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Ky ia i892 I MILT HS 4 us eal DeSMieEs ota iEAg YT | aes gt} ‘ pride "9 REIT = SS nee sarrmt a Sake Ye st Sate ae wie he, ee See = Saale SAN * :3 = te SSeS Soa ee == ——e iam fi et aw SS Te 23) Sh Pert reetontth Wien te YY ata bea at ich ba Banter bap , eal sins 554 OS QIQKINARIQIQMIOPIQONIQIOEMOIRII iy) (a) x4] 2) 9 Us 0 SCIENTIFIC LIBRARY bY x TT I Bo 9) (a) ow a 9 o xs ose 7) (ay “J cS iw) (a) ) se iy) aa Xe o) : “ a oe ry) (ay 0 & 9 (a) rw ne iy) aS as Ss 9) a) is ox gz 3 we cs LY, Q “o oy | aT Be 7 iS 7) GC Oe) = rc r Q A UNTTED STATES PATENT QFFICE & g % 3 ce % © S % 2 ms iy oS a OS bY s) SMa AM MMMrmMmMawMweA ' GOVERNMENT PRESTING OFFI0E 11—8625 THE LONDON, EDINBURGH, anp DUBLIN PHILOSOPHIGAL MAGAZINE AND / JOURNAL OE SCIENCE. CONDUCTED BY Sik OLIVER JOSEPH LODGE, D.S8c., LL.D., F.R.S. SIR JOSEPH JOHN THOMSON, O.M., M.A., 8c.D., LL.D., F.R.S. JOHN JOLY, M.A., D.Sc., F.R.S., F.G.S. AND WILLIAM FRANCIS, F.1.S. “Nec aranearum sane textus ideo melior quia ex se fila gignunt, nec noster vilior quia ex aliénis libamus ut apes.” Just. I.1ps. Polit. lib. i. cap. 1. Not. VOL. XXXIX.—SIXTH SERIES. JANUARY—JUNE 1920. 5204 LONDON: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. SOLD BY SMITII AND SON, GLASGOW ;— HODGES, FIGGIS, AND €O. DUBLIN;— AND VEUVE J. BOYVEAU, PARIS. “Meditationis est perscrutari, occulta; contemplationis est admirari perspicua .... Admiratio generat questionem, questio investigationem, investigatio inventionem.”—Hugo de S. Victore. “Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, Cur caput obscura Phebus 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 \/ FLAMMAM, CONTENTS OF VOL. XXXIX. (SIX'TH SERLES). NUMBER CCXXIX.—JANUARY 1920. Capt. A. C. Egerton on the Determination of Chemical MPSEM MENU SacI oo ch econ ool) ea ot sora e Haat ae via, Sl Soe oy ae Prof. F. A. Lindemann on the Significance of the Chemical Constant and its Relation to the Behaviour of Gases at Mayen WIPE ALCS) fos he) ote ets aol ecn eye) os dies =, dias sre o lajie ook Prof. W. C. McC. Lewis on an Unsolved Problem in the Application of the Quantum Theory to Chemical Reactions. Mr. D. L. Hammick on Latent Heat and Surface Energy.— SPEER IOLA sh ga ae sr he A ee Sere, ORR Dr. L. Silberstein: Contribution to the Quantum Theory of Spectruin Emission: Spectra of Atomic Systems containing Peoples NCES). oh aioe ss seals Wet cade cieca Major W. T. David on the Calculation of Radiation emitted in Gaseous Explosions from the Pressure-Time Curves Major W. T. David: An Analysis of the Radiation emitted MBCLASCOUS TUXPlOSIONS hc. uy eos << owmees se he gee ea Pe Mr. Loyd A. Jones on a Method and Instrument for the Measurement of the Visibility of Objects .............. Mr. Satyendra Ray on the Equivalent Shell of a Circular CTUBTEIE TF a 2 ose ely Saget Serie Bac etd GORA Rr mae A Prof. C. V. Raman and Mr. Ashutosh Dey on the Sounds of Splasitess sCibeMl ns antes ec Se ot ee Fes ee te Prof. A. Anderson on Fresnel’s Convection Coefficient Mr. F. F. Renwick on the Fundamental Law for the true Ehotographic Rendering of Contrasts. .....-....-..... Prof. I. W. D. Hackh: A Table of the Radioactive Elements much indicates thelr Structure... 4.2.0 se. on oe ees Prof. W. B. Morton and Mr. T. C. Tobin on the Construction of a Parabolic Trajectory and a Property of the Parabola PSCR Ve MECMIECU OS on Mn aiWte Le votes fo 4a thea. « Page 46 1V CONTENTS OF VOL. XXXIX.—SIXTH SERIES. NUMBER CCXXX.—FEBRUARY. Page Dr. I. Silberstein on the recent Eclipse Results and Stokes- Planck's, Atther : «4 0. hie eee ames okt. . en Sir Oliver Lodge on a Possible Structure for the Ether...... Prof. A. Anderson on the Spheroidal Hlectron............. Dr. Norman Campbell on the Adjustment of Observations. I. Dr. J. Prescott on the Buckling of Deep Beams. (Second Paper.) With an Appendix by Mr. H. Carrington...... Miss Alice Everett on a Simple Property of a Refracted Ray. The late Lord Rayleigh on Resonant Reflexion of Sound from a Perforated Walls 70.2... a Dr. D. N. Mallik and Prof. A. B. Das on the Quantum Theory of Hlectric Wisehamee” 2. 45 2... ee Dr. G. Rudorf on Latent Heat and Surface Energy ........ Notices respecting New Books :— J.de Graaf Hunter’s The Earth’s Axes and Triangulation. NUMBER CCXXXI.—MARCH. Mr. G. A. Hemsalech on the Excitation of the Spectra of 175 177 194 223 239 Carbon, Titanium, and Vanadium by Thermelectroni¢ ~ Currents. With special Reference to the Cause of Emis- sion of Light Radiations by Luminous Vapours in the Carbon Tube Resistance Furnace. (Plates II.-V.)...... 24] Mr. E. T. Bell on Parametric Solutions for a Fundamental . - Equation in the general Theory of Relativity, with a Note on-similar Equations in Dynamics’. ..-...... 5 sepa Mr. J. Tykocinski-Tykociner on the Mandelstam Method of Absolute Measurement of Frequency of Electrical Oscillations : /,:.¢ eee oo esse i ee Mr. L. C. Jackson on Variably Coupled Vibrations: Gravity- Elastic Combinations. Masses and Periods equal. (Plates WL & VILL.) 00h Se ee es cae te Mr. Také Soné on the Magnetic Susceptibilities of Hydrogen ana some other Gases. a .gu0.9 ic). 6 4 2 Mr. Gilbert Cook: An Experimental Determination of the inertia of a Sphere moving ina Elid .. 2.02... eae Mr. F. Bateson a New Cadmium-Vapour Arc Lamp ...... Mr. S. P. Owen on Radiation from a Cylindrical Wall...... Dr. L. Silberstein on the Measurement ‘of Time—a Hejoinder fowDr-cN: Capaploelly, cc cine. ev os dag et 285 289 CONTENTS OF VOL, XXXIX.—SIXTH SERIES. Vv Page Prof. J. Joly and Mr. J. H. J. Poole: An attempt to deter- mine if Common Lead could be separated into Isotopes by Centrifuging in the Liquid State .................-. 372 Prof. J. Joly and Mr. J. H. J. Poole onthe Effect of Centri- fuging certain Alloys while in the Liquid State.......... 376 Notices respecting New Books :— J. H. Jeans’s Problems of Cosmogony and Stellar RTS Fie cash Gotz ho! yok Saidipeabedar shige ghana es eats 377 Prof. R. W. Wood’s Researches in Physical Optics. Part II. Resonance Radiation and Resonance Spectra. 379 Profs. L. C. Karpinski, H. Y. Benedict, and J. W. Calhoun:s nied Mathematics, 2.24.2. 045%. (\st- 5 380 Prot. H. Lamb’s An Elementary Course of Infinitesimal I) Um SPR ese ot Pet. cobs ot «sid ial Gamaeia es Mleseart 381 Proceedings of the Geological Society :— Mr. H. Hamshaw Thomas on some Features in the Topography and Geological History of Palestine .... 383 NUMBER CCXXXII.—APRIL. Prof. W. E. Adeney and Mr. H. G. Becker on the Determi- nation of the Rate of Solution of Atmospheric Nitrogen at Oxyoeniby Water.—Part I... ug .e degen 3 385 Dr. A. D. Fokker on the Contributions to the Electric Current from the Polarization and Magnetization Electrons....... 404 Prof. J. Kunz: An Elementary Theory of the Scattering of tients by.small Dielectric Spheres... .t-s.eds bt ea 416 Prof. R. W. Wood on Light Scattering by Air and the Blue MOET Or Ge Skye en yall Mek ets ye lg Oh ve Pay ered eae) 423 Prof. F. Slate: An Alternative View of Relativity ........ 433 Mr. H.T. Flint: Applications of Quaternions to the Theory 2 TSCLULCIN TINS cake teak tere tae ROA ARIA eet IRN Ne er area Os 439 Dr. F. W. Aston on the Constitution of Atmospheric Neon. Maes Vb a UX) ee eee BAI SRA aK ee Nn 449 Intelligence and Miscellaneous Articles :— On the Equation of State, by Megh Nad Saha and Sabyendr Nath Basu 26. .h es ce eee heap elle cepa ne 456 NUMBER CCXXXITI.—MAY. Prof. W. M. Hicks on the Speetpum of, Copper ) = 2: loo PPS | log p =—pa7pyt?? bel + pay | a —0:0583aT324+C, . (2) where ihe term” =] =—8R. 5 i oo In (8/2 — 1) —In (-62T — 1) |. ® The values of —(O2= "07 ana {Ue Ue) soe aicror e values o -|-5= rT ae ‘| or different values of Bv/T are given in Pollitzer’s ‘ Die Berechnung Chemischer A ffiniiaten ’ 1912, and the corrections to be introduced due to use of Debye’s equation instead of the Nernst-Lindemann equation are to be found in ‘S¢tzber. Akad. Wiss. li. p 1176 (1912). 4 Capt. A. C. Egerton on the For the determination of the chemical constants, therefore, By, a, and two values of p must be known or one value of p and ne value of Ao. A) can be obtained from Ay and the molecular heat v Aap according to Ane roth C,dt— Cpdt, or :— Ap) e 0 nee entre ei Ao=Ae— 4903 T+ 3/2RT | oy + a 9/5 aT eat eT [a + O4e?, CC ne Unless Ap can be measured with accuracy directly, it has to be obtained from two or more measurements of the vapour sressure. Values of Ap can be obtained at various tempe- }) ratures from the vapour pressure by the Clausius-Clapeyron relation, and an average value of X, can be so derived with fair accuracy. In dealing with a liquid, the latent heat of o fusion (fm) must be added to the molecular heat (@,) serma,, the latter would be integrated to the melting-point according to the formula for the solid, and thereafter measurements Le the change of atomic heat of the liquid (¢,) must be obtained and integrated to the desired temperature at which the vapour pressure is measured. Iniscussion of Errors. On differentiating formula (2), dC No : 2°) a ; 1 a Roo IT/2), Fan, Soe rTM, es aon 7a 1 +32 (0-0583aT!?),,. if Ay and By are known with accuracy, it is clear the higher the temperature the less the error in the determination of the constant C, provided the gas laws are applicable to the vapour. On the other hand, as it may more often be convenient to eliminate Aj, and determine © from two values of p at two temperatures T, and T,, it will become of great importance to choose the two temperatures as widely separated as Le ag otherwise the error may become great. Starting with two equations of the above form, the subtraction would lead to: INS OS Alle the = ) =<, log p,—2°5 2% nt T, 7, los Pr ot 2 Po ae Ty log en l* = =| Ke i i a log ee aa oa Ts eee t-T, | : Determination of Chemical Constants. 5) and if C be differentiated with respect to T,: dC ie log? Ps 9-5 {a= log ta} at, (Ty rant os A : Paina oa ao al ea + cpr mot tO O84[ ye ! From which it is clear, that in order to reduce the error a 7 —'I, must be as large as. possible and T, as small Semible. Neglecting the last two terms, because AS A.) is small and a is a very small constant, albert 10-°, it is seen that, if T, and T, are separated by about 50° nd T, is about 250° 1 p< dU ° and log about 2, 7 will be about 0°2, and so, supposing the constant to have a value of about 1°35, the error may be 40 per cent. for a one per cent. variation in Tj. Thus the aceuracy with which the vapour pressure must be measured, particularly regarding the fixing of the correct temperature, is very important, if true values of the constants are to be obtained. Chemical Constant of Mercury. These considerations will now be applied to the deter- mination of the chemical constant of mercury. ‘There appear to be no other cases in which the vapour pressure and specific heat of a monatomic substance have been measured with the same accuracy. A number of determinations of the vapour pressure of mercury have been recorded. Those of Hertz, Ramsay and Young, Callender and Griffiths, Pfaundler, Morley, Gebhardt, Cailletet, Colardeau, and Rivicre have been discussed by Laby (Phil. Mag. iNew 1908), who concluded that the following formule summarized the work of these investigators, and gave the closest approximation to. the true vapour pressures :— From 15° to 270° G., 302; 932 log p = 15°24431 —2°367233 log 7a : and from 270° to 450° (Om D4! log p = 10°04087 — 0°7020537 log gas ee In 1909, Knudsen (Ann. d. Phys. (4) xxix. p. 179, 1909) 6 Capt. A. C. Egerton on the published a series of accurate measurements at relatively low pressures and discussed his results in connexion with those of Hertz, Ramsay and Young, Pfaundler and others. He finally arrives at the formula which agrees with that of Hertz in the value of the 2nd term, and holds up to 2(07,Ce | 3342°26 On) = — gi} | Smith and Menzies (J. Am. Chem. Soc. xxxii. p. 1434, 1910) have carried out accurate measurements of the vapour pressure of mercury from 250°C. upwards and arrive at the expression : log p = 9°9073436 —0°6519904 log T —- A On 623 x The value for the boiling-point obtained from these vapour- pressure measurements 356°95+0°1C. agrees very closely with that obtained by Callender & Griffiths 357:05 ; whereas Knudsen’s formula leads te a boiling-point 855°1C. The following table gives a few points calculated from these formule :— : Awe, 1 T° abs. Laby. | Knudsen. - Sinith & Menzies. 630 B.Pt. 760-4 Gi) | 787 | 760 473 17S) 17-41 | 17-22 373 | 0-276 (1) O-2715 02794 OT 3-4 0-00016 (1) 00001846 ~~: 00002073 234-2 M.Pt. a 0000001970 0000002354 Above 250° C. the values obtained by Smith and Menzies are probably correct within 0°2 per cent. ; between 250° and 150° C. the data somewhat conflict, and Phe nearest value probably lies between that of Knudsen and that of Smith and Menzies, while at the lower temperature it seems probable (taking into consideration also Hertz and Pfaundler’s results) that Knudsen’s figures are within at least 0°5 per cent. The extrapolation to the melting-point is reasonable, as there is no sufficient change in the specific heat of thie liquid between 0° C. and the melting-point as would lead to an appreciable change in the value of the constant of the formula. Determination of Chemical Constants. (i Haber and Kerschbaum (Zeit. Elektrochem. xx. 1914) have determined the vapour pressure of mercury at 20° ©. by the quartz-fibre manometer and obtain a value 0°:00126. At this temperature, Knudsen’s formula would give the vapour pressure as 0°00119. As Haber’s figure is the weighted mean of only two series of determinations (with values 0:00115 and 000130 respectively) it can hardly be asserted that Knudsen’s values are too low, but it perhaps may be said that the true values are not lower than those given by the Knudsen formula. Pe RT he Hie and above 250° (.: log p = 9°9073436 — 06519904 log T= ATO? and below 150° C.: log p =10°5724—0-847 log Tae ee differentiating and multiplying by RT’, above 250°C. : Nato Au-oo— 12924.) and below 140° C. : At = 15277°46—1°6812 T. Now, Ap=Ao+ (Coc) TY, if G, and O,’ are constant. The following graph gives the variation of ¢, with tempe- rature as determined by Naccari (J. de Phys. (2) vill. p. 612, 1889), by Barnes and Cooke (Phys. Rev. xvi. p. 65, 1903), and others. The mean value at the lower temperatures is about 0°0331, so that Ap=A,—(6°64—4:°96) T=A,—-1°68T, which agrees with the constant derived from the Knudsen vapour-pressure formula. Although the value of Cy decreases (to 140°C. according to Barnes), it does not do so sufficiently rayidly to account for the value 1°2942 in the equation for Ap obtained from the Smith and Menzies formula, and it seems pro- bable therefore that ci can no longer be considered nearly constant. ; PA ARE SEER HOE ORT SN a te gy a TP en en ee ae A CN ENTS 8 Capt. A. C. Egerton on the Fig. 1. aba — rT Spacific Heat of fAercury. Naccari © (extrapotsted) Koref x (mean values) Polhézer ® wh _- Atonuc Heat ee LS Barnes and Cooke ® Murbatof€ 2 19 to 234C....7-4 5 Feterson and Hedelius © ‘ WINKLEMANN 23=0-0000069 per “C. SB, Q Neo fis —— d KurbatorPr | ! | ( a ese s ao) LER: -- {00 —59 pt ‘s) 50 faze) 150 200 250 (om The following table is calculated from the above expressions for Xp :— Tasue II. T, Smith & Menzies. Knudsen. 630 eee 14161 14218 ATS | 0, unin 14364 14482 Pat ES aan We 14623 14818 2342 satan 14673 14884 The molecular latent heat at the boiling-point has been measured directly by Kurbatoff (Zeit. Physik. Chem. xliii. »p. 104, 1903), who found the value 13600 cals. This value is no doubt less accurate than that obtained as above. Konowalott (Zeit. Phys. Chem. xxxix. 1887) also quotes Regnault, who obtained the value 15500; Traube (Zeit. Chem. xxxiv. p. 423, 1903) assumes the mean value 14540 cals. at 360° C. Determination of Chemical Constants. 9 Now, 7 fe i 71 , l 1 Ae = Not { Ca. -{ Ge dt—fin _| Cai: where C, = atomic heat of vapour, Co =e - solid, Cy sr SP) 2” liquid, Jm = latent heat of fusion at melting-point. Whence Ss | n LOE ras > rs /é Ao = Ap—4:963T + ] 7 =] eee +2a1°" Un- ate where [| ean be obtained from the tables of the integration of Debye’s formula for c (Berl. Ber. lii. p. 1176, 1912), a is the empirical constant for conversion of ¢, to c,, and Ch is considered to be constant. In the case of mercury, from specific heat measurements (Moref,, Ann. d. Phys. xxxyi. p. 64, 1911; and Pollitzer, Meer: Chen. xix. 19N2) Gp= 97 3 2 Bv/T,=0°4142. and a Dias 5:083. lin = 555 (Pollitzer) 549 (Bridgman, Proc. Am. Acad. xlvii. Pung Wacoleg ebrsegne) LOM Mean = 552 *a = 21.10~-° (from Pollitzer’s measurements of specific heat). Thus AC =AG—4:963 T+ - Se al 905-2 552-2 05 —AK—A" 963 T+¢,( —T,) +1813. * As the coefficient of expansion of the solid =0:00014 at the melting-point (Bridgman, /.¢.), and vmn=14'90 (density 15:46, Bridgman, SOR : th 2 : =21.10~°T°”, the compressibility of the solid (x)s would be 2:17x10~° per ke./em.* (which lies between the values 1:66.107° for zinc and 2°3.10~° for cadmium). The compressibility of the liquid at 0° C. is 3:90.10~° per kg./cm.”, and the coefficient of expansion is 000018 and v=14°71, whence a (liquid) would become 17°35.10-°. C—C! las been determined at 0°C. by Bridgman =0°805, whence @ would be 17°85, 107’. l.c.), and. as 10 Capt. A. C. Egerton on the Between —35°6 and —3°4C. Koref found the mean c,=6" 68*; while between 0° and 200°C. the mean of Barnes and Cooke’s and Nacecari’s results Q1VE Cp ? 6°58. From the values for Ay given in Table IT. ae following values of A, can be ninco bad a dissin) JOULE lt Smith & Menzies. Knudsen. GSOR AA cee les 15452 15509 AT sare Nios swt 15402 15520 DTM coussicee ase 15341 15536 234-2 ..... hag otee 15324 15535 Above 250°, the value of ¢, is probably not constant ; the values obtained from Appz will be neglected. The values obtained from the Smith and Menzies formula vary more amongst each other than those of Knudsen, and therefore this formula probably does not express the true vapour pressure at the lower temperatures as wel] as the latter. The mean of these values, viz. 15530+5, obtained from the Knudsen formula will therefore be taken as most closely approaching the value for the latent heat of vapori- zation at absolute zero. At the melting-point :— f pe ro Oya i om IL Bet 2 aren oe 1. +o57 | T., r 3/2 1 757 A e En where ae Xr == Seuee and 92°) loo i 5924, An- Ay wer 79 1 Am— Ao Blo dar f Pei ue $133 and P51 f jeer = QC). . Boe: eae ik ae OL LO eae ESBS I 12/2 a 1( und 15° 571°" les 0:0439. p = 0000001970 mm. and log p= —5'7055 mm. = —§°986 atmos. * This value is slightly low in comparison with that of other observers (see gr aph), and the aver age variation (Winkelmann) 0-0000069 per ° C. would lead to a value about 6: 73, t. e. Ap—273 = 15538. Determination of Chemical Constauts. li Knudsen gives the probable error oi the values obtained from his vapour-; ressure formula as 0°002 p between 0° C. and 15° C., 0 007 up to 100° C. and 0-012 up to 200° C. . The measurement at 0° C. is considered accurate to less than one part in one thousand. ‘The temperature error was less than 0°-15C., and the value obtained is in all probability not subject to any serious constant error. A confirmation of this figure for the vapour pressure at 0° C. by mea- surement with great precision would be valuable, as the chemical constant could then be obtained with certainty. If the values at low temperatures are correct within +0°002 p, and if the variation in specific heat is allowed for, then the value of the vapour pressure at the melting- point may vary between 2°00 x 10~° and 1:°95x10-*% The latent heat of vaporization (Xo) has been shown to be con- sistent within one part in 3000; the variations in vapour pressure and specific heat might lead to a_ result for No —= 19930 + 15. The value of @v may vary within one per cent., so that the value of the specific-heat ‘term may be 1:779 +0: 01. The melting-point is certainly accurate within O%1 C. Henee, inserting in the formula the values obtained : +14 50740°015 + 1°823+0°01 16°330+0°025 — 5'924+0:002 8°586+0°005 — 14°510+0-:007 —14°510+0°007 C = 1:820+0:032 A al cS Now 1:5log M=3'453, and if 1:5 log M—C=—C), a universal constant, then the value of this constant would be = C633 0.032: The value calculated from the constants h and-k becomes p= —1608 ; so that the value determined is in complete roreement within the error * (see p. 19). Now if C,=1°608 the chemical constant for cadmium would be 3:076 —1:608 =1°:468, and that for zine would be & * T find a calculation on similar lines to the above has already been published by Nernst, in his book ‘ Die Theoretischen und Experi- mentellen Grundlagen des Neuen Warmesatzes’ (1918), which leads to a value C=1°83+0°03. 12 Capt. A. ©. Egerton on the 2°723—1:°608=1:115. Ina previous communication (Phil. Mag. xxxili. p. 193, 1917) results were given of determinations at low pressures of the vapour pressures of solid cadmium and zinc. Although these results were not carried out with the precision necessary to obtain an accurate value for the chemical constants, it seems worth while to inquire whether the constants, deduced above, lie within the limiting values of the constants obtained from the observed results, taking into consideration the experimental errors. Chemical Constant of Cadmium. From the mean of the most reliable observations, the following two points are chosen, which lie on the smoothed curve of the vapour pressure : T,=471'8+1° abs., log p= —3°54(6)40°02.. T,=532'5+1° abs., log p= —2°11(8)+0°02. From Lindemann’s melting-point formula : y=31.10"4/—%,, where P22 5942 abs! A124 v= T3°0): a= (i corn eal ie and.» 8y=14775 Bv TAT i A,—A, Ot —. «=== aus —=—()° Zz Nae pet = — . Q9 Taq pes 202 [ va 9°5 Bl AT ae =e. 10-23 B SoS || ip = rn : 7 —— i 0 De 9, ind T, ~ 5325 027700; 1, d if ONL THQ /9 9e?vT a 2 q Writing alT??=—_, the value of a can be estimated for (say) 291°C, «=2°25.10-° (mean of Gruneisen’s and Richard’s results), a =28°8.10-® (Matthiessen 1866) at 20°C., brent eae OF 19 x,(28:8)2 SOR OMe Oia ie o—~F18x28 10 2a | This hes close to the value found for zine. Determination of Chemical Constants. 13 The specific heat of cadmium has been measured between hitmandpe-2 Co by Gaede (Phys. ZS. iv. p: 105, 1902)- also between —186° and —79° by Behn (Ann. d. Phys. (4) i. p. 257, 1900) *. Employing Gaede’s results, Cy = O17 at 290°1 abs., Cp = 6°36 at 365°-2 abs. From Nernst and Lindemann’s formula, ¢, can be cal- culated, viz.: yp 1475 AA Mmoumie °° am” Bv 1475 53 = DAQAT Cy ono le Putting c,=c,+aT*?, Gi —vono «lime Ap = GAL Ore Meaty ua. O2d). KOT? Having regard to the need of an approximate figure only, the value 6.10-° will be taken, which is approximately the mean between the values 5°93 and 6°15. Inserting these values in the vapour-pressure formula (2), two equations are obtained from which C=1°65. By differentiation, it is found that a change of 2° in the temperature, which is the error in the temperature mea- surements, should lead to a change in the constant of 40°31, so that the theoretical value 1°47 lies within the experimental! etror 1°65+0°31. * Behn deduces from his mean values that at 194° abs. C,=s°844 (.°. C,=5'682), at 87° abs. C,=5'317 (.". C,= 5-268). Bv in the first case would be 191 and in the second 138; it seems advisable therefore to take the value obtained from the melting-point formula: 147°5. 14 Capt. A. C. Egerton on the Chemical Constant of Zinc. The chemical constant of zinc can be obtained from the vapour-pressure measurements in a similar manner. Taking the points : at 593° abs., log p= — 3°39 + 0:02, at 625° abs., log p= —1:9220°02. The temperature measurements were considered to have a probable error 41°C. The value of Sv obtained from the melting-point formula 234, and that from specific- heat measurements (Koref, ci d. Phys. xxxvi. 1911, and Pollitzer, 7.S. f. filek. xvii. p. 0, 1911) is 23: At Sven oe ae) A—A, nie sap n= ep 7 Oe ay | ee and By 234 (A Aone ee T= gas = 037440: ban | =—8°650. Pollitzer gives a from the specific-heat measurements as 04.107 From the coefficient -of expansion = 26 x 107° em. (National Phys. Laboratory); compressibility 169.107” per dyne/cm.”, and, atomic volume 9°22 c.c., at temperature 2OT° abs., a becomes 4°65. 107: The value 5:4. 107° will be used. ; »—A Inserting these values “ior @, 7, >) aidan the vapour-pressure formula, two equines are obtained, from which © = 1:23. As before by differentiation, a change of temperature 2° C. will alter the constant +0: 26, so that the theoreticsl value 1:115 lies within the limits of experimental error. Values of Chemical Constants. Both for cadmium and zine therefore, the result lies within the experimental error and there appears to be no doubt that the chemical constant may be represented by a formula Cyo+1°5 log M as shown in the table. Taspie 1V.—Chemical Constants. Atomie pa Probable eon Constant. ea Constant (eale.). Merc uy...) 200°6 1-820 +0:032 1:845 Cadmium... 112-4 1:65 +0°31 1:468 Piano. os oe Gone legs + 0:26 1115 Determination of Chenacal Constants. 15 Nernst (l. c.) has calculated the constant in two other cases :— Atomic ane Probable : AE Fee Constant. Be Constant (calc.). TA eee 39°88 O75 +0:06 0893 1 ee ee ea 2°016 —1°23 +0715 —1-151 The value of Cy obtained from all the experimental results and from the probable errors, giving due weight to the observations, is as follows :—- —C, (obs.) Weight. Mercuiy 2. 2 eGo L@) Gadus 2) 2 Gey 0-10 Zine sie 15a) 0-12 APRON) 7g). S 1°65 0:53 fivdiogen 2. . 1:68 O- 21 Cpumeaae— = Saluo22 Re This value is slightly higher than the calculated (1°608), but agrees within °85 per cent. Latent Heats of Vaporization. In the paper on the vapour pressure of mercury, cadmium, and zine already cited, values of Ay were given from the relation : Xo 8) SAS elon o. which was found to agree weli with the observed results. Xr rate ai Here log p=log K—3 log Tap and it is assumed (C, —c¢,) remains constant and eqnal to unity. This will be approxi- mately the case at ordinary temperatures where C,=2:°5R and gq=3R. It follows Ay=Ao—T’, as an approximation. The values of XX») have now also been obtained from the rational vapour-pressure formula : = +(" re i. IT+C cp =—pm Dm UA IN re ake OS P uf RT? fe p Ab RT I C7 GL OU, which may also be expressed : Art fie) 25 —-" pn i else EPA g where f(T) is a function depending on the change of atomic heat of the solid. 16 Capt. A. ©. Egerton on the T It the formula r= dag 963° — | endl —2 aT? eis 0 differentiated, it will be seen A is not necessarily a maximum at the absolute zero : dr 8 “ap = 4°963—(ep + aT), which becomes a maximum when Oya HENS.) Oy 2s (Cini It ee TO, For mereury;, 6v/1 =) 97/1595) ==), 29 aia. . jeadmium, 6y/ 1 — 147/595) — aoa oy ZN By/V = 234/195 = 1202 Orns: In order to deduce the above results, the theoretical value of the chemical constant 1s assumed and the same vapour- pressure measurements are employed, thus for zine : Ay = 1:92 + 2°881 + 2'5 log 695 08 ener iS ANY AL x D4 .107°.62541:115+0°26 — ae) ae 100): Aspe. = 31330 + 4°963 .691—5°235 . 691—2/5.a. (691)?? = SvoMil: Amax. = 31330+ 4963 .120—120. 2°684 — 2/5. a.(120)5? = (51070. The values in the last column of the following table are ealeulated from Nernst’s relation : No = Te 3-0 loo Mp. They agree closely with the values deduced from the vapour pressure *. The values for Ag, Amax aNd Ay.pe, ATe given in the table below. * It may be noted that N ernst’s other empirical ex pression, Ly = 95logT,, —0:007 Tp, does not agree nearly so well. It is obvious it is not applicable to. high temperatures. Determination of Chemical Constants. if TAaBLeey. fe No. Amax. AM.Pt. AB.Pt. Neale. Mercury...... 155305215 15646 14884 14161 14900 Cadmium ... 26770-+750 26941 26292 — 26900 PRC skit 31330+750 31570 30871 — 31200 Constants of other Metals. Measurements of the vapour pressure of other monatomic substances which have been made do not lend themselves readily to a calculation of the constant, as most of them would involve a knowledge of the specific heat of the molten metal. Langmuir and Mackay (Phys. Zeit. xiv. p. 1273, 1913, and Phys. Rey. 4, li. p. 377, 1914) have, however, investi- gated the vapour pressure of solid tungsten, platinum, and molybdenum by measurement of the loss of weight of a heated filament in vacuo. Using a vapour-pressure formula of the Hertz type, the results are expressed : Tungsten, logp(mm.) = 15°502 —47440/T—0°9 log T. Molybdenum, log p = 17°354—38600/T-—1°26 log T. Platinum, log p =14:09 —27800/T — 1°26 log T. By differentiation, Tungsten, = 216850—1-79 T. Molybdenum, = 177000—2°5 T. latinum, = 128000—2:°5T. In the latter cases, if (c,—C,) is constant throughout the temperature san ls tOape: ie then the constants 177000 ete. will represent the value of Ay; and as C,=4°963, ¢, is taken as 7°64 in the last two cases and as 6°75 in the case of tungsten. angmuir and Mackay have applied their results to the formula : +1°75 log log p= — 2 poe EST LT and have obtained values for the constants as follows :— Tungsten Pe AR Mate 5% Platinum Bt Meet eats) uli Rs Aieies Molybdenum .. 4-4 Phil. Mag. 8. 6. Vol. 39. No. 229. i 1920. C 18 Capt. A. C. Egerton on the In the case of tungsten, they have assumed a greater atomic heat than given above, about 7:5, which changes the value of A, to 218000. This is in better accordance with the observations of Corbino (Atti R. Accad. Lincei, 1912, xxi. pp. 346 & 188), who finds a value for C,=7°8, which remains fairly constant over a wide range of temperatures (2000° C.) *. It has already been pointed out that calculation of the constants in the above way leads to results without physical significance, but the knowledge of the change of specific heat at high temperature is insufficient to permit of the rational method providing a reliable value for the constants of these metals. In the case of tungsten, from the melting-point formula Bv may be taken as 301. At 2400° abs., v= 01125) and aes 14648 and oe 3-204, a beeo abs Sed 5 28) Vy elt 5 0275 agar == (06 6 I) and 0:0583aT?? = 0:0411. jp = OM Is s log p =—10°182 atmos. No 218000 and 3711 7 £571: 2400 i. on so that C= —10°1824+19°872 — 8-450 + 3°20440°041. (Cpe 4lp4ko., . For platinum, Sv has been measured from the specific heat and has a value 220. From the melting-point, the value 223 has been calculated. Taking By = 220, By ein Ap—Ag 276 15°372 1% a000 = Lt and op = ld o12 and ADIL = 3°363, 3a = 27.107° (see Nernst, Z.S. fi Hlektrochem. xvii. De Ol osu Oial). OH = 91, « = 0-40.10-2 ; * Tor platinum some measurements of Cy have also been made at high temperatures. Konigsherger (Verh. d. Phys. Ges. xiv. pp. 275, 540, 1912) gives the following figures :— 6°56 ; 6°79. “I “I os) ° Oe 0 ie) Noll Determination of Chemical Constants. Ko whence G32. 10g (at 2907 abs.) ae TE =0'207. so that At 473°, Konigsberger (/. ¢.) gives oer Wein, Xo SES, eee Uo een 187 Pm Xo = 128000 and £5711 14:000, p = 0°0001072 at 2000° C. = —6°851 atmos. Inserting these values, as in the above calculation, C becomes 2°36. Apart from the uncertainty in the value for Ay owing to the change in the specific heat, it appears that further experimental work is required before these measurements of the vapour pressures at high temperatures can be used for evaluating the chemical constants of these metals. Determination of Stephan’s Constant. Professor Lindemann, to whom I am indebted for his Kindly advice and assistance and the interest he has taken in this work, has suggested to me that a value for one of the Peto, cone anne: might be obtained from the chemical constant. Thus, Stephan’s constant C= eT uP ee C d Jap emo Ole a res log (27) = Cy+6°0056 (C.G.8.). B10 Nea As eye logo = Co—logio aa "Re 3a + 6 0056. Substituting values: Jigs SPA 6 MOY. == DONS Oy logy e = Cy — 2°6563 + 6°0056. If ie — 1°622 (see p. 15) | c= Om elo crm deca: If the value obtained from mercury, only, is employed, w1Z. 1°6330°032 : =O) st 0:30. * If R=8-306.107, N=6:062.10%, 2=6:57.10-27. Substituting (27)? RIP ; eo Ne would give 564.10—’, but this value is dependent on other deter- minations of h and N. The value —1-608 (which has been used) is due to slightly different values of the constants. (Git = —]'593, this theoretical figure, used to determine o, 20 Determination of Chemical Constants. Thus, the value deduced from the experiments is nearer the older value (as given by Planck, Vorlesungen tiber die Theorie der Weérmestrahlung) 53.1075, rather than the newer values which have ae obtained more recently, e. Y. peach (Ann. d. Phys. 30. 3, p: 209, 191679 alte “2 watt.cm.~?.deg~*. Summary. Methods of calculating the chemical constant and the. conditions necessary for obtaining accurate results are discussed. It is shown that the older formula by which most of the constants were calculated leads to a result. without physical significance. The formula which should be used requires a knowledge of the specific heat of the substance (2. e. of the value By). The chemical constant of mercury is calculated and the accuracy of the result is discussed. The constant is found to be 1°820+0°032. This value leads to a value for Co, ne universal constant of the theoretical relation —Cy=C- 1-5 log M, within the experimental error. The chemical constants C of cadmium and zine are evaluated from the vapour-pressure measurements (Phil. Mag. vol. xxxiiil. p. 193, 1917), and the values calculated using the theoretical value of Cy are found to lie within the experimental error. The value of C, for monatomic substances, the mean of all values so far determined, is found to be-— I: 622. The latent heats of vaporization of zinc, cadmium, and mercury are given for various temperatures, and the maximum value is found for all monatomic substances at a temperature —, Be = 1°99. a There appears to be need for further experiment before the data obtained by Langmuir and Mackay for the vapour pressures of tungsten, moly bdenum, and platinum at high temperatures can be used for accurate determinations of their chemical constants. Stephan’s constant for full radiation o has been deduced from the mean value of the constant Cy and is given as 5°27.10-° erg.cm:”?.deg>*.. This value, appearsiiombe definitely lower than the more recently determined values, within the experimental error, and is dependent only on the vapour pressure, specific heat of mercury, gas constant, and velocity of light. Clarendon Laboratory, Oxford. Sept. Ist, 1919, We ae [ 21 ] il. Note on the Signijicance of the Chemical Constant and its Relation to the Behaviour of Gases at Low Temperatures. By FB. A. Linpemann, Professor of Haperimental Philo- sophy, Oxford”. HE general equation for the vapour pressure of a solid, derived by integrating the Clausius-Clapeyron equation, is of the form aT eat. logp=| py Teh ae es at sae hin faa ng Ae Ey? Here X is the latent heat, \, the same at the absolute zero, ¢ and C, the molecular heats of solid and gas at constant pressure, RK the gas constant and © the chemical constant. The object of this note is to examine the question of the dimensions of C somewhat more in detail than is usually done, since it may be shown that this question is intimately bound up with the problem of the degradation of perfect gases. As will appear, the evidence of the chemical con- stants seems to be against the degradation theory if the above equation is accepted, and, moreover, the physical significance of the chemical constant appears in quite a new light when examined on these lines. In the text-books two different statements are found, namely, that the chemical constant has the dimensions of the logarithm of a pressure, e.g. Jellinck, Phys. Chem. d. Gas- reaktionen, p. 765, or that it has the dimensions of the logarithm of a pressure divided by a temperature to the power 09/2, e.g. Lewis, ‘System of Physical Chemistry,’ vol. i. p. 190. The first statement, which at first sight nr Lae number, really entails the assumption that the atomic heat of the gas becomes equal to that of the solid at the absolute zero. The second view is arrived at by assuming the gas laws to hold down to the lowest temperatures. The question may be _ stated generally as follows, If c, and C, are analytic func- ions of T one may write appears to be justified by the fact that dT is a pure T cdl =ag+a,T+al?+... ~/0 and my | CpdT=bo45,T4UT?+... a 9 * Communicated by the Author. 22 Prof. F. A. Lindemann on the Hence log p= No by — a b,— a, mM by— dy 7 RE RT + yn eS a ee ae ee 0 It is clear that every term of this series is a pure number with the exception of the third, which has the dimensions of the logarithm of a temperature to the power — One must conclude, therefore, that the dimensions of C are those of the logarithm of a pressure divided by a temperature to b;— a RS Now 2, and a, are clearly the atomic heats of the gas and the solid at the absolute zero, and there seenis little doubt that a,, and for that matter a, and a3,are zero. Therefore, if the value of b,—a, can be found it will be a measure of the atomic heat of the gas at the absolute zero. Assuming the chemical constant C to depend only on m, the mass of the atom, k, Boltzmann’s constant R/N, and h, Planck’s constant, a simple dimensional] consideration shows that it must be of the form the power TSE eagle cae) mae por 4 > R 0 —————— Sie Therefore it must be possible to represent it by an expression of the form J=K 43/2 log A+ (5/2—1,/R+ a,/R) log @, where K is a constant, A the atomic weight, and @ some un- known temperature which can only depend upon the tempe- rature at which according to the “‘ degradation” theories the ‘as laws cease to hold. Unless @ is equal for all elements *, and this is scarcely conceivable in the case of substances of such divergent characteristics as say mercury and argon or cadmium and hydrogen, a study of C should enable one to form some idea of the value of b, since a,=0. Now it has been shown that (© may be represented within the limits of error by C=K+3/2legA * This assumption was made in the case of isotopes in a recent paper (Phil. Meg. xxxvii. 1919, p. 523). It is readily seen that the question there considered, v. e. the difference of the chemical constants of BSObopes: is unaffected by which of these views is taken. Signisicance of the Chemical Constant. 23 Therefore, unless 0 by some curious coincidence is very nearly 1° for all the elements examined, one must conclude that 6;=5/2R. This, of course, is the usual value for the atomic heat of a monatomic gas at constant pressure and precisely what the classical, as opposed to the “degradation,” theory would lead one to expect. The following table derived from figures kindly placed at my disposal by Mr. A. C. G. Hegerton indicates the value of the experimental evidence. If Cc p=o/2R the column headed @ should be equal to 1°. If Gs, is to become zero near the absolute zero, one must assume that 0 has the values given in column 3, 5. e. that it happens to be very nearly 1° in all these substances. C. C—3/2 log A. 6. ee 1:820+0-:082 | —1-63340:032 | 097640030 Cue 165) 20-31 9) | 31-494 0:81. |) 1-19 £0-40 Zn... 123 £020 | -179 +026 | 111 +030 Hoe Ga 0-75 40:06. | —165 +006 | 0:962+0-06 H, ......-123 +015 | —1:68 +015 | 0-935-+0-14 It is evident that the more accurate the determination the more closely does @ approach 1°. It is always 1° within the limits of experimental error, and it seems most improbable that this approach to the purely conventional unit of the scale on which the temperature is measured should be a mere coincidence. It is true that the equation puv=RT is used in deriving Clapeyron’s equation and that a different formula would be obtained if the degradation equation were used. It seems unlikely though that “this would compensate in such a way as to invalidate the above argument. If not, one must conclude that C,=5/2R even at ue lowest temperatures and give up the degradation theories, which make it become Zero. This may be more acceptable if a further simplification is considered which puts the physical significance of the chemical constant in a somewhat different light. If the vapour pressure 1s written °T Xo —f(T) p=pTtes Tard he ae T where 7D=( cpdT, one must put a 0 (2ar) 2? m3? ke? = Po 13 ? 22 ike Prof. F. A. Lindemann on the the value (27)%? being that derived by Sackur, Tetrode, Nernst and others. Since Stephan’s constant 27° ht C= 15 @h3? one may therefore write, as mv?=3kT in a monatomic gas, Ge Se 45 VJb6r 3 oy ak wv oe dT DS I WT we Bae) AgoT* Now -;—=P is the radiation pressure of complete 3c radiation of temperature T on the walls of a containing vessel, so that the vapour pressure fun Soak eat == ) where v is the velocity of the molecules in the gas. If the velocity of sound Tene. = aA) is introduced this becomes Thro S(T) 95 S307 ul Top Lo Acne at low temperatures at which /(T) is small. If any frequency y is considered, the wave-length of the elastic wave in the V ; c gas is — and of the corresponding electromagnetic wave -. V yp Therefore the ener ey residing in the gas ina cube whose side is one wave-length divided by the corresponding energy in Xo the radiation is 1246 e ne i.e. 1°246 times the fraction of molecules whose energy is greater than the potential energy acquired when they are removed from the solid to infinity. This relation is strongly reminiscent of a well-known theorem in the theory of radiation, namely, that complete radiation is in equilibrium in any two dielectrics when the energy in a wave-length cube in one, is equal to the energy in a wave- length cube of the same frequency in the other. In com- paring the energy in the gas to that in the radiation at reasonably low temperatures the corresponding theorem would be, that radiation pressure is in equilibrium with vapour pressure when the energy per molecule capable of Significance of the Chemical Constant. 20 evaporating in a wave-length cube residing in the gas is 25 /30r 2 cube of radiation of corresponding wave-length. One may conclude therefore, since the chemical constant can be re- placed by radiation pressure, that its physical significance may perhaps be sought rather in the interaction of radiation and matter than in the subdivision of Gibbs’s N dimensional space into finite elements of of probability. If these finite cells of equal probability really exist it would seem difficult to escape the conclusion that the atomic heat disappears near the absolute zero. The evidence appears to be against this as was shown above, and one of the strongest ‘arguments in favour of such a revolutionary assumption would seem to have been removed if the value of the chemical constant can be derived from the radiation pressure. The law of complete radiation cannot be deduced without some quantum assumption of course, but it would be a considerable simplification if it could be avoided in gases. ‘equal to =1:246 times the energy residing in a Summary. It is shown that the chemical constant has the dimensions -of the logarithm of a pressure if the atomic heat of monatomic gases becomes zero at the absolute zero. In . case it should be of the form K+ 3/2 log A+5/2 log @, where @ is a _characteristic constant of the substance. It is shown that it has the dimensions of the logarithm of _a pressure divided by a temperature to the power 5/2 if the atomic heat of a monatomic gas remains 5/2 Rh down to the absolute zero. In this: case it should be of the form K+3/2log A. Experimental determinations show that the tatter form is ‘true within the limits of error. It follows either that @ is very nearly equal to 1° for all substances, which seems im- probable, or that the atomic heat remains constant ou to the lowest temperatures. It is further shown that the chemical constant may be eliminated and the vapour pressure expressed in terms of the pressure of complete radiation. It is therefore suggested that the chemical constant may express the interaction of matter and complete radiation rather than requiring that a gas can assume only a finite number of possible microphases at given temperature, pressure, and volume. Clarendon Laboratory, Oxford. 21st September, 1919. Dn | III. An Unsolved Problem in the Application of the Quantum Theory to Chemical Reactions. By W. C. M. Lewis, Professor of Physical Chemistry, University of Liverpool *. N a series of papers, entitled “Studies in Catalysis,” published during the last few years in the Journ. (Ghemn, Soc., I have attempted to apply the quantum theory to the- problem of chemical kinetics, the fundamental concept being: that the thermal radiation in equilibrium with the material system is the source of the energy required to bring about chemical change, and further, that the rate at which a reaction proceeds depends directly upon the density of the radiation of the absorbable type or frequency. By making use of Planck’s expression for radiation density, a number of results have been obtained which are in good agreement with experiment. Notably one obtains on this basis an expression for the effect of temperature upon the velocity constant of a reaction which is in good agreement with the. empirical equation of Arrhenius, and is also in agreement. with the statistical equation of Marcelin and Rice, namely dlog k/dT=H/RT?, where & is the velocity constant, and E is the critical increment reckoned per grammolecule of the decomposing substance. EH represents the additional energy which must be given to a molecule (by the absorption of radiation) in order to render the molecule reactive, for, as finite velocities show, molecules possessing the average amount of internal energy are not eorea reactive. Similar conclusions. have been reached independently by Perrin (cf. ‘Atoms’; also “‘Matiere et Lumiére,’ Annales de Physique | xi.| xi. 1919)... Further, it has been found poseble to calculate the velocity constant of a bimolecular reaction in the gaseous state in absolute measure, on the above assumptions, viz. (1) that radiation of suitable fr equency must be absorbed in order to activate the molecule chemically, (2) that one quantum of such radiation 1s required per molecule (Hinstein’s law), and (3) that when a collision occurs between two molecules thus. activated then and only then does the bimolecular reaction take place (cf. Lewis, Trans. Chem. Soe. exiti. p. £71, 1918). It is found that numbers thus calculated for the velocity constant are in good agreement with those found by experiment. Having attained a certain measure of success in the case- * Communicated by the Author. Application of Quantum Theory to Chemical Reactions. 27 of bimolecular reactions, the attempt was made to work out an expression for the velocity constant of a unimolecular reaction, based on the quantum theory. Itshould be observed that there isa fundamental difference between a unimolecular and a multimolecular reaction, in that a unimolecular reaction represents a spontaneous process independent of collisions with other molecules. The rate at which such a process occurs must in fact be determined wholly by the rate at which radiation of the absorbable type can be absorbed. Let us consider a simple case, namely, the dissociatien of the gas ABinto Aand B. By the term unimolecular velocity constant is meant the proportionality factor which occurs in the expression : dC/dt=kC, \ or = “log aa where (! is the concentration of the gas, (, its initial con- centration, and ¢ is time expressed in seconds. On applying the radiation hypothesis to such a process, it follows that the velocity constant /) (which stands for the amount of material decomposed per second, expressed as a fraction of the concentration unit, when the material is itself at unit concentration) should be obtained by dividing the rate of absorption of radiation of frequency v by the quantity hy, the latter being the amount of energy required to decompose the molecule of AB. From the standpoint of the physicist the chief interest of the problem lies in the fact that quite different results are obtained, according as we assume the radiation absorption to be continuous (Planck) or discontinuous (Einstein). First of all let us assume that the absorption is continuous. On this basis Planck has shown that the amount of energy absorbed by a single oscillator or molecule per second is Oo) We = 0 Sine where e and m are the charge and mass of an electron, and u, 1S given by 3.3 Sarhn Vv on l/kT 1 oe The simple exponential form is justifiable, as the wave- lengths which are known to be responsible for chemical changes are sufficiently short. c is the velocity of light an 28 Prof. W. C. M. Lewis on an Unsolved Problem in the vacuo, v the frequency of the radiation causing the chemical change, and n the refractive index, which, for gaseous systems, we shall take as unity. If there are N molecules of AB present in the system considered, the total amount of energy of frequency v absorbed per second is then Further, since hy is the quantity required to decompose a single molecule, it follows that the number of molecules » decomposed per second in the system is 877e7v? Tee enw y N, 3mc? This is simply the rate of the unimolecular reaction, which is ordinarily expressed : dN /dt=kyN, where ky is the unimolecular velocity constant. It follows therefore that De ome Vs ko=—s— en Av/kT *) 3 e OMLC On substituting numerical values for the constants in this expression we find: kg= 2469 x 1007? xy? Xie 0s a} We have now to compare this result with experiment. The reaction considered is the unimolecular decomposition of phosphine gas, which has been recently investigated by Trautz and Bhandarkar (Zeitsch. anorg. Chem. evi. p. 99, 1919) over a considerable range of temperature. From the observed velocity constants at different temperatures in the region 940° to 953° absolute, it is found that the critical increment E per grammolecule lies somewhere between 70,000 and 80,900 calories. We shall not be making any serious error if we take the mean value to be 75,000 cals. We then have : N,Av=75000, where No>=6'1 x 1073. It follows that y= Ox LOG (OUR on dye) Application of Quantum Theory to Chemical Reactions. 29 At the temperature 945 abs. the exponential term ea P/kT — 4°37 x NO ins: so that finally ho== 2 460.x 107? 64x 1028 x 4°37 x 107 nie or [Poastore) oY) ae The value of ky found by experiment =10°2 x 107°. The observed value is therefore 10’ times the calculated value. It may be mentioned that discrepancies of the same order of magnitude are found in other cases, such as the dissociation of iodine gas, but in these cases the value of ky used for comparison was not directly observed, and consequently, for the purposes of the present argument, less stress could be laid upon the discrepancy. In view of the above results it is necessary to try to account for the experimental velocity constant on the basis of the discontinuous view of absorption, that is, upon the assumption that radiant energy exists in small units or quanta, a view which appears to be indistinguishable from a corpuscular one. Unfortunately, in making this attempt numerical values have to be ascribed to certain quantities about which there is necessarily a very large measure of doubt. The chief of these is the dimensions to be ascribed to the quantum itself. Since an electron is capable of picking up a quantum in the act of absorption, I have assumed that the quantum itself possesses dimensions of the same order of magnitude as an electron, that is, the “ radius of a quantum ” is 2x 10- em., corresponding to the value of the radius of an electron recently given by Jeans (Trans. Chem. Soe. exv. p. 866, 1919). As a matter of fact, the final conclusion arrived at would not bealtered by ascribing to a quantum the dimensions of a molecule (L078 cm.). When a quantum and a molecule collide with one another, absorption is regarded as occurring and the molecule decomposes. On this basis we treat the problem as we would treat the collisions between molecules in a bimolecular reaction. In the case of a molecular system the number of collisions per second is given by the expression: Tire G 105 NV uy? + uy? N, No, in which o, and co, are the molecular radii, uw, and w, the root-mean-square velocities of the molecules of which there are N, and N, present per unit volume. In the analogous ease we can let o, and N, refer to the molecules and oc, and N, refer to the quanta. ‘The velocity term becomes simply ¢, 30 Prof. W. C. M. Lewis on an Unsolved Problem in the the velocity of light, since molecular speed may be neglected in comparison. ‘The rate of the reaction is then given by dN,/dt=7o,0.c N\Ng, or the unimolecular velocity constant is given by kg = TiGGeCeNGs We have now to evaluate N., the number of quanta per c.c., N, being a constant throughout the reaction, as long, in fact, as the temperature is maintained constant. Obviously the number of quanta of frequency v in one c.c. is obtained by dividing the radiation density u,dv by the quantity Ay. That is, Sav? § e 7 hu/kt ap. C [Ev =——uroricin © Setting ¢,- =2x10 ° em., and o,=2x 10-® em., we have ky= 34% 107% oy? e744 dy, eee The term dy corresponds to the width of the band at the frequency v. Certain estimates of the limiting value of dn, to which a physical significance can be attributed, are given by Schuster (‘Theory of Optics,’ 2nd ed. p. 346), from which it appears that the limiting value of dX is approximately 10-” cm. In the case of phosphine, where 2 itself occurs at approximately 375 wp, it follows that dy is of the order 10%. If we give an excessively large value to dr by setting it equal to 100 up, we find dv to be of the order 10%. Even employing this large value, it is evident that equation (2) will give rise to a far smaller value for /) than does equation (1). The discrepancy between the observed and calculated velocity constant is therefore still greater on the discontinuous view of absorption than it 1s on the continuous view. We are forced back therefore, I think, to the c ntinuous view of absorption as being the more hopeful of the two in spite of the great discrepancy which still exists. The problem which awaits solution is to account for the discrepancy factor, which, rather remarkably, seems to be about the same quantity for different gaseous reactions, and further, is apparently independent of temperature. ‘The mean value of this factor is approx. 4x10’, this being the quantity by which the right-hand expression in equation (1) must be multiplied in order to arrive at a result in agreement with experiment. The discovery of the cause of this discrepancy would be a very great advance in chemical kinetics. It ‘looks as though some modification of Planck’s expression for Application of Quantum Theory to Chemical Reactions. 31 the rate of absorption of radiation would be necessary. Possibly some physicist, better qualified than the writer to deal with the fundamental basis of the quantum theory, might interest himself in this problem, a problem of very 2 reat chemical i importance. In concluding, attention may be very briefiy drawn to an empirical expression already employed by the writer in connexion with unimolecular reactions in gases, which seems to account fairly well for the velocity “constant in such processes (cf. Lewis, loc. cit.). The expression is: 16X10" ayaa ost ee — oa Cages where p=c?/Sahn*y* (the definition of Planck), and u, has its usual significance. Applying this expression to the case of the decomposition of phosphine at T= 949, taking v=8 x 10" as before, it 1s found that eeCal C—O. Ome. whilst EOWS: == Oe 1 Digs The observed value is about eight times the calculated. The discrepancy is now small and might even be accounted for by a relatively small change in the value taken for the characteristic frequency. Whether the above empirical expression has any theoretical significance the writer is unable to decide. Summary. 1. On applying the quantum theory to a unimolecular chemical reaction, it is shown that very different results are obtained according as we assume the continuous or discon- tinuous view of the absorption of radiation. 2. On comparing the calculated with the observed ve elocity constant, it is shown that a very large discrepancy exists between the two values, the diser epancy y bein 2 much greater, however, on the discontinuous view than Tk is upon the continuous. 3. The discrepancy referred to in (2) is always in the sense that the observed velocity constant is many times greater than the calculated. The discrepancy factor (on the continuous view) is of the order 10‘, and appears to be of the same order of magnitude for different reactions. It also appears to be independent of the temperature. 4. The explanation of this discrepancy would constitute an exceedingly important contribution to the theory of physico-chemical processes. Muspratt Laboratory, University of Liverpool. Pie) IV. Latent Heat and Surface Energy.—Part I. ay D. L. Hamuicx, Chemical Laboratory, The College, Winchester *. 1 has been shown in Part I. (Phil. Mag, xxxvil. p. 240: (1919)) that the work done in getting the molecules in a gramme molecule of liquid into the surface layer is, at low ae temperatures, oa where p is surface energy in ergs X ecm.”, ¢ - V is the molecular volume, and d is tne molecular diameter calculated from properties of the vapour. This work was shown to be one-sixth of the internal latent heat of the gramme molecule, which is the work that must be done in order to move all the molecules in the volume V apart from one another against inter-molecular forces (internal pressure) until the liquid has become a vapour. The equation py tie his i pe on be used for the calculation of latent heat at € low temperatures only, because in deriving it the assumption is made of contiguity of molecules in the surface layer and 66 y Lid , 4 absence of ‘‘ vapour effects ” (loc. cit. above). The equation given by Bakker, however, (Dissertation, Schiedam, 1888), is of general applicability. Bakker’s equation has the general form CONSE) ( Mdv=,. a’ % where K is internal pressure per unit area across any section in the interior of the liquid. K dv thus represents the work done against internal pressure when the system expands V9 by dv, and | ~K dv, where x, is the volume of 1 gramme of a/ V} liquid and v, the volume of 1 gramme of vapour, gives the work done in pulling all the molecules in 1 gramme of liquid apart until the liquid has become vapour. In the absence of an external atmosphere, this work is the internal latent heat per gramme, Ay. Integrating on the assumption e ° a that K is a function of v of the form —, we get OP * Communicated by the Author. Latent Heat and Surface Energy. 35 e . e e a . Testing this expression on the assumption that —=K is Vo e . e a e identical with van der Waals’ oe and using values of a, derived in accordance with van der Waals’ equation from the critical data of the vapour, Bakker found values for a; at the boiling-point of about two-thirds of the experimental figures. He drew the conclusion (Zeit. physik. Chem. xii. p- 670 (1893)) that @ is not independent of the temperature. Traube (Ann. der Physik, v. p. 555 (1901) ; viii. p. 300 (1902)) has attempted to calculate van der Waals’ “6” for liquids by the use of the equation given by van t’Hoff (Vorles. tiber Theor. u. Phys. Chem. iii. p. 30) 2130 On = Un — 1: = (T, being the critical temperature) together with his theory of ‘‘liquidogenic”’ and *‘‘ gasogenic”’ molecules. Values of a, derived from 6, give no better results than the ordinary ‘“‘eritical” value a- when used in Bakker’s equation. By combining, however, the author’s equation, oe ; 7=L, with Bakker’s equation, it is possible to calculate a value for a, which can be shown to vary linearly with temperature up to the critical temperature. From a knowledge of a, and its temperature coefficient it becomes possible to calculate latent heats with very fair accuracy right up to the critical temperature. Referring to Bakker’s equation, it is obvious that at low temperatures v, the volume of the vapour, is very great compared with v, the volume of 1 gramme of liquid. We may write, therefore, IL A; =p : Be From the author’s equation we have, for the latent heat of vaporization of 1 gramme, 1 1 eae ay 2 Hence ag = oO site aiiot sock, au(ele) External latent heats of vaporization per gramme will now be calculated for several liquids by the following procedure :— (i.) A value of a, will be derived from equation (1) at low temperatures. Phil. Mag. 8. 6. Vol. 39. No. 229. Jan. 1920. D 34 Mr. D. L. Hammick on Latent Heat (ii.) ap will be assumed to vary linearly with the tem- perature up to the critical temperature, the temper- t ficient 2% being 2-2 ature coefficien oT. eing ToT (ii1.) Values of a, at various temperatures will be calcu- lated from the temperature coefficient and the value of a found from the surface energy from (1) and substituted in the complete Bakker equation Results are given in Table Hi The surface-ener ‘gy data used are mainly from Jaeger (Zeit. Anorg. Chem. ci. pp. 1- 214 (19L7)). ° Values of a are expressed as atmospheres pressure X (volume of 1 gramme molecule of eae at, N.T.P.)?.. The gramme molecular volume is taken as 22°41 litres at N.T.P. The temperature at which a is oalalared from p, the surface energy, is shown by the suffix after a and p. Young’s data (Sei. Proc. Roy. Dubl. Soc. xii. p. 414 (1910)) have been used for specific volumes of liquid and vapour, critical data (calculation of a,j, and latent heats. Values of d, the molecular diameter, are those used in Part I. The connexion between internal and external latent heat has been taken as N=, + oF , where M=molecular weight. TABLE I. Latent Heats from Surface Energy and the Critical Data. Benzene. @,= 03822. T,=288°°5)C) px.4=s0'0engs x em:>, (@=2 ep 0m sem From equation (1) at 5°°4 C. a5.4= "05808. ee = --'000070 or AON ERR Nasa 80 €.° 100° C. 150 C°. 200° C. 250° C. Tigh itiscs aces SOOZOO 05146 ‘04796 04446 ‘04096 Neale 2: 95°4 90:7 80:2 68°0 52:2 NYoung) =: 99°5 914 80°9 68:8 [49-5] Hither. @,,= 03951. T,=194° ©. o,—19:2 ergsxem.* -d=44 10 elem: From equation (1) at 0° C. a= 05200. or = — ‘000089. NS) 30° C. 40° C. 60° C. 100° C. 150° C. che | oc BO SOBe 04902 04862 04690 04350 ‘03925 (cale.) "077" 84:2 ee ioe OF 2) 51:5 Goin) 84:0 82°8 78°4 68°4 51:1 and Surface Energy. 35 Oarbon Tetrachloride. a= 039351, 1 — 283° ©. p,—28:'9 ergsXxem.7, d=4:52 10-8 cm. From equation (1) at 0° C. a, = 06609. Oe = — 0000946. ony Tt Ree Sea 80° C. 150° C: 200° ©. 250° C, BPE LP AN 2 05852 05190 “04717 04244 ety ee 488 39:9 83-2 24-4 2 ae 46-0 38-9 32:6 [23-2] Onlor-benzene. a= 05187. q=309° C. p,=35°9 ergsXem.? d=4:83x 10-8 cm. From equation (1) at 0° C. a, = 08684. = = — ‘0000974. 2 eee 132°C. 150° ©. 200° C. ce er eee 07399 07228 06736 1 ire eae 78:5 737 65-4 Pe ee: 742 (D 65°9 ithyl Lormate. @ = 03302" T= 235° O.* p.,=26°0 ergsxem.” d=4:09X 10-8 em. From equation (1) at 22° C. a, = "04668. oy = — 000064. or CLD fel aoa ee BOC: 100° C. 150° C. 200° C. Fay Endoe Wee Meuanes: "04488 “04169 "03349 "03529 Mediates) Je ah ena au 84°3 ae coe X (Young) i 97:9 85°7 70°5 49°35 Ethyl Acetate. a,='04383.* T.=250° C.* p,=25:5ergsxcm.? d=431 10-8 em. From equation (1) at 0° C. a, = 06328. on = — 000078. TN Goan ee eee ae 80° C. 150° C. 200° C. Xoale.) teettes 816 66°4 52-9 d-voung) Be 85:8 69°] 52°7 Methyl Formate. a,= "02291. f) eeellas CAO ihn etescem.: ad —4:035c1053 cm, From equation (1) at 0° C. Gy = "02920. ae = — 000029. IE Ree aise Pecan 40° C. 100° C. 150° C. Gn) eidecsgse< pos 02824 *02630 02585 Neale.) treet ie oa Ce \Cyoung) 007" 110°6 926 75'6 Methyl iso-butyrate. a= "04946. T.=267°6 C. p.»=25°7 ergsxem.? d=5:03 x 10-8 cm. From equation (1) at 0° C, y= "07588. or = — 000061. eres ness 90° C. 150° C. 200° C. 250° O. 25) nen ae ‘07039 06673 06368 ‘06063 \ (cale.) Saieiaet cies celeic 734 63°6 ‘ a | 37:2 Pee aoe 763 645 518 [32-0] x (Young) * Guye & Frederich, Arch Sci. phys. et natur. Geneve, ix. p. 22 (1900). + Mxtrapolated from Ramsay & Aston, Zeit. Phys. Chenu. xv. p. 91 (1894). D 2 ad 36 Mr. D. L. Hammick on Latent Heat The above examples show that from a knowledge of the surface energy of a liquid at a low temperature and the eritical data (from which a knowledge of d, the molecular diameter, can be derived), it is possible to calculate the latent heats at temperatures up to the critical temperature with very fair accuracy by means of Bakker’s equation. The agreement between calculation and Young’s “ experimental ’” values for the latent heat is not so striking as that obtained by Appleby and Chapman (Trans. Chem. Soc. ev. p. 734 (1914) by the use of a modification of Bakker’s equation. Appleby and Chapman’s expression is, however, open to: criticism (cf, Sutton, Phil. Mae. o]) xxixey age alien) that cannot be applied to the method used above. When the value of a at the boiling-point is compared with the value at the critical temperature, it is found that the ratio. Oh e e e ; —? is approximately constant and equal to 1-4. This figure Ae is supported by Traube’s results (loc. cit. above), which show that for a Jarge number of liquids the ratio of the latent heat at the boiling-point, calculated from Bakker’s equation * using the critical value a,, to the experimental latent heat is constant and equal to 1-4. Using this ratio, ae it becomes possible to deduce ec with precision the semi-empirical relationship betwee latent heat and absolute boiling-point known as Trouton’s Rule. From Bakker’s equation at the boiling - point iL j T (¥. , where V, is the gramme molecular volume of the vapour, being small) we have V, being gramme inolecular volume of the liquid and Ly, the ‘infernal Paoleculan Jatent heat. Writing a,=14a, and a Ne 2 “64Pe. we get 14x 271? 5 L,==- ‘ “64P.. Ve ° ° e . ° e (2) putting a,= , its value in terms of the critical data, * Traube used Bakker’s equation in its equivalent form RT v,—)b log— NE ame and Surface nergy. a7 A me is on the see equal to 3°7 (Young, “Stoichiometry, p.211),so that Pp -=V,.3°7. Wehave also the well-known relationship between boiling-point and critical temperature on the absolute seale, namely that ie oe (Guldbers, Zet. Phys. Chem.v. p. 3716). This, Again Viet together with the approximate value 2°7 for ~,* , gives us Vien in the above equation OZ S< IS Bed Se lis 64 x 62 He Toe: Putting R=2 calories, we get L, Wigs To obtain Trouton’s Rule in its usual form, we may put Ly = iS Lav biae or L i, pene Ths +h, whence pee Tia peo 01 ieee which is in very good agreement with the mean empirical value 20-22. It thus appears that Trouton’s Rule is merely Bakker’s equation expressed in terms of the principle of corresponding states, and liquids to which that principle does not apply approximately may be expected to give values for differing from the mean value 20-22. T;, p For oxygen at its boiling-point we have :— AM Ve RE. ds, id e779 —9-]7 * aa Daa aa T,, =l72, (ee eee PV. =3°419 }, = = 1-467, a9 L Bt. whence T =17:2. By the application of Nernst’s modi- b.p fication of Clausius’ equation, =18°0 (Nernst, Theor. Chem. p. 273). b.p * From mean value of v, in Kaye and Laby’s Tables. t+ Young’s ‘ Stoichiometry,’ p. 212. 38 Mr. D. L. Hammick on Latent Heat For nitrogen, we have :— Ww ’ ey wa ue 1p. Ma =3-491, %? 1.39 T;,, ane > Pave Qe L 2 Li 3 whence Toe 18°5. Observed (Nernst as above) ——~=17'6 bp 5 Similarly, for argon: Gale) =18°6, observed=17°3. a Dp Turning once more to the equation 6pV il ae d ow we can derive the ordinary ‘ external? molecular latent heat as bpV 1 whence L—RT 6 ih Via di rye or B60 RE 3 Vp. du Vp’ o) | Cech Neel eae (3) Now according to Walden, Vp is approximately constant fOr) a large number of liquids at their boiling-points and equal to 3°64. Looking at the right-hand terms of equation (3) it is at once apparent that approximate constancy is to be expected. The molecular diameter d has the average value of about 4°0x 107% cm. for the common organic substances and variations from this mean value are not very 1 great. Neglecting for the moment the term ——, which, Vp with R in calories, is small, we find HOP ~ the mean value Vp 3°2. When we introduce the term —— (KR in calories, p in Vp ergs xcm.”), we find that 3:2 must be increased, on the average, to 3°55. In Table II. results are given of the calculation of cals.) , liauids Va (eees for a few liqaids. * Krom v, on p. 60, Lewis’ ‘System of Physical Chemistry, and Yb.» | Ramsay and Drugman, J. C.S. xxv. p. 1228 (1910). and Surface Energy. by TaBLeE II, dx 108 Ge RE (cals:) L L Substance. aS Bon di Vip Gees) Vp Vp (found). (cale.). | 2180S ae RA a 4°41 3°26 "36 3°62 3°62 Sree accu. ccon esse 4°18 3°43 38 3°82 3°81 Mthyl formate ......... 4:09 3°51 30 3°35 3°81 Hithyl acetate....:....... 4°31 3°33 38 3°98 O71 Ethyl iodide ......:.2... 4°20 3°42 36 3°83 3°78 Methyl! iso-butyrate ... 5°03 2°85 "3D 3°68 3°20 OCI ene ean 4°52 3:18 33 3°43 3°51 CURUONS 2c As an 4°34 3°31 “36 3°78 3°67 ISGAZENE ee fecce cc eee 3h 4°88 2°94 319) 3°75 3°29 METIS 2 cee, wkacece 228 4:84 2:97 33 3°33 3°30 OU ee 4°70 3°05 33 331 3°38 O52 eee oe 321 4°47 7) 501 4°94 INTEROPENT-£.---22.26.0c0+s 3°50 4:10 ‘50 4438 4:60 The data used are from the same sources as in Table I. Hxcluding oxygen and nitrogen, the mean of each column for 2 is 3°6. The agreement between the “ observed” and calculated results for oxygen and nitrogen is particularly interesting. RY (ale) It will be noticed that the expression 7, =! is Vp (ergs) approximately a constant for the organic liquids. Its significance will be discussed in a later communication. When surface energy is expressed as ** specific cohesion,” | defined by R 2p BN, Oe oe a’ = 9-81 milligrammes weight per mm. (s=sp. gr.), we get, putting the author’s equation in the form 1 Pv Nal NS ete 6 (v=specific volume, i= internal latent heat per gramme), i aa [Ve 6v 5) DN od At Ji see u a whence Ge OSI: NG > cileieroy or = e aaah Ra AO Mr. D. L. Hammick on Latent Heat Since He OM ah )royl Vee aa Dy & i> gdje? Me ee (A) Giving as in (38) the mean value 4:°5x 107° to d, the we get 3x9°81 Ris molecular diameter, Ty becomes 15°64. Also We * die 9 8 ae os equivalent to ie ANG oie ; from Table IL, Wi is about °35 a organic liquids, so that Maz is about 1:72. Hence <= =17:4. Walden’s mean value is 17°9. Equation (4) is merely (3) altered to provide for the expression of surface energy as “specific cohesion.” Walden has, however, combined his empirical relation — with a Trouton’s Rule, and obtained Her =1:2. We can deduce Top this result at once. For Trouton’s ce we have found DS Dep T,, VI OD Me mean value Ma? Pall 17°4 has been deduced ; hence ie == ese If the freezing-point can be regarded as being approxi- mately a corresponding temperature, we can obviously 2 ae = constant. Naga The mean value found by Jaeger is 3°65; variations from the mean are, however, considerable. Thus for benzene at d°4 C. the value 1:°97is found. By calculation from Jaeger’s data and Fe x 10-§ cm., we find, on the lines of x =15°-44 at the melting-point 5°4 C. (288°°4 absolute). From the data : al Ue Rae derive Walden’s melting-point relationship © equation (A), 7p. (absolute) =2°02, Vm —— 04s PV, == Oe mp __ °05808 2 : Mx Ae 03829 oD) (p. 34) =1 2, we find Tee =O 6; a? M : : whence 7 for benzene at its melting-point comes to 1°86. and Surface Energy. 4] From equation (1) we have __ ap. d 2 p= 6 (V=gramme molecular volume), or pVi= S 5 : : at low temperatures. Expressing a, as ¢.a, and a, in terms of the critical constants we get | Ph ete Re Ae Gh alk pV'= Ge - P, ay ae ee At any (low) temperature T, we may write ile We eed ere = Ww, V =U iP WT =v, whence Pi oR ane oe ae 64 6V3" ie x and t=T,.—T, then i=: hence ae or Pewee Ree yee Nr id =e ee oa rt? According to the well-known Ramsay-Hoétvés empirical relationship the left-hand term, ae is very nearly constant over a wide temperature range for “ non-associated ” liquids. Owing to the limitation of equation (1) to low temperatures, we can draw no conclusions from the right-hand term of equation (5) at higher temperatures. Confining ourselves to low temperatures (boiling-point and under), we may notice that y is approximately a constant and that # and y will be the same for many liquids at corresponding tem- peratures ; z has been shown to be a constant (1°4) at the corresponding temperature of the boiling-point, and is probably a constant at other such temperatures. If, there- fore, it can be shown that = is approximately constant for different liquids at corresponding temperatures, the constancy of the Ramsay-Ho6tvos ratio will have been deduced for, at any rate, corresponding temperatures. In Table III., Vs and d are compared for a number of substances at their boiling-points. The necessary data were 42 - Mr. D. L. Hammick on Latent Heat taken partly from Smiles, ‘ Physical Properties and Chemical Constitution,’ p. 112, and partly from the papers of Jaeger und Young already referred to. Aare JOGE Substance. V2 ce. dX 108 cms. = yells. Benzene (shine ee enas cee 4°61 4°88 1:06 HWGherineencecsse: sess 4°73 441 1:19 OSS el Rie dates, eee 3°89 4:18 1-07 COR a ietaae tare 4-70 4°52 ‘98 CO KO) Fe mane ceeds c 4°39 4°34 1-00 Methyl formate ......... 3°99 4:03 101 Methyliacetate! 9-7 4°39 4:03 ‘89 Methyl isobutyrate ...... 5°02 5:03 1-00 thy} tormate 2. eee 4°39 4:09 “90 Bithiy/ltaeetater 2.2 y-cenene 4°75 4°31 “90 Hthyliodicdet =. eee 4-4] 4-20) 96 Anni hie: cuca tid biaces vas 4°78 4-70 “99 Polttene as. sce tenes semen 4°55 4°84 1:06 De Koy lene ys 3:05 05 wee ee 5°20 5:16 “99 Miesitylene | .)..70 eee 545 5°35 98 Chlor-benzene ............ 4°83 4°83 1-00 VAIS OL 5...65 es ee peer 5:00 4°98 1:00 Ossygenes. Jc «5374 eee 3°04 21 ‘99 INibrow en, = 4. 5: J) eeeee 3°21 3°50 1:09 ATOMS 2.27 25sec eee eneree 3°05 3°21 1-05 d Mean yi x 10°=1:005. From the principle of corresponding states it follows that a similar constancy will be found at other corresponding temperatures (unless, of course, d varies with temperature), and the Ramsay-Koétvés relation has therefore been deduced for ee temperatures. It has not been shown that = the asi should have the same value at all corresponding temperatures ; in fact, the mode of deduction of (5) would lead to the contrary expectation. Thus pV? is obviously V i proportional to c, which is equivalent to = (Gmizart t.): the variation of L, with temperature is by no means the same for all substances. And as a matter of fact Jaeger’s recent very careful determinations of surface energy show that the Ramsay-Hoétvés ‘“K” does definitely vary with temperature. The variation is, however, smal! and, as will be seen in ‘able IV., is accounted for by equation (5) 5 it Pd and Surface Energy. TABLE LY. : 100 p 5 3 dx 10° s K K* pubstance: eee. ergs Xem.? i y: se a en. sp. gy. (cale.). (observed). Hither nea: ba eter 0 192 1-71 2°80 1:40 3813 4:41 730 2:02 2°14 35° 15:9 1°52 2°66 1:33 i ae 699 2°16 alii (R. & §.)t BONZENC: .a.c.ce0 moth a8 5°4 30°9 2°02 2°04 1°52 3755 4°88 "895 2°15 215 80 20°7 1:59 2°6 1°33 ae ie ~ 813 2-24 2°10 (RB. & 8.) ft (OO) ease inser ringer Rens eee 0 28°'5 2°04 2°93 159 3°680 £02 1632 1:96 2°08 76°4 20°2 1°56 2°65 1-41 “3 ee Law 2°16 2°16 Ethyl acciate ............ 0 25°5 1-916 3°00 1-54 3049 4°31 ‘924 2:13 2°13 80 We 1-48 2°68 eet Shi ne "824 2:38 2°28 JENPROIN'S —Goaog goacddunaber —186 OOS eleyo 2°76 1:26 3°283 321 1-404 1-60 72 Nitrogen Sots. 196 8:94 1°65 2°72 1:26 3 391 3°50 8126 1:87 1°89 * Calculated from pVe=K(T,—T), chiefly from Jaeger’s data. t R. & 8.=Ramsay & Shields, cf Nernst, Theor. Chem. p. 275. § Data from Baly & Donnan, J. C.S. lxxxi. p. 907 (1902). 44 Mr. D. L. Hammick on Latent Heat also makes it unnecessary to compare liquids at corresponding temperatures. A “general” value for ue will now be calculated from (5) at the boiling-point, using the following data :— R=2 calories=2x 4:18 x10” eres, 2—1;62) 97 — 2a =14, p=37, ©, =1x10-% Ts We obtain © =K=2:17, in very good agreement with the empirical value. In Table IV. will be found some calculations of K for a few individual liquids at the boiling-point and at some other non-corresponding temperature. Here again, in the case of argon and nitrogen, theory is able to predict ‘abnormal ” values. From the equation oV wl L, as, 6 we have 2 ore pY¥: == hy Va 6 At the boiling-point - has been found to be 1:0x 107° for many liquids. We therefore have dip) 2 dL, Joe 1 lV BUvio al igen. ‘ RV ls, eolOme dl, I, cece where = = : oo = coefficient of expansion. Hxpressing L, in ergs (J =4°18 x 10’ ergs), d(pV*) ee) (6) oa ae it ee (d(pV*) _, , Ae om (GF —ra). os oa) This relation is implicit in the combination of equation (1) and Bakker’s equation used to calculate latent heats at various temperatures, and a special test of it is therefore not and Surface Energy. 45 d(pV5) Ge from latent heat data. Three examples of such calculation are therefore given in Table V. Thedata used are Young’s. necessary. It is of interest, however, to calculate TABLE V. 1 ay ane, (OM vi Substance. 2 2— d; =; OfM (ae —),2); fe) (q + Vp ot cals. X gr. or ol ol i (Jaeger). Benzene......... 80° "00125 86:5 —°28 —1°73 =o DU aoe 35 ‘00172 (ASRTL —:26 = 1X0) =F Ethyl acetate... 80 “00160 83°96 —°208 = 1/973) —1-30 Matthews (Jour. Phys. Chem. xx. p. 554 (1916)) has derived an expression for the Ramsay-Hotvoés relation by a method that bears some resemblance to that employed above. He obtains a connexion between surface energy, latent heat, and Bakker’s equation ; he assumes, however, that van der Waals’ “a” is independent of temperature. His reasoning, moreover, involves certain rather indeterminate assumptions in connexion with the intra-molecular energies of liquid and gaseous molecules, and depends on Goldhammer’s empirical relation between liquid and gaseous densities and temperature (Zeit. Phys. Chem. |xxi. p. 577 (1910). His calculations of “K” are nevertheless remarkably exact. The empirical relationships that have been shown above to have a theoretical justification have been used to calculate the association of liquids with which marked deviations from the empirical constants have been obtained. The usual procedure has been, when an “abnormal” constant is obtained, to find a number n by which M, the molecular weight of the simple or gas molecule, must be multiplied in order that the average or normal constant may be obtained. A glance at the expressions derived for Trouton’s Rule, Walden’s relations, and the Ramsay-Hoétvés “law” shows that such procedure is quite unjustifiable. Where the ex- pressions involve the molecular diameter d, the assumption would be involved that the diameter vf the associated complex is the same as that of the simplest gas molecule. This of itself might not involve serious error, since d varies. but slowly with increasing molecular complexity ; equations (3) and (4) might therefore be expected to give approximate degrees of association. Trouton’s Rule and the Ramsay- Hotvos law, as given by (2) and (5), contain, however, both d and critical data, which refer to the non-associated vapour: (Gj. Lyxer, Zen. Phys. Chem. |xxx. p. 50 (1912)). 46 Dr. L. Silberstein on the Quantum Summary. (1) On the assumption that van der Waals’ a varies with temperature, a relation is derived between ap, surface energy, and molecular vélume. eye! 6pV ae, This equation is valid only at low temperatures, By assuming that a, diminishes linearly with temperature to the critical ale a,, latent heats are calculated by the use of Bakker’s equation for several liquids. a (2) Recognition of the fact that the ratio = iat he Ae boiling-point is the same for many liquids results in ‘the derivation of Trouton’s Constant. (3) Walden’s empirical relationships between surface energy and latent heats and the Ramsay-Hotvos law are deduced. V. Contribution to the Quantum Theory of Spectrum Emission: Spectra of Atomic Systems containing a Complex Nucleus *. By Li. SILBERSTEIN, PA.D., Lecturer 1n Mathem. Physies at the University of [tome t. ale Wie. in all investigations hitherto published, or at least in all those which have come to my knowledge, the atomic nucleus (positive charge) is treated as a homogeneous sphere or, equivalently, as an ordinary point- charge, the purpose of the present paper will he to investigate, on the lines of the quantum theory, the spectrum corresponding 1/0 an atomic system with any differently shaped nucleus which will shortly be called « complex nucleus. Such would, for instance, be a nucleus consisting of two point-charges, acting as two fixed centres, —or any other axially symmetrical distribution of positive charge. The most general “ complex” or aspherical nucleus would be a charge distribution having no axis and no plane of symmetry. If —e be the charge of an electron proper, it is generally assumed by the followers of Rutherford’s and Bohr’s ae that the nucleus of a hydrogen atom has the (net) charge e, that of a helium atom the charge 2e, and * A summary of the results of the present investigation and some illustrating examples were given by the author at the Bournemouth meeting of the British Association, September 1919, Section A. + Communicated by the Author. Theory of Spectrum Emission. AT ¢ so on, generally xe, where kK is an exact (positive) integer. Now, in the case of helium and, a fortior’, of more complex atoms, it is certainly inadmissible to treat the nucleus as a point-charge. But even in the case of hydrogen there is no reason (besides a tendency to mathematical simplifi- cation) a@ priori to assert that its single nucleus-charge has a radially symmetric distribution. Under such circum- stances it.1s scarcely necessary to justify an investigation into the spectral behaviour of differently shaped nuclei. 