO Dip Fah oie aVoy Bee cme Be Maen Bde t 7-40) A fies soa db —08 Aicetonie ei haan Lots 2 ae L6OF We) 16;05 — 04 Methyl acetate ............... LS: O07 pai alas — 27 Miethiyl butyrate, .....0 sens: 26°38), 4), 2068 +°03 | Methyl valerate ............ SU SS ae eA ah Bthyl formate )....062..4.- 7-97 | 17°80 — 17 MBibiny Wacebabe\ leas sccm DOV) 22 34 +:16 ply My cahe 3). sa vkee cages |) ole 2 Gig Mine AD +16 (abrtliyl valerate’ 2.24.22 .hande2 36°07 | 33:96 —‘1l (each Ab) ob. calla h =) vee Nae peer CM inrexeaiiL cd — 34 m, Propyl acetate ............ |) 26°89 7 1.126788 — 01 Atmy | TOmmaa ts) Meese near ans ol7s) |. 31:42 —'3l amyl wvaleraben eo... sine ee men | 49-83 | 49°58 — 25 Dispersion in Relation to the Electron Theory. ATT Although the values used were simply deduced from four reference substances, a very fair agreement is shown throughout. A much better agreement may be expected if average values are worked out from a larger number of data, and for some given temperature. Unfortunately, most of the refraction data available have reference to the « line of hydrogen, and the magnetic rotations have reference to: the D line. The values of the molecular refraction P in the above table are those of Conrady *, and in many cases were obtained by interpolation. I hope to deal later with the application of the method to unsaturated and aromatic compounds, and with the recaleu-. lation of some of the values given above. If the electrons of class IV. are inoperative, the refractive value of the hydrogen atom in the ultraviolet should be zero. The value "087 is so small that a better collection of data may prove. this to be the case, but the values for the carbon and oxygen atoms can at present only be ascribed to the influence of a large number of electrons of class IV. Note 1—From a well-known theorem in inequalities we have- for n positive quantities a, b,c, .... k, (atbte+...+kyf apie (Saute eat a) The curves given by the equations (7) and (12) have been 486 Dr. 8S. Brodetsky and Dr. B. Hodgson on the plotted in fig. 4, A and B respectively. It seems that the first hypothesis, giving the curve A in fig. 4, is the more satisfactory one. A good test would be to perform a series of experiments between the same limits of #, but with Fig. 4. P different current strengths. The time occupied by the absorption should be proportional to the one and a half power of the current strength. Such experiments would perhaps be rather long and laborious. Attempts to obtain a relation between the cathode-fall and the rate of absorption from the data supplied by the results of our experiments, met with only partial success. The mathematical investigation suggests the relation to be one of proportionality, but there is no doubt that the cathode- fall as recorded was subject to considerable experimental error due to lag, especially for small and for very large values of V. fated | wl) (00__120__140___{60___180___ 200_ minutes. HYDROGEN In the case of hydrogen, quite a different type of curve was obtained, fig. 5. The absorption was at first rapid, and Absorption of Gases in Vacuum-Tubes. 487 then diminished to zero. At first sight, this would seem to be in direct opposition to the disintegration theory, but an examination of Tyndall and Hughes’ curves shows that the case of hydrogen lends additional support to this theory. Fig. 6 is reproduced from their paper. In air, the gradient Fie. 6. gration a oO” > =. f 16%) Rate of disinte N a 2 £00 400 660 800 1000 1200 1400 1600 volts Cathode fall of potential in the rate of disintegration increases as the cathode-fall increases; the rate of absorption also increases under the same conditions. In hydrogen, the gradient in the rate of disintegration soon Beeee ce arith increase of cathode-fall, tending towards zero; the rate of absorption also decreases with increasing cathode-fall, tending towards zero. The exact parallelism between lie rates aoe absorption and the gradients in the rate of disintegration in these widely divergent results is an excellent verification of the theory advocated in this paper. 8. General Discussion—The absorption is probably a phenomenon of some complexity. With metallic electrodes, the major portion is due to absorption of gas by the dis- integrated metal. Some may be due to the liberation of the alkali metals by electrolysis of the glass, and chemical combination of these metals with the gas. Hixperiments in tubes made of other materials than glass would be of great 488 Dr. 8. Brodetsky and Dr. B. Hodgson on the interest in this connexion. Many of the apparently con- tradictory results obtained by different experimenters can be reconciled by means of the dual explanation here suggested. Willows’ concluded that the absorption was chemical in origin, because he found its rate to vary according to the particular nature of the glass from which the tube was made. ‘The absorption was biggest in soda glass, less in lead glass, and least in Jena glass. IPf the glass is electrolysed this is exactly what one would expect. During the passage of a discharge the glass becomes highly charged, as was shown by Riecke ™, and may easily be electrolysed, especially if heated by the cathode rays. Warburg, in fact, used this to obtain a film of metallic sodium on the inside of vacuum- tubes for the purpose of purifying gases!*. Mey’s? results show that when sodium and potassium amalgams are used as electrodes, compounds of these metals with hydrogen and nitrogen can be formed, suggesting that the same compounds can be formed with the alkali metals in the glass. The fact that Hill obtained an absorption with the electrodeless discharge® can similarly be explained on the hypothesis that the glass is electrolysed, the metal liberated causing the absorption observed. Hill found that if successive absorption experiments are performed, the absorption gradually decreases. This is again in agreement with the theory that part of the absorption is due to the electrclysis of the glass, for as the soda glass used by him became more and more electrolysed, the sodium was gradually used up, and the absorption naturally diminished. It was also found by Hill that an experiment in oxygen followed up by one in hydrogen gave an accelerated rate of absorption in the latter. He offers as an explanation of this fact the suggestion that the cause of absorption in the oxygen experiment was oxidation, and then if we use hydrogen immediately afterwards, there is a reduction, and consequently a rapid absorption. This may be the case. But in fig. 6 the authors show the typical absorption curve for hydrogen, there being a rapid absorption at once, whether a previous experiment has been performed in oxygen * See par. 1, note 1. , Wied. Ann. ill. p. 414 (1899). 2 See par. 1, note 2. AN. CG. EnUS. x. p. 2a eau). ° See par. 1, note 6. Absorption of Gases in Vacuum- Tubes. 489 or not. Curves V., VIII., [X., X., in Hodgson’s paper 4, show the rapid absorption in hydrogen after the tube has rested awhile, an initial rapid absorption being obvious in each case. It appears, therefore, that the initial rapid absorption is due solely to the hydrogen, and does not depend upon the tube having been used with oxygen immediately preceding the hydrogen experiment. Further, the fact that the inactive gases helium aad argon are readily absorbed, shows that means other than chemical must be at work. In 1907 Campbell Swinton found that gas was occluded in the walls of the glass vacuum-tubes after discharge ’. This gas was found in the form of bubbles a little below the surface of the glass. But in 1908 Soddy and Mackenzie obtained the Campbell Swinton effect ®, and found that the gas occurring as bubbles was probably due to the electrolysis ot undecomposed carbonates and sulphates in the glass. The gas absorbed by the discharge was found by them to be not in the glass but in the film deposited trom the cathode by disintegration, and a large fraction of the absorbed gas could be recovered. They also showed that sublimated magnesium or aluminium did not absorb helium. It was, however, shown by Heald?® that many metals absorb hydrogen on sublimation in a vacuum-tube. Heald worked with hydrogen and his curves are typical absorption curves for hydrogen. The absorption in a vacuum-tube is thus not a mechanical occlusion, but is brought about by some electrical means. Skinner “ concluded that gas was evolved from the cathode and absorbed by the anode. With new electrodes there is often an initial evolution of gas especially in hydrogen and nitrogen, or in any gas with aluminium electrodes. But if the tube is used and then allowed to stand awhile, on restoring the current, no initial evolution is found in most eases. The evolution that Skinner speaks of must refer to fresh or virgin electrodes, and is probably due to the state of the surface with regard to its gas content. In the case of aluminium, gas is evolved for a long time before absorption sets in. In Hodgson’s curves XXXYV., XXXVI.4, the pressure curves for copper anode—aluminium * See par. 1, note 4. * See par. 1, note 7. ° See par. 1, note 9. *6 Phys. Rey. xxiv. p. 269 (1907). 7 Phil. Mag. xii. p. 481 (1906); Phys. Rev. xxi. pp. 1 & 169 (1905) ; Phys. Zeit. vi. p. 610 (1905). * See par. 1, note 4. 490 Dr. I. J. Schwatt on the cathode, and for copper cathode-aluminium anode aré shown. In the former case gas was evolved for a long period, and absorption did not set in at all. In the latter, absorption occurred after the passage of a few coulombs. The probable explanation is that disintegration did not set in at all in the first case, and hence there was no absorption, whereas in the latter case disintegration, and therefore also absorption, set in early in the experiment. Riecke® concluded from some of his experiments that a fraction of the atoms bombarding the cathode are absorbed. He was unaware of the changes in the rate of absorption that occur after a tube is rested, and neglected them. That they are considerable is apparent from curves VIII., IX., XXXYV., and others in Hodgson’s paper’. The apparatus used in the above experiments was obtained by a grant from the Research Fund of the University of Bristol Colston Society. Physical Laboratory, University of Bristol. LVI. Note on the Hxpansion of a Function. By I. J. Scnwarr™*. pe expand (tamed 2)22 en lta a C9) Vm oa) =2) > (—1)ytt 2 DS (1) 2 eee i Zipe— || a e’ 0 KJ oe ot ® See par. I, note 8. * See par. |, note 4. * Communicated by the Author. EHepansion of a Funetion. — 49] Letting «+a= 8, we have ae 1)2- ea 3 ats! > (— esa 78-8, 1 k=1 B=x+1 = 3 (—1)P2%-3 © B 9 A = Scape § : B=1 : c— ny 2k = Therefore ea ee, i (tan pee = (1 Pe 1 he ae : ° SAN Visine teen eres yunns (2); Again fant) =3 pan Try 7, i eee re hk =3.2! ia > (—1)«-} rs > - > (des reeds 9 K=1 2K kKi=1 AS toon SO al K @ 2) Seige. ani Sie ail 1 Pee 0 K 2) fog p-l1] «x =31\ = (—1)FaF#? & — & : dx. 2'Q B=2 n= 2 1 2k, — 1 Hence ant K } Ky Al (iam ae) = 3 312 er be bot it Sie | Lyin aS ° ° e . (3) We now assume that fee, YS (ane (a : 5) (4) | ko=1 Fe Ao | A as rae (I) = then = (tan "a)e tan~lv)eti=p! (tan 'a)P ; (tan~ 12) pi(p+1) he dz, 2ko+p—2 —1 / *a—1} =(p-++ 1)! Ss aU eo Ds a a3) VA) Ky=l + p—2a= 1 c,=17%a tp a—2 @o x > (-1)1w-2de, y=1 492 On the Expansion a Function. oy 1 sik So 7 eae 4" 3 ec asi ib Wreea i x & (—)eera m=Ko+1l =(p+1)! pe 1)™g2n+P- ‘( > 1 ) eon 2ho pe oo Es ii ; (© aeeprace)™ ee) n2m+p—1 m 1 =={ ! —_1\m—1 NG eee ins ee aly Y) 2m+p—1 =e p—-1 ;*a-1 1 ime es ae . a Ka=1 2Katp—a—2 Replacing m by « and «,-1 by x. we obtain tanta)? *= (p+)! en je (tan— P 2k p Hl eee ee 1— a—] co |. 1 ‘ a=2 Ke 1 2katpt+tl—a—2 geoxutpti—2 = (pal). —])-} pact p+l—1l Kg—] ~~ Hil ea : which is of the same form as the expression (4), p being replaced by p+1. Hence reasoning by induction (4) is true for all positive values of p. : 0 Therefore assuming II A,=1, we have ell ice) 2kqtvp—2 5h K skort P Ds (dane 2)?» ! > (— 1 )e-! wees! (= gan” 2K Se Be i Notices respecting New Books. 493 Thus : er ce =, 2% 05 ee a eee! tan w=11| 2 a + 2 ae 2 La iL) alan - SAS a 1)\ 2% glee (tan~*z) =21[5 ele ae aie - 1 a? oe! ] D | eye | ee oe. zs = = (tan~ *a 31| 55 et ae University of Pennsylvania, Philadelphia, Pa., U.S.A. LVII. Notices respecting New Books, Tubles for Statisticians and Biometricians, Edited by Prof. Kart Pzarson. 9s. (Cambridge University Press.) HIS volume contains 55 tables and abacs, with an introduction of some 83 pages giving particulars of their use, The tables fall into three groups. The first group centres round the pro- bability integral. If N is the total number of observations, m the mean value, and z the frequency of an observation of magnitude a, then ee V On where o is a constant characteristic of the distribution. Four < sal exp —(a#—m)?/2o°, oO a 1H ar tables enable the reader to find any two of z, eS cdn—( dx, o Je ae i. €. the ordinate, the abscissa, and the difference of the areas of ihe two parts into which the ordinate divides the probability curve, when G x2 the third is given. Other tables give} w*e~ 2 dx, and enable 0 the whole curve to be reconstructed from data concerning a portion of it. : The table for testing goodness of fit is founded on the above normal law of error, and may be considered to belong to this group. It deserves to be better known. Let 7’, f,', f,',... fn’ be observed frequencies corresponding to %,, %,, %,,...%n. Let z=o(x) bea AQ4 Notices respecting New Books. | theoretical frequency distribution meant to describe the above, giving __ f'\n theoretical values f,=9(“,), f,=9(@,).... Then if y*=2 (=) be calculated, P corresponding to this y” is found from Table XII, and gives the probability that in a repeated set of observations the actual frequencies will differ from the theoretical ones (this differ- ence being measured by x”) as much or more. If this probability is high, the fit 1s good. The second group is intended to simplify the calculation of the constants in the group of frequency curves introduced by Prof. Pearson as generalizations of Gauss’s normal curve. Gauss assumed (i.) the equal probability of errors in excess of the mean and errors in defect, (i1.) the continuity of magnitude in the errors, and (ii.) the independence of all the small contributions dy 0 ieee a . gs to the total error. This leads to a don ee y =e 203" Ab Co Prof. Pearson drops these assumptions and finds a. dy _ etd y de atbatex” leading to curves of seven types (five more have since been dis- covered) of which Gauss’s is one, and ma \ mb T= (1 “= =) (2 a. 3) is another. Tables XXVI-XXXI. and XLIX. include a table of Gamma functions, logarithms of factorials, the powers and sums of powers of natural numbers. ‘The latter. are especially useful for solving numerical algebraical equations of high degree. The third group deals with correlations and with probable errors of the constants employed. Briefly, if x—m.)* y—m.) 1 2 20.7 20,7 i z=Aexp — gives the frequency distribution of deviations of a and y from their means, then these deviations are independent ; but if CEs (y—m,) _ 2(~—m, )(y—™m,) € 2 2 26, 205 20,0, g=Aexp — the deviations are no longer independent: 7 is the coefhcient of correlation ranging in value from —1 to +1 and measures the dependence, and most of the tables in this section are intended to give the value of r and the degree of its accuracy with a minimum of labour from a minimum of data. Particular attention is devoted to the determination of 7 from a fourfold table, as this -ease frequently occurs in the current work of medical statistics Geological Society. 495 and experimental psychology, and requires the solution of a com- plicated transcendental equation. The tables obviate this. The Tables will be of greatvalue to psychologists and others who use modern statistical formule, and have not had the opportunity to acquire the mathematics that will enable them to estimate the significance of their results, as the tables enable the probable errors (which mean troublesome mathematics) to be read off. Considering the great amount of labour in calculating the tables and the expense in printing and publishing them, the price of the book is very reasonable. It is hoped that Prof. Pearson will be able in the near future to fulfil his promise to issue a second and larger edition of this work. LVIU. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 416.] June 23rd, 1915.—Dr. A. Smith Woodward, F.R.S., President, in the Chair. sae following communications were read :— 1. ‘On a New Eurypterid from the Belgian Coal Measures.’ By Prof. Xavier Stainier. 2. *On a Fossiliferous Limestone from the North Sea.’ By Richard Bullen Newton, F.G.S. 3. ‘The Origin of the Tin-Ore Deposits of the Kinta District, Perak (Federated Malay States).’ By Wilham Richard Jones, ise, F:G.8. Certain tin-ore-bearing clays and boulder-clays occurring in the Kinta district have been described by Mr. J. B. Scrivenor, Government Geologist, F.M.S., as being of glacial origin, and the tin-ore which they contain as having been derived from ‘some mass of tin-bearing granite and rocks altered by it, distinct from and older than the Mesozoic Granite’ (that is, than the granite now im situ in the Kinta district). These stanniferous clays and boulder-clays are stated to have furnished a more valuable horizon on climatic evidence than can be afforded by limited collections of fossils in rocks far removed from Europe, and have been correlated with the Talchirs of India and mapped as Older Gondwana rocks. It would be difficult to overestimate the importance of the origin of these clays in a country where, on the one hand, they yield a very important part of the world’s output of tin-ore, and where, on the other, they have been used as the horizon on which to base the geological age of rocks which cover about a third of the surface of the Malay Peninsula. If of glacial origin, a vast tin-field remains to be discovered. 496 Intelligence and Miscellaneous Articles. The object of this paper is to show that all the tin-ore found in these clays is derived from rocks now in situ in the Kinta district; that it is not necessary to bring in glacial action to explain any of the features which led to the adoption of the theory of their glacial origin; to point out that these deposits cannot be correlated with the Talchirs of India; and to show that a simple interpretation may be given to the geology of the Kinta district. The sources of the tin-ore here are: (1) the stanniferous granite of the Main Range and of the Kledang Range; (2) other granite outcrops known to carry cassiterite; (8) the granitic intrusions in the phyllites and schists, notably near the granite- junction ; and (4) the granitic intrusions traversing the limestone, and forming an important source of ore. The angularity of the boulders and of the tin-ore in some of these clays is due (1) to weathering 7m sztw of the phyllites and schists, which then sink on the dissolving limestone underneath ; (2) to soil-creep effecting the same result; (8) to the breaking-up of the much-weathered cassiterite-bearing boulders and pebbles in the alluvium. Over 90 per cent. of the ore worked in the whole of the Kinta district is obtained from mines situated at less than a mile from granite or from granitic intrusions. LIX. Intelligence and Miscellaneous Articles. ON MUTUAL=- AND SELF-INDUCTANCEH SERIES. To the Editors of the Philosophical Magazine, GENTLEMEN, — eS my paper on ‘“ Mutual- and Self-Inductance Series” which appeared in the Philosophical Magazine for April, I stated that the formule for the mutual induction between coaxial circles had only been completely determined in the case of circles far apart. Since the publication of the paper my attention has been drawn to two papers published by Dr. T. J. Pa. Bromwich (Quarterly Journal of Pure and Applied Mathematics, No. 176, pp- 363, 381, 1913), in which a number of the formule given by me for circles close together are obtained by a method which is substantially the same as the one I used. I wish to take this opportunity of apologizing to Dr. Bromwich for trespassing on ground already covered by him, and to assure him that I was totally unacquainted with his work in this direction until a few days ago. Yours faithfully, School of Technology, S. BurrERWoRTH. Manchester. April 7th, 1916. THE — LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. aa Ba pa [SIXTH SERINSg SYS] ‘ ~w/ | oe : he y = gs | ORE NR Se a Ste aide ys Rh » ’ 4 i aaa LX. The Variation of the Positive Emission Currents from Hot Platinum with the Applied Potential Difference. By QO. W. RicHarpson, F.A.S., Wheatstone Professor of Physics, University of London, King’s College, and CHARLES SHEARD, Professor of Applied Optics, Ohio State University *. ‘igen incentive to the present investigation was an obser- vation recorded by one of the authors f to the effect that when a fresh metal wire is positively charged and heated in a vacuum, the relation between current and electromotive force, when due allowance is made for the decay of the current with time, is approximately linear, although it is known that no measurable quantity of nega- tive ions is emitted by the wire at the temperature of the experiment. Since one would expect saturation to occur with a quite small potential under these circumstances, this phenomenon seemed to call for further examination. — The experiments have heen made with two forms of testing vessel :—(1) A glass tube in which the electrodes are a loop of heated platinum wire and a platinum plate. This tube is identical with the one figured in Phil. Trans. A. vol. cevii. p- 4, fig. 1 (1906). (2) A brass cylinder with an axial platinum wire. The wires were heated electrically, the temperature being controlled and“ measured by placing the wires in one arm of a Wheatstone’s bridge, which was * Communicated by the Authors. + O. W. Richardson, Phil. Mag. vol. vi. p. 80 (1903). Phil. Mag. 8. 6. Vol. 31. No. 186. June 1916. il Oe 498 Profs.O. W. Richardson and C. Sheard : Variation of actuated by the main heating-current in the usual way. The potentials were applied at the mid-point of a high resistance which shunted the wire outside the tube; they therefore represent the average potential of the hot wire. We aimed to work at as low temperature and with as small currents as possible so as to eliminate the time-decay of the positive emission. The currents were measured with a deli- cate electrometer. The platinum wire in the glass tube was Johnson & Matthey’s purest resistance wire; that used in the other experiments was ordinary commercial platinum and had a very low temperature coefficient of resistance (00165 per° C.). The pressure recorded on the Mcleod gauge was between the limits 0:00005 and 0-0002 in all the experiments. In the first observations with the glass tube and platinum plate-electrode the temperature was adjusted to451°5+1%5 C. Successive curves showing the relation between the current and applied voltage at different times are exhibited in fig. 1. The first current-H.M.F’, curve was not completed until about 24 hours had elapsed from the commencement of the heating, as some time was occupied in preliminary adjustments. It is marked A in the figure. The curve is slightly convex to the voltage axis, but is almost linear, in agreement with the one referred to above published by one of the writers. The approximately linear relation did not hold down to V=0, but the curves became concave to the voltage axis at low voltages as shown in curve B, which is on a different scale. All the potentials in fig. 1 are about 4 volts too high on account of the measured potential not being that at the middle of the hot wire. As the heating of the wire was continued the first effect was to rotate the curve A about O towards the potential axis. ‘This corresponds to a decay of the currents in approximately equal proportions at all potentials, With further heating the curves developed a flat region in the middle, as is shown by curve ©, which was obtained after 11 hours heating. With subsequent heating the increased current above 280 volts continues to decrease faster than the currents at lower voltages, until after about 30 hours it is scarcely noticeable. The result of heating for 274 hours is shown in curve D, The observations both with rising and with falling potentials are shown in the figure. The difference between them is not very great in most instances, but when there is an appre- ciable difference the currents with rising potentials, which were measured first, are larger (see below). It is important to remember that in these experiments the temperatures were so low that several hours were necessary for the emission to Positive Hmission Currents from Hot Platinum. 499 decay a few percent. After the stage showing saturation between 100 and 400 volts had appeared, this form of Fig. 1, 1) 10 20 30 40 : ae 1°-§5 x/O é amp _ — - Current (J O 80 160 240 320 400 Potential (Volts) -current-H.M.F. curve apparently persisted indefinitely. The only change observed subsequently, on heating at constant ‘temperature, consisted in a uniform decay of the total current at all voltages. 2L2 500 Profs. O. W. Richardson and O. Sheard: Variation of Before these experiments were made, the tube, which con- tained nothing but glass and platinum, was carefully boiled out with pure nitric acid and distilled water. The tube was dried in a current of warm air before connecting to the Gaede pump and pentoxide bulb. It was exhausted at a pressure of less than 0°0001 mm. on the Mcleod gauge before the wire was heated. The diameter of the wire was 0°15 mm. and its length 5 em. The results described above were obtained when the wire was continuously maintained at the temperature at which the emission was measured. After heating the wire, which was positively charged, to a relatively high temperature (718° C.), and subsequently observing at the original temperature (452° C.), the resulting current-H.M.F. curves were very peculiar. They were characterized by a very sharp maximum at about 240 volts, which was more marked with descending than with rising potentials. It seemed to us that these peculiarities might be due to the glass insulation becoming charged to a high potential under the influence of the copious emission of positive ions which had occurred at the higher temperature. The brass-tube apparatus shown in fig. 2 was therefore constructed in order to eliminate this possibility. Fig. 2. HIGH To HIGH POTENTIAL POTENTIAL ANDBRIDGE To|ELECTROMETER. ANDBRIDGE Seon EARTH The platinum wire W was stretched along the axis of the brass tube T which, suitably supported and properly shielded, formed the receiving electrode. The wire W was welded to two stout platinum end-pieces, which in turn were soldered respectively to the brass rod R’ and the tube R. This was done to prevent the solder from coming into contact with the hot part of the platinum and causing it to rot. Rand R’ were supported by glass tubes surrounded by earthed brass tubes, E and EH’, provided with flanges, F and F’. EF eand EH’ were supported by rubber stoppers, A and A’. The only insulation in this arrangement which the ions Positive Emission Currents from Hot Platinum. 501 could have access to consists of the rubber stoppers A and A’, and the glass tubes, and all of this is shielded from the ions by metal screens. There is thus no possibility of trouble from the insulation charging up. The joints were cemented with sealing-wax and soft wax. There was no ditfiiculty in Maintaining a. vacuum of 0:0002 mm. or less, as indicated by the Mcleod gauge reading. Broadly speaking, the time-changes with the new ap- fparatus were similar to those observed with the glass tube. A selection from a series of observations is exhibited in fig. 3. The temperature in the different experiments was Fig, 3. W =0-400 VOLTS. = 49036 VOLTS, CURRENT (= 21x 1072 AMP): 2) 40 80 120 160 200 240 280 320 360 400 PoTENTIAL (VOLTS) equal to 379+1° C. The time. which elapsed, after the heating commenced, up to the beginning of each curve was approximately as follows :—Curve 1, 0; curve 2, 14 hours ; curve 3, 2 hours; curve 4, 7 hours; curve 5, 30 hours; curve 6,48 hours. The initial curves are more curved to ‘the current axis than those observed with the glass tube under similar conditions. In curve 1 the points shown thus + were taken with rising, those shown thus ® with falling potentials. When the potential is raised the current is exceptionally large at first and vice versa. The final approximately steady values of the current would lie about half way between the sets of points, as is shown by the curve. The current-E.M.F. curves for low potentials 502 Profs. O. W. Richardson and C. Sheard: Variation of (Nos. 2 and 3) are peculiar, although similar observations have been recorded before by one of the writers*. The current is greater in the neighbourhood of 2—4 volts than it is at somewhat higher potentials, indicating that one of the effects of increasing the applied potential is to diminish the number of systems giving rise to the emission. Curves showing this maximum current at a low potential were obtained after the wire had been heated at 379° C. for 50 hours, and then at a temperature of 600° C. for 2 hours just before the observations were made. On the other hand, some curves taken at intermediate intervals showed a con- tinuous increase of current with increasing potential difference. This shows that the maximum is not deter- mined solely by some peculiarity in the initial condition of the wire. We have not discovered what the determining factor is. Fig. 3 shows that the proportional rate of decay of the current at 400 volts is greater than that at moderate potentials, for example, 200 voits, although this effect is much less marked than in fig. 1. Thus, comparing curves L and 6, the ratio of the ordinates at 400 volts is about 5 to 1, whereas at 280 volts it is only a little over 3to1. After continued heating the results show a tendency to become more irregular, but the main features exhibited in curve 6 were obtained too frequently to be likely to be due to obser- vational errors. These features are (1) a steady rise to a flat region between 80 and 160 volts, (2) a further rise from 160 to 280 volts, followed by (3) a comparatively flat region. between 320 and 400 volts. This type of curve seems to point to the existence either of two different types of ions or of two different processes involved in the emission. These particular indications of duplicity were not noticeable in the first apparatus. A later stage obtained after further heating is shown in fig. 4, curve 2. We shall now consider a number of ways in which a tube giving curves like those shown in fig. 3, curve 6, may be - made to exhibit curves like fig. 3, curve 1, again. This is found to happen if air at atmospheric pressure is admitted into the tube, and then pumped out again before the obser- vations are taken, and also if the wire is heated for a time at a somewhat higher temperature than that at which the previous and subsequent tests are carried out. The general nature of these effects is the same in all cases, so that it will be sufficient to consider one instance in detail. The wire in the brass tube after continued heating gave the curve marked. * O. W. Richardson, Phil. Trans. A. vol. cevil. pp. 10-11 (1906). Positive Emission Currents from Hot Platinum. 503 curve 2 in fig. 4 at 344° C. The heating current was then eut off, and air at atmospheric pressure allowed to remain in the tube for 22 hours. It was then pumped out to a pressure of 0:0002 mm. on the McLeod gauge and the wire again heated Fig. 4. n 2.1% 10°" AMP) W is) CURRENT q (@) 40 80 120 160 200 240 280 320 360 400 PoTENTIAL VoLTs) 1 at 342° UC. It then gave curve 1, fig.4. Our experiments show that this freshening action produced by admitting air diminished with successive repetitions of the process, but we did not succeed in reaching a stage at which the air ceased to produce an effect. The effect occurred both when the inner surface of the brass tube was left untouched and when it was thoroughly scraped with emery-paper. The increased sensitiveness could not be removed by mere pumping. In one case, the tube, after exposure to air, was pumped out for 15 hours at a pressure below 0°0002 mm. and still found to give a curve practically identical with fig. 4, curve 1. Similar effects were observed when the wire was raised for a time to a higher temperature than that at which the obser- vations of the positive current were made. Thus, after taking curve 6, fig. 3, the hot wire, under zero potential, was kept at 600° C. for 2 hours. The temperature was then reduced to the previous value (378° C.) and another curve taken. The new curve was found to be similar to curve 1 of fig. 3. The current at 400 volts had been increased in the ratio of about 4 to 1 by this treatment. Results of the same character were obtained with the glass-tube apparatus. The changes were more pronounced when the hot wire was negatively charged and discharging electrons to the cold electrode during the temporary heating. In a particular 504 Positive Emission Currents from Hot Platinum. instance, the current, after continued heating with the wire positively charged at 342° C., gave an almost constant current which only increased from 0-6 div. per sec. at +120 volts to 0°75 div. per sec. at +400 volts. The wire was then heated for 30 minutes at 595° C. and —400 volts. The negative emission was measurable at this temperature. On returning to 342° C. and +400 volts, the initial current was found to be equal to 14 divisions per sec., or about 20 times the previous value. This large current fell away fairly rapidly with time at the lower temperature (342° C.), as is shown in the following numbers :— Current in divs. persec. 