2. No matter what the shape of a nucleus may be, we will call its total charge™ xe, and we will denote by —e and m the charge and the mass of the electron moving around it. No account will be taken of the perturbations due to other electrons or planets belonging to the same system. Again, the relativistic complications (already studied by Sommerfeld tT) will be disregarded, and the electronic mass m will, therefore, be treated as a constant. In short, the system will be treated on Newtonian prin- ciples, as far, of course, as the “ stationary” orbits are concerned. It is well-known that under these circum- stances, and if the nucleus be a simple point-charge, the negatived total energy W belonging to any one of the stationary orbits is given by lt rR eer mmcomertoics. | NMeaumnnaders Nii Me (EID) ch n where » is an integer, ¢ the light velocity in vacuo, h Planck’s constant, and / the Bohr expression of Rydberg’s constant, 2. e. 27am! Mn SS eee y Fy aoe alae bhbel elt fo (2) M being the mass of the nucleus, and therefore, practically, the whole mass of the atom. If v be the reciprocal wave- length or (what is improperly cailed) the frequency, the corresponding spectrum is, by Bohr’s fundamental assump- tion, v=(W,—- W.,)/eh, 2. Cy by (Gls); ihe y= OR (sa), UA VON il G3) where n' is a fixed, and n a variable integer. In fine, * In usual electrostatic (irrational) units. Tt A. Sommerfeld, Annalen der Physik, vol. li. 1916, pp. 1 et seq. especially Part II. p. 44 et seg. ‘The latter is open to some serious objections which will be pointed out in a later publication. 48 Dr. L. Silberstein on the Quantum the spectrum is always a (generalized) Balmer series of simple, that is, of ideally sharp lines, showing no fine structure. It will be kept in mind that this Balmerian type of the series and the singleness (and ideal thinness) of its members or lines theoretically persists, no matter how small or how large the dimensions of the stationary orbits. Both are general features, being consequences of the law of the field and of the irrelevance of the diameter of the simple nucleus. Such being the well-known result for a point-nucleus, our problem will be to investigate the complications of the series, and more especially of the fine structure of its members, due to the complexity of the nucleus. In what follows the meaning of the above symbols will be retained throughout. 3. Nucleus consisting of two fixed centres (point-charges). This is the only case of a complex nucleus which ean, for any dimensions of the orbits, be rigorously solved, 2. e. reduced to quadratures, and ultimately to known elliptic integrals. This is the reason why it is here mentionedat all. But the general and complete solution of this problem, famous since the times of. Jacobi*, has in our connexion but a purely theoretical interest. (Since those orbits only are spectroscopically relevant whose dimensions are very large as compared with those of the nucleus.) It will, therefore, be enough to give here but a general outline of the theory of such a system, omitting all ihe particular results arrived at by the writer f. Let 2u be the mutual distance of the two centres, and let each carry one-half of the total charge of the arnclen, 3 a O- kxe. (The case of unequal charges is not essentially more complicated.) Take the axis of symmetry (join of centres) as the w-axis, with mid-point as origin, and denote by y the distance from this axis. Then, with the well-known trans- formation of Jacobi, 2 =acoshé cosa, 7 — a sinh. cmp the potential energy will be 9 Vice an cosh & — Ke? fee = aS cosh? €—cos? 7’ ae Ne (4) * Its integrability was already discovered by Huler. + Some of these results with the corresponding type of spectra were described at the Bournemouth meeting, without being recorded in the much abbreviated Report. Theory of Spectrum Emission. 49 and the kinetic energy, T = 4m'a?(cosh? E—cos’ n). (£2 +77?) eit aosinine es sin ms 6 (2) where @ is the angle between the variable and an arbitrarily fixed meridian plane. The moments corresponding to the canonical coordinates &, 7, @ (which are the derivatives of 7 with respect to &, 7, @) are py=m'a2(cosh? £--cos? n)E, po= mia? (idem), | (6) pz==m'a? sinh? £ . sin? n.d. DS Since the energy does not contain ¢, we haye, as one of the eanonical equations, dp;/dt=0, 2. e. P= const., an obvious result. Thus, one of the quantizing equations will simply be { psd = 2rp; = his, Be en tea } 0 where n3; is an integer. And since there is with &, 7 what is familiar as “separation of variables,” the other two equations will be, according to the principles used by Sommerfeld, Epstein, and other: S, \pidé = nyh, \ p2dn = Noh, BAe hae (8) where 2, 2, are two more integers. (For integration limits ef. infra.) Using the well-known method of separation of variables, and denoting by JW the constant value of —V—7, we have at once the two first integrals, which are Jacobi’s integrals, A= ar/Imi Wn | B cosh? £4? : *H cosh fp? feinh? é, po= ar/ 2m! W .»/ —B8 + cos? n— P?/sin? 9, ps” h?ns? 1 : alts Fe =, is an integration 2nla2 WW Sa2m'a? W’ 8 F age) where ??= constant. With the values (9) of p;. p; we have in (8) two equations for the purpose of quantizing @ and W. Thus the problem of finding the required WAGs toy 113) corresponding to the stationary orbits, and thence the spectrum Pial. Mag. S. 6. Vol. 39: No. 229. Jan. 1920. i om Ct 30 Dr. L. Silberstein on the Quantum 7, is reduced to quadratures, and ultimately serles v= il ch to finding @ and JV from the two transcendental equations (containing elliptic integrals) by successive approximations. This gives rise to a variety of intricate spectral types, whose description, however, need not—for reasons stated a moment ago—detain us any further in the present paper. It remains only to say a few words about the limits of the integrals to be applied in (8). Since, by (6), € and 7 vanish only together with p, and p, respectively, the required limits. will be determined by the roots of the equations pif) 0, pa) = 0, 2 ein the left-hand members of these equations being as on the right hand of (9). The first of (10) is a quartie for cosh €=u, say, and the second a quadratic for cos?y. Of the four roots of the former two only, say 2, us, will be: found available *, and it we require the electron not to leave the system (not to escape to “infinity ’), both of these will certainly be available. And the roots 7, 72 of the second equation will be either both complex or both real. Thus, in general, the electron’s orbit will be contained between the two ellipsoids (of revolution) €,=const., &=const., and it will either pierce incessantly all the hy perboloids n=const. or be hedged in between the two hyperboloids m,=const..,. m,=const.t Thus p, will be integrated from the inner to the outer ellipsoid and back again, and ps, either over 27, it m increases (or decreases) incessantly or between the two limiting hyperboloids (twice), if 7 oscillates. In particular, we may have €=const. throughout, and therefore &= &, ; this for instance is possible for ¢=0, when the electron describes an ellipse in the meridian plane. And if the two roots 7, 72 (are real and) coincide with one another we have 7=const.; a possible motion of this kind occurs, for example, for 6=0, when the electron oscillates along an are of an hyperbola stretching within an ellipse &=const. in the meridian plane. (<) Pm), 0 and therefore, and W\4 Thus the disturbing function, up to ia) ls F= »(< “y (3, cos? = 1), eel 3) where 4 Ke KM = 2a ° ° ° ° ° e e ° e (14) ni Quantum ee of Line Spectra,’ Copenhagen, 1918, Part II. p. Theory of Spectrum Emission. 53 It remains to average F’ over a period of the undisturbed motion and to substitute the result into (11). Let the plane of the undisturbed (osculating) orbit make with the equatorial plane (7. e. the plane through O, perpen- dicular to the axis) the angle 7; this will be the inelination. of the orbit. Further, let @ be the longitude of the peri- helion, counted from the ascending node &, (the tine of nodes: being the intersection of the orbit plane with the fixed equatorial plane), and @ the angle between the instantaneous radius vector and O28. Then, if ¢ be the eccentricity, the equation of the orbit is oe bP ecos (Oa a 5) Y where p=imi°0 = wie and our previous 7 is related to 2, 8 by ; COSI] —* Sill 175 SIMIC en cays ere CLO!) The disturbing function (13) becomes, by (15) and (16), F=6.'1+ecos(@—o) |*. [3 sin’? sin? @—1], where NU x ene Kae’ a?m =e S| SO ee al) ) ye He ieee L oS On Whence, the required average F= > | Fd, AEE 0 F = 3b.sin?i [14 3e?(14+2 sin?) ]—0(1+3e). (18) Notice that this mean value of the disturbing function contains jour of the elements of the undisturbed orbit, to wit, its parameter through p (appearing in 4), its inclination i, eccentricity e, and the longitude of the peri- helion w. Ot these only the three first can be ‘ quantized ” (2. e. fixed in terms of integers and h), while the last; , will retain its freedom of assuming any value between 0 and 27. This feature, most immediately conditioned by the absence of radial symmetry or isotropy. (replaced by axial symmetry ) will give rise to diffuse lines, 7. ¢. spectrum lines of finite breadth. The only orbits orice will give rise to ideally sharp lines will be those for which e.sin2, the coefficient. of the non-quantizable term in (18), vanishes, 2. e. all circular orbits, whatever their inclination, and all equatorial orbits, whatever their eccentricity. This will become more clear presently. Let us now quantize the three elements p, 7, ¢ by the usual D4 Dr. L. Silberstein on the Quantum principles, according to what was said above, immediately after formula(11). This will be done exactly as in that part of Sommerteld’s paper (Ann. der Phys. vol. 51) which is based on simple Newtonian mechanics, 7. e. without rela- tivistic refinements and, of course, for ane undisturbed system (nucleus =point-charge). Namely, let @ be, as in (5), the angle between a variable and a fixed meridian plane (say, that passing through the line of nedes), so that the kinetic oer GAR MENS ab pO OPE 81 a9 £2 = Zill Jy Pena Te SING. De i and let us quantize with respect to d, 7, 7. Thus Or Ve or ae AD =I 0, An =Nol i ( ~ —dd=noh \ Po! b=inyh, ‘ Py N= Noh, pd? a Pr 6 Nolt, Fes emt) Po= m'r? sin? 7 d, Pn= DUP Dee te Remembering that p=m'r’@, it will be seen at once that where Po = P CO8t, Pe being simply the projection of p from the orbit plane upon the equatorial plane. Thus, both p and z being constant, the first of (19) gives simply pcosi=7jh/27. The second and the third of (19) become at once, in virtue of the orbit equa- tion (15), 2mp(1—cos 7)=noh, and 2rp[(1—e)-2=—1]=ngh, as in Sommerfeld’s paper, the only ditference (not aftecting the value of the integral) being that @ is replaced by 0—o. Thus the quantized values of the three elliptic elements ae in terms of the three independent integers appearing in ch n Le ella P=, OS -e=(, y 206 Ny + Ns? ny + y+ ns se es (20) These are to be substituted in JW, as well as in the expression (18) for #. The sum of Wo and /' will give the required VW, as in (11). Now, W, is easily found to be equal to «etm Ae) aie and ‘hereto. by (20), and with the previous meaning of x K?Reh (my + Ng +N)” while (18), with } as in (17), becomes, in virtue of the first of (20), pe le ie Pea (hy ns)? Wa } [1+ 2e7(14+ 2 sin? w) ] sin? ?—(2 +e?) }. Theory of Spectrum Emission. DS, Let us still write (a 2, + Ny + 2s. Thus, ultimately, we shall have, for the total energy corresponding to any of the contemplated “ stationary” orbits of the electron round the two-centres nucleus, | ussebontcok at nv? ( (7, €) t A 1 chor eile cs (n—n3)8* J i ; @) where : ae aly; y= l eg (a pure number), . . . (21-1) g(@, €-) = 1+3e—Fsin?2.[1+#e(1+2sin’@)], (21-2) and, as in (20), | Mie Sw) i ee nt) Ay HO @r3) nyt ny’ n The cor responding spectrum series being given by y=AW /ch, it is not difficult to see the meaning of fhese formulee. If the two centres coincided («=0), y=0, and we should have an ordinary Balmerian series consisting of ideally sharp, simple lines, the three independent integers then appearing only through n=ny;+ng+n3, and their individual contri- butions being entirely irrelevant for the result. Owing to the complexity of the nucleus, as here con- templated, there is a general shitt of the spectrum, dependent on the numerical value of y’, and, instead of a single sharp line (which would correspond to a given pair of initial and final n-values), there is a plurality of lines, some sharp, but most of finite breadth. Whether or not at least the general Balmerian type of the distribution of these groups of lines (as we may call all the lines corresponding to a fixed pair of initial and final »- or sum-values) will be preserved, will most essentially depend on the relative separation of the centres, that is, on the numerical value of y. If y’ be but a very small fraction, we shall still have a Balmer series, although not of lines but of tightly packed groups, e each group “being a doublet or a triplet, ete., as the case may be, and each of their components consisting of several sub-components, sharp or broad,—in short, a Balmer series with fine structure of its members. But should y? mount up to more conspicuous values, even the type of the series would be entirely modified, 7. ¢., as in the case of the spectra of most chemical elements, altogether different from the Balmerian type. Both possibilities seem interesting and, perhaps, promising. 56 Dr. L. Silberstein on the Quantum A glance on the form of (21) and (21:2) will suffice to see: that the splitting and the modification of the spectrum due to the asphericity of the nucleus are of an entirely different nature from the fine structure due to the relativistic com- plications, i.e. due to the variability of the electron’s mass, as obtained in (the second part of) Sommerfeld’s paper, cited above. After these generalities let us return to formule (21), ete... with the purpose of discussing them in some detail. ‘These formulz seem to deserve some careful attention, the more so as their validity is by no means limited to a nucleus consisting of two isotropic centres (which would be a somewhat puerile assumption) but extends tou a much larger class of cases, viz.. to atomic systems with any axially symmetrical nuclei. In fact, the whole set of our formulee will continue to hold in all omeb cases without any formal alterations whatever, the only difference being that a (appearing in +) will have a different or rather a more general geometrical meaning. The exact meaning to be then attributed to a, Which is not difficult to guess, will be explained in Section 5. Thus, returning to the last set of formule, let us first note that whatever the length a characterizing an axially sym- metric nucleus, it appears only through y, (21° 1), where it is. divided by another, practically universal length. Let us estimate this length, 7 e” a [ces Reh °° that y=2«7> « being always the number of e-charges contained in the wholenucleus. Now, in c.a.s. units, e=4°7. 10™"; Conon p=(5°D) Oat Byars the Rydberg constant,” even allowing for the slightly different values of m’ as distineuished from m |cf. formula (2) ], can safely be assumed to be always of the order A— I-00 108) bus: feats Oy ° 1) +> » e or almost just one A.U. If, therefore we write, for con- venience, ‘ Ce) ING Us the value of the coefficient in question will be S26 pe Y eran 2K, ° ° e ° ° ° 22) which reads: y equal to twice the number of Angstrém units contained in @ multiplied by the total number of electronic charges contained in the nucleus. | Now, @ is certainly but a fraction (and the very validity Theory of Spectrum Emission. — . OT of the approximate perturbation method is based thereupon). But how small or how large this fraction may be, is difficult to say. If, for the moment, 2a is taken to be the diameter of the nucleus, then for the few gases involved in Rutherford’s. scattering experiments, a is certainly as small as 107" or 10-" em.} that is to say, « of the order 107% or 10°4*. But for other elements, and especially for the heavy ones, « may be much larger than this. Then, « containing many units,, y may become quite a conspicuous fraction of unity. In short, there is a wide range of values which y can assume. Tf it is near its lower limit we shall still have a Balmer series, although slightly shifted and ved fine structure of its members ; but if it mounts up, say to = or only = sp» ube very type of the series may cease Moca to resemble the Balmerian one. Within these wide limits, it would certainly be useless to attempt to prejudice the value of a or of ¥ for, say, the atoms of copper or of iron. On the contrary, one would have first to re-examine carefully and to try to disentangle the spectra of such elements and base the guesses about the appropriate value of y, and thence of a, upon the observed features of their spectrum “ series.” Having thus dealt sufficiently (for the present state of our knowledge) with the numerical aspect of the coefficient y, let us pay some attention to the equation (21) itself, together wath (21-2) and (21-3). It may be well to write that equation in a form exhibiting directly the ratio of a to the dimensions of the orbit, so as to have before our eyes the assumption of its smallness, upon which the validity of the equation rests. Such a form is easily obtained when it is remembered that (21) is but the developed form of = ae i1+ ee ch n? W Now, Wo=x?e'm'(1—e?)/2p7, and, as in (18), retaining the symbol g(?, e), 12 = Cem? 5 a ape and, whence ey Ke tie ee) O20 Wo hie pe | Lae: * If, as Sir E. Rutherford is inclined to assume, the mass of the nucleus (and therefore, practically, of the whole atom) is purely electromagnetic, then, for x=1 say, a would be about sit of the radius of an electron or a=510-" em., and therefore, e=3.10~%. But such an assumption is by no means necessary. There is, in fact, no evidence whatever for the existence of pure positive charges. 58 Dr. L. Silberstein on the Quantum But, by the orbit equation (15), p?/ke?m' is the parameter (latus rectum) of the orbit, 7. «. a(1—e), if a be the mean distance or the semimajor axis of the orbit*. Thus, the required form of (21), Wh eh a\? g(t, €) W 2B, (a) eo) a where 7 is as in (21:2). This shows us that for all not very eccentric orbits the supplementary term, due to anisotropy 9 =< re e a re ° of nucleus, is of the order of (<) , which is a fraction at any rate. At the same time we see that the coefficient y has the simple geometrical meaning [This is a constant, as it should be; for, by (20), a itself is proportional to 2. The last formula, coupled with (22), a : : : gives 2Ka= ‘ n*, which contains the warning not to expect ¢ i sufficient accuracy from our formule for those atoms for which the traction 2«« may acquire 2 comparatively huge value.| From (21') we see also that even for smal] = the deviation of the series from the Balmer type can become dominant if the orbits are strongly eccentric, that is to say, when 73 is a large number in comparison with n, +n. Returning once more to formule (21), (21:2), (21:3), let us consider them in connexion with the various shapes and orientations of the orbits. | From the last set of these formule we see that all meridian orlits (2. e. those contained in any meridian plane, cosi=0) are given by 0 3 the corresponding value of ¢ is 7 Ra Ae 3 Ne" | g{=,e)/=——+-e(1—6sin’o), @=l1— =e (NE “ &G ‘ ectae ); (no+ ns)” ) * Thus, the smallest non-vanishing value of p being h/27, the smallest «stationary ” orbit has the parameter or, if circular, the radius IieeueelOS = 2 — = —— 10-* em., 2x Reh * 2k inversely proportional to x. It will be important to keep this well in mind, especially for the heavier atoms. | Theory of Spectrum Emission. ag OF these only the cireular orbits (nzs=7,=0), for which g=—tHt, will give rise to ideally sharp lines, furming series 2 of the type pi Gea Ml y=eR | (1 or 75) ~2(14 all on ale a ‘The remaining elliptical orbits (n3>0) will give rise to more or less road lines; the greater the eccentricity, the broader the lines. The part of y responsible for this effect is C 22S 2 Sie a Ang 3 ranging therefore, for the different individual atoms, from 8) 4 4 All equatorial orbits ((=0) are given by tg) and the corresponding value of g being independent of the perihelion longitude, r Zin, D a(0, €) = i Saye EA et hoy (I) all these orbits will give rise to sharp lines, no matter what their eccentricity. For the sub-class of circular orbits (7g=1,=0) we shall have g=1, so that the corresponding lines will form the series y=er|—, —*4-¥(a-35)]- wee a) Compare this with (M,). All circular orbits (e=0), of any inclination, are given by lg= Q. ~The corresponding vaiue of g being : Be oe (Zit) — Ns) — Ne. . GeO) tein IME co.cc PSI] independent of the perihelion, all these orbits will give rise to sharp lines, of which, in general, a plurality will enter -into one group. (Thus, for instance, if n=n,+n.=2, we shall 20 for 0+2, 1+1, 2+0, the ree pee es g=—tor —ior +1, ‘and for? n=3, g= —4, —4, +t or +1, so that even the group 2, 3 would contain far iG circular i -components, so to call them shortly. Which of these are to -be rejected as being more or less “ improbable” is a further 60 Dr. L. Silberstein on the Quantum question.) The energy belonging to any circular orbit is. given by | Ws eke | erin kool. whence also the corresponding frequency formula. The orbits responsible for broad “lines” or bands will be all those and those only for which both sinz and e differ pies . ; 2) from zero, 2. e. No, Ng FO. There is no harm in ealling such orbits (diffuse or) broad orlits*, and the remaining ones, sharp orbits. In this nomenclature the passage from a broad to a sharp orbit, or vice versa, will give rise to a line of finite breadth, and a band will be still broader if it is due to the passage from a broad to a broad orbit. If it is assumed, for instance, that, for any fixed e, 72, all the longitudes of the perihelion are equally probable (which need by no meaus be the ease), then, . the mean value of sin’ being 3, the mean value of g will be, by (21:2), g=1 +26) 1 —s5in22); 2 eee) and its two extreme values, corresponding to sinw=0 and 1, 2 gece sate, ie oe The mean energy belonging to a broad orbit will be obtained by substituting g for g in (21), and the position of the centre of a band vail be given by v=(W’— W)/ch, while the breadth of a band due to the passage from, say, a broad orbit (7) to a sharp orbit, will be given 1 by QO 2 Rey? : Sy ee (24) and similarly for the passage from a broad to a broad orbit by combining the appropriate extreme values of g and 9’. In the case of an equatorial final orbit, for instance, (24). becomes, with y =2«a, as in (22), Wererlay OV eee (n—ns) To form an idea of the numerical relations take, for- example, 1 =3, and the least eccentric orbit compatible with iN Oe (ey idea c= Then Sv= g, kta? R = 8600«42? em.-}, * A “broad orbit” will thus stand for many orbits, all having the- same @, 7, €, but all possible perihelion longitudes o, from 0 to Dare belonging | to the individual atoms of the emitting substance. = & (2£e)- Theory of Spectrum Emission. 6L cand if, for instance, we require the breadth of the line to be [dd | =0° 01 A.U., to which corresponds in, say, the red region of the spectrum, 6v=0'023 cm.~', we Bnouid have Ka= 00952, | that is to say, the value of a required to give such a line, sid A.U. broad, would be 200 9- OGG: Ke? 4. e. for e=1 (as in the case of hydrogen) a=5°2.107~" cm., and soon. Notice ao the breadth of the line w an coterts paribus, increase as «‘a’, Thus, if we had, say, only a==5.107-" cm. for such atoms as “those of uranium en e)) the “line” in question would be drawn out into a very broad, band, in tact, a continuous spectrum extending over 7160 A. U.,and even a=107* would still givea band stretching over little less than 3. A.U. And it will be kept in mind that in these heavier atoms larger a are more likely than in the light ones. 5. Any axially symmetrical nucleus. As has been already mentioned in the preceding Section, the set of formule (21) ‘continues to hold for any axially symmetrical nucleus whatever. The reason is that the disturbing function F is for an atomic system with any such nucleus exactly of the form (13). The only difference is that the previous seml- distance a of the two centres acquires a more general meaning, and the whole expression may have one or the opposite sien according as the nucleus is “oblate” or “‘»rolate,” in a generalized sense of these words. In fact, let the whole charge «xe of the nucleus be distri- buted in any manner whatever over a volume tr. Let O be the “centre of mass” (electric analogue of ordinary mass- centre), and A, B, C the principal *‘ moments of inertia’ of the whole charge, each divided by the charge, 7.¢., if v,, 71, 21 be the coordinates of a charge element de along the principal axes, with O as origin, let t= bflgtees HESCUC aa w a) tee (a) Let the Miewiagn be at je. and let A be the “moment of inertia” of the nucleus about OP, divided by «e. Then, if OP =r is large compared with the ddnensions of the nucleus, the negative “potential energy of the system will be, by a well-known result of the ordinary potential theory, p2 fl ES f ul ON AE ae a ys AtBr C aK), 62 Dr. L. Silberstein on the Quantum Thus our disturbing function F’ will be Px (A BO SK. Here 4, B, C are constants, characterizing the nucleus,. their dimensions being, by (25), those of a squared length, and A’ is a function of the orientation of OP with respect to the principal axes, to wit, if 7,, 72, 73 be the angles which OP makes with these axes, to which belong A, B, C respectively, K=A cos’ n,+ B cos? ny + C cos? ns, where, of course, cos? ,+ cos? + cos?73=1. Thus the disturbing function will be pak a :{ A(A—3 c0s? 91)+ B(L—3 cos? np) +\O(1 —3.cos?o) |: ee J for a nucleus of any shape and of anv charge distribution whatever. In particular, tor any qaially symmetrical nucleus, with say the A-axis as axis of symmetry, 7. e. with C=B, and with 9 written for, the disturbing function of the atomic system becomes at once a F=5 a =A) ous? p= ll). which is exactly of the form (13), the squared semi-distance of the two centres, a’, being here replaced by B—A, Thus, as was announced, the set of formule (21), etc. continues to hold for any axially symmetrical nucleus. If B> A, we will say, shortly, that the nucleus is “ prolate,” and if A> B, that it is “oblate” (for such it would be in the ordinary oeometrical sense of the words if its charge were, for instance, uniformly distributed). In view of these two sossibilities, affecting the sign of the perturbational term, it will be better to rewrite here the generalized equation (2 Me thus ‘ap (i Wo ge »n*q(t, €) ae a eye a Gea) id) ie a r (n—ns) op ee) eR 1/2 where ye ele A_B\ ee Gen a the positive sign to be taken for an oblate, and the negative Theory of Spectrum Emussion. 63. sion for a prolate nucleus*. The value of g(2, €) is still as mae 21:2), wath? (213). This set of formule determines the spectra corresponding to large electronic orbits round any axially symmetrical nucleus. T,, for instance, the nucleus be a homogeneous rotational ellipsoid of semi- axes 2 and b (ie former being the axis of symmetry), then A=?b? and B=} (a? +b’), so that Ala |e a If this ellipsoid be oblate, and if ¢ 7 the eccentricity of its generating ellipse, then 2elvek \. OG Oe NES so that our previous a is to be replaced by b&/\/5, and the positive sign is to be taken in (28). And if it be a prolate ellipsoid then a is to be replaced by af/\/5, and the negative sion is to betaken. In either case the effect on the spectrum is seen to be, cvteris paribus, proportional to-€?. 6. Nucleus of any shape. In the most general case the disturbing function is given by (26). The coefficients A, B, C being all different from one another, the longitude ce ae node oO: hitherto irrelevant and physically meaningless, comes to its rights and, in addition to the perihelion, becomes a fresh source of broadening of the spectrum lines. For, as in the case of w, there is no way of quantizing the longitude of the node. If the orbit is written as in (15), if 2 be the inclination of the orbit to the BC-plane (so that 7, becomes our previous 7), and if the longitude © of the ascending node be counted from the b-axis, we have cos 9, = Sin? sin 8, COS ny= cos 8 cos Q— sin O sin O cosi, eos 73= cos 8 sin O + sin @ cos 0 cosi. These values are to be substituted into (26). It will, for the present, be enough to write down the result for the case in which all node longitudes are cane lly distributed among the atoms. Thena jena containing sin 20, disappears in the * The previous two centres formed, of course, a nucleus of the latter kind. ‘64 Dr. L. Silberstein on the Quantum average taken over all atoms, and the relevant part o£ the disturbing function # is given by 273 || COS: 0+ Csin? O— (B sin? 04+ C'cos? Q) cos? 2 | — A sin?7| sin? 6 +1(A+B4+C)—(B6 c03?04+ Csin? QD). 2 . (29) - If, keeping in mind the finite breadth of the lines due to Bre? ©, we desire only to deal with the position of the centres of these “lines”? or bands, then it is enough to retain the average of # taken over all emitting atoms, which we may -denote by /,. It is manifestly legitimate to take first this average, and then the average, ta over a period of the undisturbed motion. ‘hus (29) reduces at once to 2 OY) ae h ee ae: sin?2 sin?O—1)) 2 (29%) which is exactly of the same form as (13), with a? replaced ‘by VBS OA. Thus, taking the time average, the corresponding energy will again be given by (88), with the only difference that the coefficient y, (281), will now be replaced by the more general one, ~ ye A 3B C)ie The value of gi,e) will still be as in (21:2), and the eccentricity € will, manifestly, be quantized as before. It yemains, however, to see whether 72, the inclination *, does not now in this respect behave differently. Returning to equations (19) and to what immediately followed upon them, we shall see that the only difference is that the relation tan @= tan @.cos7 has now to be replaced by tan (6—Q) =tan 8. cos. This, however, does not affect the results. The geometrically -obvious relation Po=P ©O8 2. -continues to hold, and therefore pcosi=n,h/2m. And since it will be remembered that z was taken with reference to the Bont and this is the reason why 4 is privileged in the last two or -three formule. Theory of Spectrum I’mission. 65 p, is as before, we have also 27(1— cos7)=n2h. In short, all the equations (20), including ny cos i= ee retain their validity. Thus g(t, €) is, also in terms of the integers 2, 19, 73, exactly as before. But with equal right as BC we could have taken CA or AP as reference planes for the inclination, and every time 2 would be quantized in exactly the same way. Thus, the asymimetry of the nucleus gives rise to new “ stationary ” orbits and, therefore, to new spectrum lines or bands. Ultimately, borane. we shall have for the determination of the centres of the bande writing W, instead of W in (28), the equation WW ac Pity or EG oe ae aa) ch n (2 —73)°® where G6. Ube: il —s sine a) ye fA(e0 1) with ¢,7 as in (20), and where y has any of the three values 2 | aa j 1/2 awl a aa | B—3(C+-A) |", eA 2 ea OV 630.9) The positive or the Pai, sign is to be taken in (30), according as A is greater or smaller than $(6+C), and similarly for the remaining two values of ¥. Thus the spectrum belonging to a nucleus without any axis of symmetry would be much richer in lines or com- ponents than that of an axially symmetrical nucleus, for which we have had only, as in (28:1), A—B|!? instead of the three values in (30:2). But looking back upon that case from the point of view of the present more general case, 7. e: putting in (30°2) C=, we should have for an axially symmetric nucleus not only the previously described lines or “components” but also those corre- sponding to T= |H(B- A)? silane agers ag with (30) and (30:1). In fact, the former lines were obtained by taking the equatorial plane as the plane 7=0, and the reason was that this (or its normal, the axis of Phil. Mag. S. 6. Vol. 39. No. 229. Jan. 1920. iy 66 Major W. T. David on the Calculation of symmetry) was uniquely privileged. But from the present, more general point of view there is certainly no reason for rejecting a meridian plane as the reference plane (:=0), and this, 7.e. any such reference plane, will precisely give, in addition to (28°1) the new coefficient (28'1’) and thus also new lines or components. Still, the asymmetric nucleus will be richer in possible lines or components. Which of these are to be rejected, can be decided only on the ground of spectroscopic experience. One might put forward, for instance, as a restricting eri- terion that out of the three y-values (30:2), that only is to be retained to which, evterts paribus, corresponds the greatest W,, or perhaps the smallest W,, or what not. But all such guesswork, unsupported by observation, would be pretty useless. The chief purpose of the present paper has been to show the numerous and broad possibilities opened up by a non- spherical nucleus both with regard to the fine structure of lines or groups and to the very type of the spectrum series. All numerical applications of the general formule and their discussion in connexion with experimental spectroscopic knowledge are necessarily postponed to a later opportunity. Research Dept., Adam Hilger. London, October 18, 1919. VI. The Calculation of Radiation emitted in Gaseous Explosions from the Pressure-Time Curves. By Major W. T. Davin, M.A., M.Sce.* INTRODUCTION. IL, lhe this paper an attempt is made to build up formule by means of which the radiation emitted in explosions of inflammable mixtures of coal-gas and air may be caleu- lated from the pressure-time curves. A large number of photographic films on which were traced curves of pressure and of radiation emitted | have been examined, and it has been found possible to establish some simple equations which would seem to apply within fairly wide limits of mixture strength, density, and volume. * Communicated by the Author. + Curves taken from a considerable number of these films have previously been published in the Phil. Trans. (A. vol. cexi. pp. 370-410). Reference should be made to this paper for full details of the experi- ‘mental methods employed. Radiation emitted in Gaseous Explosions. 67 2. It will be convenient to describe here very briefly the experimental arrangements under which these films were taken. The gaseous mixtures were exploded in a cylindrical cast-iron vessel 30 cm. in diameter and 30 cm. in length (shown in fig. 1). The pressures of the gaseous mixtures INT SS = zz ————= — ——- re —VLLLLLLLLL LLL LL WK Scale during explosion and subsequent cooling were measured by means of a Hopkinson optical indicator which threw a spot of light on to a revolving photographic film. The radiation was measured by means of a platinum bolometer connected with a reflecting galvanometer which also threw a spot of hight on to the same revolving film. The bolometer was protected from the hot gaseous mixture by means of a plate of fluorite as shown in fig. 1. The fluorite transmits almost exactly 95 per cent. of radiation of the wave-length emitted by an exploded coal-gas and air mixture. After making an allowance of 5 per cent. for the absorption of the fluorite and 5 per cent. for reflexion from the blackened surface of the bolometer, it is considered that the measurements from the films give radiation values of a high degree of accuracy. F 2 38/07 CF EF. PEL SEO Fr. S pers OF Evry orf thet fore jee) Curve #'AP~ fe PE OVBEICS? - ECGhOTi ft e eS “TVe (scale La? 68 Major W. T. David on the Calculation of 3. The radiation emitted was measured in three positions: on one of the end-curves of the explosion vessel, viz. at the top (position A), at the bottom (position B), and at the centre (position C). The radiation measured at position C was a little greater than that at either A or B; and that measured at A was a little greater than that at B. The mean value of the radiation received at A, B, and C gives a fairly accurate estimate of the average radiation over the whole vessel. 4, When the bolometer was placed close up to the window of fluorite the radiation it received per sq. cm. of its surface: was equal to that received by a sq. cm. of wall surface in the immediate neighbourhood ; and when the bolometer was placed some distance away from the fluorite plate the radia- tion it received came from a cone of the gaseous mixture of small solid angle (N, as shown in fig. 1). Throughout this paper the radiation measured with the bolometer in the latter position has been divided by the solid angle of the cone so as. to give the radiation from a cone of unit solid angle. The rate of emission from a cone of gaseous mixture of unit solid angle has been called by Prot. Callendar the intrinsic radiance. 5 [i ee ~ ~ TT OTE yh ims vw ie \" 7 e | | * Eh UD = _ a “Al, — Curve RES-Radratren per SE. Cf7. Curve FPDE-Gas Pressure (abs). 6) A GE Al Time x+*r060R Ig NtEron ). A list of the symbols used in this paper is given below. Tn ce them it will be convenient to refer to the curves. in e272) which have been t taken from a typical record. nme BY ! a a evita ES UE AIT NN - te ae PP max. P P max. é a poke Rr Radiation emitted in Gaseous Haplosions. 69 —time from ignition (OA). —time of explosion (OC), i. e. time taken by gaseous mixture to develop its maximum pressure. —pressure of gaseous mixture at time ¢ in lb. per sq. in. (abs.) GX): —maximum pressure (CD). —rate of change of pressure at time ¢. —the maximum rate of change of pressure (which occurs during the explosion period). —mean absolute “temperature in °C. of the gaseous mixture at time ¢ calculated from the pressure curve by means of the equation pu=Ré after making a small correction (of the order of 3 per cent.) ‘for contraction of volume which occurs on the combustion of the coal-gas. ax, —-Maximum mean eee a developed in the gaseous mixtur —rate of change of mean time ¢. gas tem perature at . —maximum rate of change of mean gas tempera- ture (which occurs during the explosion period), —quantity of coal-gas present in explosion vessel (measured in litres at atmospheric temperature and pressure). —volume of cylindrical vessel in c.c. of dimensions ° vie 2 / cm. diameter and / cm. long= rae (In the explosion vessel used in the experiments /==30.) —area of interior surface of vessel in sq.cm. (In the explosion vessel used in the experiments a=4380 sq. cm.) —density of gaseous mixture in atmospheres. —total radiation received by walls of explosion vessel per sq. cm. of surface at time ¢ measured in calories (AH, when bolometer is close up to fluorite window). —final value of R, 2.e. the value of RK registered after a time when the gaseous mixture has cooled to such an extent that it emits no further radiation (KJ). For all practical purposes this is the radiation registered at 1 sec. after ignition, for after this time the radiation emitted was insignificant. 70 Major W. T. David on the Calculation of Rh —differential coefficient of R with respect to ¢, i.e. the rate at which the walls receive radia- ; tion per sq. cm. of surface (AF). Ryax- —the maximum rate at which the walls: receive radiation per sq. em. of surface (HG). This takes place during the explosion period. R, —total radiation at time ¢ received by the bolometer per sq. cm. of its surface from a cone of gaseous mixture of unit solid angle (AH, when bolo- meter 1s some distance away from fluorite window). ! Riv —final value of R,, 2.e. the value of R; registered after a time when the gaseous mixture has cooled to such an extent that it emits no further radiation (KJ). For all practical pur- poses this is the value of R; registered at 1 second after ignition. R; —differential coefficient of R; with respect to ¢. This is the intrinsic radiance (AF). Rinax-—the maximum value of R; which occurs during the explosion period (HG). — Tar Maximum Rate or Emission oF RADIATION DURING EXPLOSION. (a) Cylindrical ELezplosion Vessel 30 cm. diameter and 30 em. long. 6. It has previously been shown ™* that the rate at which the gaseous mixture emits radiation is a maximum some little time before the attainment of maximum pressure. From an examination of a large number of films it would appear that the maximum rate of emission occurs at or near the second point of inflexion in the pressure curve during the explosion period. The curves in fig. 3 have been taken from three typical films. The A curves relate to a 10°5 per cent. mixture of coal-gas and air at atmospheric density; the B curves to a 10°2 per cent. mixture at 1°24 atmospheres. density ; and the © curves to a 9:2 per cent. mixture at 1°37 atmospheres densityt. The dotted curves are the dif- ferentials of the radiation curves, and they show clearly that the maximum rate of emission occurs near the time when the gas-pressure or temperature curves undergo their second * Phil. Trans. A. vol. cexi. p. 381. + All the gaseous mixtures referred to in this paper were originally at atmospheric temperature. 2600 Radiation emitted in Gaseous Explosions. 71 inflexion, which, of course, takes place immediately after the period during which the rate of rise of gas pressure or temperature Is a maximum. t Rate of Emission of Radiation | Calories per sg. C77, per Sec. Time-seconds. A curves: 10°5 % mixture of coal-gas and air at atmospheric density. B_ curves: 10:2 % mixture of coal-gas and air at 1:24 atmospheres density. C eurves: 9:2 ¥ 7. It occurred to the writer, therefore, to make an attempt to connect the maximum rate of emission (R max.) With the maximum rate of rise of pressure (7p max.) or of temperature (6 max.)> and it was discovered that for mixtures containing the same quantity (Q) of coal-gas R ax, was proportional to the square root of 6 max. This will be seen on comparing columns 1 and9in Table I. It wili be noticed that the ratio R max i is practically the same in all cases where Q has the G 0 max. same value. Further, this ratio increases as Q increases and it seems to be very approximately proportional to the square root of Q. The tigures in column 10 are the ratios of the figures in column 9 to \1/Q—1.e. they are the ratio R max. MOO cac , mixture of coal-gas and air at 1°37 atmospheres density. Aadiation-cealories Sg. Cm, ar --OLXOO1 [7 avery . Es in: Sas 3 zo cee = See Seen 2 Z0-1 6-G §-9 81-6 CL-y O88¢ 96 G O¢-T 0-¢1 GO. Ss & 80-T 0:6 8:6 TFl 00-6 OCT ¥8-1 GET 0-01 LE-§ ~ Q0-T 0. 9-F SEG 99-¢ 09cE t8-1 66-1 Gol LE-€ 5 60-T 0-4 BG L9G FLL OF9E €8-T 0-T¢ 0-¢1 TE-8 iss) SS 80-1 8.1 C1 92-0 FL-0 0¢G L9.T 86-1 G6 ORG S 96-0 9.1 CG ge. OF-G 06c1 L9.T FE-t GO | 08-6 S 06-0 9.1 L&é 16.1 88-€ OLGG LY: 1 S1-T é-I1 08-¢ 2 ZO-T Dit 0-F OF-G CLG O0C6G 19-1 0-1 C.cf | 08-6 = 06-0 Gat 9.F CO GL-6 O8té L9-1 CLO LOL | LLG = | F0-T 9.1 ZS €6-1 Cue 0061 6¢.1 0-1 CrOl — |= Ck A 00-1 on G1 cg. ley | 008 GF Di | ea 16-3 So | _: | 8-OLX€60 | 3-OIXZ1 G-¢ z2OT X 99-6 pO X TL OOST 62-T ¢-0 0-S1 | 29:1 = | | a] — eS cS -xeur “XeUr : ; ‘gas aad .| *atngx e | pon : a UL ue: (assed Qa) | ee ay) DA “(ado ydsornyr) anon: (any) 4 xeUh XP Uy ca 7) “XEUL : | a “quedteg | w) ° | "| Hav, Radiation emitted in Gaseous Haplosions. ies for the various mixtures (which it should be noticed vary in density as well as in strength). These ratios are all pretty much the same—indeed, having regard to the large varia- tions in Rynax, Omax Q, and D it is somewhat surprising to find them agree with one another so closely. 8. It would therefore appear that certainly within the limits of mixture strength 9°2 per cent. to 15 per cent.” and of density = atmosphere to a little over 1} atmospheres (as shown in Table I.) R max, in the explosion vessel used in the experi- ments is proportional to the square root of the product Q x @max.. Expressed in the form of an equation R max. — KV/Q x Gn BB aia ho Vis) yrs vale (1) where K is a constant equal to 1:0 x 107? or 0°01, 2. e. the mean value of the figures in column 10. It may be noticed that as 6 is proportional to P (the ratio D @xD —— being equal to 19°6) this equation may be written i () x i max. R max, = Ky AUR Bane: cipal Sunt ° (2) where K,=0:01 x \/19°6=0°0445. 9. Some confirmation is lent to the proportionality between - R ax. 20d 1 Q X 6 wax. from the writer’s measurements of the intrinsic radiance of coal-gas and air mixtures at various densities from 4 atmosphere to 14 atmospheres f. These are set out in Table II. (which contains similar information to that given in Table I.). It will be noticed that the ratios in column 10 are again very much the same, though of course . . UM ryy they differ from those in column 10 in ‘lable I. because the values of Rimax, are different from those of R,.. The equations for the maximum intrinsic radiance are therefore Ferner ONIA/ OX Grace «Pe dunys C3) and Bemus. =0-0524 / ® aa * A 15 per cent. mixture is the strongest that will burn completely t. €. 1t uses up the whole of the oxygen of the air. The weakest mixture that will explode is probably between 7 per cent. and 8 per cent. The limits of mixture strength in these experiments are therefore very wide. + Phil. Trans. A. vol. cexi. p. 395. / von O Major W. T. David on the Calculat 74 FOIL Te = UNOTy ier | 9% ol OL 09-4 «O88 00-6 S31 0-61 6L-+ 0z-T rate oR) 163 03-8 | 00¢F 3-1 0-1 0-CT FEE LUT 0:6 Nesp G8-G GPG O9LG IL-1 0:1 (05) a A Gel 8-1 9.G 02-8 1-6 OSPS 6¢-T cL.0 0-S1 GO-G e-OLX9L-1T | g-O1X@1 8.8 sOLX0G% — O01 XL0-9 0091 6a-1 G.0 0-G1 19-1 POX ot) N ‘XUUE 2,7. ‘xe | (as ted “9 9) (oes aod 4 ‘(eraydsow ye) Sw Jo *(So.ty1]) | *xeUl a. “XUOlI, a | ‘xeut ‘ur ‘bs aad qq) | “O/ a Nees o- “a a 0 “XBU “quale d TI @tavy, Radiation emitted in Gaseous Explosions. 75 10. In considering these formule it is suggestive to regard Q as being proportional to the density of distribution of chemical energy in the inflammable mixture and 6 as being proportional to the rate of conversion of chemical energy into thermal energy. Ina previous paper it has been sug- gested that the chemical energy in process of conversion first passes (either wholly or a considerable proportion of it) into the form of internal vibrations of the combining molecules, and it is reasonable to suppose that the radiation to which these vibrations give rise and which comes from the whole mass of the burning gas should be a maximum towards the end of the period during which combustion proceeds at its maximum rate. (b) Cylindrical Explosion Vessel 1 em. diameter and | em. long. 11. It will be realized that the radiation values in the pre- ceding sections have referred to explosions in a single vessel 30 em. in diameter and 30 cm. long. Some indication of the effect of size of vessel on the radiation values may, however, be obtained trom the writer’s experiments on the diathermancy of coal-gas and air mixtures exploded in a vessel with silver- plated walls which could be made reflecting by polishing or absorbent as regards radiation received by them by coating with a thin layer of dull black paint *. 12. In fig. 4 an attempt is made to plot Re (which is. proportional to R max.) for mixtures of various densities for a cylindrical vessel / cm. diameter and / cm. long against .. Confine the attention first to the thick curve which con- nects Ree with / for a 15 per cent. mixture at atmospheric density. The-value for /=30 (viz. 6°6 calories per sq. cm.. per sec.) is, of course, directly obtained from the experiments, for the cylindrical vessel used in these experiments was of dimensions 30 cm. by 30 cm. It is clear, too, that kR;=0 for !=0(, so that there are two definite points on the curve. Two otner points, one for /=15 and the other for /=59, may be roughly fixed in the light of the following considerations.. That for /=59 will certainly lie somewhere between eee for 59 cm. in the polished vessel (which is 10°9) and Rimax. for 59 cm. in the blackened vessel (which is 8-4) = Ph. irans, vol. CCx1. p. odo. 76 Major W. T. David on the Calculation of and probably will lie rather nearer the latter *. The other for /=15 will in all probability lie between R; max, from 15 cm. in the blackened vessel (which is 4°7) and half that from 30 cm. in the blackened vessel (which amounts to 3:3) +. Fig. 4, IS 30 45 60 Cylinder dimensions — Cl - cr7. Maximum Intrinste Radian cé - Catortes per $9. Ci? Per Sec. ™ The thick line then is a fair curve drawn through the two ‘definite points (0, 0) and (30, 6°6) and between the two pairs of points (15, 3°3-4:7) and (59, 8°4-10°9). * There seems little doubt that the radiation from a layer of gas increases not only witb its thickness but with its lateral dimensions as well. The radiation from a cylindrical mass of gas 59 em. x59 cm. will ‘therefore be greater than that from a cylindrical mass 59 cm. long and 30cm. in diameter. On the other hand, the radiation from 59 cm. in the polished vessel approximates to that from a mass of gas 59 cm. in thickness but of infinite lateral dimensions. + Ri max. from 15 em. in the blackened vessel will be greater than that ‘from a cylindrical mass 15 cm. by 15 em., because in the former case the lateral dimensions are greater than in the latter. Radiation emitted in Gaseous Hxplosions. Ee The dotted curves relate to 15 per cent. mixtures at 4 atmosphere, ? atmosphere, and 14 atmospheres density, and they have been drawn from similar data. It is not suggested that these curves give a very accurate relationship between Rj; max, and cylinder dimensions, but it is considered that they give a very definite indication of the shape of the true curves. 13. These curves (with the exception of that for $ atmo- sphere density) indicate that eye ae (and therefore R nag!) varies more or less as V/. The curve for 4 atmosphere density is practically a straight line up to/=30. This seems reasonable in view of the known higher transparency of the gaseous mixture at this density. Assuming the relationship een /1 to be reasonably near the truth, eyuations (1) and (2) may be amended to cover varying volume in the following way :— : owe ase =0°01 Vie VQ xO max. = OGNS3N/A le Ox Oy elsinan 6 (5) And equations (2), (3), and (4) will become— ry 4 Lx Q x Dine R nox. =0 oosi2.4 / ae = aa aa naa BA 00204 A/ OOK Oli ek CE) 7 s ~ l x () xp max. R; max. = 0 0095.4 / Wo ee 14, These equations it is thought may be expected to give values for Ryax and Rj max, correct to within perhaps 10 per cent. when applied to mixtures of coal-gas and air ranging from 9 to 19d per cent., within the limits of density 2 atmosphere to perhaps 14 or 2 atmospheres and within the limits of cylinder dimensions 15 cm. x 15 em. to 60 cm. x 60 cm. Further experimental work will be required in order to decide whether they apply approximately outside these limits. 78 Major W. T. David on the Calculation of RATE OF EMISSION OF RADIATION DURING COOLING. (a) Cylindrical Explosion Vessel 30 em. diameter and 30 cm. long. Gaseous Miatures at Atmospheric Density. 15. It is not possible to build up an accurate formula for It at any temperature under wide conditions of mixture strength, density, and volume until much further experi- mental work is accomplished. The whole question is complicated greatly by the fact that the transparency of the gaseous mixture varies with the time after ignition, and also (to a much less extent) because the vibratory energy of. the molecules is dependent upon other factors as well as temperature”. It is important, however, from the point of view of gas-engine calculations to be able to estimate the radiation loss at various points during the expansion stroke, and it is believed that an empirical formula sufficiently accurate for this purpose can be established. 16. It has been previously shown that the radiation from the gaseous mixture when corrected for absorption varies with the temperature approximately in accordance with Planck’s formula for a wave-length of 3°6u Tf (which at high temperatures varies very nearly as the square of the absolute temperature). Itso happens, however, that when uncorrected for absorption R in a vessel of the dimensions used in the experiments 1s approximately proportional to 64, and it is considered that a formula based upon this law will give gas- engine designers fairly reliable information as to radiation loss when applied to cylinders within the limits of dimensions 15 em. x 15 cm.to 60 cm. x 60 em.f * See Phil. Mag. Feb. 1913, p. 267. + Phil. Trans. A. vol. ecxi. p. 402. { See Phil. Trans. p. 386, fig. 9. The 6' curve there was made to coincide with R at comparatively low gas temperatures when the radia- tion is small. By increasing the scale a close agreement may be obtained at the high temperatures (2400° C. abs. to 1600° C. abs.), which is the important part of the curve from the point of view of the gas-engine. The agreement between the 6! curve and R in the neighbourhood of the maximum temperature in the 9°8 per cent. mixture is not so good, but a fairly correct value for R in this epoch for weak mixtures may be obtained by calculating the mean of R given by the 6! formula (equation (9) § 17) and I max. as determined by the equation in the preceding sections. The statement made in the paper referred to (footnote p. 402) to the effect that the 6‘ law would not hold in the case of a cylinder of widely different dimensions from those of the vessel used in the experiments requires some modification. ‘There seems little doubt that this statement Radiation emitted in Gaseous Explosions. 79 17. In a vessel 30 cm. in diameter and 30 cm. in length GMO Ors a et (9) where the constant 1:75x10-" has been determined from the writer’s experiments *. (b) Lect of Density and of Cylinder Dimensions. 18. These experiments also indicate that R varies as v D+, ‘and as we have seen in § 13 of this paper it also varies as V1. In order to cover density and cylinder dimensions equation (9) thus becomes Be 7/7 =| G7lix =) ave ites R=175 x 10 6 vDN =0°32x10-M6!vixD. . . . . (10) 19. This equation has been used to calculate R in a large number of cases for the vessel 30 cm. by 30 cm. within the limits of mixture strength 9:2 to 15 per cent. and of density 4 atmosphere to 15 atmospheres, and it has been found that the calculated value agrees with the observed value to within 15 per cent. 20. It will be clear that equations (1) to (10) apply only in the case of a cylindrical vessel whose length is equal to its diameter. In a gas-engine, however, the length varies throughout the stroke, and some modification of these equa- tions is necessary in order that they may be applied directly to gas-engine calculations. This problem will be considered in another paper. is true in the case of a vessel whose length is varied but whose diameter remains fixed at 30 cm. (Compare curves C and JD, fig. 14, p. 396.) When, however, the diameter is varied in the same proportion as the length it is probable that, owing to the fact that the transparency varies os the lateral dimensions, R would more or less follow the 6! law. (Compare curves A and D in the same figure. The former gives the intrinsic radiance for a cylindrical mass of gas of effective leng th 59 cm. and of effective diameter greater than 59 cm., while the latter gives the intrinsic radiance for a cylindrical mass 380 cm. in diameter and 30 cm. jong. Both curves follow the 6! law approximately.) **Phil, Trans. A. vol. ccxi. p. 386, fig. 9. + Phil. Trans. A. vol. ecxi. pp- 388 & 405. 80 Major W. T. David on the Calculation of ToraL RADIATION EMITTED DURING EXPLOSION AND SUBSEQUENT CooLina. (a) Cylindrical Vessel 30 cm. diameter and 30 em. long ; Gaseous Mixtures at Atmospheric Density. 21. In fig. 5 are given mean gas temperature curves and total radiation curves per sq. cm. of wall surface for mixtures of coal-gas and air of atmospheric density varying in strength from 9°7 per cent. to 15 per cent. The radiation curves were all taken when the bolometer was in position C, in which * ANA Radiation — calories Per 5G. oT a i = - Time — seconds. A curves: 15 %Y mixture of coal-gas and air. B curves: 15 YY mixture \, i (taken at a later date when the cal. value of the coal-gas had probably altered). C curves: 13 mixture of coal-gas and air. D curves: 10°2 % mixture KE curves: 10:0 ¥% mixture F curves: 9°8 % mixture /-~ 79o og 99 9? 9? 7 ”? 99 position the radiation measured was very approximately » per cent. greater than the mean of that measured in positions A, B,& C. The curves cover a period of one second after ignition, after which time the radiation from the gaseous mixture is very small. Radiation emitted in Gaseous Livplosions. 81 22. A glance at these curves shows at once that a close rela- tionship exists between the total radiation emitted and the maximum gas temperatures developed in the various mix- tures, and in fig. 6 the total radiation received by the walls per sq. cm.* has been plotted against the maximum gas temperatures. It will be noted that the total radiation is a linear function of the maximum temperature. The equation to this line is tu O0-000560(Cro.— C00)... «| (11) (b) Variation of Total Radiation with Density and Cylinder Dimensions. 23. The writer does not possess sufficient records to enable him to plot similar curves to'that in fig. 6 for other densities. Fig. 6, 1600 i800 2000 | 2200 2400 2600 Maxitnum Gas Temperature —“C 24s. Teta! Radiation received per sg. cin. of wall surface * The radiation values in fig. 6 are those shown in fig. 5 multiplied by 0°95 so as to give the mean values over the whole vessel, Piul. Mag: Ser. 6. Vol..39. No. 229. Jan. 1920. G 82 Major W. T. David on the Calculation of He has, however, two pairs of records for various mixture strengths *, the one pair at 1} atmospheres density and the other pair at 15 atmospheres density. Calculations from these records give the information shown in Table III. TasBLeE III. ‘D Percent. strength g R (atmospheres). of mixture. max. eyed 1:24 10°2 1800 0-7 1:24 12:2 2250 0:94 1:50 15:0 2400 UAL 1°56 10:0 1840 0:87 The equation derived from the first pair at 14 atmospheres density is Rp =0:00053 (Omac — 480)... ee and that from the second pair at 14 atmospheres density (approximately) is Rq=0-00053 (Omax.— 190). . 0 SINAN (13) 24. In considering these equations it must be remembered that each is derived from two points only and the constants are therefore not to be relied upon. They show fairly definitely, however, that the greater the density the smaller the constant within the brackets, and it would appear that this constant is dependent upon the shape of the cooling curve. 25. An examination of the records for 15 per cent. mix- tures shows that between the limits of density ? atmosphere and 14 atmospheres Ry and Ry vary approximately as /D. An approximate relationship between Ry, 0, and D is there- fore given by Ry=0-00056 (Cua: —700) D9 anaes * Taken in position fen + The form of equations (12) and (13) indicates that it would have been better to express equation (14) in the form Ry=0-00055/@max. — 700/(D) |] where ol when D=1:24 or s when D=1'53. But as has been stated equations (12) and (.3) have been derived from two pairs of oie Radiation emitted in Gaseous Leplosions. 83 26. These records also show that Rj varies with cylinder Illumination is measured by a special type of photometer, Phil. Mag. 8. 6. Vol. 39. No. 229. Jan. 1920. jel S being the area. 98 Mr. Loyd A. Jones on a Method and Instrument usually referred to as a lumeter or illummometer, these being calibrated to read directly in some suitable illumina- tion ‘units. The coefficient of reflexion or total reflocliae power of a surface is defined as the ratio of the total reflected luminous flux to the total incident luminous flux. In most practical work this value is not of great importance, the value desired being that of the reflecting power of the surface measured under certain specified conditions, such as the angle of inci- dence of the flux and the position from which the surface is viewed. The term ‘‘ Reflexion Factor,” R, is used to indicate this particular value and is defined as the ratio of the reflected to the incident flux. Reflexion from a surface may be either specular, diffuse, or a mixture of the two. In the case of pure specular reflexion all of the incident flux is reflected in such a way that the angle of reflexion is equal to the angle of incidence ; while in the case of completely diffuse reflexion the reflected flux is equal in all directions regardless of the angle of incidence, the distribution being in accord with Lambert’s cosine law. Very few cases of pure specular or diffuse reflexion are found in practice, there being generally a superposition of the two. The reflexion factor is measured 2 the use of a reflectometer, a photometer of special design, care being taken that conditions of illumination ana angle of “hon are such as to give correct values for application in the particular case under consideration. This value is purely numeric and is usually expressed as a percentage value. If, with a specified condition of illumination, the reflexion factor, R, and the brightness, B, of a surface are measured from the same position, then B=E.R, and hence the value of B may be determined ; or in any case where two of these factors are known the third can be computed. The quality factor of the luminous flux is that property which depends upon the spectral distribution of that flux, colour being defined as the subjective evaluation ‘as expressed in terms of hue and purity or saturation. Hue is that pro- perty of colour which depends upon the variation in the sensation due to the variation of the wave-length of the luminous flux, while saturation expresses the proximity of the colour to a condition of monochromatism. Mono- chromatic spectral light has a saturation of 100 per cent., while pure white light has a saturation of zero. White, therefore, is a limiting colour having no hue and zero saturation. In practice it has been found convenient in many cases to express the saturation factor in the inverse order, that is as impurity rather than purity. The term used for the Measurement of Visibility of Objects. 99 in such expression is called the “ per cent. white,” for which the symbol I is used. Thus a colour for which I=100 per -cent. is equivalent to zero saturation and if [=O per cent., saturation is 100 per cent. It has been demonstrated experimentally that any colour can be matched by the mixture, in the proper proportions, of white light with monochromatic spectral light of the proper wave-length. In this way a direct measurement of the fundamental sensation properties of a cclour may be made.: The hue is specified by the wave-length of mono- chromatic light used (wave-length of the dominant hue). The saturation is specified either as the purity (per cent. hue) or as the impurity (per cent. white), the former value being obtained from the ratio of the intensity of the mono- chromatic to the total intensity (monochromatic plus white) of the mixture, while the latter value (per cent. white) is given by the ratio of the intensity of the white to the total intensity of the mixture. These values are pure numerics. The usual unit used in expressing the wave-length of light is the millimicron, which is equal to ‘0000001 centimetre and is designated by the symbol py. In the foregoing paragraphs have been defined the various terms that will be used in the following discussion of the subject of visibility. These are summarized briefly in the following table for convenience of reference :— Symbol. Quantity, Unit. hoe ee Luminous Flux. Lumen. 1B) feta ees Tilumination. Foot Candle LS 5 Brightness. Lambert. ee Reflexion Factor. Per cent. gees... ee ebue: Wave-leneth (up). Sa ieee Saturation Purity. Per cent. Hue. Impurity. Per cent. White. Theoretical Analysis of the Visibility Problem. In general it may be said that non-luminous objects are visible by virtue of the light reflected from them. However, any particular object in the field of vision becomes visible as such only by contrast with its surroundings—that is, when the light emanating from that object (either by re- flexion or emission) differs in some respect from the light H 2 100 Mr. Loyd A. Jones on a Method and Instrument flux which enters the eye from the projected space imme- diately surrounding that object. The sensation caused by i the incidence of radiant energy, which we call light, upon tT | the retina of the eye may be said to consist of two factors, brightness and colour, the former being dependent upon the intensity and the latter upon the quality of the incident. radiation. This second factor of the sensation may be said | also to consist of two parts, hue and purity or saturation. | Hue refers to the position, in the spectrum, of the dominant wave-length, and saturation expresses the proximity of the colour to monochromatism. It is evident, therefore, that a sensation due to the impingement of radiant energy upon the retina may vary in three respects, that is, with respect. to brightness, hue, and saturation. A contrast in the visual field resulting in the visibility of an object may be due, therefore, to brightness contrast, to hue contrast, or to i saturation contrast; or to a combination of any two or all three of these factors. For the purposes of the theoretical treatment of this problem it will be necessary to make certain simplifying hypotheses. Begin first with the problem of the deter- mination of the visibility of an object uniform in colour and brightness viewed against a background also of uniform Hh Hit i Fi 1 | anions | Hi} a een fe sath — — — _ A io | oees —_— or paki — ———_—= SSS Se _—— rs _ — — Bo ak — Been es — — —— — k, and one for the 2 B case where —! h, for the Measurement of Visibility of Objects. 109 as asymptotes the lines y= +50 and e=0; while Case IT.., — ik. 2 is represented by a series of straight lines, all passing through the point z=0, y=+50. ‘The slope of the line for any particular assumed value of W is given by the expression k EE 0) a es Wd)’ 110 Mr. Loyd A. Jones on a Method and Instrument Case IT., ie ale leads to family of straight lines all passing through the point a=0, y=—51, the slope of any one being given by the equation > \ e OS NY aN ee Vv iN WS fel ear Ve (N27) ZiAAXK A LEER -3O 50 3 d : : - REFLECTION - FACTORUCRa) - Visibility as a Function of the Reflexion Factor. Tt will be noticed in solving for the value of tan a that in B Case I., where 7 >k, 2 the value is negative, while the value of tana in Case II. is positive. This offers a convenient method for indicating for the Measurement of Visibility of Objects. eh whether the object to which any visibility value applies is lighter or darker than the background against which it is measured. ‘The visibility value itself must from the very nature of the term be always positive, but by specitying the sien of the first derivative of the visibility function ata given point it can be determined whether such visibility is due to the object being brighter than the background or vice versa. The first derivative of the function V = (iy) is aN a R, ; which, since the function is a straight line, is equal to the tangent, thus dV == Fane Fy eas alV 5 é Ce hits B, LE TR, is negative it indicates that B, >k and therefore that the object is darker than the background. In case Oe is positive it indicates that _ ) the atomic weight, by the addition of the number of He (4:00) to w resp. t ; (ce) the valency, by the number of valency electrons (E,) ~ written at the end; (d) the number of metastasic electrons (H,,), by simply adding the inner and valency electrons together. The relationship of this system of radioactive elements to the periodic system is established by attaching figure 1 to the lower part of the new periodic table ft. Whether or not. it is possible to extend this scheme of isotopes to the non- radioactive elements is a problem of the future. Berkeley, Cal., ~~ July 26th, 1919. XIV. Note on the Construction of a Parabolic Trajectory and a Property of the Parabola used by Archimedes. By W. B. Morton and T. C. Tosin ft. dite a particle is projected, in a given direction, from a | given point A with such a velocity that it hits a second given point B, then any number of points on the path may be obtained by the simple construction shown in fig. 1. AH is drawn to any point on tho vertical through B, HI is parallel to the direction of projection, then the vertical IP meets AH in a point of the path. This construction involves a simple property of the parabola which does not appear in the ordinary text-books. If chords are drawn * Phys. Rev. loc. cit. t Loe. cit. t Communicated by the Authors. 158 Messrs. W. B. Morton and T. C. Tobin on the from a point A on the curve to any other two points B, P, and if the diameters through B, P are drawn to intersect AP, AB respectively, then the line joining the points of inter- section is parallel to the tangent at A. tesa: Referring to fig. 2 in which the parabola is placed in the more usual posture, assume that the points A, B and the tangent at A are given, and construct P. in aie manner indicated above. It can easily be shown that P lies on the curve. For AN : AK =PN:HK = PN: BM and AK :AM= 1K : BM = PN: BM AN : AM = PN? BM?. It is interesting to notice that this property is really a special case of Pascal’s theorem about a hexagon inscribed ina conic. Let the angular points 1, 2, of the hexagon be ay Aewomeb B, 4,5 at the point at infinity on the axis of the parabola, md) at ie hen side d2ms the tangent at A, 23 is AB, 34 is BH, 45 is the line at infinity which touches * This way of looking at the matter was pointed out to us by Mr. F, M. Saxelby. Construction of a Parabolic Trajectory. 159 the parabola, 56is Pl and6lis AP. The line of collinearity of intersections of opposite sides is HI and the tangent at A meets this at infinity. To continue with the properties of the diagram fig. 2, oem join HM meeting the diameter through P at W. Then evidently XW = IP and BWU is parallel to AP. Again get the point V by joming HU, then VW = WX and BV is parallel to the tangent at P. For, since NA = AT, the igure BX WV is similar and parallel to PNAT. From these results it follows that Va =) ele — lial toe =. VICK KOA, This brings us to a theorem used by Archimedes in the course of his investigations of the positions of equilibrium of a floating paraboloid of revolution, contained in the second book of the work on Floating Bodies. In the sixth pro- position of that book he proves that a paraboloid, the length of whose axis has to tie latus rectum a ratio lying between 3 See: : : : the values 7 and =, if placed with a point on the circum- ference of its base in the surface of the liquid and then released, will turn, under the action of its weight and the 160 The Construction of a Parabolic Trajectory. buoyaney of the liquid, towards the position with axis vertical. In the course of the proof he quotes, as known, a property of the parabola which is an extension of that just obtained. ‘The source from which Archimedes derived it is unknown. Using the letiering of fig. 2, which agrees with that adopted. in- Heath’s edition of Archimedes, the theorem in its most general form is as follows :— From a point B on a parabola ordinates BM, BV are drawn to any two diameters AM, PV. Through any point K of AM a line is drawn, parallel to the oidimane BM, tomeet.PY ind. Then PY: Pl = or yikes Archimedes refers only to the special case where A is the vertex of the parabola. A proof of this case, on somewhat algebraical lines, is given by Dr. Heath. The alternative geometrical proof now given exhibits, perhaps, in a clearer light the connexion of the theorem with the fundamental properties of the curve. It has been shown above that the two ratios compared are equal when I is the intersection of AB with the diameter through P. It remains to prove the inequality in other cases. Let K'I’ be another position lying, say, to the left of KI, and let it meet AH in H’. Join MH’ meeting UB in W’, PW in W", and BH in H’. We want to compare UW” with IW. We have WW" HEY = AP: AH = ANG Are HA”; BE = 0M UW =A Reese Ee” IT’ and so 'W" < IW. If I’K’ is taken to the right of IX it will be found in the same manner that WW’ (SIXTH SORIBS.] FEBRUARY 1920. XV. The recent Eclipse Results and Stokes-Planck’s ther. By L. SivBerstew, Ph.D., Lecturer in Mathem. Physics - at the University of Rome”. 1. TT is well known that, in 1845, Stokes proposed a theory of aberration (Phil. Mag. xxvii. p. 9), which was based on the assumption that the luminiferous ether surrounding our planet is dragged along in its annual motion so that the velocity of the ether relative to the Earth is nil at its surface, and, increasing continuously, becomes equal and opposite to the Harth’s velocity at very large distances from the Harth or, to put it short, at infinity. The purpose of this hypothesis, as opposed to that of Fresnel’s stagnant zether, was to give a rigorous independence of all purely terrestrial optical experiments from the Earth’s annual motion (combined with that of the solar system). In order to account for the semi-terrestrial phenomenon known as astronomical aberration, Stokes had to assume that the motion of the ether, between the Earth and the stars in question, is purely zrrotational. But, by a well-known theorem of hydrodynamics, this assumption was not com- patible with the zncompressibility of Stokes’s cether and, at the same time, with the absence of slipping over the Harth’s surface. 2. In order to overcome this essential difficulty Max Planck has suggested that the incompressibility could be * Communicated by Sir Oliver Lodge. Phil. Mag. 8. 6. Vol. 39. No. 230. Meb. 1920. M 162 Dr. L. Silberstein on the recent Eclipse Results given up * and replaced by the assumption that the ether is condensed round the Earth, and other celestial bodies, as if it were subjected to the force of gravitation and behaved more or less like a perfect gas. Lorentz, in spite of his personal preference for a fixed ether, took up Planck’s idea and worked out the problem under the special (but by no means the only possible) assumption that the esther density p and pressure p obey Boyle’s law, p=ap, where a=const. If M be the Earth’s mass, in astronomical units, this gives p= pee. a ean where p, is the density at infinity and r the distance of any external point from the Harth’s centre. The maximum velocity of slip at the Earth’s surface (r= /), in the direction opposite to that of its motion becomes ft 3 (by (on E Ve= 2 e REMAN NOD ID aea MEATGTS PEA aT Co pL GN es e e e 9 4 e7—(1l+a+40")’ (2) where c=2M/F, and v,, is the velocity of the ether, relative to the Harth, at infinity. To account for the astronomical aberration within the limits of experimental error it is necessary and sufficient to make v=y)ov,- This gives, by (2), with sufficient approxi- mation (since the required o is manifestly so large as to make the second term of the denominator negligible), a0 040%. so that the said requirement is amply satished by Ga VO ko ee (E) This means, according to (1), a condensation { of the ether amounting at the Harth’s surface to little less than = ee 7000; and gives at the same time for the (lower limit of the) coefficient « the value 10°-2//J/, to which we may return * Cf. H. A. Lorentz’s paper on Stokes’s theory of aberration in A mster- dam Proc. for 1898-99, p. 443, reprinted in vol. i. of his Abhandlungen. + A short deduction of this formula will be found in Loreniz’s ‘Theory of Electrons,’ 1909, p. 314. ; t What is commonly called ‘‘condensation”’ would in our case be ? _1. But it will be convenient to use this as a short name for P| Roos Pa which will henceforth be denoted by s. and Stokes-Planck’s Avther. 163 later on. In order to reduce the slip to $ per cent. of v, a condensation of about 60000 would be required *. In view of this considerable condensation, required by the theory of Stokes-Planck, Lorentz made in 1909 (‘ Theory of Electrons, pp. 173-4) the following characteristic remark :— “Tn this department of physics, in which we can make no progress without some hypothesis that looks somewhat startling at first sight, we must be careful not rashly to reject a new idea, and in making his suggestion Planck has certainly done a good thing. “Yet I dare say that this assumption of an enormously "condensed ether, combined, as at must be, with the hypothesis that the velocity of Tight 3 as not tir the least altered by it, 1s not very satisfactory.’ {The last words are italicised for our present purpose. | In fact, such a condensation, introduced aid hoc and serving only the negative purpose of not upsetting the theory of ‘aberration, did not seem very satisfactory, and the present writer has as recently as 1914 expressed the same opinion in his book on Relativity (p. 63), not so much to defend Fresnel’s and Lorentz’s fixed zether, as to prepare the reader’s mind for the complete abolition of the ether and thus to introduce him to Hinstein’s “special” relativity of 1905. Such has been the position of things until recently. 3. Now, it so happens that, stimulated by the desire to test Einstein’s generalized relativity and theory of gravitation, the astronomers participating in the last Eclipse Expedition have found an undoubtedly positive effect, the bending of rays passing near the Sun. As I have pointed out on previous occasions, it seems premature to interpret this result as a verification on Hinstein’s theory, not merely in view of the small outstanding discrepancies, but chiefly in view of the failure of detecting the spectrum shift predicted by the theory, with which the whole theory stands or falls. But the Heclipse result proves at any rate that there is an “alteration,” a change of light-velocity all around the Sun, which thus invalidates the words of Lorentz italicized in the quotation above. The condensation claimed by Planck’s modification of Stokes’s theory, for the Sun as well as for the ‘ Harth and for all other material bodies, is no longer devoid of influence on observable phenomena. It suddenly acquires physical life, so to speak. * Notice that the aberration is a first order effect, while such phenomena as that expected by Michelson-Morley are second order effects (v*/c*), so that the above condensation suiting the aberration up to 1 per cent. will reduce the Michelson-Morley effect to one ten- thousandth of its value, and thus practically annihilate it. There is thus no need for making o larger than 10-2. M 2 164 = Dr. L. Silberstein on the recent Eclipse Results In other words, the discovery made at Brazil naturally suggests the idea that the observed deflexion is due to the condensation of the ether around the Sun*, and although one has been an implacable enemy of any ether at all, for the last fifteen years, one does not hesitate to point out this. possibility—a last glimpse of hope, perhaps, for the banished medium. Let us imagine for the moment that Hinstein had never published his debatable, though undoubtedly beautiful, new theory—not even that of 1905. Then it is almost certain that the Eclipse result would readily be acclaimed as an evidence of the condensation of the ether near the Sun, as required by the theory of Stokes-Planck, and would encourage. the physicists to work out in detail the optical and associated. consequences of such a condensation. But even though Kinstein’s theory has been published, and is being made. popular in a most sensational way, we cannot help clinging to the said idea. I just learn from ‘The Observatory’ for August that Mr. Jonckheere suggested some months ago that. refractions may, inter alias, be caused by “a hypothetical condensation of ether near the Sun.” My point, however, is that such a source of refraction acquires a particular interest 2f it is treated in connexion with the half-forgotten theory of Stokes-Planck, when it ceases to be a detached hy pothesis. It is in this sense and in such an organic connexion that T should like to draw attention to this aspect of the subject. Of course, the quantitative details of the suitable modifica- tion of the optical, or the electromagnetic, properties of the: ether due to a radially symmetrical or any other condensation have to be worked out carefully. It is not the purpose of this Note to give a complete investigation of this kind, but only some hints at its possibility. Such hints, together with some remarks on the possible advantages of the advocated a see will occupy our attention in the following sections. If, merely to fix the ideas, the Boyle law is still adhered to, aie condensation Slo. oniale a radially symmetrical gravitating mass is given, as in (1), by logs= ee Sa (BY J£ we assume, for places near the Harth’s surface, not more and not less than what is just needed for the theory of * The logarithm of this condensation would amount, at the Sun’s surface, by (1) and (1), to the enormous figure o = log s=31100. Cf. the following footnote. and Stokes- Planck’ s Avther. 165 aberration, 7%. e. ¢= log s=10°2, we shall have at the surface of the Sun, as already mentioned in a footnote, o=logs=102",, 10'=31000, . en (S) which means, no doubt, an enormous condensation *. The corresponding relative velocity of slipping v/v,, will, by (2), be almost evanescent ; the drag will be almost complete. On the other hand, at the surface of a hydrogen atom, assumed for the moment to be a homogeneous sphere (and | the only existing body), we shall have log s=1°7. 