14 10 8 S1 34 26 18 1:25 1:05 1:00 Time inminutes’ ....:/... 910" 5 10 925° 35 45 "G0" “7a ead The negative currents which were used to stimulate the tube in these experiments were quite small. They varied a little, but were always of the order of 10°*% amp. ‘The relation between current and potential for the positive leak in the stimulated tube appeared to depend a good deal on its previous history. The curves obtained at first showed an almost linear relation between current and potential from 40 to 400 volts, after allowing for the time factor. With successive stimulations the curves became more concave to the potential axis, until finally they were almost saturated all the way from +40 to +400 volts. It appears therefore. that, with successive treatments, the tube gets into such a state that the passage of a negative discharge from the wire at a higher temperature produces an increase in the posi- tive emission, subsequently measured at the original lower temperature, which does not increase with the applied potential at high potentials. This increase is still of the order of magnitude, even at +400 volts, of that which is observed in the earlier stages and dies away with time ata rate which is of the same order of magnitude. The progressive increase of the currents, with increasing potential difference, which has been recorded in the various cases tested, might be due to a variety of causes. Recom- bination, which is the most important factor preventing the attainment of saturation in the usual cases of ionization of gases at moderate pressures, does not occur here, as there is no current, at the temperatures at which the positive emis- sion was measured, when the wire is charged negatively. Ionization of the surrounding gas by collisions would not be expected to be a serious factor on account of the smallness of the pressures. We are thus limited either to an effect of Infra-red and Ultra-violet Absorption of Sulphur Diovide. 505 the electric field on the emission at the hot wire, or to an effect which it may produce indirectly by the impact of the ions on the negative electrode. Some experiments which we made led us to conclude that the second of these alterna- tives was the correct one *. In these experiments it appeared that the large currents at high potentials couid be cut down by the application of rather small transverse magnetic fields, indicating that the carriers of part of the discharge were negative electrons liberated at the cold electrode. However, further experiments showed that this influence of a magnetic field was not always to be relied on ; so that the results of this test cannot be regarded as decisive. At this stage we were compelled to discontinue the experiments, which were made in the Palmer Laboratory at Princeton University. At the suggestion of one of us the further investigation of these phenomena was undertaken by Mr. H. H. Lester, whose results will shortly be published in this journal. LXI. The Infra-red and Ultra-violet Absorption of Sulphur Lnoxide and their Relation to the Infra-red Spectra of Oxygen and Hydrogen Sulphide. By C. Scorr GARRETT, BWSe.t a a recent paper on the absorption of sulphites and sulphur dioxide, the equilibrium conditions existing in the aqueous solutions of these substances were investigated, and it was intimated that further data on the absorption of sulphur dioxide would be published separately t. The present paper deals with these data and contains a detailed study of the gaseous absorption of sulphur dioxide, which was undertaken more especially to test the applicability of Baly’s theory of the connexion between ultra-violet and infra-red absorption §. This theory, which is based on the energy quantum theory and Bjerrum’s work, has found considerable experimental support in the absorption spectra of benzene and certain other organic compounds, and it was thought probabie that it could be put to a more extended test by an investigation of the absorption system of sulphur * Phys. Rev. vol. xxxiv. p. 391 (1912). + Communicated by Prof. E. C. C. Baly, F.R.S. t+ Trans. Chem. Soc. evii. p. 1335 (1915). § Phil. Mag. xxvii. p. 632 (1914); xxix. p. 223 (1915); and xxx. p. 510 (1915). 506 Infra-red and Ultra-violet Absorption of Sulphur Diowide.. dioxide. The apparatus employed consisted of a quartz- ended glass cell of special construction connected by glass tubing to a reservoir containing the dry gas and to a mercury manometer. As the light source, the condensed spark between nickel electrodes was used and the absorptive. power was determined by photography with the help of a quartz spectrograph and a Hilger rotating sector photo-. meter. As is well known, the broad absorption band exhibited by sulphur dioxide in the liquid and solution phases is resolved. into a number of small band groups which, at any rate in the less refrangible ultra-violet region, are symmetrically distributed around a central band group of maximum absorption. By varying the pressure and the thickness of the absorption layer of the gas, the temperature being kept constant, I have been able to trace out the whole of the- complex absorption and to determine the molecular extinction coefficient for each component band group. In the first place, it may be noted that series of constant. frequency differences occur between the centres of the band groups, and, although with the relatively small dispersive: power of the spectrograph it was found to bea matter of some difficulty exactly to determine the centre of any one band group, yet the averaging of these frequency differences has. enabled me to arrive at their real values with considerable- accuracy. In the second place, the whole region of absorption can be divided into three portions, namely, a less refrangible portion in which the average difference between the wave-. numbers of the centres of the consecutive bands is 22°4, an extreme ultra-violet region where the average wave- number difference is about 35, and an intermediate region where combinations of these differences occur together with a third difference of about 12%. In the following tables are given the wave-lengths and wave-numbers of the centres of the small band groups together with the differences. In Table III. the numbers. (1), (2), (3) after the differences refer to the values 22-4, 35, and 12 respectively. The molecular extinction coeffi- cients are given at the more important points. * For the sake of convenience wave-numbers (reciprocals of wave- lengths) are used, expressed in four figures; that is to say, the number of waves in 1 mm. Tasie I.—Absorption of Sulphur Dioxide in the Extreme: Ultra-violet Region. Wave-lengths in Wave Wave-number Mralecntar a gstréms. Numbers. Differences. Extinctions. 2334 4303 39 2307 433A 39 (1) 2295 4357 34 2277 4391 36 2259 4427 36 2241 44638 3] 2225 4494 39 180°3 2206 4533 34 551-1 2189 4567 36 2172 4603 38 2155 4641 36 2140 4677 34 2123 4711 36 2107 4747 Mean Wave-number Difference = 35:1 (2). TaBiLE I].—Absorption of Sulphur Dioxide in the Near Ultra-violet Region. Wave-lengths in Wave Wave-number Molecular Anes Beis. Numbers. Differences. Hxtinctions. 3182 3143 24 40°8 3157 3167 99 52:1 3136 3189 04 104:7 3112 3213 93 180°3 3090 3236 25 2268 3066 3261 26 3150 3042 3287 22 3546 3022 3309 26 354°6 2999 3035 93 402°2 2978 3358 20 402-2 2961 8378 20 (Optical | Centre) 402°2 2943 8398 99 402°2 2924 3420 i 402-2 2906 244] O4 354°6 2886 3465 20 354°6 2869 3485 21 ; 3546 2852 3506 23 3546 2833 3529 90 3546 2818 3549 20 3546 2802 3569 3150 Mean Wave-number Difference = 22°4 (1). 008 Mr. C. Scott Garrett on the Infra-red and TasLE I11.—Absorption of Sulphur Dioxide in the © Intermediate Region. Wave-lengths in Wave Wave-number | Angstréms. Numbers. Differences. | 2789 3586 11 (3) | 2780 3597 91 (1) | 2764 3618 a | 2757 3627 30 (2) : 2734 3657 20 (1) oe Te 48 (2)+(8) 2685 3725 ey 9 2669 3746 12.) 2661 3758 14.(1) sri ae 43 (2)x(2) 2621 3815 foes 2614 3825 5010) 2594 8854 19 (1) 2582 3873 ue 2572 3889 Sales ae : fon 27 (1) oe Hi 64 (+) ag aan 52 (2)+(8) 2464 4058 an 2456 | 4071 ee 2427 4121 Mean of all differences (1) = 22°4. 99 by) 99 (2) = 344, ” » » (8) = 126. It would seem obvious from the above tables that the whole absorption region in reality consists of two principal band groups, the centre of one of which lies near A\=2960, while the centre of the other lies beyond 71=2107 which is the working limit of the spectrograph. In the first of these bands there is a constant difference between the wave- numbers of the centres of the sub-groups of 22°4 and in the second band there is a mean constant difference of 34:4. As regards the less refrangible band, although in Table II. five band groups have been given with a molecular extine- tion coefficient of 402°2, the trace of the absorption curve Ultra-violet Absorption of Sulphur Diowde. 509 shows clearly that the true optical centre lies at A=2961 or 1/A=3378. According to Baly’s theory this number, 3378, should be an even multiple of a fundamental band in the short-wave infra-red region. The absorption of sulphur dioxide between 3y and 11y has been observed by Coblentz, and the most pronounced band lies at \=7°4y or 1/A=135-1.. Now 135:1 x 25=3377:5, which is exceedingly near to the observed ultra-violet wave-number 3378, which entirely supports Baly’s contention. Moreover, Baly further postu- lates that the constant differences in the wave-numbers' in. the ultra-violet are compounded from certain basis constants which are characteristic of the molecule, and that these basis. constants also determine the infra-red absorption of the molecule. Now, the wayve-numbers of the six infra-red absorption bands of sulphur dioxide observed by Coblentz* can _ be- expressed as multiples of 96°32, as shown in Table IV. TaBLE LV. Infra-red Absorption of Sulphur Dioxide. eee Wave Wave-lengths Wave-lengths| Differences amore | ‘Numbers. Calc. Obs. Obs. —Cale. 9°632 10 96°32 10°38 u | 10°37 p —0°01 9°632 x 12 115°6 8:65 87 +0:05 9632 x 14 1848 7°42 74 —0-02 9°632 x18 173°3 577 5°68 — 0:09 9°632 x 26 250°4 3°99 397 —0:02 9°632 x 32 308°0 3°23 318 —0:05 We therefore have three numbers which appear to be- characteristic of the absorption of sulphur dioxide, namely, 22°4, 34:4, and 9°632, and if Baly’s contention is valid these must be derived from certain basis constants charac-- teristic of the oxygen and sulphur atoms. Now 22°4="2 x 4x 2°38 x 10 34°4=°2x 4x43 x 10 9°6382='2x4x 28x43; and therefore it would seem that there are four basis. * Publications of the Carnegie Institution, Washington, 1905, No. 35,. p- 02. -510 Mr. C. Scott Garrett on the Infra-red and -constants, 0°2, 2°8, 4, 4°3, of which the product of the first three multiplied by 10 gives a wave-number difference -occutring in one ultra-violet band in sulphur dioxide, the product of another three multiplied by 10 gives a wave- ‘umber difference in the second ultra-violet band, while the product of all four determines the infra-red spectrum. Finally, the centre of the first ultra-violet band of sulphur dioxide is a multiple of the product of all four constants. ‘On these grounds Baly’s theory would lead to the expec- tation that the four basis constants, 0°2, 2°8, 4, 4°3, must yhave their origin in the oxygen and sulphur atoms. That this expectation is justified is shown by the fact that 4°3 is characteristic of the sulphur atom, since the whole of the vinfra-red spectrum of hydrogen sulphide can be expressed rin terms of this constant, and further by the fact that the infra-red bands of oxygen can be expressed in terms of the product of 4 and 2°8, namely 11°2, as is set forth in ‘Tables V. and VI. TaBLe V. Infra-red Absorption of Hydrogen Sulphide. | | atin Wave Wave-lengths Wave-lengths| Differences eae Numbers. Cale. Obs.* Obs. — Cale. 4°3X21 90°3 11:08 10°95 —013 4°3 x 22 94°6 10°57 10°6 +0°08 4°3 X23 98:9 10°11 10:08 —0:03 4°3 X25 107°5 9°30 9°65 +0°35 43X27 1161 861 . 846 —015 4:3 X30 129-0 77d (rls) +0:03 43x33 141-9 705 | 7:12 +007 43X41 1763 BGT ae) oe —0:07 43X55 236°5 4°23 4°24 +0°01 TasLe VI. Infra-red Absorption as Oxygen. | raat Wave Wave-lengths macwans Differences aati Numbers. Cale. = eObs.* Obs. — Cale. SOLE PS ae —| 2 11219 212°8 4°7 | 47h 0-00 11:2x 28 3136 3°19 | 32 +0°01 * Coblentz, Joc. cit. Ultra-violet Absorption of Sulphur Dioxide. 511 The differences between the calculated and observed values in Tables LV., V., and VI. show how well the theory con- forms to experiments. It should, moreover, be remembered that the order of accuracy in infra-red measurements is smaller than that in the ultra-violet regions. The whole gaseous absorption of sulphur dioxide can, therefore, be analysed as follows :— Basis Constants: 0:2, 2°8 (oxygen), 4 (oxygen), 4°3 (sulphur). Fundamental Frequency: (F) 0.2x2:8x4x43=9°6382. Infra-red Spectrum: (F) x10, 12, 14, 18, 26, and 32. Centre of First Ultra-violet Band: (F) x 14x25. Constant Wave-number Difference in First Ultra-violet Band: 02x 28x 4x10. Constant Wave-number Difference in Second Ultra-violet Band : 0:2x4x43x10. I hope in later work to pursue this search for the basis constants characteristic of the elements. Reference may be made to Miss Lowater’s analysis of the completely resolved ultra-violet absorption spectra of sulphur dioxide, in which she determined the wave-lengths of all the absorption lines composing the sub-groups of the whole band groups*. Altogether 590 lines were measured by her, and these were arranged in 44 arbitrary series with constant wave-number differences. The mean of all these wave- number differences was 22°3, which is practically identical with that now found between the wave-numbers of the centres of the sub-groups over the region dealt with by Miss Lowater. Owing, however, to the great dispersion used, she did not recognize the composite nature of the spectrum dealt with above. Moreover, the intensity of her lines was estimated by inspection on the conventional basis of 10 for a very dark (absorbed) line. By such a method it is impossible to estimate the persistency of the lines, and hence the real optical centre of the resolved complex band system of the gas remained undiscovered. Finally, as the wave-number difference 34:4 characteristic of the extreme ultra-violet region of absorption of sulphur dioxide is derived from the specific sulphur bases, it may be concluded that it is the sulphur atom of sulphur dioxide which is mainly responsible for the production of this second ultra-violet band group, a portion of which only has been observed on account of the limited transparency of quartz. By similar reasoning the oxygen atoms in the molecule must be the chief determining cause of the typical band system characteristic of the first absorption band group in the near ultra-violet region. ; * Astrophys. Journal, xxxi. p,.311 (1910). [ples 4 LXII. The Ultra-violet Absorption System of Suleeee Di- omde. By H.C. C. Baty, MSe., F.R.S., Grant Professor of the University of Liverpool, and C. 8. Garrutt, B.Sc. * lle was shown in the preceding paper that the ultra-violet -absorption band group of gaseous sulphur dioxide ean be resolved into a series of sub-groups and that constant differences exist between the wave-numbers of the centres or heads of the sub-groups. The whole region over which absorption is exhibited can be divided into three portions, namely, that between \=3182 and X=2802 where there is a constant difference of 22-4 between the wave-numbers of twenty successive sub-groups, that between A= 2324 and X=2107 where there is a constant difference of about 34:4 between the wave-numbers of fourteen sub-groups, and the intermediate portion with twenty-one sub-groups where both constant differences occur together with a third difference of about 12. It was further shown that these differences are compounded from four basis constants, 0°2, 2°8, 4, and 4:3. Further, the wave-numbers of the infra-red absorption bands of sulphur dioxide can be expressed as multiples of 96°32, whieh is the product of all four basis constants, while the product of 2°8 and 4 gives rise to the infra-red absorption of oxygen and 4:3 gives rise to the infra-red absorption of hydrogen sulphide. Again, the wave-number of the centre of the less refran- gible ultra-violet band of sulphur dioxide was shown to be 25-times that of the most important infra-red band of this gas, namely, 96°32 x 14. Mention was also made of the accurate measurements by Miss Lowater of the component absorption lines which are associated together in the sub-groups ft. Miss Lowater only investigated the region between A=3118 and A=2707; that is to say, she only measured the component lines of the less refrangible ultra-violet absorption band of sulphur dioxide. She arranged the lines in 44 arbitrary series of constant wave-number differences, and the mean of all 44 differences was 22°33. Although 42 lines were found not to fit in any of her series, it is obvious that the same constant differences occur between the individual absorption lines as was shown in the preceding paper to exist between the heads of the sub-groups. * Communicated by the Authors. + Astrophys. Journ. xxxi. p. 31] (2910). Ulira-violet Absorption System of Sulphur Diowde. 5138 Owing to the great dispersion used, Miss Lowater was able to measure the wave-lengths of the component absorp- tion lines with great accuracy, and it should at once be possible to arrive at far more accurate values of the basis constants from her numbers now that the central wave- number of the less refrangible band is known. From a knowledge of these it should also be possible to calculate the wave-number of every single component Jine in the absorption band group, and thus put the theory brought forward by one of us in previous papers* to a far more rigid test than has yet been possible with the compounds already studied. Owing to the very great accuracy of Miss Lowater’s measurements the wave-numbers in the following pages will be expressed in five figures, and, moreover, note must be made of the fact that her values were obtained in air and were not reduced to vacuum. This correction is necessary before any calculations from them can be made. It was not possible to apply the general theory to sulphur dioxide until an accurate determination of the central wave-number of the ultra-violet band group had been made. As stated in the preceding paper, it has now been found that this central line lies at X=296L.or 1/N=33780, but it must be remembered that the accuracy of this measurement is far below that obtained by Miss Lowater. There are three lines of maximum intensity observed by her near this position, any one of which might be the true central line. Their waye-numbers in air are 33770°3, 33767°2, 33761°2, and the question arises as to which one of these should be selected. ‘The whole essence of the Bjerrum conception lies in the fact that the component absorption lines in any one band group lie symmetrically distributed about the central line, and it is a matter of simple calculation to prove that the greatest symmetry is obtained when the wave-number 33761°2 is taken as centre. This number when reduced to vacuum becomes 337516, and this may be taken as the true value for the centre of the less retrangible ultra-violet absorption band of sulphur dioxide. The relationships found by Garrett may now be dealt with, since the central wave-number is known and the basis constants caleulated. The wave-number of the most impor- tant absorption band in the short wave infra-red region of sulphur dioxide is given by 33751°6/25 = 1350-06, corre- sponding to a wave-length of 7-407 4. Again, the least * Phil. Mag. xxvii. p. 632 (1914); xxix. p. 223, xxx. p. 510 (1915). Phil. Mag. 8. 6. Vol. 31. No. 186. June 1916. 2M 514 = Prof. Baly and Mv. Garrett on the Ultra-violet common multiple of the basis constants is given by 1350°06/14 = 96°433, and, further, the wave-numbers of the infra-red absorption bands of sulphur dioxide are multiples of this least common multiple. This is shown im hab ven: TABLE I. Infra-red Absorption Spectrum of Sulphur Dioxide. Factors. WING A Cale. d Obs. Difference. 96-433 x 10 964°33 10°37 ps 10°37 pw 0:00 96°433 x 12 1157-20 8:64 87 +0:06 96°433 x 14 1350-06 7°41 T4 —001 96433 x18 1734-79 5°76 5°68 — 0-08 96°433 x 26 2507°26 3°99 397 —0:02 96°433 x 35 3182°29 314 318 +0:04 As regards the basis constants, Garrett has pointed out that these are approximately 2°8, 4:3, 4, and 0°2, and that the first and third are due to the oxygen atom, while the second is due to the sulphur atom as shown by the infra-red absorption wave-numbers of oxygen and hydrogen sulphide. He showed that the wave-numbers of the intra-red absorption bands of hydrogen sulphide may be expressed as multiples of 4°33x10. As a matter of fact, they are still better: expressed as multiples of 4:32 x10. Now, two of the most pronounced infra-red bands are at X=8°46 w and 4°24 pw, and their wave-numbers, 1182 and 2359, are very nearly equal, to 43:2 x 27 and 43:2 x 55 respectively. On the other hand, these two bands are clearly harmonic, and this suggests at once that their wave-numbers are due to the convergence. frequency of two bases. The most probable value of the second basis constant therefore is 2°73, and the wave- numbers as calculated are given in Table II. Two of the basis constants are now known accurately, and the third, which is the product of Garrett’s remaining two. constants, may be found at once, since 96°433 is the con- vergence frequency of them all. The third, therefore, is. found to be 96°433/(4°32 x 2°73)=8:177. Finally, the two. — infra-red absorption bands of oxygen can be expressed in, terms of the fundamental 96°433, as shown in Table III. Absorption System of Sulphur Dioxide. 515 TaBLe IT. | Infra-red Absorption Spectrum of Hydrogen Sulphide. | Factors. 1/n. d Cale. d Obs.* Difference. 43221 907-2 11:02 p 10:95 p 0:07 43:2 x 22 950-4 | 10°52 10°6 +0:08 43-2 x 23 9936 10:06 10-08 +0-02 | 43-2 x24 1036'8 9:65 9:65 0:00 432x273 | 11794 | 848 8:46 —0-02 432x380 | 1296-0 7-72 7-78 +0-06 432x383 | 14256 7:02 712 +0-10 Boat the A75TR «| SOB 56 —0-05 432x546 | 23588 | 4:24 4-24 0-00 * Coblentz, loc. cit. Tapen TLL, Infra-red Absorption Spectrum of Oxygen. Factors. | 1/X. |} ACale. | A Obs. | Differences. 96-433x22 212153 471g 47 Oil bel : 96°433 x33 ; 3186229 | 314 32 +0:06 We have now, therefore, three basis constants, namely, 2°73, 4°32, 8:177, and from these three, combinations may be made as follows :—8°177 X 2°73 =22°3225, 8:177 x 4°32 Jo.024, and 2-713 4°32—=11-794. . Of these, 2273225 «10 is the value of the constant difference in the less refrangible ultra-violet absorption band of sulphur dioxide, 35°324 x 10 is the constant difference in the more refrangible band, while 11°794 x 10 is that found in the region intermediate between the two bands. It is now possible to calculate the wave-numbers of all the lines in the less refrangible ultra-violet band of sulphur dioxide. The true wave-number of the central line is known, and also the whole band group is known to he sub-divided into a series of sub-groups with a constant difference of 223°225 between the wave-numbers of their central lines. 2M 2 516 ~— Prof. Baly and Mr. Garrett on the Ultra-violet The general structure of the whole ultra-violet band group will be as follows :—From the general theory as laid down in the previous papers the wave-numbers of the principal lines in the band—that is to say, the heads of the sub- groups—will be given by 33°751-6+nK, where K is one of the fundamental bands of the infra-red, andn=0, 1, 2,3.... It has already been found, as shown above, that K is the least common multiple of two of the basis constants, 2°73 and 8177, and has the value 223:225. The wave-numbers of the heads of the sub-groups, therefore, will be given by 33751°6 +n x 223°225. Again, the fundamental infra-red band with the wave-number of 223:225 according to Bjerrum is itself complex and consists of a group of lines the wave-numbers of which are given by 223°225+v,, where v, stands for the two basis constants 2°73 and 8177. Since, however, 8°-177=3 x 2°73 almost exactly, the structure of this fundamental band will be given by 223°225+n x 2°73, where n=0, 1, 2,3.... The upper limit of n is defined by the overlapping of the band with the next infra-red band, and it is not possible to state with any exactness how far the two consecutive bands overlap. From what follows it will be seen that on the long-wave side the upper limit of n is about 44, while on the short-wave side the limit is about 37. In the fundamental infra-red band, therefore, there must be 82 lines, namely, the central line together with 44 lines on one side and 37 on the other. The lines in the ultra-violet band are due to the com- bination of the central wave-number with the infra-red vibrations, and therefore the component lines of the central ultra-violet sub-group will be expressed by 33751°6 +n x 2°73. The other sub-groups are due to the combination of the wave-numbers of the lines in the fundamental infra-red band and their integral multiples with the central wave-number 33751°6. Thus the first sub-group on each side of the central sub-group wili be given by 33751°6 + 223°225 +n x 2°73, and the second sub-group by 33751°6+2 x 223°225 +n x 2°73, and so on. There appear to be seven sub-groups on the less refran- gible side of the central sub-group and thirteen on the more refrangible side. When each individual sub-group is considered and the wave-numbers of its component lines calculated, it appears, on comparison with Miss Lowater’s list of lines, that the group is slightly asymmetric, for there are 44 lines on the less refrangible side of the centre and 37 on the more refrangible side. | Absorption System of Sulphur Dioxide. 517 The general formula, therefore, expressing the whole system of lines in the whole ultra-violet band group is 33791°6 + p X 223°225+7n x 2:73, where p=—7, —6,....0 -.e. $12, +13, and n=—44, —43,....0.... +36, +37. The whole band group, therefore, consists of 21 sub- ‘groups, each containing 82 absorption lines. It is obvious from this arrangement that there must exist in the band group 82 series, each containing 21 lines with a constant difference of 223°225 between their wave-numbers, and we thus arrive at the true physical explanation of Miss Lowater’s discovery that the lines can be arranged in series of constant wave-number differences. Miss Lowater, however, arranged her observed lines in 44 such series, but it must be remem- bered that there were 42 lines which she was unable to include in her series. Moreover, also the values of the mean wave-number differences she found seem to vary far more than her experimental error would permit. For example, in her 42nd series the first and nineteenth members have wave-numbers 32202'1 and 36231°6 respectively, and these give a mean wave-number difference of 223°8. The mean wave-number difference calculated from all 44 series is about 223°3, which would point to an experimental error of about 4°5 in the wave-numbers of each of the above two lines, an error which is very much greater than is possible from the accuracy of her work. In general it would appear that, although Miss Lowater’s discovery of series of lines with constant wave-number differences is perfectly sound, yet her arrangement of the lines in these series is not the best possible. We have calculated the wave-numbers of all the lines which the above formula establishes to be present in the whole group, and they amount to 1722. These we have arranged in the 82 series of constant difference, and com- pared them with Miss Lowater’s observed values after reducing the latter to vacuum. The agreement between the observed and calculated values is exceedingly good, but it cannot be claimed that this agreement by itself is an absolute proof of the validity of the formula. The difference in the wave-numbers of any two consecutive lines in the spectrum is 2°73, and this is not much more than twice Miss Lowater’s maximum experimental error, and therefore the difference between any observed line and the nearest caiculated line cannot be much greater than the maximum experimental error. On the other hand, the agreement between the observed and calculated values is on the 918 ~=Prof. Baly and Mr. Garrett on the Ultra-violet average very much better than this, and would seem to give great support to the formula. The most important . result obtained, however, is the arrangement of the observed lines into 82 series of about 21 members each, and, although no such series is complete, yet the mean wave-number differences calculated from these series are far more com- parable amongst themselves than those found by Miss Lowater. There is no need to publish the whole of the calculations, but in Table IV. are given the number of the observed lines in each of our series and the mean values of the wave-number differences that we have found, together with those of Miss Lowater’s series. Our series are numbered from the central line of the sub-group in each case. Only one line has been observed which belongs to the 23rd series on the blue side, and therefore no wave-number difference can be found from this series. The superiority of the new arrangement of the lines is manifest from Table IV. Not only are the extreme values of the differences, 223°45 and 222:94, much closer together than those of Miss Lowater’s table, namely 223°9 and 222-8, but 64 per cent. of the new mean differences lie within 0:1 of the mean value, while, according to Miss Lowater’s arrangement, only 34 per cent. lie within the same limits. Then, again, the means calculated from the red and blue series are the same, 223°23, and equal that calculated from the infra-red basis constants. Further, the interval between any two members of any of the new series is never greater nor less than the calculated interval by an amount greater than can be accounted for by the experi- mental error which Miss Lowater gives in her paper. As ‘was shown above, Miss Lowater’s arrangement of the lines can be criticised from this point of view. Hvery single observed line finds a place in our series between the limits 1/A=32070 and 1/A=36646. Six lines outside these limits have not been included, two of smaller frequency (1/A=32065 and 32061) and four of larger frequency (1/A=36633°3, 36712°5, 36885:1, and 36922). The former two lines no doubt belong to the 8th sub-group on the red side noted by Garrett, while the four latter lines lie in the intermediate region between the two main absorption band groups, and therefore we have not included them since it has been shown that in this region the fundamental interval varies. Some reference may be made to the fact that only 586 lines have been observed out of a total calculated number of 1722. This cannot: be considered as an argument against the theory, for there can be no doubt that by a still further Absorption System of Sulphur Dioxide. og TABLE IV.—Series of Constant Difference in the Absorption Band Group of SOs. Rep Serizs. Brive SERIEs. Miss LowateEr’s SERIEs. Medes Nee tse (EB! gree 8 eet Ve cea 8) 52.) See) ee ga 8 ~ n Se = mo n ==) S ~ nm ea SS ives (tee Sta FEE) | a eee Ess ere aie! .S. Sy abe | Sees hee So @ Sees GS Gano eS Gl area, seve cS A = a le = Aw = Central Central Series. 11 223:23| Series. 11 223°23 } 7 223°18 1 4. 223°25 t 8 223°1 2 8 223°3: 2 13 23°24 2 14 223°3 3 8 22328 3 3 223°27 3 13 223°4 4 9 223°23 + 5 223-10 4 9 223°5 5 10 223°15 5 9 223°22 5) 9 223°9 6 8 223 27 6 7 223°28 6 13 223.2 7 7 223°16 7 10 223-28 7 10 223°1 8 al 223°25 8 6 223°25 8 10 223°1 9 6 ad | 9 12 223°24 9 10 222-9 10 10 23°21 10 6 223°26 10 8 223°4 11 8 22322 11 9 223°21 11 12 223°4 12 8 22305 12 9 223-30 12 8 223°] 13 1 223:19 13 6 223°37 13 12 223°0 14 a 223°26 14 8 223°36 14 13 222-9 15 10 223°21 15 +: 293714 15 15 23:0 16 5 223°19 16 9 223°28 16 12 23°4 sii 6 223-40 Dy 4 223-00 Vi. ll 223°5 18 11 223°36 18 8 223-30 18 6 223°4 19 8 223°18 19 6 223°22 19 1) | 223°1 20 7 223°12 20 3 223°27 20 9 223°0 21 9 223°21 a a 223°21 21 10 223°0 22 4 223-10 22 7 223°25 22 12 222°8 23 10 223°26 28 1 ? 23 14 223°3 24 7 223°17 24 rt 223°28 24 if 223°4 25 9 223°20 25 3 223°40 25 12 223°1 26 6 223°28 26 8 223°18 26 13 223°1 27 8 22318 27 3 223°29 25 TA. ys 22358 28 5 223°30 28 6 223°23 2S 13 223°0 29 9 223:19 29 7 223°30 29 14 223'2 30 9 223-17 30 5 22294 30 12 222°9 31 7 223-28 31 9 223°16 31 13 223-0 32 13 223° 24 2 7 223°19 32 16 23°0 33 7 223°24 33 6 223°12 33 10 223°3 34 3 223°25 34 8 223°27 34 i 223°3 35 8 223°14 35 5 223°17 35 14 223°3 36 4 = 223°24 | 36 5 223°14 36 16 223°4 37 7 223 19 37 4 223:30 37 14 =. 2223-2 38 5 223°24 38 14 223°1 39 5 223°32 39 9 223°9 40 8 223°27 40 16 223°6 4 2 223°45 41 16 223°7 42 4 223°17 42 15 223°8 43 6 22323 43 16 223°8 44 5 223-16) 44 15 223°7 Short Series 5 223°7 Mean...... 223°23 Mean...... 223°23 Meanie. ec. 223°26 The mean values of the wave-number differences in the various series were calculated from the observed lines in the series and are therefore com- parable with those given by Miss Lowater. 520 Dr. J. R. Airey on Bessel and Neumann improvement of the experimental conditions more lines would be observed. One additional point may be noted, namely, that the interval between any consecutive pair = observed lines is never less than that calculated from the theory. ' It may fairly be claimed that the results brought forward in this paper entirely support the theory of the relation between infra-red and ultra-violet absorption advanced in the previous papers, and there is little doubt of the validity of the formula 13951°6+ p x 223°225 +n x 2°73 for the less refrangible ultra-violet absorption band of sulphur dioxide. The whole of this band group therefore has accu- rately been calculated from the infra-red absorption spectra of sulphur dioxide and hydrogen sulphide, the only additional fact being used of the central wave-number of the ultra- violet band which enabled the true wave-number of the fundamental infra-red sulphur dioxide band at 7°407 p to determined. The University, Liverpool. LAIIT. Bessel and Neumann Functions of Equal Order and Argument. By Joun R. Atrey, VW.A., D.Se.