10~*, that is to say, s= Po =~ 141-7,10-%4, Px indistinguishable from unity. Notice that for small o the denominator in (2) reduces to 4o*+540'+ higher terms, so that the relative slip becomes toy a, ee oo For such bodies, therefore, as a hydrogen atom, or in fact any other atom, the ratio in, question will be exceedingly nearly equal its limiting value 3/2, which is well known to be the maximum relative slipping for a sphere moving in an incompressible liquid. In short, for such small bodies there will be practically no drag at all. The more so for electrons, if one wished to attribute to them gravitational properties. This behaviour will be important in connexion with some such electrodynamic theories of ponderable media, as is that proposed by Lorentz, which require a complete slip. But even a sphere of the mass of 1 kg. and the radius of 10 cm., for which c=1:09 . 1071, will practically have no “‘ grip upon the zether.” This will readily be seen to account, among other things, for the negative results of Sir Oliver Lodge’s ingenious experiments with the Ether machine, even if its whirling part were made much more massive. As a mere curiosity notice that even the Moon would have only a partial, weak erip upon our rehabilitated ether. In fact, at the Moon’s surface we shoald have c=10-2 x 0:094=0:96, and therefore, by (2), = =1:15, which differs only by 0°35 trom tlie full slip. Thus the Selenites would obtain with a * Such fantastically large condensatious need not frighten us. They can be reduced if Boyle’s law is replaced by some other appropriate form of relation between pressure and density. Boyle’s law, which is by no means necessary, is here used only, as the simplest one, for the sake of illustration. v a 166 = Dr. L. Silberstein on the recent Kclipse Results Michelson-Morley experiment a pronounced positive effect. But enough has now been said in illustration of the formulze for the condensation and for the slip. 5. Before passing to consider the Keclipse result it may be well to generalize the condensation formula (3) for the case in which Boyle’s law is replaced by any relation between the pressure and the density of the ether. The corresponding generalization of the slip-formula (2), not required for our present purposes, may be postponed to a later opportunity. Let the pressure p be any function of the density p alone,, and let there be any distribution of oravitating masses.. Introduce the function, familiar from hydrodynamics, dp Die @ia) — \“ «8 e e e ° e (5): cy P j Then, in the state of equilibrium, and with dm written for any mass-element in astronomical units, » =| i en) a where 7 is the distance of the contemplated point from dm, and the integral, representing the total gravitational poten-. tial, extends over all material bodies. © being a knowm function of p, formula (6) gives the required relation. It will be seen from the definition (5) that the dimensions of ® (work per unit mass of ether} are those of a squared velocity. In order to bring this into evidence, let us recall that 99) Ae a, ° ry e e (7) de is the velocity of propagation of longitudinal waves in any compressible non-viscous fluid *. This velocity is, in general, a function of p, and becomes a ‘constant for the special case of Boyle’s law, namely, our previous 1/Va. Using (7) and writing, as before, mes A log s, we have DP { v-diog s, atelia peer Hehe coe) the required form. The integral is to be extended from * This result, known as the formula of Laplace, holds also for the most characteristic kind of waves—to wit, for a wave of longitudinal dis- continuity (Hugoniot, Hadamard), for which it follows directly, without integration, from the hydrodynamical equations of motion. See, for instance, my ‘ Vectorial Mechanics,’ p. 169. and Stokes-Planck’s Avther. 167 s = 1 (or log s = 0) to the actual value of the condensation. Thus the condensation formula (6) becomes { v?.dlogs = 9, BEAT Mens pata (CO: where © has been written for the total gravitational potential at the place under consideration. For constant » (Boyle’s law), and for a single spherical body, the previous formula (3) reappears. Tt will be kept in mind that although the ether is assumed to behave in this way (say, like a gas) with respect to slow processes, it can still propagate rapid transversal light- disturbances as if it were an elastic solid (like the fameus cobbler’s wax of Lord Kelvin) ; but it will be best to think of light as of electromagnetic disturbances. The normal velocity c of propagating them is another property of the ether, independent of that which is represented by v, and subjected only to slight variations with condensation, as will appear presently. The ratio of v to ¢ will be of importance, but as to the longitudinal] waves themselves, they are of no physical interest for the present and, on the other hand, are not likely to become a nuisance. Tor it is not in our power to produce them to any relevant extent, and even if they are generated and maintained by some gigantic natural pro- cesses, their only effect would be to alter very slightly, here and there, the normal velocity of light- -propagation. If we wish to form an idea of the numerical value of v, or at least of its upper limit, for the case of Boyle’s law, say, it is enough to take the value of o given above for the Sun, and to remember that M/c?=1°5 km., and, in round figures, R=7.10’km.- Then the result will be : Oe Ome bhiitels to say, v equal to about 2°5 km. per second *. This is quoted by the way only. But the ratio of these two velocities will be seen to acquire a particular interest in connexion with the recent astronomical discovery. } 6. Let c,as before, stand for the propagation velocity of light in uncondensed ether, 2. e. in absence of, or far away Cen gravitating masses, and let ¢! be the light velocity at a place where the ether has undergone a condensation s. The question is: How are we to correlate c’ with s? In other words: On what are we to base the optical behaviour of the ether modified by a condensation? The only reasonable * If so, then the condensational disturbances due to the Earth and other planets, whose velocities exceed », will be confined to conical regions as in Mach’s famous experiments, 168 Dr. L. Silberstein on the recent Kelipse Results answer is: On experience. Tor, clearly, we cannot deduce a relation, which is essentially electro-mechanical, from me- chanical principles alone, or from electromagnetism alone. Nor can we imitate the usual dispersion theory (which makes use of both kinds of principles), for we are interested in those portions of the sether in which there are no atoms and no electrons. Jn short, as was announced in section 3, let us write down the required relation by utilizing the observational result obtained by the Eclipse Expedition. In other words, let us see what that relation must be like in order to give the observed effect. » Now, if we disregard the small discrepancies (which may be either due to accidental errors or, perhaps, due to a superposed slight ordinary refraction), the observed total deflexions of the rays passing near the Sun are represented by Einstein’s formula (quite apart from his theory) where 7 1s the minimum distance of the (undeflected) ray from the Sun’s centre, and it can easily be shown that such will be the case* if the refractive index n=c/c' at any distance +> from the Sun’s centre be determined by 4M Ne == Wt or, denoting the potential by ©, and generalizing to any distribution of gravitational matter, 40, =1 a = ° of e e e e (9) Ce [ This, in fact, is the formula which would follow at once from Einstein’s approximate line-element 2 ds* = c’dt?(1 — == (da? + dy? + dz”) (1 + = for a “static” field. | In order to obtain the required relation, that is to say the assumption to be made on the optical behaviour of the con- densed sether, it is enough to combine equation (9) with our last equation (8), which gives mois ale 2 logs: 27 ie) Ge * Approximately, that is, for small AO, and consequently for a refractive index but little ditferivg from unity. and Stokes-Planck’s A’ther. 169 Such, then, would be the required refractivity of the condensed ether, obeying any law p=/(p). In particular, if it obeys Boyle’s law, we have 2 Bie bt Ae lok sy e-1 4715) (10a) which is of a surprisingly simple form, and reads: n?—1 equal to four times the logarithm of condensation multiplied by the squared ratio of the two velocities of propagation charac- terizing the ether. Notwithstanding this temptingly simple form of the relation, I shall not try to “deduce” it from things more familiar. I prefer to regard it as an assumption, dictated by observation. If the reader so desires, he can write n?—1=4w/c?, where w is the work, per unit mass of matter, done by the gravita- tional field in condensing the ether. The small fraction n*—1 being known from the Hclipse results (for any 7), the numerical value of this work is determined without any further assumptions. If we agree to the lowest estimate of log s at the Sun’s surface, as required by the aberration theory, we can also evaluate separately the ratio v/c, as already mentioned. This, however, is only a secondary matter. 7. Some details and further implications of the Stokes- Planck ether theory, supplemented by assumption (10), must be postponed to a later opportunity. Here it will be enough to add only a few more general remarks. It will be kept in mind that the proposed theory would account not only for the observed astronomical aberration and for the older terrestrial optival nil-effects, but manifestly also for the nil- effect of the Michelson-Morley experiment. The bending of rays round the more massive celestial bodies would be onlya by-product of the theory. Again, in view of the exceedingly small condensation of the ether round single atoms or cor- puscles there will be no difficulty in working outa satisfactory electromagnetic theory of ponderable media. The proposed theory would also have the advantage of not predicting the obstinately absent gravitational shift ef the spectrum lines. It might also react, in part at least, upon the 1905 relativity, depriving it of its indispensability in most cases, but by no means banishing it from the whole domain of physico- mathematical investigations. Finally, the just objections raised by the advocates of the physical principle of causality against the fixed and homogeneous ether of Fresnel-Lorentz would not apply to Stokes’s modified ether. For this 170 Sir Oliver Lodge on a Possible latter would by no means be a mere framework of reference axes and, as such, illegitimately privileged. For in referring wu class of phenomena to the wether here advocated we rate ultimately refer them to assignable physical things, namely those most massive gigantic ‘bodies w hich, so to speak, have the ea orip upon that medium. It is, among other things, this latter -remark that | hope to make particularly Alene at an early opportunity. London, December 22, 1919. XVI. Note on a Possible Structure for the Ether. By Sir OttvEr Loper*. R. SILBERSTEIN’S communication ‘gives me an opportunity for calling attenticn to a paper of mine on many points in connexion with the ether which mast surely be of interest even to those who are contemplating thie abandonment of that medium. In that paper an estimate is made of etherial density, and an attempt to measure experi- ental its lower limitis described; there are also comments of interest from Sir Joseph Larmor oa Sir J.J. Thomson. The paper is in the Phil. Mag. ser. 6, vol. xii. pp. 488-506, and is of date April 1907; though among other things it relates experiments conducted in and about 1893. The transmission of transverse vibrations like light shows that the ether cannot be a mere structureless fluid; and if it is to be treated dynamically, which at first is surely a legiti- mate attempt, it must have properties akin to what we call, in matter, Rigidity and Inertia. Its inertia must be something fundamental, which underlies and accounts fer the inertia we perceive in matter, possibly in a way having some analogy with a motion of a solid through a perfect fluid. EFor when an electric charge is moved, a magnetic field in the shape of an ether vortex-ring is generated ‘(with an energy of circula- tion per unit velume equal to p(ewsin @)?/87r*), and this confers upon the charge its observed momentum if the podium has the requisite density (see Phil. Mag., April 1907, vol. xiii. p. 492). The rigidity may be explicable hydrodynamically by a vortex Toenltaon a turbulent motion having a circulatory velocity of the same order as. that of the waves which the medium is able to transmit. In Lord Kelvin’s laminar vortex arrangement the velocity * Communicated by the Author. Structure for the Ether. iA of wave-propagation comes out 3./2 or °47 of the average velocity of turbulent motion Gis Phil. Mag. for October 1887, p. 350). In all investigations the ‘two velocities come out of the same order: and in FitzGerald’s collected papers, No. 53 and No. 91, the two velocities can be identical for a certain arrangement of turbulence (cf. pp. 259 & 256). On page 457, FitzGerald expresses his tentative opinion that the hypothesis that ‘‘ the ether is a turbulent liquid has great possibilities underlying it.” And, again, on p. 486, ‘ there seems very little more besides interpretation of sy mbols to make a turbulent liquid a satisfactory explanation of the structure of the ether.’ Some assurance of stability may also be needed. Many things show that any granular structure which may thus be possessed by the ether ‘must be of a fineness incom- parably minuter than any dimension associated with the material units on which we can experiment. In fact, the ether may quite well contain a linear dimension of the order 10-*° or 107*8 centim., and an energy of 10% or 10* ergs per cubic centimetre (Phil. Mag., April 1907, p. 493). The calm self-sufficient way in w hich it sustains all our stresses, and transmits all our mens shows that anything we can impose upon ether is as far from perturbing it, or calling out even second orders of small quantities, as the slight bias of an ordinary draught of air is from perturbing the normal motion of the molecules which compose it. A bullet in air and an electron in ether can, however, attain perturbing velocities ; and the fact is bound to be instructive when increase of mass with speed is fully assimilated and its mechanism understood. As said on p. 490 of the Phil. Mag. for April 1907, retaining the meaning but slightly improving the wording: The reason for the concentration of magnetic intensity at “the equator of an electron, moving with something approaching the velocity of light, is ‘that the flow associated Sith and indeed constituting the magnetic field is then no longer a small fraction of the intrinsic rotational velocity of the ether itself (see also loc. cit. p. 494). To explain gravitational and other facts, we must assume that the very formation or existence of an élestron sets up a radial strain or tension all round it, varying as the inverse distance, and likewise redtices the prea ot y energy in its immediate neighbourhood; not necess arily causing any change of density, since electrostatic facts (2 otably the Cavendish experiment) show the ether to be practic ‘ally, and probably actually, incompressible, but affecting its elastic or dielectric constant in such a way as to modify “the velocity 172 Sir Oliver Lodge on a Possible of light in the neighbourhood. (Compare Lord Kelvin, Baltimore Lectures, p. 465.) An electron might, in fact, be a small region in which intrinsic circulation has ceased, so that it possessed inertia only. The tension or reduction of pressure set up in the neigh- bourhood of such a centre of force could explain gravitational attraction, and a change of rigidity would also suffice to explain the very minute reduction in the velocity of light. ‘The refractive index needed at any point is 1+ 2yM/rc?, or i+u?/c’, where wu is the velocity of free fall from infinity, which is just what the light has done. The dielectric constant would be modified so that K/K,=1-+4yM/re? ; the second term being Hinstein’s deflexion. It may be taken as representing the deficiency of etherial circulation-energy near a massive body, as compared with the unmodified circulation-energy in free space. Just outside an electron this deficiency is of the order 10-* ; though just outside the Sun it is of the order 10~°. This note is hardly germane to Dr. Silberstein’s paper ; so I may just add that the complication of introducing compressibility, and not only compressibility but an enormous gravitational compression, in order to evade rotationality in a hypothetical ether dragged by moving masses—for absence o£ velocity-potential is well known to complicate unduly the theory of astronomical aberration—does not commend itself to me. An incompressible ether, not viscous at all, is far more simple; and astronomical aberration then follows, as easily as on the corpuscular theory, without any ingenuity. But, as speculation in these unconquered regions is a legitimate preliminary to exploration, 1 may say that I am fully prepared, as Dr. Silberstein in one part of his paper seems also prepared, to accept a gravitational influence on the Ether’s dielectric constant, and, therefore, on the square of its index of refraction: though I should like to see this done without postulating any increase of density in a medium of which space is already completely full. It is also highly desirable to avoid the frictional and therimal considerations, accompanied by dissipation of energy, in- separable from any sort of viscosity. ‘These imperfections are appropriate to a secondary or derived cosmic ingredient, like matter; they are not appropriate to the fundamental substance itself. If the ether has demonstrated anything, so far, it has shown us, by its very elusive character and complete Structure for the Ether. 1a: transparency, that its properties are of the simplest and most uniform kind, and free from any imperfections which would accumulate waste energy in particular fractions of itself. If we attribute all locomotion to matter in its most general sense, including electric charges ; and all elasticity to ether, regarding the latter as the uniting and potentially strained medium responsible for every force which holds. atoms together ; we shall be on sound and simple lines. This is not to deny that potential energy may be susceptible of ultimate kinetic explanation, in terms of the postulated fine- grained vorticity. How the ether can be tied into the knots which we call electrons—in other words, how the peculiar regions or singular points characteristic of electric charge are consti-. tuted—remains to be discovered. The small second-order tension responsible for gravitation will, I feel sure, be ‘accounted for as soon as the electric structure is made out. The fact that a luminous disturbance simulates the funda-. mental properties of matter is mving us a broad hint. A wave-front is an evanescent kind of matter—a sort of attempt of an accelerated electron to reproduce itself : the question is how such a peculiarity, when generated, can be made permanent and its violent locomotion checked. We must find out how to disturb the ether in such a way that the modification shall remain concentrated, and not instantly rush away and disperse itself with the speed of light. The. electric and the magnetic components must be separated, the one kept and the other annulled. In this connexion, I take permission to make a few extracts from the 1907 edition of ‘ Modern Views of Electricity,’ so as to bring the- suggestions before those who may be interested inthem. I quote from pages 830 onwards :-— “Wherever electrons and atoms exist, they modify the ether in. their immediate neighbourhood, so that waves passing through #. portion of space containing them are affected by their presence, as if the ether were more or less loaded by them; because the electric displacements which go on in the unseparated and still perfectly united constituents of free ether [in a beam of light] are also shared to some extent by the separated peculiarities .... All those charges which possess externally-reaching lines of force must share in the motion of the waves, without having the requisite amount of resilience to compensate for their inertia.” “The positive and negative constituents, when they combine or cohere, do not destroy each other and revert into plain ether again ; on the contrary, they retain their individuality and persist, in either- a combined or separate state. We do not know how to produce or to destroy these peculiarities .... {for whereas] matter can be: 174 On a Possible Structure for the Ether. dissociated with extreme ease, the dissociation of ether is unknown and hypothetical, save as represented by its apparent results. “‘ Nevertheless, it must be the case that the slight, almost infinites- imal, shear, which goes on in a light wave, is of the nature of incipient and temporary electrical separation.... It appears possible that a sufficiently violent It.M.F., applied to the ether by some method unknown to us at present, must be the kind of influence necessary to shear it beyond the critical value and leave its components permanently distinct; such constituents being opposite electric charges, which, when once thoroughly separated, oniv combine to form matter, and do not recoil into ordinary ether again.” Let me make one more quotation, immediately following, relating to gravitation :-— ‘“Kvery attempt at separation of this kind, even if no stronger than exists in ordinary light, [is] accompanied by a longitudinal foree—Maxwell’s pressure.... If the disturbance could be made so extreme as to result in permanent dislocation, this pressure might leave behind it, as permanent residue, a longitudinal pressure [or tension] extending throughout space.” There seems to be a necessary connexion between transverse and longitudinal stresses, the one being 3w/v times the other. If we re-estimate Maxwell’s data for luminous vibrations, as given in his article ‘‘ Mther ’ in the Kney. Brit. (Collected Works, vol. 2, p. 767), on the basis of a reasoned high estimate of ether density, ignoring the guess of that period that in the brightness near the Sun the amplitude of a light vibration might possibly be as great as one-hundredth of a wave- length, for this was only an upper limit and it is surely bound to be much smaller, we can proceed thus :— Let a be the maximum amplitude of shear of a light wave, y =a cos pix —vt), near the Sun; where the luminous energy is nearly 2 ergs per c.c.; and let w be the maximum speed of elastic recovery ; u a then —=paz2r—., v y IN The energy }pu? = 2 ergs per c.c., so,if p=10!2, w=2x107%em. persec., and a= 107%, Hence, expressing conditions near the sun in Maxwell’s manner (oc. c7t.), Energy per cubic centimetre = dpva*p’ =2 ergs. Greatest tangential stress per sq. cm. =pv’ap =6 x 1LO'dynes, Coefficient of rigidity of ether =O ea IS Oye G. Density of ether = 17) ===)! It will be observed that pv*ap is the same as puv, which is an expression for the travelling momentum of a light-beam. XVII. The Spheroidal Llectron. By Prof. A. ANDERSON *. N the supposition that the shape of an electron in motion is a spheroid, the direction of motion being along thie axis of symmetry, and the charge on the correlated electron being distributed on its surface as if it were a conductor at rest, the values of the momentum and energy in the ether can be calculated. The length of the semi-axis in the direction of motion is 6,and that of the semi-axis at right angles to this is a: b is thus the contracted length, or the length of the semi-axis in the direction of motion after it has suffered the Lorentz-FitzGerald contraction. As usual, 8 denotes the quantity (1-5) , where v is the velocity of the eiectron and ¢ the velocity of light. ‘The results are, if Bb>a, momentum i eRe 2pr? — a? 1,80 + (Bi? —a?)* 16me? (62D? —a?)?L B'L?—a? ©” Bb —( 8%? — a2)? a ee ( 82h? — ol and energy e 22 P (+8) (4498 32m BB mali? a 2) a — Bb+ (B70? —a?)? a ee} Joo oe goae} 5 p—(g—a)' ere FY) ibe Bba, and we obtain e vt Gea He | v? lo cS uonome | C2 aC e+y are Saale} which are the momentum and energy associated with the Abraham electron. Both electrons are, therefore, particular cases of the general spheroidal electron. The transverse mass is M/v, and well-known experiments. have been made to determine e/m and v, or, which is equi- valent, e/M and v. Thus it would, no doubt, be possible, though perhaps the mathematical work would be tedious, to determine the value of the ratio of 6 to a for which the theoretical value of e/M would agree most closely with the experimental results. A determination of this ratio would be of interest. We may, however, remark that if the ratio of 6 to a tends to zero, the corresponding value of M tends to ZI Moaez that is, the ether momentum associated with an electron whose shape is a plane circular disk moving with uniform velocity in a direction perpendicular to its plane is equal toe times the momentum associated with the Lorentz 8 electron moving with an equal velocity. The ratio of its transverse mass to its mass when v=O is the same as for the Lorentz electron, and the experimental results could not decide between them. In the case of a very elongated prolate spheroid moving in the direction of its axis of symmetry, both the momentum and energy become very great. ema a, col XVIII. The Adjustment of Observations. I. By Norman CampBELL, Sc.D.” iL. Ro more than fifty years the method of adjusting | observations affected by experimental errors has always been that originally proposed by Gauss. The rules necessary for ifs application are embodied in the formalism of the “ Method of Least Squares.” Against the method and the rules by which it is applied two main objections have often been urged: it is said that the theory on which the rules are based is not true and that, even if it were true, the rules are not an accurate expression of it. No serious inquirer pretends nowadays that the method can be completely defended against these objections: its use is justified partly on the grounds of practical convenience ‘and partly on the ground that any method not open to these objections would produce practically the same results. The second contention is probably valid, but it provides a justifi- cation for the use of the method only if the first is also valid, and if there is no method equaily convenient which is not open to more serious theoretical objections. I believe that the first contention is not valid and that in some cases—and especially in those cases of most importance in physics— there is a method of adjusting observations which is at once more convenient in practice and more sound in theory. The object of this paper is to explain and support that view. Perhaps I may be pardoned for insisting at the outset that my remarks deserve some attention. The theory of errors has great intrinsic interest, but it is not a matter to which physicists, even if they are in the habit of using it, generally pay much attention. Its later developments are extremely complex and highly technical, and most of us do not study carefully the memoirs dealing with it which appear from time to time in scientific journals. Those memoirs do not generally pretend to subvert the accepted rules for adjusting observations, but only to extend them to somewhat unusual examples or to provide additional support for them. But I do wish to subvert those rules : I contend that the Method of Least Squares is an intolerably cumbrous method for obtaining quite misleading results, that there is a method which is incomparably simpler and gives results which are not misleading, and that the only persons who have any adequate reason for continuing the * Commuicated by the Author. Phil. Mag. 8. 6. Vol. 39. No. 230. Feb. 1920. N 178 Dr. Norman Campbell on the use of the older method are the members of the National Union of Computers (if there is such a body) who might be thrown out of a job if the proposed method were adopted. Sinee the only justification for the older mathod which has so far stood the test of criticism is that it is practically convenient, I maintain that the mere proposal of a more convenient method throws the onus probandi on those who refuse to use it. 2. The three problems. There are three sets of circumstances in which the need may arise for adjusting “‘ inconsistent ” observations. (1) A number of measurements which do not agree completely are made directly on a single magnitude, for instance, the length of some definite rod or the time of some definite process. It is required to determine from them the “true value.” The rule universally adopted is that the arithmetic mean of the measurements should be selected as the true value*. We shiall see that its validity might be established directly by experiment. It is doubtful whether the necessary experiments have actually been per- formed, but I shall assume that the universal acceptance of the rule shows that no experiments conflicting with it have been made, and that, therefore, if it were suitably tested, it would be established directly. The matter is exceedingly important because on the acceptance of this rule are based, either explicitly (as by Gauss) or implicitly, the rules for solving the two remaining problems. Any theory of error which is to lead to practical rules must assume that in this case some rule is known for determining the true value from the inconsistent obser- vations. If we did not accept the arithmetic mean as the true value, we should have to accept some other mean if any progress was to be made. (2) A number of measurements have been made on several’ magnitudes between which a relation is known. The arithmetic means of the measurements made on each magnitude do not obey this relation ; consequently they cannot be the true values. It is required to adjust the observations so as to obtain true values which do obey the relation. For example, the magnitudes may he the three angles of a plane triangle: their sum must be 180°; * Some modification of this statement may be necessary if “ systematic error” is suspected. Such error will be discussed in the sequel. I do not think it can arise if the conditions contemplated are fulfilled strictly, and the measurements are made directly on a perfectly defined system. Adjustment of Observations. VE9 and yet it may be found that the arithmetic means of the measurements made on each angle do not add up to 180°. This problem is of great importance in some of the practical applications of science, such as surveying It is not of much importance in pure physics, for we very seldom require to know with great accuracy and certainty the value of any directly measured magnitude; it is derived magnitudes that are important ”*. (3) Measurements have been made on many sets of magnitudes (2, y, z,...), which are known to be re- lated by a numerical law of which the equation is Wt 0, 222 - @ One...) — 0, the forny ‘of the function, £ being known, but not the values of the constants a, b, c,... For example, measurements have been made of the activity of a pure radioactive substance (1) at various times (¢). It _is known that I and ¢ are related by the equation [=I).e~™. Tt is found that no values can be assigned to the constants which are such that all the measured sets actually satisfy the equation. It is required to determine those values of the constants which are to be regarded as the true values. This third problem is of immense physical importance, and the solution of it is involved in almost every expe- rimental research. It is often solved by graphical methods : numerical computation is used only when the number of constants is too great to be represented on a plane diagram, or when it appears that graphical methods do not utilize fully the accuraey of the observations. But it is desirable to discuss methods of computation applicable even to those cases where graphical methods can also be used ; for it will be admitted that both methods should be founded on the same principles. 3. The principle of solution. The accepted method of solving the second and _ third problems, which is embodied in the Method of Least Squares, depends on the assumption that the true values of the mea- sured magnitudes in the second problem or of the constants in the third are those which make the sum of the squares of the residuals a minimum: the residuals are the differences between the measured magnitudes and those calculated from * The reason is that pure science is not concerned with the investigation of the properties of individual objects, but: only with the establishment of laws. A magnitude which is determined by a law, and therefore important for pure science, is always a derived magnitude. N 2 180 Dr. Norman Campbell on the the true values. The rule for applying this method to the third problem may be stated as follows :— It is assumed that the equation which the measured. magnitudes have to satisfy is linear and of the form az +t byt cz+.2). ov = OF emo If (as in the example of radioactive decay) the equation is. not of this form, certain methods (which will be accepted for the present without inquiry) are available for reducing it to this form. It is ‘clear also that, without) loss or generality, we may always put a=1, and this procedure will be adopted in what follows. If there are n variables. (@, y, 2 s- .) and, consequently, n constants (0 uau ien). and if N sets of the variables have been measured, we have N equations of the form a+ by, tes +... $m=0,) au + byn + cen +... +m =0. N is greater than n. To obtain unique values of (4, ¢,... 2) we have to reduce these N equations to n equations. We form one of these equations by multiplying the rth equation by w, and forming the sum of all the equations so multiplied ; another by multiplying the rth equation by y, and forming the sum; another by multiplying the rth equation by <, ; and soon. We thus obtain n “normal” equations relating the n constants (b, c,...m) to sums of squares and products of the variables (w, y, z,...). In these equations (6, ¢,...m) are now treated as variables; the solution of them gives the true values of (6, ¢,...m). The rule for solving the second problem can be expressed in a form very similar ; but since, as has been noted already, this problem has not much physical importance, it will be left on one side for the present. 1t will concern us only in so far as we have to determine whether any other principles proposed for solving the third problem are, like those of the Method of Least Squares, also applicable to the second. Regarded apart from the theory of errors on which it professes to be founded, the rule given is merely a device for reducing the N equations for n unknowns, the solution of which must be indeterminate, to n equations for n unknowns, the solution of which is determinate. But there is a much simpler method of effecting the reduction. We may simply divide the N equations into n groups, and add all the equations in each group: we thus arrive at » “ normal” equations. Adjustment of Observations. 181 The procedure is so obvious that it would be the first to occur to anyone to whom the problem was presented. It has doubtless not been adopted mainly because the alternative method of Least Squares was held to be the only one that is justifiable by the theory of errors. But this procedure can also be based on theory. When we select a group of the equations and add them to obtain a normal equation, we are assuming that this equation is absoiutely correct, and that ihe sums of the measured magnitudes are the same as the oS sums of the true magnitudes to which those measured (=) magnitudes approach. In other words we assume that, if we take the sum of a group of measured magnitudes, the errors of measurement cancel out; we assume ‘that the sam of the errors in any group is zero. Now if the group is sufficiently large, this assumption will be true, even if we believe in the Gaussian law of errors, but it will also be ‘true if we adopt any other reasonable law of errors ; for it is an assumption more fundamental than those on which the Gaussian law is based that positive and negative errors are “equally probable.” Accordingly, if the groups into which the equations are divided are sufticiently large, the assumption that their sum will be free from error is based on theory much more firmly than the assumption of the Gaussian law ; for the first assumption is part of the second, and the part which is least dubitable. The only question which can arise is whether the assumption, and the pro- cedure founded on it, is justifiable when the groups which are added are not very large. In a later part of this paper I shall argue that, even in this case, the procedure, though not capable of complete theoretical justification, has more theoretical justification than any other, and a great deal more than that of the Method of Least Squares. The proposed procedure may, therefore, be called the Method of Zero Sum (Z.8.) in coutradistinction to the Method of Least Squares (L.8.). But even if its theoretical basis is accepted, two further objections may be urged against it. ‘The first arises in connexion with the second problem of the adjustment of observations. We have measured the three angles of a triangle and find that the measured values do not add up to 180°. The method of adjustment proposed is to choose true values such that the sum of the errors is zero. But it is at once apparent that it is impossible to choose such values which will at the same time add up to 180°; for the sum of the true values ‘must be the same as the sum of the measured magnitudes : in this example then the method will not work. I cannot 182 Dr. Norman Campbell on the help thinking that it is the failure of the method in this important application which has prevented its serious con- sideration. I shall argue later that in this example the Gaussian method also will not work, and that its application in the ordinary manner leads to a result which is directly contradictory to the assumptions involved in it. The Gaussian method, like that of Z.S., involves the assumption that there is no systematic error, and in this case the assumption cannot be maintained. We must base our procedure on a theory which recognizes systematic error. However, at present we are concerned only with the third problem, and in connexion with this an objection may be urged against the method of Z.S. It may be said—and of course the statement of fact cannot be dispated—that the result which will be obtained will depend on the manner in which the observational equations are grouped to obtain the normal equations: if one grouping is adopted, one set of values will be obtained; if another grouping is adopted, another: Z.S. does not, lke L.S., lead to an unique solution. Herein lies, to my mind, one of the chief advantages of Z.S. For the apparent uniqueness of the solution by L.S. is altogether misleading. It is true that the introduction of ‘probable error” admits implicitly that solutions other than that given by the normal equations are admissible; but that admission is so important that it ought to be stated explicitly. When we discuss in detail the theory of the matter we shall see that there is not the slightest reason of any kind for selecting one of the admissible solutions rather than another. However, for the purposes of practical con- venience, it is certainly desirable to have some standard method of selecting a single value to represent the obser- vations, even if that which is selected is not really different in importance from many others, simply in order that no scope may be left for personal choice and that all persons who consider the same observations may arrive at the same single value for expressing them. But if it is admitted that the choice of that standard method is to a large extent arbitrary, there is no difficulty in devising one that is suitable. Accordingly in the application of the method of Z.S. it is proposed that the groups should be selected in the following manner. There is always at least one of the measured magnitudes («, y, z,...) which may be assumed to be free from error: let it be y. (This assumption is also involved in L.8.) The observational equations are to be Adjustment of Observations. 183 arranged in increasing (or decreasing) order of y. If there are N equations and n magnitudes (a, y, ¢,...), and if N=pn+q, where p, g are integers, then the first normal equation is to be formed by adding the first p equations in this order, the second by adding the second p equations, and so on until n—g normal equations have been formed; the other g normal equations are to be formed by adding p+ 1 observational equations taken in order in the same way *. The basis of part of this rule is obvious. The method provides that the number of observational equations added to form a normal equation is (as nearly as possible) the same for each normal equation. Since the assumption underlying the method is only true if that number be large, it is desirable to prevent it being smaller than it need be in any one case ; that result is obtained by making the number equal in each case. The basis of the remainder may be .seen by considering the case when there are only two magnitudes, w, y. Then we are practically taking the mean of each of two halves of the observations and deter- mining 6 and m from these two means. ‘The determination might be made graphically: we might plot the points representing the two means and draw a straight line through them. It would then be obvious that the accuracy with which the straight line could be drawn would be greater the greater the distance between the two points. It is desirable therefore that the difference between the two means should be as great as possible: this condition is obtained by arranging the observations in the order proposed for the purpose of forming the normal equations ; for one mean is that of all the smallest values of y and of all the smallest (or largest) values of x, whereas the other is that of all the largest values of y and all the largest (or smallest) values of a. 4. Krrors. It will be convenient also to express the matter ana- lytically. The normal equations will be Xy = b6Y,+¢Z,+ ee e ML, ] XG = bY,+cLy+ 50 mH” > y ab) J * It may be observed that if g is not 0 the result obtained will be slightly different according as an increasing or decreasing order of y is adopted. But the differences arising from this latitude of choice are completely negligible. 184 Dr. Norman Campbell on the where (X, Y, Z,...) are the means * of a group of observed (2, Y, 200+). The solution is b6=D),/D; C=DIDsy. . 23. i ee where -1o; iD, Na Ai xe : D ee, x. a Reh | oun Now suppose that only one of the measured magnitudes, «, is liable to error and that all the measurements on the others are absolutely correct. This assumption is practically true in a large number of important cases ; it is moreover essentially involved in the method of L.S., so that we are not introducing any new error by adopting it. Then if db, dc,..: dm are the errors caused in the calculated values of, c,...m, by errors db = 1/D(dX,.D,;'+dX,.D,?+...), a (5 dm = 1/D(d&, D,+dX,. Dee.) where D, is the minor of X, in D,. The mean error of L.S. (from which the probable error is calculated by multiplying by 0°6745) is the mean of all the errors which might be expected if the observations were repeated a large number of times and the quantities calculated from each set of observations. In deducing it, it is assumed that the mean square errors dX,7, dX,”,... are all equal and that the mean product errors dX,, dX_..- are all zero. Adopting this assumption, we find de? = aAX*/p,; de? =aX*/p,; ...; dm? =a) where ip oa Me ashe’ OES) * Strictly speaking, they are the sums, not the means. Ifgq is 0 they could be converted into means by dividing each of the equations by the same number p—without, of course, any effect on the result. If gis not O the results would be slightly different if some equations were divided by py and some by p+1, but again the effect of neglecting this difference is quite inappreciable. In practice the normal equations will always be sums, not means ; but in discussing them generally it will be assumed that they are means, not sums: no appreciable error is thereby introduced. e e e ‘oad Fe Adjustment of Observations. 185 In accordance with the usual terminology, we may call dX? the error of a single mean X, and pp, P...-+ Pm the weights of 6,¢,...m. For a given accuracy a the observations, the calculated values of b, c,-..m willbe the more accurate the greater the weight. If there are only two magnitudes a, y and only two constants b, m we have iY) (¥,-Y2) p= 5s p= Yoav 7 @ If there are three magnitudes and three constants 6, ¢, m we have _ (Vidp— Vo Voly—TaVs+Vols—WsYaye ey (Ly— L;)? + (LZ3— Ze)? + (LZ — Zs)? | — (¥1Z,—Z,Y.+ YoZ,;—Z.Y;+Y3 Z;—43¥;)? ie he (Y,—Y,)?-+ ( (Y; —Y,) 24-(Y,—Y3)* ) r (7) = Zp— L,Y .+ Yo4,—L2Y3+ Y,4,—Z;Y,) | ae (¥yz —L,Y>)? "+(Y, ie VER hey 2+ (Y, Li— ZV, ye From (6) we reach again the conclusion that the calculated values will be made most accurate by making Y, and Yas different as possible, which is effected by grouping the obser- vations in the manner proposed. In (7 ) the matter 1s more complicated, and the most accurate way of grouping the observations depends on the values of 6, ¢; but in many important cases the rule which has been eS sed gives the calculated values the greatest possible weight, and in no case does it seem to give them a weight very much less than the greatest possible. When ease of application is taken into account it is improbable that any more suitable rule of general validity could be found. By the aid of (4) and (5) probable errors of the calculated values can be found in a manner exactly similar to that of the method of L.S. We shall inquire later what is the significance of such probable errors according to L.S. or Z.8., but for the present we shall use them merely as a rough method of comparing the results obtained by the two methods. It should be observed that when (4) is used in L.S. to obtain the probable error, da? occurs in place of dX”, where da? is the mean square error of a single observation and dX? the mean square error of the arithmetic mean of p such single observations. In our estimates of probable error by Z.8., 186 Dr. Norman Campbell on the we shall assume the usual results of L.S., namely, that 9 Yv? : he ae where Xv? is the sum of the squares of the ca Hestuals, amd qi Ne Examples. It will now be well to show how the proposed method works out in practice. ‘The chief difficulty is in the selection of material. I have a large amount of matter of my own to which it might be applied, but the use of that matter might not inspire contidence. Observations are not usually published in sufficient detail to enable them to be recalculated; so three examples have been taken (although they are not wholly suitable, because the number of observations are so small) from the 8th edition of Merriman’s ‘ Method of Least Squares, pp. 126, 132, 138. The advantages of the method in the saving of labour increase rapidly both with N and n, but even in these simple cases they areenormous. In L.8. 3Nn(n+1) multiplications have to be performed, and then 4(+2)(x+1) columns, each of N entries, added; in Z.S. there is no multiplication, and only N(n+1)/n columns, each of N/n entries, are added. Using a calculating machine, to which I am thoroughly accustomed, omitting all ‘“ checks”’ (and the omission wastes — time on the whole), and reducing writing to a minimum by keeping figures on the board of the machine, I found that the mere writing, quite apart from calculation, involved in the formation of the normal equations of L.S., occupied longer than the complete formation of these equations by Z.S. The solution of the equations takes the same time in either case: it takes longer than the formation of the normal equations by Z.S., but not nearly as leng as that by LS. In each example there is given (1) the equation which the observations have to satisfy; (2) the observations; (3) the normal equations and solution by L.8.; (4) the normal equations and solution by Z.S.; (5) in the last four columns of the observations, the residuals and their squares according to L.S. and Z.S. The observations which are added to give the normal equations of Z.S. are bracketed. In the third example the equation (1) is not linear: for the purposes of calculation it was reduced to linear form, in accordance with the usual practice, by taking logarithms of both sides ; these logarithms, used in the calculation, are given in the table. Adjustment of Observations. 187 EXAmMPLe 1. x= by+m. LS. ZS. wv, y. at SRS ae Fes “SS Raa (1) >) 3921469 09688402 Bee todl 4394 1030 (2) | 39-20335 09289304 eG 1 Lees 0 (3) | 3919519 08904120 Le 24 EYe 18 (4) § 39:17456 0°7929544 98) 790 277 767 (5) | 3913929 _ 0°6127966 __ 96 7 93 5 (6) | 3910168 — 0-4254385 cae a 0 06 0 (7) | 3903510 —0:0948286 4 95 6 4 28 5 (8)) 39-:02425 —0-0505201 Hines) Meo “167-29 (9) | 3901884 00341473 sy, Wisse 377-1498 (10) | 3001907 0-0218033 i 1 Sips 2. (11) F 3902410 0-0190338 4457 2088 4454 2061 (12)! 39:01214 —-0-0019464 Wings ae 396 1568 (13)J 3902074 0-0000515 4505 2550 4503. 2530 v4 1306 ¥v29689 Ev, 1305 229706 Sv_ 1308 Sy_ 1300 (1.8.) 508:18390 = 13:000000 m+ 4°848702 b 189:94441 = 4:848702m+3°804394 b * m = 39°01568+0-00077, 6 = 0°20213+0-00142. (Z.8.) 274:06386 = Tm SLY AL AOL 234:12004 = 6m + 0:-127501 6 m = 39°01571+0:00089, 6 = 0°202044+0-00184. Promvequs. (1-1) Mane —939-01530 b> = 0:°20265 = (GS-13 ascot > = 0-208 + This equation is misprinted in Merriman. In Examples 1 and 3 there is no material difference between the results obtained by the two methods; they agree well within the probable error. The probable error in Example 3 is actually less for the result by Z.S. than for that by L.S. Indeed in that example the comparison does not appear as favourable to Z.S as it ought to be. For the residuals are calculated for log x: strictly they should be calculated for x. If they are so calculated, Sv? is slightly less for the result by Z.S. than for that by L.S.: the Least Square method does not actually produce the least squares. 188 Dr. Norman Campbell on the EXAMPLE 2. B= by+cc+m. LS. 2.8. x. Y. zZ Ca air “aN a oe ma) # vx104, 210%, UxXAOLE, TOGO? (1) | 81950 0 0 Ei 1 aes 625 (2) 3°2299 Ol 0-01 --18 324 —14 196 (3) | 32532 02 0-04 i 4 2A 121 (4) 32611 0:3 0:09 +21 441 — | ae (5) | 3°2516 0-4 0°16 +19 J61 — 6 36 (6) | 32982 O05 0:25 131 961 ag 64 (7) 3°1807 06 0:36 —44 1936 — 59 a481 (8) 3'1266 O-7 0:49 —39d 1089 —33 1089 (9) | 30594 O8 0-64 0 0 4.99 484 (10) | 2°9759 0-9 0:81 +24 576 +72 5184 Svy 97 Lv? 5693 Su4 127 Sv? 11261 Sv 96 Sv_ 129 (L..8.) 31°761600 = 10-00m + 4:500 y + 2°8500 z 14-089570 = 4:50m+2°850y + 2°02502 8°828813 = 2°85m+2:025y 4153332 m = 3°1951+0:0015, b = 0-4425+0-0077, ¢ = —0°7653+0-0081. (Z..8.) 26731 53m +039 -ROwae 97409 = 3m +124 74D 902 12°3426 = 4m 4+3°0y +2°302 m == 3:1925 +0:0034, b = 0°4678+0-0072, ¢ =—0:7960 +0-0081. Prom eqns. (1-5), m= 3:219590, 0b = 030025 oO ene. 9 (6-10) m = 3°20265,- b = 0:26705" ef — 0 30h. This discrepancy is, of course, due to the fact that it is the sums of the squares of the residuals of log v2, and not of «, that have been made a minimum: this process, though almost always adopted, is not justifiable by the theory on which the method is based. On the other hand, it is legitimate to apply the method of Z.S. to the logarithms ; for, so long as the errors are small, the solution which makes the sum of the errors of # zero will also make the sum of the errors of f(x) zero, whatever may be the function 7 (so long as it has no singular points in the neighbourhood). Here is another advantage of the method of ZS. It can be applied directly to an equation that is not linear, so long as that equation can be reduced to Adjustment of Observations. 189 " EXAMPLE 3. c= my. L.S. ZS. £. y. | log x. log y. ae ~ - a~ — | UXNOE ae x LOT Oe KOR 2 >< 10°. 1731 O-1144 0°23830 —0°94157 +7 49 +9 81 18538 01312 | 0:26788 —0-88207 —38 1444 —37 1369 1984 01445 | 0:29754 —0-84013 +18 324 +19 361 2081 O1579 | 031827 —6-80162 + 5 25 + 5 25 rm Wy oe Oe C0 033666 —0°76930 +4 16 + 3 S) 2°258 01813 0°35372 —0-74160 +15 225 +15 25 2-326 0:1925 | 036661 —0-71557 2315 25 = 6 2-397 0:2026 | 0:37967 —0-69336 Ba | 1 = 9 2460 0-2123 0:39094 —0:67305 to 25 eT Sv+ 49 Sv? 2134 dv+ 51 Sv22159 Su 49 SOLS 2 (L.8.) 279496 = 9:0000 log m—7:0583 0 —2°2758 = —7:0583 log m+ 5°6008 b Maron 000. b= 07>127220;0045. (Z.8.) 145865 = 5 log m—4°23471 6 149094 = 4 log m—2°82358 b Mm = GU0ESO025, 9b = 0:5743+0-0023. From eqns. (1-5) m=6:07, 6=0 a. (GO oem 573, b = 0-546: the linear form by a change of variable ; whereas strictly the method of L.S. ought not to be so applied. If the method of L.S. is adopted the process of successive approxi- mation, which has to be used when the equation cannot be reduced to the linear form by change of variable, ought always to be applied to a non-linear equation, even when it can be so reduced. On the other hand, in Example 2 there is a difference between the values of b and ¢ deduced by the two methods which exceeds the probable error of either method. But an examination of the residuals at once shows the explanation. The equation to which the observations are to be fitted is empirical; the residuals show a systematic variation which indicates that the formula is not strictly true. At the bottom of the table are given the values of b, c, m calculated by L.S. first from the first 5 equations and 190 Dr. Norman Campbell on the second from the last 5. The differences between the resulting values of 6 and ¢ obtained from these two parts of the observations differ by far more than the probable error; this difference is an indication that the formula applicable to the observations with small values of y is not applicable to the observations with large values. It is not strictly legitimate to apply the same values of 6, c, m to all the observations. And if it is not legitimate, of course the method of Z.S., which divides the observational inaterial into two parts and assumes that the equation is equally true for all of them, will give a result different from that obtained by the methud of L.S. which treats that material as a whole. If our object is merely to represent the observations as nearly as possible by a con- venient empirical formula, it may certainly be better in such cases to employ the method of L.S8.: a closer fit is likely to be obtained. But then the problem is not one of pure physics, which is not concerned at all with merely empirical formule; neither method of adjustment has in such cases any true validity at all, for the adjustment itself is fundamentally false. And if the object is merely to obtain an empirical formula it may be urged that, though L.S. gives a closer fit, it is scarcely worth obtaining at the expense of the enormously greater labour. It may be noted that, for comparison, the observations of Examples 1 and 3 have been divided similarly into two parts, each of which is adjusted separately by L.S. The results are given at the foot of each table. It will be seen that in these examples, where the equations do fit the observations (although the equation of Example 3 is also empirical), there is very much less difference between the values given by the two parts of the observations. (The value of m for the second half of Example 1 is subject to a very large probable error, as may be seen by examining the observations on which it is based ; the differences are not at all inconsistent with the applicability of the same constants to the whole of the obser- vational material.) 5. Some further considerations. These examples—and a great many others have been examined—show that the method proposed is perfectly practicable and that it does not lead to results differing very greatly from those of the method of L.S. And that proof, as I hold, throws the task of Justifying their action on those who continue to employ a method which is admittedly invalid theoretically and exceedingly cumbrous in practice. Adjustment of Observations. 191 I maintain that, until it is shown to lead to misleading results, the method of Z.S. holds the field against any other, merely on the ground of simplicity. However, I am pre- pared to admit that in certain cases it loses much or all of) its advantage, namely in those for which N is not very much greater than n, and the number of observations not much greater than the number of variables. It is then found—as might be expected from the fact that p is little if at all greater than 1—that values obtained for the constants vary greatly with the precise grouping of the observations in the formation of the normal equations, and the probable error of the result is much greater than can be judged significant according to the criterion which will be developed at a later stage. Of course the best method of dealing with such cases is to make more observations and so cause N to be much greater than n ; but if, for any reason, that course is impossible, and if some single value must be obtained, then it is probably better to employ L.S. unless N is at least as great as 3n. But the use of that method is a mere matter of practical convenience : I deny altogether that, in general, the results obtained have any greater theoretical significance than the widely differing results obtained by the method of Z.S. There is simply no theoretical ground for any single value whatever within very wide limits. These considerations have a bearing on the second problem of adjustment of observations, namely that in which it is required to determine true values of the measured magnitudes and not constants of an equation which they satisfy. The true values are now values such that they satisfy some equation of which the constants are definitely known. In one form of the problem, this equation contains, besides the variable magnitudes, a constant term to which a definite numerical value is assigned. An example of this form is the problem of the angies of a triangle, and we have already noted that the method of Z.S. (and that of I..S.) must fail when applied to that problem. But in a second form, the ‘equation of condition ” does not contain a constant term, but relates only the measured variables. An example of this form is the problem of adjusting the results of a levelling survey : here it is known (e. gy.) that the height of A above C must be the sum of the heights of A above B and of B above OC, but there is no numerical constant known apart from the observations. In this problem it is possible to find true values such that they satisfy the equation of condition and make the sum of the errors zero; and rules for applying the method of Z.S. E92 Dr. Norman Campbell on the can easily be devised. But since the equations of conditions ure never much more numerous than the observations (they are usually much less numerous), N is never large compared with n. Accordingly ZS. gives results varying widely as different methods of grouping are adopted ; and though there is no reason to believe that all these results are not admissible, it is difficult to fix precisely one method of grouping, so that the first requisite in problems of surveying, namely that a definitely unique set of values shall be obtained, is fulfilled. It is certainly better to adjust by the method of L.S. And it is fortunate here that the method loses most of its disadvantages. The coefficients of the equations of condition are usually small integers, so that the calculation is easy. Moreover it. will appear in the sequel that this is the one form of problem to which IS. is strictly applicable on theoretical grounds. It is here that there is most evidence that the Gaussian law of error is true, and here that the method is an adequate expression of the Gaussian theory. I believe indeed that the Gaussian method was originally elaborated to deal with just this problem: if so, it was completely justified. It is only its extension to the other form of the second problem (where there is a constant term in the equation of condition), and to the third problem, that is both theoretically illegitimate and prac- tically inconvenient. i It is admitted then that, in this direction, room still remains for the method of L.S. This admission may seem to weaken somewhat the case for its replacement elsewhere by Z.S. Accordingly, before proceeding (in a subsequent paper) to a discussion of the validity of the two methods according to the theory of errors, it may be wel! to point out that there are examples to which L.S. is as clearly inapplicable as Z.S. is to that which has just been discussed. These examples occur when the equation (1) reduces to the simple form 2=by, as happeus when we have to determine a density (b) from measurements of a mass (#) and a volume (y). | An elementary student, when he had measured several sets of associated values of # and y, would doubtless take the ratio /y in each set, and take for b the mean of these ratios. A more experienced worker would realize at once that such a procedure gives undue weight to the ratios derived from the smaller values of x,y, which are likely to be less (relatively) accurate. He would probably add all the «’s and all the y’s and take the ratio of the sums. This is exactly the procedure of the method of Z.S. But nobody, I believe, would adopt the Adjustment of Observations. 193 7 Ar method of I.S., according to which b=". In this case actual practice would always follow Z.S. and not L.S. And I do not think the reason is merely simplicity. This example calls attention to a fundamental ambiguity in L.S. If the relation between 2 and y were written av=y, then L.S. indicates that a= Se," But then, in general, it would not be found that a=1/b; in our example the specific volume would not be the reciprocal of the density. On the other hand, according to Z.S. that relation always would be fulfilled. It is probably the realization of this ambiguity which prevents anyone using L.S. in dealing with such observations. But the ambiguity is not confined to this simple example. If, in place of PROCS leer mee S13) is ioe (8) we write Gaye er 1.. ie ms oes 7 tle CQ) we shall not find in general that (a, ¢’,... m’) =(1, ¢, ... m)/b, if we calculate the constants by L.S. The reason is that the method of L.S. assumes that only one of a, y, z,... is affected by any error at all, and that this one is z, of which the constant coefficient is1. We get different results by using (8) and (9) because in one case we are attributing the residuals to errors in #, in the other case to errors ot y. It is generally re- ecognized that L.S. is only applicable strictly when one variable alone is liable to error—though it is often applied when that condition is certainly not fulfilled. But it appears not to be recognized generally that, even when it is known that only one variable is liable to error, it is very seldom known which of the variables is this one. For instance, IT am observing the position of a pointer at various instants of time. Are my errors due to observing the wrong position at the right time or observing the right position at the wrong time? Indeed is there any sense in asserting one of these statements rather than the other? But according as I adopt one statement or the other I shall obtain different values for the constants of the equation relating position and time, if I calculate according to the method of L.S. This appears to me the most fundamental objection to L.S. as a means of obtaining unique values in the solution of problems of the third kind. It is important to realize that it does not arise in dealing with problems of the second kind. To ascertain how the ambiguity is introduced, we must Phil. Mag. S. 6. Vol. 39. No. 230. Feb. 1920. O 194 Dr. J. Prescott on the examine the method in further detail ; but since I do not wish to found my plea for Z.S. rather than L.S. on theoretical considerations, the examination will be left for a subsequent occasion. All that [ am concerned to prove here is that Z.S. is much more convenient than L.S. in the problems which arise in physics, and that since L.S. is certainly not theoretically unobjectionable, there is nothing to outweigh its practical inconvenience. Summary. It is urged that, for the adjustment of observations in those problems which are of importance in physics, there is a method much simpler and more convenient practically than the con- ventional Method of Least Squares. This method, which is called that of Zero Sum, depends on the principle that the sum of a large random collection of errors is zero. It is not maintained that the method of Z.S. is completely . valid theoretically, but it is maintained that it is no less valid than the Method of Least Squares. Accordingly the burden of proof rests on those who continue to use a method which is neither practically convenient nor theoretically unobjectionable. However, in a subsequent paper it will be maintained that the method of Z.S. has the advantage in theoretical validity as much as in practical convenience. Research Laboratories of the General Electric Co. Ltd. Nov. 1919. XIX. The Buckling of Deep Beams. (Second Paper.) By J. Prescott, W.A., D.Sc., Lecturer in Mathematics in the Faculty of Technology of Manchester University ; with an Appendix by H. Carrineton, 8.Se., M.Sc.Tech., A.MI.M.E.* N the first paper on this subject t it was shown thata deep beam may fail by a sort of torsional instability, and the particular load at which this instability occurs was. calculated for beams under uniform loads or under concen- trated loads with various methods of support. In this paper more general forms of the differential equations will be deduced than in the earlier paper, and some of the results will be extended. Moreover, the appendix contains a veri- fication of one of the formule by means of experiments, * Communicated by the Author. t+ Phil. Mag. October 1918. Buckling of Deep Beams. 195. carried out in the Manchester College of Technology by Mr. H. Carrington. The same notation as in the first paper will be used, and it will be useful to explain the notation again here -— H= Young’s modulus, EC=the flexural rigidity of the beam for horizontal bending, n=the modulus of rigidity, Kn=the torsional rigidity of the beam, 7=the angle of twist of the beam at anv point D, y=the deflexion of the central line of the beam from the naturally straight state at the same point, v=the abscissa of D referred to an w-axis taken along the unstrained central line of the beam. The coordinate y is a horizontal one, the assumption being made that the depth of the beam is so much greater than the width that the vertical deflexion is negligible compared with tiie horizontal deflexion when buckling occurs. More- over, the vertical plane containing the central line in the unstrained state is supposed to be a plane of symmetry of the beam. In the first paper the differential equations for each beam were worked out separately. It will be shown here that they all come under one general form. Suppose ODB is the plan of the central line of a uniform beam which is bent sideways as shown in fig. Sra the Fig. 8. 0 x B same time the beam is twisted so that, when 7 is positive? the lower edge of the beim is further from the straight line OX than the corresponding upper edge. The force on the part of the beam on one side of D may be considered as made up of the following four actions at D :— (1) a bending moment in the vertical plane touching ~ the central line of the beam at D due to vertical forces 5 2) a torque about the central line ; 3) a vertical shearing force at D ; 4) a possible ee due to fixing the ends. 2 = ( ( (4) 196 Dr. J. Prescott on the Let G denote the bending moment at D in the vertical plane, G+dG the bending moment at D'. Let T and T+aT denote the torques at D and D! causing the twist in the beam. Representing these couples by vectors perpendicular to their planes we get the system of couples shown in the next figure. 7 Fig. 9. GtdG a We denote by @ the angle between the z-axis and the tangent at D to the central line of the beam, so that dd is the change of the angle between D and D’. The vectors for G and (G+dG), being normal to the beam at D and D’, contain the angle dd. Resolving the couples on the beam about the tangent at D' we get, to first order, Gdo+dT=0, . 2 eee whence aT fo dd oe G Aa But ; = =tan d=¢ nearly. Therefore an as qe&y di. dace” (71) The moment about the tangent at D’ of the shearing force at D, as well as of whatever load there is on the element DD’, nas been neglected in the preceding equations because, assuming that the load is on the central line of the beam, this moment is a quantity of at least the second order. The correct equation when the load is not on the central line is given later (equation 101). Now the bending moment G can be resolved again into a pair of components about lines respectively parallel and Buckling of Deep Beams. 197 perpendicular to the twisted depth at D (see fig. 10). The former component, of magnitude Gr, bends the central line in a plane perpendicular to the depth of the beam, and this plane of bending is everywhere nearly horizontal. If there Fig. 10. G G7 is an end couple M on the beam acting in a horizontal plane, as in clamping the end, then the total couple at D causing bending in a horizontal plane is Gr+ M. Then, since the curvature produced by this couple is the ‘ cause of the deflexion y, we get dy adx* Oe = Gr Me oa es 6 12) Equations (71) and (72) are the general differential equations, which, together with the end-conditions of the beam, determine the buckling load when there is no tension or thrust in the beam, and when the load is applied at the centre line of the beam. Zz Substituting in (71) the value of “7 from (72) we get dx? dT G 9 de = 7 EG MGT +. . . ah tet (an) From the meaning of Kn we have Kn x (angle of twist per unit length) =torque, that is, Kn aL eae NE nas Sener Therefore (73) becomes finally BOKnS4=—G(Gr+M). . . . (175) It is clear that a tension in the beam would help to stabilize it, and that a thrust would make it less stable, for the beam could buckle under a thrust alone. Ifa thrust R is applied at the ends of the beam, then an extra term — Ry 198 Dr. J. Prescott on the occurs on the right of (72). Thus Cy, EUs =Gr+M—Ry, ° e e e (76) the term Ry being the bending moment in Euler’s theory of struts. Case 8.—Beam of length / under a pair of balancing couples, each G, at the ends, together with a thrust R (Gove) 1b); ; Fig. 11. OQ | B\ Rp x R G Elevation & O 8 x D y Plan of Central Line It is understood that the section of the beam has at least one symmetrical axis which is assumed to be vertical, and the length of this axisis several times as long as the greatest horizontal breadth of the section. The end couples men- tioned in this problem are in a vertical plane parallel to the length of the beam. It should be noticed that the direction of G, and therefore of 7, are contrary to their directions in Case 1 in the first paper. In this case M is zero. Then the equations applying to this case are (71) and (76). Since G is constant (71) gives Nyy dy T= G7 s, aiht) cethaeaepstemma (IE)) or SCL dy Kn = —C + N. e ° ° e e (738) But at the middle of the beam 7 and y haye both maximum or minimum values, and consequently = and a are both zero. It follows that the constant N is zero. Buckling of Deep Beams. 199 Integrating again we get Gi Oe te el a oh eC O) no constant being added in this case because both 7 and y are zero at the ends. he negative sign on the right side is due to the fact that 7 is negative by our convention. Now equation (76) gives, since M is zero, a Gann tae HO 4=— Kath fy lee a8) which is the same equation as for a strut under a thrust Kn theory of struts instability occurs when Ge tm (= SRN SECHY Vo. . (81) If the couple G is zero then we get Huler’s value of the thrust, namely +R), By exactly the same reasoning as in Huler’s 2 T ive ECy, and if R is zero we get the value of the end couple G that causes instability when no thrust acts, namely C= 7 VEnCK, which result has already been obtained in Case 1 in the first paper. If the end force is a tension of magnitude R’ we need only put —R’ for R, and then we find (GG? J aN ; : KR = ECT ee eee then.) From this it follows that a very big couple could be applied at each end and the beam would remain stable provided a suitable tension R’ is also applied. That is, the tension wholly or partially neutralizes the effect of the end couples in producing instability, while the couple G weakens the beam when used as a strut. Case 9.— Beam under the same forces as in the last case with the addition of a pair of couples in horizontal planes applied at the ends to keep - zero at those points. This corresponds to a strut with clamped ends, and differs 200 Ded. eescoih a whe from Case 2 only in having a thrust in addition to the couples, 2) Equation (79) is true in this case as in the last. In equation (76) M is not zero for this case, and therefore the equation corresponding to (80) is Bos = Ms fe enh ya utes) This is the same equation as for a strut with clamped ends and a thrust (i +k) . Also the rest of the conditions for the beam are the same as for the strut. Therefore, by analogy with the strut, x Ag? HC | rege pe Pee ut) (O45) If a tension R’ is applied instead of the thrust R the equation becomes G? 7 4 Kage Gd It should be observed that the two cases just worked out can be regarded as solutions of the strut problem. Suppose the two thrusts at the end are each applied at distance p from the centre of the end section in the direction parallel to the depth as in fig. 12. Then the couple G is Rp and the Fig. 12. rz ) : thrust at which instability begins for a pair of free ends is R given by equation (81), that is, by the equation Rép? aoe =HU7, ibe aD If Rp? is small compared with Kn we may use the approximate BO HGuOR | R+ nenot— 2! (ac%) : Sit /? Kn BC ie a NTE ee mp? HC) Gal ee reres: Buckling of Deep Beams. 201 Without assuming that R is small we see that equation (85) gives a pair of roots with opposite signs. The negative root indicates a tension, and thus we see that the beam could buckle under a tension applied in a line parallel to the unstrained central line but not along it. Since the sum of the roots of the equation is negative, it follows that the tension that would cause buckling is greater than the thrust that must be applied in the same line. Moreover, if p is. small then the tension at which buckling occurs is very great, and an approximate value is obtained by dropping the term on the right of (85). This approximate value is This is a very remarkable result in that this tension is independent of the flexural rigidity of the beam. In the first paper the loads were all taken on the centre of the sections of the beam. ‘'T'wo cases will now be worked out to show the effect of taking a load slightly off the centre of the beam. Case 10.—Beam built into a support at one end and free at the other where a load P is applied. Fig. 13. Elevation Plan of Central Ling a _ The load P, before the beam is strained, is situated at (0, p, q) reterred to three rectangular axes through the free end, as indicated in fig. 13. End view 202 Dr. J. Prescott on the The origin being at the free end of the beam the bending moment at x is Gen and therefore equation (75) gives 2 ECKn = Bt eee, 2 De S = ET, |. eG) where revea Pe 37° This is exactly the same equation as for Case 3 in the first paper. The only difference is in the end conditions. ‘These conditions are now t=0 where w=1, Ok ae (88) ~ Kae = torque —P(qr+p) where 2=0) 92739 89) The negative sign is necessary because 7 decreases as w increases, The solution of i (86) in series is mia? a m* 0 T=4 4 B— : {: 158 Om 8.8 ie te das eres eC) Now condition (89) gives Kna=—P(qb+p), .). 72 7 sean) and condition (88) gives mil4 m8 Qe ae ae 37 mil? mils 04 th re ry | 2 ee GOD Substituting the value of a from (91) in equation (92), and writing s for m*/*, we get ! Pl(qb + p) oa s° oe ee $ 5 (ithe ace }= +0) teats =0. (93) Buckling of Deep Beams. 203 Now the value of P which makes 0 infinite will make + infinite except at the fixed end. That is, the beam is unstable for that value of P which makes the coefficient of b zero in equation (93). The condition for instability 1s, therefore, : : Plq S s? mea a5 ta ae \ 22 ane +3437} =0. . (94) ee 3 2. Bae, 8 It should be observed that this equation is independent of p, which shows that the stability is not affected by putting the load a litttle to one side of the centre. The only result of displacing the load laterally is to put a torsion on the beam, thus giving a new equilibrium state of the beam from which instability begins. Let us write, for shortness, S ? Oe reer has ag s s? DN menslL 3.2.7.8 o- ‘Then our equation for the load is TAS) te (3) Or asi reste ary (OD) An approximate solution of this is the solution of the equation nh (S) Ol 45, eeene as S96) since the other term is small because g is small. Let s, denote the solution of (96), and let P, be the corresponding value of P. P, is, of course, the value of P found in Case 3. In the small term in equation (95) we may use the approximate value P, for P. ‘lhen writing 6 l for ae and (s;+2z) for s, equation (95) becomes —6f(s;+2)+ F(s,+2)=0. Since z is small, F(s,+2)=F(s,)+2F’(s,) approximately =0+-F'(s,). Then taking account of the first powers of 6 and z only we get — 6f(s}) +2F'(s,)=0. 204 Dr. J. Prescott on the Now 1 2s ie Ty ea ao atl G)= 7 3a ee Tea oe j See Tr oie { U7 47 8 Je and, from the result of Case 3, . IP , i The results of tedious arithmetic are As) =0°3577, F’ (s,) = —0-04344. Therefore a Us 5 7 004544 5; Pel — — 89: 16g 8:23 Kn- Hence s=S, +2, — (1+ = Ne 8] $5] and as » S =—]+ : itor Sy} a 8) it B = eee ——_— y) / 81 A/S) 1 1 Ss 8:23P lq V Hn “\ + 2x 4012 ( P,EKn q OF 1102 cal 1—1:°0 a7 Ka Finally PPR = Vsx V¥BnCK =4-012 { WV EnCK — 1-025 7 EC ; me i, The critical Euler load for this beam, when used as a strut and supported in the same way (fig. 14), is tT KC Ra oe. Ce Buckling of Deep Beams. 205 Now it is worth while to note that the correction to the buckling load due to putting the load at height g above the centre of the end instead of at that centre is qHC 4012x1025 x4 IR hal? 1 l =16667R, . NEE LO thus showing once again the intimate connexion between the strut problem and the buckling beam problem. 4°012 x 1:025 Fig. 14. Case 11.—Beam under a total load W distributed as a uniform load w per unit length, the load at x being situated, before the beam is strained, at (2, p,q). The beam is fixed at one end and free at the other as in the last ease. We have now to extend equation (71a) to the case where the distributed load is not on the central line. Dealing with the element in fig. 9 we find that the load wd, when the beam is twisted, has a torque wde(gt +p) about the tangent at D’. Then, instead of (71a), we get Gdb+dT + wde(gr+p)=0, . . . (100) and instead of (71) ay ey | a as TOP... (101) which, when M=0, as in the present case, is equivalent to dr (¥? Kn 7 (aac nee le sc 3 LOD) 206 — Dr. J. Prescott on the This is the general equation when the load is off the centre and there is no couple M at the ends. In the present problem, the origin being taken at the free end, G=4we2", and therefore 2 Kno" wat na =— (GG +wq)1—wp. 5 ; (103) Whether p and g are functions of « or constants the solution has the form T= af (2) ta, liz) babe), jee wee) where (wx) is a particular integral corresponding to —wp. The conditions to be satisfied at the ends are A | ata where ~=0, and +=( where v=l. These give O=a)f"(0)+a,F’"(0)+¢'(0), and O=aof(l) +a, F (1) + (2). Eliminating a, from these we get O=aoff'(O)FU)—/AOE (0)} +(0)F(l)-—P@() EO). . . . . (105) Now the analytical condition for instability is that ay should be infinite, and this can only occur if the coefficient of a in (105) is zero; that is, the condition for instability is FOF O—fOF(=0,. . . - (106) which is independent of ¢ and therefore of p. : We have now shown, as in the last case, that p has nothing to do with stability. : Since p does not affect stability we can drop it from our equations. Then, assuming that q is constant, equation (103) becomes, when p is dropped, 7 aa (mat + 67) 7, >. i ee ela) where Atos Ae YAO K (108) Fe em (0!) Now the assumption is being made that q is small, and Buckling of Deep Beams. 207 the problem is to find + when q is zero and then correct for q (or c”). When ¢ is zero the solution of (107) that satisfies all the conditions of the ae is ma me? ¢l2 si a {1-3 Ue One y: = ere m being given by the following equation, taken from Case 6 in the first paper, Mey =aAe BOE ties My echgen | hey ls MOLLY) The value of + in (110) being denoted by 7, the equation we have to solve is approximately dr Pinion emy). . . . (12) the term on the right being now regarded as a known function of x. This process amounts to treating c* as negligible while c? is not negligible. The particular integral of (112) is “2 a ( itt a ) = { GO 5.6 mia “rapa | iL i 1 ) ae Bp) ale: ee | iL se 1 120 (aa Ty DY Hoe Cc co) alley ea 1 1 | Spe on eee ee ne ee (iL eM ge aa, ELT) Then the complete value of 7 that satisfies the condition that the torque is zero at the free end where 2=0 is T=71—ayc7a? F (m*e®) Gj Were ayce km)... . . (l14) where /(m*®x®) is the series in the brackets in equation (110). The other condition that has to be satisfied is that r=0 when z=l. Therefore Camm yeaca ney. sk (NLD) 208 Dr. J. Prescott on the Let s be written for the value of m°/° satisfying Fak) SO, ee eee lat) that is, s=41°30,. 