* : te Bessel and Neumann functions J,(<), G,(<), and Y,(¢) of equal or nearly equal order and argument. occur in many physical problems, and various authors have given formule expressing their behaviour under these conditions. Graf + found that when n is very large, (4) W2In(n)= 5S. 2 atte Re a (1) ah Further and more general formule, to order _> Were found from the relationship between the Bessel and Neumann functions when the argument and order are nearly equal, * Communicated by the Author. + Graf u. Gubler, ‘Theorie der Besselschen Funktionen,’ Erstes Heft, 96-104. ¢{ Nicholson, Phil. Mag. Aug. 1908, pp. 276-279. Functions of Equal Order and Argument. 521 and Airy’s Integral, viz. in the case of the J,(z) function, Be) / BN * eos w+ (n—2z) A gine w\z Ne = ah p =) Dar, p* e) oa = Vn( Joost + PT / cose + GAP 3 COs | +... where p=(n -2)(2)°; ie 2) or, using De Morgan’s expression * cos (w®—mw) dw, ae) 5 eee 6m i a4 eae) marl) Us)+ ten) G+ ba) yr(g)]; The third and following terms are correctly given in (14). on. When — — is not very small, the error in (2) becomes 23 considerable. From Table II. in the papert+ on the “Relation of Airy’s Integral to the Bessel Functions,” it would appear that the first root p; of Jn(Z) is given by pr=n+11814n8. Ps aa Sat se ame sea, Pal) From this result, p; for Jyo-9(Z) is 1011°8, whereas the- correct value is 1018°62. It can be shown from other | ’ Dany? : - considerations that when X is written for a , Pp, 18 more accurately represented by Ve O9e a Brey | 3 py=n+t a0 ae CS, EVA ag) ] =n-+ todas AOL a58. This gives 1018°51 for the first root of Jyoo0(<). The function J,(<) has also been expressed asymptotically } in terms of Ja and J_1 when the argument and the order 3 * Airy, Camb. Phil. Trans. vol. viii. pp. 596-599. t See also Lord Rayleigh, Phil. Mag. Dec. 1910. t Nicholson, Phil. Mag. ‘July 1909, p. 14. Fortables of the Ji and J _ functions, see Dani, Archiv der Math. u. Phys. 18 Band, 8. 337, 338. (3): 522 Dr. J. R. Airey on Bessel and Neumann do not differ widely : 1 Z(z—n =) ee oii) Ce RS (6) the J functions of fractional order having the argument 2o a 3 ; From a consideration of Sommerfeld’s Integral, Debye * has given the following formula for J,(z) when n and z are where o= —p. nearly equal. When - =l—e, ~ Sea! st] 1(ge = % By(e)(z)* TCF-)sin +5, (7) T s=0 where Bo(ez) = 1, Bi (ez) = ez, Cem eek B,(ez) = ? —_ () 5 . es Pies ea — oO pass 24 -- 280° ee The Bessel Function Jn(z). These results can be obtained very easily by evaluating the integral (3% F J.= 2 cos(zsinw—nw)dw. . . . (8) 0 In the case where n and z are large and equal f, 2 3 5 7 Jn(n) = =| cos n (sin w—w) dw= = cos n( re ) dw. 6. 120 "5040 Changing the variable by writing T 0 2 w w? wi! 6a 3 Ww? w! or ——— s/h) Dm ta 0. 502088 Fs —~ 90 + 840°"? = ae” w? w n 60 T $400 8400 * Math. Annalen, |xvii. p. 557 (1909). 7 Lord Rayleigh, Phil. Mae Dec. 1910. Functions of Equal Order and Argument. 523 Reversing this series, we get he aes, _ (6a\t 1 (62 1 oy wart + Tapp =) * oC) + ato “ps 0 Bie) Val Ley6Vr 4 ] . dw= Ee (2\* 4 Lv + aan) + ag 840 (5) oe dx. Hence 3 6 Ty foi\a J(n)= a [(c) sale )+an(-) ..| Cos wide.» (Q) But 4 2-1 cos edax=VT(p) oe. EERO) 0 Therefore ioaas ell Gla an) r()- amo (5) Lgeeagl P(g) 4, (11) The coefficient of the last term in the bracket, oniae , 1s approximate. Even for very small values of n, this formula gives J,(n) with considerable accuracy. J,(1) is given correctly to three places of decimals, viz. 0°43995 instead of 0°44005. A closer approximation is obtained as the value of n increases, J¢(6) to six, Jyg(48) tu nine, and J759(750) to fourteen places of decimals. nN. J(n)- 6 0°2458 37 48 Or 250 son oe 750 00492 3244 5583 97 Although n must be an integer in (8), the formule (11), (13), and (14) appear to be valid even when n does not fulfil this condition. When z and n are nearly equal, say z=n+k«, J (Z) = =| cos | 2(sin w—w) + kw] dw 0 2 =) cos 2 (sin w—w) cos kw dw — =I) sin z(sin w—w) sin kwdw. (12) a; 0 ie 0 524 Dr. J. R. Airey on Bessel and Neumann Taking the first integral, ih (~ as fui \d ey hae ou Kw Kt wt ? . = Kw cosz(sinw—w)dw= — et op We aseenlommg Es W zs 9 - + 21 (sinw—w) dw “0 As in the preceding nae Al Fre) i a 1. 6\3 6\3 P =A) cos 2 (sin w—w)dw= 95 ( yr (5)- saa.) (8) i | Similarly,. We 207 il Ke Vp iy (6 Veer Ms | 2 cos z(sinw—w) de ga Fit 5) E(3)+ )+ alc) r(3) | and so on. The terms arising out of the second integral of (12) can be found in a similar way from (co p= ue vaaell oP sun e d= igo) sine” Sr 0) Collecting the terms, the following expression for J,(<¢) is obtained. when <=n+4« :— sonst) 1C)emC)E) IOP “i00) 9G) a) 0) BPG 2x0). i a or where Bo= Ll: Bix, Ke ok Cam le Kit kee 1 B= 5a— 74 + 230° cna ea m cere 2, 720. 1420" 2SSi i aCOes Pe K! a as ‘s 19! ne 13K 5040 900 ' 12600 = 31500’ pee K Ki (als Reauc® 7939K 369880 ~ 30240 + 604800 ~ 907200 * 232848000 Functions of Equal Order and Argument. 525 In the important case where the order of the function is n—1 and the argument n, ee 6 op Jn—a(1) = 27/3 (2) P(5)+ +) r(5)— 30 so(s) r(s) Ti 2 1 5 r (3) ete el aim) 'G 23 (6\8 (2 947 (Vip i + aa00(s y r(3)+ T 74344000 \n (3). 2 (4) The Peel values occurring in the above formule are log T (3) =0-42796 27493 1426: r (5) =2'67893 85347 077 log (5)=0-13165 64916 8402: P(5)=1-35411 79394 264 The formule (11) and (14) give the values of J,(n) and 2 Jn—1(n), and the recurrence formula Jn1— — Int+Iny1=0 enables one to find functions of higher or lower order. In this way the behaviour of J,,(z) in the neighbourhood of its first root can be investigated. For example, in calculating the first root of Jyoo(z), Jicg(109) and Jyo(109) are first found, and from these values the functions J4;(109), &e., down to Jy99(109) as follows :— nN. J,(109). nN. J,,(109). 109 0:093639 104 0°134966 108 0-111472 103 0°115672 107 0-127260 102 0:083643 106 0°138378 101 0:040870 105 0°141879 100 —0°007901 Bessel’s addition theorem * for the J functions, viz., va oth J a (+h) =(——) [Iu@)— FpInle) + F Insel)... ], (18) _ h(2z+h) +h) rawr re. . then gives the value of h, (—0°164...), which makes Inizth)=0 when n=100 and z=109. The first root of Jio0(z) is therefore 108-836. By the same method, and by where * Abhandlungen der Berliner Akademie, 1824; Nielsen, ‘ Theorie der Cylinderfunktionen,’ 8. 266. 526 Dr. J. R. Airey on Bessel and Neumann the one indicated at the end of this paper, the first roots of Jio(z) and Jyoo9(z) are 14°4755 and 1018-62. The value of J,(n+«) can be found directly from te) when « is larger than unity, if n is large. For example*, J13(20) =0°25109. This result is more easily and accurately obtained from Jy9(20) and Jo9(20). The Neumann Functions G,(z) and Y,(z). The function T G(2) 7 pr — cos ntJ ,(z) | 2 sin n7r Te ghee. Q J_»(n) = es (2) cos n(5 +n)— 450) rG eos n( ~ +n)...], (16) and making n integral, we get Gu60=§[(7) 8(3) + gaol) 2(3) ~ sao) 2G) ~ sgaa00 (a) P(3)+--]- aD The following asymptotic series represents G,(z) when the argument z=n + K :— a aC ()8G)-3 hee +n) nC) sa? )2() M0} 2 Sih el eal SE the coefficients By, B,, &c., having the same meaning as before. In the special case where n is the argument and n—1 the order of the function, a al; y r(3)- (i) F 3)-3 Al I r(3 5) + Bol “) F(5) mB 4 @r0)- ral! r(5) on (Exo ul | ; a -) r(5).-|- mee Ge .* Phil. Mag. July 1909, p. 17. t Gray & Mathews, ‘ Bessel Functions,’ p. 242. Functions of Equal Order and Argument. 527 The following values calculated from these formule ma be compared with those * found from Gp(n) and G,(n) by 2n means of the recurrence formula, G,41= — G,—Gy_i. z n. G. (2). G0): 7 0°636755 0:313070 8 0608951 0°314261 9 07585445 0:°314077 10 0°565195 0°313060 ca: 0°547485 0°311526 12 0°531806 0:309676 13 0°517782 0°307632 G,—1(v) has a maximum value near n=8°316. To find the first root of Gyoo(z), the values of Gy,(104) and (4493(104) are calculated, and from these results functions. of higher and lower orders are readily obtained. n. G (104). Ne G,,(104). 100 0°025594 104 0°258794. 101 0-091802 105 0°309836 102 0152713 106 0°366837 103 0-207751 107 0°437957 Bessel’s formula (15) is equally applicable to the G and Y functions, giving finally 104°380, as the first root of Gioo(Z). Since the Neumann function f Y,(z) is given by Y,()=(log2—y)Iil2)—Galz), .. (20) expressions for Y,(m) and Y,_1() can be found by substi- tuting (11) and (17) or (14) and (19) in (20). The first root of Yyo9(z) calculated as in the previous examples is. 104°133. Tt can be shown that pm, the mth root of J,(z), is given very approximately by the formula Pin SOCOM Te GAO sine). (21) 4m—1 where tan 6— d= Met MELANIN Ww fa C2) * British Association Report: Calculation of Mathematical Tables, 1914. + Gray & Mathews, ‘ Bessel Functions,’ pp. 14, 242. 528 Prof. V. Karapetoff on Divergence and This formula and others giving still closer values of the ‘roots are especially useful when m is small in comparison with n. The formule given by McMahon ™* and others for the roots of J,(z), Y,(z) are limited in their application, and cannot be employed in finding the earlier roots of J,(z), &c., ~when n is large. Similar expressions are found for the roots of the two Neumann functions. -LXIV. Divergence and Curl in a Vector Field in terms of curvature and tortuosity. By V. KARAPETOFF ft. rs awe usual expressions for divergence and curl, in Car- tesian coordinates, convey a somewhat indirect picture -of the physical nature of these quantities, particularly to the beginner. ‘The expressions given below are based directly upon the characteristics of a field near a given point, namely, the geometric shape of the lines of force, and the rate of -change of the field density along a line of force and in the directions perpendicular to it. No fixed origin of coordinates or projections with respect to this origin are introduced into the discussion. or the sake of simplicity, and as a sort of introduction to the method, a two-dimensional field is considered first. 1. Two-DIMENSIONAL FIELD. Let AB, A'B’, A''B” (fig. 1) represent lines of force of a ~vector field in the vicinity of a point O. Let these lines lie ‘in the plane of the paper, and let all the lines of the field lie in parallel planes, so that the field may be called two- -dimensional. The field at and near point O is determined by the following quantities and characteristics :— (1) The magnitude and the direction of the flux density D at O. (2) The scalar rate of change of flux density, 0D/ds, along the line of force AB. (3) The scalar rate of change of flux density, 9D/dn, along the normal to AB. (4) The radius of curvature, PO=p, of AB. (5) The radius of spread, QO=R, at O. * Annals of Mathematics, vol. ix. (1895); Proc. Physical Society, ~vol. xxiii. part ui. (1911). + Communicated by the Author. Curl in a Vector Field. 529° The latter radius is defined as the radius of curvature of the curve MN, orthogonal to the lines of force at and near point O. The centre of spread is denoted in the figure by Q. Fig. 1. Curve MN is not necessarily an equipotential line, because the discussion is not limited to a lamellar field. The field is assumed in the general case to be neither lamellar nor solenoidal. | La. Divergence in two dimensions. Let dn=OO’ be an element of the normal to the line of force ; then the flux through OO’, for unit thickness of the field in the direction perpendicular to the plane of the paper, is Ddn. By definition, divergence is the limit of the ratio of the excess flux through an infinitesimal volume to the volume itself ; thus dh (D dn}ds : Os | Cig Dice PP ae tere ; SAR Cid Sa ant y ge (1) or aD 2 (dn) Phil. Mag. S. 6. Vol. 31. No. 186. June 1916. 2N 930 Prof. V. Karapetoff on Divergence and Since we are dealing with infinitesimal dimensions, OB may be taken to be a straight line, and O'B’ coincides with the tangent O'K’. Drawing a line O'L’ parallel to OB (or to OK) triangle O'K’L’ is obtained, in which angle da represents the rate of increase of dn with s, or _ 0(dn) da= ae But the same angle obtains at Q, where de=dn/R, so that we have Q(dn) _ dn OS cade Substituting this expression into eq. (2) we finally obtain oD. D div D= a + ae a) fen! Stemi : (3) In other words, the divergence at a point in a_ two- dimensional field depends only upon the flux density, the rate of change of this density along the tangent, and the radius of spread. The divergence is independent of the radius of curvature of the field and of the rate of increase of the flux density along the normal. Special cases: (a) Solenoidal fux—For such a flux Ddn is the same at OO’ as at BB’; thus, Ddn . a ess cs ae: or D dn=const. This means that for any two points, 1 and 2, along a line of force, pie D, wi dn, Hq. (3) shows shows that div D=0, when | aD: ae Hi. Ciel Wh eu By analogy with the potential function for a lamellar flux, a flux function, yr, may be introduced for the solenoidal flux, This function is defined by the equation p=, e e ° e P e (5 a) and the existence of such a function is a condition for the Curl in a Vector Field, 531 flux being solenoidal, We have Ddn=dy, and since dy is independent of s, o(dy) Os =(). e e . ° e e (5 b) Eq. (5) becomes git oe 4 5 SF a0. 85 0) The physical meaning of function wy is as follows: the numerical value of is equal to the flux comprised between the line of force under consideration and some other arbitrary line of force selected as a reference ; the flux is understood to be per unit of thickness in the direction perpendicular to the plane of the paper. (b) Lines of force are parallel near O,—In this case R=, and divi. ab ew sw 4 SD) (c) Lamellar jield.—Let ¢ be the potential function. Then D I YS ~~ ~] SY and eq. (3) becomes op , 1 Od.” Seay hake ° qt he ° (8) this is equivalent to Poisson’s equation in rectangular co- ordinates. (d) Solenoidal lamellar field : o*p , 1 Ob _ An op a OR) Mah nie eee taco This equation is equivalent to Laplace’s equation in rectangular coordinates. 16. Curl in two dimensions. By definition, the curl of D at point O (fig. 1) is the limit of the ratio of the line integral of D taken (say) along OBB’'O'O to the area enclosed by that path. But the line integrals along BB’ and OO’ vanish because the field there is perpendicular to the path, so that Dds)dn = ( dD D d(ds) Cerrar man * di on on) ee 2N2 932 Prof. V. Karapetoff on Divergence and To express the last term through the. radius of curvature, we draw KM’ parallel to OO’. Angle K’KM’ is a measure for the rate of increase of ds along the normal, or o(ds) On - But the same angle df obtains at the centre of curvature P, where dp= ds dg=—- ens Thus Olds) | ds | Sh ee a he 2-6 (11) Substituting this value into eq. (10) we get . oD D | curl D= 5 ete ee It will thus be seen that the Ls at a point in a two- dimensional field depends only upon the flux density, the rate ‘of ‘change of this density along the normal, and the radius of curvature of the field. The curl is independent of the rate of increase of the flux density in the direction of the field and is also independent of the radius of spread. es expression (12) for the curl together with q. (3) for the divergence, it will be seen that these two. a dale together “contain all the five characteristics enumerated at the beginning of the article. Moreover,. expressions (1), (2), and (3) are formally similar to eqs. (10) and (12), and. this fact perhaps helps to see why in vector analysis the same Hamiltonian operator is used for both divergence and curl. Special cases: (a) Lamellar fluw.—For ‘such a flux there is a potential function ¢@ such that D=dd¢/ds. In eq. (10),. D ds=d¢, and since dd between two equipotential surfaces. is independeni of 2, (Dds) _ Bn or ) curl D0; Kq. (12) becomes grt =0,. e+ C8) on Curl in'a Vector Field. . yi) which is the condition for the flux being solenoidal. Intro- ducing the potential function we get — | oo _ lod_ | eldwake On Meer ye sien) Cla) This expression is analogous to equation (5c) for the solenoidal flux. (b) Lines of force are parallel near O,—In this case p= and curl D= 9°. ea (ce) Solenoidal fux.—tIntroducing again the flux function vr we get es Saas |) This is analogous to Poisson’s expression for the divergence of a lamellar field, in the form (8). (d) Solenoidal lamellar field : On) lone an? Bone =o) ae Mae company i) ) This corresponds to Laplace’s equation in the form (9). The curl of a two-dimensional field is a solenoidal vector, because its lines of force are straight lines, and the flux density remains the same from layer to layer of the original field. Thus, in expression (3) both terms on the right-hand side are equal to zero separately. 