7 2 oe aint) as given in equation (111). Then let the solution of equation (115) for m°l® be written melS=s+v. Actually equation (115) has to be solved for w which is involved in both ¢ and m, but since the term involving c is ‘small, we can use, in the expression for ¢, the approximate “elms. of w given by (111). Now equation (115) becomes 0=f(s+v)— CPE (s +v) = (8) + of") - CFF (s) as far as terms of the first dimension in v and c?. But f(s) =0 by (116). Therefore. | LO) Ve Fiisye (118) ‘Since s Se nase )—1~ 55° 5,611.12, 5.6, 01 noe ‘therefore f! (S)= nae 1 ae 2s tie es + ‘ B65 6 IN 2 5 6s 1 eae ‘With the value of s given in (117) we find that . 04890 Oe and F(s)=0°'1888. Therefore a 0-1888 x 30 2p 7 i 04890 = ele wg? 1b. Q —l Ste Mere a (US) ‘where w, is the value of w given by (111) ; that is, wl =2 V41-30EnCK, Buckling of Deep Beams. 209 Then a Be 2 v= VES OK 3 3% (120) : 93-08 17-2 and 818 = 41-30 — 23°08q 4] 3HC l Ku 23°08 g EC } = 41-30 {1 -— 4 rent (121) Consequently, W being the total load, WP= ni V4EnCK TLECT AWWA TEN Ala PES a ANG —9 V/A1- Mn(3 pa bite die BOS panel 2 / 41-30 V EnCK i = Kat 13-85 mek —93-12HC. .. 2... (122) l Thus the correction to WI? due to the distribution of the load along a line at height g above the centre line of the beam instead of along the centre line itself is 23:17 HC. If the load were below the centre by an amount q the term involving g would be added instead of subtracted, the beam being in that case more stable than with the same load on the = el line. The problem of the stability of a beam fixed at one end and free at the other was worked out in the first paper for the following two cases: firstly, with a load P at the free end and no other load ; secondly, with a uniform load per foot and no load on the end. Now we will try to combine these two loads. Case 12.—Beam fixed at one end and free at the other, and carrying a load P at the free end and a small unifor mly distributed oad in addition. To find the condition for instability. It is assumed that the load on the end is much greater than the uniformly distributed load. Let w be the uniform load per unit length. Then the bending moment G at distance 2 from the free end is Cae Snesumetme., 2 CL23 ) yp? Therefore, neglecting a G2 == P22? + Pw. pee ys, (OA) Phil... Mag. 8.6. Vol. 39. No. 230. Feb. 1920. P compared with unity, 210 Dr. J. Prescott on the Consequently the differential equation for the twist t is K d*7. ena Oe 2 i, EC GM pay wa =~ Fo (1+p)s - es Let TST 4p, 6k 2 ee where 7, is the solution of the equation 2 92 Kenton ee dx? Ho that is, 7, is the value of 7 for the case where w=0, which is the problem solved in Case 3 in the first paper. In the present case, are Ee | a Se U dx? Pe a ae eon: Now w is a small quantity and p, He due to w, is therefore also small. Then neglecting the product wp we get Cy a a eee? E HS da, daw ih) NG [ute mt sea) By using equation (127) this last equation becomes ide ae Wk Knge=— po [et pape + + (29) . 2 that is, EG varie eee (RO) daz where peas Mee (131) En’ w ; ra’, (132) and, by equation (22) in the first paper, 451 mia? ae - m8 iia { Dee, aia aaa a The particular integral of (129) is ‘ mx Sma? Pe VO BS o 101m-Pa!? emer Buckling of Deep Beams. 211 The condition to be satisfied at the free end of the beam is that the torque is zero, that is, dt a viererE— 0. du This condition is satisfied by the value of 7 in (126) if p has the value given by (133). The other condition is that T= where v=l. Let mahi), = —rbak (m?2"). Then we have to solve the following equation for m?/? bpmeP 7 In? )=0. 1. 2... (134) _Let ml7=s+v where AC Ie iaaaains Wire te iny C1) that is, from equation (25), Sere SaaS PAU tie ts eaty Teta a0) Then f(s+v)—rlF (s+v) =9, or JS Tl h(sy—07)).. 2) (1387) on neglecting v?, rv, and small quantities of higher orders. Equations (135) and (137) give After the necessary arithmetie we arrive at the results F(s)=0°1014 x s, #"(s) = —0°0869 x s. Therefore v= —1'1677! W being the total distributed load, and P, the buckling load when W is zero. ; a) ~ IN, Dr. J. Prescott on the Finally, WwW PH Pee yes 4| 6 eae n?=s—1:167 P, 2 or jo ae VW HnCKk Po = 4-012 {1-0-2015} Py cere Pa ae ds e291 P P, Ww ap }1—0 215 t =P,-0-201W, . . ee mesy whence 4-012 P+0:291W=P,= p V/ HinCK. (5a ay aE) The above is the equation that holds just when instability occurs provided W is very small compared with P. Now the buckling load when P is zero is, by Case 6, ye OVS AMUN Me sek W= ne / HnCK. Therefore the equation Pr Wi? ah tae is true in two cases: (1) when P =0, (2) when W=0. Moreover, this last equation can be written Fe 0312 W = 258.0 2) a eerie “which does not differ much .from (139). It seems very probable then that equation (140) will be a good one for all values of the ratio of W to P. If the ratio between W and P is fixed it is possible to find the actual values of these loads when buckling occurs, but the problem is very awkward unless one of the loads is small compared with the other. It is worth while, however, to work out the case where P is small compared with W, so as to see if equation (140) remains approximately true in one more case. ‘This we wil! now do. Buckling of Deep Beams. 213 Case 13.—The same problem as Case 12 except that the load P on the end is small compared with the uniformly distributed load W. Here, as in the last case, G= Pet swe’. But Ge =dotet(14 ie WH = wx" (1 af =) approximately. Therefore CE We Pe ee dx? 4Kn ok! + we =—m'at(14+ =r, eee (CL 4)} where re Meme ne CK: io 8 ree os (143) Now let es es ts)! aaa eel Baer (CLARY where moy® mizz' pa ets Tapes en ei . > and therefore d*7, iS a ee Ae. capt aterum EG) Then equation (142) becomes d?r, d’p _ 4P i? ee” ese 2) 4) ia 4P =—Mwx a ae ante, neglecting the product Pp since both factors are small. By means of equation (146) this last equation gives 2 6 oe + mitp= — ar, EN ake enone (8 eG Now if we differentiate through (146) we get d?7,' 1 de + moat7,/= —4m'n*7,, . . . (148) dt t,/ being written for —+ dx SE ———EEEEE——— 914 Dr. J. Prescott on the A comparison of (147) and (148) shows that a particular integral of the former equation is Pel Ba ae ar p= aa! = Ate ° ° ° Bis (149) Then it follows that (iit P dt, i ey, ee ae a . e 5 2 e (150) is a solution of equation (142), and this solution satisfies the condition that the torque is zero at the free end where «x=0; that is, dae 0 where «=0. ax The only other condition that it is necessary to satisfy is that 7+=0 where z=, that is, P dt, Tet aay ae 1" ww dz =() where 20,0) 9) ee) and from this equation m is to be found. Let iy Ad Baie 62 (CLS) Then, since me is small, a P hd P hd j(metn =f (mu) +m a) (mz) ty ote be ale eam legen) It is now clear that equation (151) can be written in the form j(mltm,) s= Qo 5.) a eaen ee clbeney But we found in the first paper (equation (55)) that the solution of the equation Fnl) —0 nel = 643.6090). ee lesa It therefore follows that the solution of equation (154) is 3 mn? ( L+ ) = G74 35 Ww was given by that is, Ba On Ie a A a : mi (1+ w) — 6°43 approximately Buckling of Deep Beams. 215 Consequently Wie a 7 3P 2 BnOK \ W or Wr IEE TT Tee ig VECK. 1. 2. (156) Now we see that, whether W is small compared with P, or P small compared with W, the result expressed by (140) is nearly true. Then it is sure to be nearly true for all other positive values of the ratio W:P. The result may be expressed roughly in the following form:—A load on the free end of the clamped-free beam has approximately the same effect in buckling as three times that load distributed uniformly along the beam. _ _ There is one assumption in the working of the last case that should not be passed over without justifying it. It is the assumption, made in the squaring of G, that P is small compared with wa. This, of course, is quite true everywhere except where «is small. But if we consider that the actual term neglected, namely P?2?, is itself small in the region where there is any possibility of error, and that the assump- tion is wrong only over a very small range of values of z, it is clear that the error made is negligible. There is still another point of view that will show the Justification for this assumption. The actual method of solution consists in dropping a term from G%, and if we had added a term of the same order we could have got RNG G?=—1 ae zim) \ = Ww (+ =) 6 | Then, by changing the variable to (« Ga =| our differential equation would have reduced to the same form as in Case 6 where P was zero. It is easy to show ty this method that the solution of Case 6, with (2+ =) for w and (i+ *) Ww \ Ww for /, applies correctly to the present case provided powers PEOE = beyond the first are neglected. But this is precisely the solution we have obtained by taking a different value of G?. Then it follows that the term neglected in G? does not affect the result to our degree of approximation. Case 14.—Beam carrying a small uniformly distributed load W and a much larger concentrated load P at the middle, —— 216 Dr. J. Prescott on the and supported at the ends with just the necessary forces and couples to keep the depth of the end sections upright. This is a combination of Cases 4 and 7 in the first paper with the condition that the ratio of W to P is small. In this case, taking the origin at one end, G=4(P+Wye- 4 a? =3Qe— 3) ae 22) Sel)s | eRe nme (157) Q being written for (P+W). Therefore d?r aay Qe 1 Wea.” dz? 4BnCK WR QL 2We = — mia? {1 Bera U ~ | T approximately, (148) where rae ny = ABnCK (159) If we also write » for the small quantity yr , then 72 ~ eg = eT. een elon.) Now let Tat + Oy. ag (161) where ' m2? mex? 7 T= a) tm oe 6 39 ee (162) and therefore au maT OL) Se elie) da Then (160) becomes, after making use of (163), tmetp=—nira ry, . 1) eee elite the product of the pair of small quantities 7 and p being neglected. A particular integral of (164) in series is ‘Dele mixt mene oe 8 { TS) Mee ye mi2y12 ron 13am ao Bian If we use this value of p in (161) we get a solution which Buckling of Deep Beams. ee satisfies the condition that r=0 where «=0, which is one of the conditions of the problem. | Another condition is that the torque is zero at the middle. This follows from symmetry, for clearly neither half can be exerting a torque on the other. ‘Then dip. dp ve: = eat 5 ° e 166 a of 7 0 when x=3l, (166) that is, writing w for ($ml)‘, iE he - Ww 2 ees ates i ; mets (45.99.12 ° rin 6 10u | lor meg 4.9" 2.89.13 ae ,+}=0. Vr clon) Now let u u ae = = ile aye F(1) aie A .= 10u? 143 eee | As 513 then our equation is rl 60 Let us put U=S +H, Mee (yO 1 > 5 68) where s is the solution when W=0;; that is, (9) Ue aie oe aie CS) and, by equation (33) in the first paper, Ose yo a Ge CL EO) Then, retaining only the first powers of v and r and no products, equation (168) becomes f(s) +07/"(s)+ 7 Bs) =0, or rl vf (s)+ 60 G0: Therefore pee eh (Ss) = RFS (171) 218 Dr. J. Prescott on the After the usual arithmetic we find that F(s)=21°64, f(s) =—0-1974, whence cS 2 OA ae) 2W and consequently me ony gn =s + v=4°482 oe e827 “Oe = 4°482 2 ae jmeP= VW 4482 ! si ((jp 407" ah or, OF W : GK = vias {1+o407d }. Therefore, dividing by (140-407 9] and treating © as a small traction, 1?Q } 10-407 ot =8 4482 / HnCK, Q or (Qi? —0°-407 Wi? = 16:94 VY EnCk, whence P?+0:593WP=16°94 /BnCK. - This result can be written in the form BP vee GO Bee vHnOK, 2, = gine) Now when P is zero the present problem reduces to Case 7, the solution of which is ee 28-31 But this differs very little from what we should get by putting P=0O in (173), which has been obtained on the widely different assumption that W is small compared with P. Then it is very likely that equation (173) is nearly correct for all values of the ratio W : P. The constants EC and Kn, which are involved in the buckling loads, occur merely as constants in equations for bending and torsion respectively ; that is, there is no = /BnCK, . . 2s: Buckling of Deep Beams. 219 assumption in this paper as to what Kn means except that iy torque ~ angle of twist per unit length’ Kn There is no assumption then that K has any particular form such, for example, as is given by St. Venant’s theory of torsion. The only assumption is that torque is proportional to twist and that Kn is the constant expressing the quotient obtained on dividing one by the other. Similar remarks apply to EC, but there is no need to lay the same stress on this as on the torsion coefficient because the bending coefficient is better established, although even here C is not rigorously the moment of inertia of the section if E is Young’s modulus. The point of the preceding argument is this, that EC and Kn are a pair of coefficients which should be obtained by experiment for any particular beam ‘that is to be used for testing the theory of this paper. It would not be right to calculate EC and Kn, even if E and n were known accurately, for that would be burdening the theory of buckling beams with whatever errors are contained in the theories of bending and of torsion. Of course, when the buckling formule have to be used in practice it will usually be necessary to be content with calculated values of C and K and assumed values of E and n, for that will be the best that can be done. But, in the testing of the results in the experiments carried out by Mr. Carrington, HC and Kn are found experimentally and their values substituted in the expressions for buckling loads, and these are compared with the actual experimental buckling loads. In using the preceding results in practice, where it will usually be necessary to calculate everything, the value of C is the same as in the bending of beams and the best value of K is the one given by St. Venant’s theory of torsion. For a beam of rectangular section, breadth 6 and depth d, these values are C= bed, K=4ld (1-0-6307), ¢ the latter being the approximate value of St. Venant’s torsion coefficient when 6 is less than 3d. 220 Whey Jel, Carrington on the Appendia by H. CARRINGTON. Experiments were performed to determine the degree of accuracy with which the buckling loads for cantilevers loaded at one end, as calculated by the mathematical ex- pression in (1) below, agreed with those obtained by experiment. Five steel strips were used, each of which was straight, free from dents, and accurately ground. The expression tor the buckling load is 4-012 4/EnCK . P= P ste Bie ah : ° (1): where / is the length under test, EC the least flexural rigidity, and Kn the torsional rigidity, so that in order to calculate P it was first necessary to determine these rigidities. The flexural rigidity of a strip was obtained by arranging ii as a cantilever and loading it with weights suspended from the free end by a fine wire. The deflexions (6) of the free end corresponding with increasing increments of load were noted and plotted one against the other. The lines so obtained are given in fig. 1, and the flexural rigidity (HC) is Eien 1 2 3 4 5 a tested tested tested tested ferngth 20° Jength 10", length 10", length 20” p / : yi Ys f : Zz : y, Vd (oa) uo Load (76s) = ra) Ww bh ee Derlection (/ division = O-/1nch) Woe 1 W “sg Where -— is the slope of the line and / is the length under test. The thicknesses of the strips (see table below) were small compared with the lengths under test (10 in. and over) and the deflexions due to shear were accordingly negligible. In order to obtain the torsional rigidities the strips were given in each case by the expression HC = Buckling of Deep Beams. 221 fixed in a delicate torsiometer which was capable of trans- mitting small torques of known amount, about the longi- tudinal axes of the specimens. The angles of twist corre- sponding with different gradations of torque were noted and plotted one against the other, and the curves are reproduced in fig. 2. The torsional rigidity Kn is given by Kn= a ip d Se where > is the slope of the curve and / is the length of the 6 corresponding strip under test. Vo 2 Fig. 2. 28 | : e S) a 5 tested | tested tested tested, testea | 26 | fength 20 length 20 Jength 20° ¥- fength2o length20 o t 22 f | é | 20} { , | / Torgi? (15-105 ) y A © OO ie tO et y | i 1 2 ly H { 73 ae ot Se OL, : (a) ! 2 3 Om! Came Se ter ye 167 i Ar) Veena 4G; eB IO (6) 2 4 6 0 | (2 $) é 277 ; Angles of twist - (/ = j99 radian) The values of the rigidities for each of tle five strips are given in the table (below) and also values of P calculated by the expression (1) and corresponding with two lengths for each strip. It may be noted that no dimensions of the strips other than the lengths under test are involved in equation (1). The buckling loads were obtained experimentally as follows :—each strip was firmly gripped in a vice and pro- jected horizontally outwards with its sides vertical. The joad was transmitted by weights suspended by a fine wire which passed through a small hole drilled through the mid- depth of the specimen near its end. When the buckling load was almost reached the specimen could vibrate slowly from side to side and finally came to rest in the central position. When the buckling load was just past, the The Buckling of Deep Beams. oN | ON NI (‘popuno.a adam ‘Gg pue ‘% ‘T ‘son sdi.qs JO sespo eo L— aN) | Vier ee | y= Z9— | ge— Oe | OG io z-0 vee Eee eee " “quao tad eousseyiq 0-8 0-28 G-¢] 0-84 8:31 0-1 LT C96 9:66 OTS “GT ‘prop Suiqyong [rquomadsy; | | | | | oe oes 5 oF = ¢9-L 9-0€ GFI 0-16 OG) 1 ber FFI G.cz 8.86 LI¢ moun © Oe | WoL “GT pRoy Sulpyonq pejepnoyre,) OT G ie Ol i OL G.), | 0 | gi | pote (SUL) SUlTYon 20) pojsa} YJ Us'T 18% O89I ILE GIF Oece ie a "WU qy (wy) S4tpisiyy [wuols.o J, | LCT 00ZT | 1¥G FOE | G1SS ee cUl-"G] (OW) APIpIoIyy [VNxXs] | | 86F0-0 OO0T-O G0¢0-0 10¢0-0 0660-0 [eee ee es "* (SUL) sBOUYOUT 60-0 91¢-0 LLL-O G16-0 1¢0.T pos ee eee GSU AUDEN c - e 7 | | Retracement drys jo ‘ony On a Simple Property of a Refracted Ray. 223 specimen could still vibrate slowly from side to side, but came to rest in a position displaced on one side or the other of the central position. The buckling load was taken as that which would cause the beam to come to rest so that the displacement from the central position was the smallest appreciable. The results of all the experiments can be seen at a clance from the accompanying table. It was found of importance that the sides of the strips should be exactly vertical, for if they were very slightly inclined, a displacement was noted at comparatively small loads. A delicate adjustment was necessary, so that when the buckling load was reached the strips would remain at rest, very slightly displaced, on either one side or the other of the central position. In every case the buckling loads were determined experi- mentally before their values were calculated by equation (1) ‘in order that the determination of the experimental loads should not be influenced by a knowledge of the calculated values. This was necessary because it was difficult to decide, within 2 or 3 per cent., at what load buckling began. XX. Note on a Simple Property of a Refracted Ray. By Autce Everert *. | ae AP be a ray incident at a point P on any refracting surface, BP the refracted ray, CPN the normal, C the centre of curvature in the plane of incidence ; ¢, ¢’ the angles of incidence and refraction, and y, yy’ the angles CAP, CBP made by the rays with a transversal CAB through C cutting them at A and B. Then ¢4+w’=¢'+W=y, say. The angele y plays a leading part in some modern optical generalized formule (see Optical Society’s Transactions, vol. xx. pp. 28-31, Nov. 1918, where it is pointed out that, for a spherical refracting surface, y is also the angle made with the axis by the line joining C to the points of inter- section of the rays with the aplanatic surfaces), but objection has been raised that this angle is not readily visualized. ‘The following simple property which, curiously enough, seems hitherto to have escaped notice, may be of service in this respect. * Communicated by the Author. From a communication to the- Transactions of the Optical Society for March 1919, p. 203. 224 On a Simple Property of a Refracted Ray. If PT be the tangent at P to the circle circumscribing the triangle APB, ‘then the angle TPN, between this tangent and the normal produced, is equal to d+ =, since the angle APT between this tangent and the chord AP is equal to the angle at B in the opposite segment of the circle, and the angle APN is ¢. Without actually drawing the circle APB, the direction of the tangent can usually be judged by inspection. If now the transversal rotate about CU, while the rays remain fixed, the tangent will rotate about P at tne same rate, but in the opposite direction, the angle swept through in either case being A= increment of wr =increment of yr’. When Ay attains the value 7—vy, the tangent to the circle coincides with the normal PC to the refracting surface, and if CB'A’ be the corresponding position of the transversal, then / BCB'=7—y4, Me, Jet). LPA'C=¢’, UN Sa yin 518. CBl ier yw, pw being the refractive indices, and r the radius of eurvature. | XXI. On Resonant Reflexion of Sound from a Perforated Wall. By the late Lord Rayueteu, O.1L, R.S.* [This paper, written in 1919, was left by the Author ready for press except that the first two pages were missing. The preliminary sentences, taken from a separate rough sheet, were perhaps meant to be expanded. Prof. Wood+ had observed highly coloured effects in the reflexion from a granular film of sodium or potassium, which he attributed to resonance from the cavities of a serrated structure of rod-like crystals. | HIS investigation was intended to illustrate some points discussed with Prof. R. W. Wood. But it does not seem to have much application to the transverse vibrations . of light. Hlectric resonators could be got from thin con- ducting rods 4 long; but it would seem that these must be disposed with their lengths perpendicular to the direction of propagation, not apparently leading to any probable structure. The case of sound might perhaps be dealt with experi- mentally with bird-call and sensitive flame. A sort of wire brush would be used. The investigation follows the same lines as in ‘ Theory of Sound,’ 2nd ed. §351 (1896), where the effect of porosity of walls on the reflecting power for sound is considered. In the complete absence of dissipative influences, what is not transmitted must be reflected, whatever may be the irregu- larities in the structure of the wall. In the paragraph referred to, the dissipation regarded is that due to gaseous viscosity and heat conduction, both of which causes act with exaggerated power 1n narrow channels. Tor the present purpose it seems sufficient to employ a simpler law of dissipation. Let us conceive an otherwise continuous wall, presenting a flat face at w=0, to be perforated by a great number of similar narrow channels, uniformly distributed, and bounded by surfaces everywhere perpendicular to the face of the wall. If the channels be sufficiently numerous relatively to the wave-length of vibration, the transition, when sound impinges, from simple plane waves on the outside to the * Communicated by Sir Joseph Larmor. + See Phil. Mag. July 1919, p. 98-112, especially p. 111, where a verbal opinion of Lord Rayleigh is quoted that in certain cases the grooves of gratings might possibly act as resonators. Phil. Mag. 8. 6. Vol. 39. No. 230. Feb. 1920. Q 226 The late Lord Rayleigh on Resonant waves of simple form in the interior of the channels occupies a space which is small relatively to the wave-length, and then the connexion between the condition of things outside and inside admits of simple expression. On the outside, where the dissipation is neglected, the velocity potential (@) of the plane waves, incident and reflected in the plane of zy, at angle @, is subject to d/dP =a (Pdjdx?+d*o/dy?), .... CQ) or if 6 x e'”, where n is real, Uolda? + d7bjidy kb —0.) a) ae (2) k being equal to n/a. The solution of (1) appropriate to our purpose is ae eh ey ain Aer cos De ea : } (3) the first term representing the incident wave travelling towards —w, and the second the reflected wave. .From (3) we obtain for the velocity wu parallel to x2, and the con- densation s, when 2=0, w= OP 2 poeriysn 832 cos OCA Ry Ax pes cll 1 do ae m i(nt +ky sin 6) 3 as= 2 a Y (A+B), Q airs (5) so that U B—A ah = COS 6 Rau 4 6 5 5 e ° (6) For the motion inside a channel we introduce in (1) on the left a term hdd/dt, h being positive, to represent the dissipation. Thus, if ¢ be still proportional to e, we have in place of (2) db/da? + d?p/dy? + @d/d2+k?bd=0, . . (7) where k is now complex, being given by k= k?—inh/a*.)\.)) 2 SO ee)) If we write k’=h,—iks, where k,, ky are real and positive, we have ko kek, kyko=tnh/a7. Soe eo) At a very short distance from the mouth of the channel a?d/dy*, d*/dz” in (7) may be neglected, and thus oe {A cosk «+B sink a}. Wo) ean ane) Refleaion of Sound from a Perforated Wall. 227 Tf the channel be closed at c= —J, A'sin k/l+ B' cos k1=0, and we may take ae cose (at lie oy aha MCLE) From (11) when 2 is very small, pe—imon— eA ‘sink jem, yg). (12) as=—a'ddb/dt= —ikA" cos k'l.e™, . , (13) so that ea! / as ten Fl SN Sn tl ta A ML) Now, under the conditions supposed, where the transition from the state of things outside to that inside, at a distance from the mouth large compared with the diameter of a channel, occupies a space which is small compared with the wave-length, we may assume that s is the same in (6) and (14), and that (o +0')u in (6) =ou in (14), where o represents the perforated area and o! the unper- forated. Accordingly, if we put A=1, as we may do without loss of generality, the condition to determine B is B-1 _ oid k' tan k'l (15) BP G+ c')cos0 tk | If there be no dissipation in the channels, h=0, and k =k. In this case _ (@+0°) cos @ cos kl—iosin kl ~ 16) ~ (o+0') cos @coskl+io sin kl” ( Here Mod. B=1, or the reflexion is total, as of course it should be. If in (16) c=0, B=1, the wall being unper- forated. On the other hand, if o’=0, the partitions between the channels being infinitely thin, cos 8 cos kl—i sin kl ~ cos Ocoskl+isin kl reper a - In the case of perpendicular incidence 6=0, and Bem ih acid aly sey CLS) the wall being in effect transferred from «=0 to «= —l. We have now to consider the form assumed when &’ is complex. In (15) cos k'l= cos kyl cos tkyl + sin kyl sin kyl, ’ (19) sin k'l= sin k,l cosikgl— cosk,l sin ikal. § Q 2 228 The late Lord Rayleigh on Resonant Before proceeding further it may be worth while to deal with the case where h, and consequently ka, is very small, but kl so large that vibrations in the channels are sensibly extinguished before the stopped end is reached. In this case cos thol=4e*2’, sin tkel =40e"2!, so that in (19), tank’"I=—i. Also by (9), k/k=1, and (15) becomes B-1 o Balm) (ee \cs6 * ae) making B=0 when, for example, o’/=0, cos@=1. The reflexion may also vanish when the obliquity of incidence is such as to compensate for a finite o’. In examining the formula for the general case we shall write for brevity Ce eels Beanie ron (25L). and drop J, so that k;, ko, & stand respectively for k,l, kal, Al. This makes no difference to the first of equations (9), while the second becomes | kykg==tnhl? ja... 2473 ee ee co ato)) Thus _ kS cos k' —2k' sin k’ \ ~ kS cos k’+ik' sin kh’ © RN cy 2) Separating real and imaginary parts, we find for the numerator of B in (22) ky tan tks hs tan hy cos ky cos tk, [as — +2 \#8 tan fe tan ky + ey) -1 @a) The denominator of (22) is obtained (with altered sign) by writing —S for 8 in (28). In what follows we are concerned with the modulus of B. Leaving out factors common to the numerator and denomi-. nator, we may take Mod.? Numerator = {iS— pe —hk,. tan ah +4 (is tan tk, wh 2 —,) tan ky + EL (24) Reflexion of Sound from a Perforated Wall. 229 The evanescence of B requires that of both the squares in (24), or that iS= = tan y= ik; cob ikp—ky cot hy, (25) or again with elimination of 8, ik, (tan tke + cot tk) = ky (tan ky + cot ky) 3 whence : Asin 2k, 7k, Sin 2iko—O0} 46 6 s (26) or in the notation of the hyperbolic sine Meoum 2ey=— eo, similn 2ieael Nia) wh is es CG) If this equation, independent of co, o', and cos @, can be satisfied, it allows us to find 4, from an assumed fg, or con- versely, and thence & bv means of (9). The next step is to calculate S by means of one of equations (25). If $,so found, > cos #0, we may choose o'/o so that B shall vanish ; but if S < cos-@, no ratio o’/o will serve to annul the reflexion. If the incidence be perpendicular, S must exceed unity. If S were negative, the reflexion would be finite, whatever may be the angle of incidence and the ratio o’/c. It is natural to expect an evanescence of reflexion when the damping is small and the tuning such as to give good resonance. In this case we may suppose k, and 7—2k* to be small, and then (27) gives approximately so ; 7 6k? By (25) / kS=h, tanh ky +h, tan hy, = ‘ tanh kes + k,/ tan (2khq?/t7), = 35, (L+ hy? +...), so that as ht 4 1+ 1+ 3) i+. ae Since § is large and positive, the condition for no reflexion can be satisfied by making the perforated area o small enough. For a more general discussion we may trace the curves * So that wave-length is 4 times J. 230 _ The late Lord Rayleigh on Resonant (B, A, fig. 1) representing the two members of (27), regarding k, and hk, as abscissee and taking as ordinates y=kysin 2k, y'=hksinh2k,. . . . (30) If k; and k, be both small, y= 2h (1 — 2h"), G2 eal + 2h,*), . . (31) so that at the origin both curves touch the line of abscissz and start with the same curvature. Subsequently y’>y and increases with great rapidity. On the other hand, y vanishes whenever f, is a multiple of 477, although the successive loops Reflexion of Sound from a Perforated Wall. 231 increase in amplitude in virtue of the factor ,. ‘The solutions of (27) correspond, of course, to the equality of the ordinates y and y'. It is evident that there are no solutions when y is negative. The most important occur when fy, is small and 2k, just short of 7. But to the same small vaiues of fk, correspond also values of 24, which fall just short of 37, 5a, &c., or which just exceed 27, 47, &c. More approxi- mately these are 4 cos ma . ko" 2hy=m7 + aes iar) en (32) where m=1, 2, 3, &c. In order to examine whether these solutions are really available, we must calculate 8S. By (25) 2 jhe IS=h (1 5h°)("" + cos mT “) 2, mir mir . 2cosmm .k,” + ky tan ( + ——__——_}. MVIT If m is odd, we have approximately KS = 5 (1+) SMe er Cae) and if m is even, i mks Meus 1) poy lth? ae emailed Since k is approximately 4mz, we see that when m is odd, S is large, and the condition of no reflexion can be satisfied, as when m=1. On the other hand, when m is even, 8 is small, and here also the condition of no reflexion can be satisfied, at any rate at high angles of incidence. It should be remarked that high values of m, leading to high values of £, correspond with overtones of the resonating channels. A glance at fig. 1 shows that there is no limitation upon the values of the positive quantities 4; and k,. And since fy is always greater than ky, k, as derived from &, and fo, 1s always real and positive. So far we have supposed that the values of k,, corresponding with smaJl values of k, are finite, as when m=1, 2, 3, &c. But the figure shows that solutions of (27) may exist when f,, as well as 4, is small. In this case we obtain from (31) pee len py. 5) 2 CS MnO GE) making k?=h,?—k 232 Reflexion of Sound from a Perforated Wall. Hence by (24) kS =k, tanh ky + kp tan ky =2h.?(14+2h7), . (87) and S=4/3.( + he?) 09. On Here again the condition of no reflexion can he satisfied, whatever the angle (@) of incidence, by a suitable choice of o'/o. But the damping is no longer small, in spite of the smallness of /,, since ky is not now small in comparison with ky and k. On the contrary, &; and k, are nearly equal, and B is small in comparison with kg, so that this case stands apart. Not only is 16 always possible to find a series of values of ky satisfying (27) with any assumed value of ko, but the values so obtained makeS positive. For in (25) ky, ho, tanh k, are positive, and so also is tan 4;, since tan k= sin 2k,/2 cos? k,, and sin 2k, is positive. It is a question of some importance to consider whether when co, o', and 6, determining S, are given, the reflexion can always be annulled by a suitable choice.of k; and kp. It appears that the answer is in the affirmative. Let us consider the various loops of fig. 1 which give possible values of k,. The ranges for 24, are from 0 to 7, from 2a to 3a, from 47 to 57, and so on. As we have seen, the intermediate ranges are excluded. In the first range hetween 0 and 7 we found that 8 may be made as great as we please by a sufficiently close approach to v7. At the other end where k,=0, the value of S was ,/3, or 1°7320. This is the smallest value which occurs. When 2k,;=47, it appears that k,=°5656, k=:5449, and S=1:947. And again, when 2h =3a, ky ='5795, S=1:964. We conelude that within this range some value of k, with its accompanying k, can be found which shall annul the reflexion, provided 8 exceed 1°7320, but not otherwise. In each of the other admissible ranges, 8 takes all positive values from 0 too. At the beginning of a range when 2h, slightly exceeds 2a, 47, &c., S starts from 0, as appears from (34); and at the end of a range, as 37, 5a, &c. are approached, 8 is very great (33). Within each of these ranges it is possible to An the reflexion by a suitable choice of ky, ky, whatever oa, o!, and 6 may be. Tf the actual value of S differs from that calculated, the reflexion is finite, and we may ask what it then becomes. If we denote the value of 8, as calculated from 4, ka, by So, (24) gives Mod.? Numerator =k?(S —So)?{1 + tan? /, tanh? ko}, The Quantum Theory of Electric Discharge. 233 and in like manner (by changing the sign of 8), Mod.? Denominator = k?(S + So)?{1+4 tan? k, tanh? ko} ; and hence 8— Sy , y sae Mod2 B= (s =). (39) where N= CosOla +o )ia. 9 =. i (21) If o,the perforated area, is relatively great, it makes little difference what its actual value may be, but if o is relatively small, as in the case of strong resonance, it is otherwise. It would be preferable to suppose 8 fixed at So and to calculate the effect of a variation of & with h given. The resulting expressions are, however, rather complicated, and it is evident without calculation that the reflexion will be very sensitive to changes of wave-length when there is high resonance as a consequence of small dissipation and accurate tuning. The spectrum of the reflected light [in the corre- sponding optical circumstances | would then show a narrow black band. XXII. The Quantum Theory of Electric Discharge. By Dr. ). Ne Mari and Prof. A. B. Das, M.Sc.* 1. ()* the quantum theory, the energy of a source of radiation varies in a discontinuous manner by equal quanta. If this is to be accepted, since electrons give out radiation when their velocity is changed, the quantum theory would seem to suggest that these changes are effected by a series of jerks, that is in other words the changes involved are due to impulses only. Moreover, since a material atom is known to be made up of corpuscles and positive ions, having configurations satisfying conditions of dynamical stability, under their mutual attractions and repulsions, it is necessary to admit that they are revolving doublets or higher complexities, In view of their rapid movements and collisions, such systems must be admitted to have complex nutational motions as well. It stands to reason therefore, that the energy of such a system cannot be a continuously varying quantity. It must either remain constant or, if it changes at all, it must change by finite * Communicated by the Authors. Paper read before the All India Science Congress, 1919, Bombay. 234 Dr. D. N. Mallik and Prof. A. B. Das on the amounts, in view of the fact that the changes are effected by (generalized) impulses. In any casea variation of energy in a discontinuous manner is consistent with many well-known phenomena. In the present paper we have attempted to interpret the behaviour of electric discharge in a vacuum-tube in terms of this theory. 2. In a vacuum-tube, as we know, the discharge is non- luminous or silent unless the difference of potential is greater than a certain amount, so that there is a minimum difference of potential for the passage of a spark. Now assuming the formula ch 1 i = Nee Nice? eR A__] where c¢ = velocity of light, A = wave-length, h = Planck’s constant, N = Avogadro’s number, Dy - for a gas, and Hy is the energy required for electronic oscillation of wave-length A, we observe that this quantity has a certain constant value, depending only on the nature of the gas. In other words, so long as a gas retains its specific properties, so as to yield a characteristic spectrum, the total energy required for luminosity has a constant value. That. is, the energy corresponding to the ionization of the gas when attended with luminosity has a value which is. constant and independent of the pressure of the luminous. gas, so long at any rate as the gas continues to yield its characteristic spectrum. 3. If, then, E is the energy of a corpuscle which is in a condition to ionize, yH will be the energy that will be transferred to the ionized gas, where y is a proper fraction. And for luminosity, yE must be equal to E,. It seems. reasonable to suppose, therefore, that the minimum energy of the ionizing corpuscle must be Ey. And if we further admit that E is proportional to spark-potential or at any rate of the form a+bV, where V is the spark potential, the minimum spark potential Vg must be proportional to Ho, Quantum Theory of Electric Discharge. 2395 or of the form a+6H,. The minimum spark potential must be then independent of the pressure. This has been found | to be the ease. 4. Now, it is well known that as the pressure of the gas decreases, a dark space is formed next to the cathode called the Crookes dark space. And it is obvious that until the energy of the ionizing corpuscle attains to the minimum value required, it will not be in a condition to ionize. This explains the formation of the Crookes dark space, and further suggests that the difference of potential between the cathode and the negative glow, called the cathode-fall of potential, must be approximately equal to the minimum spark potential and independent of pressure. ‘This is also in accordance with known experimental results. As the electric intensity in the Crookes dark space is very high, it will follow that the breadth of this space will be very small, increasing as the electric intensity decreases with pressure. It is seen further that the cathode- fall of potential must be proportional to Ho, or at any rate be of the form a,+0,Kp. 5. Again, the energy of a corpuscle when it collides with a molecule of the gas is evidently proportional to Xel, where X is the electric intensity and | the mean free path. Hence Xel=E. Moreover, it can be shown that if a is the coefficient which determines the rate of increase in the number of corpuscles, then, when @ is equal to zero, Xel is constant. We find now that this constant value must be = Ky. Since, further, [<5 where p is the pressure, - we find that when @ is zero, — =const., a result which was RS obtained otherwise in a previous paper (Phil. Mag. Oct. 1912). 6. Returning now to the Crookes dark space, we observe that it is limited by the point at which, through the action of the electric force, the energy =p, and, accordingly, this also marks the beginning of the negative glow. As the corpuscle moves through the negative glow, its energy decreases on account of collision, according to the law given (as J. J. Thomson has shown) by H?=EH,?—267 or E= Ky (1-f 2) nearly, where « is the (small) distance 236 Dr. D. N. Mallik and Prof. A. B. Das on the traversed and #8 is a function of the masses of the col- liding ions. It will happen, therefore, that if X is the electric force, B=E)—2%+{Xde, so that ionization will QO \ no longer be possible, when [Xde< Fi : This will then mark the beginning of the next or Faraday dark space. 7. Now, the energy at the extremity of the negative glow is evidently of the form E,)—/2, where & may be taken to be constant as a first approximation. It is, moreover, known that # increases as pressure decreases. Assuming A that vac(— +B), this energy may be written equal to i: caine ee aL a |, De If further, X, is the average electric intensity through the F araday dark space and d is the length of this space, we must have By— 3 iy X d= Kp. 1 U : We have moreover X,;x« ~- and d=a'+ ~-, approximately? Pp Ie during a certain range of pressure. We have thus, on this theory, a means of coordinating the various quantities associated with the cathode dark space, the negative glow, and the Faraday dark space. 8. Experiments show, however, that the linear law both as regards the negative glow and the Faraday dark space holds only for a limited range. Obs. In both these experiments, the distance between the electrodes was 14°6 cm. Experiment I.—Voltage in the primary about 11 volts. (Curve I.) Scale-reading Scale-reading corre- Length Pressure corresponding to sponding to the beginning _ of the inmm. the cathode terminal. of the positive column. dark space. 27 14°9 14°7 7 Git 15 14-9 14-15 a5, D 14°9 13°48 1:42 ,, 3 14°9 3325 iGo ss) 14°9 Oe) onl es Quantum Theory of Electric Discharge. 237 After this stage, the cathode glow spreads up to the positive column and gradually overlaps it, practically the whole tube being illuminated. Experiment I1.—Voltage in the primary a little above "25 em. [One E076 12 volts. (Curve II.) (Illumination stronger than in the previous case.) 35. 14°9 14°65 17 14°9 14:3 7 14:9 13°85 5 14:9 13°4 2 14°9 13°45 aS eee As the pressure is further reduced, the cathode glow appears to extend up to the positive column and overlap it. 40 Ww [o) => Pressure. (niet Oa MeReley, => length of the dark space in c.m. 238 Dr. Rudorf on Latent Heat and Surface Energy. 9. Again, with an induction coil, the line of demarcation between the Faraday dark space and the positive column is never well defined nor steady. This is in accordance with theory, for owing to a rapid variation in the potential of the induction coil, the energy of the field must vary in a rapid manner. 10, The peculiar behaviour of the striz indicates that they are similar in character to the Faraday dark space and are of similar origin. On the Quantum theory, accordingly, they are regions over which the energy of discharge is Jess than Ey. At very low pressure, before the disappear- ance of the strive, both the Faraday dark space and the striae continue to retain their breadth, showing that the corresponding change of energy is negligible. XXIV. Latent Heat and Surface Energy. To the Editors of the Philosophical Magazine. GENTLEMEN,— [ a recent number of the Phil. Mag. (vol. xxxviil. p. 240), there appeared an interesting paper by D. L. Hammick on arelation between the surface tension (y), the interna! latent heat (1), the molecular diameter (c), and the molecular volume (V). For argon, the data used were as follows :-— | y=11:00 dynes/em. at T=87° abs., i, —1284 cals: oo 2x 107? cme d=M/V=1°'404, whence . De = 226, the agreement between these two numbers being considered good. It does not appear to be generally known that the values of y for liquid argon given by Baly and Donnan (Journ. Chem. Soe. Ixxxi. p. 907 (1902)) are incorrect. The correct Notices respecting New Books. 239 values are as follows (cf. Rudorf, Ann. d. Physik, xxix. p. 751 (1909)) -— Mees te eeee 6) 8% 1 881.890) 1.90) | || 13-45 | 13-19 | 12-93 | 12-68 | 12-42 | 12917 | 11-91 | As regards a, the best values for the monatomic gases are, undoubtedly, those of S. Chapman (Phil. Trans. ecxvi. p. 279 {1916)), who found for argon o=2°84x10-* cm. Using these corrected values for y and o J fis ney = OO Areas: ao J On the other hand, L; is almost certainly more than 1284 (cf. Nernst, Ztschr. f. Elekirochem. xxii. p. 185 (1916)) and approximately 1440, whence ae 240. The agreement between 302 and 240 can hardly be considered good. Yours faithfully, 52 Cranley Gardens, G. Ruporr. Muswell Hill, N. 10. SG Bai. XXIV. Notices respecting New Books. The Harth’s Awes and Triangulation. By J. DE Graar Hunter, M.A., Mathematical Adviser to the Survey of India. (Survey of India Professional Paper No. 16.) Pp.viii+219, with 6 charts. [Printed at the Office of the Trigonometrical Survey : Dehra Dun, 1918. Price Rs. 4 or 5s. 4d.] HE first triangulation series of the Survey of India was com- menced as long ago as 1831. At that time, the best values for the lengths of the axes and for the ellipticity of the earth were those derived by Everest. Consequently the observations were reduced using Hverest’s data as a basis, and in order to avoid discon- tinuities, subsequent triangulation series have been reduced using the same data, although it has for long been known that Everest’s values are substantially in error. Moreover, the values used for the latitude and azimuth of Kalianpur, the origin of the survey, have also been found to be in error owing to a considerable local deflexion of the plumb-line. The necessity for freeing the results from the errors due to these incorrect data has become more and more pressing. ‘To recompute the whole triangulation was ot course entirely out of the question; the labour involved would 240 Notices respecting New Books. have been too enormous. The natural thing to attempt was to determine differential corrections to the coordinates of each point. This is the problem which Mr. Hunter has attempted and suc- cessfully solved, and the volume under review contains the results of his labours which have extended over several years, together with much other interesting matter arising directly out of, or suggested by, this research. The problem i is not such an easy one as it might appear. The complication arises from the fact that the observations have been adjusted to fit the Everest Spheroid, and consequently will not fit any other spheroid without readjustment. As a result of this, the corrections obtained for the coordinates of any point depend upon the route by which that point is reached from the origin. Various routes are discussed in the memoir, and after a detailed discussion it is concluded that, in view of the methods by which the observations of the triangulation in Jndia have been reduced, the method of calculation along geodesics through the origin is the correct one to use. The discussion is of considerable theoretical importance and the conclusion arrived at appears to be justified. In the latter part of the volume the general questions of the adjustment of a triangulation and of its “strength” are discussed. A criterion of the strength of a triangulation is determined which ~ is superior to General Ferrero’s in that it takes account of the length of the sides and the general formation of the series. This quantity enables probable errors of length of side and azimuth and also of latitude and longitude of any point of the triangulation to be expressed. Numerical values of the strength of all the Indian series of triangulation are given. The more difficult question of the assignment of probable errors es adjustment is considered in some detail. An interesting by-product of the investibabion is a pretty method for the solution of a large number of normal equations in the particular case when the equations can be divided into groups in each of which certain of the variables have zero co- efficients. A well-ordered method of solution is illustrated in detail by a numerical example. The volume embodies the results of a long and laborious research, and the results obtained are of the greatest practical importance. The author and the Survey of India are to be congratulated upon it. Ser THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. a te ™ & “ARIS 'SRIXTH SERIES.] = BA RCE 1920. * % XXV. On the Excitation of the Spectra of Carbon, Titanium, and Vanadium by Thermelectronic Currents. With special Reference to the Cause of Emission of Light Radiations by Luminous Vapours in the Carbon Tube Resistance Furnace. By G. A. HEMSALECH *. pelates Th—V-] CONTENTS. § 1. Introduction. § 2. Experimental methods. § 3. Temperatures of graphite plate for various potential gradients of heating current. § 4. Luminous phenomena observed in the vicinity of an electrically heated plate of graphite. Bluish vapour and red fringe. § 5. Influence of a transverse magnetic field upon the visibility of the red fringe. § 6. Origin of the red fringe. Thermelectronic current. | § 7. Spectroscopic analysis of the red fringe and luminous vapours. § 8. Cause of the sharp outline shown by the red fringe and of the abrupt cessation of its spectrum emission. § 9. Displacements of red fringe emission by transverse magnetic fields. § 10. Possible cause of excitation of red fringe spectrum. § 11. Discussion of results and their application to the case of the electric - tube resistance furnace. § 12. Probable cause of disagreement between Dr. King’s results and mine. § 13. Summary of results. § 14. Concluding remarks. * Communicated by the Author. Phil. Mag. 8. 6. Vol. 39. No. 231. March 1920. R 242 Mr.G. A. Hemsalech: Huacitation of Spectra of Carbon, $8 1. Introduction. B* directly comparing the spectra of iron as emitted by flames of different temperatures with those given by an electric carbon tube-resistance furnace heated to corresponding temperatures, I arrived at the conclusion that the electric furnace gives out two different kinds of emis- sions caused by two separate and distinct modes of excitation. The spectrum of one of these emissions was found to be identical, at all temperatures up to 2500° C., with that observed in the mantles of various flames, being indeed com- posed of class I. and II. lines only. The spectrum of the second emission, which was observed in the furnace at temperatures of over 2500° C., showed in addition also class III. lines, the group at 14957 being particularly pro- minent. 6 535), ARO) 5, se 1 ly Rash eran weet 1G5) 3 528 , PASTS, down 1S joes Ne eno TRO), 473, 2400, fe ND PPS Cee ak 950k LOR, AO oa 25208 up. PAD ei bin cara soll PANO) 5, 5°64 ,, 2780 _,, — 724] sean a ari ae Z10> 5°64 ,, AIA) i. down, Jai al ENE aU Una 2105 5°64 ,, 2780 ,, up. PSAs pat Se Ee PAO) S00 2 2300). a AH MAES al OPO a i a 2000 |; S07. 4: 3200 ,, — as dca MM aise cia 26058, 7:28 ,, 2480 ,, as PAO AMA em le en 260° 3; SiO x 3300 ,, — DA tei RN ae 125-05 3°64 ,, LOTOme — 743) isc OO ED en TOS. Fp eh es 1690, = AS) se eve IGS ARID) las 1730 ,, — Pee sete k as oss LEAS) 6 4°73 ,, 2310 ,, — 0 We eo SoA Peer 145. 418 ,, 21200 5 — DE SHAN CORR TC EEN 145 ,, 418 ,, 2090 ,, — Danae aahee saewtesees 125 i SY fete 1990 _,, a DAL Pieces sepapcte 125g) 364 ,, 1910 _,, a With the help of these results the accompanying tempera- ture curve has been constructed (fig. 2). This curve is approximately a straight line. Those points which were Titanium, and Vanadium by Thermelectronic Currents. 249 obtained with the magnetic field on are specially marked— 2 completely shaded circle, when the magnetic force was 3400 3200 » 3000 ny) @ 12) ) “N N ae) o oO @) (2) 9 Temperature in degrees centigrade. N N ° ° 2000 1800 Potential gradient along plate in volts per centimeter. Temperature curve of graphite plate. acting upwards, and a half-shaded circle, when the force acted downwards. There seems to be no definite effect due to the presence of the magnetic field, for although Nos. 19 and 22 lie rather much above the line and seem, if anything, to indicate a rise in temperature, when the magnetic force is acting upwards, they are yet too near the limits of possible errors to allow of any positive conclusion being derived. But the result, as it stands, is of the utmost importance for the interpretation of certain phenomena to be discussed in § 9. A few additional temperature determinations were made with plates which were not protected by a layer of car- borundum powder, so that their upper surfaces were left exposed and free to radiate heat. Under these conditions the temperature of the undersurface, with a potential gradient along the plate of 5:9 gay cm. J450°C. was found to be only namely, about 250° ©. less than when the plate 250 Mr. G. A. Hemsalech: Excitation of Spectra of Carbon, was protected. The upper surface, which naturally loses : ©. : It heat more rapidly, gave, with a potential drop of 5°63 —— Cite > only about 2250° C. These figures emphasize the great importance of the protecting layer of carborundum, and it deserves to be mentioned that some of the most important results achieved in the course of the present investigation were directly due to tlis precaution. § 4. Luminous phenomena observed in the vicinity of the electrically heated plate of graphite. Observations were made by forming an image (about twice actual size) of the heated plate upon a white cardboard screen.. This plan proved not only very convenient, but it also had the further advantage over direct observation through dark glasses, of showing the phenomena in their true natural colours,—a most helpful adjunct for interpreting the meaning of the luminous effects displayed. As the temperature of the plate is gradually raised, yeliowish vapours begin to form along the under sumbice and a continuous stream of similar rapours is seen to rise upwards from above the plate. At temperatures of from 2300° to 2500° C., the coloration of the vapours beneath the plate changes to bluish gréy and, furthermore, they present now 4a sharply- -defined outline as sketched in fio. 3.) In the region above the plate, where the temperature is “much lower, the colour of the rising vapours continues yellow, with here and there a greyish streamer or patch. In the neighbour- hood of the plate, principally just above it, are also seen red-coloured regions. All these colour effects are due, no doubt, to the various light radiations emitted under the pre- vailing temperature conditions, by the vapours driven out from the graphite plate and the carborundum. In order to account for the sharp demarcation of the incandescent vapours beneath the plate two factors must be taken into consideration—namely, the continuous effusion, from the undersurface, of vapour which is being forced downwards into the protected Space and the upward draught of air, as explained in §2. At the boundary sur- face of the protected space the hot vapour comes into contact. with the air current, is cooled with consequent changes in its radiating properties, and is then immediately carried away upwar rds. Since it is reasonable to assume that, whilst the temperature of the plate remains constant, both the quantity of vapour passing through the protected space and Titanium, and Vanadium by Thermelectrome Currents. 251 the velocity of the uprushing air will not vary appreciably, we may expect that some stable régime will become estab- lished:in which there are no rapid variations in the position Rising vapours “ Carborundum \\ AWAY powder \\ \ Bluish vapours Plate temperature : 2300-27009 C. of the boundary surface between the vapours and the sur- rounding air currents. The sharply defined outline of the bluish vapour might accordingly be caused by a steady state of the actine forces and by the continual clearing away of the superfluous vapours through the upward rush of air. Although the existence of appreciable ionization currents through the luminous vapours, under these temperature con- ditions, was easily shown by means of a pair of exploring electrodes, the spectroscopic results did not, however, indi- cate that the light radiations emitted by these vapours were entirely governed by them. A most remarkable effect, both luminous and spectroscopic, was however observed when the graphite plate was raised to a temperature of about 3000° C. In immediate contact with the undersurface of the hot plate and suspended from it as it were, there ap- peared a sharply defined luminous band of pinkish hue, stretching right across the space between the clamping bars 252 Mr.G. A. Hemsalech: Fwvcitation of Spectra of Carbon, and extending downwards to a distance of from 1 to 2 mm. (fig. 4). Tene convenience sake, this luminous band will henceforth always be referred to simply as the red fringe, although there is a distinct violet shade in its colour. Relative position oF spectrograph s/t. Plate Temperature: 3000° C. A preliminary examination of the spectrum of the red fringe, which was found to be entirely different both as to character and composition from that given by the luminous vapours situated beneath the fringe, at once suggested thermo-electrical excitation as the cause of its emission. It was therefore provisionally assumed as a working hypo- thesis that, owing to the very great degree of ionization prevailing at this high temperature, part of the heating current had passed out of the graphite plate into the space below and that its path was revealed by the formation of the red fringe. Titanium, and Vanadium by Thermelectronie Currents. 253 $5. Influence of a transverse magnetic field upon the visibility of the red fringe. If, as has been assumed, the red fringe were caused by the passage of an electric current outside the plate, it should be acted upon by a transverse magnetic field. In order to test this graphite plates were mounted between the hollow field coils of an electromagnet in such a way, that the heating current flowed at right angles to the direction of the lines of magnetic force. ‘The temperature of the plate was raised to about 2700° C., at which stage the red fringe is ordinarily not yet visible. As soon, however, as a magnetic field of from 125 to 175 0.4.8. units was put on with the force acting upon the heating current in a downward sense, the red fringe immediately appeared. On taking the magnetic field off, the red fringe disappeared. On reversing the direction of the magnetic field, no red fringe was observed. One would have expected, in this latter case, to see a red fringe appear above the plate, but undoubtedly, owing to the prevailing convection currents, the vapours do not become sufficiently ionized under these temperature conditions to allow of the passage of an electric current. But when the plate was raised to a temperature of about 3000° C., so that the red fringe was well visible beneath the plate even without the aid of a magnetic field, then the application of an upward acting magnetic force of 1300 c.a.s. units brought out a red fringe above the plate, whereas the fringe beneath the plate practically disappeared. On taking the field off again the red fringe above disappeared, whereas at the same time that beneath the plate reappeared. By applying Fleming’s hand- rule it was easy to show that the displacements of the red fringe were quite in accordance with the laws of electro- magnetic induction, which is a direct proof that the red fringe is governed by the flow of an electric current. Furthermore, the appearance of the red fringe cannot be governed by temperature alone, for otherwise it would mean that when a magnetic field of from 125 to 175 units acts in the downward sense upon the plate at 2700° C. its tempera- ture should increase by about 300° C., which, as the results of my temperature determination ($3) clearly show, is not the case. But, in addition to these facts and arguments in favour of the electrical nature of the red fringe, a most important factor, which also gives strong support to this view, is the following observation made when the red fringe was acted upon by a transverse magnetic field in a downward sense :— luminous streamers were seen to pass out of the fringe into 254 Mr.G. A. Hemsalech: Hxertation of Spectra of Carbon, the space below, forming bright and most sharply defined spiral or other paths, such as might be expected to result from the action of a magnetic field upon a stream of luminous particles carrying electric charges. $6. Origin of the red fringe. It may be useful, at this stage, to briefly inquire into the nature of the vehicles which convey the electric current in the red fringe. It is weli known from the work of many physicists, in particular from Professor Richardson’s extensive researches, that at temperatures much below that at which the red fringe -is formed, electric currents are passing through the ionized vapours or gases in the neighbourhood of an electrically heated carbon “rod or metal wire. It is generally assumed that these currents are caused by the dis- placement of ions under the influence of the acting electric field, and Professor Richardson has therefore proposed to call them thermionic currents. There is no doubt that the electric currents, which I was able to register between two exploring electrodes held in the luminous vapour beneath the heated plate, at the lower temperatures (see § 4) were of this nature. Now, these currents do not seem to have any appreciable influence upon the character of the spectra emitted by the vapours, which, as will be shown later, are practically identical with those given by these same vapours in the outer mantles of flames. But also in other respects the thermionic currents differ greatly from the current which causes the red fringe—namely in a_ transverse magnetic field they are much less acted upon than the latter. Whereas the red fringe is extremely sensitive to the magnetic force and a field of a very few units suffices to produce a distinct spiral or helical path, a field of at least 500 units is necessary to form an ill-defined and relatively undeveloped curved path in the luminous vapour with ther- mionic currents. It seems to me that this difference in behaviour between the thermionic current and that causing the red fringe might possibly be accounted for by assuming that the particles which convey the electric current are of different masses in the two cases. ‘Thus, whereas the ther- mionic current is probably due to the motion of relatively heavy particles, in this case perhaps carbide molecules, the current in the red fringe would arise from the displacement of relatively light particles. These are perhaps constituted of free radiating atom-ions formed by the electrons which emerge in large quantities from the hot graphite plate, and the paths of which are under the control of the magnetic Titanium, and Vanadium by Thermelectronic Currents. 255 field set up by the heating current, as explained in § 8. The origin of the red fringe emission would thus be, in some way, connected with the process of generation of these atom-ions. As it will be necessary, especially for spectroscopic purposes, to make a clear distinction between the thermionic current and that producing the red fringe, I propose to call the latter the thermelectronic current. § 7. Spectroscopic analysis of red fringe and Juminous vapours. 5 plate and the luminous phenomena below it was accurately focussed on the slit of the spectrograph, already described in a previous communication *. The image was adjusted in such a way that the slit passed through the middle part of A two-fold magnified image of the incandescent graphite a the plate and perpendicularly to it, as indicated by the dotted line on fig. 4. By this means the spectral changes occurring along the distance from the hottest part in the immediate vicinity of the plate, down to the region of com- plete extinction of the luminous vibrations, could be observed at a glance or recorded photographically. The times of exposure for the photographic records varied from 3 seconds at the highest to over half a minute at lower temperatures. The spectrum of the vapours which form beneath the plate is of course due to the elements contained in the graphite and carborundum as impurities, probably in combination with carbon as carbides. At lower temperatures appear the lines of Na, K, Li, Sr, Ca, Mn, Al, and Fe, and the whole spectrum is observed to grow progressively in in- tensity and development as the plate temperature gradually rises. In addition to the line spectrum there is also seen a continuous spectrum which is particularly strong near the lower edge of the bluish vapour and extends to the same distance downward as the latter. Now, the lines emitted by the various impurities expelled from the graphite and carborundum generally pass well below the border of the continuous spectrum ; this indicates that their emission centres travel to a greater distance from the plate than the bulk of the bluish vapour. Hence there appears to be no direct connexion between the line emission observed and the sharply bordered cloud of bluish vapour described in § 4. Tt is possible that the bluish vapour is formed only along the boundary surface of the protected space, constituting as it were a kind of envelope, and that the emission centres, to * Hemsalech, Phil. Mag. vol. xxxiii. p. 7 (1917). 256 Mr.G.A.Hemsalech: Excitation of Spectra of Carbon, which are due the impurity lines observed under these con- ditions, are located within the enclosed space—namely, in the hot region extending down from the central part of the plate. In order to account for the greater extension down- wards of the line emission as compared with the continuons emission due to the bluish vapour, we may suppose that the emission centres, which are probably constituted of charged particles (perhaps carbide molecules, see § 6), travel along lines of force under the action of the electric field established along the graphite plate by the passage of the heating current. As it will be of advantage for the sake of clearness to desig- nate the particular vapour which gives out the line emission, as distinct from the bluish vapour, it will always be referred to as the luminous vapour. With regard to the tron lines it will be shown hereafter that their relative development is in accordance with that, which I have previously observed for the same element in flames and in the tube-furnace. From this we may conclude that, also in the present case, their emission is caused by thermo-chemical excitation—namely, the action of heat on a chemical compound of iron, in this case probably a carbide. A characteristic feature of all the lines emitted by the luminous vapour is that they die out gradually on passing downwards. ‘This is, of course, to be expected, if these radiations are, as Il presume, controlled by the plate tem- perature, for the latter naturally decreases with increase of distance from the plate. In this respect these lines behave very similarly to what they do in flames, in the mantles of which they are likewise observed to die out only gradually on passing into cooler regions. When the plate temperature is raised to about 2700° C. a new spectrum begins to develop in the immediate vicinity of the graphite plate, with the appearance of the carbon * bands at 3883 and 4216 and of numerous lines due to titanium and vanadium. As the temperature is gradually increased this spectrum gains in prominence and more carbon bands appear in the visible part until nearly all the bands of the Swan spectrum are out. Finally, in the red part is seen a group of most intense, hazy, and broad bands, which when the spectrum is fully developed constitute its most brilliant feature. Now the very striking and dis- tinguishing character of this spectrum is, that the lines * Runge and Grotrian have recently concluded that these bands are due to nitrogen. As a result of my own experiments, which were like- wise made at atmospheric pressure and of which I hope to give an account on a future occasion, Iam unable to endorse their view. Nor do my experiments indicate that the presence of N is always essential for their emission. Titanium, and Vanadium by Thermelectronic Currents. 257 and bands which compose it, unlike those emitted by the luminous vapours previously described, pass only a short— distance down from the graphite plate and stop quite abruptly, as though the exciting agent had suddenly ceased to acts Evidently then, this spectrum cannot be entirely controlled by the plate ‘temperature, as otherwise its lines and bands, like those of the luminous vapour emission, would die out only gradually. We must therefore trace its origin to some other cause, and it is only natural to connect the emission of this spectrum with the formation of the red fringe. A direct experimental proof for the reality of this connexion will be given in § 9. Thus, the emission of the carbon bands and of the lines of titanium and vanadium is in some way caused by the thermelectronic current which, at these high temperatures, passes through the strougly ionized vapours in the immediate vicinity of the graphite plate. The emission would therefore be due to what I have previously called thermo-electrical excitation *, and it is no doubt of the same nature as that which I had preconceived to exist in a high temperature tube resistance-furnace. The spectrum of the red fringe and of the luminous vapours are reproduced on Plate If. The narrow strip of sharply defined and strong continuous ground, which forms the upper edge of each section, is due to the Juminous emission by the exposed edge of the incandescent graphite plate. The ultra-violet end a was obtained with the special furnace arrangement in which the vapours beneath the plate are better protected from air eee and sections 6 and e with the clamping bars described in § 2. In this latter case the vapours are more exposed to air draughts and therefore the spectrum lines do not pass down quite so far. In the zone of the red fringe the two emissions are naturally seen in superposition, but by reason of the abrupt extinction of the lines and bands which compose the red fringe spectrum, the separate existence of each emission is clearly “brought out. Some of the low temperature lines are seen in absorption upon the continuous spectrum due to the edge of the graphite plate. This is of course caused by the ‘constant stream of excess vapour which is being carried upwards in front of the plate by the aforesaid air currents. It is also well to draw attention to the behaviour of the H and K lines of calcium. These lines are seen to pass down a much shorter distance from the plate than neighbouring iron lines of equal or even less intensity, such as the group at 3920. They are * Hemsalech, Phil. Mag. xxxvi. p. 295 (1918). Phil. Mag. 8. 6. Vol. 39. No. 231. Warch 1920. S 258 Mr.G.A.Hemsalech: Hweitation of Spectra of Carbon, therefore not emitted at so low a temperature as iron lines. This fact is indeed quite in harmony with M. de Watteville’s and my observations regarding the flame spectra of these elements. In the mantle of the air-coal gas flame the iron lines are well developed, whereas no trace of H and K is: seen. But the latter are quite intense in the oxy-coal gas flame at 2400° C. It would therefore seem that also in the present case the H and K radiations are caused principally by thermo- chemical excitation, and they do not appear to be appreciably affected by the thermelectronic current. The following tables contain all the lines and bands of the spectrum of the red fringe, but only those of the luminous vapour which were observed by me likewise in a carbon tube resistance-furnace. Lines to which an asterisk is affixed are marked on the plates to the nearest Angstrém unit. Titanium. Relative intensities. Relative intensities. a PRET NR TCLs eo aaa IS =) r. Red fringe. Arc. Spark. | » Red fringe. Are. Spark. 3635°47 000 15 3 407847 000 8 4 3642°68 000 15 3 43008 tie 15 3 3653°49 OO 15 4 4305°91 1 20 8 3671:°66 000 4 3 4314°80 000 5 3 3689°89 000 3 2 4393°93 00 5 2 371739 000 5 2, 4512-74 00 15 4 3729°77 1 8 4 451803 00 15 4 *3741°14 1 15 2 4522-80 00 15 4 375287 5) 5 4527:°31 00 15 4 3753'63 00 3 3 *4 500 2 ) 1 20 5 *3771-64 2 4 3 34:78 15 4 3900°53 00 5 50 *4po> 92 Q 8 3 390477 0 10,1516 oot aa ae 3994°52 1] & 3 4544-70 0 10 3 3947-75 1 10) ae eee See 3948-66 PA On AOU) : 104 eS 3956-28 12 15 4 4681-91 0 20 10 3962-86 0 8 3 4999°51 at 20 10 3964-27 is 8 3 5007°22 ay 20 10 *3981°77 1 15 3 *5014 26 1 20 8 3982:54 O 8 3 5039°96 4 10 3 #3989-77 9 20 6 5064°66 1 10 4 *4009°14 1 8 4 5192-97 0 20 8 400968 4 000)1r) 4 2 *5 21089 aa eo *4024'57 3 10 3 Titanium, and Vanadium by Thermelectronic Currents. 259 Wave-lengths are given in international units and the scale of intensities is that outlined in a former commu- nication *. Some lines situated in the bright tails of the carbon bands have probably escaped attention. The relative intensities in are and spark were obtained from the observations of Hxner and Haschek. A comparison of these results shows that the red fringe spectrum of titanium is composed of the brighter are lines, and only one so-called enhanced line has been recorded—namely, > 390053. As to the character and origin of this spectrum very little can at present be said or suggested, because the flame spectra of this element (chemical and thermo-chemical excitation) are as yet practically un- known. It is, however, of interest to mention that none of these lines were observed by me in an ordinary are between . electrodes made of the same kind of graphite ; but they ap- peared near the cathode in heavy current ares of from 20 to 80 amperes. No trace of them was observed in the ordinary condensed spark between graphite poles, but a few of the brighter ones were detected in the self-induction spark. It is therefore remarkable that they should constitute such an important part of the red fringe spectrum. Vanadium. Relative intensities. Relative intensities. (Cia Bir Ta CS : OR Se TN r. Red fringe. Arc. Spark. r. Red fringe. Arc. Spark. 3902-25 0 4 2 4128:10 a 10 10 4092-68 OU 15 3 | 4131°98 4 10 10 4099:80 00 20 2 4134-47 000 10 10 4105-20 00 10 4 *4379 24 1 30R 30R 4109°78 000 1b tO 4384°73 Z 30 30 R 4111-80 2 301k 2 2 4389-98 0 20 20R 411517 00 Rene *4408'50 x 10 15k 411650 000 15 5 4586:37 00 10 8 4123-65 000 5 3 4594-09 00 10 10 This spectrum is likewise composed of the brighter arc lines only. It is of course possible and even probable, that some lines have been overlooked, as in the case of titanium ; this is, however, inevitable on account of the presence of a strong continuous ground and of the carbon bands, which fact, coupled with the relatively low dispersion of my * Hemsalech, Phil. Mag. vol. xxxiii. p. 9 (1917). S 2 260 Mr.G.A. Hemsalech: Hxecitation of Spectra of Carbon, spectrograph, would naturally tend te mask or obliterate some of the lines. | Also in this case it is impossible to form an adequate opinion as to the true character of the spectrum emitted by vanadium, on account of the absolute lack of information with regard to its flame spectra. Carbon. Red edge of band. = ToT TEN DN Xr. Relative Intensity. *3861-71 4 q > *3871:39 6 P | i pee Wy J | So-called *4167-61 1 i \ Cyanogen *4180°82 1 | bands. *4197-08 2 f | *4215:96 3 J J 4684-76 00 in ; 4697-29 a | 4715:13 2 r Group IV. | *4737-00 3 ae | : } wan 5129-24 1 tl Group IIL. Spectrum. *5165°12 4 ) cae 5585:28 v \ Group IT. | *5635°21 1 J The spectrum is composed of the so-called cyanogen bands and three groups of the Swan bands. I have no doubt that group I. of the last-named bands—namely, that with head at % 6188, would likewise show were it not for the presence, in this part of the spectrum, of one of the hazy broad red bands. (Onda, +b da, + bisdxz + bude), "2 a) 6=ll subject to the condition that the determinant of the system bi day tbiedx, + bisda, t+ biuday=;(u) (@=1, 2, 3,4), - (4) in Which dz, da, dz, d&, are the unknowns, does, not vanish. If (1) is such that this reduction is impossible ; viz., if the a; are such that no 6,;; exist such that the deter- minant of (4) is different from zero, a slight and obvious modification of the algebra leads to a parametric solution of essentially the same sort as that now given. 3. The determinant of (4) not vanishing, we can on for the dz;, getting dx; = Babi) + Biodo(w) + Bishs(v) + Buda(w) (W=1, 2, 3,4), , () in which B, are functions of the b;, hence of the aj, and therefore of the w; alone. Hence, integrating, v; =\(Ba i (uw) + Bodo (u) + Bi ds(u) + Birdy (u) du; . (6) or when the B,; are independent of wu, 4 => By) 6; (ujdu -G='1 12, 3,4) ea) j=l We shall now determine s and the ¢;(w) in terms of the functions (2) so that on substituting in (6), s and the resulting «; reduce (1) to an identity. There are many ways of doing this; one of the most obvious is Ss = Cae + E,° SF E,? + &,") du, cor (1) — Byes ee) ot Gin Gilt) =28F,, da(u)=2Fbs, $4(u) = 2Esks. 7 Fundamental Equation in general Theory 07 Relativity. 287 For, on putting these values of the $;(w) in (6), and then going back successively through (6), (5), (3), we get for the right-hand member of (3), (4 Sy —€.° +.) + (26,83)? + (Eck)? + (2E1Es)?, which is identically (Get oo es° + Ex), that is, ds?. Hence (3), and therefore (1) which is equiva- lent to (3), is identically satisfied by the indicated values of S, Uy, Ug, V3, V4. 4. The same device of making the solution depend ulti- mately upon an identity connecting sums of squares, can obviously be applied to find parametric solutions of other equations occurring in general dynamics. It will be suffi- cient to indicate the identity applicable to ds?’ = Ya;,dx;da; i= ie 2, ssi-5 n). Cae (8) Suppose that the quadratic form on the right of (8) is algebraically reduced, in the ordinary way, to a sum of nm squares, ds? => (bada,+ bedayt ... +bmdan)?, . .« (9) (=I such that the determinant |;,) does not vanish. As before, special cases in which this reduction is impossible may arise ; but they present no essential difficulty. Then, for n>1, we resolve n—1 in any way into a pair of factors 7, s so that n=rs+1; and put ? s A,= > coe B, = 2 ne where &, n; are arbitrary functions of a parameter uw. Then the identity leading to a solution of (9), and hence of (8), is (Axes b,) + 4A.b,. ." . , €10) For, on multiplying out Js doe the right of (10) is a sum of rs+1l=n squares, Viz., (AL ee) + > > en)? s- - (11) == Gal \ > 288 Fundamental Equation in general Theory of Relativity. and hence on putting s = \cae +B,)du, bia t birtg t+... +bintn=\i(u)du (i=1, 2,...,n), (12) where the functions ¢;(w) are identical in some order with the 2&;7;, (A,—B,) in (11), we get, on solving (12) for the x;, a solution of (9) and hence of (8). 5. The cases of (8) in which the quadratic differential form on the right is reducible to a sum of 4«+3 squares, where k=0, 1, 2,3,..., are in many respects remarkable. In those cases it is always possible, in several essentially distinct ways, to obtain types of general parametric solu- tions in a form free from all quadratures. The same property holds also when the right of (8) is reducible to a sum of two squares. When the number of squares in the reduced form is three, the most immediate interpretations of (8) being to the theory of curves in three dimensional space, whether Euclidean or not, the solution has a particular interest. The cases of reductions to 2, 3, or 15 squares present in addition many properties not shared by other forms. The dynamical interpretation of the general case is evident in terms of the generalized velocities, momenta, and kinetic energy of a system, the last either in the Lagrangian or Hamiltonian form. Hence it may be expected that if the generalized coordinates of a system are 4<+3 in number, the system will have special dynamical properties. In addition to these quadratic differential forms, there are many other classes of forms of degrees higher than the second possessing a like property that the solutions of equations between several such forms of the same kind may be very readily obtained in parametric form, not, however, free from quadratures. An account of all the cases men- tioned in this note will shortly be published elsewhere. It may be of interest to remark that all of the general solutions free from quadratures, and those relating to forms of degree higher than the second, first presented themselves in some work relating to the theory of numbers. University of Washington, Seattle, Washington, U.S.A. XXVIII. The Mandelstam Method of Absolute Measurement of Frequency of Electrical Oscillations. By J. TYKOCINSKI- TyKkoctrner, late Manager of the Radio Department of the Siemens and Halske Works at Petrograd*. URING the summer of 1915 a considerable number of wave-meters for radio-stations had to be calibrated in the Radin Department of the Russian Siemens and Halske Works in Petrograd, taken under the control of the Russian Government. ‘The comparison of wave-meters calibrated by usual methods by different firms or by the Russian Chamber of Measures and Weights showed large discrepancies. To enable a larger number of stations to work with each other without interference, not only has sharp tuning to be applied but the precise setting of the radio apparatus for a _given wave-length is of paramount importance. A reliable method of wave-measurement reduced to the use of the simplest standard becomes of great necessity. Dr. Mandelstam, chief expert of the Works’ Research Department, investigating the behaviour of high-toned buzzers used at that time for generating high-frequency oscillations, for measurements and testing purposes, found that oscillations in a circuit energised by a buzzer do not depend solely on its capacity and inductance, but depend also upon the frequency of the pulsating current delivered by the buzzer, and to a large extent upon the character of its interruptions. Mathematical analysis of the phenomena showed that a buzzer, because of the steepness of the curve A (fig. 1), ies. ae / characterising the interrupting current, can be made, in con- nexion with another circuit, a source of trains of oscillations possessing a wide scale of frequencies. ‘The amplitudes of the variety of oscillations obtained are net. equal for all * Communicated by Dr. L. Silberstein.” Phil. Mag. 8. 6. Vol. 39. No. 231. March 1920. U 290 Mr. J. Tykocinski-Tykociner on the frequencies, but depend upon the ratio of the frequency of generated oscillations to the number of interruptions per second the buzzer is operating. Those oscillations, the frequencies of which represent exact multiples of the number of buzzer interruptions, have the largest amplitudes. Basing himself on this result, Dr. Mandelstam devised (July 1915) and developed the following method of absolute measurements of frequencies used in radio work. A buzzer B (fig. 2) giving regular interruptions and working from a_ battery of accumulators HE excites an Ce aperiodic circuit I consisting of a resistance R and in- ductance L. This circuit is a source of oscillations of all possible frequencies in accordance with Fourier’s analysis of the curve A into sinusoidal components. Another circuit II, capable of performing free oscil- lations, with its variable capacity C, and inductance Ly, is inductively connected with the generating circuit I and with a circuit III, containing an indicating instrument D, as for instance a thermo-element with galvanometer or a detector with a telephone. By variation of the capacity of the condenser C, a great number of maxima of the oscillating currents in II can be observed, arranged in definite positions all along the scale of the condenser C,. Changing the number of interruptions per second of the buzzer produces the effect that the maxima come closer to each other if the number of interruptions decreases, or become widely separated if the number of interruptions increases, ‘The use of a detector with a telephone in the indicating cireuit IIT coupled with If gives a means of hearing a pronounced musical tone, corresponding to the frequency of inter- ruption of the buzzer only in positions of the condenser C, which form circuits of multiple natural periods to that of Measurement of Frequency of Electrical Oscillations. 291 the period of interruption of the buzzer. Every position of the maxima on the scale C, defines thus the frequency of a certain harmonic of an oscillation, whose fundamental is given by the number of the buzzer interruptions per second. Let x be the number of the interruptions per second of the buzzer, v, the natural frequency to be measured of the circuit II in a certain position of the condenser ©,', vz a higher frequency of the circuit II in another position of the condenser C,", « a number indicating which harmonic y, is in relation to the fundamental n, and s the number of maximas observed while moving the condenser C, from the position C,’ to C_’:— Cy SS TL Re eee wee 5a. a an eer Ly) W(t) op 1. shh ee teins ao co oes (29 Sot 2 “(1 Vo K+8’ V; From (1), nya Os 7) key (F ) ie a TRG Wea his MCS) By counting the number of maxima heard in the telephone while moving the condenser Cy, from (C,’ to Cl’ s is found; nm can be measured by any known acoustical or optical method; it remains to find the ratio = The most practical way is to choose the second 1 position of the condenser (©,”, not arbitrarily, but te determine it so that the natural frequency of circuit IT for that position shall be a known multiple of that to be measured (C,’). This position can be easily found by excitation of the circuit Il from a separate source of oscillations energised by an electric are or thermo-ionic valve and containing usually harmonics of its fundamental frequency. The absolute measurements were accomplished in the U2 292 Mr. J. Tykocinski-Tykociner on the following manner (fig. 3). [Mr. John W. Perry was kind enough to draw the figures after the author’s rough sketches.—L. 8. ] Fie. 3 C4 . —+ Ly lv Vv az e Aue U c> Ric Lis { L, Ww The circuit II (wave-meter) to be calibrated is placed between the aperiodic buzzer circuit I, the aperiodic indi- - cating circuit III, and the harmonic circuit IV with its inductance L, and variable capacity C,, energised by an are V or valve v. In the buzzer supply circuit the coil W of a telephone is inserted, the diaphragm M of which closes a tube F containing a piston P and rod Kk sliding over a millimetre scale S. After the condenser C, of the circuit II is set in a position C,', for which the natural frequency is to be determined, the circuit IV is brought into resonance with it by variation of the condenser Cy. Then the condenser C, is gradually turned to a position of smaller capacity oe until the indicator D (hot-wire instrument, heterodyne receiver, or ticker) again shows resonance, corresponding to the first harmonic. Thus a frequency of the circuit II twice greater than that in the position C,' is established. The next step is to close the buzzer circuit and to count the number of maxima while turning gradually the con- denser Cy from the position C,'’ back to C,’. Finally, the piston P is moved, the length of the tube F between the diaphragm M and piston P determined for a certain | number of maximum sound intensities heard in the ear- piece H, and the wave-length in air of the sound calculated- Measurement of Frequency of Electrical Oscillations. 293 Thus the measurement of the natural frequency of the wave-meter II for a certain position C,’ is reduced to the measurement of a length contained between two maxima of sound intensities on the scale S. The number of interruptions of the buzzer is determined from the known e U e e e e e relation n=s, if v is the velocity of sound in air corrected for the diameter of the tube used and XA the wave-length of the sound in the tube. Choosing the position C,'’ of the condenser to correspond with the first harmonic of the oscillation in position C,’, the relation (3) is reduced to V; = Sr. : SOAS 72) se nitaes ane Me (4) For the excitation of the circuit I a buzzer was used, the vibrating part of which consisted of a pair of steel wires g bridged by a soft iron plate ¢ forming the armature of the electromagnet vu. By means of a screw the tension of the steel wires could be regulated and the number of interruptions changed from 500 to 1500 per second. To check the number of interruptions of the buzzer an interference method was used additionally sometimes. A telephone was connected to the secondary winding of a transformer, supplied with two primary windings—one inserted in the buzzer supply circuit, the other connected to a small alternator, generally used’ for telephone mea- surements, the speed of which could be exactly measured. By counting the number of beats per second heard in the telephone, the frequency of the buzzer circuit could be calculated. The above method was used for calibrating and checking wave-meter standards. Wave-leneths from 7=3000 m. to = 20,000, corresponding to frequencies from 100,000 down to 15,000 per sec., were measured directly by counting the number of maxima s, while shorter waves were then deter- mined by using harmonics of resonating circuits. An accuracy of 0°5 per cent., sufficient for the purpose, was easily obtained. The buzzer could be regulated with such exactness that in continuous work during two hours and more its frequency remained practically unchanged. To spare time necessary to find C,” by tuning the circuit I] to the first harmonic and egrinmiine a ‘large number of maxima between two positions of the con- denser C,, the second corresponding to a frequency double of that of the first position, the following simplification was adopted in particular cases. 294 Mr. L. C. Jackson on In the relation (3) = can be replaced by 1 Kors Qar/ Cy! Ly =f i fe 1/19 Ts KG 7) ¥y, 0 2m/ Cy" Le Db ey Jgn For the first harmonic = =4, By the use of a calibrated 2 ! variable condenser C, any suitable smaller relation e 2 can be chosen. Thus the number of maxima to be counted is reduced and the operation with the circuit IV spared. The application of this method to closed circuits con- taining known capacities, and inductances gave results in good agreement with those calculated by Thomson’s formule. The above method was also used for the exact deter- mination of wave-lengths emitted by vertical antennae and other radiating circuits, with distributed capacity and inductance, used in radio-telegraphy. On board the ‘ Lorraine,’ Aug. 30, 1919. XXVIII. Variably Coupled Vibrations: Gravity-Elastic Com- binations. Masses and Periods equal. By L.C. Jackson, FPS L., University College, Nottingham * [Plates V1. & VII.] I. InrRopvucTrIoN. NHE present paper is a continuation of the work of Prof. Barton and Miss Browning (Phil. Mag. vol.xxxiv. p. 246, vol. xxxv. p. 62, vol. xxxvi. p. 36) on the subject of coupled vibrations. The coupled system treated in the following pages consists of an elastic lath pendulum and a gravity pendulum, which can beattached to the lath at different points along its length. The degree of coupling of the pendulums thus depends * Communicated by Prof. I. H. Barton, I’.R.S. Variably Coupled Vibrations. 295 on the position of the point of suspension of the gravity pendulum™. The paper includes 32 photographic traces of the motion of the gravity pendulum under various conditions of coupling and starting. IT. THrory. Equations of Motion and Coupling. For the gravity and elastic pendulums, let the masses of the bobs be M and N, let the length of the simple pendulum equivalent to PM be 7, and at time ¢ let the linear displace- ments of M and N be y and z respectively ; further, let the Fig. 1. linear displacement of P be ez. It should be noted that the displacement of the bob N of the elastic pendulum has always the positive sense as shown in fig. 1. Then for small oscillations, and considering PN as straight, the equations of motion may be written 2. PAA Mo 4 Mg (==) = 0, dt? az Noa * The bob of this gravity pendulum carried an electric lamp and a lens, and left on a plate below it a photographic record of its motion. r +Nn?z = Mga( ==) ‘ 296 Mr. L. C. Jackson on These may be re-written : d*y Maat Mm’y = Monza, aiimig SaeKeR) daz Niet (Nn? + Mm?a?)z = Mm’ay, emerge’! 5 (2) where m and n are derived from the free isolated vibrations of M and N respectively, viz. : y= a sin, Sandan = bemene and ee Following the analogy of electrical practice, we may write the coefficient of coupling y as given by Min?? 2— ; % 1 Na Min” = 2) Solution and Frequencies. To solve (1) and (2), try in (1) y= ore.) This gives Ts a ‘ ek (5) NVA Then (4) and (5) in (2) give the auxiliary equation in wz: Nw’ + 2?(Nm?+ Nn? + Mim’?e?) + Nn?m?=0.. . (6) This may be re-written in the form Be 07( p? + 9?) +79? = 0. ee ip) Hence c= pi or ot. . | re From this point we shall treat the case in which M=N, mM=N, Thus the equation (3) for the coupling now reduces to pee es a el + a Hence, on inserting the usual constants, we may write for the general solution and its first derivatives, y — Wisin (pit+c)+ FP sin(gt+o), <2... 7) eee) EE ; i E sin (pt te) +— Gare , F sin (gt+¢$), (0) ae ma Variably Coupled Vibrations. 297 es ee omer aay) (Od) dz ae — mm? at (12) It will be seen that these are the equations of the Cord and Lath Pendulums*, but the values of the various quantities are different, as will be seen below; but this difference might he expected, since the elastic pendulum oscillates entirely on one side of the vertical. From the comparison of (6) and (7), and yet MN and m=n, we see that LGR = Wie aoe ; a SW Nea ees te! Fn (1S) ge =n _ Eliminate g from equations (13), thus obtaining the quadratic in p’, ema tmea:)\ pms 4 >.) (14) Thus, calling the larger root p? and the smaller g?, we have » mt (2+a?)+[ (2+)? 4]? u (15) 'S (16) 2 , _ m{(2-+4+a2)—[(2 + a2)—4]#} G= ; 7 whence ge On A a Q {(2+07)—[2+a%)—4yF Initial Conditions. G.) Lower bob struck. We may here write ee, he dy _ EE OES ne VAN ap). se tas Deter © Os oh (Alte) These inserted in (9) to (12) give equations satisfied by e=0 =(0, E= Meg NY yr Gel ati) 19 my (p?—9?)p’ (P= gq 2? ~* Phil. Mag. vol. xxxiv. no. 202, p. 260. 298 Mr. L. C. Jackson on and these values put in oh and eo give the special solution _ (mi—e)u (p?—m*) sin pé+- ~~ sin gf wp) ' @-fa 4 | Lee (p? —m*)(n? —g")u (m?—q?)(p?—m?)u . ee La ge Vp sin DE ints pig sin qt ; so BY OM SG a ay Cale rae Eee and rime (21) where G and H are the quick and slow z vibrations. (i1.) Upper bob struck. Here we may write ayy dz —_ = . (= 22 a = ()) ai o for ¢ =—W0Nie seman ) These conditions put in (9) to (12) give equations satisfied by 7 V0 ce—— a —- m7av ne miav (23) De Say 1: 4aan So inserting these values in (9) and (10) we have the special solution e= 05 6 = 0,7 hi— — mav mav : 5, Sin pt +78 at Be Op faq ae pan )O (m?—q’)v . ~— gin pt-+-—.—+,— singt; . . (25 == Sop (p-g)q 7 ” $0 Wid (p> —m*) q 9 p= and H Gesu (26) Note the contrast between (21) and (26). (iii.) Lower bob displaced ; upper free. Let the displacement a of the lower bob M be produced by a horizontal force. Then the corresponding displace- ment < of N when at rest can be found statically. We thus obtain ara dy dz y= a4, Cima tice —_ 3 = 9 for t= 0. ° (27) Variably Coupled Vibrations. 299 These conditions inserted in (9) to (12) give equations satisfied by i aa CSO ia. ca (1+.2?)(p?—gq’) a t (28) aa Wh (2a? +1)(m?e) £ | i? ae (1+a)( p?—q’*) : These values put in (9) and (10) give the special solution : _ Pa + (pag)a= (Ra? + adn? 1 Paar) (2a? + 1)(m*a) (p= @ (is: 2”) te taka (p?— m?) {p? — 9G? + (p?— g?) a? — (203 + a)m?} “2 (1+ 2)(p?— @°)(m? a) oe (m? — q?) (22? +1) (1+ 2')(p'—q’) So the ratio of the amplitudes of the quick and slow vibrations of the y and z motions are given by acos pt @iCOS Jes. 8. (ae) WREOS Gia ty no Naat ey des: ICBO) eo Hp ae ee em? B (2a? + 1) (ma) te meng ore) (OL) G _ =(p—m'){p— 9+ (pm a— (Carta) ogg EE 7 (2a? +1)(m?a) aii ) (iv.) Upper bob displaced ; lower bob free. This case may be represented by dy _ iz Ne a ee — = QO for t=0. a) (33) w= a0, 22d; oF These put in (9) to (12) give equations satisfied by Se 4) 300 Mr. L. C. Jackson on These values in (9) and (10) give the special solution yi G2 cos pl +e 7 COS gt, p 35 fa PHP og ong OPO PD oo fe (pi—qi)m* C27 So EK — —@q G (p?—m’)¢? K pe and H = Gaerare et aes (36) III. RELATIONS AMONG THE VARIABLES. Vig. 2 is a graph showing the relation between y and «, the couplings being ordinates and the values of « abscisse. The data for the graph are given in the following table :— Coupling. Values of a. Frequency ratio p: ¢. per cent. 0 0 1-000 5 0:05006 1-051 10 01005 1-106 15 01517 1121 20 0°2041 1:226 25 0°2582 1:294 30 0°3145 1:362 35 0:3737 1479 40 0°4364 1-542 45 0°5039 - 1:656 50 0'5780 1-767 D9 0-6586 1-910 60 0°7499 2:083 65 08553 2°296 70 09802 2:572 Tt will be observed that, while the general solutions for the present system are the same as those for the Cord and Lath Pendulums, the equations for the couplings are not the same, there being no term in a in the equation for the gr avity- -elastic arrangement. Fig. 3 is a graph showing le relation between y and ©, q Variably Coupled Vibrations. 301 Fig. 2. i aes _ EaBawa ae Savane 0 Ci) O2eee See O:5 .0:'6) O27) O76 0:9 i -TV. EXpErRIMENTAL ARRANGEMENT AND RESULTS. The actual experimental arrangement used is illustrated in fig. 4. It consists of a lath L clamped at A and carrying a bob N. The gravity pendulum M is attached to L at P, P being movable. The bob M is of special construction. It enables a spot of light to be focussed on a naked photo- graphic plate B, which is moved by hand in a frame between 302 Mr. L. ©. Jackson on guides perpendicularly to the motion of M, the room in which experiment is made being dark. The box part of M contains Fig. 3. -Q|"o- 0 is 30 45 60 75 per cent an electric battery which lights a small lamp inside C. The light then passes through a pinhole and is focussed in B by alensin D. By this means a trace is obtained directly for the motion of M. If the apparatus is to be used for demon- stration purposes for a class of students, the bob M is replaced by a funnel and sand, as in arrangements used in papers previously mentioned. Variably Coupled Vibrations. 303 Length of lath pendulum == 400 -ems. Length of gravity pendulum = 65°) cms. Fig. 4. N Figs. 1-32 (Pls. VI. & VII.) are photographic repro- ductions of traces obtained for the motion of the lower bob under various conditions. Figs. 1-16 (Pl. VI.) are traces obtained when the upper bob was struck, the coupling ranging from 5 per cent. to 65 per cent. The first figures, 7. e. those for the smaller couplings, show very well the phenomenon of beats and the slow surging to and fro of the energy of the bobs. The series shows in a marked manner the effect of a progressive tightening of the coupling until, in fig. 14, the case of the note and its octave (frequencies p: g=2:1) has been reached very nearly. It will be seen that a coupling intermediate between that of fig. 14 (58 per cent.) and that of fig. 15 (62 per cent.) would give the «atic p:g=—2:1. By com- parison with the table in Section III., it will be seen that 304 Variably Coupled Vibrations. the experimental result is in good agreement with the theory. Fig. 16 shows the effect of frequencies nearly 5:2; theory indicates that for 65 per cent. coupling p:q=2°4: i Figs. 17-24 (Pl. VII.) are traces obtained when the upper bob was displaced, the lower bob hanging free. A com- parison with Pl. VI. brings out the dependence of the details of the traces on the initial conditions. Thus fig. 20 (58 per cent.) is a trace for the ratio p:q nearly equal to 2:1; but the characteristic ‘‘ kink” of the 2: 1 curve is hardly visible, low down in the troughs of the curve. Fig. 21 (58 per cent.), on the other hand, shows the “kink” very well ; but the trace is slightly distorted because the bob possessed, at the time, a small transverse motion as well as the correct longitudinal motion. Fig. 22 (approx. 59 per cent.) shows the effect of a combination of frequencies rather greater than 2:1, producing a wandering of the “ kink,” with a definite period up and down the main curve. The figure shows rather more than a complete cycle of this wandering. Figs, 25-32 (Pl. VII.) are traces obtained when the lower bob was displaced, the upper bob being free. This was effected by a horizontal thread, which was burnt when all was steady. The coupled system here described thus presents a fairly close mechanical analogy to the case of coupled electric circuits, as will be seen by the foregoing theory and experi- ment. On account of the simplicity of the arrangement it is a convenient model by the aid of which the somewhat abstruse subject of coupled eiectrical vibrations can be demonstrated to a class where visible results are needed to satisfy the non-mathematical student. SUMMARY. 1. In the present paper the mathematical theory of a coupled system consisting of a gravity and an elastic pendulum is developed and confirmed experimentally. 2. The paper is illustrated by 32 photographic repro- ductions of the traces obtained for the motion of the lower bob under various conditions of starting and coupling. 3. The system here discussed gives very similar results to those previously obtained with the Cord and Lath Pendulums by Barton and Browning, and can be used as an analogy to the electrical case of circuits sof equal inductances and frequencies. In this mechanical case, as in the electrical one, the motions of the components of the systems are not interchangeable. 4. It is hoped later to deal with the more general case of the same arrangement 1 in which both masses and eros are unequal. Physical Department, University College, Nottingham, June 1919. XXIX. On the Magnetic Susceptibilities of Hydrogen and, some other Gases. By TAK& SoNE™. INDEX TO SECTIONS. 1. INTRODUCTION. 2, METHOD OF MEASUREMENT. 3. APPARATUS FOR MEASUREMENT. (a) Magnetic balance. (6) Compressor and measuring tube. 4, PROCEDURE FOR MEASUREMENTS. (a) Adjustment of the measuring tube. (6) Determination of the mass. (c) Method of filling the measuring tube with gas. (d) Electromagnet. (e) Method of experiments. 6G. OXYGEN. 7. CARBON DIOXIDE. 8. NITROGEN. 9. TlypDROGEN. (a) Preparation of pure hydrogen gas. (6) Filling the measuring tube with the gas. (c) Results of magnetic measurement, (dj Purity of the hydrogen gas. 10. CoNCLUDING REMARKS. §1. INTRODUCTION. “EN the electron theory of magnetism, it is assumed that the magnetism is due to electrons revolving about the positive nucleus in the atom; and hence the electronic structure of the atom has a very important bearing on its magnetic properties. The models of the atoms or molecules hitherto proposed are so constructed as to explain only the phenomena of light; but the question whether the nature of atomic or molecular magnetism, due to the system of the revolving electrons, agrees with the results of observation or not, 1s in most cases not touched at all. For example, Bohr’s model f of hydrogen molecules explains very satis- factorily the light dispersion of hydrogen ; but its magnetic polarity is paramagnetic in contradiction to the observed fact that hydrogen gas is diamagnetic. Xp & Xd° Since y, depends on the configuration of the atoms in a * P. Langevin, Ann. de chim. et de phys. viii. p. 70 (1965). if ISG Honda, Ann. d. Phys. xxxii. p. 1027 (1910). it KK. Honda, loc. cit. § K. tlonda, Sci. Rep. iii. p. 171 (1914). Oo —— of Hydrogen and some other Gases. 307 molecule, it may change with temperature, cr by the change of states, etc. ; and hence the observed susceptibility ~ may change in a similar way, as actually observed. The con- tinuous change of the susceptibility-atomic weight curve from the paramagnetic elements to diamagnetic above referred to is also explained on the same basis. According to the above theory, the molecules of a para- magnetic substance must therefore possess a definite magnetic moment, while those of a diamagnetic substance have only a small magnetic moment or none. In Bohr’s model of a hydrogen molecule, x, 1s decidedly greater than y,*, and therefore x or x¥,+X, is positive in contradiction to the observed fact. A In a recent paper, Professors Honda and Okubo f proposed a new theory of magnetization of the gases. According to the kinetic theory of gases, besides translational motions, the molecules of a gas are continuously making rotational motions about their centres of mass; and in their theory, these molecules are treated as gyroscopes. Since the axis of rotation of the molecules does not in general coincide with the magnetic axis, the magnetic moment of the molecules is supposed to be resolved in the direction of the axis of rotation and that perpendicular to it. Under the action of a magnetic field, the paramagnetic polarization results from the former component and the diamagnetic polarization from the latter, and therefore a resultant polarization is the sum of thesetwo. The resultant may be positive or negative, according as paramagnetic polarization ¢ diamagnetic polarization. The theory proves that the sign of the magnetization of a gas depends on the shape of the molecules, and not in the least on their magnetic moment. In fact, a gas whose molecules have a definite magnetic moment comparable with those of iron may be diamagnetic, provided the axis of rotation is perpendicular to the magnetic axis. This kind of diamagnetism is not dealt with in any of the previous theories. In order to test these theories it is necessary to have the correct values of susceptibility of different guses, which are at present scarcely known. ~ One of the chief difficulties which we encounter in the determination of the magnetic susceptibility of gases, lies in the preparation of pure gases, 2. e. those which are perfectly * J. Kunz, Phys. Rev. xil. p. 59 (1918). + K. Honda and J. Okubo, Sci. Rep. vii. p. 141 (1918); Phys. Rev. viii. p. 6 (1919). 2 308 Mr. Také Soné on the Magnetic Susceptibilities free from air, and the other in the extreme smallness of the volume susceptibility of gases. In the present research | paid special attention to the preparation of pure gases, the removal of the air contained in the generators and purifiers being the constant object of my endeavour. As for the measurement of the magnetic susceptibility, I succeeded in overcoming the difficulty by constructing an apparatus, by means of which I could seal gases in a glass tube at a very high pressure without the least fear of leakage. This apparatus enabled me to use in each case a quantity of gas sufficient for the determination of its magnetic susceptibility and density. A special magnetic balance of high sensibility was constructed for measuring the magnetic force acting on the gas which was sealed in the glass tube and placed in a strong magnetic field. The details of the method and the arrangement of the experiments are given in the following pages. . § 2. MerHop or MEASUREMENT. The method of measurement is based on the following principle :—A cylindrical rod made of the material to be tested is vertically suspended between the horizontal pole- pieces of an electromagnet from an arm of a magnetic balance specially constructed, the lower end of the rod being placed in Fig. 1. the axial line of the pole-pieces and Toon Ee the upper end ina place where the | magnetic field is negligibly small. Suppose at first the balance to be in equilibrium, with no mag- ~_ netic field acting on it, by applying aa we the field the rod is supposed to (i undergo a slight upward displace- a a ment 6a, but in equilibrium acted ~ on by a force f arising from an inclination of the balance beam. Then this force is just equal to the lifting force due to the magnetic field H. The work done by the force is equal to the change of the magnetic energy of the rod, that is — pes / [sa= SS H*S 6a, where 8 is the cross-section of the rod, « and «' are the susceptibilities of the test specimen and the surrounding medium respectively. oj Hydrogen and some other Gases. 309 ! K—K Hence i — 9 H’s, Di : pastel MNea The force f is measured by a small deflexion of the balance beam, which causes a vertical rotation of a small mirror suspended by a bifilar system, the upper ends of the fibres being attached to the lower end of the pointer and to a fixed stand. This rotation of the mirror is measured as usual by a scale and telescope. The above method is due to Lord Kelvin. | For a very small displacement of the suspended tube the force is proportional to da, and consequently to the deflexion of the scale 6; hence ¢ being a proportional constant, we get | j=60. Introducing this relation into the expression for «—x’, we get ae 2c mi Hes: If the intensity of the field remains constant in the range K PAN of the displacement, the factor ie is also constant. Let p denote this factor, then the above expression becomes K—K' =o. In the present experiment the measurements of the sus- ceptibilities of the gases were always made relatively to water or to air ; that is, for the case of air, the comparison was made with distilled water, while for other gases the comparison was always made with dry air. In the present day the susceptibility of pure water * is accurately known, its value —0°720 x 10~° was assumed in the present experi- ment. The susceptibility of air was determined relatively to water. The upper half of a glass tube separated by a glass partition in the middle was filled with the gas or liquid under examination, while the air in the lower half of the tube was evacuated and its lower end sealed. The tube was then vertically suspended from the arm of the balance * P. Séve, Jour. d. Phys. (5) iii. p. 8 (1913); de Haas u. Drapier, Amn. d. Phys. xiii. p. 678 (1918); A. Piccard, Arch. de Genéve, xxxv. p. 209 (1913). -310 Mr. Také Soné on the Magnetic Susceptibilities between the pole-pieces of a Weiss electromagnet, the upper surface of the partition being placed on the axial line of the pole-pieces (fig. 2). The field was then applied and the corresponding deflexion of the eee scale observed. Since the tube Be alone produced some deflexion of the scale it was necessary to eliminate the effect by making two similar observations, first after evacuating the upper half, and secondly after filling it with distilled water. | If the virtual susceptibility of i the system below the glass par- | ExG tition be denoted by x’, we have ove the following relations for the ye three cases, ‘when the upper half of the tube is evacuated, and when it is filled with the gas to be tested and with Seater respectively : cil Ko—K =po, Kg—K' =p 8q, ! Ky—K ==) On where ko, «,, and x,, denote the susceptibilities of the empty space, the gas, and water respectively, and 6,, 6), and 5, are the corr esponding defiexions of the scale. Eliminating «' from these equations and putting « equal to zero, we vet the following relation between the suscepti- bilities of the gas and water : K, Og —%o Op — Oo In the actual case, since the glass tube is not placed in a perfectly symmetrical position with respect to the axial line of the pole-pieces, the term «, is not zero thoug! it isa small quantity, and it is the susceptibility due to the glass tube itself. In the terms 6, and 6,, the above quantity x, due to the glass tube is involved, and the differences 6, 6,=d, and 6,—6,=d, are the true deflexions due to the glass and water respectively. And finally we get Kw Gis If the densities of the gas and water be respectively Py of Hydrogen and some other Gases. dll and p,,, we have for the ratio of the specific susceptibilities x, and x,, of these two substances Xy _ Gl Py oe Pw pe d,/ Mg as at, Al tity where m, and m,, are the total masses of the gas and water occupy ing the same volume of the tube respectively. The same formula may also be used when water is replaced by air, in which case we obtain the susceptibility of a gas relative to air. §3. APPARATUS FOR MEASUREMENT. (a) Magnetic Balance. The magnetic balance and its accessories used in measuring the magnetic susceptibilities of several gases are diagram- matically shown in fig. 3. B ae ean | : ae - i ee fal ae | ian | \ 1 Ba,” ul eal | SH ra a Gar | S ee YU Yi hy YY PY), VT Tf, WD Wy fy 777, aa | | | a uy J AB is an aluminium arm of the magnetic balance, 80 cm. long, C an agate knife-edge resting | on a smooth plane of steel, and HD an Aiea pointer, a similar counterpoise being attached to the same arm, but on the opposite side of the beam. DE shows the side view of the bifilar system 312 Mr. Také Soné on the Magnetic Susceptibilities consisting of two Wollastone wires 0:015 mm. thick, M is a small mirror attached to the lower end of the bifilar system and facing at right angles to the plane of the figure, and F a copper vane damper dipped in vessel containing a mixture of petroleum and machine oil. I is a rectangular brass: pillar with a slide-arm carrying a fixed suspension of the bifilar system. Q is a Quincke microscope with an ocular micrometer whose smallest division corresponds to 1/60 mm. With the microscope the breadth of the bifilar on its upper end can be observed. 8 is a trifilar system consisting of two horizontal Y-shaped wooden frames with a_ small aluminium Y between them, these frames being connected by three fine copper wires, as shown in fig. 4. The aluminium frame can be moved upwards or downwards, so that one can make a minute adjustment of the height of the suspended tube. T is the measuring tube suspended between the pole-pieces of a Weiss electromagnet of inter- mediate size. ¢ is a thermometer placed near the measuring tube to determine the tempe- rature of the specimen under examination. P is a microscope (shown in section) readable to 0:005 mm., with which one can adjust the measuring tube toa correct position. K isa collimator tube of a spectroscope used for the purpose of clamping the arm of the balance by means of its vertical slit. This is necessary when it is required to take the measuring > tube away from the arm, or when the tube is Fig, 4. to be adjusted to a correct position by means of the trifilar system without giving the least disturbance to the balance. L is a pan in which a balancing weight is to be placed, and G a metal damper dipped in a vessel of machine oil. W is a glass cup containing water, into which is dipped a fine class tubing, connected with a fine copper tube U, both being filled with water. The whole arrangement was set upon a stone foundation and covered with a case to prevent the disturbances due to air currents. The case has a large glass window, through which we could observe the deflexion of the mirror with a scale and telescope. To make the finer adjustment of the orientation of the mirror from outside, an arrangement shown in fig. 5 was used. AB is a kind of reservoir made of glass filled with water and mercury; the left end of this reservoir is connected with the copper tube U described above and the other end ‘of Hydrogen and some other Gases. 313: oha with a fine capillary tube C, 0-1 mm. in diameter throug three-way-cock. One of the three wv ways in connected with a mercury reservoir D, and to the left end of the capillary tube, a calcium chloride tube with a long rubber tubing is connected. By this arrangement we can pour into or draw out any desired quantity of water from the glass cup W shown in fig. 3, by forcing the air or sucking it at the end of the rubber tubing F, The reservoir D serves in the case, when a large quantity of water is desired to be supplied or extracted from the cup. (b) Compressor and Measuring Tube. The compression of the gas was made with a Cailletet hydraulic compressor of an ordinary type. The compressing cylinder was replaced by a cast-iron cylinder, specially designed for the present purpose and having a capacity of 1300 c.c. The pressure-gauge was also replaced by another capable of measuring up to 70 atmospheres and graduated to one atmosphere. ! The glass tube, in which the gases are to be filled ata high pressure, has the followi ‘ing construction (fig. 6) :— “the mechanism of the valve of the measuring tube is similar to that of the pressure-gauge commonly used for compressed-gas-bombs. A is a hollow brass cylinder having two bores in it, a narrow straight hole, 1 mm. wide, is bored through from the left end to the central hole termin: iting in acone. In this hole a small brass cylinder P lightly fitted to the hole is pressed on a rubber plate on the right end of the hole by a weak spring. On the left end of this cylinder P, a small piece of ebonite is imbedded. The right opening of the cylinder A is provided with a cup C, which can be screwed into the cylinder, till it firmly presses the rubber plate covering the right opening of the central hole with a thin metal ring. Within the cup ©, there is a piece of 314 Mr. Také Soné on the Magnetic Susceptibilities metal Q which can be moved smoothly along the axis of the cup by means of a screw H. To the left end of the cylinder A, a brass piece B, and to its lateral surface, another piece D is screwed in. The former piece is tightly connected with the measuring tube, and the latter with the glass tube forming the projecting neck of the compressing cylinder. Fig. 6. (Scale 1: 1,) This connexion between the glass tubes and the brass pieces is made airtight by sealing- wax mixed with a small quantity of linseed oil. In order to prevent the flowing of wax into the interior of the tube, when melted wax is poured into the interspace between the brass piece and glass tube, a thin rubber plate is placed at the end of the glass tube. All the packings for the screws are made of thick lead plates. If the measuring tube with the brass pieces are connected with the compressing cylinder by means of the screw D and the inside air is evacuated by a pump connected with the compressing cylinder, the piston P is displaced to the right by a spring, so that the interior of the glass tube and of the cylinder A communicate with each other. If after the gas is compressed into the measuring tube, the screw Hi is turned aud the piston P made to press on the rubber plate, the piston tightly closes the left hole of the cylinder A. The screw D is then disconnected, and now the measuring tube with the compressed gas inside can be suspended from the beam of the magnetic balance between the pole-pieces of the electro- magnet for the determination of the susceptibility of the gas. of Hydrogen and some other Gases. a Ls, If the volume of the compressed gas is to be measured, a screw-head similar to D with a short piece of glass tube is screwed in the side hole; this tube is then placed under a receiving vessel for the gas. By turning the screw lH very slowly, the compressed gas can be let into the receiving vessel. § 4. PROCEDURE FOR EXPERIMENTS. (a) Adjustment of the Measuring Tube. The position of the measuring tube suspended from the beam of the balance was accurately determined by means of the micrescope. ‘The forward or backward displacement of the tube from the central line joining the centres of the pole-pieces was adjusted with an accuracy of 0:1 mm. by aid of the lines marked on the pole-pieces. ‘The height of the tube could easily and quickly be adjusted by means of the trifilar suspension system, with an accuracy of 0°05 mm. The breadth of bifilar carrying the mirror was observed with a Quincke microscope and adjusted with an accuracy of 0'01 mm. The distance of the mirror from the scale and telescope was 190 cm. When the breadth of the bifilar suspension was 0°3 mm. the deflexion of one seale-division corresponded to a vertical displacement of the measuring tube by about 1x 107? mm. (b) Determination of the Mass. The mass of the gases subjected to the experiment was de- termined by two different methods and the results compared with each other. The first method was to determine the mass directly by weighing, that is, from the difference of weights, first when the tube was filled with the compressed gas under examination, and afterwards when it was evacuated. The evacuation was always made after the tube was repeatedly filled and evacuated several times with pure hydrogen by means of a Gaede auxiliary pump, and then the weighing was conducted with the utmost care. The second method was to determine the mass by measuring the volume of the gas collected in an eudiometer. Knowing the pressure and temperature of the gas, we calculated the volume of the gas at the standard pressure and temperature, and by multiplying the density of the gas at the standard conditions by the volume thus obtained, the mass of the gas was obtained. These two methods were used in most of the experiments and the results were found to agree very satisfactorily with each other. Bat, for the reduction of the observed results, either 316 Mr. Také Soné on the Magnetic Susceptibilities of these results was used according to the kind of gases under examination. Thus the mass of carbon dioxide was always determined by the weighing method, and that of hydrogen chiefly by the volumetric method. The mass of air sealed in the measuring tube at the atmospheric pressure and temperature was obtained by calculation, the conditions of the atmospheric air and the volume of the tube in which it was sealed being known. The volume of the measuring tube was accurately determined by filling it with mercury. (c) Method of filling the Measuring Tube with Gas. The measuring tube is connected with the neck of the compressing cylinder E by means of a screw D (fig. 7). The lower end of the glass cylinder E is bent upward, and connected by a short rubber tube with a glass tube coming from a three-way-cock T. One of the three ways com- municates with a Gaede auxiliary pump and another with the gas generator or reservoir. The lower part of the cylinder EK is dipped in a mercury bath as shown in fig. 7. By means of the three-way-cock, the cylinder is first con- nected with the pump, evacuated, and then the cock is turned, the gas is introduced into the cylinder. Next the cy linder is again evacuated and the gas introduced ; these processes are usually repeated four or five times. Then the air previously contained in the cylinder is removed, and the cylinder now contains the pure gas at a pressure of abou one atmosphere. ‘Then the cylinder HK and the glass tube 1 disconnected under the surface of the mercury in the bath” 07 Hydrogen and some other Gases. oe A quantity of mercury then flows into the cylinder and partially fills the bottom of it. In this state the cylinder is transportable to the Cailletet compressor without any risk of introducing other gases into it. (d) Hlectromagnet. The magnetic field was obtained by a Weiss electromagnet, the pole-pieces being always 1 cm. apart. The end surface of the pole-pieces was a circular section of 1 cm. in diameter. The magnetizing current of 10 amperes produced a field of 22,000 gauss in the place where the magnetic measurement was to be made. In the case of air 4 amperes were used and the corresponding field 12,500 gauss. During the magnetization a slight convection of alr was produced by the heating of the electromagnet, and this made the obser- vation of the deflexion of the mirror somewhat difficult. To avoid the disturbing effect the coil of the magnet with the exception of the pole-pieces was entirely covered with a winding of lead tubing and the water mantles, water being constantly circulated through the lead tubing and mantles during the observations. The intensity of the field was measured by means of an exploring coil and a ballistic galvanometer as usual. (e) Method of Experiments. In the magnetic balance a delicate knife-edge rests on a smooth steel plane, so that a very minute gradual displace- ment of the knife-edge, either translational or rotational, can never be absolutely avoided. This gradual displacement is usually accelerated when the field was repeatedly applied, causing the gradual displacement of the zero point on the scale. However, by comparing the results obtained when such a gradual displacement of the zero-point occurred and when it was absent, it was found that this displacement did not affect the final results, provided the mean of the successive zero-points in each observation be taken as the true zero of the deflexion. In making the observations, we first passed a current of 10 amperes in the electromagnet, and the maximum deflexion of the scale was observed. It teok usually 30 to 60 seconds. Then the current was quickly reduced to zero, and the final deflexion or the zero-point was observed. These processes were repeated usually ten times, and the mean of these deflexions was taken. The whole process required about ten minutes. The temperature of the gas under examination 318 Mr. Také Soné on the Magnetic Susceptibilities was carefully observed at each set of observations with a thermometer suspended near the measuring tube in the space between the poles of the electromagnet. A current of water was constantly passed around the electromagnet during the observation. 6/5) eAuiRe | The magnetic susceptibility of the atmospheric air has been determined by many investigators, such as Faraday, Becquerel, (Juincke, and more recently by Curie. But other investigators such as Du Bois, Hennig, and Piceard, have deduced the susceptibility of air from that of oxygen by neglecting the susceptibilities of nitrogen and other gases present in the atmosphere. In the foilowing table, the values of the volume suscepti- bility of air and also the ratio of the susceptibilities of air and oxygen in four cases, in which the susceptibilities were independently determined, are given :— TApuE dT, Date. Observer. Kr LOe Eu x_/ KO,. POH ar Mere Uae Faraday * 0°024 ie 0:184 US Du Meee Becquerel t 0:025 ii 0-208 TSS meee Quincke t 0-032 16° 0:248 SOD seve oe Curie § 0:027 20° 0:232 SSS yar wes Du Bois || 0-024 15° a AoW Vie a tease Hennig 4 0-024 25° NG een keane: Piccard ** 0:029 920° M. Faraday, Exp. Res. of Elec. (3) p. 502. E. Becquerel, Ann. de Chim. et de Phys. (3) xliv. p. 223 (1855). G. Quincke, Wied. Ann. xxxiv. p. 401 (1888). P. Curie, Journ. de Phys. iv. p. 197 (1895). H. Du Bois, Wied. Ann. xxxv. p. 137 (1888). R. Hennig, Wied. nm. 1. p. 485 (1893). A. Piceard, Arch. de Genéve, xxxv. p. 408 (1918). KS etm st OK * In the above table we see that the magnetic suscepti- bility of air as obtained by several observers shows a large discrepancy, and also the ratio of the susceptibilities of air and oxygen directly determined is not constant. Hence it is to be concluded that the susceptibility of air and its relation to that of oxygen are not yet correctly known. In the present research the susceptibility of the atmo- spheric air was determined relatively to that of redistilled water. The air to be examined was introduced into the compressing cylinder after passing through the tubes con- taining solid potassium hydroxide, calcium chloride, phos- of Hydrogen and some other Gases. d19 phorus pentoxide, and cotton ; thus the air in the cylinder was entirely free from Som dioxide, moisture, and dust. The air was then compressed into the measuring tube at a pressure of about 30 atmospheres and subjected to the magnetic measurement. The masses of the gas and the water were determined by weighing. In this case the magnetizing current was usually 4 amperes, and the breadth of the bifilar 1 mm.; but by changing the bifilar distance and applying a weaker or stronger field, a moderate deflexion of the scale could be obtained. In the following table one set of observations is shown as an example :— TasLeE II, Sept. 5, 1918. Air (p=80 atm.) Vacuum. He5O™ Par, ¢—=24°°5 C.,6=600, 25 35™ p.m, ¢=24°°8 C., 6=60°0. Mm, +t, =89'0927 gm. ee 9438 gm. C. S. é. C. 8. 0. amp. em. cin. amp. em. cm. 0 56-90 17-55 ) 39°40 0-15 4 74:45 WAS) 4 39°55 0°35 0) 56°75 WEDS 0 39°20 0:20 4 74:30 17°60 4 39°40 0:20 0 56°70 17°60 0 39°20 0:20 4 74:30 17°70 4 39°40 0:24. 0 56°60 17°60 0 39°16 0:24 4 74:20 17°70 4 39°46 0:20 0 56°50 17-62 (mean) 0 39°20 0:22 (mean) WATER. CALCULATION, 3h 35™ p.m., £=24°2 C., 6=60°0, d,=17-62-0'22=17-40 cm., - M+, =93-1178 gin. d,,= —14-64-0:22= — 1486 em., C. 8. - mM,=0':1429 em., m,,=4:1740 om., amp. em. em, TOK 0 61:29 —14 59 i 91489 — 1169 KG 4 46°70 —14°70 0 61-40 —14-69 ye eS Te 4 46 71 e479 ee ge HED. 0 61°42 —14°56 va 1169 as 4 46:36 —1456 e = Ta oe 0 61:42 —1462 . 4 46:80 —14:70 Noah U0 0 61°50 — 14°64 (mean) In the above table 0 is the breadth of the bifilar expressed in the scale division in the ocular micrometer, 60 divisions corresponding to 1 mm. mo,, mo, and mw me the masses of oxygen, the tube evacuated, and the water respectively, 320 = Mr. Také Soné on the Magnetic Susceptibilities do, and dw are the deflexions of the scale due to oxygen and water respectively. C is the magnetizing current, 8 the scale reading, and 6 the deflexion of the scale. K is a constant which depends on the sensibility of the apparatus and the intensity of the field. Vifteen sets of such observations were made at a mean temperature of about 25° C.; reducing these observations to 20° C. by assuming Curie’s law, we obtain the following result :— 10° .y,= 23°95, 23°52, 24°20, 23°58, 23°58, 24°15, 23 69, 24°04, 23°65, 24°37, 23°81, 23°42, 24:06, 23°66, and 24°13, the mean value is then N= Doone LU wat 20.02 ‘with a mean error of +0°07x 107°. Multiplying the density of air at 20° C. and 760 mm. pressure, we get as the magnetic susceptibility per unit volume of dry air at 20° C. and at the normal pressure, 6, — 00257; x10" 3, 0:00009 ta § 6. OXYGEN. Oxygen is the only gas whose magnetic susceptibility has been determined with a fair accuracy ; yet the values obtained by different observers differ so widely from each, other that the extreme values deviate from the mean by more than 10 per cent. The following table contains the values which have been determined by several investigators :— TABLE III. Date. Observer. Koy: LO®: be SSS Wiese Faraday * 0143 as SSD euaee: Becquerel t 0°149 Hp TESS ese ae Quincke t 0:129 16° Ufeket oy asuinelea Du Bois § 0-117 152 NSOS ccna Hennig || 0-120 25° TSO Siok! alana Curie { 0-115 20° OTS) hi a Piccard ** 0-141 202 MONG cn Roop tt 0:146 16° * M. Faraday, loc. cit. | R. Hennig, doc. edt. + E. Becquerel, Joc. ct. q PR. Curie, Toe. t G. Quincke, loc. cit. xx A’ Piccard, loc: ed. § H. Du Bois, loc. cit. tt W. P. Roop, Phys. Rev. vii. p. 529 (1916). In the present experiments the pure oxygen was obtained of Hydrogen and some other Gases. O21 by the decomposition of a solution of potassium hydroxide by electrolysis. The electrodes were of sheet nickel attached to stout nickel rods, the strength of the solution being two normal, The cell consisted of two concentric glass eylinder S, the inner one having no bottom ; one of the electrodes was placed in the inner cylinder and the other placed in the space between these two cylinders. The current density was 0°05 ampere persq.cm. The generated gas was collected in a large glass reservoir by replacing water. Before intro- ducing the gas into the measuring tube, the gas was purified by passing through bottles and tubes containing strong sulphuric acid, solid potassium hydroxide, and phosphorus pentoxide. The susceptibilities of the gas thus purified and the air treated likewise, as described in the last Section, were compared at the ordinary temperature and pressure; the ratio of the mass of the gases being determined by calcu- lation by knowing the temperature and pressure of the atmo- sphere at the time of filling the gases in the measuring tube. In the following table an example of the results of the measurement is shown :— i) Ae March 19, 1918. Arr, Vacuum. C— 1293 C1 p— 148-0 mime, <—13"5 C.) 7h 15™ pw, Z=15°°5 C., b= 200. 7 45" pa, £=15°°3 C., b= 20-0. C. 8. é. C. S. O. amp. cm, em. amp. em. cm, 0 41°50 10°70 0 50°75 22°90 10 52°20 10:20 10 73°65 22°40 0) 42-00 9°60 0 51°25 22°30 10 51:60 10°30 10 73°55 2215 0 41-30 10°10 0 00°80 22°60 10 51-40 10°40 . 10 73:49 22°40 0 41:00 10°22 (mean) 0 51-00 22°56 (mean) OF CALCULATION. Gia C..9=1 748-0 Mis, Go. -o.C,) 7 55" pau, += 14°75 C., b=20-0, mq mae c e i d_,=22-56-10-22-= 12°34 om., alp. clin. cin. 0 13°60 72:00 (¥0, _ 59°67 _ 0:001210 _ 287-3 10 85°60 69:20 X1/ 20° 0, 12°34 “* 0°001335 3. 0 16°40 69:00 = 4-365. 10 85°40 69-50 10 15°90 69:30 (XO) 909, = LF 12 x 107". 10 85:20 69°80 0 15°40 70°15 10 85°55 70°15 0 15°40 69°89 (mean) Phil. Mag. 8. 6. Vol. 39. No. 231. March 1920. Y 322 Mr. Take Soné on the Magnetic Susceptibilities Hight sets of such observations made at various tem- peratures ranging from 14° C. to 18°C. give the following values of the susceptibility of oxygen, which are all reduced to the value at 20°C. 10°. yp =104:00, 104°48, 104:00, 104°10, 103:39, 104-12, 104°60, and 103-86, the mean of cine is Yo, = 1041 x 10-8 at 20° C., the mean error being +0 09x 107%. Multiplying the density of oxygen at 20°C. and 760 mm. pressure, we get as the susceptibility of oxygen gas per cubic centimetre, Ko, = 0°138, x 107° +0-0001 x 107°. The above result is in fair agreement with Piccard’s value, which seems to be the most reliable among the values of the previous investigators, but as he used a commercial oxygen, the error which might arise from the impurity of the gas would probably make the mean error of his result larger than he had believed. He also determined the mass of the gas by the absorption method, which is accompanied by some uncertainty. In the present case the gas was obtained by electrolysis and carefully purified, so that there is no uncertainty about its purity. At any rate the error in my case, if there is any, may arise from the determination of the susceptibility. If the magnetism of air is due solely to that of oxygen present in it, the ratio of the specific susceptibilities of air and oxygen should be identical with the ratio of the mass of the oxygen in air to the total mass of air, that is, the ratio Xa _ 23°85 Ne LOT must be equal to 0°2315, the difference being 0:0024. The difference amounts to about 1 per cent. of the total value and is certainly beyond the experimental errors of the measurements of the susceptibilities. The explanation for this discrepancy will be given later on. = (2200 \§ 7. CARBON DIOXIDE. About ten years ago Professor K. Honda* found an anomalous behaviour of tin in that its magnetic property * 1X. Honda, foc. cvt. of Hydrogen and some other Gases. 323 changes from a paramagnetic to a diamagnetic during melting. Since that time no other substance showing a similar change has been found. Meantime Mr. T. Ishiwara in our laboratory found in the course of his researches on the magnetic susceptibility of chemical substances at low temperatures, that solid carbon dioxide has a diamagnetic susceptibility per unit mass of y=—042x10-° in the temperature range between —100° C. and —170°C. In the literature we have only a few data for the magnetic susceptibility of the gaseous carbon dioxide. The earlier investigators, such as Faraday * and Becquerel f, agreed in the view that the magnetism of gaseous carbon dioxide is too weak to be detected by their experiments. Quincke { found, however, that its specific susceptibility is y= +0°017 x 10~°; more recently Bernstein § found its volume susceptibility to be x= +0°0002x10-®. If these two results be true, at _ least in sign, then gaseous carbon dioxide is paramagnetic, _and we have, besides tin, one more example of a magnetically anomalous substanee. In this respect an exact determination of the susceptibility of the gaseous carbon dioxide seemed very interesting. In the present experiment the carbon dioxide was obtained by the reaction of dilute hydrochloric acid on calcium carbonate. Pieces of pure marble previously boiled in hot water for about 24 hours in.order to drive off the air occluded, were put in a Kipp apparatus with boiled distilled water, care being taken not to expose the pieces to the air. No air bubble was allowed to remain in the bottle. Then the strong hydrochloric acid was poured into the apparatus through the upper opening, and at the same time, by expelling the water from the exit, we could easily replace the water with hydrochloric acid of a moderate strength without introducing any trace of air into the apparatus. By this means we were able to obtain the carbon dioxide entirely free from air. Before introducing the gas generated in the Kipp apparatus into the compressing cylinder, it was first passed through a bottle containing water, and then bottles containing strong sulphuric acid and pieces of calcium chloride. In the following table one set of observations is given as an example. *M. Faraday, doc. evt. 7 E. Becquerel, loc. cit. tG Quincke, Joc. ert. § Bernstein, Diss. Halle, 1909. Y 2 824 Mr. Také Soné on the Maynetic Susceptibrlities dan. 13, 1917. VACUUM. TAREE 12°15™ pin., 2=17°-0 C:, 6= 20:0. 7, = 89'3657 gm. C. S. amp. cm. 0 53°30 10 44-80 ) 54:00 10 46°10 0 55°80 10 700 0 55°90 10 48°00 0 OT 20 10 49-20 0 57°60 10 49°50 0 58°80 10 49°90 0) 59°60 10 51°90 J. cm. — 8:50 —9°20 —7:90 — 9-70 — 8-80 — 3:90 —7:90 — 9°20 — 8:00 —8 40 —8 10 — 9°30 —8:90 —9-70 =O — 868 (mean) Air (p=1 atm.). 5b 40™ pi, C= 167 OC. 62000: M, +m, =59'3706 gm. C. 8. amp. ci, 0 53°70 10 — 54:50 0 53°10 10 54:60 0 53°70 0. cm. 0:80 0°80 0°90 0:90 0°85 (mean) V. CO2z (p=28 atm.). 3) 15 Pat. C= 199: OMe a Oe TC, +m, = 89: 5867 gm. C. 8. 0. amp. em. em. 0 65°60 — 15°60 10 50:00 —17:00 ie) 67:00 — 15°40 10 51°60 — 16:60 0 68°20 — 16°40 10 51°80 —18-10 0 69:90 — 13°50 10 56°40 ~ 15°50 0 - 7190 * —17°90 10 54:00 —18°60 0 72°00 -~ 15°90 10 56°10 — 15°90 0 7200 — 16°51] (mean) CALCULATION. doo, = —16°31+8°68= — 7-68 cm., _ d,=085d+8'68=9'53 cm., Mog, = 02210 gm.,7 ,==0°0049 gm., Koo eS = — 345 K, So eee = 1945 K, "60, SS ane AT LS | The values of the susceptibility obtained in three sets of observations which were made at a mean temperature 18° GC. are 10°. ¥¢o,= —0'423, —0°429, and —0-416. The mean value is Xco.= —0°42,x 107°, In the above calculation the susceptibility of air is corrected for the temperature at the time of measurement. As the of Hydrogen and some other Gases. ene) diamagnetic susceptibility is considered to be independent ot temperature, we can get by multiplying the density of the gas at any temperature and pressure into the specific susceptibility above obtained, the volume susceptibility of the gas at that temperature and pressure ; thus we get as the volume susceptibility of carbon dioxide at 20° Cand 760 mm. pressure, Koo, = —0°000775 x 10-°. The same consideration is used in the calculation of the susceptibilities ef the diamagnetic gases investigated in the present research. The value of the specific susceptibility of carbon dioxide obtained above is in fair agreement with the value Xco,= — 0°42 x 10-8 for the solid carbon dioxide obtained by Mr. T. Ishiwara mentioned above. This shows that the specific susceptibilities of carbon dioxide in the solid and gaseous states are almost equal to each other. Putting for the moment this delicate question of magnitude out of consideration, the close agree- ment of these determinations seems to give strong confirm- ation of the conclusion that the value obtained in the present experiment for the gaseous carbon dioxide is not far from the true valne. At least we can assert that the magnetism of carbon dioxide is diamagnetic, contrary to all previous determinations. The paramagnetic result of the gaseous carbon dioxide obtained by the previous investigators comes probably from the impurity of the gas examined, such as a trace of air mixed with the gas. Of the three sets of measurements quoted above, the last one was made by a method which was in some respects different from the other two. The method employed was the null-method described in the last section. Namely, we measure the force / in terms of the volume of water which is to be supplied or to be taken out of the vessel hanging from the arm of the balance, in order to bring the measuring tube to the initial position against the magnetic force, or which is equivalent, to bring the deflexion of the mirror to its initial reading on the scale. The volume of water can be read from the volume of mercury thread in the capillary tube. The following table contains the data of the measurement for carbon dioxide by means of the null-method. Here r is the reading of the head of the mercury thread in the capillary 326 Mr. Také Soné on the Magnetic Susceptibilities tube, & is the difference between these readings, and 6, the mean for each set of observations :— TasLe VI. May 28, 1917. CO, (p=35 atm.) Vacuum. t=1609 COO b=10°0, m,=98'7750 gm. M EQ tM, = 990673 gm. C. r. é. Ons C. t. 0. One amp. cm. em. cm. amp. em. em. em. QO 27:32 —2:48 — 2:65 QO 29:00 —1453 —1473 10 29:30 —2:82 10 43:53 —14:93 0 26:98 0 28°60 0 2676 —2:34 —2°52 0 2860 —14:55 —14:35 10 29:10 —2:70 10 48:15 —1415 Q 26°40 0 29-00 0 26°25 —2:30 —2°55 0 29:35 —14:35 —1463 10 2855 —2'80 LO Vato Oko 0 25°75 0 28°80 — 14°57(mean) 0 25:70 —245 —~2°73 10 2815 —300 QO 25°15 — 2°61 (mean) Arr. CALCULATION. t=182:0' C2 6=10;0; doo, = —1457+2'61=—11°96 cm., Mo +m =98°7795 gm. d,=7:02+261=9-63 em., C. r. 6. Ons =0°322 I; ==()" 5 ‘ amp. cm. cm, em. Loe nee » M,,=0°0045 gm., 0) 2457) (6.876 86 X ap oe ee ake oa C02” 03223 : LO On) ie S ae 0 24-75 ge Oe i Xa= 90045 40K 0 24°76 7:00 7:08 Xoo aera 109 17-75) gab ae = O10 we 0 2490 702(mean) ~~ The fact that the results of these different methods are in close agreement with each other not only indicates that the deflexion method is practically equivalent to the null- method, but also that the intensity of the magnetic field coming into play is constant, at least in the range in which the measuring tube displaces itself by the magnetization. The deflexion method is, however, much simpler in operation than the null-method, because the latter method requires much time, and consequently many difficulties are likely to occur during observation, such as those caused by the convection current due to the heating of the electro- magnet. Hence after it was ascertained that the deflexion of Hydrogen and some other Gases. 327 method always gives a correct value, it was used throughout the following experiments. § 8. NITROGEN. The literature regarding the magnetic susceptibility of nitrogen is scarcely known. Faraday * first found that the susceptibility of nitrogen is paramagnetic. Becquerel + examined the magnetism of the gas, but he could not detect it. Quincke t found a minute paramagnetic effect for gaseous nitrogen. Pascal § concluded from the study of the susceptibility of some organic compounds containing nitrogen that this element is diamagnetic in its gaseous state. Their values are given in the following table :— Raprr Vif: Date. Observer. fee AKO 18aa) FA: Faraday +0:0021 1855) eee Becquerel 0 NSSSee ces. Quincke +0:001 TOP Pascal ~—0:0005 Thus all the previous investigators except Pascal agree in the view that the susceptibility of gaseous nitrogen is paramagnetic. In the present experiment special attention was paid to the preparation of nitrogen gas, so as to avoid contamination with air and nitric oxide. Three different methods of preparation were employed, and the susceptibilities of the gases obtained by these methods were compared with each other. The first two methods consisted of the chemical preparation of pure nitrogen gas, of which the second one was com- paratively imperfect and served only as a check on the first, and the last one the preparation of the so-called atmospheric nitrogen, e. g. the nitrogen accompanying argon and other inert gases in the atmosphere. The method of preparation and the experimental data obtained in the magnetic measurement of nitrogen gas thus produced are described below in order. The first method is the process first used by B. Coren- winder || in 1849. Though this is an old method, it seems to be an excellent one for the preparation of nitrogen gas of high purity. Lord Rayleigh] proved that the nitrogen * M. Faraday, loc. cit. + E. Becquerel, Joc. cit. t G. Quincke, /oe. ert. § P. Pascal, Ann. de chim, et de phys. viil. p. 1 (1910). || B. Corenwinder, Ann. de chim. et de phys. (8) xxvi. p. 296 (1849). { Lord Rayleigh, Travers, “Study of Gases,” p. 48. | 828 Mr. Také Soné on the Magnetic Susceptibilities gas thus obtained was free from any trace of the nitric oxide which is more or less present in the nitrogen gas obtained by most of the other methods, and the removal of which was very difficult. A solution of ammonium chloride and potassium nitrite was gently heated in a flask on a water bath. After all the air in the flask had been expelled, the gas generator was connected with a large glass reservoir and then the gas collected in it. Lefore introducing the gas into the com- pressing cylinder, the gas in the reservoir was passed successively through a ‘red-heated copper gauze, bottles containing a solution of. ferrous sulphate cooled with ice, strong sulphuric acid, and tubes containing soda lime, ello chloride, phosphorus pentoxide, and calcium chloride. The ends of the tubes and of the bottles were brought so closely together that the rubber tubes connecting them were exposed to the gas as littleas possible, and all the connecting parts were covered with cellodion films. The following table contains an example of the data obtained in the magnetic measurement :— Taste VIII. Dee. 26, 1916. N, (p=25 atm.) Vacuum. 2h 45™ p.m., ¢=16°°7 C., 6=20°0, 35 25" p.m., 2=18°:0 €/96—20:0) my, tin, =91 -40868 em. m,=91°31585 gm. C. S. 0. C. S. 0. amp. em. cm. amp. em. em. 0 62°80 — 5°90 0 49-80 —4°50 10 56°90 —590 10 45°40 —3°80 0 62°80 — 590 0 49-20 —4°45 10 56°90 — 5°80 10 44°75 —3°85 0 62°7 —5'88(mean) 0 48°60 — 4°15 (mean) Arr (p=12 atm.) CaLCULATION. 4 33” p.m., 2=199°5 C., b=20°0, dy, = —5'88+4415= —173em., mM, +m, =91'3748 gm. d=98:00+4:15=102°15 em., C. S. é. my, =0°09278 gm., m,=0:0590 gm., amp. em. em. yaa —hisK 1 ae K 0 2:20 98:00 mae 10 100-20 98°60 102-22 K — —__ = 1732 K, 0 1:60 96:90 ta “90-0590 a oF x 12-65 10 98°50 98°50 IN, BlSiGo ae 0108. ) 0:00 98°00 (mean) x, 1732 Three sets of such measurements gave as the specific susceptibilities of nitrogen gas ata mean tempera ie OKs 10° - Xy, c= —0°258, —0'272, and —0°265 of Hydrogen and some other Gases. a29 the mean value being Xn, = — 0°26, x 107°. The volume susceptibility of nitrogen at 20° and 760 mm. pressure is Kx = — 0°00030, x 10-°. The second method of preparation was due to Mai*. To a mixture of ammonium nitrate and glycerine contained in a flask, a few drops of strong sulphuric acid were added, and the mixture was heated in an oil-bath at about 160°C. The generated gas was collected in a reservoir after passing it through a strong solution of caustic potash. Before the gas was introduced into the compressing cylinder, it was passed through the trains of purifiers, which were the same as those used in the former case, except that in this case the solution of ferrous sulphate was replaced by a strong solution of caustic potash. ; One example of the experimental data of the measurement for the nitrogen prepared by the second method is given in the following table :— TABLE IX,. dan; 17, lobe N, (p=25 atm.) Vacuum. AM pay <= 199 OC. 620 0; 4h 50™ p.m., ¢=19°°5 C., 6=20°0, in, +m, =89°4828 gm. m0 =89°3651 gm. C. S. 0. ite @; S. é. Onn: amp. cm. em. em. amp. em. em. em. Oi 57-70% — 9:35) ) = 10-40 0 59:80 —650 —7-00 10 48°35 —11:45 —10-47 10 5330 —750 —7:90 0 59°80 — 9:50 —10°38 Q 6080 —830 —815 10 50°30 —11:25 —1028 10 5250 —800 —7-‘85 0 61:55 — 930 —10°32. 0 6050 —T770 —7'65 10 52°25 —11°35 ——— 10 5280 —760 —7-60 0 63°60 —1037(mean) O 6040 —760 ——-————~— 10 52°80 — 769(mean) Air (p=latm.) CALCULATION. G) par 7 — 18 Co — a0) dy =—10:37+769= — 2°68 cm., M+ M,=89'3100 gm. d,=1:60+7:69=9'29 cm., C. 8. 0. au" my, =O 1177 gm., 7,=0'0049 gm.. amp. cm. em. em. 9-68 K 10 71:20 2°20 1-70 an iE 0 69-00 1:20 1:55 renee 00045 =1895 K, 10 70:20 1°90 1:70 0 6830 1:50 X qn _ 22°76 _ “Na = = — _0:0120. 10 69:80 160(mean) x, 189 * J. Mai, Ber. d, Deutsch, Chem. Ges. iii. p. 8805 (1901). 330 Mr. Také Soné on the Magnetic Susceptibilitres Two sets of such measurements were made for the nitrogen obtained by the second method ata mean temperature 18° C.; the results are Oe Xw,= —()'288 and —0°287, the mean value being X,= —0°288 x 10-°. The volume susceptibility at 20° C. and 760 mm. pressare is — 0100033, x 10a Thus the value of the susceptibility of nitrogen obtained by the second method is about 8 per cent. more diamagnetic than that obtained by the first method, but it was sufficient as a check and no further study was made of the cause of the deviation. In the third method the atmospheric air was introduced into the reservoir through the bottles, containing a solution of caustic potash, calcium chloride, concentrated ammonia solution, and red-hot copper gauze. Before filling the gas in the compressing cylinder it was passed through bottles containing strong sulphuric acid and calcium chloride, and ‘then through the trains of purifiers, which were exactly the same as those used in the preceding experiment. The following are the data in the determination of the susceptibility of the atmospheric nitrogen :— TABLE) (XG Jan. 11, 1917. N’, (p=30 atm.) Vacuum. Gl 10M Pac, C= 1590 Cc 0= 200; 8b 40™ p.m. 2=14°°7 C., 6=20°0, My, +i, =89-4916 gm. mM, = 893518 gm. C, 8. On C. 8. 0. Por ae amp. em. em, amp. em. cm. em. 0 60°00 —11°80 0 56:00 —650 —8-00 10 48°20 —11°80 10 49°50 —950 —8&35 0 60:00 —12°50 W) 59:00 —720 —7-70 10 47°50 —11-70 10 51:80 —820 —810 0 59°20 — 11°40 0 60:00 --800 —8:35 10 47°80 — 10°80 10 52:00 -—870 —860 0 58°60 — 12°10 0 6070 —850 —875 10 46°50 — 11:80 10 52:20 —900 —8:35 0 58°30 —11‘74(mean) 0 61:20 —810 —76E& 10 5310 —720 —765 0 60°30 —810 —T7-95 10 52:20 —780 ——-——— 0 60-00 —8'15(mean) of Hydrogen and some other Gases. > gal AiR (p=1 atm.). CALCULATION. 108 5™ p.m., £=12°-0 C., 6=20°0, dy = —1174+815= —3°59 em., m_-+-m,=89°3566 gm. : C. at Me S. : 3. d,=0°27 +8:15=8'42 cm., amp. cm. em. M x71, =0'1398 gm., m,=0:0048 gm., 0 52:70 0°20 & 10 52:90 0:20 ye cai on gave 0 52-70 Few ORES ) 0 53°40 0°30 349K 2 10: 53°70 Xa= o0048 71 /ot K 0 64:70 0°30 x E 10 65-00 0-30 CAS Se ees 0 64°70 0:30 Kuen toe 10 65°00 0°27 (mean) The values obtained in two sets of observations made at a mean temperature 18° (’. are 10°. vy, = —0°363 and —0°356, the mean value being Xw,= — 9°360 x 10°. The volume susceptibility at 20° C. and 760 mm. pressure is Nw, = —0°00042, x 10-°. Taking the mean of the results obtained for the gas prepared by the first method as the magnetic susceptibility of chemically pure nitrogen gas, we have, as the specific susceptibility of pure nitrogen, Xan, = — 0°265 x 10-8, and if we consider the result for the last case as the sus- ceptibility of the atmospheric nitrogen containing about 1-7 per cent. of inert gases, of which argon is a chief constituent, we have i Xqr,= — 0°360 x 10-®. The large discrepancy between these results can never be explained as an experimental error; it must be due to the magnetism of the inert gases, and we may assume for the present that the result is due only to the magnetism of argon, as the quantities of other gases are exceedingly small. Now let us calculate the magnetic susceptibility of air from the susceptibilities of its constituent gases, assuming 32 Mr. Také Soné on the Magnetic Susceptibilities that the additive law holds. We have then the following relation :— Xa Po Norma Xv, 5 PAC ce X Ont eum ICY. | pau 02 Po, ! WHEL Vo, Xo. Na» and x, are the specific susceptibilities of ar, oxygen, nitrogen, and argon, and p,, py, and p, are the ratios of the masses of these gases in air to the total mass of air respectively. Then the value of the susceptibility of argon can be deduced in the following way, taking the percentages of nitrogen and argon in air as follows :— 79°5 and 1:3 per cent. Hence the percentage of argon in atmospheric nitrogen is 13x 100 ‘ see = 17 per cent. (od + 13 But the presence of argon in the atmospheric nitrogen produces an increase of the diamagnetic susceptibility from Ny,= ~ 0:263 x 10° to xy, = —0°360 x 10°*. Hence the susceptibility of argon will be = 0:360)xl05® | 0:983:x (026 ae ee A OL ae 0-017 sated oleh < NO Taking as the weight percentage of oxygen in air 23°15 and introducing the values for y,, vy, and y, in the equation for oxygen, we get 23°85 x 107") 0°755 x (—0 265) x 10m: Xo.= 0.9815 1235 8 1 OS x (F218) EO | 0°2315 =(103°0-+0°94 0:3) x 107°. OS x 10a: But the susceptibility of oxygen directly determined being Nos 104°, x 1078, these two are in fair agreement with each other. of Hydrogen and some other Gases. 339 It becomes now clear that the volume susceptibilities of the nitrogen and argon present in the atmosphere are not negligibly small, as was believed to be the case ; and that the sum of their values amounts to a little above one per cent. of the susceptibility of air. § 9. HypROGEN. Hydrozen is one of the gases the magnetic susceptibility of which has been comparatively well studied. Nevertheless, owing to the difficulties which accompany the experiments, we have as yet no reliable experimental value of its magnetic susceptibility:; even its sign was not decided till quite recently. But recent investigators seem to agree in the view that hydrogen has a diamagnetic susceptibility, and now the determination of an exact value of its susceptibility becomes an outstanding problem. Quineke * first Ae amined the magnetic susceptibility of hydrogen, and found as its susceptibility a value R= +-0:0003 x 105& Bernstein + published the results of his experiments on the magnetism of some gases in his dissertation at Halle, and cave as the susceptibility of gaseous hydrogen, K=— 0005 x 10°. Assuming the additive law, Pascal t calculated the atomic susceptibility of hydrogen fom the study of some organic compounds ; the result ‘of his calculation being Mod x 1078. Kammerlingh Onnes and Perrier§ measured the magnetic susceptibility ‘of liquid hydrogen and found as its volume susceptibility, c= —0'186 x 107°. Taking for the density of liquid hydrogen a value obtained by Dewar 0:07, he deduced as the specific susceptibility of liquid hydrogen, y= — 2 ( x107° No account of the experimental details was given in their * G. Quincke, loc. cit. * Bernstein, doc. cit. t P. Paseal, loc. cit. < Hi Kammerlingh Onnes and A. Perrier, Amsterdam Proc. xiv. p. 121 (1911). 334 Mr. Také Soné on the Magneite Susceptibilities paper, but the authors did not claim too much weight for their result and were satisfiel with the fact that their result roughly agreed with the value obtained by Pascal. More recently Biggs* carried out the same investigation of gaseoushydrogen. He tried to overcome the experimental difficulties by utilizing the large absorptive power of palladium for hydrogen, and found too large a value for the susceptibility of the gas. The chief difficulties met with in the determination of gaseous hydrogen are two: the first is the preparation of pure hydrogen free from oxygen, and the second the deter- mination of the magnetic force exerted upon the gas, owing to the smallness of its volume susceptibility and density. In the present experiment these two difficulties were over- come, and a highly trustworthy value for the susceptibility of hydrogen gas was obtained. The details for the prepar- ation of the pure gas and the determination of its magnetic susceptibility are given in the following pages. (a) Preparation of pure Hydrogen Gas. In the present investigation the pure hydrogen was prepared by the method which Morley 7 used in the deter- mination of the volumetric ratio of hydrogen and oxygen in water. The chief differences between his and the present case were in the construction of the decomposing cell and the combustion tube. In my case an open vessel was used as a decomposing vessel, and platinum electrodes were introduced into the cell by insulating the leading wires with glass tubes. Oxygen which was generated at the anode, was allowed to escape into the atmosphere. ‘The electrodes were separated from each other as far as possible in order to lessen the diffusion of oxyzen from the anode to the cathode. The open end of the vessel was covered with a mica plate ; besides, the upper portion being wholly wrapped in cloth in order to prevent any contamination of liquid. In the early period of the present experiment, I used for a long time a combustion tube of hard glass, which contained pieces of reduced copper gauze, both ends of the tube being vround and joined to the rest of the purifying train. But in the course of the experiment it was found that in a long period of time the joints became gradually loosened by repeated heating and cooling, allowing the diffusion of air * H. F. Biggs, Phil. Mag. xxxii. p. 131 (1916). + E. W. Morley, Amer. Journ. Set. iii. xli. p. 220 (1891). of Hydrogen and some other Gases. 335 into the tube: hence finally I used palladium asbestos instead of copper; and in this case the heating temperature being below 250° C., an ordinary glass was used as the combustion tube, both ends of the tube being fused together to the rest of the apparatus. In consequence of this change of arrangement, not only was the measurement of suscept- ibility of hydrogen greatly facilitated, but the results of the experiment became quite consistent. The hydrogen generator, purifying train, the reservoir, and the compressing cylinder, together with the measuring tube, are shown in fig. 8. a is a large cylinder having a | ee =a ee capacity of 4 litres. About 3 litres of pure dilute sulphuric acid was put into it; the concentration of the solution was the same as used by Morley, e. g. it contained one-sixth of pure sulphuric acid in volume. ¢ is a glass tube placed concentrically with the cylinder ina. The anode is placed in the upper portion of the cylinder, while the cathode is placed at the lower part of tube 6. Eacli electrode has an area of 2) sq.cm. C is a bulb for preventing the passage of the acid fumes upwards. d isa large glass bottle filled 336 © Mr. Také Soné on the Magnetic Susceptibilities halfway with a 50 per cent. solution of caustic potash and placed ‘oblig uely. eisa long horizontal tube containing fine pieces of glass wetted with the above solution. fis a bulb for preventing the mixing of the solution of caustic potash with sulphuric acid in a vessel g; h is a soft glass tube loosely filled with palladium asbestos, and 2 a thermometer eraduated to 360°C. 7 is a wash-bottle containing strong sulphuric acid. & is a cock dipped in a mercury bath, which prevents the diffusion of hydrogen to the outside and that of air to the inside. J is a tube containing molecular silver, which serves as a sensitive indicator of hydrogen sulphide. m is a large cylindrical tube in which rods of caustic potash are placed, and » a cylinder containing phosphorus pentoxide. o is a large glass bulb having a capacity of about 3 litres. p and q are two glass cocks, ‘the latter being attacked to a side tube leading to a Gaede auxiliary pump. All the stopcocks deseribed above are lubricated with Vakuum-Hahnfett and dipped in small mereury baths. The end of the delivery tube r is brought in contact with the tapered end of the compressing cylinder s at v and covered with a thick rubber tube, and this connected portionis dipped ina merenry bath uw. The portion extending from the generator to the end of the delivery tube is wholly made of elass having no ground joint, except the portion of the stop-cocks, and is therefore entirely safe from the leakage of gases for several months. As the rubber connexion between the compressing cylinder and the delivery tube was used only when the compressing cylinder was to be filled with the gas, the diffusion of air through the rabber tube was negligible ; and the gas in the compressing cylinder was proved by an experiment—which will be described later—to be perfectly free from air, so that [ could safely use this connexion throughout the whole experiment. (b) Filling the Measuring Tube with the Gas. The method of filing the compressing cylinder with hydrogen is as follows :— To drive oft the air contained in the purifying train and the reservoir, a current of hydrogen Baten ake! in the decomposing cell, at the rate before described, is passed for about two days, while the palladium asbestos is heated to 900° C. After almosi wll the air has been driven out of the of Hydrogen and some other Gases. 337 apparatus with the hydrogen, the stop-cock & is closed, and the gas contained in the bulbs m,n, and o pumped out by the Gaede auxiliary pump, and then the cock q is closed and & opened to introduce the hydrogen into the bulbs m, n, and o. When they are filled with gas the evacuation is again made, and then the new gas introduced. After the same process. has been repeated several times the cock p is finally closed and the reservoirs are filled with gas till the pressure of the gas reaches about 2 cm. higher than the atmospheric pressure. This pressure is attained when the surface of water in the cylinder 6 of the decomposing cell reaches the lower position, as shown in fig. 8. To fill the measuring tube and the compressing cylinder with gas, the tapered end of the latter is brought in contact with the end of the delivery tube, and then the air in them is pumped out. When the cock q is closed and the cock p opened, the gas stored in the reservoir rushes into the evacuated cylinder, filling it almost instantaneously at a pressure of about one atmosphere. Next the cock & is opened and new gas supplied to the reservoir, till the former pressure is again attained in about 15 minutes. Thus the cylinder and the measuring tube are always filled with hydrogen at about one atmospheric pressure ; they are in an evacuated condition only for a moment, when the communi- cation to the pump is stopped by the cock g and the cock p opened to introduce the gas into the cylinder. Thus any diffusion of air from outside through the packings and connexions is completely prevented. The same process of washing the interior of the measuring tube and the com- pressing cylinder with pure hydrogen is usually repeated tive times and the gas finally introduced into the tube and the cylinder is employed for the measurement. The method of measurement of the mass of the hydrogen was to determine the mass by replacing water in a eudiometer with the hydrogen stored up at high pressure in the measuring tube. If the pressure and ’ temperature at the time of “Avemeeie! of the volume are known, then by taking the vapour pressure at that temperature into con- sideration, we can calculate the volume of eas at standard conditions. Multiplying the density at 0° C. into the volume obtained above we obtain the mass of the gas. In some eases the determination of the mass of hydrogen was also made by weighing and comparing the weight with the results of the Siete nae: measurement. Phil. Mag. 8. 6. Vol. 39. No. 231. March 1920. Z, 338 Mr. Také Soné on the Magnetic Susceptibilities (c) Results of Magaetic Measurement. About thirty independent observations were made for the pure hydrogen gas; the following table contains the data for the magnetic susceptibility of the gas obtained in twelve sets of measurement which give the most reliable results. In the table, m,.t,, Xu,:"ay tq, and Yq denote the masses, temperatures, and specific susceptibilities of hydrogen and air respectively. The experiments are arranged in a chronological order. TaBLE XI, INGE (9G ay Oo Hy T Hy’ qs. Mm» Cs aaa H|Xa nee 10°. 