2. THREE-DIMENSIONAL FIELD. Fig, 2. n A three-dimensional field at and near a point O (figs. 2, 3, and 4) may be described with reference to the following 034 Prof. V. Karapetoff on Divergence and three mutually perpendicular directions which are defined by the line of force AB through O :— (1) The tangent ¢ to the line of force ; (2) The principal normal n to the same curve ; (3) The binormal 6 of the same curve. Fig. 3. These three axes determine the following three planes ;— (1) The osculating plane, én, in which fig. 2 is drawn ; (2) The rectifying plane, tb, in which fig. 3 is drawn ; (3) The normal plane, nb, in which fig, 4 is drawn. Curl in a Vector Field. 535 The curves drawn in these figures represent, therefore, the projections of the line of force AB upon these planes. The field is determined by quantities similar to those men- tioned above for the two-dimensional field, only a greater number is required to account for the changes in three dimensions. 2a. Divergence in three dimensions. Consider an infinitesimal tube of flux near O, of a rect- angular cross-section, having a width dn in the direction of the principal normal, and a width db in the direction of the binormal. The flux comprised in the tube is Ddndb, and the volume of the tube of a length ds in the direction of D is dndbds. Thus, by definition oO 5, (D dn db)ds diy DS hae oe et Meret) Her! kis (18) or 0 0. yy dn) ~~ (ab : On) Os * Os \ F divD=$>+D°—+D2_. . . (a9) The last two terms on the right-hand side are similar to those in eq. (2), and by a similar reasoning we obtain : ae a 9 Pan is nee where Ry is the radius of spread in the osculating plane and R, is that in the rectifying plane *. It will thus be seen that the divergence does not depend upon the rate of change of flux density in the normal plane. For a solenoidal flux, (a) ames ier 9 Loge Os seg bes mio ° e e . . (21) For a solenoidal and lamellar flux, if @ is the potential function, 0d eal Vo : = —e- —— a ° e . . 2 which is equivalent to Laplace’s equation in the usual form. * A similar formula may be deduced by considering directly the radiz of curvature of a surface orthogonal to the flux. See W, v. Ignatowsky, Die Vectoranalysis, 1909, vol. i. p. 84. 236 Prof. V. Karapetoff on Divergence ana 2b. Curl in three dimensions. The vector of a curl, generally speaking, is not normal to the original vector D. It is therefore convenient to deduce expressions for the projections of the curl upon the three axes of coordinates ¢, n, and 0b, referred to above. The expressions so obtained may in some cases give a better insight into the physical nature of the curl than the usual expressions of its projections upon a fixed system of coordinates. (a) The component of the curl along the binormal.—This component is perpendicular to the projection of the line of force shown in fig. 2: its value may be deduced in exactly the same way as is done for the two-dimensional field above. We thus get an expression similar to eq. (12): aD (curl D), = oa Pa? (23) The subscript } indicates a component of the curl along the binormal, and p is the radius of curvature of the line of force at O. This component of the curl represents the whole curl in a two-dimensional field. (b) The component of the curl along the normal,—This component is perpendicular to the projection of the line of force shown in fig. 3. By analogy with eq. (23) we have owe» (curl D), = Sa Pe ees) where 7 is the radius of tortuosity * of the line of force at. (c) The component of the curl in the direction of vector D atself.—This component of the curl is perpendicular to the projection of the line of force shown in fig. 4. Since it is not possible to speak of the curvature of the line of force in the plane of fig. 4, no expression similar to eqs. (23) and (24) can be written for this component of the curl. It is convenient to apply to this plane the usual expression for the component of a curl in Cartesian coordinates. Using the directions of 6 and n as the axes of coordinates, we get D, oD | (curl D)= 99" — 90, ke, * The term tortuosity is used in this article in place of the more usual term torsion, in order to avoid a possible ambiguity in application, for example, to the theory of elasticity, where the term torsion has a meaning of its own. Curl in a Vector Field. 537 where D, and D, are the projections of the vector D upon 6 and n, at points near O. Consider points O' and Ol’ (fig. 4) infinitely close to O, and situated upon the m-axis. At these points the total components of D upon the normal plane of the vector at O are directed along the n-axis, as is shown hy heavy arrow- heads. This is because the osculating planes at O' and O” form an infinitesimal angle with that at O, and D has no projection upon the n—b plane at O. Thus, the component D; at O' or O"' is an infinitesimal of the second order as compared with the component D, which is an infinitesimal of the first order. Therefore, in eq. (25) we may put Ds. : en ==) ° ° ° e . . (26) and consequently D, (curl D);= a his Ra Me (27) Consider now points M’ and M"' infinitely close to O, and situated upon the b-axis. If the component of the curl according to eq. (27) has a finite value, the components Dn at these points must be in opposite directions, because D,=0 at point O. Thus, the existence of a curl in the direction of the vector itself is due to a lack of symmetry of the field with respect to the osculating plane nt. A two- dimensional flux is always symmetrical with respect to its plane, and consequently there is no component of the curl in the direction of the vector itself *. Take a point T in the plane nd, with coordinates dn and db. The component of the vector D at this point, parallel to n, is mio WD ! oD, ; Di= Sp db+ Sdn... . . (28) * The foregoing expressions for the components of a curl may be also derived as a specific case from tke general expressions for ‘ rotation” in curvilinear orthogonal coordinates. See, for example, A. EK. H. Love, ‘Mathematical Theory of Elasticity,’ second edition, pp. 54-56. It is believed, however, that the simple derivation given above is better adapted for the needs of physicists, and perhaps gives more insight into the nature of curl. See also J. Spielrein, “ Geometrisches zur elektrischen Festigkeitsrechnung,” Archiv fiir Elektrotechnik, vol. iv. p- 78 (1915). 538 Divergence and Curl in a Vector Field. Let point T be so selected that D,=0. Then oD, oD, ah db + SF dn=0, or an. 0D,/0 = SDajpnt tt In other words, the points for which D,=0 determine a plane NN inclined by an angle y to the rectifying plane. This angle is determined by eq. (29) and characterizes the amount by which the flux is skewed. Without tortuosity, that is in a two-dimensional field, y=0 and plane NN ~ coincides with ¢. In the preceding discussion, the derivative QD;/00 1s left. out of consideration, because it does not enter in the ex-= pression for the curl. ‘The flux may spread to any degree in the rectifying plane, without affecting the component of the. curl in the direction of vector D. tan y= — SUMMARY. Divergence. j Sis ‘$0 Di SD In two dimensions: div D= ae + R?: In three dimensions : div D= ee 2 +D(q+ x): 0 ive Curl. aD D. In two dimensions: curl D= aa + — In three dimensions the projections of the curl are as follows :— DUE along the binormal, (curl Dy= $> +—; p ab D- ob along the vector itself, or along the tangent, oD, ae along the normal, (curl D),= rit (curl D),;= F B39 LXV. On Wave-Patterns due to a Travelling Disturbance. By Prof. H. Lams, F.R.S.* 1. “JXHE procedure of a former paper | may be adapted to the case of propagation in two dimensions, as. when a pressure-point advances over the surface of water. Although nothing very novel is to be looked for in the way of results, the generalized treatment of a somewhat intricate problem may be acceptable. The method, it may be recalled, depends ultimately on integrals of the type F(z) I) taken reund suitable cis lie The functions F(z), f(z) are usually algebraic, and f(z) has one or more simple roots of the form «+7, where « is positive and s,, which depends. on an assumed frictional coefficient, is in the end taken to be infinitesimal. For simplicity of statement it is assumed for the most part that there is only one root of f(z) of the above type. First suppose that z is positive. We put z=k+im, and integrate round the boundary of the infinite quadrant in which , mare positive. If u,>0, this region contains the singular point «+7im;,, which must be excluded. Thus CEO Bh Nin Wide ial ke pe ~ F(k) she Qrik(«e+ip,) . .(°-F (am) dk=—,_ ~~ e IKI — 44x e7 mez ; ba FOG itil til Rim Cay The latter integral diminishes rapidly with increasing z. In particular, if iF (im) Fae ACY Nase ae be kee its asymptotic value is is ies Oy aun Bee =o se ee anid at) Hence, when p;->0, we have, for sufficiently large values of 2, QniF (Ke) . etek OE) fie | 5) ee Tk 7 («) ? * Communicated by the Author. t Phil. Mag. (6) vol. xxxi. p, 886 (1916). 540 Prof. H. Lamb on Wave-Patterns If, on the other hand, ,<0, the first term on the right- hand side of (2) is absent, since the region considered has now no singular point. We have then F(&) 0 SA) with the same approximation as in (5). Next, suppose x to be negative. We now take ag our contour that of the infinite quadrant for which & is positive and m negative. Thus, if “<0, e@dk=0, . ) ann NE) QriE («e+ ipy) (2 R( 2a ena tdk an - UKE —— Ry : : J pmx A OT Pet Oy or, in the limit, ‘ F(A) kx jp — 27ri F(x) pik i" Hk)” dk F(x) Me ek TS) when 2 is considerable. If u,>0, we have “EE) Jo J) in the same approximate sense. It is to be remembered, however, that the integrals which we have neglected may be important for small values of a, and may even become infinite for <=0. The infinity may be avoided by the insertion of a factor e~*° in the value of F(z); this leads to a factor e~*” under the integral sign in the last term of (2), and so secures convergence by fluctu- ation. This artifice has moreover a physical justification, as enabling us to represent the effect of a force which is diffused about a point, instead of being absolutely concentrated. The integrals, as thus modified, will still be suificiently important to be taken into account when «is small. In the application to one-dimensional wave-propagation this is of no great consequence, as the matter only relates to the state of things in the immediate neighbourhood of the source of disturbance. In the problems now to be considered, however, the point requires attention. dk = 0), .” cms > tue 2. Proceeding now to our question of wave-propagation in two dimensions, an impulse of the type Jo(ka@), where a denotes distance from a point Q, will give rise to an annular wave-system of the type taJy(ka)d(kje . . . . . (10) about Q as centre. The form of (Xk), and of o as a function (7) due to a Travelling Disturbance. BAL of &, will be determined by the nature of the medium. For instance, in the case of water-waves we have OEP UG OCs tela Lab CLL) where p is the density, whilst | Be egh yes) a me? if the depth be great. It follows from (10) by a theorem of C. Neumann that the effect of a unit impulse concentrated in a point at Q will be ENE ree Guedes. sk US) 20 0 Since Jo(ka) =5,\ Cp ay. es a) MEAD) this may be regarded as made up of rectilinear wave-trains whose directions of propagation are distributed uniformly in azimuth about Q. Suppose now that we have a source of disturbance. travelling in the negative direction along the axis of #, with the constant velocity c, and that at the instant under con- sideration it has reached the origin O. Let Q be its position at any antecedent time ¢, so that OQ=ct. If x, y be the coordinates of any point P, the distance of P from a recti- linear wave-front through Q will be (ct —x) cosw—ysin vy, where wy denotes the angle which the normal to the wave- front makes with Ox. This expression takes the place of @ cos x in (14), and since the integrand is periodic in respect to w, with the period 27, we may replace y by W as. independent variable, with the same range of integration. 542 Prof. H. Lamb on Wave-Patiterns The effect of an impulse 6¢ delivered at Q is therefore given by oC= eb he eHttuo—ke cos Y)t+tk(w cos t+ y sin Wh (k)kdk dy, (15) the factor e~“t being introduced as in the former paper to represent the effect of slight dissipative forces. Integrating from t=0 to t=, we have ie il © ("ap etzeos ty sinWh(k) kdkdup =a, rere pea 3. The approximate integration with respect to k is -effected by means of the formule of §1. Let « bea value -of k satisfying the equation o=ke cosy, . ss re ‘where yy must of course lie between +7. That is, 27/k is the wave-length corresponding to the wave-velocity ccosy. It may of course happen in particular cases that there is no such value of k, or there may be more than one, .as in the case of waves in superposed liquids treated in the former paper, or in that of waves under the joint influence of gravity and capillarity. In the latter event each such value wil] give rise to a separate term in the value of €. Taking for definiteness the case of a single root of (17), the denominator in (16) will vanish, if w be small, for k=k+ip,, approximately, where pu= ej (ecosyr—U), 2 2) eee U being the value which the group-velocity (do/dk) assumes for k=«. Hence, referring to (5) and (9), we have, if UO) . 9:7). ae) If, on the other hand, U>ccosw, we have Bele eix(zcoswty sin Wi h( dr = Fl U-ccosy ee the range now including such values as make “costr-ysinar<0..- , . 5 yee due toa Travelling Disturbance. 543 The frictional coefficient has been finally put =0, but it may sometimes be necessary to retain it, in order to secure the determinateness of the integrals. This necessity will arise whenever U—ccosy vanishes within the range of +p, as for instance in. the case of water-waves subject to gravity and capillarity combined, if c exceeds the mininum waye- velocity. It is to be remarked, moreover, that we have so far imagined the disturbing influence to be concentrated at a point, with the result that the integrals with respect to m which we have discarded become in some applications infinite when wcosy+ysiny=0. This may be remedied by the introduction of a factor e-* in (13) and subsequent formule. The effect of this is that the distribution of the disturbing force is now given by the formula La = Th = b 93) * the factors being adjusted to make the integral amount unity. The degree of concentration varies inversely as b. The modified integrals with respect to m may still be important when xcosf+ysiny is small ; but if any pre- scribed standard of smallness be imposed the range of w for which this is satisfied becomes narrower the larger the value of /(a?+y). It results that the terms in question, after integration with respect to x, become negligible at a sufficiently great distance from the origin. It is not easy to do more than indicate in this way the course of the argument in the general case. The particular case of gravity waves will be examined more in detail presently (§7). 4, The definite integrals in (19) and (21) may now be evaluated approximately by Kelvin’s methodf, If we write, for shortness, K(acos+ysinv)=f/(v), . . . (24) * Obtained by differentiating with respect to 6 the identity 2 rae dk = de De : { J (kw)e “'d (Ge) + Proc. Roy. Soe. vol. xlii. p. 80 (1887); Math. & Phys. Papers, vol. iv. p. 8303. Reference is however due to Stokes who had briefly indicated the method, as an alternative to one which he actually employed, in a footnote to his paper of 1850 on Airy’s integral and other functions. (See his collected ‘ Papers,” vol. ii, p. 341.) O44 Prof. H. Lamb on Wave-Patterns and put ~=0-+.o, where @ is a root of i) A a FQ) =0, 0 ee we have AD =P0) + 50°F" (0) +-0- U/c. The equation (32) which determines the con- figuration of the wave-ridges becomes (37) p= =" (ecos@—-U) 2M eh 0d! AGB) The ridges therefore form a system of circular arcs, of radii 27nU/o, lying to the right of O and convex to it, and touching two straight lines drawn from O at angles + sin~!(U/c) to the axis of 2. 6. In the case of a pressure-point moving over the surface of deep water we have, if capillarity be neglected, (39) ae a c? cos? The equation of the “isophasal” lines, as they are called by Kelvin, therefore takes the form. Qn? Piet ree bh aoe lene ay * Schuster, ‘ Optics,’ 2nd ed. pp. 333-4. Phil. Mag. 8.6. Vol. 31, No. 186. June 1916. 20 546 Prof. H. Lamb on Wave-Patterns Since the group-velocity is always one-half the wave- velocity, we have U=sects@,. . so that the formula (27) applies. The proviso azcos@+y sin 0>0 shows moreover that the wave-system lies to the right of O, 2. e. it follows the disturbing agent. The formula (40) leads to the well-known wave-pattern™ of transverse and diverging waves. _ Again, from (35) and (89) f''(@) = 2nw(1—2 tan? 6). 