1 —4:63 0°0134 18°5 13°47 0:00327 13°:0 0-0840 2°052 2 —273 00076 15°:0 14:92 0°00332 15°2 0-0803 1-945 3... —5°34 00127 15°°5 17-22 0:00333 15°92 0:0815 1-975 4 —4:44 00143 18°°0 1825 0:00468 17°22 00791 1-903 5... —513 00161 18°0 1832 0:00464 18°:2 0:0805 1-931 6... —495 0:0156 20°°0 1734 000455 19°-2 00834 1994 1.. —4£:99 O0157" -20°:0 17,00) (000456 26°°0 10,0555 2-039 8... —422 00129 22°:0 18:28 0:00450 21°'5 0:0805 1-910 9... —3°85 00124 23°°0 16°51 0:00450 22°°0 0:0846 2004 10... —485 00149 26°°2 16:34 0:00488 27°°0 0-0872 2-030 11... —453 00144 22°°5 1664 0:00454 22°0 0-0860 2°038 12... —468 0°:0156 25°-°0 16:19 0:00450 23°°5 00834 1:965 (Mean) 1:982 +0°015 As we see in the above table, we get as the mean value of the specific susceptibilities of hydrogen obtained from twelve measurements at a mean temperature of about 16°C., Xq, = — 1°98, x 10-5 + 0°01; x 10-8, the mean error amounts to 0°76 per cent. of the total value. Multiplying the density of hydrogen at 20° and 760 mm. pressure, we get as the susceptibility per unit volume of hydrogen at the normal pressure and 20° C., Ky, = —9°000165, x 10~° + 0:000001; x 107°. The following is an example of the data of the magnetic measurement of gaseous hydrogen :— of Hydrogen and some other Gases. Ba Papin XL. June 24, 1918. EL, (p=00 atm): Vacuum. 2h 49™ p.m, £=20°°0 C., 6=10°0. 4h 427 pw, b6=10°0. C. S. é. C. S. é. amp. cin. cm. amp. cm. cin. 0 50°50 3°60 0 48°50 9°10 10 54°10 310 10 57°60 7°10 0 51:00 3°60 0. 50°50 9:00 10 54°60 2°88 10 59°50 7°08 0 51°72 3°58 0 52°42 9:08 10 55°30 2°80 10 61:50 715 0 52°50 3°40 0 54:35 8°95 10 5590 272 10 63°30 CANS: 0 53°18 3°21 (mean) 0 56°15 8°08 (mean) H,, (ditto). VACUUM. 3° 5™ p.m., 5=10-0. 5h 3m p.u., 6=10°0. C. S. 0. C. S. é. amp. cm. cm. amp, cm. cm. OU 44-50 3°25 0 46°85 8°35 10 47°75 elo 10 55'20 7:90 0 44-60 3°12 0 47°30 8°40 10 47-72 2°92 10 55°70 760 ) 44-80 3°00 0 48:10 8°80 10 47°80 3°00 10 56°90 7°60 ) 44:80 - 3:00 ) 49°30 8°90 10 47°80 3°05 10 58°20 745 0 44°79 3°45 0 d0°75 8°15 (mean) 10 48-20 3:08 0 45°12 3°10 (mean) Totalican 2.2 J23.0- o1l6 Motalimeanryssss4.5: 811 Volume of gas=186'l c.c. at 219°2 C. Baro. press. =753'25 mm. at 21°°6 C. Air (p=1 atm.). CALCULATION. t=21°°5 C., 5=1000, dy, =3'16-—8'11= —495 em., Baro. press. —752°0 mm. at 219-3 C. tg ir one nae d,=25:45—-8:11=17 34 em., C. S. é. itz, =0°01556 gm., amp. em. em. 0 44-95 24°75 m_,,=0°00455 gin., 10 69:70 25-05 = 4: 5 K - 10 70°10 25°10 0 45-00 26:00 184K), 10 71-00 25:00 o> qe eo eae 0 46-00 26:30 Ny ae 10 72°30 25-05 eee 6) 0834. 0 47-25 26-75 ie ESA 10 74:00 25-00 0 49°00 25°45 (mean) Z 2 340 Mr. Také Soné on the Magnetic Susceptibilities (dl) Purity of the Hydrogen Gas. Morley * stated that the nitrogen which was present in the hydrogen obtained by his arrangement was. less than 1/20000 of the hydrogen, and that the hydrogen is quite free from oxygen. In my apparatus constructed on the same principle as that of Morley, a similar result might have been expected. As the volume susceptibility of the atmo- spheric nitrogen is twice as large as that of hydrogen, the presence of that amount of nitrogen can produce no sensible change of susceptibility. But the oxygen which might be present in the hydrogen owing to an imperfect purification may be expected to have a serious effect on the value of the susceptibility of the hydrogen. If we take as the volume ee of oxygen and hydrogen, the values Ko, =0'189xX10~° and «Kz = —0°000166 x 107° respectively, we see that the susceptibility of oxygen is 837 times as large as that of hydrogen, and hence a 1/837 volume of oxygen present in hydrogen will cance! the diamagnetism of hydrogen, and a 1/150000 volume of oxygen will diminish its diamagnetism by 0°56 per cent. This is the the same order of magnitude as that of the experimental error in the measurement of the magnetic susceptibility. Hence at least the upper limit of the oxygen content must be known. From the fact that the molecular silver placed in the path of the hydrozen did not show any change in appearance, we know that no sulphur compound was present in the gas. For this purpose an apparatus as shown in fig. 9 was constructed. a is a eudiometer with platinum wires sealed near its closed end for the electric discharge, and 6 a glass tube forming a U-tube, the shorter arm being inserted into the eudiometer with a rubber cock. The longer arm is connected to a glass tube e with a rubber tube ; the tube eis provided with a bulb / containing calcium chloride, which is connected to a long rubber tube g. The audionerss is first | filled with mercury, ‘and then replacing it w ith the hydrogen to be examined, the connexion of the eudiometer with the tube ) is made in a mercury bath. The eudiometer with the tube is then brought in the tank ¢ and supported firmly in the vertical position. The tank has a capacity of about 160 litres and is filled with water. The euciometer and the tube are immersed in the water, leaving only a few centi- metres of the upper end of the eudiometer above the surface. * HK. W. Morley, loc. cit. of Hydrogen and some other Gases. 341 Another eudiometer, which is also filled with hydrogen, is placed side by side; it serves as a control for the change of the atmospheric pressure and the temperature of the tank. The connexion of tubes 6 to tube e is then made by a rubber tube d. By means of the microscopes Q, the head of the meniscus of mercury in 0 is observed ; the smallest division of the ocular micrometer corresponds to 1/66 mm. By repeatedly blowing the air at the open end of the rubber tube g, we can easily make the height of the meniscus in tubes b settle to a correct position. The height of the meniscus is then read with microscopes Q. When the tem- perature of the tubes b becomes equal to that of the water in the tank, an electric discharge is passed for 6 minutes through the eudiometer with an induction coil (30 cm. maximum spark distance) working at 12 volts. After an hour, when the temperature of the gas becomes again equal to that of the surroundings, the reading of the meniscus in the tubes is again taken. If the readings of the two micro- scopes undergo the same amount of change, it is to be concluded that the hydrogen does not contain an amount of oxygen which can be detected with the present apparatus. It, however, some difference in the readings of the micro- scopes be observed, another electric discharge is passed, and 342 Mr. Také Soné on the Magnetic Susceptibilities after an hour the observation is repeated. These processes are repeated so long as there is no relative change in the readings in these microscopes. The length of the eudio- ineter is about 50 cm. and its inner diameter 17 mm.; while the inner diameter of the tube 6 is 4 mm, Let us now consider the change of the Figs no! height of mercury in the tube 6, when a change of volume of the gas in the eudiometer takes place by electric dis- charge. Let S and s be the sectional areas of tubes a and 0), P and p be the initial and final pressures of the gas in the eudio- meter respectively (fig. 10) ; then if the meniscus of the mercury in tube a is raised by 6H, while that in tube 6 is lowered by 6h, we have the following relations :— SdH=séh, and P—p=dH+6h. Hence we have s P—p=bh(1+<), P—p__ oh 3 Me es If the quantity of oxygen previously contained in the hydrogen be very small compared with the total quantity of the gas and the hydrogen was perfectly dry before «ischarge, the water formed by discharge will remain as vapour, and © since one volume of oxygen and two volumes of hydrogen form two volumes of water vapour, the total volume change of the gas due to the discharge is just equal to the volume formerly occupied by the oxygen gas. Hence the term in the left side of the above equation is the ratio of the partial pressure due to the oxygen to the total pressure of the vas. Suppose 6h to be 1/66 mm., which is the smallest division of the scale of the microscope, then we have, from the above equation, P—p 1055 Lo ea Pe 665760 wm 27500 50000) Hence we see that if the height of the mercury meniscus in tube 6 does not change by more than 1/3 division of the of Hydrogen and some other Gases. 343 micrometer ocular, the partial pressure due to the oxygen contained in the hydrogen before the discharge is almost less than one-one hundred and fifty thousandth of the total volume. The volume susceptibility of this amount of oxygen is therefore OSS 3 LO Sane Lt cre This corresponds to about 0°56 per cent. of the suscept- ibility of the hydrogen gas. In the actual test for the content of oxygen in the hydrogen gas used in the magnetic measurement, no relative change of the readings of the microscopes exceeding 1/3 division of the ocular was observed. Hence we can safely conclude that the error due to the oxygen, which might be present in the hydrogen gas, is at most less than 0°6 per cent. of the susceptibility of hydrogen (Table XIII. B. &C.). Hence we may take as the correct value of the susceptibility of pure hydrogen gas at the ordinary temperature, the value obtained in the present experiments. The following tables contain the data of the discharge experiments for the test of the purity of the hydrogen :— TaBLE XIII. A, : Hydrogen before passing the Hydrogen after passing the palladium | palladium tube. tube heated at about 70° C. Reading of micro- Reading of micro- : meter. \Relative| -: meter. Relative mest change] mest | TT change : | oO : 0 | | mab: | Discharge |Control reading. vation: | Discharge Control reading. | | * tube! tube. tube. tube. 1b 5™ p.m. | 50 50 | 15 30™Pp.M. 50 50 | Qh 305 P.M.| 8 83 | gh 30™ P.M. 65 92 \Ghange of ge .6) Male) = ) |e) | reading 42 +33 | 15 +15 +42 Daf ( i j 344 = =Mr. Také Soné on the Magnetie Susceptrbilities B. ae after passing the palladium Hydrogen after passing the palladium tube heated at 150° C. | tube heated at 200° C. | | a | Reading of micro-| | Reading of micro- | neter. ivel on: meter. i Mgmecoh pi ion Rel imotot ( eee obser- Se | . ee obser- = | . Be Lat Discharge|Control \reading. | yanu Discharge|Control reading. tube. tube. | tube. tube. | | | eT Tn ie ) | | 6® 30™ p.m. 50 HO (0 45™ p.m. 50 50 | | | | | 7 30™ P.M. 10 | LOW (2 >5™ pow. Oi, 27 ES a bs LON RACE SE Speed et) | | | | Change of | Change of ae Seat | | reading. mee | Oe ee reading. me ae | ° C. | | Hydrogen treated likewise when it is subjected to magnetic measurement. ‘ Vice Reading of microscope. | | ‘ Relative | | es . change of avon: | Discharge | Control | reading. | tube. tube. | 62 3™ P.M. 90 90 | 7m, 90 90 | | | | Change of 0 0 0 | reading. The data contained in table B was obtained for the hydrogen directly filling the eudiometer from the delivery tube of the purifying train. The last example was obtained for the hydrogen which was previously compressed in the measuring tube at atmospheric pressures and then delivered into the eudiometer. The sign of the change of reading is positive when the volume of the gas increases, and negative when it decreases; and therefore the negative sign in the last of Hydrogen and some other Gases. 345 column indicates that some volume contraction took place in the tube which was caused by the electric discharge. From the above results we see that the hydrogen obtained by the electrolysis of dilute sulphuric acid containsa small quantity of oxygen diffusing from the anode to the cathode ; that the hydrogen passed through the palladium tube heated above 150° ©. does not contain more than 1/150000 volume of oxygen, and by table C., that no air was allowed to enter into the hydrogen during the process of compressing the gas into the measuring tube. § 10. ConcLtupine Remarks. The present experiment is a relative measurement of the susceptibility of gases, in which the susceptibility of pure water is taken as —0°720x10~-°. In the measurement of the susceptibility of air, redistilled water was used as the standard substance; for the cases of other gases the air at the ordinary or at some high pressures was used as the substance for comparison. The specific susceptibility of gases is assumed to be inde- pendent of pressure, at least in the range of pressure (1 to 68 atmospheres) in the present experiment. This assumption was tound to be correct within the accuracy of the experi- ment by the fact that the value of the susceptibility of each gas was the same irrespective of the pressure applied, The values of the magnetic susceptibilities of the gases investigated in the present experiment are summarised below :— Taste XIV. GaAs X- 10". «10°. 1 aeRO Lp SAD ACE E Si Bein Ae is 8 aes +23°8.. +0:0308, (85 6 f12) 1 RRR Nees och Ea a ae +104, +0°148, Carbon dioxide: -cesespeeeenase ee tee —0'42, | —0-00083, ’ Nitrogen (chemically pure) ............ — 0°26, — 0:00033, Nitrogen (atmospheric) .................. —0°36, —0-:00045, (ATO ON », (22.-inaipgasen sess t ee eee tay oe —5'8, —0:010, Ely GrOGOM sss doccsecseeoeeaeeeaeee ee — 1:98, —0:000178, In the above table the specific susceptibilities are referred to the state at 20° C., and the volume susceptibilities to that at O° C, and 760 mm, pressure respectively. 346 Mr. Také Soné on the Magnetic Susceptibilities The fact that the susceptibility of air directly determined shows a close coincidence with the value calculated from the susceptibilities of its constituent gases, proves not only the correctness of the values of the susceptibility of each gas, but also that the additive law holds for the susceptibility of a gas mixture. Hence, in deducing the susceptibility of air from that of oxygen, it is not correct to neglect the magnetism of nitrogen and argon. The magnetism of these two gases contributes about one per cent. to the total magnetism of air. It has long been believed by many investigators that nitrogen is paramagnetic, with the exception of Pascal * who obtained a diamagnetic susceptibility for this element by an indirect method. His value corrected for the susceptibility of water 1s Xx, = 220-39 filled with cadmium alone. The map * of the spec- truin of gallinm given in figure 2 is interesting. The wave- -lengths and intensities of the lines are given in “Table I. in this connexion. It will be observed that there are but five lines in the visible spectrum and that from practically 4200 A to 6400 A there are no lines. 6000 5000 4000 3000 * der & Valenta, Atlas Typischen Spektren. es Or Cfo On a New Cadmium Vapour Are Lamp. (Bin Visible Spectrum of Gallium. Wave-length Intensity. J. 3020°61 5 4033°18 10 4172°22 20 6396°99 8 6413°92 6 When the lamp is operated at a temperature sufficiently high to bring the quartz to a cherry-red colour, and there is danger of softening the lamp, several gallium lines become faintly visible. The investigations of Uhler and Browning * indicate the possibility of two gallium lines 5353°81 A and 5359°38 A. However, these lines, if present, are so faint at the highest temperature at which the lamp can be operated, that they cannot be identified. The cadmium spectrum is thus obtained in a condition exceedingly favourable for those purposes for which an intense monochromatic light source is indispensable. No gallium lines are found between 4200 A and 6400 A, and the gallium lines which are detect- able have so low an intensity that they are wholly nee in polarimetric and other fields of work. There are now available practically no dependable intense monochromatic red light sources. Any source to meet modern demands must permit of continuous operation with minimum amount of attention and an absence of flicker. The very pure red line (1=6439 A) of cadmium seems to be the only possible source of sufficient intensity in this region of the spectrum. It is believed that the cadmium-gallium lamp will make this much needed source, as well as other lines of the cadmium spectrum, available for many lines of endeavour. The writer desires to acknowledge his indebtedness to Nik. iP. Phelps for valuable assistance in the experimental work. Bureau of Standards, Washington, D.C. * Amer, Jour. Sci. xlii. p. 889 (1916), XXXII. On Radiation from a pe wcal Wall. By 8. P. Owen, B.Sc. Wales * ‘o the experiments conducted by Todd (Proc. Roy. Soc. A. vol. Ixxxiii. (1909)) on the thermal conductivity of gases, the problem of the effect of the radiation from the cylindrical vertical wall on to the horizontal lower disk arose. The effect was not calculated but was eliminated by experimental means (T.¢..p. 20). In the following paper an expression is deduced for the amount of heat radiated from a cylindrical vertical wall similar to the insulating ring in Todd’s experiments, to a horizontal circular plate ‘placed near the bottom. The conditions are slightly different from those obtaining in Todd’s experiments, in that here the temperature of the wall is assumed constant whereas in the insulating ring there is a linear gradient of temperature from the top to the bottom. Very little alteration is needed in the calculation to fit these latter conditions. The result is tested by obtaining values of the Radiation Constant by a simple experiment “which makes no claim to great accuracy. The values obtained agree closely with the accepted value. The experiment is described in the second part of the paper. 8 lL. Theoretical. Let a=radius of cylinder, b=AO=radius of the copper plate. Consider two elements of surface, one at G and the other on the plate at F. Let 6 be the angle between two vertical planes, one con- taining GN the diameter of the cylinder through G and the other GH, a line in the same horizontal plane as GN, the plane cutting the plane of the plate in the line DF. Since the plate is symmetr ically placed with respect to the cylinder, evidently EDC=8. GF is the line j joining the two elements at G and F making anole @ with GH. Taking D as the origin let DF =r and DG=.: * Communicated by Prof, G. W. Todd, D.Sc. 360 Mr. 8. P. Owen on Radiation If N=normal radiation from the element at G, the total radiation received from G at F _ Neos@cosdsing.r.drdé fe de A ’ where ».drd@=area of the element at F, dude =area of the element at G; de being an ele- ment of the circumference of the cylinder, and d =KG. This will be absorbed by the disk if the latter be a perfect absorber of radiation. The total radiation received by the whole plate from the whole cylinder aay) er, ( le B sess ma raz an 2 (CPFN cosOcosd.sind.rdrdé.dx.de i 2 « 0 2, 0 2’ DA pe ge ner) whereiw—= DIK and ¢,— DM, from a Cylindrical Wall. 361 In I., taking OAB=a, DA=a cos 0—/ cos 2, DB=acos6+h cos a, and bsina=asin 6. DA =a cos 0~\/B?—a? sin? 0 and — DB=acos 6+ /l?—a? sin? 0. rake ae : Se tales, In L., cos = [= Tent sin d@ = ae: Hence (1) becomes h PD) v ti z 2m ay gtr es Necos@.ardr.dé@.dz.. dc (a7 nt) ty eI Integrating with respect to ¢ and then to r we get b Sythe "25 Vsin, ling > x cos OT. DB DA laraN vd v is | tan ay Gi ~ 7th el ) uw : _DB DA —sin2 tan7' a + 4sin2 2 (tan =) | dé. Using the above values for DA and DB we get after reduction , DB DA. Qe/b?— a? si sin? re (SR Se ee ie ee v BY a ae = ian « therefore as part of the integral with respect to 6 we get L FESiN ere VA cos 6 tan7 eV i? — a sin® °8 1 « 0 4 be 24 2— fp? eed ("% a Ad sin? 6 cos @ 16 we si nest RG ye nf Gee —«a? sin? mz OS1 rf A262 — a” sin? 20)} by integration by parts, Qe where A> az? +.q?—b S. P. Owen on Radiation 362 Mr. Putting asin @=bsin ¢, this reduces to 2 AD? sin? | bcos b.do bcos (1 + A26? cos? @) (Ch a a, let Alb eos ae Fe {1/ VSRAe ee TT a == Teed (2 +te—PyP 4 4h? ee +0 — ?)b ae Sun Sera DB Taking the part { cos @.sin 2. tan7* — dO, us 0 eee. ERO mee ON a cos 8 +y/U? —a’ sin? @\ ct —— nea e dé, tgin a t. @. cos 6 sin 2. tan7 ue . : e/ (9) and putting a cos 0 +4/ 1? ~a? sin? O=y, the integral reduces to Vath y + @? 2 2 ( ae | 2a ly : dy. Ngee V 4074? — — (i + a2— b?) ae 2y Car ih)) aoe \ Similarly \ cos @ sin2.tan7’ — .dé 0 U a—b Deny on ie aly 24 =| il (y- a ean ty Veen 402? — (4? + a? — BP)? by ae 0P) we SILI ; B af (sin 2. tan” a sin 2. tan7' oP eos d.dé 0 Ww Ho by Lal aa 2ay (ye +e? — 2") ; = + dy app a(v? +y") )V/4a%, y—(y+a—b ia ay + a? —b)? F 2 ia Jars ay(2? + y)r/ day? — (y?—a? +B) from a Cylindrical Wall. 363 In the first part put y?—a’—b?=2ab sin 0, and we get —, SS Se ih). Mia tact 6 2absing Tr ( 22x a’? +ab sin @ which gives after ana, mal oeee ae it. eS) Tae +6)? — dab? Using the same substitution in the 2nd part, 7. e. : Ya—h “ly? aL a—b?)? Se 3 seh 2 5) 2.9 9 >, 7a\9 dis Jars ay(a? +y?)v/ Lay? — (y?—a? + 07)? we get a (° — dat + 8a°b sin 0+ 4a°b? sin? 6. = 3a 2@ +12 +2absinO)(a?+a°+l? 4 yaismg) Lae i ( Pea. ey C an = s 18 ae + 6? + 2ab sin 6 x? +a? +b? + 2ab sin at i 2 where B= Coie 2 ate and C=— — which gives after integration eae se, u 2a | a? — |? J (2?- +02 +0? ae | The whole integral (1) _4naN un sf POE INI AD 9 irl wept 9 : Te: V/ (a +a —b ) + 4h tb — (a +a — b*) f Td SS phe abet oh SIA NOSE -+] t] a Ee V2 +o: 24 4? 4q2h § Te B C 2a 1 a* — 6? NV (e +a2 + B2)?- — 40h? isi Now (a? +a?—b?)? 4 4022? = /(e? +02 +02)? — 4070", Using this and substituting the values for B and C the integral reduces to —N ae (ie ee a(e* + a? + U*) et Us Le We +024 07)? —4a7b? Ga 364 Mr. S. P. Owen on Radiation Putting w?+eC+b?=y, the first part =+ ergs li 1e first pat Vp ae = 4 [4/7 he? | . Total Inte oral l N or? Peis Bz ene Tg! oa ee +a? +b7)?—4a?h* When the wall and disk are “ full” radiators then New aed, ] 1), a) an ; where R is the total radiation, « the Radiation Constant, T, and T, the absolute temperatures of the cylinder and (hele respectively. .. Amount of heat received by the disk TO Sey TTA / alte +e Pier —/ (a9? +0? +07)? — 4¢25? — aye t+. we |b Bo (5)) § 2. Huperimental Verification. Using the above result, the radiation constant o was determined by the following experiment : The apparatus consisted of an ordinary steam jacket, the inside of which was covered by an even layer of lampblack obtained by the burning of camphor. This was fixed in a vertical position. The amount of heat radiated to a copper plate .was measured by means of a modification of Bunsen’s Ice Calorimeter. The plate consisted of the bottom of a copper calorimeter, enclosed in a block of wax, the bottom being exposed, Taps with the surface of the wax. A capillary tube, previously calibrated, was fitted through a rubber stopper to the top of the calorimeter. The inside of the vessel was filled with crushed ice and water, and by fitting in the stopper the capillary tube could easily be filled with the water from the vessel. By observing the rate at which the meniscus in the capillary moved, the amount of contrac- tion and hence the amount of heat given up to the ice in unit time could be calculated. from a Cylindrical Wall. 365 In order to eliminate the heat radiated from the sur- roundings, a guard-ring filled with crushed ice was used. This was made so that the plate could be completely cut off’ from exposure to the cylinder, and by means of an aperture equal in size to that of the plate, it could be exposed to the cylinder at will. . Observations were taken at half-minute intervals, a few minutes with the plate covered, then exposed, and finally with the plate covered again. The mean of the first and last set of values was subtracted from the mean of the second set and thus the heat radiated from the cylinder was obtained. In the experiments, which were conducted with two cylinders of different heights, the diameter of the plate was practically equal to that of the cylinder, thus putting in the expression (5) w=, it simplifies to: Ly lal =o(T'—T.)5 | n/a? + dae ie \ This expression was used in the calculation : Average radius of plate and cylinder ......... == 739) OMe Volume of ¢: ipillary per em. length ............ = OD Oilierc: Mean contraction per minute lien plate was covered ........ caged tials ail PN ea Oe ===! C7) CIM. Mean contraction per minute when plate wa EE pOSed) Shatummereer Mas eee ue ==1C5) CHM T.. T. v4. X,. C,. Cy e—e,. oX105, 373° abs. 273° abs. L-lem. 23°3cm. 2°38cm. 2°88cem. ‘d8em. 4:88 wee oe) | 265, 176.4, Ones eto oa TS. N12 ar 38. ys AO The accepted value of c=5°32 x 10~* ergs/cem.? see. dew.! In conclusion. I must express my great indebtedness to Dr. G. W. Todd, who suggested the problem and whose advice and eriticism has been inv: aluable in the experimental work. lioval Granumar School, Newcastle-on-Tyne, sept. 1919, Te 86o. XXXII. On the Measurement of Time—a Rejoinder to Dr. N. Campbell. By i. SILBERSTEIN, Ph.D., Lecturer in Math. Physics at the University of Rome * FYXHE paper on “A Time-Scale, etc.” by the present writer, published in the September issue of this Magazine, opens the investigation by a statement that the principle of common time-scales amounts to this :—A certain kind of motion (translatory or rotatory) is declared to be a uniform motion; the path is then cut up by means of compasses etc. into a series of equal seyments (or angles) and the instants of passage of the mobile through the divisions f this metrical scale are taken as £=0, 1, 2, 3, and so on. Dr. N. Campbell, in the November issue of the Phil. Mag. (pp. 652-4), believes “this statement to be untrue.” It will be my duty to show that it is true. In the second place, Dr. Campbell believes ‘still more untrue” (as if truth were liable of gradations) “the statement implied, that time- measurement is impossible except by some such artificial and elaborate method as he [Silberstein] proposes.” Now, concerning this second point, | have not said nor meant to imply that other methods independent of space-measurement were impossible. I simply proposed one, without excluding the possibility of other methods being invented by others. Thus I have nothing more to say about this second point. A third point, however, is that Dr. Campbell offers us his own views on the meastirement of time, and these are so palpably unsatisfactory as to require but a few words to be refuted. But let me first attend to the first point. Now, my state- ment, quoted at the outset, is not only logically true (tl rat is to say, that a theory of chronometr y based on *‘ uniform? motion apd paths or angles carved up into “equal parts would be a possible logical theory), but also, which is of great importance, historically true, the two principles, uniformity and rigid subdivision or transfer, being the dominant and basal features of ever y practical chronometry since times immemorial and up to, and including, our own days. In fact, the most ancient measurement of time, as practised by Babylonians, Assyrians, Egyptians, and whom not, was based on the assumption of uniformity of rotation of the heavenly sphere round the Earth, and on a rigid, metrical subdivision of the angles invo olved in this pheno- menon. Massive columns were erected and carefully kept * Communicated by the Author, On the Measurement of Tome. 367 for this purpose; later on, up to our days, sun-dials were constructed, and improved with the aid of Kuclidean geo- metry. And the jirst step (not the last, as Dr. Campbell thinks) was here emphatically the picking out of some ar andiose phenomenon and declaring it to go on or to evolve “uniformly,” ‘“equably.” Nor did these principles of chreno- metry suffer any serious shock from the great Copernican reform. Somehow our forefathers chose to declare the Harth’s revolution round the Sun and its spinning motion about its own axis as ‘S uniform,” and continued to subdivide the associated angles. Manifestly the sun-dials, or their prototypes, continued to show the hours in spite of the modified standpoint. Yet these natural solar clocks had their bad side, which perhaps is best expressed by the old and beautiful words to be still read on some sun-dials in Italv : Horas non numero nisi serenas. Other time-keepers were, therefore, invented and con- structed in very early times, thatis to say, even much before Copernicus, whom we vieninemed only incidentally— and in all of them the said two features play ed a dominant role. I do not propose to enumerate here all such old chrono- metrical devices ; nor have I the required historical erudition. Bui one such device attributed to Alfred the Great, who ruled over the West Saxons (871-901), I cannot pass here in silence, since it seems particularly characteristic in relation to our subject. According to what my little boy heard in his school *, Alfred the Great had good tall candles (of what stuff I know not) made for him, and, confiding no doubt in the uniformity of their burning down, divided them into equal segments, and thus knew the time in day or night. But apart from the “nisi serenas”’ condition, the solar clocks had the defect of not being applicable to short tine spans (certainly not to our “s econds,” and not even to our ‘‘ minutes’), and the other \panatre kind of natural time-keepers, the human heart or ‘ pulse,” was too often atfected by passion or disease to retain permanently the title of uniform (here uniform succession of discrete pulse-beats). Thus the medizeval physicist and astronomer had recourse to a variety of artificial chronometric devices. Even a long time before the Renaissance complicated wheel machines were constructed as clocks, but none of them was * well regulated ” until the times of Galileo and, more especially * [have no other means at the moment to verify the historical truth of this report. 368 Dr. L. Silberstein on the Huyghens. Properly speaking, these “ chefs-d’ceuvre (’agencement cinématique de mouvements,” as Jules Andrade calls them, were not ‘“ regulated” at all, t. e. were felt not to be worth the name of ‘ ‘uniformly going,” not keeping pace with the heavenly clock. The now undisputed merit of con- structing the first clock in the modern sense of the word is due to Huyghens, although it was Galileo’s discovery of the isochronism” of small. pendulum oscillations which he utilised in such an ingenious way. Yet, before Huyghens’s invention, Galileo, who was the first to measure compara- tively short time-intervals, constructed his own clock for the sake of his famous investigations on falling bodies, a water- clock that is, but more precise than the water= or sand-cloeks and the“ mechanical” clocks which he inherited from his predecessors. (ralileo’s own clock is, in the present con- nexion, as instructive as the burning candle of Alfred the Great. It consisted of a vessel or water-basin of large section having a very small hole in its bottom, to ensure, no doubt, the ‘ uniformity ” of the outflow of that liquid. This was his first care. The remainder of the procedure was again in full harmony with our statement ; Galileo measured the volume of the water (by weighing it, that is , but this only to make the vole measurements more precise), and he spoke of t=1, 2, 3, ete., as proportional to the number of equal volumes of water; this is equivalent to measuring lengths along the axis of a w ‘ell-calibrated and narrow ey inden vessel, if he had one. Galileo’s times (the ¢ in his great law s~#?) were proportional to these volumes or ultimately lengths, read on a metrical scale. That the same principles can ane instantly traced in all our modern clocks, watches, and chronometers, needs scarcely to be insisted upon. But they occur perhaps in their purest form in those modern instruments which serve to measure very short times, even down to one-millionth of a second, and perhaps a little less. I have in mind Siemens’s high-speed spark chronograph. It consists, in essence, of a little revolving drum of good steel driven by a carefully finished clock-w ork. Against this drum, which we used (1897) to cover tightly with a strip of paper blackened with a turpentine lamp » Zoot, is mounted an isolated platinum electrode. Sparks corr elated with the events in question pass between the platinum point andthe spinning drum and leave marks (little craters) on its blackened surface. The ae ork is then stopped and the drum turned round slowly by a micrometric screw, while the marks are viewed through an appropriately placed microscope. Their angular distance, as read on a subdivided Measurement of Time. 369 circle of the hand-screw, gives the time-interval between spark and spark. This is the ‘‘space-measurement” ; and the ‘‘ uniformity 7 was most emphatically expressed in a letter of Siemens and Halske accompanying the SOS to this effect': If you wish to obtain satisfactory results, do not start the sparks at once but only after the drum was already spinning for a good while, a prescription, no doubt, based upon the makers’ dynamical knowledge of the driving machinery, but at any rate a direct appeal to what had to be trusted to be uniform beforehand, without in this case the least possibility of checking the uniformity by investigating, as Dr. Campbell says in his concluding paragraph, whether the “body covered equal distances in equal times,” the ‘equality ” of these minute intervals being in this case not otherwise actually definable, unless one appealed to yet another uniformity, viz. that of the propagation of electro- magnetie waves along the wire-systems*. I have dwelt upon this example, not only because it shows the two prin- ciples in their neatest form, but also because in writing my first paper on the time- scale, I had this spark-chronograph incessantly in my mind. The same remarks can literally be repeated with regard to all the familiar devices in which the drum is replaced by a light rotating mirror used as reflector. Very minute time-intervals are thus being measured and give well consistent results. But intervals still much shorter, the periods of light- oscillations, are measured again on the same principles. The pr opagation of light is declared to be uniform, and then linear segments (translated more or less indirectly on a magnified scale into an interference pattern) associated with this propagation are measured in the Euclidean fashion. And there is even an ever-growing « priori confidence into the uniformity of light-propagation and a tendency to make it the highest court of appeal for all properly mechanical uniformities. In short, every precise chronometry is kinematical (motion of bodies or propagation of light), and the foremost concept of kinematics is that of “uniform motion,” exactly so as is that of “straight line” in geometry. Both are, theoretically, undefined terms, and in application things are declared to be good approximate samples of either or - pointed out (with yes as it were)—this or that is unitorm, this or that * The interval between the sparks was, in the application of the chronograph I have in mind, due to the difference of the two corresponding circuits, one very short and the other about 3 km. lone. Phat, Mag. S. 6. Volaad, No: 231. March 1920. 2 8B 370 Dr. L. Silberstein on the is straight. There is no defining of “straight TO neriiot ‘‘ yniform.” All so-called definitions of these terms are but apparent, each of them containing a vicious circle. A definition of uniform motion such as Dr. .Campbell repeats after the naive little text-books [to wit: “‘ we define uniform motion as that of a body which covers equal distances in equal times,” p. 654 | would be exactly as bad as: a straight line is that which slopes down or up (relatively to another straight!) by equal heights in equal horizontal distances. It is precisely such a ‘definition’? which prevents most people from seeing the possibility of non-intersecting, Lobatchevskyan straights and the hypothetical nature of Euelid’s parallel axiom. And the kinematical correlata of these things are made manifestin my first paper, showing the possibility of a generalized (hyperbolic) system of kinematics. ‘The analogy between “uniform” and “straight” becomes still more manifest if one thinks of the modern relativist’s four-world, in which a “ straight’ stands for a space-straight as well as for uniform motion or propagation. But there is no need to appeal to that famous “* union ” of space and time to show the fundamental, irreducible character of uniform motion ; this character belongs to it historically, since times immemorial until our days. Both the assumption of uni- formity and the rigid subdivision of the paths or angles are inherent in all the more precise chronometric methods ever devised by man. | This settles the first and chief point of the present note. It is scarcely necessary to add that in declaring such and such a phenomenon to go on uniformly the physicist’s, or the astronomer’s, choice is, among other things, based upon reasons of convenience, aiming at a certain kind of simplicity of laws or differential equations, such as I attempted to explain in the introductory chapter of the “'Theory of Relativity.” The second point being already settled at the very beginning, let us pass at once to our third point, concerning that is Dr. Campbell’s own views on the question of the measure- ment of time. Dr. Campbell is under the fatal misappre- hension that he requires but ‘‘ three definitions ” in order to set up a system of measurement of time | nay, of any other magnitude }. These are his (1), (2), and (3), p. 653. - The second concerns only the equality of two time-intervals whose both the beginning and the ends coincide, and the third fixes only, in the usual way, the meaning of the sum of two adjacent (consecutive and gapless) intervals. They need not detain us any further. The whole burden is loaded Measurement of Time. Bild upon (1) which reads: “The period occupied by the happening of some definite process in a definite system is defined to be 1.” Now, if this ‘definite process ” stood for a single particular process covering a single Ree te of time, obviously nothing could be done with (1), (2), (3). It would amount to as much as giving on a ca a pair of points, O, A, calling OA the “segment 1,’ and declaring any segment ‘LN to be the sum of the (adjacent) segments LM and MN. This would never enable us to say what is a segment 2, or 3, and so on. ‘But, if I eal grasp his meaning, Dr. Campbell under- stands by “definite process,”’ a process such as a fall or a complete oscillation, happening now, or five minutes hence or to-morrow, and so on. This, however, amounts not to solving the problem of measurement, but to massacring it at its very root, or else it amounts to a concealed assumption of uniformity of (in this case) the succession of a discrete set of events. To make my meaning clear, let that standard process be a complete oscillation of a pendulum (to-and-fro) marked by an audible click at its beginning and at its end. Then Dr. Campbell defines all the intervals between a click and the next click as equal to one another, zeroth to first = first to second=fifth to sixth, and so on. But this means either the setting up of an indefinite series of entirely arbitrary time-scale divisions, or else contains the tacit belief in the uniformity of the succession of the clicks. Such a procedure per se would not deserve the name of chronometry ; it would be chronoscopy pure and simple. I say, per se, v. e. without relating the pendulum-oscillations to some fundamental kinematical and dynamical principles. With such support the scale would cesse to be merely chronoscopic ; but then it would indirectly rest upon some continuous uniform motion as the fundamental concept of the very science (mechanics) which is its support. If so, however, then it is preferable to utilize directly a uniform motion (instead of a uniform succession of discrete clicks), say, a uniform spinning—which brings us back to both of our old principles. In_ fact, Huyghens, who certainly pre- ceded every bosy in applying the pendulum to chronometry, used it only as an auxiliary, intervening in his mechanism at discrete instants, and he milived for rigid subdivision the continuous spinning motion of his wheels. (Such also is the only réle of the pendulum in cur modern clocks.) That our last remarks are by no means superfluous can be seen from Dr. Campbell’s embarrassment when, having dealt very rapidly with “integral numerals,’ he looks for fractional ZB2 312 Prof. Joly and Mr. Poole: Attempt to determine intervals. ‘‘ The fractions ”’—he says—‘ can be obtained by other pendulums,” with my italics. Thus, other and other pendulums are to be declared as fresh standards (for instance, for t=4 we should require a smaller pendulum, such that its three oscillations just fill out the interval between two clicks, ior t=, yet another, and for what Siemens’s chronograph yields, a pendulum of ultra-molecular dimensions) ; thus the postulate (1) would have to be extended and enriched without any end. (Moreover, the ‘fractional’ pendula could only be found by endless trials, for Dr. Campbell’s set (1), (2), (3) does not yield a method of constructing the required sub- divisions. Nor is, of course, such a scheme adaptable to any somewhat refined chronometry.) Is this satisfactory? Is such a set as Dr. Campbell’s (1), (2), (3) satisfactory as the basis, logical or physical, of a theory of time-measurement ? I think not. Moreover, Dr. Campbell believes (1), (2), (3) to be good enough for a theory of the measurement, not only of time, but also of any other “magnitude.” He quotes time only as a little example. Now, temperature is certainly an example of “ magnitude,” and better still, length or distance is another, and it would be extremely interesting to see Dr. Campbell setting up an intrinsic scale in both of these cases, most especially in the latter one. The psychological clue to all fallacies of Dr. Campbell is contained in his concluding sentence: ‘‘Of course, this is all as elementary as ABC.” If this were so, gigantic mentalities, such as was Cayley and many of his successors here and abroad, would never have devoted so much time to what is known as the Theory of Distance. : November 4, 1919. XXXIV. An attempt to determine if Common Lead could be separated into Isotopes by Centrifuging in the Liquid State. By J. Jouy, F.R.S., and J. HW. J. Poonn, BATE NINCE it has been discovered that both the Uranium and Thorium radioactive families yield elements which are isotopic with ordinary lead but differ from it slightly in atomic weight and density, it has often been suggested that common lead itselfis not a homogeneous elenient, but consists of a mixture of isotopic uranium and thorium lead. This view of the constitution of common lead is based on the fact that * Communicated by the Authors. if Common Lead could be separated into Isotopes. 373 both its atomic weight and density are found to be inter- mediate between those of its two isotopes, and that therefore an appropriate mixture of the two isotopes would have the same mean atomic weight and density as ordinary lead. If this idea of the real nature of lead were correct, it would seem to be possibie that some separation of its two constituents, which would differ by about 1 per cent. in density, might be effected by centrifuging the lead while in the liquid state. Such a separation could be most easily detected by determinations of the density of the lead from the top and bottom of the centrifuging tube, and this was the method adopted in these experiments. The centrifuge used was one constructed by Leune of Paris, which runs at about 9000 revolutions per minute. The lead was contained in steel tubes, which were fitted with steel lids to avoid oxidation of the lead as far as possible. Quartz containing tubes were first tried, but were found too weak to stand the strain of centrifuging. The steel tube containing the lead was heated by means of a coil of asbestos-insulated nichrome wire wound round it. This coil was kept in place by two collars turned on the steel tube, one at each end, so that in effect it was really bobbin-shaped externally. The whole tube with its heating coil fitted into the usual outer metallic holder of the centrifuge, which hung from trunnions in the ordinary way. The heating current was supplied to the coil in the following manner :—One end of the heating coil was con- nected to the outer metallic holder of the centrifuge which made contact with the main rotating spindle of the centri- fuge through the supporting trunnions of the tube. As it was not desirable to pass a current through the bearings of the centrifuge, a copper gauze brush was used to make contact with the vertical spindle of the centrifuge, and this brush was connected to one pole of the source of current. The other end of the heating coil was connected by a flexible connexion to an insulated horizontal copper disk which was fixed on the iop of the vertical spindle of the centrifuge. A vertical carbon rod was used to make contact with this disk. This rod was insulated from the main body of the centrifuge and held in contact with the revolving copper disk by a simple adjustable spring device. Itwas connected to the other pole of the source of current, and thus the circuit through the heating coil was completed. As the carbon was arranged to be concentric with the axis ot rotation of the centrifuge, the minimum amount of power was wasted by the brush. | — 374 Prot. Joly and Mr. Poole: Attempt to determine The method of procedure in conducting an experiment was as follows :-—-The steel tube was first carefully cleaned and then filled to the requisite height with lead which was melted in a small porcelain crucible. ‘The lead used was pure lead obtained from Johnson, Matthey & Co. The small lid was placed on the steel tube, and the three other earriers of the centrifuge carefully balanced against the one containing the steel tube and lead. The carriers were then replaced in the centrifuge and the lead melted by turning on the heating current for about ten minutes, after which time the motor driving the centrifuge was started, and usually the centrifuge was kept running for about an hour before being stopped. It was found impossible to run the centrifuge for longer as the motor was inclined to overheat after this period. When the centrifuge was stopped the lead was removed, while still liquid, in six lots by means of glass pipette arrangement. The density of the top and bottom portions of the lead was then determined. The density of the lead was determined by casting small spherical bullets from it in an iron bullet-mould. These bullets were weighed first in air and then suspended in methylene iodide. Methylene iodide is especially suitable for this purpose, both on account of its high specific gravity (about 3°3) and also its small surface-tension. Fortunately it only attacked the lead very slightly, producing a very slight tarnish on the surface. Some trouble was experienced at first in obtaining sound castings from the lead, but it was found that by allowing the mould to cool slowly from the bottom upwards, bullets free from all cavities could be obtained. It is essential that the conditions under which the bullets are cast should be as nearly identical as possible, as unfortunately the density of lead is largely affected by the heat treatment it receives. However, the results show that this source of error was eliminated. The balence used was sensilive to 75 mgrm., and the weights were standardized against a new set by Becker & Co. which had a certificate from the N.P.I.. certifying them as correct to 7y mgrm. Summary of Results. Norte :--W = weight of bullet in air, B = loss of weight in methylene iodide. Hence i is proportional to density. if Common Lead could be separated into [sotopes. 379 Wi 8: Hxperiment. —— oS ee eS Remarks. Bottom Bullet. Top Bullct. IV sca sn a 34015 34033 Top Bullet 0°05 por cent. denser. ae ee 3°4092 34094 Top Bullet 0-006 per cent. denser. 2) ee 34112 S4111 Top Bullet 0-003 per cent. lighter. 2.2.4 0 aes 3°4090 34090 ~—-No difference. o,f AS eee 3°4£065 34069 Top Bullet 0:01 per cent. denser. It will be seen from these results that there is absolutely no evidence for any separation effect. The density of the upper and lower bullets agree as well as could be expected, as a difference of 7y mgrm. in B which weighed about 3 grms. would cause an error of 0-003 per cent. in the density. The slight variation of the density from day to day is probably due to the variation in the temperature of the methylene iodide, and may also be caused by small variations in the casting conditions. This point was not fully investigated as the method is essentially a comparison one. It is rather difficult to form an idea of the separation we might hope to obtain on theoretical grounds, owing to our ignorance of the equation of state for a liquid. Since these experiments were inaugurated, however, Drs. Lindemann and Aston have shown, in a paper entitled “The Possibility ot Separating Isotopes,” that if we neglect compression, and assume equal atomic volumes for both leads, and then treat one lead as simply dissolved in the other, we might expect to get a concentration of thorium lead at the edge nearly 50 per cent. greater than that at the centre, if the peripheral velocity was about 10° cm. per sec. In our case, however, a peripheral velocity of only about 104 cm. per sec. could be attained, which would only lead io a difference in con- centration of about } per cent. This would only give a difference of ‘005 per cent. in density, which is too small to be detected by the method of determining the density used. On these grounds, then, it is not surprising that with the centrifuge at our disposal no positive results were obtained. It would seem, however, certainly possible that with a specially constructed centrifuge some definite result might be obtained. Iveagh Laboratory, Trinity College, Dublin. iy 876. ul XXXV. On the Effect of Centrifuging certain Alloys while in the Liquid State. By J. Jouy, F.R.S., and J. H. J. Pooxs, BOARS URING the course of the experiments described in the previous paper as to the effect of centrifuging liquid lead, certain alloys were also dealt with. The results obtained are appended. It will be seen that in the case of the silver-lead alloys no definite separation could be obtained. The silver-lead alloys were specially dealt with, as, if silver and lead could be separated by centrifuging, the process might perhaps be cheaper and more expeditious than the existing cupellation method. Unfortunately the method appears to be a failure, at least with the velocities we were able to employ. In the case of all the other alloys an undoubted separation was effected. In all cases, there is a considerable difference in density between the constituents of the alloy, and no very large amount of separation was effected. It is of interest that a definite alloy like lead-tin alloy, which is of the composition PbSn, should be capable apparently of being separated by the action of the centrifuge. Silver-Lead Alloys. RY RB’ Exp. Composition, §=——~—-+~——~ Result. Bottom. Top. >. JUDE aan Pb 97 per cent. 33761 3°3746 Positive. Top bullet about VS OU ip 0-045 per cent. lighter. XVIII. ... Pb95 percent. 3:3937 3:3947 Negative. Top bullet about PASTA? | a 0:03 per cent. denser. ©. EIS fb 90 per cent. 3°3892 33833, Positive. Top bullet about Ag lO ie. 0:17 per cent. lighter. XXIV. ... Pb 90 percent. 33922 33923 Negative. Top bullet about Ae LO he 0:003 per cent. denser. Other Alloys. XT sae: Pb 63°6 per cent. 2°8308 2°8129 Positive. Top bullet about Sn 36°4 ° 4,0"); 0:63 per cent. lighter. OL Ditto. 2°8530 2°8114 Positive. Top bullet about 1:5 per cent. lighter. DO. CVE Baas Pb 82 per cent. 3°0594 3:0328 Positive. Top bullet about Smee ries 0-9 per cent. lighter. DOG nah Ditto. 31078 3°0519 Positive. Top bullet about _ 18 per cent. lighter. XOX ee Pb 32 per cent. 2°9979 2°9679 Positive. Top bullet about SG us, 1 per cent. lighter, IBiFo2 iy) lveagh Laboratory, Trinity College, Dublin. * Communicated by the Authors. ee oRT XXXVI. Notices respecting New Books. Problems of Cosmogony and Stellar Dynamics. By J. H. Juans, M.A., F.R.S. Being an essay to which the Adams Prize of the University of Cambridge for the year 1917 was adjudged. Cambridge: at the University Press, 1919. 293 pp., 5 plates. (THE essay is a daring attempt, in continuation of previous attacks initiated by Maclaurin, Kant and Laplace, and followed up by Roche, Jacobi, Kelvin, Poincaré, and G. H. Darwin, to continue the investigation of possible configurations of a rotating gravitating fluid mass, and of its stability, and to carry it on to a gaseous conglomeration, as of the spiral and other nebule. The book falls then into two parts: in the first six chapters a homogeneous incompressible gravitating liquid is postulated, and the shape investigated which it can assume, starting from a spherical form, and then endowed with rotation gradually in- creasing which causes it to assume a variety of shape, passing through the Maclaurin spheroid into the Jacobian ellipsoid, and this again into the Poincaré pear-shaped figure, finally breaking cataclysmically into two parts, a main body and its satellite, or the state of a double star, the final object of Darwin's research. In these last two investigations the mathematical difficulties are almost insurmountable, and extraordinary approximations are required to arrive at any definite result, and even then the methods are not of universal acceptance, and much controversy has been excited. The difficulty of the existence of a free surface, and its stability, is the chief impediment to progress; and the various stages are very instructive in revealing the branch points where the class of surface changes place. Throughout the subject the angular velocity w appears involved with the density p, in the form of the fraction = , so that p is here ™p the astronomical density, of the dimensions of the square of an angular velocity, or (time)—?. Astronomical density p is converted into U.G.S. density é(g/em*) by the factor G, the constant of gravi- tation, G=666 x 10-1°, according to the experiments of C. V. Boys, and then the fraction becomes =a The astronomical unit of 1 mass is then q@ = 10" x 1°5, g, or 15 metric tons. This fraction can be made more intelligible physically by intro- ducing Maxwell’s idea of the grazing satellite of the stationary spherical globe; then if K is the grazing velocity, andr, the radius 378 Notices respecting New Books. of the globe, ° “ 4 —=g= 57Gor,; and if © denotes the angular i Keo velocity of the satellite, Q°= —; = 57G6, which is independent of the radius 7,: and then the fraction 0° 2 2 2 35 be OT @ we 9 f{w Qrp.) pL ao se 1S (G) f I 52 4 to whicha definite physical meaning can be attached. If the globe could retain its spherical shape when the angular velocity was raised to Q, bodies at the equator would be lively on the surface, like the mud particles on the top of the wheel of the old hansom cab seen through the side-window, and everywhere else the plumb-line would be parallel to the polar axis. For the Earth this must be 170-fold, Q=17w. Maxwell suggested as the universal unit of time, for the Solar System, and ail space beyond, the period of the grazing satellite of a sphere of water, instead of such a parochial unit as our terrestrial mean solar day; this new unit proves to be about 200 minutes. ‘ It would be impossible to go into details here of the extra- ordinary audacity of the mathematical attack ; a mere summary of the results must suffice, considered under the heads of I. The Tidal Problem. IT. The Rotational Problem. III. The Double Star Problem. Starting with the gravitating globe at rest, in the Tidal Problem the motion is through a series of prolate spheroids: in the Rota- tional Problem the motion is first through a series of oblate spheroids (Maclaurin’s spheroids) and then through a series of ellipsoids (Jacobi’s ellipsoids): in the Double-Star Problem the motion is through a series of ellipsoids. The second half of the essay undertakes an additional difficulty in developing a general theory of the configurations of equilibrium of a compressible mass, in its departure from the state predicted in an incompressible model. Here the difficulty is great enough when rotation is absent, and the gas is stratified spherically, and various plausible physical assumptions must be made to allow the equations to be integrable. Dr. Schuster’s results from the limiting case of y=1-2 are of very great importance, but acloser examination seems required to show that the agglomeration would be unstable at the core, if a rotation was imparted. The object of this investigation of a compressible mass is to frame some theory of the internal state of density in the Spiral Nebule visible in the telescope, conjectural Solar Systems in the making; a feeler into Space, like Relativity, but without abandoning Newtonian Dynamics. The whole essay is a direct frontal attack on impregnable pro- blems, and will require to be reinforced by outflanking equations of related problems that will yield to solution, Notices respecting New Books. 379 Researches in Physical Optics. Part II. Resonance Radiation and Resonance Spectra. By R. W. Woon, LL.D., Professor of Experimental Physics, Johns Hopkins University. New York: Columbia University Press, 1919. 184 pp., 10 plates. [Publication number eight of the Ernest Kempton Adams fund for Physical Research. | Part I of this Research, published 1918, was devoted to the radiation of electrons. An adequate description in detail of this monumental Research in Part II would take up more space of the Magazine than can be spared between the multiplicity of subjects. A mere outline of the scope must suffiee for the reader. The author is the best known exponent of the experimental side of the Science of Light,in Physical Optics, and his contribu- tions to the applications of Theory in recent warfare will it is hoped be allowed to see the light now, for the general benefit of Science. Of the whcle gamut of the light spectrum only a fraction can be apprehended by the human eye; but the author has succeeded in devising apparatus for picking up an impression of the part beyond the visible rays, and utilising them in operations such as heliograpbic work; the signals can then be received without attracting outside undesirable atteution, as of an enemy. We have been hearing much lately of the new Theory of Relativity, as revealed in the Gravitation of Light, so that itis no longer a paradox to say that Light is Heavy. In utilising the dark rays, the author provides a discussion of the Light that is Dark. ‘he Research is chiefly a careful description of the delicate apparatus required in the experimental work of the spectroscope. Nothing is recorded that has not been observed directly, and that is capable of being redetermined from a description of the apparatus. No appeal is made to new theories of the ether, and there are no elaborate mathematical developments, founded on conjectural hypothesis; nothing to spoil the pleasure of the physical experimenter, and to interrupt his manipulative skill and interest. A Table of the Contents may be cited to show the scope of the investigations. . Plane Grating Spectrographs of Long Focus. The Resonance Spectra of Iodine. . Resonance Spectra of Iodine. The Series of Resonance Spectra. . Band and Line Spectra of Iodine. . Zeeman-Effect for Complex Lines of Iodine. A Photographic Studyof the Fluorescence of Iodine Vapour. . The Magneto-Optics of Iodine Vapours. “TO OH OO WO @ 0) 380 Notices respecting New Boeks. 9. The Fluorescence of Gases Excited by Ultra-Schumann Waves. 10. A Further Study of the Fluorescence produced by Ultra- Schumann Rays. 1]. Scattering and Regular Reflexion of Light by an Absorbing Gas. 12. Separation of Close Spectrum Lines for Monochromatic Illumination. Unified Mathematics. By L. C. Karprnsxt, Harry Y. BENEDICT, Joun W. Catnoun, Professors in the University of Michigan and Texas. D.C. Heald & Co., 1918. 522 pp. Prrry’s ‘ Practical Mathematics’ would be our equivalent for the scope of this book, intended to show, here and in America, that the old plan is obsolescent of keeping a school-boy marking time for years over arithmetic and algebra, and then rushing him through some Calculus and Coordinate Geometry in his last year. But the essentials of the Cartesian geometry are inculcated here in the use of squared paper, for drawing the simple graphs, and the illustrations follow of the geometrical applications of the Calculus. The logarithm is introduced at an early stage and its use exemplified in multiplication and division in applications of real interest to large numbers and decimals by the aid of a compact four-figure table. But there is no mention of the Slide Rule, equivalent of a three-figure table, and amply accurate for ordinary purposes. Formerly the only table to be found was a seven-figure table, never hardly to be seen. It makes us groan when we have oeeasion to turn to it, to think how late in life the use of it was introduced to our attention. In a first introduction to the logarithm, no base should be mentioned except 10, and then the definition y=107=al a, w=log y, with the abbreviation (al) for antilogarithm, as (log) for logarithm. 482 23 o RUS ale 20): 490) 398 Prof. Adeney and Mr. Becker: Determination of Rate of (b) Heperiments with Oxygen. The oxygen was prepared by heating potassium per- manganate ina hard glass tube, and washing the gas with caustic potash to remove any traces of carbon dioxide which might be formed. The apparatus was exhausted several times with a water-pump to wash out the last traces of air. The gas was collected over water which had been boiled for ‘ some time and cooled out of contact with air. A series of experiments was made over a range of temperature of 35°, and the results treated by the two methods as mentioned in the case of nitrogen. ‘The results are contained in Table VIII., and the variation of 6 with temperature is shown in fig. 7. TaBLe VIII. Results of Experiments with Oxygen. Values from | Values from W | NG 3 A . OF Saturation Values. empera- | log graph. graph. Mean ture, —- -——- —-—— |- Wg Git eC. | Adeney t. b. a Wo. ob. | and Winkler. Bobr. ae A ie Becker. 2:5 Ti e202) M28 2G) 4-450) 4:600 4690 “264 88 64 -310 | 1:230 3910 -315 | 3814 3970 4-080 5 (3 15°5 59 336 | 1140 3:830 342 | 3254 3372 3450 | +339 90-2 | 54:5 -356 |1:150 2950 -390 | 2970 38-:040 3-110 | :373 25°2 Fea Oe, 11-150 2810) 410 ) 28120) 2.310 2880 | “411 30'°3 | 46 -4387 |1:080 2450 +432 2485 2-510 2575 434 35'1 | 13 ‘477 |1:120 2250 -498 | 2:323 2°355 2-400 “487 In the above series of experiments the water in the tube at the end of each experiment was analysed for dissolved gases, using the extraction pump and measuring apparatus described in Part I. of this communication. The solubilities of oxygen and nitrogen at the given temperatures as cal- culated from these analyses are given in Table IX., as are also the values obtained by Bohr and Winkler by absorptio- metric methods. Solution of Atmospheric Nitrogen and Oxygen by Water. 399 TABLE LX. Nitrogen. Oxygen. Adeney Adeney Becker. ie Becker 3°55 :02189 -02200 02203 25 04540 -04625 -04590 14-2 - -01788 01890 01820 | 88 03866 -03965 -03710 150 :01654 01757 ‘01701 155 =—03823 08405 = 03206 201 . 01505 -01598 -01549 20°2 03019 -03089 -02955 248 01392 -01461 -01456 25°2 =:02783 =-02805 3902732 304 °01276 -01312 013822 || 303 ‘02488 -02552 -02465 oot - O1183 “01200 —-01220 35:1 =-02302 »=6-02347 = 02270 cae Winkler. Bohr. and iS ce. Winkler. Bohr. and | Values of b. Temperature in degrees centigrade. Values of 6 for Oxygen plotted against temperature. Formula :—b=:00672(T—236'5). VII. Repucrion oF REsuuts to Unrr AREA AND VOLUME TO OBTAIN FUNDAMENTAL CONSTANTS. The results have been shown to be in agreement with the general formula dw A SSA) yp a lt wy C 400 Prof. Adeney and Mr. Becker: Determination of Rate of where w=total quantity of gas in solution at any moment, S=the initial rate of solution per unit area, f=the coefficient of escape of the gas from the liquid per unit area and volume, A=area of surface, and p=pressure of the gas. The values of 6 for different temperatures and different gases have been found for various temperatures using a volume of water of 101°8 c.c..and an exposed area of 71°3 sq. cm.; hence since A b= fy we can calculate the values of 7. The values are given in the second column of Table X., and when they are plotted against temperature in each case, three straight lines lying very close together are obtained, as shown in fig. 8. Values of F. Temperature in degrees centigrade. Values of f plotted against temperature. Formule :—f=:0096 (T —286°5) for Oxygen. f=:0103 (T—240-0) for Nitrogen. f= 0099 (T —239°3) for Air. Solution of Atmospheric Nitrogen and Oxygen by Water. 401 When the values of f for any gas are multiplied by the corresponding solubilities, the product gives the initial rate of solution in each case, since S=/s. It will be seen by reference to Table X. that the value of S is practically a constant over the range of temperature given. The value of S is approximately proportional to the solubility, being about twice as great for oxygen as it is for nitrogen ; and if + of the value for nitrogen be added to } of that for oxygen, a value for air is obtained which agrees fairly closely with the actual figures thus :—# of :0083 ++ of "0160 =:00665 + :00320 = "00985, while the mean experi- mental figure is ='0100. TABLE X. Oxygen. Temp. °C. WP S. Sis. from analysis 2S 3713 "04390 0164 88 434 "03710 “0161 15°95 “499 "03206 "0160 20-2 *D45 "02955 ‘O161 25°2 591 02732 ‘0161 30°3 641 "02465 "0158 35°1 687 02270 °0156 Nitrogen. 3D “372 "02203 0082 EE 448 701820 -OC81 150 “490 ‘O1701 ‘0083 20°2 548 01549 ; “0084 24°2 093 701456 ‘0086 30°4 647 "01322 "0085 30°1 696 "01220 0085 aes (Dittmar.) 36 302 02700 "0095 11-4 "441 02260 "0099 15:6 “476 702120 *0100 20:0 “52D "01538 "0101 25°0 "O74 01780 °0102 29°6 623 °01660 "0103 34:3 672 °01550 0104 Phil. Mag. 8. 6. Vol. 39. No. 232. April 1920. 2D 402 Prof. Adeney and Mr. Becker: Determination of Rate oj VIIL[. Statement oF RESULTS. From the figures given in the previous section it is possible to calculate the rate of solution of the gases dealt with, for any conditions of area exposed, depth, or degree of saturation, provided that the water is kept uniformly mixed. The expression can be put either in the form dw Pe =a—bw, which gives the rate of solution at any instant, or in the form w= (wo—tw ,)(l—e-"), which gives the amount dis- solved at the end of any given time when w)=saturation value and w,=amount of gas in solution initially. For practical yampases it is most convenient to work in per- centages of saturation ; hence the latter equation becomes w= (100—w,) (1 —e-it), and since bal by substitution Lge t =(100—w,)(l-—e * Vv) as the general equation for any given temperature, and since j varies with temperature according to the equations Oxygen f= 0096 (T—237) Nitrogen /f='0103 (T— 240) Aa f= "0099 (T—239), the corresponding general equation for each gas by sub- stiluting these expressions in the formule is obtained, thus :— for Oxygen w= (L00—w,) | 1 _ 9 mvs 297) 44 » Nitrogen w=(100—2)) [1 9 URE 200) 529 mi oAir w= (100 —20,) [1 — et, As an example of the use of these formule, consider the question of the dissolved oxygen in 1000 ¢.c. water, area exposed being 100 sq. cm., temp. 2°°5 C., and initial gas- 403 Solution of Atmospheric Nitrogen and Oxygen by Water. Percentage of Saturation. Time in Minutes. Curves for Oxygen, showing the variation of the gas-content of the water with time, Volume of water 1 ¢.c, Avea exposed 1 sq. cm, 404 Dr. A. D. Fokker on the Electric Current from content=40 per cent. of saturation. How much gas will be dissolved in an hour ? t=60 minutes, 313 w=60(1—e 1") =60(1—e- 2) = 60(1 —-8009) =60 x 1991 =11°8 per cent. saturation. Hence after an hour the water will have risen to 51°8 per cent. of saturation. These equations can also be used to calculate curves showing the rate of solution in water of the different gases under different conditions, and as an example the curves for oxygen between 0° C. and 30° C. have been calculated in percentages of saturation, and are shown in fig. 9. It is noteworthy that when expressed in percentages of saturation, the curves for the three gases lie very close to each other, those for oxygen and nitrogen being practically identical. The authors desire to express again their indebtedness to Dr. Hacket (Lecturer in Physics in this College) for the interest he has taken in this investigation, and the valuable assistance he has generously given in the mathematical treatment of the subject. Chemical Department, Royal College of Science for Ireland. AXXXIX. On the Contributions to the Electric Current from the Polarization and Magnetization Mlectrons. By Dr. A. D. FoxKer (Leiden) *. A” important question in the electronic theory of matter is the evaluation of the electric current due to the motion of the electrons of electrically neutral atoms. To Minkowski the idea is due to put the question as a variation problem of a current by small virtual displacements, in a manner to be described hereafter. Born has worked out this idea after Minkowski’s death Tf. * Communicated by the Author. Abstract from a paper offered to the Kon. Akad. v. Wetensch. at Amsterdam. + Hermann Minkowski—Max Born, Line Ableitung der Grundgleich- ungen fiir die elektromagnetischen Vorgiinge in bewegten Kérpern, Math. Ann. lxviii. p. 526, 1910. Polarization and Magnetization Electrons. A05 I venture to offer a new development of the same idea, which distinguishes itself by the extreme simplicity of the means employed. No use is even made of any theorem from the theory of relativity. After completing the deduction, nevertheless, it is easy to show the covariancy of the equa- tions obtained in the sense of the theory of general relativity. In addition, one hits on a contribution of the bound elec- trons hitherto not yet signalized, so far as I am aware (§ 7). §1. Minkowski’s Idea. Consider a stream of neutral atoms. [or simplicity’s sake we shall take them to consist of a positive nucleus and one accompanying electron, both of them carrying the elementary charge. ‘The motions of the heavy nuclei will be identified with the motion of matter, and we shall assume that neigh- bouring atoms will be very nearly similar and similarly situated, so that the functions defining the positions of the electrons relative to their nuclei, though not strictly constant, will vary but slowly from one atom to the next. Of course the stream of positive nuclei will constitute an electric current, and the stream of electrons another. As a result of the displacements of the latter relative to the nuclei their current will not have the same intensity as the positive current from the nuclei. The combined effect will be the current required in the field equations for ponderable matter. It will be clear that if, given the displacements, we succeed in finding the resulting variation of intensity of the stream, our problem will be solved, as soon as we shall have inter- preted the result in terms of physical quantities such as polarization and magnetization. The displacements can be regarded as depending ona varia- tional parameter @. It turns out that the terms in the result proportional to @ are connected to the polarization, and that the terms proportional to 6? express the effect of magnetiza- tion mainly. § 2. The displacements. We imagine a stream of particles moving through a space which will be described by the co-ordinates x, y, ¢. Let there be N of them per unit of volume, moving with velocities dxdt, dy/dt, dz/dt, which, after adding to them as a fourth 406 Dr. A. D. Fokker on the Electric Current from quantity dt/dt, we shall for symmetry denote by w?, w, we, a0), respectively. In the same way we shall often for convenience sake write v2, 2, «®, «, for x, y, z,t. The components w” are assumed to be continuous functions of the co-ordinates and the time. The stream-components are seen to be Nw", Nw®), and Nw®), to which we add a fourth Nw®=N. They will be altered when the particles suffer displacements as defined in the following. We take @ as a variational parameter and suppose a quaternary vector given with components 7%, 7, 7®), o. If the parameter increases by dO, the particles shall shift from the positions (a, y, z) occupied by them at the instant ¢t to the positions vetrde, ytrdd, 2+7%dd, to be occupied at the instants br dd, The components 7%, r®, 9, 7 are assumed to be con- tinuous functions of the co-ordinates and time. For each particle the values of r* must ke taken which are actually found in the place and at the instant from where the infinitesimal shift begins. It will be seen that the total displacements and shift in time of the particles from the point-instants of their un- disturbed motions (@=0) will be PDO, 729, 789, 46, in a first approximation, and, taking account of terms with 6° in a second rs aie or Ave “ds r+ 3c) — 75), ~pe “0 Ou = 18-4 33() 26, (a=1,2,3,4) . . A) 4 where 7* and Q7r%/Q@x° have values corresponding to the point-instants of the undisturbed motion. In this and sub- sequent formule the summations are to be extended over all values from 1 to 4 of the index put in brackets. In consequence of these displacements the stream- com- ponents will change to Nw’ +dNwt+40?Nwi+. . Polarization and Magnetization Electrons. 407 It will now be our business to find the first variation 6Nw2 and the second variation 6?Nvw*. $3. The first variation of the. stream. The following conception of the stream-components will greatly facilitate the evaluation of the variation. We keep our eyes on the content of a space-elemént dV, situated at the point 2, 2®, 2, at the time 2. Though physically infinitesimal the element is supposed to contain a great many particles so that NdV isa great number. In an interval dé these particles will in a four-dimensional space-time exten- sion describe their so-called world-lines, that will fill up an infinitesimal extension dVdt. Now sum up the components of these lines in the direction of X“,say. We tind obviously NdV dx. Dividing by the four-dimensional extension dV dt we can say that the stream-component in the direction of X%is the sum of the components described by the indimdual particles per unit of volume per unit of time : N@V dat Bee es NG We hardly need say that the fourth component represents the number of particles per unit of volume. It is obvious that these components will satisfy the condition of continuity: ON HN CLM S(b) Ovens eka Con By the displacements the component of each individual world-line will change to dat + S00 or dat, if we neglect 0”. The sum of the components will thus grow to Ht Oma. Nav { ae +X(0)0 do! } On the other hand we must be aware that the extension occupied by the world-lines has changed also. We can find the increase with the aid of the functional determinant of the 408 Dr. A. D. Fokker on the Electric Current from at + Axe with respect to 2%: O(at+ Axe) 0(e++ Axe) Ox ou? 0(2?+Ax®) A(v?+ Aa?) (dV dy'=| Bae MMe ce or, consistently neglecting @? : ore 9% 1 We Des go” ore _ oe : Dest i > a Verde: cn Fa zqe2 orev dt. Thus we find a the displacement the sum of the individual components described per unit of volume pee unit of time to be: Nw? + ANwe= [Newt + 3(6) Nw S| [1-055], =Nwt+>(d 64 Nes oe —_ Now ork. Now it must be borne in mind that this value of the new stream-component is found in the point-instant 2+ Ax and not in the point-instant «¢, where we wanted to knowit. To obtain the latter value we obviously ought to choose our starting-point in the point-instant #¢—6r¢, and take Nwt— 20) 7 Or instead of Nwe. Thus we o Rene ourselves of the equation of continuity (2) also: Nwt+dNwt=Nw2+2 (6 {rawr Nwe } and the first variation is sNwa=¥(0)0-2, ; rN wh — rb N we \ es (3) Polarization and Magnetization Electrons. 409 § 4. The second variation. The second variation may be found without calculation by a formal process. Indeed, we only have to substitute 6Nw¢ for Nw in formula (3) which gives ONw? to get: SNe =HO)OL| SNe |, oO? N wt = > (bc) a, [ 2 ; rONwe— reN wh ; = of rtNwe—reNeoe bY. 2 (4) It is, however, very important to state that this formula for the second variation implies the definition of the dis- placements with an accuracy up to terms with @” as given in formula (1). This can be verified by a direct calculation following throughout the same line of argument as in the preceding section, taking account of the terms of second order everywhere. We shall not give the calculation at full length, we may restrict ourselves to the indication that at the last step, viz., in choosing the right starting-point from where the displacement will carry us to the point-instant «4 under consideration, we have to be careful to take 12S) 0" at — Ore + 67 (c) ae me, and not #*—Az*,as might be thought erroneously at first sight. § 5. The simultaneous displacements. As yet the displacements considered have been accompanied by a shift in time. In view of the physical interpretation of the formule obtained, it will however be necessary to realise the simultaneous positions of the electrons relative to their nuclei. There is no objection to simplifying our formule by drop- ping @ henceforth. Now, in a first approximation, we find the electron belonging to the nucleus, which at the instant z® is in the point 2, 2), v®, shifted to the point at the instant 410 Dr. A. D. Fokker on the Electric Current from Thus we see that its position at the time # will be given by 240, aM +p™, a 4 p®) where ptr — ep) a ee a For an obvious reason p“=0. Next, to obtain the second approximation, consider the nucleus at the instant On!) pe — 2 = yer (4) Ox? f 35.) (4) _. »(4)_ 1 2 hi pos 3 (0) | when its co-ordinates are 7) Oxe This line implies the preceding as a special case, for a=4. Then the displacements of the nee will be (ee a 15 (0) ne 308 so that its position will be given by 1 dw ren -E po— wey) itn oie pully a) (4) + v dre( 9 dt > (« :) 2 ee — wr) —Lyad(c) (4) mee — ? iB wer Lag + i Ei Hy, Ox? 2 7p), Oi 3 ae fe) ve F (4) Owe Owe Taking a=4 it is easily seen that this formula yields the positions just at the instant «, 2. e., the simultaneous posi- tions and displacements ; for w=lisa constant, and all terms vanish except the first. We can simplify the ere considerably. Writing d ZA we 2 Se SS 2(¢) Owe dt’ and introducing a notation well known in three-dimensional veetor-analysis : Ore oe Syn Oni Oe rae (p. Vv): our expression reduces to Pap a o\p:\/ pt ar grt} ar — (p. Vw} and the simultaneous Pn are p? +3(p-V)p o— rio fe dt —(p. (9. V7) } Le (a=li2ave ey Polarization and Magnetization Electrons. 411 § 6. The interpretation of the first variation. If the negative charge of an electron, the elementary charge, be denoted by e, the current carried by the stream of the positive nuclei will have the components —eNw, and the stream of accompanying electrons will carry a current | eNwt + edNwt + 4ted’Nwe, the resultant current from the charges bound in the neutral atoms amounting to : eONwt + 4e0°N we. Now let us consider the first part, originating from the first variation. It contains what was formerly called the contribution from the polarization-electrons. We know by formula (3) that eOSNwt=X(b) 2 i ert Nwb — ov?Nuce : We shall consider the tensor : Pab— eyaN wo — er Nw. This is the same as ee a aa =] So = Neptw?’ — Nepiw*, where p® is the principal term in the expression (6) giving the simultaneous displacements. Introducing for the prin- cipal part of the polarization the thrée-dimensional vector Pa = Nep*, we recognize in the (14)-, (24)-, (84)-components of the tensor Pa components ion and in the (12)-, (23)-, (31)- components the components of the well-known. Rontgen- vector which is the three-dimensional vector-product of the material velocity w into the polarization. Collecting the components of P¢® in the scheme Pee ie My 1) Pe Pab( =) —[p.wi, [p.w], Pp, [p-w],, See Wile P. ae —P~, —p, 412. Dr. A. D. Fokker on the Electric Current from one may see at a glance that the first three of the com- ponents of the variation c6Nw* are the components of rot|p.w]4p;. 724 ee ee) and that the fourth becomes —div.p. oS os em The first variation thus furnishes a polarization-current p and the corresponding Réntgen-current rot|p.w]|; besides it shows a polarization-charge. § 7. The wnterpretation of the second variation. The second variation too is connected by a differential operator to a tensor, M%: ab 1eNoe 3 Oyo Owe ’ M% =1t(ertdN wo —erbd Nw). We divide this into two parts: Me =LedN (rew? — row) + eN (12d? — rdw), the first of which is nothing but a correction to the polari- zation-tensor P¢. We can put it into the form LedN (148 — ro wt) =tedN(p2w> — pou), where and as in the preceding section we find contained in it a polarization 4edNp*, and the corresponding Réntgen-vector. Ata closer inspection, taking ON from (7) we see seONp4*= — tex aay, N p%o¢ = _o® PrP) + WN (p.V)p" The latter part is in good agreement with a term of the exact expression (6) for the displacements. The former part implies a correction that is grasped in its meaning as follows. Imagine that we want to know the polarization and that we therefore choose an arbitrary closed surface, taking the sum of the electric momenta of the atoms within and dividing by the volume. Then our correction amounts to saying that an atom will be reckoned to lie within the surface only when the centre halfway between nucleus and electron lies within. Polarization and Magnetization Electrons. 413 Turning to the second part of M# we require the value of . we, This we get from the known values of d6Nw* and SN =dNw (form. (7)). It turns ie to be Sit = SP —>(c)p° oer Now write down Zan Swe —rb§wr) = LeN (p2dw? — pdw*) | —LeN (7802, wb —r Sw? .w), and we see that the latter part: gives a Ce again with the oneal bes vector. This last correction, —teNr® {a —(p J) t : brings the polarization into complete agreement with the value given in (6) for the electrons’ exact displacements. These second-order polarization corrections are very inti- mately.connected with the magnetization, and always come into play whenever magnetized matter is moving. This is well known. It remains to investigate the nature of the part SEN (p4dw? — phdw?). If we write 4eN (p81? — pl3w*) = ye eN(o*“T, os i = 33 (EN (orp S — pipe WU), we notice that both expressions vanish for a value 4 of one of the index-numbers a or b. We recognize that - wel ot a a , dp a p ior are the components of the magnetic momentum of an atom. Thus, writing m,, m,, m, for the magnetization we evidently have — LN o ae p’ i Hel, e 414 Dr. A. D. Fokker on the Electric Current from In the second expression appear the quantities 3ep"p’, being the quadratic electric momenta of the atoms. These quantities figure in recent investigations of Debye and Holtsmark on the broadening of spectral lines of luminous gases under increased pressure*. They seem to afford a measure for the electrical extension of the atoms, and so it is proposed to call their sum per unit of volume (provisionally) the electric extension : Kae = seN pipe. In a form of three-dimensional vector-analysis we can con- tract the three components under discussion into a vector ke CK) .wil; where the number 2 added to the left of *K has to remind us that 7K is a symmetrical tensor and therefore (?K.V) is a differential operator with vector-properties +. — This vector k is analogous to the Rontgen-vector. It accounts in its curl for the second order influence of the motion of polarized matter on the electric current. Gathering the various corrections of the polarization into a single vector n, we can collect the result of the second variation into the scheme for our tensor M2: m+k,+[n.w], Syn [n.w], 2 aE —cm,—k,—[n.w], m,+k+([n.w], 2, Meb(=) cm ot cle mw soem, late wl | : ny —n, it, —i, whence we get by the. formula , SONT a) SS oMeé 4e3°Nwt= 3 (b) Sar a current : erotm rot k+ rotin .w] +.) 2) aan and a charge: divin. ea ey) We notice a polarization-current n, the Réntgen-current * Debye, Phys. Zschr. xx. p. 160; Holtsmark, zb¢dem, p. 162 (1919). + Cf. the notation of Prot. J. A. Schouten in “Die directe Analysis zur neueren Relativititstheorie,” Transactions Kon. Akad. v. Weten- schappen, Amsterdam, xii. No. 6, 1919. Polarization and Magnetization Electrons. 415 corresponding to the complementary polarization un, and the well-known magnetization-current crotm. As stated above the current rot k originates from a second order influence of the motion of polarized matter. Itis neglected because of its smallness in the deductions of Lorentz and of Cunningham*. In the paper of Born cited above it is not separated from the magnetiza- tion-current. Perhaps its action might be detected experi- mentally if a rotating sphere of insulating material were surrounded by a fixed circuit about its equator and placed in a strong homogeneous electric field with the lines of force parallel to the equator’s plane. An oscillating rotation of the sphere should induce an oscillating current in the circuit. § 8. Remarks concerning Covariancy. In the introduction allusion was made to the covariancy of the result of the variation. Indeed, a reader familiar with Hinstein’s theory of general relativity may easily con- vince himself that equations (3) and (4) are invariant in the general sense. From the definition of Nw¢ in §3 it is clear that Nw is “gx a contravariant vector, where Vg is the well-known factor in that theory ; for in the numerator NdV is a definite’scalar number and dat a contravariant vector, in the denominator gdVdt would have been a scalar. Nw being /yx a contravariant vector, we see that rN wb —7N we is “gx a contravariant anti-symmetrical tensor, and SNue: = g Von —7dN ONwt= =(0) Sp re Nw —r? Nw) is seen to be Wax the ae vector-divergency of the tensor and therefore /gX a contravariant vector itself. } The same may be said, mutatis mutandis, of the second variation 6?Nw-. Knowing the character of P# and M# as yx contra- variant tensors, it is easy to deduce the ordinary transforma- tion-formule for the polarization and the magnetization. This, however, will not be done here. * Lorentz, Line. d. Math, Wiss.; Cunningham, ‘Principle of Rela- tivity.’ Prey XL. An Elementary Theory of the Scattering of Light by Small Dielectric Spheres. By Jakos Kunz *. Tee problem was solved a long time ago by Lord Rayleigh f, who used spherical harmonics. In the present analysis it will be shown that the same fundamental result can be obtained by elementary considerations without Fig. 1. NS Ftadius PR x & the use of spherical harmonics. We considera plane polarized beam of light, proceeding from above in a vertical direction, Z, so that the plane of polarization is perpendicular to the % Communicated by the Author. + Scientitic Papers of Lord Rayleigh, vol. i. p. 87; vol. iv. p. 897. Scattering of Light by Small Dielectric Spheres. 417 plane of fig. 1, which contains therefore the electric force, H, oscillating in the z direction. In the path of this beam of light is placed a sphere of dielectric constant & and of radius R small compared with the wave-length >». In each moment we may consider the sphere as surrounded by a uniform electric field, E. Now, if a dielectric sphere is placed in a uniform field, there will be induced in the sphere a uniform field also, and the original field outside the sphere will be disturbed as if the sphere were replaced by a doublet of moment where g represents the charge in one pole and I the distance between the two poles of the doublet. If the field is alter- nating according to the equation e=E sin 2ant, the doublet will oscillate according to : (2 e HH sin 2arnt or p=qgl sin 2rnt agi le es —— > 2 3 Y ee . and Te (27) ET sin 2arnt (1) If the doublet oscillates it will emit electric waves which have been studied by H. Hertz. In the neighbourhood of the doublet the oscillations are fairly complicated, but at large distances we find simple spherical waves, in which the electric and magnetic forces can be studied by the pulse method as follows :— The variable moment yp is either equal to A= ql sin 2rnt=ge or equal to p=qsin 2rnt .l=q,.. In the latter case the charge is considered variable, and the length / constant; in the former case the charge g is con- sidered constant while oscillating in simple harmonic motion through an interval 2/. This corresponds to the oscillation of a charge around a centre of attraction, and we have NE COTS VGH oe oe oe 5 a are aE ee Ce) where / is the acceleration. Phil. Mag. 8. 6. Vol. 39. No. 232. April 1920. 23 418 Prof. Jakob Kunz: An Elementary Theory of the Now, if a charge having a velocity v is abruptly brought to rest at O, of fig. 2, a spherical pulse of strong electric and magnetic forces will spread out with the velocity of light c. lig. 2. The ratio of the two components as of the electric force Ei in the pulse will be equal to fi, } fy fea et sins R q' — So — ________ ° u a = . iii AC 6 : / 9? 2 gut sin 3 h,= teue= =. because a6 ctrd yusin 3 : gusin 3 or UH L = - and the magnetic force H=ch,= d ad ae is vertical on the plane of the last figure in the point A. Now let the decrease of the velocity of the charge g be Av during At seconds, then we have 6=cAt and E AvgsinS — Avg sins ie cont CANE ic ee Scattering of Light by Small Dielectric Spheres. 419 In the limit - = —/, we find a fasinS 4. = ge asin P E . A 3 (3) 24 sin $ or by (2) ral oe ae 4 2D or by (1) H=n) oe ue LEN But An=e, hence eae aah ee Rs 4qr* sin sin 2arnt, pte’, Ne or, if we write e; for E, the secondary wave can be repre- sented by h— aus sin 3 e,= a 7 oe ama tS sin 2an(t—t’), ! © where ¢=-, n= c Me peas cin 3! Qa =H, 2 ae ts ae pay Oh —— A : és=Ls. yh are (ct—1r) (4) . 295 or e,= EB, sin 1 (f=); k—1 R 47” sin 3 REO on? where =f while the primary wave is represented by Ts, e=Tisin ee om) See aan ee ae For $=0, the electric and magnetic forces vanish and no light is given out in the direction x of the oscillation of the doublet. For += ay the electric and magnetic forces be- come a maximum on the surface of a sphere of radius r. No light is given out in the direction a, which is perpen- dicular to the original plane of polarization; the light emitted in the vertical plane yz is plane polarized ; it is also polarized in every other direction so that the plane of polarization in every beam is perpendicular to e; (see fig. 1). The energy per unit volume and the intensity of the ‘ight 2 2 420 Prof. Jakob Kunz: An Elementary Theory of the scattered in any direction is proportional to H,? and therefore proportional to sin? 9. The energy dE contained in a ring of volume dt=27rr sin 3rd3 .1 due to the oscillating doublet ; ii’? d is equalto dH= ea aT SA Nee OF 9.0) or equalto dH= i (5) ee = we sin Sd3, where V = “TRS is the volume of the scattering sphere. The energy E, in a spherical shell of unit thickness is therefore 2 W2V2 /h—1\?2 i jy le ( 5) -2.{ ane ods SAO Na Jo ORE fl 2 We) \ RESO == Sar but the electric energy E, per unit volume of the primary 2 beam is equal to H,= = , hence T V7? (k—1\? H,= 240°H, 54 h+2 For the light scattered by N particles per unit volume, arranged in random order so that energies may be sumnied without considering phase differences, we should have Ve (kl? (S)N . ee) at Ae oes et Eee an expression first given by Lord Rayleigh by a different method. The energy radiated from a layer of thickness dz and of unit area is therefore OD ea Beas? i SRS SS Wile HK, fe cites | Nd Gre) BE, = Bye, Nh 4 ema | N22 where h= 24773 Me G ee INI: k+2 Scattering of Light by Small Dielectric Spheres. 421 If, however, we investigate the scattered light proceeding in the direction z of the original beam, we have to take into account the phases as well as the intensities of the electric forces. Let us consider a layer dz of equal particles, of Fig. 3. \ \\V / MY index of refraction y, and calculate the resultant electric force e; at any point P on the path of the original beam where sin3=1, due to all the doublets in the stratum. The volume of an infinitesimal ring will be (fig. 3) dt=dz27pdp=dz2nrdr as Prva’. For sinS3=1 and H=1 we get from (4) e a re NE mee Ge SNe r ; The resultant electric force due to all particles in dr is therefore Hey Naz" —l3rV k+2 2ardr 1) ‘ Fee, 2a sin — (ct — x ( 422 Scattering of Light by Small Dielectric Spheres. and the resultant electric force due to the infinitesimal layer is equal to k= sa Vi ea (e,)»==N dz foo op (ct—r)dr.27 me Vi uy Or "(es) = tn ae ads cos’, (et (ct—<),. k+2 while that of the original beam is equal to yell. sin (ct—2) and the sum is equal to 2a k-1l30V Qn e=e,+(e,),=3in a Oe aes 7 Cos (cf —2) (7) 2 or e=sin = (ct —<—6d) 2 9 2, or exsin (et =2) 003 (#2) y eee a CS) because the path difference 6 is a small quantity. Com- paring (7) and (8) we obtain 278 Lo pS 1 387rV un ee r a ON or o=N 5 Oe (9) On the other hand, Huygens’ theory gives for the path difference 6=(y--L)dz, 3Vk—-1 hence | Me Sa ae = ee a) o({k-1 or V2.N2 (=) =(y— ape 9: Tf we substitute this expression in (6) and assume E,= we obtain ne 32 m3 (y—1)? oe Nae an important relation first established by Rayleigh. | (433. J XL. Light Scattering by Air and the Blue Colour of the Sky. By R.W. Woon, For.Mem. R.S., Professor of Experimental Physics, Johns Hopkins University®. | ies scattering of light by dust-free air was first observed by Cabannes (Comptes Rendus, clx. p. 62), who focussed the image of a quartz-mercury arc at the centre of a dark box lined with black velvet, viewing the scattered radiation through a glass window in the wall of the box on a line formed by the ‘prolongation of the image of the arc. Strutt, working independently and without knowledge of the investigation of Cabannes, observed the same phenomenon and made a very comprehensive study of the relative scat- tering power of different gases, its dependence upon the density of the gas and its state of polarization (Proc. Roy. moc, vol. xciv. p: 45a, vol. xcv. p. 155, 1918). The results of certain investigations of atmospheric trans- mission appear to justify Lord Rayleigh’s theory that the blue sky is completely accounted for by the scattering of the air molecules themselves, independently of the presence of any foreign matter. So far as I am aware, however, no attempts have been ~ made, up to the present time, to compare the scattering exhibited by dust-free air in a tube with the direct light of the sky (as to intensity), or to compare the scattering power of the air near the surface of the earth on a very clear day with the average scattering power of the whole atmosphere. Abbot’s work on the absorption of light by the atmosphere showed that the loss of intensity due to passage through the lower mile of air above Washington was practically equal to the total loss resulting from passage through the entire atmosphere above the first mile (roughly four miles of homo- geneous atmosphere). This indicates that its scattering power is considerably greater than that of the higher atmo- sphere, where the foreign matter is present in much smaller quantities. There would doubtless be less foreign matter in country locations. The present paper deals with the scattering power of the air close to the ground, and the photometric comparison of the intensity of the light scattered by dust-free air, when illuminated by a concentrated beam of sunlight, with the intensity of the blue sky on a very clear day. We will commence with the second subject. The air was contained in a tube of black fibre 10 em. in * Communicated by the Author. 424 Prof. R. W. Wood on Light Scattering by Air diameter and 70 em. long, provided with two lateral tubes of brass furnished with glass windows for the entrance and exit of the concentrated solar beam. ‘The further end of the tube was lined wich black velvet, which is far superior toa smoked surface. The observation window was carried on a brass tube 2 cm. in diameter and 15 em. long, which was soldered into a hole which perforated the end plate of the tube. This was to shield the observation window as completely as possible from light reflected from the edges of the lateral tubes. The tube was filled with air filtered through cotton, and a beam of sunlight reflected from a silvered glass mirror focussed at the centre of the tube by means of a double convex reading- glass 15 cm. in diameter. The scattered beam was easily visible, even in a well-lighted room; it was bluish in colour, and was practically extinguished by a nicol properly oriented. No motes were visible. The diameter of the solar image formed by the lens was 4 mm., which gives us an area ratio of image to lens of 1/1400, 2. e. we have a layer of air 4 mm. in thickness, illuminated by a beam of sunlight 1400 times as intense as normal sunlight at the earth’s surface. This is to be compared with the intensity of the blue sky near the zenith, considered as due to the illuminaticn of five miles of homogeneous atmosphere by normal sunlight. It is to be remembered that the entire atmosphere if brought to normal pressure would form a layer five miles in depth. | As the sun was at 60° from the zenith at the time of the experiment, I chose a point 30° beyond the zenith for obser- vation, in order to work with the rays scattered in a direction perpendicular to the sunlight. A small flake of silvered plate glass with a razor edge was mounted in front of, and close to, the observation window. This reflected to the eye the.light from the selected part of the sky, which was reflected from a large silvered mirror placed in the shade just below the mirror which reflected the sunlight to the condensing lens which illuminated the tube. A rotating disk of black cardboard 35 cm. in diameter, mounted on the shaft of a motor, and furnished with a very narrow radial slit near its rim, was mounted close to the glass sliver. By means of this the intensity of the skv light could be reduced by narrowing the slit until it matched the intensity of the scattered light in the tube. If the glass sliver is viewed from a suitable distance its razor edge is seen in focus projected against the cone of scattered light in the tube, the edge disappearing when the match is secured. This occurred with a slit 0°2 mm.in width. The reduction in the intensity of the sky light is given by the ratio of the slit and the Blue Colour of the Sky. 425 width to the circumference through which it moves, in this case 1/4867. As we now know from Strutt’s experiments, the light cattered by dust-free air is almost completely polarized. The light of the sky exhibits, however, not much over 60 per “cent, of polarization in a direction perpendicular to the exciting rays. It seems reasonable to infer from this that about 40 per cent. of its light is due to secondary scattering (scattering of light coming from the rest of the sky and ‘the earth together with a certain amount scattered by the larger particles: forming the haze found at lower levels). This means that the sky as observed in the experiment had an intensity about 1°7 times as great as would be shown by a column of air five miles in depth. illuminated only bythe rays of the sun, and viewed end-on, which is really what is to be compared with the tube “Teanitnevsion. We have therefore effected a reduction of intensity with the rotating disk 1:7 times as great as would have been required if the conditions were as just specified. Applying this correction alters our ratio of 1/4867 to 1/2860. This is to be compared with the ratio calculated for the 4mm. of air illuminated in the tube and the five miles of air forming the sky. Sunlight at sea-level according to Abbot’s tables has a value for the blue-green portion of the spectrum of about 50 per cent. of its value in space, when the sun is at a distance of 60° from the zenith. This has been increased 1400 times by the lens, and we can therefore represent the scattered illumination in the tube (on an arbitrary scale), if we call the intensity of sunlight in space unity, by 1400 x 4x 1/2=2300. We must now compute the scattered intensity which we should expect from the atmosphere on the same arbitrary scale. Since we are observing a point 30° from the zenith, the effective depth through which we are observing is about 1:2 times the zenith depth, or six miles of homogeneous atmosphere. If we consider the sky as due to the illumination of this depth of air by sunlight of its full intensity in space (unity), the illumination will be represented by the number of millimetres in six miles, or 9,600,000, while our tube illumination was 2800. The ratio of these two calculated numbers is 1/3430, Snic the corrected ratio measured with the photometer was 1/2860. The agreement is remarkably good considering the enormous difference between the two intensities compared experimentally, and the uncertainty about just what values 426 Prof. R. W. Wood on Light Scattering by Air to take in the calculations. For example, the six miles of air are considered as illuminated by sunlight of the full intensity which it has in space. The light loses intensity as it penetrates the air, and is reduced to about one half of its value when it reaches sea-level. On the other hand, the scattering power of the lower atmosphere appears to be abnormally high, due to the presence of foreign matter, and there is in addition secondary scattering ; there is as “well probably some true absorption in the lower air. These effects compensate to a certain extent, and on this account it seemed best to consider the full intensity of the sunlight available for the production of scattering, in the case of ie sky. It would be far better to make. the experiment on the top of a ligh mountain, or even at one of the mountain obser- vyatories, and with the data given as to the dimensions and disposition of apparatus, the whole thing could be done in a day or two. The easiest way to make the slit on the disk is to paste two strips of very thin black paper on a microscope cover-glass and then paste the whole over a larger slit cut in the pasteboard disk. When the proper width has been found the cover can be detached and the slit width measured with the microscope. This was the method adopted in my summer laboratory with very limited facilities. An adjustable slit would be more convenient of course. The colour match was very perfect, which alone indicates that the light of the blue sky comes chiefly from the air molecules; for, as will appear presently, the light scattered . by the foreign matter in the lower atmosphere is yellowish in comparison with the colour of the clear sky. It is perhaps open to question whether we are justified even in considering the sky illumination as represented by the number of molecules in the line of sight (or in other words, the thickness of the homogeneous atmosphere) mul- tiplied by the intensity of the illumination. The relation holds undoubtedly for small thicknesses of dust-free air, but Abbot’s observations indicate that it does not hold at all for the distances concerned in producing the sky light. For example, he finds that withthe sun at an altitude of 46°, the sky 3° above the horizon is less than double the brightness of the sky at 57°, though the mass of air under observation in the line of sight for the sky near the horizon is thirteen times greater than in the case of the sky at an altitude of 57°. He shows conclusively that the intensity ° of the scattered light increases rather slowly in comparison with the increase in the number of the scattering molecules. and the Blue Colour of the Sky. 427 In this case his value found at 57° was at a point of the sky only 10° away from the sun, and is undoubtedly somewhat too large, as the sky in the vicinity of the sun shows an excess brightness due, as we shall see presently, to diffraction by foreign matter. I have a record of one measurement made by Mr. Nietz with an illuminometer from an air-plane at 3000 feet, which gave the zenith sky an intensity of half that of the horizon sky. Molecular Scattering in Directions nearly parallel to that of the Lwciting Beam. The most elementary theory shows that the intensity of the scattered light in directions nearly parallel to that of the exciting beam should be only Aeatle the value observed in perpendicular directions. ‘This follows from polarization considerations, for the light scattered in the parallel direction will be unpolarized, all of the components in the incident beam contributing to the ulumination. It is well know n, however, that the light of the sky, even ona clear day and on the top of a mountain, 1s enormously brighter close to the sun than in distant regions of the sky. At sea-level on a very clear day the light of the sky at 45° from the sun has only about 5 per cent. of the intensity shown close to the solar disk. This estimate was made by holding a mirror which reflected 5 per cent. of the incident light against the blue sky, and then observing the reflected image of the sky close to the sun, the solar disk being just hidden by a chimney. A very wood intensity match was secured at about half a solar diameter from the sun, though the colour match was imperfect, the light from the region close to the sun appearing yellowish white in contrast to the blue sky. The mirror was an acute prism of glass with a knife edge, one surface being painted with black paint. The reflexion was observed in the glass surface, and measurements with a photometer showed that, at the angles commonly employed, the intensity of the reflected light was roughly 5 per cent. of the incident intensity. ‘This device was employed subse- quently in other experiments, and will be referred to in future as the 5 per cent. prism (4 per cent. reflexion from the front surface and | per cent. from the back). While there appears to be no doubt but that the great intensity of the light in the vicinity of the sun is due entirely to diffraction by small particles, it appeared to be wor th while to examine the scattering by dust-free air in a direction nearly coincident with that of the exciting rays. The expe- riment was made with a tube of galvanized iron 3 metres TMNT TANTO \ a a a a = | LPE ALS tessa emai kp pAl gt Ue Me os a gg aT 428 = Prot. R. W. Wood on Light Scattering by Air long and 12 cm.in diameter, painted black inside. The ends were closed with wooden caps lined with black velvet; they are shown in section in fig. 1, The front cap was per- forated by a hole 3 cm. in diameter near the edge of the tube, ign te VELVET TR STAINS Oa a a 8 SS ea ee ee a OS eS es a ee ey and a lens of 3 metres focus cemented over it. This lens forms an image of the sun at the opposite end of the tube of the same diameter as its clear aperture, consequently the tube is traversed by a cylindrical beam of sunlight of constant cross-section with sharp edges. The cap at the back of the tube was also perforated witha hole, which was covered with paper painted black with the exception of a narrow strip near the edge of the hole. A bole 3mm. in diameter was burnt through the centre of the paper disk with a hot glass tube. denotes the angle between r and the direction v. Thus ry’ =r—vrcosrz (1- 3). Thus 1 UENO It immediately follows that 7’? = 77(1 —v? cos? 2d). Again from fig. 1 R?=7?(1—2v? cos? A + v' cos? dN) and R r sinX sind From these equations we derive 7 Y= 3" (1 —a? sin? Ay). On the Constitution of Atmospheric Neon. 449 Thus eR, Be 8?R2(1 —v? sin? aN R, is the unit vector along eP’, and H=vE sina. These results are of course well known, but [ think it will be admitted that the above is a particularly easy way of obtaining them. By extension of the principle described to quaternionic operators it is evident that the whole of the theory of Relativity can be very conveniently expressed in this notation. In conclusion I should like to express my thanks to Dr. Silberstein for reading my paper and for his interest Wat. XLIV. The Constitution of Atmospheric Neon. By F. W. Aston, ALA., D.Se., Clerk Maxwell Student of the Uni- versity of Cambridge™. [Plates VIII. & IX. se periodic tables of the elements arranged in order of their atomic weights the part lying between Fluorine on the one hand and Sodium on the other is of considerable interest. Soon after the discovery of argon and while the mon- atomic nature of its molecule was still under discussion, Hmerson Reynolds, in a letter to ‘Nature’ (March 21, 1895), described a particular periodic diagram which he had used with advantage. In this letter, referring to the occur- rence of the groups Fe, Ni, Co: Ru, Rh, Pd: and Os, Ir, Pt, the following passage occurs :— eC _ the distribution of the triplets throughout fhe whole of en abe known elements is so nearly reoular that 1t is difficult to avoid the inference that three elements should also be found in the symmetrical position between 19 and 23, 1. e. between F and Na,.... of which argon may be ONG hee” In 1898 neon was isolated from the atmosphere, in which it occurs to the extent of -00123 per cent. by volume, by * Communicated by the Author. Phils Mag. Ser. 6. Vol-39.-No. 232. April-1920. 2G AHO | Dr. F. W. Aston on the Ramsay and Travers, and was accepted as an elementary monatomic gas of the helium oroup lt density was mea- sured with extreme care by Watson (J.C.S. Trans. vol. 1. 810 (1910)), and found to correspond with an atomic weight 20°200 (O=16), making it the lightest element to diver ge from the whole number rule in an unmistak: me manner. Neon has many very remarkable properties, its com- pressibility, viscosity, and dielectric cohesion are all ab- normal ; but the first suggestion that it might be a mixture “ae ile alseenne Mon tin Ode by Sir J. J. Thomson of a faint but unmistakable parabola at a position corresponding roughly to an atomic weight 22, in addition to the expected one at 20, in positive ray photographs, whenever neon was present in the discharge-bulb (v. ‘Rays of Positive Hlec- tricity, p. 112). The first plate which showed this was obtained from a sample of the lighter constituents of air supplied by Sir James Dewar ; other specinens of impure neon gave a-similar result. So also did a portion of the gas used by Watson in the atomic weight determinations, whieh fact, together with the complete invisibility of any parabola at 22 on hundreds of plates where neon was known to be absent, was very strong evidence that the line was ascribable to neon and to neon alone. These facts led the author to undertake a_ searching investigation on the constitution of the gas by two distinct lines of attack, firstly attempts at separation, secondly accumulation of the evidence obtainable by positive rays. Hvidence of Separation. The experiments on fractional distillation and fractional] diffusion through pipeclay have already been described (F. A. Lindemann and IF. W. Aston, Phil. Mag. vol. xxxvii. May 1919). The former were completely nevative and only succeeded in confirming Watson’s value of the density already referred to. It has recently been shown (F. A. Lindemann, Phil. Mag. July 1919) that this negative result was theoretically inevitable. The diffusion results were more hopeful, an apparent change of density of about ‘7 per cent. being obtained in the first set of experiments. On the other hand, the more elaborate automatic apparatus started in 1914 has given very disappointing results, a difference of only °3 per cent. being obtained. This is doubtless due to the initial mistake in designing the apparatus to work at atmospheric pressure, Constitution of Atmospheric Neon. 451 under which conditions the mixing is very bad. It may therefore be said that the diffusion results are positive but too small to be conclusive. Kvidence of Positive Rays. This is available on three distinct counts: the character of the lines, their position and their intensity. Character of the parabolas. Plate VIII. shows a dark and a light print taken from a negative obtained in 1913 by Thomson’s method of analysis from a gas containing a large percentage of neon. The line due to the lighter constituent which will be called Ne* can easily be recognized as the brightest on the plate, the Ne? i. e. 22 line being the fainter one immediately below it. It can easily be seen that the latter possesses characteristics identical in all but intensity with those of the former. As has already been pointed out (‘ Rays of Positive Electricity,’ p. 111) the prolongation of the lines towards the vertical axis indicates that the particles causing them are capable of carrying more than one charge; multiple charges not occur- ring on molecules but only on atoms, one is led to infer that both lines are due to elements. Position of the parabolas. Measurements of plates obtained in this way indicated that it was probable that the lighter constituent did not correspond in mass with the accepted atomic weight of 20:2, but the accuracy was not sufficient to make this certain. Intensity of the parabolas. The relative intensity of the Ne* and Ne® parabolas ob- tained from atmospheric neon untreated by diffusion has been estimated by three different observers as about 10 to 1. Its apparent invariability is corroborative evidence against the possibility of the 22 line being due to the presence of other gases in the discharge-bulb. It will be seen that although by Thomson’s system of analysis the presence of two isotopes in atmospheric neon was indicated by several lines of reasoning, none of them can be regarded as quite conclusive, and it was realized that, failing separation, the most satisfactory proof would be afforded by measurements of atomic weight so accurate as to prove beyond dispute that neither constituent corresponded with the accepted atomic weight of atmospheric neon. 452 Dr. F. W. Aston on the Mvidence of the Positive Ray Spectrograph. The ‘‘ mass-spectra”’ yielded by the new method of positive ray analysis recently described (Ff. W. Aston, Phil. Mag. Dee. 1919) supply these measurements in an entirely satis- factory manner. Plate 1X. A, B, C, D, are prints trom negatives obtained by means of this apparatus. Hach contains a number of spectra taken with different electric and magnetic fields; the following table of values of P the potential between the electrostatic plates in volts, I the current passing through the magnet in amperes, and 7 the time of exposure in minutes, is given for reference :— ein: A B ee ec So Sain ecamenatag a Ee a 1 2 3 Os 1 2 3 4 5 P= 240 240 240 320 320 320 320 360 240 240 I='130 -450 :600 -600 -800 ‘$51 -600 -600 600 173 PA LOU Oey MOM ene 15) bo ore ls 4, C D a ath eo eae eee — See || ee ——- LO) So AS eG 102 62 ets F= 240 240 280 320 360 360 320 320 320 320 320 320 I='380 -550 °550 °550 -550 -700 482 -520 -554 606 -701 °798 P= 15 215) poe eae ao 10 110! “10 MOy aiOmaar On the left of each spectrum can be seen the small circular dot photographed on the plate just before or during the exposure, this is used as a register spot for measuring purposes. Plate A was taken with carbon monoxide. That is to say, the vacuum in the discharge-tube was maintained by continual pumping with a Gaede rotating mercury pump against a small leak of CO. It must be understood that this does not imply that the contents of the discharge-bulb were pure CO, since the use of tap-grease and wax joints necessitates the presence of hydrocarbons, etc., but at least one can be certain that the quantity of neon present was negligible as none had yet been put into the apparatus. The electric deflexion is away from the register spot, the magnetic towards it, so that the heavier masses are to the right of lighter ones. Spectrum AJ. was taken with a very small magnetic field showing the lines due to the hydrogen atom and molecule. In AIT. the field has been increased and a group of five lines are seen. These, which may be called the C, group, are 12-C, 13-CH, 14-CH, (or N), 15-CH;, 16-CH, (or O). Constitution of Atmospheric Neon. 453 They are important lines of reference and are certainly of the relative masses given above to the order of accuracy (one-tenth per cent.) claimed in the present experiment. In ATV. the deflexion has been still further increased and a new group of lines, the Cy group 24, 25, 26, 27, 28, 29, 30 containing the strong reference line of CO (or C,H,), have come into view. In AIII. of the ©, group only 15 and 16 are visible, and in A Y. the C, group has moved to the left and the strong line 44, CO, is seen to the right. Plate B was taken with CO to which about 20 per cent. of atmospheric neon had been added. Considering the spectrum B III. it will be seen that four unmistakably new lines have made their appearance, one pair between the C, and C, groups, another weaker pair to the left of the C, eroup. ‘The first pair are (Ne*)* 20 and (Ne*)* 22 singly charged, the second pair are the same atoms with double charges 10(Ne*)+* and 11(Ne*)** respectively. The other spectra consist of lines already mentioned brought into different positions to increase the convenience and accuracy of comparison and, in addition, there are on C I. two other valuable reference lines, O** apparent mass 8, and on the extreme left just visible CT™ apparent mass 6. | Method of comparing masses. It will be noticed that although the lines are broad (the best focus was only obtained by a series of trials after these results were completed) their edges, particularly their left- hand edges, are remarkably sharp, so that measurements of a reasonably good line from the register spot repeat to a twentieth of a millimetre with certainty. Hence for accurate determination of unknown lines only two assumptions need be made. Firstly, that the masses of the reference lines are known, and secondly that, whatever the function connecting displacement with mass, any two positions on the spectrum being taken, the ratio of any two masses giving lines in these positions will be constant. This being so, by moving a group of reference lines into overlapping positions along the spectrum it is clear that the whole length can be plotted out and calibrated. Fortunately there is an easy method of testing both these assumptions, for although it is impossible to measure the magnetic field to one-tenth per cent., it can be kept constant to that accuracy while the electric field is altered by a known ratio. But, for constant deflexions, mv? « X and mv « H=const. Therefore m« X71, so that, to take a ADA Dr. F. W. Aston on the typical case, the position occupied by carbon with a field of 320 volts should be exactly coincident with the position occupied by oxygen with 240 volts when the magnetic field is constant. Over the range of fields used in the case of neon, all such coincidences when expected have been found to occur within the error of experiment whatever the position on the plate. Kor some reason, by no means obvious, connected with the geometry of the apparatus the relation between displace- ment and mass is very nearly linear, a fact which lightens the labour and increases the accuracy of calibration very considerably. Numerical results. In the case of plate B the masses of the neon lines were estimated by carefully drawing the calibration curve repre- senting the relation between displacement and mass by means of the known lines 12, 13, 14, 15, 16 checked by that at 23. With plate D another mode of procedure was adopted. A linear relation was assumed and a table of corrections made by means of reference lines, which correction when subtracted from the observed displacement gave an exactly linear relation with mass. A correction-curve (apparently parabolic) was drawn, from which the appropriate corrections for any displacements could be written down and the masses corresponding to those displacements obtained by simple proportion. The following table gives the results :— TaBueE II. Plate B. | @evai (Ne®)t *, (Ne')t. (Ne®)*, 9:98 11-00 20-00 20-00 10:02 10-99 19-95 29-01 10-00 10-99(5) 19-97(5) 22, 00(5) Plate D. 10°01 11-06 20:00 21:90 9-98 10-98 19:98 22°10 9-98 11-01 20-00 99-08 ma an 19:90 21:98 9:99 11-01 19:97 29-00(5) The method of measurement combined with a slight halation of the plate tends to make the edge of bright lines appear a little too near the register spot, This is enough to account for the reading of the very bright Net line giving a Constitution of Atmospheric Neon. 455 mass a little too low. The above figures therefore can be accepted as fairly conclusive evidence that Atmospheric Neon contains two isotopes of atomic weights 20°00 and 22:00 respectively to an accuracy of about one-tenth per cent. In order to give the accepted density the quantities required are 90 per cent. and 10 per cent., which is in good agreement with the estimated intensity of the lines. Possibility of a Third Isotope. On the clearest spectra obtained with neon there are distinct indications of a line corresponding to an isotope of mass 21. This line is extremely faint, so that if this con- stituent exists its proportion would be very small, probably well under 1 per cent., and it would not affect the density appreciably. Attempts to bring this line out more distinctly by longer exposures have not succeeded owing to the fogging from the strong neighbouring lines, but it is intended to return to this point when further improvements of the method give hope of more conclusive results. This matter is interesting in connexion with the suggestion by Kmerson Reynolds already quoted”. ! In conclusion the author wishes to express his thanks to M. Georges Claude, who kindly supplied the neon used, and also to the Government Grant (‘ommittee for some of the apparatus employed. Summary. A brief account is given of facts which lead to the idea that atmospheric neon may be a mixture of isotopes. The results of attempts of separation are suminarized. The several lines of evidence adduced from the parabolas obtained by Thomson’s method of Positive Ray analysis are considered and shown to be consistent with the above theory but hardly conclusive. Mass-spectra obtained by means of the Positive Ray Spectrograph are produced. Measurements from these are given which prove conclusively that neon contains two isotopes having atomic weights 20-00 and 22:00 respectively to an accuracy of about one-tenth per cent., their proportions being therefore 90 per cent. and 10 per cent. by volume. The possibility of a third constituent is indicated. Cavendish Laboratory, December 1919. * Though at the time this was made isotopes were not thought of, and the modern idea of atomic members has since precluded the possibility of three distinct elements. a ree XLV. Intelligence and Miscellaneous Articles. ON THE EQUATION OF STATE. To the Editors of the Philosophical Mugazine. SIRS,— | a paper published in the ‘Philosophical Magazine’ for August 1918 it has been shown that from Boltzmann’s theory of entropy we can arrive at an equation of state ) at pa—sy log 1-7 =o New 1. Lapa ee _ Amongst other applications of this theorem it was shown that 2e the value of the critical constant ca would be = 3°08, perve Crm 8 Ned instead of van der Waals’ 3 and Dieterici’s 5 — sha) A table was given showing that in the case of the most of the elementary gases the value 3°53 corresponded better with experi- ° 8 zy mental results than either ao 3°69. The list comprised He, N,, O,, Xe, bus not Hydrogen, about which the experimental data were not satisfactory. _ Recently the critical data for Hydrogen have been re-determined in the Laboratory of Prof. K. Onnes (vide Proe. K, Akad. Weten- schappen, Amsterdam, vol. xx. 1917). It has been found that 6¢=33" 18K, dp =:0310, pe =12°8 atmospheres ; from these data, Ka 330. | It is superfluous to add that this value of K is in much better agreement with the value 3°53 than with either the value 3 == 0601 = 369, thus corroborating the belief expressed in the aforesaid paper that the equation (1), though not final, marks a step in the right direction. Mucu Nap Sana. University College of Science, SATVENDRA Natu BAsv. Calcutta, India, THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. ee f | [SExTH SERIES. ] \ WKY CY ar won XLVI. The Spectrum of Copper. By W.M. Hicks, P.2.S.* N the following pages similar methods are applied to the discussion of the copper spectrum as have already been successful in dealing with those of silver and goldt. The are spectrum of copper differs from those of the other metals of the group in that it is very rich in lines. It is, however, similar to them in general plan, and the similar conclusions arrived at in each receive greater support in combination. Strong well defined sets are given for the orders m=2, 3 in both the S and D sets and for the first order of P. Those for S(2) are in the red and in step with those for silver. The others show the Zeeman patterns proper for their respective types. These may therefore be adopted definitely as normal series lines. For the higher orders, however, the intensities fall off more quickly and irregularly than usual and there is also evidence, as will be shown below, for numerous displacements. In fact these are apparently the cause of the low intensities and the large number of weak lines. In consequence the determination of the S and D limits with any exactness from the series lines alone is rendered impossible. The doublet separations of the S (2) and D(2) sets are the same within at least 01. For S8(2) * Communicated by the Author. + Phil. Mag. Sept. 1919-——July 1919. Plul. Mag. Ser. 6, Vol, 39, No. 233. May 1920. 2H 458 Prof. W. M. Hicks on the measures of Meggers * and Meissner} give 248°43, Eder and Valenta 248°42, whilst Meggers’ estimates for his probable errors for the two lines give for dn -03 and °01, probably less on a difference. The separation for D(2) as calculated below from Fabry and Perot’s interferometer measures of D,,, Do: is 248°44. The separations for the S and D doublets therefore agree within one unit in the second decimal. Now a 6 displacement on the limit alters the separation by ‘06 and the limit itself by 4°94. Consequently the two limits S(~ ), D(«) cannot possibly differ by more than a shift due to one oun, z.e. by 1°2, and are practically, if not absolutely the same. If, however, the observed sets for m=4 as shown in Tables J., II. be taken as normal, the limits as calculated from the first three lines are S,(% )=31536°29+2 and’ D,(o )=31515-48+10, and they cannot be the same. At least one set of the fourth order must be abnormal. In the case of 8,(4), Kayser and Runge and Eder and Valenta (spark) agree and Crew and Tatnall give dA='02, whilst in 8,(4), C. T. give dA=:04 on K. R. and E.V. have not observed it. We may be justified therefore in taking observation errors as small. ‘The ob- served separation (K. R.) of 248°71 is thus abnormal. It is ‘27 too large by K. R. and ‘13 by C. T.{ The observations are sufficiently exact to prove that a negative displacement has taken place, but not so exact as to determine how much. If the displacement be —w6, e=1 or possibly 2, vis increased by ‘062 and the limit by 4°94a. In other words the ob- served lines must be regarded as being affected by a limit 4°94 larger than the others. Treated in this way the true limit 8;( ) calculated from m= 2, 3, 4 becomes 31536°29 — 12°'462+2. The data for the D give a wide margin of variability for D(«), chiefly due to uncertainty in D, (4). K. R.’s measures give n=27103°00, C. T.’s 27104°62 with separation 252°14 instead of a value less than 248-44. Also i. V. give a spark line n=27109°23 which gives a separation of 245°66 with D., (4), or a satellite separation of 2°78. The data are too uncertain to secure acloser value to the true value of D(«). It remains to see what evidence can be obtained from summation lines. The material from the summation lines is given in Tables I., 11. TheS give direct indications from §,(2),8,(3),8.(4) for a limit 81523°47+ 2°55; ..23°48+°7; ..24°8241°6 and from the 8, (4) of ..28°35+1°5. But we have already seen that * Bureau of Standards, Washington, No. 312 (1918). + Ann. d. Phys. 50. p. 718 (1916). + But Hasbach gives v=248-42, the Spectrum of Copper. 459 the S, (4) hasa limit 4:94a larger than the others. If there- fore 37169 is the true summation line for observed 8, (4), its limit should also be 4°94z larger. This is clearly indicated with 2=1 making the normal 8, (7) =31523°41. This is a remarkable agreement supporting the previous reasoning of (oO ) = = 3153629 —12°460-+2 = 31523" 83+2 with «=1. The set S.(4), S.(4) give an intermediate limit which, how- ever, is explained by t “takin g 37410 as the normal 8, (4), not displaced as is the S,(4 ue The mean would therefore he Lx 4-94 =2°47 preter than the true, or S,(« )=31770°79 “1 7 giving S, (x = olo22-30 a: 17, the same as the others within error limits. There appears no §,(2) but a line observed by McLennan X= 1925 or n= 51931-8413 in which all values within +13 are equally probable is linked to it. In fact e.51931-8+13=50932:0+13 as §,(2) gives S.(o )=31766°68+6°5=31771-92 so far as the wave length is known. The evidence from the combined difference and summation lines, considering normal lines alone, is thus quite decisive in making Ss (2) quite close to 31523-48. The D material is not so definite alth ough it supports this value. The sets under D,,(3), Do.(3) give 31524:17+1 ; ..21°56+3°4. Also the doubtful D,,(5) is related to a D,,(5) which gives D,(@ )=31522°39+°8. Butinm=2 the lines AXN2276, 2263 as the summation Dy, Doz in the table give D, (0) =31537" 59+°5. There is however a set con- Heed be the u-link to Dip, Doo Which agrees with the others. The general agreement in favour of S; («© )=31523 shows that the lines AA 227. 2263 cannot be au D(2) set. Their separation as a Dy, Dy set should be 245°44+4 6°60= 255-04. Tt is 254:13+2 as observed. Ii they represent (—526)D, the separation should be 52x-06=°35 larger or 255°39 which the observation errors allow, and the normal D set be 53 x 4:94= 28°41 less for Dj; and 52x 4°99= 28-70 less for Ds.. The means then become 14°20 and 14°35 less or 31523°39, 31771°31 agreeing with the others. They may therefore be this displaced set, but they must not be used for deducing the limit. The result is that the limit is within a few decimals of 31523°4. The closest limits of variation are those given by m=3, viz. 8,;(«© )=381523-48 in which the possible. error 1s +'7 but the actual error, judging from the combined results, is probably much less aaiihons 102. (he same value is also giv en by certain p(1)—f, p(1)+/ combinations considered below (p. 465). For one of these sets—/ (4)—the observa- tion errors are small and give 8,(« )=31523°475. In taking then the limit as 31523°48 + & the value of & will not be more +han a few decimals. Using this as limit and the accurate ed Be 460 Prof. W. M. Hicks on values for m==2, 3, the formule for 8; and Dy, are e OR Dy) S 3152348 —N/{ m+-a31291— "ob 9 “9 2 D 31523°48—N { m-9s9023— 20778 } ; or if the order in § be taken as m—4, ( ies $=31523-48-N/4 m5 719 A i a : The calculated 8, (5) =27783°86 is sustained by two linked lines, viz. ue (6n)3512°19 = 2846424 — 680°68 = 2778356, (2n)3688°60 .uw=27102°99 + 680°68 = 27783°67. The latter is the line allocated by K. R. to Dy (4) which we have seen cannot be correct. ‘The calculated value for Dj (4) is 2 = 2710052. An are” lime) Msg Oe iien - n= 27104°60 would be (2 6,) Duy, also by C. T. at n=27117-17 is 9°63 ahead and as 26 shifts 9°88 it would be (—26)Dy. These are therefore possibilities. Hasbach gives 7 = 3687°5 L. A. or n=27110°717—7:3 dr R.A. His excitation may have produced the normal set. The P series—There is a clear set for P(1), strong, with the normal separation, showing reversals, and the Zeeman patterns belonging to Pj, P2. With the value of 8, (# ) just obtained the limit P (# ) is known with the same exactness. But although the observed region should include several succeeding orders it is difficult to allocate them with certainty. There is a large number of weak lines where P (2) should occur. Like the 8 and D series the P series appear as if frayed out and any summation lines lie far down in the ultra-violet beyond any observed region. There is a line at. 49363°7T0+4°9 (X= 2025) showing reversal and in about the proper position for P, (2). P,(2) should be about 60 above it, but there is no trace of it. It gives for the combination p (1) —p, (2), n=18580'93+4°9 which may be the observed spark line (H. V.) (3)18579°61. Another spark line at- n=(3)18772°87 might correspond to p.(1)—ps2 (2) giving the separation of the P (2) lines as 55:10, in order but rather small. The formula calculated from this line as P, (2) with P(1) and the known limit gives lines for succeeding orders which have not been observed, nor does the calculated sequence tor P (3) give any p(1)—p(38) combinations. If, however, the weak line n=(1)49069'64+2°4 be taken as P, (2) with P, (2) not observed, the resulting formula gives. the Spectrum of Copper. 461 lines for m=3....6 whose wave numbers are 55031'92+1°30, 57710°49+°75, 59141°05+'5, 59994:15 in which the am- biguities are due to possible errors in P, (2). These all lie in the ultra-violet region of Handke. In comparing observed with calculated it should be remembered that Handke gives his measures to ‘1 A., whilst dn varies from 28dd to 40 dX. Consequently any value of dX between +:05 is equally probable or of dn between limits varying from 1°4 to 2.. In addition are observation and standard errors of which no estimates are given. The formula for P, is ef\< 2 n=62306-25—N] | 89742 Lae } m=2. 49069. This gives p, (1) —p, (2) =18286°87+2'4. This is not observed, but an observed line n= (2) 18475°93 treated as p,(1)—p.(2) gives the P (2) separation as 59°38+, the expected amount. m=3. |55031:°92+1:30]. There is an observed line at (1) 55026-69—30dr. Also (2n) 24245°24 (6) as pill) —pi (3) gives P; (3) = 5502801 +°6. m=4, [d1710°87+°75]. There is an observed line at (4) 577234 — 33 dn. Also (2n) 26931°74+°3 as p,(1)—p,(4) gives P, (4)=5771 45143. (2n) 27188°69 as (1) — (4) gives P, (4) = 5772323 4-4. The former supports the calculated P,(4). The second is the observed, but it is curious that P, should be seen and P, absent. m=95. [59141°05+°5|. This is not observed, but it is connected by u-links to lines on both sides. Thus ue (8)99822°9 = 59142°24—36d), (6) 58459'0.u = 59139'70—34 dn, an exact series inequality. Also (2n) 2836638 as p,(1)—p,(5) gives P,(5)= 99149°15-°8, probably P, with no observed P; as in the previous order. m=6. {5999415}. This is not observed, but there are v-linked lines for P,, P., viz. (6) 59297°9 .v = 59989:'9—35 dr (8) 59290°9 .v = 59982°9—35 dn. These would give the P(6) separation as 7 —35(dd,—d)q) =(+3°'0 as against a calculated value of 4. It thus agrees within equally probable errors. Also (2n)29212°61 as p,(1)—p,(6) gives P,(6) = 99995°38+:8—also to the violet of the linked lines. 462 Prof. W. M. Hicks on The next two calculated lines are 60543, 60917. There are strong lines at 60536—36°6dA and 60908 7—37d),. but they are too strong and moreover show evidence of being summation lines (see later, p. 473). The formule reproduc- tions combined with the observed combinations would then seem to support the allocations given as at least one of the frayed out fragments of the system. The whole region round the observed combinations is full of lines of the same nature (2n) and may possibly be combinations for some of the other P fragments. The combinations considered above are collected in Table III. at the end. The establishment of the hypothesis of the break-up of a normal series into a large number of displaced, and linked or otherwise shifted lines is of fundamental importance. The laws regulating this break-up can only be discussed when a large mass of material for comparison has been collected. As a contribution to this some instances are considered in an appendix. Spectral constants—With the establishment of the 8 limit as 31523°48 + Eit becomes possible to apply the same methods. as were employed in the discussion of the spectra of Ag and Au to determine the value of v more accurately and to deduce therefrom the value of the oun and of the various links. There are nointerferential measures giving v directly but, as in Ag, Au, Fabry and Perot give such for D,,(2) and D.(2). K. R.’s values are very accurate and give definitely _ that A=506 and that the satellite separation for D(2) is due to 236,. The separation of Dy, Do. given by F. P.’s measures, AND153°251+ 001 2,, 5218°202+-001 2, in I.A.,. is 241°4632 —:00376 2.+°00367 2,. The calculation carried out on the same lines as in the case of Ag and Au gives tm ye AS vy = 248°44402 —-0038 (a. — 2), A = 7307:087 —°3310 E—-118 2 +110 2, = 146°1419 —-00662 €—-0022 y where y=a.—4. From these and s(1)=31523°48 + E—P, (1) =62306°254 & the calculated links are Gh) Psy yA C==2 oles b = 248-44 d= 2a4io9 Q == DSR i u = 680°68 Vi OG 20 2e With ambiguities of €=+°l and y of 1, the spectral the Spectrum of Copper. 463 atomic weight of copper in terms of that of Ag comes to 63°5569+°006. Brauner’s value of the chemical atomic weight is 63°56 +:01. With this approximate value of A it is now possible to test for D lines corresponding to the order m=1, in which the D satellite sequence or d,(1) has its mantissa a multiple of A. The actual values will depend on the true value of N but the deviation from the truth on this account will not amount to more than 7 or 8 units in the sequent, and will be passed over for the present. The extrapolated mantissa of d,(1) ts 967250. The actual value for d,(2) is 977297. That of d,(1) must be less than the latter, and the multiple therefore be in the neighbourhood of 132. Any evidence of the real existence of D(1) lines must be based on com- bination lines with the inverse D! set, and on the presence of summation lines. It may be said at once that no such evidence can be found for 131A. With the values above 132 A=964535—43°6 &—14y. The d,(1) is N/(1+132 A)? = 28417°664+1°26&4-41y. Consequently Dyp (1) = 3105°82 —-26 &, D,.(1) — 59941°1442°26€. There is a line at (1) 99916°12 —36dr of Handke’s but no other in this region. If it is the surviving strongest Dj, line the D set, should be Di. = 59941:14+4-2°26 E, Di, = 59916°12— 36ad)nX, Do» = 60189°62 + 2°26 E. Eis probably not greater than a few decimals so that the satellite separation is c=25+ 36dn, but this may be modified up to +7 by the unknown error in N. As it stands how- ever, the separation of 25 would be due to 246,=66, as usual slightly larger than for the m=2 set (236)). The inverse D’ lines are in wave numbers in R.A. Dj.’ = —(8) 17537°66, Dire = —(1) 17439°40, 9 = — (8) 17289-22. It should be noted that the D,, line instead of being the strongest of the set is extremely weak, and seems to indicate that very few of the d, configurations have split up to form 464 _ Prof. W. M. Hicks on those giving d,. The calculated and observed combinations then give Cale. . Obs, oN « a3 (1)20640°47 Exner and Haschek ee Ure 18) ta) 20639°69 Hasbach d!(1)—dy(1) = 2054522 (2n) 2054350 +°85. There is also a line by E.H. (2) 20574:25+°85 which if it is the combination d;'(1)—d,(1) gives c=30°75. With 133A, d,(1) = 28207:-434+1:25€, D,.(1) = 3316°05 —°25€ Dyo(1) = 59730°914 2-25. There is a line (7) 59719°32 —35-6dX which would serve as a D,, giving c=11:64+35dd. It is so strong that we should expect that at least Dy. should be visible, but there is nothing which can possibly stand for it. The combinations are Cale. Obs. d,'(1)—d,(1) = 20853'94 (2n)203852°66, dy'(1)—d,(1) = 20755°61 none ; and again 20767°6 (a spark line observed by Hemsalech) can serve as d,/(1) —d,(1) giving c=12, or the same as by the direct summation lines. Litile weight perhaps should be given to this case, but so far as it goes it points to a second system of D series. That part of Handke’s observations which lies beyond X=1770 shows clear evidence of connexion as summation lines with D(1) sets, depending not only on different dy sequences, but on displaced d sequences and ‘displaced D(« ) limits—a short discussion of them is given in the Appendix. The EF series—So long as D(2) was believed to be the first set in the D series it was natural to attempt to allocate F series with limits =d,(2), d,(2), with therefore a separation of 6°98. No such doublet series were found. Nevertheless Randall proved the existence of certain se- quents with all the appearance of belonging to the F type. The proof depended on the existence of lines satisfying the conditions for combinations p(1)—/(m) d'(1)—/f(m) and one doublet set d(2)—/f(3) far up in the ultra-red but with a separation 10°5 instead of 7, thereby indicating a satellite effect in the / sequence. We will consider the Spectrum of Copper. 465 these, with extensions, before proceeding to the discussion , of the true F series d(1)—f(m). It may be noted in passing that the existence of the D(1) set explains why no F set was found on the old supposition, as only a few lines of the combination d(2)—/(m) would be strong enough to appear. The material is given in Tables V-VII. We will now discuss them in order, beginning with Randall’s allocations for m=3. The data for the constant sequents are pi(1) = 31523°48, dy’(1) = 4896291, (2) = 12365-67. po(1) = 31771°92, dy'(1) = 4906137, da(2) = 12372°65. m=3. The p-f (Table VII.) give /,(3)=6880°31 fo(3) = 6877°16. d(2)—f (Table VI.) ,, (3) =6880-06 f(3) = 687652. dQyef » Ff (3)=6880'12. 33 99 The agreement amongst the sets renders the allocations practically certain. But it may be noted as exceptional that the jf, sequent is less than the /,—1. e., it is a positive displacement on f; contrary to the universal rule in d satellites though not unknown in f. Further, in the d,'—f we should expect f, and not f, as here. It is natural to suppose that there is no / satellite and that the d,(2) receives an extra displacement instead of ff, but this is negatived by the fact that f, occurs also in the p—f line where this explanation is excluded. The appearance of the f; in the d,’ may be due to the fact that in copper the, D,,; line is scarcely formed and _ that its place is taken by the undisplaced D,,. On this sup- position we should expect a double line dn=3'2 or dN="2. Lines are also found corresponding to p.+f. and d,' +f). The first, a spark line, gives f,(3)=6876°08; the latter fi=6885°8—31 dx. The separation of 7,(3), f,(3) appears as 3°54, 3°15, but the measures are not very exact. A displacement of 616 gives a separation 3°14. m=4. The p—f are displaced by the electric field and strengthened by it, as is the rule with this type of com- bination. Stark’s measures for zero field are used. These give f,(4)=4402-00, f,(4) =4399°56. The summation gives f,(4) =4402:00, f.(4)=4399°68. The difference and sum- mation thus agree with great exactness. They give as the mean for p,(1), 31523°475, thus closely supporting the 406). Prof. W. M. Hicks on already obtained value. We may therefore regard the & as probably <‘02. The d,’—f gives f(4)=4402°56+1, that is as in m=3 il gives cn The lines 44576 and 53446 may be put in evidence because they appear to satisfy a set d\/—X,d.'+X. d,/—X would be 98°46 larger and = AAG TAG. This with the second as dA gives dy' = 49060°56, X =4386:1, or a corresponding set with a large sequent displacement as in f(3). m=. Only the p—f combinations are observed. They are the samé as for D(5), 1 e: fO)=dO), and) ain J1(9) = 3059-20 4-4, 7,(0) = 305m 10 1-2) intend Bey ie a an observational lacuna and no d’—f is seen. m—6. The p,f, ‘ive j,=22d07008 andy e250; 0maiuN mean limit 31523-85. Nevertheless there may be some doubt as to their belonging to this system since their mantissee as given below are not in step with the others. Also the p+/ line is excessively streng. If this /(6) 1s —2A displacement on the normal it would be more in line. The line under p,—f is also probably not the real line. It is 1°56 too small and moreover is the exact linked line ¢.P,(1).46800 as d,’—f gives f(6)=2261°22 and 46834 gives f=2227:44. These look like equal and opposite displacements of 76, in the limit. Their dif- ference is 83°78 and 146, shifts 33°58. This would give f,\(6) =2244-°33 whose mantissa 992044 is quite in 1 step with the other orders. m=71, d’—f gives f(7)=1733-70. It is not in step with ue normal. f. If it is analogous to the second in m=6, i.e. (76) d,’—f, then fH1Tls 9 and comes into line with the ieee An attempt may now be made to determine the values of f(1, 2). The run of the mantissee of the higher orders as seen below shows that the denominators of these sequents must be near 1:992 and 2°992. Moreover we have to expect satellites depending on displacements of about 646, which as has been seen is that for f(3). This means for Mba fe ( oy abont 12250 and satellite separation 7, and for m=1, f(1) about 27450 and satellite separation Zoe The fact that the lower orders of F series are very sus- ceptible to displacement and consequent weakening must also be borne in mind. Several representative sets corre- sponding to the same order m may therefore be expected, and the allocation of a suitable set does not mean that it must be the normal one. the Spectrum of Copper. 467 m=2. The p.f give f,(2)=12257-10, /,(2)=12246-01 with satellite separation 11:09. In the d’.f there now appears a representative of d,’—f;. They give Fit2) =12204-15(1:34) and f,(2)=12249-53 (1°34). The denominator of 12257:19 is 2°991400, that of 12249°45 is 2°992313, which differ by 646. Also 12257°10 and 1226415 differ by the same within limits. We are here clearly in the presence of successive satellite displacements. 1t may be noted that again the p.fand da’. 7 combinations both give f. O10, Ons and such a condition never can be verified from (5), which for great values of p, gives at most p=? In other words we can also say inasmuch as the sun has an apparent density equal to 1:41, the coeficient of absor ption h cannot be greater than 7°65. 10-2, The experiment gives 6°73 .10-1*; theretore the facts agree up to now ith the BB iced theory. I bring these considerations to a close, ¢ calling attention to the fact that if we admit the hypothesis of gravitational absorption, the calculation worked out for the sun with the simplification of constant density cannot lead us to very erroneous results. -In fact, if we substitute for this hypothesis of constant density, another law of variable density, this will be, as a matter of course, greater at the centre than at 504 Haperimental Researches on Gravitation. the surface. Therefore, on the one hand, the fact that the matter would accumulate itself towards the centre would have for consequence that the absorption of the gravitational force of the greatest part of the matter would be accomplished across greater thickness, as the gravitational action would have to pass first chiefly from the deeper layers to the surface, then to the exterior ; but, on the other hand, the exterior mass has a reduced density, hence the absorption itself is diminishing. Therefore these are two contrary causes that in general will not balance each other, but the effect of one is subtracted from the effect of the other, leaving the mean true density not very different from the one established by my experiment and calculation. Summary and Conclusion—Hxamining Newton’s Law, I have come to think that the force of gravitation can weaken itself by absorption due to ponderable matter. Following other arguments, J have come to suspect that the matter which shows the torce of gravitation might heat itself. Although such conception might resolve in a new way the old controversy on the origin of the sun’s heat, I state it with all reserve. I have undertaken afterwards to treat theoretically the case of a spherical mass with constant density, subject to the absorption of its own force of gravitation, and from this work [ have deduced the elements necessary to carry out an experimental control of my hypothesis. I have carried out this experiment by weighing in vacuo a leaden ball whose weight was 1274 er. symmetrically surrounded by 104 kg. of mercury. Having previously avoided all the possible causes of error, I have been able to conclude that the leaden ball loses 7°77 . 107" of tts weight by the presence of the mercury. Such result causes the determination of the quenching constant (factor of absorption) per unit of density and length, as Gua 0 e. Applying finally this result to the sun’s case, I calculate its true density as 4°27. The importance of this research is obvious, and I do not think that reasons for criticism can easily be found. Anyway, as I am the first to wish to test in all possible ways the results [am publishing, ] may mention that it is my intention to repeat my experiments with far bigger apparatus. For the purpose, in the Laboratorio di Fisica del Politecnico di Torino (Italy), an apparatus is being built in such propor- tions as will render possible an experiment with 10,000 ke. of lead. On the results that I shall obtain with it I shall report in due time. [ 505 ] XLIX. The Hot-wire Anemometer: its Application to the In- vestigation of the Velocity of Gases in Pipes. By J.S. oF Tuomas, M.Se.(Lond.), B.Sc.( Wales), A.R.C.S., ALC [Plates X.—XIII.] ss ae possibility of utilizing the cooling effect experienced by a fine heated platinum wire, when immersed in a stream of fluid, asa method of practical anemometry has been placed on a sound theoretical basis by King f. Morris ¢ has examined the characteristics of wires of various kinds for use in this connexion, and has described a number of methods of employing the hot wire for the same purpose. Both these investigators§ and others || are responsible for types of so-called hot-wire anemometers to be employed for the measurement of the velocity of air-currents. The author has recently had occasion to examine the possibility of ‘ the use of such instruments in an investigation connected with the flow of gases through pipes and orifices, upon which he is at present engaged, and the present paper contains an account of certain interesting results obtained as the result of such examination. Of the various types of hot-wire anemometers available, the type due to Morris and described by him in Eng. Pat. 25,923/1913, was found on examination to be the most suitable for the purpose of the investigation. This type of anemometer is constituted of four equal wires of the same material—platinum by choice—composing the four separate arms of a Wheatstone bridge. One pair of alternate arms of the bridge is shielded by means of sur- rounding tubes; the resistances being all adjusted to equality initially ; at any temper ature, the bridge remains balanced at any other temperature. The bridge-wires being inserted into a stream of fluid, the balance of the bridge is upset, the unshielded arms alone being subjected to the cooling effect of the fluid current, and the galvanometer deflexion serves, after calibration of the instrument, to measure the velocity of the stream. Morris, in the salves of his instruments, employed air-currents produced in a vertical wind channel, the stream of air passing vertically downwards therein, at * Communicated by the Author. Tt Phil. Trans. Roy. Soc., A. 520, 214, pp. 373-482 (1914). Phil. Mag., 1915, p. 570. 7 British Association, Dundee, 1912; Electrician, Oct. 4, 1912, p. 1056; Engineering, Dec. 27, 1912. % King, Eng. Pat. 18,563/1914; Morris, Eng. Pat. 25,923/1913. || See e.g. King, Phil. Traus., loc. cit. p: 404. Phil. Mag. 8. G. Vol. 39. N . 233. May 1920. 21 506 Mr. J. S. G. Thomas on right angles to the anemometer bridge-wires. The heated wire in King’s calibrations was attached to a radial arm which was capable of being rotated in a horizontal circle. The wire in these latter experiments was used horizontally, ver- tically, and inclined at 45° to the vertical, respectively, The conditions of calibration in the experiments both of King and of Morris differ essentially from the conditions ruling in the case of the flow of gas in a tube or pipe. The anemo- meter employed in the present experiments is shown dia- erammatically in fig, 1, and a transverse section across the ioe ale tube at the exposed platinum wire is shown in fig. 2. Two fine platinum wires (the one AB exposed, the other CD shielded within a surrounding copper tube) were inserted, as shown, in the tube, through which a current of air or other gas flows. The manner of inserting the wires is best seen in fig. 2. The ends A and B of the fine platinum wire are affixed, by means of the smallest amount of silver solder affording a secure junction, to portions of considerably thicker copper wire AH and BF as shown. These copper wires pass tightly through holes bored through plugs of ebonite K and L inserted into brass tubes joined to the main tube at right angles as shown in figs. 1 and 2. Precautions ae oe ce the Hot-wire Anemometer. 507 were taken that the ends of these plugs were diametrically - placed with respect to the section of the flow-tnbe, and that the continuity of the transverse section of the latter was not disturbed by the insertion of the ebonite plugs. The position of these plugs was secured by means of screws Oand P. The copper wires AE and BF passed through fine holes in the rods G and H, and by tightening up the nuts M and N the positions of the copper wires were secured. ‘The ends of the copper wires were secured as shown between nuts on the rods, which also conveniently served as terminals for inserting the fine platinum wire in any desired circuit. The protected wire CD was inserted and secured in like manner. The wires AB and CD were cut from the same sample of platinum wire secured from Messrs. Johnson and Matthey. The sample was aged by the passage of a current of 1°5 amp. for 2 hours or so. Their respective diameters were measured by means of a high-power microscope carried on an accurate micrometer- serew measuring to0°01 mm. and by estimation to 0-001 mm. Portions were chosen of as nearly as possible the same radius, and whose surfaces were pitted as littie as possible. The lengths of the wires employed were, as nearly as possible, equal to the diameter of the flow-tube in which they were inserted. The wires were located in the tlow-tube at such a distance apart that disturbances of the flow set up by the presence of CD produced no effect at AB*. The two wires constituted twoarms of a Wheatstone bridge, the remaining arms of which were formed of resistances unplugged from a resistance-box. The individual resistances of the box were composed of man- ganin of negligible temperature coefficient, and the individual resistances were found to be correct to 1 part in 5000. The battery and galvanometer were connected tothe bridge in the manner indicated in fio. 3. AB represents the protected wire and BC the uncovered wire. AD and DC represent resistances unplugged from the resistance-box. In general, AD was adjusted to either 1000 ohms or 2000 ohus. The battery is shown at E. By means of the rheostat R the total current in the bridge, as indicated by the ammeter M, could be adjusted to any desired value and maintained constant. The ammeter was:a direct-reading Siemens and Halske millivoltmeter provided with shunts. Readings could be made to 0°002 amp. by estimation in the region of 1 amp. Calibration of the instrument against a standard Weston instrument indicated that its readings were correct * Jt can be shown, see e. g. King, Phil. Trans. A. 520, p. 405 (1914), that the disturbance at a point distant from the inserted tube equal to 10 times its radius amounts to only 1 per cent. of the velocity thereat. 97, 2 508 7 Mr. J. 8S. G. Thomas on to within 0:2 per cent. The drop of potential along AB could be measured by means of the Weston voltmeter of resistance about 1000 ohms, which could be inserted for the purpose by means of the key K. The resistances of the wires Fig, 3. 19) M R es AB and BC when heated by the currents employed were of the order 0°5 ohm. It is evident that to 1 part in 2000, the . whole of the current in the bridge when balanced, passes through the arms AB and BC. A bridge connected and em- ployed in this manner may be described as a constant-current bridge, possessing maximum sensitiveness in accordance with Maxwell’s rule, and with the maximum generation of heat in the sensitive branch. The bridge employed in King’s expe- riments was of the constant-temperature ty pe, the balance of the bridge after disturbance due to the cooling effect of the current of fluid being restored by the passage vof additional current through the wire. The gene ral mode of employing the anemomeree in the present | series of experiments is shown in fig. 4. The air or the Hot-wire Anemometer. 509 other gas is passed through the gas-meter M, and thence to a wide tower loosely packed with lumps of calcium chloride. A thermometer T is placed in the stream at a point just prior to the gas entering the brass pipe P in which the anemometer A is inserted, a gas-tight junction between the ends of the anemometer tube and the remainder being effected by means of carefully made spigot unions, constructed so that no dis- turbance of the stream is introduced thereby. No appre- ciable error could be attributed to irregular rotation of the drum of the meter. Even at low velocities extremely steady velocities were indicated by the hot-wire anemometer. lt is essential to employ water free from any dissolved coal- gas or similar gas in the meter as the indications of the hot-wire anemometer in the form described are extremely erratic in the presence of small quantities of coal-gas de. The meter was therefore thoroughly washed out sori water. The stream of air was derived from a gas-holder having a ‘volume of 5 cubic feet. The pulley over which passed the supporting rope of the bell, the other end of which was weighted, was provided with a cam over which passed a rope carrying a compensating weight, so that the same pressure was maintained throughout the fall of the bell in its tank. Any desired pressure could be obtained by suitably weighting the bell. It was found that the desired constancy of pressure could be readily obtained by the use of this pressure device, without the introduction cf any further pressure governors in the circuit. The precautions already detailed were taken to ensure the freedom of the water in the holder from any dissolved coal-gas. In calibration of the anemometer, air was passed through the apparatus, and the rate determined by observation of the pointer of the gas-meter, this obser- vation extending over a period varying with the velocity of flow in the pipe, so that. with low velocities the period of observation was proportionately longer. A length of about five feet of tube was situated on either side of the anemo- meter wires, and the internal surface was throughout the whole length of the same made as smooth as possible by Eolehnug: Usually the whole length of tube was wrapped with asbestos cord. A number of fine copper gauzes was introduced into the flow-tube at G, so as to effect distribution of the current of gas across the section of the tube. The meee Mi ayaa ataahe type customarily employed in the technical practice of gas measurement. It was made by Sugg, and one rotation of the drum corresponded to the passage of 1/12 cub ft. of gas. This volume was adjusted accurately to the desired amount in the usual manner by the 510 ~Mr. J. S. G. Thomas on use of the 1/12 cub. ft. bottle specified in the Notification of the Metropolitan Gas Referees. The volume could be adjusted with certainty to 1 part in 400, and with very little change from day to day. The movement of the indicating hand of the meter was very eu even when revolving at extremely low velocities. A definite current—in different experiments adjnered to -values between O°6 amp. and 1°5 amp.—was maintained in the bridge, the constant-ratio arm being adjusted to 1000 or 2000 ohms. The bridge was first balanced with zero flow of air, the current being adjusted to-the specified value. The air current was now passed through the tube, and the deflexions of the galvanometer observed on both sides of the zero after adjustment of the current to its initial value. The galvano- meter employed was either a unipivot pointer memes by Paul of resistance 60 ohms, or a suspended-coil reflecting galvanometer whose sensitiveness was suitably adjusted: by the use of shunts, or by the insertion of resistances in series with the coil. . The general nature of the results was the same in all cases. The deflexion having been read, the drop of potential across the shielded wire was determined as already explained, and the bridge restored to balance by adjnsting the resistance in the fourth arm of the bridge, the electric current being maintained constant throushott: The temperature of the air-stream was determined and likewise the temperature of the air in the gas-meter, and the barometric pressure. The excess pressure.at the meter outlet was observed. Inno case did this exceed 0°8 inch of water. The method of reducing the readings is deter- mined by the fact that, as pointed. out by Kine, his own theoretical inv estigations, and the experiments of Kennelley and Samborn t show that the hot-wire anemometer measures the mass-flow of the gas. The readings of gas volumes were therefore throughout reduced to 0° C. and 760 mm. pressure. The necessary correction of velocity for pressure has been shown by King to be given by GV dD. 5 and for ordinary variations is negligible. Nie eee Do +> ¢ Amongst other disturbing factors which might affect the anemometer readings, mention may be made of the effect of variation of temperature and the transverse vibration of the wire upon the observed reading. Variations of tempe- rature of the stream of fluid are not entirely eliminated by * Phil. Mag., xxix. p. 570 (1915), + Proc. American Phil. Soe., vii., (1914). t King, Phil. Mag., loc. ect. the Hot-wire Anemometer. Hild the compensating device employed. There is still a possible outstanding source of error due to the fact that the passage of heat from the wire to the stream of fluid is determined by the difference of temperature of the two. The bridge being of the constant current type, for ordinary variations of atmospheric temperature, this difference will be practically independent of the latter temperature, so that the correction necessary for variation in atmospheric temperature is, in any case, very small. In fig. 17 (Pl. XIII.) are plotted separated observations for two calibration curves, the mean temperature of the air stream in the one case being 20°2 C. and in the other 17°9 ©. These are seen to be practically coincident. With regard to the vibration of the heated wire, the magnitude of the necessary correction can be deduced by the formula developed by King*. Asthe wire employed in the present experiments was not heated to a very elevated temperature, the maximum amplitude of vibration was small, of the order of 0°01 em., and as, moreover, the experiments were confined to measurements at comparatively low velo- cities, the correction due to this cause was small. The correction due to conduction along the leads is reduced by the consideration that the eden ion of velocities across the cross-section of the tube is such that the wire is cooled most at the centre and least at the wall of the tube. The investi- gations of Kingt have shown how the necessary correction can in any case be calculated, and he directs attention to the fact that no correction is necessary if the anemometer is employed under the same conditions of air velocity in the neighbourhood of the terminals of the wire as those ruling during calibration. A preliminary series of experiments showed that the ratio of the resistance of the unprotected wire to that of the shielded wire was not constant for different values of the current in the bridge, The variation in the ratio is shown in Table I. It is seen that in the case of zero air-flow the ratio of the resistance of the unprotected wire to that of the protected wire increases with increasing current. This is no doubt partly due to the fact that the cooling of the pro- tected wire is facilitated by the presence of he protecting sheath, an effect analogous to that described by Porter ft This point will be returned to later. It will be observed from Table II. that the resistance of the protected wire was * Phil. Trans. Roy. Soc. A. 520, pp. 214, 898 (1914). + Loc. cit. p. 597. { Phil. Mag., Sept. 1910, pp. 511-522. Gr = bo ‘Mr. J. S. G. Thomas on TaBLe I, Rate. Resistance of unprotected wire. ce oR oa . . . Resistance of protected wire. Current in Bridge. | Voltmeter not inserted | Voltmeter across across covered wire. | covered wire. 0-02 0-983 0-983 0:2 0-983 0:985 0:3 0:988 0-990 0-4 0-992 0-995 0:5 0-997 1-090 0-6 1-000 1-004 0-7 1-002 0:8 1-004 1010 0:9 1 007 10 1-012 1-020 tal 1-019 1:2 1028 1-040 1:3 1-038 14 1-048 1-061 15 1:057 1-070 constant when the intensity of the air-current passing over the sheath was varied, the bridge-current being maintained constant. The method of calibration is best illustrated by the consideration of the reduction of a set of observations as recorded in the laboratory record :— Page 48. Anemometer No. 2 (a). Length of wires.