2). eee Taking the value of $(£) from (11), it appears that along an isophasal line the elevation due to a travelling pressure of integral amount ee is sec’ 0 (ust ¢ ) im e 5: at Maes aa ee a/— tnt - of which the real part is of course to be retained. The upper sign in the epee) relates to that branch of the curve (40) for which tan?@<4, and the lower to that for which tan?@>4. There is accordingly a difference of phase of a quarter-period between the transverse and diverging waves in the neighbourhood of the cusps. This was first remarked by Kelvin, in 19057. The law of height which he gave in the paper referred to differs, however, somewhat from (43), owing to the fact that his source of disturbance was constructed by superposition of lines or narrow bands of pressure distributed uniformly in azimuth about a point. This does not give a strictly localized pressure. Mathe- matically the effect is equivalent to the omission of the factor k in (13) and subsequent formulee. The formula (43) makes ¢ infinite for tan?0=4, i.e. at the cusps where Si two systems of waves coalesce, and again for 0=+4 The former infinity is a mathematical accident, due to. me failure of our approximation through the vanishing of f/(0) in (26). The calculation might be * ‘ Hydrodynamics,’ Art. 256. The pattern was first investigated by Kelvin in 1887; see his ‘ Popular Lectures,’ vel. ili. p. 482. + Math. and Phys, Papers, vol. iv. p. 412. ‘ due toa Travelling Disturbance. 547 amended by continuing the expansion in (26) a step further, and using the formula * ie ae dw = eye ya fm - . ° (44) The infinity which occurs when 0= +47 is due to the concentration of the source. If we introduce a diffusion- factor e~™ in our formule, as already explained, the second factor in (43) is replaced by e79hie . sec? 9seact Q (49) 1 a 2 —tan? 7 which vanishes at the places in question. 7. It is possible in a definite problem such as the present to estimate the importance of the terms which were omitted in the integration with respect to kin § 3. Putting Ooi ween po." oF =o. ho GD in (16), we have to consider the integral Lain. iP, dx . \ 1 —g-2ce? cos pr’ taken along the imaginary axis, p being written in place of «cos w+ysinwy, and w put=0. This is equivalent to Wise mam Th legrteGmii cos v7 a The important part of this integral is due to comparatively small values of m, so that the second term in the denominator may be omitted. We thus obtain 1 1 (p +2b)?’ x (wcosw+ysin +26)?’ Sag this being, in fact, the first term in the asymptotic expansion of (48) bythe usual method. ‘The result has to be integrated with respect to yy. Since (47) SV apt - (ete - roy Ot ~ecosp+ysiny+ib- }_,rcosntib = V(7?+0?)’ aa ey * Havelock, Proc. Roy. Soc. A. vol. lxxxi. p. 422 (1908). t+ The equality of the second and third members is established by putting ¢= tans, and integrating over the contour of the upper half of the ¢-plane, having regard to the singularity at ¢=e%, where a= tan—1(4/r). 20 2 048 Wave-Patierns due to a Travelling Disturbance. where r=+/(x?+y7?), we have, by differentiation with: respect to 6, i ai __ = 51) _-(@cosptisinyr+ib)? (v7? +67)2° * (51) Inserting the various constant factors that have been omitted, we find that the part of the value of € which has been omitted has the value . a b Qaap (+0): 5 c 5 5 cl ° (52), This is negligible when the distance r from the origin is large compared with b. 8. When the velocity c of the travelling agent is sufficiently small the influence of capillarity predominates over that of gravity. Taking account of it alone we have 9 Frees, . . | eet RS where T is the surface-tension. ‘he forms of the isophasal lines are accordingly determined by p= : 5786070, oie) ea giving a quasi-parabolic shape. Since = 36 cos'@,. . ss.) Ae the formula (27) is now to be replaced by that derived from (21) in a similar manner. The approximation makes ¢ vanish unless acos0+ysin@ in negative, so that the head (9=0) of the wave-system lies to the left of O, 7. e. it precedes the disturbing agent. Again, by (35) we have, along an isophasal, f'(@)=—6nr(14+2tan?@). . . « (56) The final result takes the comparatively simple form C= 1 1 (2044) im ~ Vnrk* V/(3+6 tan? é) °° (57) r 549 J LXVI. The Variation of Thermionic Currents with Potentials. By Horace Lester, Ph.D., Instructor in Physics, Princeton University *. NITIAL thermionic currents from hot surfaces differ from the currents found after steady conditions have been obtained. The peculiarities of these currents have been studied by several investigators. Richardson t found that the positive emission was not permanent, that it did not ‘saturate’ easily, and that the currents increased with the temperature according to an exponential law, at least for a limited range of temperature. He found the carriers to be of atomic magnitude, and considered them to be produced by thermal dissociation of impurities in the hot anode. A further investigation of the initial positive emission has been undertaken more recently by Richardson and Sheard. They were prevented from completing their experiments, but a preliminary account ¢ of the results obtained has been pub- lished in a note in the ‘ Physical Review.’ They found that the positive currents decayed with time, the rate of decay depending on the temperature, and that the current-potential relation at constant temperature followed a steep curve up to about 40 volts, at which point a bend occurred towards the potential axis, and beyond which there was a nearly linear increase up to 400 volts. According to Richardson’s theory referred to above one would expect saturation to occur at much lower potentials than those mentioned, since the number of negative ions or electrons present is negligible. Saturation for positive currents does not ordinarily occur, although Richardson and Sheard found that with time the slope ot the curve above 40 volts decreased ; so that approxi- mate saturation was obtained eventually. At the suggestion of Professor Richardson, the writer undertook the continuation of this investigation. Asaresult, the above-mentioned peculiarities of the current-voltage curves for the initial positive emissions have been confirmed, and similar phenomena have been found to occur with the initial negative currents. The further conclusion of Richardson and Sheard that the increase in the current beyond 40 volts is due to secondary ionization caused by the bombardment of a film at the opposite electrode has not been verified. It is possible that such an effect may occur, but it seems clear * Communicated by Prof. O. W. Richardson, F.R.S. + Phil. Mag. July 1903, Sept. 1904. t Phys. Rev. vol. xxxiv. p. 392 (1912). Dr. H. Lester on the Variation of Fig. J. 590 x : ae Ee 8 oa 8 : : Renee tase = S Gea! OLULLLLILLIALULULLLELL LLL ULLUL UAE e ae a agp LOLI LE LIL DL LAL OAL LANL LITT ITIL ITIL IT ITT LT IIIT TTT o : GLEE LITE E se ty NS 3 SS % er =) is G a 0 ay a a 0 A OF OD a wT OP OD OD OO NLL, S Filament. H /L= Thermionic Currents with Potentials. 551 from the experiments that it is not a necessary cause of the increase, and that it cannot have caused any considerable fraction of the increase observed in the present experiments. A number of other points connected with initial thermionic emissions have been examined in the course of the in- vestigation. The experimental investigation involved observations of thermionic currents in good vacua for various temperatures and potentials, and with various other conditions to be de- scribed below. The currents were obtained from platinum wires mounted axially in cylindrical vessels, which contained the receiving electrodes. Figs. 1 and 2 show two types of A Se a Glass ring- as AIAN \ ZZ vessel used. The currents were measured by an electrometer in the ordinary way. Jor large currerts the electrometer was shunted across a resistance, and in some cases a micro- ammeter was used. Generally speaking, the currents were chosen so as to be as large as possible subject to the decay being negligible during the course of the observations. The vacua were produced by a rotary mercury-pump, and were usually about 3x10-° mm. of mercury on the McLeod gauge. Oo2 Dr. H. Lester on the Variation of Experimental Results. In the beginning it was found that lack of saturation characterized both positive and negative currents apparently without distinction. The potential-current curve showed a sharp bend towards the potential axis at about 40 volts. The increase at 400 volts over the value at 40 volts divided by the Jatter represents fairly the difficulty of attaining saturation. The lack of saturation will be represented by the expression # where 7 is the current at 40 volts and Az is the increase in 2 between 40 and 400 volts. This extra current may be due to ionization at the surface of the cold electrode, to ionization of a gas or vapour layer near the hot surface, or to phenomena at the surface of the hot electrode. In order to test the first possibility, an apparatus was constructed whereby it was possible to rub the inner surface of the receiving electrode with emery-cloth. Readings could be obtained within 30 seconds after rubbing. This treatment showed no appreciable change in the value of a Another apparatus was designed in which it was possible to heat the cold electrode to approximately 700° C. While the electrode was being heated the platinum wire was cold, and was drawn completely out of the experimental chamber. Repeated heatings, some of several hours’ duration, showed no change in the lack of saturation. Another test was carried out as follows :—Experience has shown that after considerable heating the value of - becomes very small, at least in many ‘cases. IEf s is due to the ionization of the film at the surface of the cold electrode, then the film having been destroyed by one wire, = should be permanently small for a second ‘wire not previously heated. ‘Three or four tests of this sort showed that the value of a for the second wire was inde- pendent of the value of = for the first wire. The series of experiments seemed to show conclusively that ionization at the cold electrode of a volatile film could not account for the observed lack of saturation. Thermionic Currents with Potentials. 55a To test the second possibility—viz., that lack of satu- ration was due to ionization in a dense vapour near the hot surface—a wire was mounted in a glass cylinder containing a cylindrical receiving electrode and a quantity of coconut- charcoal in a space below the open end of the cylindrical electrode. Liquid air could be applied to the charcoal and also to a trap in the pump-connexion just outside the glass cylinder. Observations were taken with liquid air off and on the charcoal, off and on the trap, and off and on both together. There was no detectable change in the value of = The pressure on the McLeod gauge was 1x 107° before applying liquid air, and no change occurred upon applying the liquid air. No change in the gauge-reading was expected, however, because the vapour, if it existed, would condense on the colder portions of the tube before it reached the gauge. The result of this experiment indicated that there could be no considerable amount of vapour around the hot surface. The fact that there was no appre- ciable cooling of the hot wire when liquid air was applied was a further indication that there was very little gas present. Since the first two hypotheses concerning the location of the phenomena causing lack of saturation were apparently untenable, there remained only the hypothesis that the effect was due to phenomena at the hot surface. To get at the nature of these phenomena, a detailed study of the pecu- liarities connected with initial emissions was undertaken. In all twenty-three different wires were studied under various conditions of purity, surface-cleanness, pressure, &c. It was known that water vapour actively modifies thermionic currents, so water vapour was in all cases excluded from the apparatus. In parts of the experiments the wires were sprayed with salts, and in some cases they were heavily coated with CaO or BaO. The results of the series of investigations may be sum- marized as follows:—In addition to the well-known time decay and variation with potentials, there were two features that characterized all of the emissions studied. The first of these was an initial growth in the value of =o This growth was rapid at first, and more gradual later. n, and a?—4b<0, into partial fractions, using the division method. Let SO. w-?-K denote the quotient and Ajv+By the Kk=0 remainder (which will, at most, be linear) obtained by nN dividing & mea”-* by 2? +ax+, k=0 then s ; ss Met" Or yn 2-k (eo tae +b (War tbye* Ge paeroe Clearing of fractions and equating like powers of w, we have (Q,g=0, for negative values of 8), Oi teQ) 7-14 09) «2—m, (c—0) TO 2 | Ag + 4Q1,n-2+ 6Q) n-3=Mn—1, Bo + 6Q)., n-2=™Mn- { We then derive (2) Qio=mM;3 Qu=mMm—am) 3 Qye=m,—am, + a?) —bmy 5 133= M3 — AMg + A?M4 — A?My — bm, + 2am ; Qiu=m,— ams + a?m,—a?m, + amy — b(m_— Zam, + 3a?my) + 07M $ Qis=m5— am, + a?m3— a? m2 + am — a’ nig— b(m3— 2am, + 3am, —4a?mo) + 67(m,—3am). We now assume [3 2p Qi ae igs Pak C1) ran (* e Me—2p-a, + (3) where lel is the integral part of a and we shall prove aw that this form holds for Q, «4. * Communicated by the Author. On the Partial Fraction Problem. 561 From (2) Q: 41 = M41 — AQ ee OQ e-1 [5] a = MNe+1— a > (—1)*b8 > (—1)°a* (é Me —28—o B=0 a=0 = k—-1—28 2 <9 S (—1)Po" & (1K ali he |e (4) Let « be even, then bos —28 a+ Qy ep1=Meqi-+ & S (—1) pe > (—1)sFart (4 8 ) rena B=0 a=0 ae K—2B—1 + eye ea toee? 2 (—1)%a (a) real (5) Designating the first double summation by s, and the second by se, then 5 K—2B+1 ee $;= > (—1,08 > (-1yar(**5 Me +1—28—a B=0 a—I ; K+1—28 me == > (—1)808 pa (—1)*a* ee i Mye41—33—a p=0 a =0 3 ag — > (—1) BULB fie i) Me+1—2B9 p=0 but the last summation is equal to m,+4,, since if B40, ep ("2 ) =. Therefore 5 K+1—28 ay) . ee LyPGe = (—1)*a ae *) tet tp-2— Met (6) Phil. Mag. S.6. Vol. 31. No. 186. June 1916. yA 562 Prof. I. J. Schwatt on the and 5 K+1—28 EN | so= % (—1)PUF & (ye ("Bey \o a B=1 a=0 . = k+1—28 es = > (—1)Pb* > (—ya(*8 ©) ict oagee B=0 a=0 K+1 ae | it se a ( i \ine nies a= The last summation is equal to m,.41, since if a0, (a) ut Hence 5 K+1—28 fe = S, (—1)#b8 = (-- yale *) Metta (7) B=0 a=0 Substituting (6) and (7) in (5) we have 3 K+1—28 = qT aia a6 nae Qi «+ Ri, (— 1)? = eee 6 ng Vesa. 5 K+1—28 ee | + > (— 1) 8b8 S) (—yra(*8 ) 19a Me (3=@ a=0 . kK+1—28 7 +6-—1 at+tB—-1 —Lom Soret] Neat PH Gs ke ( Js Z Me41—2B—a— Mp4}. ° ° (8) Now Ce) =a But if «<=8=0, (9) is equal to 2. Taking into account : , ‘ 1 — M4, and remembering that if « is even 5 = ome (8) becomes ae K+1—26 Oe = (—1)8)F ba (—1)*a (FOS) ete (10) which is of the same form as (3). Partial Fraction Problem. 563 By a similar method it can be shown that (3) holds also for odd values of x. It follows from (2) that Ay is formed in the same manner as Q: »-1 (if it existed). Therefore are = 2 n—1—28 \ SS 2 (— 1)8)8 > (— ilajpate es e gy Mn —2B—a—1 B=0 a=0 and | oad (aula) ees n—2—28 r a+ Bo=m,—b > (—1)8F & (—1)*a* 8 ) Mnp—283-a—2 B=0 a=0 n—4 Again, let > Qo,v”-4-* denote the quotient and A,x+ B, K=0 the remainder (which will, at most, be linear) obtained by n—2 dividing } Q, ,v”-?-* by 2? +aaz+ 6, then «k=0 n—2 n—4 > a as y Oe oii K=0 Kk=0 Aywt B, (2? + ae to)? (et fau+b)P-? i Core NV ae 2) where Q,, is the same function of Q,, that QQ. 18 of me. ae ; Clearing of fractions and equating coefficients of like powers of w, we have Qo = Qe —-AQae-1 — OQo ea (x=0) E 2, odds n—2), A= Qin-3— AQon-1— OQon- 55 ae (Le } By =Qin-2—OQon_4 We then obtain Qeo=m9; Qo =m,— 2am, ; Qog=my— 2am, + 3a7mg— Qhing ; (Qo3 =m3— 2am, + 3u?m, — 4a? my— 20(m, —3am) ; Qos = my — 2am; + 3a?m,— 4a?m, + 5atmy — 26(mo — 3am, + 4a?nw) + 3L? mo. We now assume K [5] x—2B \ au eve (EP) Sane CHEE mee ae shall now prove that this form is true for all values of «. yay 564 Prof. I. J. Schwatt on the From (13) follows Qe = Qien es —9(PEY Saye (BEET) ae 1)6)8 Ga Se 1° a y Me, open (19) or k+l a aA YS are “at(* 5" ym ae 1 ecpm(PP) SE Ag eye) [FI] K—1—2B a =, ee nyerigari (PTT) Seas oe (16) a=0 Let « be odd, then aS Qo i= iG LDP Gale le 1)"a iC 2 2 B+1 at) ain e eo + > ( = 1)F) ea = . re LtHq a B41 Ny, ogee =i +30 —nenyen (BETS tae Cc Ie ("ae My tesgaeed (Lt) Changing in the second double summation & into «—1, and 8 into 8B—1 in the third double summation, we have —_ x+1—28 ara Costs Qe = a6 1)PUP = (qe ( a) M+ 1—2B—% —1 xk+1—2p > a ic a+ ae ra Lue & (—1h Ce ( 1) Met 2p—e Karel) ei k+1—2B $3 (—190° ES (—1tat(* BP) Bmesiope - (18) p=0 Partial Fraction Problem. 565 Adding to the second double summation ad oa a = je a ey bt Sh (1) | ma 2 a=0 | we finally obtain _ k+1—2B QZ (— 1)8)8 = (—1)°u*nr,¢ +1—28—« lH") +(GECE)+ C49) Ctl | K+1—28 i i x Bp SoD nen ee is of the same form as as In a similar manner it can be shown that (14) holds also for even values of x. It follows from (13) that A, is formed in the same way as Qin—s (if it existed), therefore n—3 [S n—3—2B fete Sok i a = iL A,= —1)8}8 4) > —1)%a"( Jmeatone | = 2 Cp(PT) = pe (By )mcae and ay) Seine [Om or n—4 : 2 ] 4\ 2—4-28 z B+1 a alt@tB+l —b oe (—1)90( 1 ) = (—1) a ( Bat )tant-29-« | Continuing the process of division we arrive at Qe = Qe — AQ 1 — VQe 25 At=Qin-1-2¢— AQ 41 n-2-2¢ —VQu4in—3-2 P» (21) Be= Que AQee-1 — VQi 2 We now assume K [5] i S ( | aes B= 2 (— oe lar a —1)% ana : Mop 15 ee and Ae that this form holds ia Opa Qiry and Oya 566 Prof. I. J. Schwatt on the Substituting t—1, t—2, t—3, ..., 8, 2, 1 for ¢ in Q,. pel Qi -1, nti — AQ, e—UQ:, K—ls and adding the resulting equations, we obtain } if t Qi, c$1= M41 — AY Qy, e—F & Qy, e-1 wes) =) ee b)8( —a)*Mx-28- af s (B49 Ca a Seeseore tas spon (Fe ee ): Now Guage heref £ (#9 1)(e829-2) =(“4°) 3(* ous = (“ a x coefficient of «tT in 3 (Ltayrteers a = ) x coefficient of att! in [(14+.x)*t8t*— (14 a)2*F] ia eae ee +t eX on bles : eee = 2 —- b)8( — a)" Ger Cae Me +1—28—a en i ee ey (eee 1 sii If a=0 in the first double summation, (**3~*) = 0. The term corresponding to a=8=0 is m41. The upper limit of 8 can be written Ea . If B=0 is the second 0 = double summation, (* a i ‘| == \(). Partial Fraction Problem. 567 Therefore Qian SSA b)®(—a)* pales ){ ae oie eae Mr+1—2B—a+ CECE A (58) SS a abate ) al Tine | GBR Gas \S Now and Hence P= | >) | «+1—28 eee 3, = SO a yee us Therefore, if ¢ is fixed, the assumed form of Q,, is true for all values of «. We shall next prove that the assumed form holds also for Qy41, Ke We may write [E] 28 Qi41, c= a> (—8)(0)"(*5P)Q, «26 B=0a=0 or Qree= exe b)8(—a)" ae) a Seen _ aye, e(ytt—-1\(ot+yt+i-l fe ee, oe ( ai “) ( Hi yt 6 ) X Mg—2B—a-2y—6+ Letting z=x—2B—-X, LS] 26 amie) at hee | ee ig ( Pp ) ie : 2 3 (8a (vt Saat aig Jmrnt y—06=0 568 Prof. 1. J. Schwatt on the . and letting 8=~—2y—p, | oe 3 xe b)8( —a)<—?F- ore hy py ype (Vt -1)\ 9 ae ‘olen ( Das a . ( Y )( yti-l ) Ma Since ell [£] sededs we a y= an y=0 A=2y therefore Qi4i, k= RS (—b)B+¥( —a)«-28-2y B=0 =) r¢ K—28)—2 Pig Wa ON oe Ke eG es eee a Now eS (-o)- 7S ing sei art ae cae \ Naha ©; yt+t—1 mM aia a ne 2 = —a\- my i (45) > (—a)em Al ies 2B—r)\ A —2y — ), (*) =a(?-2-7), C8 TET )-o 2a = (-— ">" x coeiicient Of a qe'r se mie c) io ee “(gf ee and since Therefore Ls EI Re K—2P—2y Qc ye COIR” FOS = gee B=0 y=0 u=0 ytti—1\(«e—B-—yt+t—p)\_ (YEE (Ge) Partial Fraction Problem. 569 Next let 8+ y=’, then [SIL] k—2a = pea Ls fe Olle oe Pt | “ a les mu( t—1 \(“ a’ +t i k k—2a' mel ites = 8-1 Earn (EI NEE): Lu e (e'—6+t—1 7 it ) = coefficient Of 27 lin’ (Lt ae)" 8 ee 6=0 B=0 fe a) =e tls Geena B(— DOE (—ayem-n (APNE EH my =0 a therefore Letting BG anne be , then [s] —2a' 5 Interchanging a! and f! and dropping the prime signs, we have Be —28 a. > SS = Mee BM ta | tee B=0a=0 a which is of the same form as the expression for Qs, x We finally show that (22) hoids for Qe4ic4i- Now Qr4ictl= Qect1 — AQ 41 ie bQr41 feat [Ss] +1-28 | +t-1\"S Ey ibaa fol al ape mee (—1yee(P EE ) = (=1)%e rae )mr1-ap-e [<| k—2B = +f . aa t oe (op ) als a aan ) ra26- =e 2 bir reas +t —b > (—1eoe( ' ) S (—1)*u* 8 ) im. o1-26-« (23) lO On the Partial Fraction Problem. k+1 2S ys Saye ("S88 F9 (PED metal 1c eS (- LHe a CT ata Leet y, C 1)*a eC) (24) Let « be even, then xk+1—2B a af t@tPt+ti—l aero Quicr= 5 Gs ae ~ alan 1)*a eee ee | mies K xk+1—28 arias) P | ale Sco = (lye aoa Wee ) Medi-26—e Pe ee a (25 Geen ce B+t—1 t eh a2e as OD since the second double summation gives zero terms when a=(), and the third double summation gives zero terms when Adding the three double summations in (25) we have K+1 ie § IHS aye mn nl HET) RG HEE NED) CET IOD) as But the sum within the brackets equals (2+B+t)! (4+ B+t)!(B+t)!_ (at+B+t\(B+t Deak (sya a =( B+t di ).. Pints see: Therefore [=] per t Oe a= 2 S(epee(PP re Mp Ree (28) Geological Society. Fal In a similar manner can be shown that (22) holds for odd values of x. It follows from (21) that A, is formed in the same way as Qi+1n—1—21 (if it existed). Hence ea) n—1~2t—2B ) 2 B+t—-1 dra A;= = (—1)90°( yal SO ste a=0 6 oe ire Mn—21-28-a-Is | and — 2t—2B8 2 B+t—-1\""— B= > — 108 ( | ) Bi eo t om ) pth a ) ihn) Aeon a+B+t—l B+t—1 ) n—2e—28—« E= n—2—2t—2B —- Action, on the quantum of, 156. Airey (Dr. Weak. on Bessel and Neumann functions of equal order and argument, 520. Alpha particles, on long-range, from thorium, 379. Anderson (Prof. A.) on a method of measuring surface-tension and ancles of contact, 143; on an optical test for angles of contact, 285. Angles of contact, on a method of measuring, 143; on an optical test for, 285. Appaswamaiyar (S.) on discontinuous wave-motion, 47. Atomic structure, on the scattering of X-rays and, 222. Batya ero. H. C. C.). on light absorption and fluorescence, 417 ; on the ultra-violet absorption ays- tem of sulphur dioxide, 512. Barkla (Prof. C. G.) on the scatter- ing of X-rays and atomic structure, 922; ; on experiments to detect re- fraction of X-rays, 257. Beam, on the strength of the thin- plate, 348. 3essel and Neumann functions of equal order and argument, on, 520. Beta rays, on the energy of the secondary, produced by partly absorbed gamma rays, 440. Biggs (H. KF. .) on the energy of the secondary beta rays produced by partly absorbed gamma rays, 430. Bilham (E. G.) on a comparison of the are and spark spectra of nickel produced under pressure, 163, Bismuth, on the high-frequency spectrum of, 403. Books, new :—Pierpont’s Functions ofa Complex Variable, 88; Briggs & Bryan’s Tutorial Algebra, 406 ; Thompson’s A New Analysis of Piane Geometry, 407; Couturat’s Algebra of Logic, 407 , Findlay’s Prac ical Physical Chemistry, 408 ; Parker’s Elements of Optics, es: al {OS ae f 408 Carslaw’s, PINS EAsicoh: see ate A Colwell & Russ’ Radium, X-rays, and the Living Cell, 410 ; More’s The Limita- tions of Science, 410; Pearson’s Tables for Statisticians and Bio- metricians, 493. Boswell (P. G. H.) on the lower eocene deposits of the London basin, 411. Bowen (J. E.) on a method of measuring surface-tension and angles of contact, 143; on an optical test for angles of contact, 285. Bragg (Prof. W. H.) on the structure of the spinel group of crystals, 88. Brodetsky (Dr. 8S.) on the absorption of gases in vacuum-tubes, 478. Butterworth (S.) on a method for deriving mutual- and self-induct- ance series, 276, 496; on the coefficients of mutual induction of eccentric coils, 443. Channels, on the establishment. of turbulent flow in, 322. Child (C. D.) on the production of light by the recombination of ions, 139. Coils, on the coefficients of mutual induction of eccentric, 443. Cole (Prof. G. A. J.) on a composite gneiss near Barna, 414. Conductor, on the wave-length of the electrical vibration associated with a thin straight terminated, 96. Cooling of cylinders in a stream of air, on the, 118. Cosecants, on the sum of a series of, at: Crystal size, on the electrical and magnetic properties of pure iron in relation to the, 357. Crystals, on the structure of silver, 33; on the structure of the spinel group of, 88. Curl in a vector field, on divergence and, 528. 574 Currents, on the positive emission, from hot platinum, 497. Cylinders, on the cooling of, in a stream of air, 118; on the elec- trical capacity of approximate, 77; : Deeley (H. M.) on the theory of the winds, 399. Dempster (A. J.) on the ionization and dissociation of hydrogen mole- cules, 458. Dewey (H.) on the origin of some river- gorges in Cornwall and Devon, 571. Divergence and curl in a vector field, on, 628. Dunlop (Miss J. G.) on the scatter- ing of X-rays and atomic structure, 222. Electrical capacity, on the, of ap- proximate spheres and cylinders, IEC properties of pure iron in rela- tion to crystal size, on the, 357. vibration associated with a thin. straight terminated con- ductor, on the wave-length of the, 96. Electron theory, on magnetic rotary dispersion in relation to the, 232, 454. Electrons, on the velocities of the, emitted in the normal and selec- tive photo-electric effects, 100; on free, in gases, 186; on the determination of the work func- tion when, escape from the surface of a hot body, 197. Evans (KE. J.) on the absorption spectra of the vapours of inorganic salts, 55. Expansion of a function, note on the, 490. Ferguson (Dr. A.) on the variation of surface-tension with tempera- ture, 37. Flicker photometer, on the theory of the, 290. Flow, on two-dimentional fields of, 190; on the establishment of tur- bulent, in pipes and channels, 322. Fluorescence, on light absorption and, 417. Frequency distributions, on the formula for testing the goodness of fit of, 369. INDEX. Friman (E.) on the high-frequency spectra of the elements gold- uranium, 403. Function, note on the expansion of a, 490. Gamma, rays, on the energy of the secondary beta rays produced by partly absorbed, 430, Garrett (Mr. C.8.) on the infra-red and ultra-violet absorption of sulphur dioxide, 505, 512. Gases, on the relation between the thermal conductivity and the viscosity of, 52; on free electrons in, 186; on the absorption of, in vacuum-tubes, 478. Geological Society, proceedings of the, 411, 495, 571. Gold, on the high-frequency spectrum of, 403. Greenhill (Sir G.), skating on thin ice, 1, Haines (W. 8B.) on ionic mobilities in hydrogen, 339. Hall effect, on the, 367. Harvey (Miss E. J.) on an applica- tion of nomography to a case of discontinuous motion of a liquid, 130. Hicks (Prof. W. M.) on a notation for Zeeman patterns, 171. Hodgson (Dr. B.) on the absorption of gases in vacuum-tubes, 478. Hot body, on the determination of the work function when an elec- tron escapes from the surface of a, ISVs i Hughes (Prof. A. Ll.) on the veloci- ties of the electrons emitted in the normal and selective photo-electric effects, 100. Hughes (J. A.) on the cooling of cevlinders in a stream of air, 118, Hydrogen, on ionic mobilities in, 339, molecules, on the ionization and dissociation of, 438. Inductance series, on a method for deriving mutual- and self-, 276, 496. Induction, on the coefficients of mutual, of eccentric coils, 443. Inorzanic salts, on the absorption spectra of the vapours of, 55. Tonic mobilities in hydrogen, on, 339. INDEX. Tons, on the production of light by the recombination of, 139. Iron, on the electrical and magnetic properties of pure, in relation to _ erystal size, 357. Ives (Dr. H. E.) on the theory of the flicker photometer, 290. Jones (W. M.) on the most effective primary capacity for Tesla coils, 62. Jones (W. R.) on the origin of the tin-ore deposits of the Kinta district, 495. Kam (J.) on Van der Waals’ equa- tion, 22. Karapetoff (Prof. V.) on divergenco and curl in a vector field, 528. King (Prof. L. V.) on the establish- ment of turbulent flow in pipes and channels, 322. Kingsbury (E. F.) on the theory of the flicker photometer, 290. Lamb (Prof. H.) on waves due to a travelling disturbance, 386, 5389. Lankester (Sir E. Ray), obituary notice of H. G. J. Moseley, 173. Laws (B. C.) on the strength of the thin-plate beam, held at its ends and subject to a uniformly distri- buted load, 348. Lead, on the high-frequency spec- trum of, 403. Leathem (Dr. J. G.) on two-dimen- sional fields of flow, with logar- ithmic singularities and free boun- daries, 190. Lester (Dr. H.), on the determination of the work function when an electron escapes from the surface of a hot body 197 ; on the variation of thermionic currents with po- tentials, 549. Light, on the production of, by the recombination of ions, 139. absorption and fluorescence, on, A417. Liquid, on an application of nomo- graphy to a case of discontinuous motion of a, 1380. Liquids, on the constitution of the surface layers of, 260, Magnetic properties of pure iron, on the, 357. rotary dispersion in relation to the electron theory, on, 232, 454. Mennell (I. P.) on the rocks of the Lyd Valley, 416. 575 Mercury, on the high-frequency spectrum of, 403. Morton (Prof. W. B.) on an appli- cation of nomography to a case of discontinuous motion of a liquid, 130. Moseley (Henry Gwyn Jeffreys), obituary notice of, 173. Neumann functions, on Bessel and, of equal order and argument, 520. Nickel, on the are and spark spectra of, 163. Nomography, on an application of, to a case of discontinuous motion of a liquid, 130. Partial fraction problem, note on the, 108, 560. Pearson (Prof. K.) on the funda- mental formula for testing the goodness of fit of frequency distri- butions, 369. ' Photo-electric effects, on the velo- cities of the electrons emitted in the normal and selective. 100. emission, on the complete, 149. Photometer, on the theory of the flicker, 290. Pipes, on the establishment of tur- bulent flow in, 322. Platinum, on the positive emission currents from hot, 497. Pollock (Prof. J. A.) on the relation between the thermal conductivity and the viscosity of gases, 52; on the wave-length of the electrical vibration associated with a thin straight terminated conductor, 96. Quantum of action, on the, 156. Radioactivity, on the variation of the, of the hot springs at Tuwa, 401. Raman (Prof. C. V.) on discontinuous wave-motion, 47. Rastall (R. H.) on the granitic rocks of the Lake District, 415. Rayleigh (Lord) on the propagation of sound in narrow tubes of va- riable section, 89; on the electrical capacity of approximate spheres and cylinders, 177. Reynolds (Prof. S. H.) on the igneous rocks of the Bristol district, 414. Richardson (Prof. O. W.) on the complete photo-electric emission, 149; on the variation of the posi- tive emission currents from hot platinum with the applied potential difference, 497. 516 Richardson (S. 8.) on magnetic rotary dispersion in relation to the electron theory, 232, 454. Rutherford (Sir E.) on long-range alpha particles from thorium, 379, Schwatt (Prof. I. J.) on the sum- mation of certain types of series, 75; on the partial fraction pro- blem, 108, 560; on the expansion of a function, 490. Series, methods for.the summation of certain types of, 75. Sheard (Pref. C.) on the variation of the positive emission currents from hot platinum with the applied potential difference, 497. | Shorter (Dr. 8. A.) on the consti- tution of the surface layers of liquids, 260. . Siegbahn (Dr. M.) on the high- frequency spectra of the elements gold-uranium, 403. Silver crystals, on the structure of, 83 Smith (A. W.) on the Hall effect, 367. Sound, on the propagation of, in narrow tubes of variable section, 89. Spectra, on the absorption, of the vapours of inorganic salts, 55; on the arc and spark, of nickel, 163; on the high-frequency, of the ele- ments gold-uranium, 403. Spheres, on the electrical capacity of approximate, 177. Spinel group of crystals, on the structure of the, 88. Springs, on the radioactivity of hot, 401. Steichen (Prof. A.) on the radio- activity of the hot springs at Tuwa, 401. Sulphur dioxide, on the infra-red and ultra-violet absorption of, 505, 512. Summation of certain types of series, methods for the, 75. Surface layers of liquids, on the con- stitution of the, 260. Surface-tension, on the variation of, with temperature, 37 ; ona method of measuring, 143. Thallium, on the high-frequency spectrum of, 403. INDEX. Thermal! conductivity, on the relation between the, and the viscosity of gases, 52, Thermionic currents, variation of, with potentials, 549. Thin-plate beam, on the strength of the, 348. Thompson (F. C.) on the electrical and magnetic properties of pure iron in relation to the crystal size, 357. Thorium, on long-range alpha par- ticles from, 379; on the high- frequency spectrum of, 403. Travelling disturbance, on waves due to a, 386, 539. Tryhorn (F. G.) on light absorption and fluorescence, 417. Turbulent flow, on the establishment of, in pipes and channels, 322. Uranium, on the high-frequency spectrum of, 403. Vacuum-tubes, on the absorption of gases in, 478. Van der Waals’ equation, on, 22. Vegard (Dr. L.) on the structure of silver crystals, 83. Viscosity, on the relation between the thermal conductivity and, of gases, 02. Warren (S. 1.) on the late glacial stage of the Lea valley, 413. Watson (Prof. G. N.) on the sum of a series of cosecants, 111. Wave-length of the electrical vzbra- tion associated with a thin straight terminated conductor, on the, 96. Wave-motion, on discontinuous, 47. Waves, on the propagation of, in water, 1; on, due to a travelling disturbance, 386, 539. Wellisch (Prof. E. M.) on free elec- trons in gases, 186. Wilcockson (W. H.) on the granitic rocks of the Lake District, 415. Wilson (Dr. W.) on the quantum of action, 156. Winds, on the theory of the, 399. Wood (A. B.) on long-range alpha particles from thorium, 379. X-rays, on the scattering of, and atomic structure, 222; on expe- riments to detect refraction of, 257. Zeeman patterns, on a notation for, 171. END OF THE THIRTY-FIRST VOLUME. Printed by TayLor and Francis, Red Lion Court, Fleet Street. SMITHSONIAN INSTITUTION LIBRARIES wut il ——