—Proteeted 2°0140 cm. ee i ” Unprotected 2:0066 em. f °° ~ ; Mean diameter of unprotected and protected wire 0:00784 cm. Temperature coefficient of unprotected wire=0-00291. Ryz7-5=0°282 ohm, when c=0:01 amp. Mean diameter of tube 2°0534 cm. Current 1:3 amp. Dry air flowing in tube. Ratio arm 1000 ohms. 513 the Hot-wire Anemometer. TUNBInbelel Temperature. a oc i eget ane |< Air | HES) MUILe han ans | Poe esi Galanok. One | ah ae | Reading. Protected i nateat 1 Stream. Meter | _ passed p. hr. 6° C. & 760 wm. | | S Wire. Wire. & aa. Sake: Ins, Cub. Feet. Mins. Cub. Feet. Ohms, Ohms, Volt. Ohi. Ohm. = a er 0 ae 0 0 1042 1042 0-971 0-747 0:778 188 65:0 29°76 4°90 2 7 2:24 5°5 1042 1023 0-971 0:747 0-764 19:0 65:0 29°76 6°55 2 299 Hotel 1042 1003 0-970 0'746 0748 19:0 65:0 29-76 784 2 3°58 20:1 1041 973 0-970 0°746 0726 19°4 65°5 29°76 978 2 4:46 30°5 1010 936 0:970 O-746 0:698 19'5 66-0 29°76 11°83 2 5:37 40°7 1040 902 0-970 0-746 -0:673 19°6 66:0 29°76 14:20 2 645 50°7 1039 870 0 670 0:746 0-649 19°7 66:0 29°76 16°85 2 7°66 60:0 1039 839 0-970 0:746 0626 199 66°5 29°76 20°50 2 9°31 70:2 1039 805 0-970 0-746 0-600 20°0 66°5 29:76 24°94 2 11-08 80°8 1039 ial 0-970 0-746 O70 20°2 66°5 29°76 3 31-0 2 14:08 91:0 1038 736 0-970 0746 0549 20-7 66°5 29°76 37°95 2 Wie22 Ord 1038 705 (0-970 0-746 0-526 514 Mr. J. S.°G. Thomas on These results, together with similar results obtained, forming a series in which bridge-currents equal to 1:1, 1:2, 1:3, 1-4, and 1°5 amp., are shown in the accompanying diagram (PL. X. fig. 5). The abscissee in the figure denote galvano- meter deflexions, and the ordinates the number of cubic feet of air (at 0° C. and 760 mm.) passing per hour. These latter are readily converted to the corresponding values of the mean velocity of the air in the tube, in cms. per sec., by multiplying by 2°374. The relative Sencti vanes: of the anemometer employing various currents is seen from the diagram. In general, a more open scale of deflexions is secured at the higher velocities by employing a larger eurrent. .In-all the curves, the existence of a point ot inflexion is clearly seen. The existence of these points of inflexion is, of course, connected with the existence of what is termed ‘the .“ free convection’ current from the heated wire. The total cooling effect to which the wire is subjected is the resultant of the cooling effect due to the air current passing down the tube, and that due toa convection current cf heated air rising from tne wire. A mathematical investigation of the - magnitude of the convection cooling effect has been made by King *, who has interpreted the experimental work of Langmuir t on the subject in the light of the results obtained. In the present experiments, the heated wire being fixed horizontally, the air-current and the free convection current are at right angles to one another, and the resultant cooling current to which the heated wire is subjected is equal to VV .7+v2=V, say, where V, denotes the mean value of the velocity of the horizontal air stream, and v, that of the ‘““free convection ” current. The value of v, diminishes as the impressed velocity V, is increased, owing to the jowering of temperature of the wire produced thereby. Assuming that the diminution of v, is proportional to v, or some positive power thereof, it follows that the free convection current will influence the form of the calibration curve at a higher impres ssed velocity, the higher the initial temperature to which the wire is raised, 2. e. the larger the electric current employed. With a view 4a) the investigation of the effect of the free convection current upon the form of the calibration curve, a series of experiments was carried out using currents 0°9, 1-1, 13;. 1:4, and 1-5 amp., respectively, im conjmunexion using a reflecting galvanometer of resistance 60 ohms by * Ning, Phil. Trans. Joc. cit. p. 425. + Langmuir, Phys. Rev. 1912, xxxiv. 415. the Hot-wire Anemometer. RRS Paul, the galvanometer sensitiveness being appropriately re- duced either by means of a shunt or by a resistance in series. The results obtained for the series in which the galvano- meter was shunted throughout by a resistance of 6 ohms are shown in Plate X. fig. eS Exactly analogous results were obtained when 1000 ohms was employed in series with the - galvanometer. The results show that, in. the region of low air-current velocities, where the effects of the “free con- vection’’ current are considerable, the sensitiveness of the anemometer is greater the smaller the heating eurrent employed. Thus, examining the respective deflexions for the same air-current amounting to 1:5 enbic feet per hour, it is seen that the deflexion ahah a current of 0:9 am». is employed -is. considerably greater than that when a current of 1:5 amp. is used in the bridge. The values of the deflexions for intermediate values of the current are seen to range between these two values, being greater the smaller the heating current employed. ‘This inversion ‘of the order of sensitiveness as related to the heating current employed, compared with what occurs when higher values of the air- current velocity are employed, is accounted for by the fact that the balance of the bri idge ‘‘ with zero flow” is effected under the influence of the “free convection” current. This ‘“‘ free convection ’ current is greater the larger the heating current employed in the bridge, and the pr oportional cooling effect due to any definite impressed velocity is, of course, smaller the larger the “free convection ” cooling effect. The matter can be regarded mathematically thus:—The proportional change in ‘the effective velocity of the cooling stream) when a small velocity v due to the air-current ic impressed upon the free eee on current is V1.2 - c ie roy Ue UF This is greater, the smaller the value of x, 2. e. the lower the temperature to which the wire js initially heated. A com- parison of the curves for 0°9 amp. and 1:1 amp. shows that for higher velocities of the air-stream, the latter affords the greater sensitiveness. The point at which the inversion from the less to greater sensitiveness occurs, in the case of any two currents, is seen to occur at a greater velocity the greater the current employed. In the figure, the point of inversion of the relative sensitiveness ie not been reached in, the case employing values: of the current 1:3 and 516 Mr. J. S. G. Thomas on 5 amp., but an examination of £8: 5 shows that such inversions do occur in this case also * In Plate X. fig. 7 the ordinates are proportional to the square root of the velocity of the air-current. It is seen that in every case there is a region over which the graph is a straight line, the extent of this straight portion being greater the larger the current employed. In the case employing a current of 1°5 amp., this straight-line portion is seen to extend over the region included between the points P and (. The curves become concave towards the axis of deflexions in the region of the origin, the value of the de- flexion at which this occurs being “greater in proportion as the electric current employed is increased. The point at which this occurs has not been attained in the case where a current of 11 amp. was employed. The straight portion of the several graphs is succeeded by a portion convex to the axis of deflexion. With the use of still larger electric currents, within the limit of destruction of the filament, the straight portion can be considerably extended. The resistance of the exposed wire was determined in every case when exposed to the various air-currents, the electric current in the bridge being adjusted to its appro- priate value when balance of the bridge was restored by ‘unplugging resistances from the box. ‘The resistance of the protected arm is immediately calculated from a know- ledge of the drop of potential occurring across it, and the resistance of the exposed wire calculated therefrom in the usual manner. The results obtained are shown in Plate XJ. fig. 8. It will be observed that the effect of the free con- vection current is very pronounced at the lower velocities, and that, moreover, the velocity at which the change at curvature of the respective graphs occurs is greater the larger the heating current employed. In Plate XI. tig. 9 the abscissze represent resistances, and the ordinates are “the logarithms to base 10 of the volumes of air passing through the pipe measured in cubie feet per hour. It is seen that within the range of mean gas velocities employed, and where the effect of the free ponvechion is small, if V denote the volume of air passing through the pipe in cubic feet per hour, and R the resistance of the wire, the relation between V and R is, outside the region of low velocities where the effect of the free convection current is of importance, of the form :— R=a—b logy V. * Although the curve for c=1'4, as shown, cuts the curve for c=1°5 8 before that for i°3 cuts the latter, it is Me dent that such order of cutting may be attributable to experimental error. 2 the Hot-wire Anemometer. ay iy ee The table herewith gives the values of a and / for the various values of the electric current employed :— TaBLE III. Current. C (amps.). R=a-—dblog,, VY. Hey R=0°'6556 —0°1816 log,, V. Lez R=0°7642 — 0:2345 log,, V. 1-3 R=0'8866 — 02932 log,, V. 1-4 R=1:0140—0°3467 log,, V. 15 1 et NV; If, instead of expressing the results in terms of the volume of air passing through the pipe, results are expressed in terms of the mean velocity v in cms. per sec. of the air across the section of the pipe, then the relations assume the forms :— TaBLe IV. Current. C (amps.). R=a—blog,, v. Pel R=0'7238 — 0°1816 log, , v. 1-2 R=0'8522 —0°2345 log, , v. 1163} R=0'9967 —0°2932 log, , v. 1-4 R=1'1442 —0°3467 log,, v. 15 R=1-2760 —0°3654 log, , v- In Plate XIII. fig. 10 the respective values of a and 6 in the relation R=a—b logy V are plotted against the respective values of the electric current as abscissze. ‘The relation of a and of 6 respectively to the number of cubie feet of air flowing per hour in the tube is seen to be a linear one in each case. he relation of u to V is given by a=1172 C—0°6361, and that of 6 to Cis: b=0°5040 C—0:3670, so that, R the resistance of the wire is related to the electric current flowing in the wire, and the current of air V flowing in the pipe, expressed in cubic feet per hour, by the relation :— R= (1172 C—0-6361) — (0°5040 C—0-3670) log V. The expression for the resistance of the wire in terms of the electric current, and the mean velocity of the air across 518 Mr. J. S. G. Thomas on the section of the pipe, is readily obtained from the above, as already explained. It is seen therefore that, in general, within the range of values of the electric Cnet aaE employed, and within the range of mean velocities where the free convection effect is somewhat reduced, the resistance R of the wire is related to the current C amd the mean velocity v by a relation of the form :— R=(aC—B)—(yC—S) log» =C(«a—-y log v)—(8—6 log v), where a, £, =) and 6 are constants readily determined experimentally. In the preceding experiments, the heated wire was mounted horizontally, and the pipe itself was horizontal. In order to study the magnitude. of the free convection current, the pipe was now placed vertically, the heated wire being still horizontal. Arrangements were made whereby the current of dry air could he passed over the heated wire in the pipe either in an eae or downward direction. In the former case, if V, denote the impressed velocity of the air stream at any part of the wire, and y, the velocity of the npward- flowing free convection current therefrom, it is obvious that in the one case the effective velocity of fe air-current is V,+ 7, and in the other V,—7,. ‘The method | of carrying out the series of experiments was exactly similar to that detailed above. The pipe in which the air flowed was set up vertically and had a total length of 5 feet. (;auzes were arranged close to the entrance to the pipe to distribute the stream across the section of the pipe. When the gas flowed in a.downward direction, it was found that disturbing effects were reduced to a minimum by passing the air from the Hlow-pipe through a wide glass tube bent twice at right angles and of length about 30 inches. The exit limb of this tube was arranged vertically, and the gas passed therefrom in an upward direction. It was found impossible to secure the very steady electrical conditions with low values of the velocity of flow, such as could be obtained when the flow-tube was arranged horizontally. The spot of light in general oscillated from 2 to 5 divisions on either side of the equilibrium position, the maximum deflexion obtained in the present series of observations being about 500 divisions. In all cases, however, where the velocity of flow exceeded a cer- tain value, which varied with the electric currént employed, and was oreater the greater the electric current employed, the readings - were extremely steady. With velocities in excess mG the Hot-wire Anemometer. 519 of these critical values, the hot-wire anemometer can be employed equally well with streams flowing in an upward or downward direction, as is the case with horizontal streams. In the latter case, however, the steadiness of readings is remarkable throughout the whole range of velocities, and wherever possible hot-wire anemometers should be installed in horizontal streams rather than in vertical streams of gas. Morris * employed a hot-wire anemometer immersed in a downward-flowing stream of air, and with regard to its calibration remarks: ‘“‘an unstable part of the curve will be noticed at a velocity of perhaps half a mile an hour; this is probably due to the unstable w ay in which the ‘upward centle natural convection current is met by the downcoming air-current due to the fan.” The author’s experience leads him to believe that this instability is less when the air- eurrent is directed upward than when directed downward, although even then there is still some little instability oF reading, but the instability disappears at a lower velocity of the air-current than in the case of a downw ardly directed flow of air. The instability with vertical air- current is net to be attributed entirely to the difference in the velocity of the air-current at different points in the cross-section of the pipe, as this condition also holds in the case of the horizontal flow experiments where remarkably steady readings are obtained, and, as pointed out, the instability i is also present in Morris’s experiments carried out ina wind channel, It is of interest to note that in the case of variable velocity across the section, as in the case of flow in a pipe, as the stream flow decreases towards the boundary of the pipe the cooling of the wire due to the stream necessarily diminishes in the same direction. The free convection current therefore increases from the centre towards the boundary of the pipe. The result, therefore, is that, in the case of an upward air-stream flowing in the pipe, the possible difference of temperature existing in the wire at the centre and boundar i is diminished by the existence of the free convection current, the opposite effect being produced in the case of a downwardvy directed air-current, he dimimished instability in the case of an upwardly directed current of air, compared with that in the case of a downwardly directed current, is probably largely due to this cause. It seems probable that the instability. present in the case of a vertical flow is partly due t» the fact that the vertical tube is subjected throughout its length to the varying conditions of the surrounding atmosphere. It * British Assoc,, 1912; Engineering, Dec. 27th, 1912. 520 Mr. J. S. G. Thomas on was found that the effect was somewhat reduced, though not entirely eliminated, by providing the tube with a double lagging of thick asbestos cord throughout its length. The reneral instability was greater the larger the heating current employed, and on this account it was not possible to employ the whole of the heating currents employed in the present series. Hxperiments were, however, carried out employing currents of | 0: 6,1078,-0;9, 1.0) a, and 1-2 amp, respectively in the bridge. The sensitiveness of the galvanometer was suitably adjusted in each case, so that the maximum possible deflexion occurred with a finned) Gon OF aor ot about 4 cub. ft. per hour. Low rates of flow were alone examined, and the results for downward flow of air are shown in Plate XI. fig. 11. Owing to the varying sensitiveness of the galv anometer employed, the various series of deflexions are not strictly comparable one with another. The results in the case of 0°6 amp. and 0°7 amp., however, are strictly comparable, as are likewise those for 1:1 and 1:2 amp. respectively. In every case it was found that on increasing the air-flow gradually from zero, the deflexion first in- creased, attained) a maximum value, and then decreased until zero deflexion was again reached. Thereafter the deflexion was reversed and increased continuously. Com- paring the results for 0°6 and 0:7 amp., it will be seen that, initially, the bridge employing the larger current is more sensitive than that employing the smaller. Ultimately, however, a point is reached at which the inversion of the relative sensitiveness occurs, the smaller current then affording the greater sensitiveness, This is clearly seen by considering the respective deflexions corresponding to flows amounting to O'4 and 1°8 cubic feet per hour. The re- spective points at which the maximum defiexions oe on the left of the origin are indicated by P, Q, R, 8, T, U, V. The significance of these points is obvious. The maximum deflexion occurs when the cooling effect due to the free convection current is exactly negatived by the downward flow of air. We may therefore conclude that the velocity of the free convection current from the wire at the appro- priate temperature is in any given case equal to the mean velocity of the air-stream at which the maximum deflexion occurs. Considering the portion of any one of the curves to the left of the axis of volumes, it is seen that the ordinate through any deflexion such as —100 cuts the curve in two points. These two points represent a condition of affairs in which the temperature of the wire is the same. The free convection current from the wire is therefore the same, v; say. the Hot-wire Anemometer. yA In the case of the smaller impressed velocity 1, the effective cooling velocity is ve—v,, and with the larger impressed velocity vo, the effective cooling velocity is v3—ve. As the wire is brought to the same temperature by these two effective net Une? a The free convection current corresponding to any deflexion to the left of the axis of volumes is thus seen to be represented by the point. midway between the two points at which the curve is cut by the ordinate through that deflexion. This affords an accurate means of determining the exact positions of the points P, Q, R, &e., and also enables the variation with temperature of the free convection current to be very accurately determined. ‘The temperature of the wire corre- sponding to any defiexion to the left of the axis of volumes increases as the deflexion increases. ‘The free convection current from the wire therefore increases in the same direction. Bearing in mind how the free cenvection current is determined from the curve, this fact explains the want of symmetry of the respective curves about the horizontal lines through P, Q, R...respectively. The upper half is blunt compared with the lower half, as the lines joining the mid- points referred to above slope upwards to the left. In particular, reference may be made to the fact that the length of ordinate intercepted at the origin represents twice the free convection current from the wire in the absence of any impressed velocity. In Plate XII. fig. 12 are given the form of the air-flow deflexion-curves for “values of the electric current equal to 0'6 and 1*1 amp. respectively, in the case both of an upward and downward air-current. The galvanometer shunt was different in the case of the two currents employed. The curves for other values of the electric current ranging from 0°6 to 1-2 amp. are omitted for the sake of Arenmavece. They all show the same characteristics as those illustrated. The characteristics of the portions of the curves to the left of the axis of volume have been already discussed. To the right of this axis, it is seen that, for the same value of the electric current, commencing with a maximum difference between the respective ordinates for upward and downward flow corresponding to zero deflexion, the difference thereafter continuously diminishes, the curves approaching more nearly with increasing flow in the tube. This approach is attributable to the fact that corresponding to an impressed velocity V, of the air-stream, the actual velocity in the neighbourhood of the wire is in the case of the upwardly directed stream V,+»,., while in the case Phil. Mag. 8. 6. Vol. 39. No. 233. May 1920. 2M velocities, we have, numerically, ve— vy =v2—U,,1-€. Ue = 522 Mr. J. S. G. Thomas on of the downwardly directed stream the effective cooling velocity is V,—v-, where v, and v,’ are the appropriate velocities of the free convection current in the two cases. Were the wire maintained throughout the series of experi- ments at the same temperature, then v. would be equal to ve, and the difference of the respective ordinates corre- sponding to the same deflexion would be a measure of twice the free convection current appropriate to the temperature in question. ‘To the right of the axis of ordinates, the two curves would in that case be parallel. In the present case, however, v, is necessarily less than v,’,and both diminish with increasing velocity of the impressed air-current. The differ- ence of the ordinates corresponding to any definite deflexion being proportional to v.+v. therefore necessarily diminishes with increasing deflexion, and the appreach of the respective curves to one another is to be attributed to this fact. Hvidenee will be given later that this difference of ordinates persists with comparatively high values of the impressed velocity of the air-current. } In Plate XII. fig. 13 are shown the values of the resistance of the wire for upward and downward flow, the wire being heated by electric current of values ranging from 0°6 amp. to 1°2amp. In the case of the downward streams, the re- sistance in each case attains a maximum value which occurs when the cooling of the wire by the free convection current is neutralized by the effect of the impressed air-stream. Utilising the results shown in fig. 12 and fig. 13, we are able to determine the velocity of the free convection current from the wire at various,temperatures. The temperature * of the wire is deduced from its maximum resistance in every case (points P,Q, has; 2) U,V, fel 3)e andiiine respective values of the free convection currents from the similarly marked points in fig. 12. The temperatures have been reduced to the scale of the nitrogen gas thermometer, by employing the correction table for 6=1:50 given by Harker (Phil. Trans. 1904). The resistance of the wire when conveying a current 0:01 amp. at 17°5° C. was 0-2820 ohm, giving a value of Ryp=0:2683 ohm. The diameter of the wire employed was 0:00784 cm. The results obtained are shown in ‘Table V., the velocities of the free convection * Owing to war conditions prevailing at the time, the author was unable to obtain a sample of platinum wire of the purity usually em- ployed in platinum thermometiy. The sample of wire employed was de- clared by Messrs. Johnson and Matthey to be of the purity 99-5 per cent. Its temperature coeflicient was 0-00291, and this value has been employed in deducing the temperature of the wire. : the Hot-wire Anemometer. 523 currents being calculated on the assumption that the tem- perature of the ascending convection current is the same as that of the wire. TABLE V. Heating Temperature Velocity of free Current. of Wire. Oonvection Current. amp. ae. ems, per sec. 0-6 135 ol OF 191 4-0 08 245 5-4 0-9 315 70 10 420 Bell Ae 03d 12-2 1:2 677 5s") From a comparison of these values with the values of the free convection current as calculated by King * for wires of diameters 0°:00691 em. and 0:01262 em., on the basis of Langmuir’s + observations, it is seen that the present experi- mentally determined values for the velocities agree within the limits of estimated experimental error with the calculated values. The Double Exposed Wire Anemometer. A calibration curve was obtained employing two flow- tubes set up in a vertical position adjacent to one another. An exposed anemometer wire was inserted as already explained in each flow-tube. Owing to the close juxta- position of the tubes, the employment “of subsidiary shielded temperature-correcting wires was unnecessary. ‘The two exposed anemometer wires were made from the same sample of wire, and were adjusted as nearly as possible to the same resistance. They constituted two arms of a Wheatstone bridge, the other two arms of which were formed of re- sistances unplugged from the resistance-box. One of these latter was made equal to 2000 ohms throughout the experi- ments, and the ether adjusted so that balance of the bridge was obtained when no air-flow was established in the flow- tubes, the bridge-current being adjusted to 1:0 amp. The flow-tubes were connected in series, and the air-current passed upward through one tube and dow nward through the other. Now, with this arrangement the effective velocity * Phil. Trans., loc. cit. p. 424, ¢ Langmuir, /oc. cit. p. 415. 2M 2 24 Mr. J. S. G. Thomas on of the cooling current of air in the upward flow tube will be Vit. where V, is the impressed velocity of the air-current and v, the velocity of the free convection current from the wire. In the downward-flowing tube, the effective velocity is +(V,— vv’), where v,’ is the velocity of the free convection current from the anemometer wire therein, the + or — sign being chosen according as V, is greater or less than wv, respectively. A little consideration will show that for alk values of V,, the galvanometer deflexion due to an impressed air-velocity V, in the anemometer tubes is that due to a velocity equal to the difference between v.-+v. and its initial value. The deflexions respectively due to effective currents V.tv, and V,—v,’ in the anemometer tube are shown by the curves for upward and downward flow in fig. 12, and the resultant deflexion corresponding to any value of the air-flow, with a pair of anemometer tubes employed as explained, is obviously represented by the horizontal distance between the points on the respective upward and downward flow curves corresponding to the air-fliow in question. The actual calibration in the present case confirms this theory. The deflexion of the galvanometer increases with increase of the impressed air-flow starting from zero, attains a maximum value, and thereafter decreases. The maximum air-flow in the present experiment corresponded to the passage of 23°7 cubic feet of air per hour, and the deflexion of the sus- pended coil galvanometer shunted with 20 ohms, 2000 ohms being unplugged in the constant ratio arm, was 55 divisions on the seale. It is obvious, therefore, that at the velocity corresponding to this relatively high flow, and the corre- spondingly low temperatures of the wires, the effect of the free convection current is quite appreciable on the form of the calibration curve. A consideration of the curves for upward and downward flow in fig. 12 and others affords an immediate explanation of the characteristics of the calibration curve obtained in the present case. The horizontal distance between points on the respective upward and downward flow curves increases from zero with increasing air-flow, attains a maximum, and thereafter diminishes in accordance with the experimental result obtained. This form of double exposed wire anemometer is especially suitable for use with low velocities, owing to the increased sensitiveness obtained by its use. The instability apparently inherent to all hot- wire anemometers employed with vertical How-tubes is somewhat reduced, though not entirely eliminated, and stability of deflexion is certainly secured at a lower velocity than that characteristic of a single flow-tube anemometer employed in a vertical position. the Hot-wire Anemometer. 525 Double Exposed Wire Directional Anemometer. This type of anemometer was introduced by the author * for indicating the direction of flow of gases in pipes. As shown in fig. 14, this type of anemometer consists of two fine platinum wires, parallel to one another and separated by about 0°5 mm. They are inserted into the flow-tube in the manner already described, so that the wires are at right angles to the direction of flow of the gas-stream. The wires form two arms of a Wheatstone bridge, the bridge being completed in the manner already explained. As before, a constant current is maintained in the bridge. The indications of the anemometer depend upon the tact that the heated wire, upon which the gas-stream is first incident, exercises a shielding influence upon the cooling effect experienced by the second wire, the gas incident upon the latter having been already somewhat heated by passage over the first wire. It is obvious, therefore, that the direction of deflexion of the galvanometer will be to. right or left, according to the direction of flow of the gas in the tube. Such an anemometer constitutes probably the most sym- metrical type as regards free convection current that it is possible to realise experimentally. The temperature com- pensation in the bridge is all that can be desired, the wires being so closely apposed, and being both exposed to the stream without any separate shielding device. Calibrations of a directional anemometer of this type were made with the flow-tube arranged vertically and horizontally. With a vertical flow-tube, the air-stream was directed in a downward and in an upward direction in separate experiments. The * Journ. Soc. Chem, Ind, xxxvii. pp. 165 T-170 T (1918). 326 Mr. J. S. G. Thomas on calibration curves are shown in Plate XII. fig.15, the ratio arm employed in the bridge being 20 ohms, and the suspended coil galvanometer shunted with 12 ohms resistance. In the case of this type of anemometer employed with either an upward or downward flowing current of air, there is, as in other cases, a region of comparative instability corresponding to the portions of the calibration curves extending from zero flow to a flow of about 2°5 cubic feet per hour. Thereafter the readings obtained are remarkably steady. The steep inclination of the succeeding part of each curve shows that very high rates of flow of the gas would be necessary before the instrument would cease to function efficiently as an indicator of direction of flow. A deflexion of 248 divisions was obtained with a rate of flow corresponding to the passage of 25:4 cubic feet of air per hour, the flow-tube being mounted horizontally. With a horizontal flow-tube, owing to the symmetrical nature of the bridge, the zero remains extremely constant for various values of the heating current, and no adjustment of the balancing resistance in the br idge was found necessary during a series of calibrations. For the investigations of low velocities, this type of double-wire anemometer possesses a further advantage over the type employing one exposed wire and a second provided with a protecting shield. The great sensitiveness of the double- wire type at low velocities when used with horizontal flows is shown by the appropriate curve in fig. 15. An examina- tion of the variation of the values of the resistances of the respective exposed wires with varied rates of flow showed that while that of the wire first meeting the air current diminished continuously from 0°5619 ohm to 0°4160 ohm, ‘with in- creasing rate of flow from zero to 23°95 cubic feet per hour, under similar conditions that of the second wire increased from the value 0°570 ohm with zero flow to 0°594 ohm, corresponding to flow of 1°702 cubic feet per hour, there- after diminishing contini rae to the value 0450 ohm, corresponding to a flow of 23°95 cubic feet per hour. The explanation of this phenomenon is as follows. By its passage over the first wire the air in the immediate neighbour- hood of the wire is heated, and this heated air being trans- ferred to the neighbourhood above the second wire reduces the natural convection loss from the latter. Some heat is of course convected from the latter by the stream of air, but on balance, the resistance of the latter wire increases, and sucht increase becomes greater for small values of the flow as the velocity of the flow is increased, owing to the fact that with such greater flow more heat is transferred from the first wire the Hot-wire Anemometer. Doe to the neighbourhood of the second, and, moreover, this heated current of air suffers less fall of temperature in transit from one wire to the other, on account of the diminished time of transit. With still larger rates of flow, the cooling eftect due to the stream more “ine Srcabaleamecs the increase of temperature due to convection of heat from one wire to the other, and thereafter the resistance of the second wire falls. It is clear, therefore, that for low velocities of flow, owing to the heating of the second wire as explained above, a double exposed wire anemometer of this type may be a far more sensitive instrument than the type in which the second wire is provided with a protecting shield so that it experiences no heating or cooling from the stream of air. This point is discussed in detail in the Proceedings Phys. Soc. 1920. Use of the Hot-wire Anemometer with Gases other than Air ; the Glass-coated Hot-wire Anemometer. The measurement of the velocity of a stream of air ina pipe, although frequently occurring in technical practice, is by no means that most frequently called for, at least not in the author's experience. The hot-wire anemometer in the original form described on pp. 505-507 can be readily applied to the case of the anemometry of gases, such as COs, No, Oz, ete., which are not adsorbed by the heated platinum wire nor catalytically decomposed thereby. The use of a bare heated platinum wire, particularly if the temperature is somewhat elevated, is practically impossible with a gas such as by- drogen, carbon monoxide, and gaseous mixtures such as coal-gas, ete. The readings ohiained in such cases are extremely unsteady and unreliable. The sphere of useful- ness of the hot-wire anemometer is considerably extended by employing as the sensitive wire a fine platinum wire on to which is fused a fine coating of glass*. This is easily effected by drawing down a piece of glass tube so that it is very fine in the walle and so that its internal diameter is very slightly i in excess of the wire. The glass is slipped over the wire, and can then, with practice, be readily fused on to the wire so as to forma surprising oly uniformly ‘thick coating by heating the two cautiously 1 in a small blowpipe flame to just past the softening point of the glass, care being exercised that the glass is not heated to too high a temperature and co) not exposed to the elevated temperature for such a length of * Thomas, Journ. Soc. Chem. Ind., lac. cit. 528 Mr. J.S. G. Thomas on time as to cause the glass to break up into a number of separate droplets—a tendency on the part of the softened glass, which is explained by well-known considerations relating to surface-tension phenomena. The fusing of the glass on to the wire can also be readily effected by heating the glass and wire cautiously to a predetermined tempera- ture ina small muffle. Thereafter the wire and coating are annealed cautiously, and the wire aged in the usual manner by the passage of a current of about 2 amps. for some hours. An anemometer wire so prepared can be employed in the usual manner io gases such as hydrogen, methane, coal-gas, ete., and yields particularly steady and consistent results. Before passing to a consideration of the results obtained by the use of such an anemometer wire, and of an anemometer wire of the usual type in streams of air (dry and moist), carbon dioxide, hydrogen, nitrogen, and coal-gas, it is of interest to direct attention to a phenomenon differentiating the one type of wire from the other. It has already been remarked (see p. 511) that using an exposed uncovered anemometer wire, the resistance of this exposed wire in- creases more rapidly than that of the protected wire of the pair when the current thr ough the wires is increased. The results obtained by the comparison of the resistance of the exposed wire with that of the shielded wire in the case of the bare wire and glass-covered anemometer wire are seen in Plate XIII. fig. 16, the results being g given in the case of air for the bare wire anemometer and in the cases where the glass-coated wire is immersed in (1) air, (2) hydrogen, and (3) coal-gas. The wire employed was cut from the same specimen of wire as that employed in anemometer No. 2 (a), and had a diameter of 0°00784 cm. The mean diameter of the glass-coating was 0°08 cm. It will be seen that in the case of the bare wire surrounded by air, the resistance of the exposed wire increases with current more rapidly than that of the shielded wire with increasing supply of heat to the wire. With the glass-covered exposed wire in air, it is seen that, with increasing supply of heat to the wire, the resistance of the exposed wire decreases compared with that of the shielded wire, the ratio attaining 4 minimum value and increasing thereafter. With the glass-coated wire sur- rounded by coal-gas, the decrease in the ratio of resistances is still more marked, and attains a minimum value at a rate of supply of heat larger than is the case when the wire is surrounded by air. The result with hydrogen is similar to that in the case of coal-gas, the minimum value of the ratio being, however, smaller and occurring at a still higher value — a eo ee ee ee the Hot-wire Anemometer. 529 of the rate of heat-supply. The difference of behaviour of the bare and coated wires when surrounded by air, so far as the increase in the ratio in the former case and the initial decrease in the latter, is explained by the simple: theory explained by Porter*. It is recognized by Porter that the simple theery developed by him is inadequate to explain the whole of the phenomena observed in his work, and the discrepancies between the ratios of the temperatures of tne bare and coated wires in his experiments is attributed to the variation of emissivity and thermal conductivity with temperature, and of emissivity with the radius of the wire. The results of Porter and the results obtained in the course of the present research can possibly be more rationally ex- plained by a modification of Porter’s fundamental equation, taking account of 2 possible additional heat loss /(@,) from the wire, proportional to some higher power than the first of the temperature difference between the wire and _ its surroundings, the fundamental equation given by Porter, r) H=— har 9" = 270) 10, becoming H=—é&. amr =¢.27b.0,+/( 8%), According to King f, the free convection loss is given by 2m kyo 1+ cA] /[logb/a] and is seen to include a term proportional to 63, the square of the excess temperature of the wire. Looked at physically, the interpretation of the minima obtained with air, coal-gas, and hydrogen presents itself thus. So far as losses proportional to 6; are concerned, the glass-coated wire loses heat more rapidly than the shielded wire. With small heating currents, the loss of heat J(@) from both wires is small. As the temperature of the wires is increased the term /(,) becomes of increasing importance and its value for the bare wire becomes greater than that for the glass-coated wire owing to its higher temperature. The point is reached at which the decreasing ratio of the resistances reaches a minimum; thereafter an inversion occurs in the decreasing ratio between the re- sistance of the coated wire to that of the shielded wire. This ratio then, instead of decreasing, increases owing to the greater loss of heat occurring from the shielded wire arising from the term /(@,). The occurrence of the minimum * Phil. Mag., Sept. 1910, pp. 515-518. t Phil. Trans,, loc. cit. p. 403. 530 Mr. J. 8S. G. Thomas on value of the ratio at higher values of the rate of heat-supply in the cases of coal-gas and hydrogen than is the case with air, is to be attributed to the relative effects in their cases of the loss proportional to @ being somewhat greater compared with their respective losses /(@,) than is the case with air. ‘This is explainable in terms of the relative diathermancies, conductivities, specific heats, and densities of the respective gases, but it is unnecessary to pursue the point further here. The results obtained for the calibration for a bare hot-wire anemometer in the cases of dry air (three results), air saturated with water vapour at atmospheric temperature (two results), carbon dioxide,and oxygen are shown in Plate XIII. fig.17, the bridge-current being 1:3 amp. in each case. Jn Plate XIII. fio. 18 the results are given tor the calibration of a glass- coated anemometer wire in streams of dry air, coal-gas, and hydrogen. ‘The curves obtained with nitrogen using a bare wire anemometer were practically the same as those for air and are omitted from the diagrams for the sake of clearness. The curve obtained for the calibration of bare wire anemometer No. 2 in air using same current 1°3 amp. is added to fig. 18, for sake of comparison. The various gases passing through the flow-tube were sampled and their compositions ascertained with the following results :— per cent. Sampleol CO; ee co, 97-0 O OD N (diff.) 2:5 Sample Of /N, (0c ate awe et anaes Co, 0 : oO); i 02 N, 99°8 Sample of Oy venis5. tee eres CO, beens) O, 96:0 iN, — 40 Sample of H, BRE Score ae lals 99°8 Sample of Conleoa sa). eae nee co, 2 O, 0-2 CnHm 3-2 Co 76 CH, 30°5 A, 48-0 N,, (diff) 8:5 . } the Hot-wire Anemometer. 531 An examination of the calibration curves given in fig. 17 shows that the deflexion corresponding to any constant flow of air was consistently a little greater when the air was dry than was the case when the air was saturated with water vapour at the temperature of the experiment. The difference in deflexion was of the order of from 0°5 to 1:2 per cent., and was obtained consistently in a series of experiments, two of which are shown in fig. 18. The difference is not attributable to the slight difference in pressure at the meter outlet in the two cases. King * was unable to detect any effect of variation of humidity upon his final readings. In the present series of experiments the variation of humidity was much greater than that recorded by King, and the result obtained shows that a variation of about 25 per cent. in the humidity of the atmosphere is the minimum that could influence the results in King’s experi- ments. Jt is not stated whether such a variation occurred. The relative sensitivities of a glass-covered hot wire in air, hydrogen, and coal-gas when the same current is used in the bridge is shown in Plate XIII. fig. 18. The comparative insensitiveness in the cases of hydrogen and coal-gas is directly attributable to the much greater thermal con- ductivity of these gases compared with air. A reference to fig. 16 shows that with zero flow, employing an electric eurrent equal to 1*3 amp., the resistance of the wire when in hydrogen is less than that when immersed in coal-gas, and very considerably less than when immersed in air. On this account, the initial temperature of the wire for zero flow was much less in the case of hydrogen and coal- gas than was the case with air. A fair basis of comparison of the respective sensitivities is only afforded when the wire immersed in the several gases has the same resistance in the absence of flow. The results in Plate XIII. fig. 17 enable a comparison of the thermal conductivities of the various gases detailed to be made. Expressions for the heat convected from the wire have teen given by Boussinesq t, King }, and Rayleigh §. _ Boussinesq’s expression, H=8(sohkVa/7)?4, where s is the specific heat at constant volume of the gas, o its density, and & is its thermal conductivity, V the *— Phil. Trans.,,doc. cit. p. Alf. + Journal de Mathématiques, |. pp. 285-832 (1905). { Phil. Trans., oe. cit. p. 381. § ‘Nature,’ xcv. p. 66 (1915). It may be remarked that the final expression given by Lord Rayleigh is obviously in error due to a misprint, and in place of ha 1@x/(bvc/i:) should read h x 10 (bvck).. 532 Mr. J. S. G. Thomas on velocity of the stream, a the radius of the heated cylinder, @ the temperature of the cylinder above that of the sur- rounding medium at a great distance, or the similar expression deduced by Rayleigh (see footnote on p. 531) is the type of expression most readily applicable to the present series of experiments. Considering the curves for air and carbon dioxide showm in fig. 17, it may be remarked that the resistance of the exposed wire under zero flow in the respective gases was 0°7906 ohm and 0°7869 ohm, so that the initial temper- ature of the wire in the two cases was very approximately the same. In the cases of oxygen and nitrogen, the initial temperatures were still more nearly equal to that in the case of air. Consider two points P and Q on the respective curves for carbon dioxide and air having the same value of the _deflexion. The total heat convected from the wire is made up of that due to free convection from the wire, and that con- vected away by the stream. The respective resistances of the wire corresponding to the deflexion 100 (at this deflexion the proportional etftect of free convection is the least in the present sequence of experiments) were 0°541 ohm in air and 0°536 ohm in carbon dioxide. The corresponding temperatures of the wire are 358° C. and 352°C. It is therefore legitimate to assume that for corresponding points P and Q on the same ordinate, the heat convection from the wire is very approxi- mately the same in the two cases. The subscript ,; referring to air, and » referring to carbon dioxide, we have for points such as P and Q, Hi, = Jalge= 8(s10,k, V1a,/7)2001 = 8(soooksV oao/7) Dao. Now, 6,002; dj)=d2: and V, and Vz are proportional to the respective ordinates at P and Q. We have therefore very nearly syoyky Vy = seookoV o, he me $10, Vy de é. 7; = aici ky Sera Before applying this relation it is essential to refer to the temperatures at which the respective values of 51, 04, 52, 9, e (ox e e are to be taken. Now the ratio — remains practically oO . 2 unaltered over the possible range of temperatures employed in these experiments. The equipartition theory requires 5] constancy of the ratio The ratio remains practically s 2 the same if values deduced from the quantum hypothesis are SS ee pon — the Hot-wire Anemometer. 533 used in the case of O, N, COs, air. The value of s, is given by Joly (1891) as s=0" 1715 +0: 02788p, where p=density of the air in germs. per c.c., and s,=0°165+4 0: 2195 0-+40-34p2 (Joly, 1894). Applying the appropriate values of p, it is seen that it is quite sufficiently accurate to use the values s;=0°1715 and s,=0°165. We have therefore, since Y,=87°5 and V.=100°6 (the numbers are relative and are taken from the diagram), Kare mst Hi = cur) Al oees)) a my e OA es . a = SP a on Sat aie Ua (4) where §,= ne in the same notation. Dynamical Motions of Charged Particles. 541 Let 1, 92, Y3 be three generalized coordinates defining the position of the particle. The components of r, are then Ou aoe %r known functions of the gs and mee for any com- Oy og ponent of r, and any of the q’s. Take the scalar product of (4) by a Then ; Gol). dm, ‘ ))= d fm i) _ mm Or, i) lag’ dt B, dt t By ys By \ 0g’ : dem Ol. aes ii (0) Ae ey = di-t..B). 0g2 aS Bi 0g 2 and putting in the value of ;, this reduces to Q, (--mC7A,), ao fo) lt Oy fold] pression —m(C?8, has an obvious connexion with the “ world line ” of a particle in relativity theory. The next term in the equation is where S, = the Lagrangian operator. The ex- =e A¢)= =, 5 =e: Dyp. For the remainder we simplify by writing out one com- ponent of the vector product, [r1, curl a ae <= +4 Om eck) foe for os j2as DA. ) ae eee. and the second factor is es oe, where 14 denotes the _ total rate of change of A at the moving particle. Thus 1G), Bb roma) =3(4 34) 2, 48 ey ° BSG, A); So the equations of motion can be derived from a Lagrangian L= —m CB, —ed -b A (Cae ry) 5 ‘ . (d) 542 ane Mr. ©. G. Darwin on the This expression is valid for any fields, including explicit dependence of @ and A on the time. In the case of a con- stant magnetic field it is a matter of indifference what par ticular. integral is taken in finding A from H. For the general value of A is given by the addition of a term AQ to any particular value, where ©, is a function of v, y, 2. This adds on to La term (t,, AG) = —, andif O is any function d. | ae of wv, y, z and ¢t whatever dQ Wd OOF 0 dO : a dt ~ dt Og Ta ayeh ai ae = © so that the extra terms will be without effect on the equations of motion. » 4. We next find the Lagrangian for a number of freely moving interacting particles. Suppose there is a second one of char "ye é) and ‘mass my atYg. This particle is in motion, but at first we imagine it outside the dynamical system, that is we suppose Ty to “be known in terms of the time. Then the motion of e, is governed by (5), where ¢ and A are to be ealeulated from the position and motion of e,. The potentials are given by Qs A=” 2 ae r+ (To, Pg—T1)/C ret,’ Crt Gh tr Ces In these expressions 7?= (r,—1,)? and (fr, ry—r,)/7 1s the component of velocity of e. away from e,. The quantities are all to have retarded values. If the effect reaching e, at time ¢, left e, at time t—7, we have Solving by approximation we find fit Yo 7) T= Cs as (2 2 a Ob + (f. 95 ocmame —r,)+ (5, Tey i eles ee : ane Ce) 7 Saia(ee Substituting in (7) Y es 7 - = (Yo, Yo—T1) _ (tp, eae T1)” NS ee Ms aly If PAN Ta eS ae Dynamical Motions of Charged Particles. 943 The solution of A is only carried to this degree, because of the further factor C71 in L which multiplies it. Substi- tuting in (5) we have 12 €1@2 bby (Eo + (Yo, YT, =f) = 2(r1, Tp) L=—m,C a= 5 2 ee : {fs 2C, ? Ay ee (Xo, 1) owe \ r? j i ae Add to this the expression —2C?Bo+ 5, = (To, t2— 11) r The first term is without effect because it is a pure function of the time, the second by (6). The result is L=—m,C?8, — m,C?B, €1@o , 102 § (i, fy) C2} a i oe aa aoe = (10) From its complete symmetry (10) will also give the motion of eg when r, 1s regarded as known in Boe of the time. Thus the equations of motion of e, are ©,,L=0 and of e, are QD, L=0. It q be any generalized coordinate involving both r; and r, we have ON sD ) Om =) ae & a 1. Meg as may be seen by writing out the values.of S,, and ®,, or directly from the covarianée of the Geisler oa tor point transformations. ‘Thus (10) is the Lagrangian for the simultaneous motion of the two particles, Phen can now be regarded as both belonging to the dynamical system. The last term in (10) is only accurate to the terms in C~%, so for the sake of consistency the first two should only be expanded to this degree. They are then of the form — mC? + smut? + sag muti. il 8C Thus we have the complete Lagrangian for any number of charged particles in any field in the form : : 1 : = Lamyr,? +i ae Wits — eid snes (ft A)— =233 ae) Tete Pigg “fr fp fat +22 502 1 Cp Te) mn COS 1) ( ao Te v) 12 Tins Ai) 544 Mr. CG. G. Darwin on the The double summations are for each pair of particles counted once only. Finally, a (11) without the vector notation Le: é L= >i Pe as M101- — Xe) aie 2 ‘ae cos Xa ae : ‘12 + >> a ae (cos Wy2—cos 6,” cos 0,'), CLs) DE iB where ¢;, 1, v, are the charge, mazs, and velocity of the first particle ; g@, and A, are She scalar and vector potentials at ¢ due to external fields ; x, is the angle between the line of motion of e, and the vector potential ; 7, 18 the distance between e, and ey ; Wi. is the angle between their lines of motion ; 6,7 is the angle between the line of motion of e,; and the line joining it to eg. There is a certain interest in knowing how far wrong the approximation (11) will be according to the classical theory. This is done by calculating the next term for 7 in (8) and evaluating the corresponding terms in (9). These are then oa _ substituted in (5). ‘The force on e,; from e2 is found to be DS ea ne Thus the total force on e, is 308 summation will include e,, as well as the rest, as this term is the reactive force of an electron’s radiation on itself. Krom the point of view of generalized coordinates we have Or. o0L : a —z. = <—, so that the equations of motion can be put in the { ; il ae ; form l= ae where F = 3(8 (Ye, t,)”. If Fis neglected altogether, it is easy to see that the ratio of terms omitted to those included is of the order v/C. 5. When the Lagrangian for any problem has been found, the transition to the Hamiltonian follows in the usual way. : OL We find the momenta p= =— and solve for the g’s in terms of them. The Hamiltonian is then H=Spg—L expressed in qs and p’s, and the equations of motion have the canonical OF oe form p=— Thus if be the momentum l Og’ Op Pi ; —— Dynamical Motions of Charged Particles. O45 corresponding to each component of Y,, it is easy to see that, extending the use of the vector notation, we have H=sP py’ Pars A x 2162 om, ios aoe Leip — ie 7, (Pi ) le pp 19 102, (Pi, Po) , (Pi, Y2—T1) (Do, t2—11) A 35 (fe {Oe Pd 4 Geet Pe on) C712» T19 Tio All the developments of general dynamics (such as the Hamilton-Jacobi partial differential equation etc.) follow at once, with the exception of such theorems as depend on the kinetic energy having a quadratic form. For many problems it will be quicker to work in the Lagrangian form direct. When gq, does not occur explicitly é OL in L, we have an integral a ee constant, and when 18 this coordinate is “ignored” the modified Lagrangian is L'=L—p,g,. The energy integral will exist syne? the ex- ternal fields ¢ and A do not contain the time explicitly and 2 oll is then of the usual form Aes —L=const. Applying this to (11) we have the integral 1 eo DV bmyty?+ S 2myry* + Leip+ 2a eo § (Ti, Ye) _ Yy—Y}) (Te, ¥y—T}) a] ae sao A eae Pi aa iter == OOM ee 6 ary A) The first two terms can be obtained either direct in the expanded form, or else from the fact that : Ps) iL T,2 —1/2 a ee —— = —— = ea: Pia a Bi (1 os) y) agreeing with the known fact that the kinetic ener ey of an electron i is mC?/B. 6. We now apply these results to the “Problem of Two 39 Bodies.” Take e,=—e, my =m amd a= Hee mn, = Mi Phe motion is supposed to take place in a plane and the particles 546 Mr. ©. G, Darwin on the are at (2, 41) (ve, y2) at any time. Then from (11) we have ae He | E ni e, 2 MC Ber as pe) ae it Sey se 3 U Lr "(hid Ae) + 9Y2) + [ti(a1— az) + gee [tet 2) + Holy Yo) | \ (14) The first transformation is m=X+ME(M+m), #2.=X—mé/ (Mm), a with similar expressions for y;, y 2. Then X, Y may con- veniently be called the centroid, though except for low velocities it has none of the properties ordinarily associated with the name. Then 1 Mm a 3 Mam 2 7 ) i: Mm 2C? M+m ( L=4(M+m) (X27) + tM Fm) ( Xe ee 8 a 2 = Mm 2M +n 1 Mm(M—™m) SANG + 92 (M+ m)? (EX +4Y) (+77) 1 Mm(M? — Min +m’) ee (M+m)? EX +7Y)? — (X24 Y?) (6? +77?) : E (E497)? + ~ aa (XP + Y*) + (EX 4+)? M—m + Mpm en EX + 9Y) + (EX +0) (C6 +77) Min : : ~ arpa Bea) + eet on]. eh) As P= +7, X and Y do not occur explicitly in (15), and so we have integrals OL OL —= =P, ew ne ee (LD aX ay | (16) Dynamical Motions of Charged Particles. DAT For considering quasi-elliptic orbits we naturally take pz2=p,=0 for the integration constants. If this is not done it will be found that the modified Lagrangian deduced from (15), if expressed in polar coordinates, con- tains @ explicitly. Under these conditions the ordinary integral of angular momentum does not exist. But any such case could be worked out easily by taking p,=p,=0 and when the complete solution has been found, applying a linear relativity transformation to give the system the proper motion of translation. Such a transformation would be expected to introduce the time explicitlyinto the formule. So it appears that the angular momentum integral would be replaced by a complicated integral involving both @ and ¢. The study of such an integral might have an analytical interest, but it would appear that in any specified case where pz does not vanish (and the same applies to motions of the particles which are not in a plane) the required results could be quickest attained by relativity transformations. Thus, in studying the collision of a moving electron with a stationary, we should work out the orbit with both moving in such a way that pe=p,=0, and afterwards apply the transformation which would reduce one of them initially to rest. Taking p,=p,=0 in (16) we have equations of which the - solution is + iL Mes: Sa Sah SEO ae (M +m)? E( +77") Ke M-— 7 + 90,8 (Wea? 2E + E(EE +7) |. . Ek) Next form L'=L—p,X—p,Y. This is given by simply omitting X, Y from (15), since X, itself of the order .C-?, occurs ‘everywhere either squared or else multiphed by Q-2. In polar coordinates we then have oe Min ieee 1 Mm(M?— Mm +m’) is > OM a ez (M+ m) Ee Ee Mm 272476 ewer t+ aie (18) where w?=i7 +7767. The integral of angular momentum is Mm Ui? oC” (em? % * OM am) J *? (19) 548 Mr. C. G. Darwin on the and of energy is 2M+m rol Om (M+ m)° pee He Mm 2724762 rs Ore ne es DE AS 8 . 9 T 90: (M+)? r = oe Ws age) The integration constant is taken as —W, so that W may be positive for elliptic orbits. Following the usual pro- cedure we eliminate the time between (19) and (20), and express the orbit in terms of 6 and u, where w=1/r. The result is an equation of the form f 2 Ke ex Dre M?— Mim + m? Oa) Cp a He Nini Mere ae) de ieee C2 Mm(M +m) 7° fee 2W . Mm op! W M—Mn-4+m? oe oe nee 2C?, Mm(M-+m) ) The solution of this equation to the same order of approxi- e . e - mation as before is where a= ub — 9 +S COSNO 4 Cos ZAG. ene | (22) where A=1—fZaq—3P, g=g9 +t qa(39°— k) + Bg, e=P—k+aq(4¢? — 3k) + B2P7—h), l=—la(g’—khk). The last three expressions cannot be much simplified, but He? ~ 20%, which is independent of the masses and depends only on the angular momentum. It is the same as Sommerfeld’s result and implies an advance of perigee by wH?e?/C?p? each revo- lution. The term in / makes a slight increase in the radius at the apses, and a decrease at the ends of the latus rectum. The solution of the relative orbit is completed by finding the time from (19). The formula is complicated and of no special interest. A=1 Dynamical Motions of Charged Particles. 549 We next solve for the motion of the centroid. As X and Y are of the order C~*, it will be sufficient to use large order values for &, etc. Then changing the independent variable to @ and making use of the known value of w’, (17) gives pe il eae a d—& éEdr Meare: 28 OLE a ag ce “(Gat de) } and this is directly integrable in the form 1 M— eee =F » (Heu— If a is the half major axis a=o/(q?—s?) and W= He/2a and we have a i M—m > Ke ah ee cay: : (GEO COsiy Fs Gas (2a) If the motion of the centroid is to be valid for many revolu- tions, we must replace cos @ by cosd\@. Both are the same to the degree of approximation considered, but cos A@ will enable the centroid to keep pace with the motion of the apse. From (23) we see that both particles and their centroid always remain collinear with the origin, which is the in- variable point of the system. Observe that this invariable point cannot be calculated by taking the centre of mass of the particles as though each had its mass increased separately by the effect ot velocity, Such a process would give X proportional to w?€ or (r—2a) cos @) iene is im fact no simple definition for the invariable point. Expressed in polar coordinates the centroid describes the curve eS IWYE. Ke M—m e+cos 4 Mame eT)? 1+ cook. (24) where e=,/(1—s?/q’) is the eccentricity of the relative orbit of the particles. It M>m, R is negative at perigee, that is to say, the centroid is towards M. At apogee it is an equal Mcienue towards m. If the time average of R be taken it is found to be Ke M—m 22 (M +m)? 26 _— faa ho On 4 that is, on the average it is towards m. As the velocity of the lighter particle is the higher, so its mass is the more increased by the motion, and so (25) is directly contrary to 990 Dynamical Motions of Charged Particles. what would be expected at first sight. Observe that (24) shows that the centroid is at rest at the invariable point in the case of any circular orbit, as well as for the obvious cases M=m and M infinite. 7. Finally we apply these results to Bohr’s theory of spectra. To do so we use Sommerfeld’s* quantum relations, so as to determine the integration constants p and W. These relations are nh = \ pe oh ih \prdr, where the integrations are carried round a complete period of the variable in each case. Then é nh={“pdd = 2mp i eee (26) 0 and MICE ioe (AN i Q — ad. If the values be taken from (18) and (22), the last gives after some partial integration “27 scos @ +4l cos 2 so cine u'h=pr a on a Aiaa , ee d@. Ne q-+scos@ +1 cos 2h qgtscos¢d The evaluation of the integral is rather long, but by taking advantage of the smallness of / and a it can be reduced: to / 1 : n h= 27 pr | (g—s)'? Tag I a5 4] x Ts) 7 OND SAG 375; ) Putting in the values from (22) we have (4g*—6q?s? +3s*) + a [¢ —(g—s?)!?] ; wh =2np 4 a —l1+#aq+tB—« vil. 3 vk: d This is to be solved for W by using the values given in (21) and (26). The result is Wo 2a? lire? Mm if Pe: (els =] \ 97) {ake (n+ n')? E an (M+ im)? 3 i Bete ( where p=2a He/Ch. The spectrum lines are given by p={W(ins, no )— Wy, 2) fA. * Loc. evt. The Specific Heat of Carbon Dnoade and Steam. 551 It was not of course to be anticipated that our work should give any effect perceptible experimentally for the distribu- tion of lines in the hydrogen spectrum, but it is interesting . to observe what extremely little difference the finite miss of the hydrogen nucleus does make. In the first place there is the factor M/ (M +m) in the large terms, corresponding toa slight alteration in Balmer’s constant. This comes out of ordinary dynamics and was given by Sommerteld.. In addition, we have a minute shift of the whole position of I Mane 4 (M +m)?" But the fine structure of each line, which is given by the term in n’/n, remains absolutely unaffected by the mass of the nucleus. the composite lines, represented by the term in LU. The Specific Heat of Carbon Diomde and Steam. By W.T. Davin, M.A* 1. FENHE specific heat of many gases, notably carbon dioxide : and steam, increases very considerably with tem- perature. In this paper the suggestion is put forward that the specific heat of these gases “depends to an appreciable extent upon volume and density as well as temperature. 2. Some of my experiments upon the emission of radiation in gaseous explosions indicate that the intrinsic radiance from thicknesses of gas containing the same number and kind of radiating molecules does not depend upon the temperature alone, even atter correcting the radiation for absorption. This implies that the vibratory energy of the radiating mole- cules is not solely dependent upon the gas temperature. It depends upon the volume and the density of the gas as wellf. I have suggested an explanation of this in terms of the kinetic theory of gases which it will be convenient to repeat here briefly. A radiating molecule as it describes its tree- path loses energy owing to the emission of radiation and gains energy owing to ‘the absorption of energy from the wether. Its vibratory energy will thus increase or decrease according as the absorption is greater or less than the emis- sion. During collision with another molecule there will be a Pee rerence of energy between the vibratory and the rotational and translational energies, which, as Mr. Jeans * Communicated by the Author. 7+ Phil. Trans. A. vol. ccxi. (1911) pp. 402 & 406. {.Phil. Mag. Feb. 1918, p. 267. 502 The Speeific Heat of Carbon Dioaide and Steam. has shown, will be very rapid if the duration of collision is comparable with the periods of vibration of the molecule. In the case of carbon dioxide and steam at high temperatures (1000° C. and over*) the duration of collisions between mole- cules is probably short in comparison with the periods of their low frequency vibrations, and the vibratory energy of the molecules will therefore tend to take up during collisions a value such that the energy in each of the vibratory degrees of freedom f equals that in each of the rotational and trans- lational degrees. During collisions therefore the vibratory energy of the molecules will tend to take up a value which is proportional to the absolute temperature ; but, as we have seen, during the free-path there may be considerable depar- ture from this value if the energy density in the ether is above or below a certain value and the time of description of free-path is not very short. It is clear therefore that the value of the vibratory energy of the molecules averaged over a time which is greater than that of the description of free- path may depart considerably from. a value proportional to the absolute temperature of the gas. The extent of this departure will depend for any given gas at any given tem- perature upon the energy density iu the ether (which, in view of the high transparency of a gas to its own radiation, is dependent in a very large measure upon its volume) and also upon the time of description of free-path (which, of course, depends upon the density of the gas). 3. The specific heat of a gas is dependent upon the trans- lational, rotational, and vibratory energies of its constituent molecules. In a gas in thermal and chemical equilibrium the translational energy and the rotational energy are proportional to the absolute temperature of the gas; but, as we have seen, the vibratory energy may depend upon volume and density as well as temperature. The specific heat would therefore appear to depend upon volume and density as well as temperature. At any particular tem- perature the greater the volume of the gas the greater its specific heat, and the greater the density the less its specific heat. 4. It is clear from the theory suggested in § 2 that the vibratory energy of the molecules during the free-path will vary to a considerable extent only when the eas radiates * See Phil. Mag. Jan. 1920, p. 93. + Or, rather, ihiose vibratory degrees of freedom which share in the heat motion of the gas. In the case of steam and carbon dioxide at tem- peratures of 1000° C. and ores these degrees of freedom are those which correspond to radiation of 2°83 » and longer. Dk Period and Decrement of Oscillatory Electrical Circuit. 553 strongly. My experiments* show that carbon dioxide and steam emit radiation in appreciable quantity in the neigh- bourhood of 1000° C. and very strongly in the neighbourhood of 2000° C.- The specific heat of these gases may therefore I think, be expected to vary slightly with density and volume at temperatures in the neighbourhood of 1000°C., and at temperatures in the neighbourhood of 2000° C. the variation with these factors may be marked. LILI. On the Period and Decrement of an Oscillatory Elec- trical Cireuit provided with a Short-crcuited Secondary. By Joto Jonss, B.Se., University Research Student, Un- versity College of North Wales, Bangor f. HERE are several cases in which one of two coupled electrical circuits possesses inductance and capacity, ‘and the other has its terminals connected through a resistance or is short-circuited. One example of this kind is afforded by an induction-coil immediately atter contact is made at the interrupter. In this case the secondary coil, supposed to have no discharge between its terminals, has capacity, in- ductance, and resistance, and the primary is short-circuited. Another example is that of an induction-coil after ‘ break,” when the secondary terminals are connected by the are which usually follows the passage of a spark. In these circum- stances the primary circuit includes the condenser, and the secondary is short-circuited by the arc. Of the same nature is also the case in which a single circuit, including inductance and capacity, is provided with a tubular metallic core, the circular paths of the eddy currents induced in the core forming the short-circuited secondary. In the present communication we shall consider the effect on the period and decrement of the oscillation of the system, of varying the resistance of the short-circuited coil. It will be assumed that a primary circuit, having self-inductance Ly, capacity C,, and resistance R,, is coupled with a ciosed secondary coil having self-inductance L, and resistance R,, the mutual inductance of the two coils being M. It will also be assumed throughout that the capacity of the secondary is negligible ; the resistance Ry, may be assumed to be varied by altering the thickness or the specific resistance of the secondary wire. # Phil. Trans. A. vol. cexi. (1911) pp. 375-410. + Communicated by Prof. E. Taylor Jones, D.Sc. Phil. Mag. 8. 6. Vol. 39. No. 233. May 1920. 20 O54 Mr. Tolo Jones on the Period and Decrement The fundamental equations for such a system are di, dio L, ii + M a -b Ri, + We —=()), o ° 40 a (1) 28 ee : (2) ie 2 dt at EES a5 a aaa and the current flowing through the condenser is given by Sans y= C, ar : A 5 5 6 3 . 2 (3) Substituting this value of 7, in (1) and (2), these equations become — : d?V, dis ~ dV, Thin as , 4 L,Ci—7-+M 3 7 Cl, +V,=0, ees) dig Nene ean : le 2+ MC, +Ri=0. - - 2 2 5) Assuming V,;=Ac!, 2.= Be, substituting in (4) and (5) and eliminating the ratio A/B, we arrive at the cubic : e ; L. (db, SWE BESGL Pein fee + (G i R\R,) ++ 2 ng. i i . and on introducing the coupling Te represented by £?, this 4} 19 cubic becomes eli; Ly 1—k? (oe s Ibs ‘ tba il I oul 3 i-F i, Oe The solution for V, is known to be of the form ~3 Raich | (eae + y ae—* eos (2a4rnt — 6) + be~ 6, where 4; represents the decrement and n the frequency .of the oscillations, so that the roots of the cubic (6) are given by a= — hy + 2ani | sg — hy —2arni a eer} <3 ==. 5) For the purpose of tracing the effect on k, and n of varying the ratio R,/L, a number of numerical cases have been worked out. In these the following numerical values are assumed— of an Oscillatory Electrical Circuit. D900 values which may be taken as typical of a fairly large induction-coil :-— i, Cp oG. 10> > c.G:s. The coupling 4? is varied from 0°5 to 0°95, and the ratio R,/L, from 100 to 2000. In each case the three roots of the cubic (6) were calcu- lated by Cardan’s formula, and hence the values of n, 41, and 6 found by (7). The results of these calculations are shown in Table I., the first column giving values of 1”, the second those of ~R,/L, the third, fourth, and fifth columns the values of ky, n, and 6 calculated as explained above. TABLE J. bay bp — ib fC — 1206 GNOmec E-Se N. © rt) =~ 2, Re. ky. ; 0-5 100 100-00 812°91 100-00 : 500 297-56 807-92 50486 x 1000 530-00 79298 1040:00 Es 1500 729°43 768 78 1641-13 M 2000 873°81 736°84 235236 06 100 137-48 908°71 100-04 500 43458 ONESIO NS 505°63 1000 78810 878:76 1048-80 . 1500 1098°27 841-47 167847 ee 2000 1322°85 789-01 2479°33 0:7 100 199°97 1049-10 100-06 e 500 663-25 1037-88 506°87 1000 1220-58 1001°62 1058°85 1500 1729°57 939°77 1725-51 5 2000 208821 848:03 265492 0-8 100 324-96 1284°38 100-07 500 1121-05 1263°26 507-91 1000 2090°73 1198-24 1068°54 1500 298530 1080-74 1779°38 4 2000 3649°52 884°52 2950-96 OG 100 699-96 1814°59 100-08 s 500 2495°52 1758°10 508°94 1000 4709:98 1581-05 1080-02 1500 6825°84 1228 04 1848°32 * 2000 Imaginary ~ 3919°21 0-95 100 1449-95 2560:32 100-09 500 524528 240702 509-45 1000 995712 1887°70 1085-75 1500 Imaginary 4057°57 2000 3679°45 225°17 33641°21 bo © bo ' ~ 0 Mr. Iolo Jones on the Period and Decrement The results given in Table I. are also shown graphically in figs. 1-4. Fig, 1.—Variation of ky with R,/L,. The general effect upon the damping of the oscillations caused by increasing R,/L, with the coupling kept constant is shown in fig. 1. With a low coupling such as 0:5, the damping increases rather slowly with increase of Ro/L», whilst for greater values of the coupling, the increase is more rapid and also more nearly uniform, the curve for the value k?=(0°95 being almost a straight line. The effect of the variation of the coupling upon the de- crement, R,/L, being kept constant, is shown in fig. 2. The decrement increases with the coupling, more rapidly the closer the coupling. Figs. 3 and 4 show the corresponding effects upon the frequency. At any constant value of the coupling the frequency diminishes as the ratio R,/L, increases, the diminu- | tion being more rapid the greater the value of the coupling. With a constant value of R,/La, the effect of increasing the coupling is to inerease the frequency, the rate of increase being greater for the smaller values of R3/Lp. of an Oscillatory Klectrical Circuit. Fig. 2.—Var.ation of k, with /°. ao0 558 Mr. Iolo Jones on the Period and Decrement Fig. 4.—Variation of 2 with 7. Mee ea ka The Condition tor Real Oscillations. As shown in the Table, for certain values of 4? and R,/L; the corresponding values iO the frequency become i imaginary, which means that no oscillations are produced at these values of k? and R,/L,. This occurs when the coupling is 0-9 and 0-95 and when the ratio R,/L, is also lar ge. It is clear however, that there must be a real oscillation when R, is infinite, in which case there is no induced secondary current, and consequently there must be, at the larger values of 2, a certain limited range of values of R,/L, over which no real oscillations are produced. The extent of this range depends upon the coupling, and it can in fact be shown ‘hen there is & limiting value of 42, below which there are real oscillations for all values of R,/L., and above which there is a finite range of values of R,/L, over which no real oscillations are produced. Thus, considering the cubic in z given by (6) and writing it in the form : 2+ 362? + 3cz2+d=0, ) of an Oscillatory Eectrichl Circuit. 559 the condition that the roots of this cubic should be all real is (d—3be + 263)? +4(e—1?)° =0 or d? — 3b?c? — 6bed + 463d + 42 =0, which, on transformation, reduces to the quartie in R,/L, 4 RY a. Mg LACS IE ie ook? TEs [ei(4K ~6K*) +6K*, + Be —nelhl s 09 Ke = 61K) = ok oe ; a aie L, a (L,C,)? Reg Ry Ry 1 > gee sal [2 (-8K?— as ie uk us R,? ore: EL 93) gy ul eeatiee 1 4K ——_.. =0, ‘ (L,C)° Bei W here : 3K = Te . We shall here confine our attention to the case R,;=0. The condition that all the roots of the cubic (6) should be real, or which is the same thing, the condition that » should be imaginary, Is given by the quadratic in (Re/Le)? R,* a yr r 1 1k eerie 9 ——— SK —3 Kk? aye on Jee — The condition that a range of values exists for R»/Ly, over. which there are no oscillations is that the roots of the above equation should be real, ie. (9—-18K—3K?)?—192K?>0, or =3K*—28K3490K?2—108K 4 2750, or (K—~—3)3(K—1/3) >0, Of) Koa. or Th? > -8898 Hence for all values of k? below °889, the oscillations are real whatever be the value of R»/Lz, whilst for values of ? 560 Mr. Iolo Jones on the Period and Decrement above *889, there exists a range of values for R»/le over which no oscillations are produced. | Table II. shows this range of values for various values of k? between this limiting value *889 and unity, R, being kept at zero. Tase IT. Ja = 0. Coupling vole oO R,/L, WIIG Ly ey Extent of =e if corresponding to corresponding to Bane ma Lower Lt. of Range. Higher Lt. of Range. pe 0°889 = Limiting Value 2086°06 208606 0 of #?. 0:90 2020°04 2043°70 22°96 0:93 1765°79 1956:08 190-29 0°95 . 1580-23 190767 BIT 44 0:97 1212°84 1864°37 651°53 0-99 715°33 1825°12 1109-79 0°999 228°10 1808°42 1580°32 1:00 0 1806°58 1806°58 Tables III.-VII. give the values of 4, n, and 6 for various values of the coupling, a little below and a little above the limiting value, R»/L. being given various values ranging from 2900 to 6000. TaBLeE III. k? =O Te Ry lO. = 14 C) == Te66.. Wace: R,/L,. (ee nN. O. 0 0 1049-90 0 900 531-20 1038°67 50684 1000 1137-52 1003-19 1058°30 | 1500 163432 944°04 1731°38 2000 2009°67 85464 2647°33 2500 2043°47 743-08 4159°47 3000 180656 659°59 6386°87 3500 150403 620:95 8660°61 4000 1268°75 606°28 10795°83 5500 87700 588°36 16579°33 6000 79679 083°83 18406°45 of an Oscillatory Electrical Cireuit. R,/ty — hy. 0 1412-46 2796°25 4106-10 5076°44 5113-92 4898°89 4185°53 3211°43 2268°19 1530-56 Ry ‘Ly = 0, Ge 0 1995°36 3960°80 5832°85 7241-11 5438°93 3031-10 2265°51 1570°21 R,, i 0), ky. 0. 2244-72 4460°17 6636°45 3123-82 4648-09 3:)26°35 2260°31 1579-42 TABLE LV. k?=0-85. ie 7-66. LO>* Gress. 2. 1485°09 145517 136115 1182-34 856-67 T5155 623-12 533°40 526°05 545°45 560-60 TABLE Ve hk? = 0-889. 1253°96 597°95 128-25 469°03 517-05 547-55 VApnne Vv 1. (9. é. 0 508°41 107415 1787-79 317926 377216 4867-87 6962-27 10243-80 15463°64 23604°56 a 0. 0 509:28 1078°40 1834-30 3517-78 8022715 16439°80 22468:99 3285959 ier —7-66 .L0—* Gs. nN. 1818-48 1766°65 1599°63 1269-05 301°40 60°52 411°30 509°74 543°91 18947°30 25479°38 36841:15 D6L 262 Mr. Iolo Jones on the Period and Decrement TasLe VIL. ke =) 95) Rijly=0, In0;= 766. 100" ¢.e's: R,,/L,. ore nN, 6. 0 0 OaleTl 0 500 4547-28 2433-20 509°43 1000 9457°28 1956-50 1085-44 1500 14059°33 516°67 1881:34 2000 3780-90 214°35 32438°19 2500 2777°40 421:14 44457°21 3000 2223-06 481°66 55553°88 4000 1611-92 527°65 7677627 Figs. 5-8 show the curves of variation of » with R»/Le for different values of the coupling in the neighbourhood of the limiting value 0°889. Fig. 5 is the curve for k?=0°7. At this coupling the frequency diminishes, but not uniformly so, as R»/T» increases. At 4°=0°85 (fig. 6) the curve shows a 0 1000 =2000 3000 4000 5000 6000 R Tit5 minimum frequency at R2/L, nearly equal to 2400. The bend in the curve at this point develops into a double point in the curve for k?=0°889, the limiting value, as shown in fig. 7. For values of k° above this limiting value, the curve breaks up into two distinct portions as shown in fig. 8, the curve for k?=0599. His, 6. Comparison with the equation for @ single circuit. For a single circuit, the frequency of the oscillations is given by 1 1 5 Ne n= Dar Tee —h, ° ° ° . . (8) When R./L, is very large, the values of » given in the Tables should approximate to the values given by the formula. Table VIII. shows bow this approximation becomes closer the higher the value of R,/L», 4? in this case being 0°95. It appears therefore that the simple expression (8), which is strictly applicable to a single oscillatory circuit possessing resistance, cannot generally be used when the damping arises from the resistance of a closed secondary coupled with the circuit. As already remarked, the above results are applic- able only to the case in which the capacity of the secondary coil is negligible. In certain actual cases, when the secondary Our for) ie) | Light Absorption and Fluorescence. Tapunev ILL, 12=0°95. R,/l,=0, 1,0;=7°66.10-° c.G.s. n n B/,. from Tables. from formula (8). 500 2433°20 Imaginary 1000 - 1956°50 Imaginary 1500 516-67 Imaginary _ 2000 214°35 Imaginary 2500 421-14 367°82 3000 481-66 453°32 4000 527°65 514-16 5000 546°68 -538'32 6000 555°54 550°19 resistance is varied by means of a rheostat across the secondary terminals and the secondary capacity is not a negligible quantity, the system will possess two oscillations if NE resistance of the rheostat is greater than a certain value. The Physical Laboratory, University College of North Wales, Bangor. LIV. Light Absorption and Fluorescence.—V. The so-called Molecular Rotational Frequencies of Water. By E. C. Copa | Crbebweilaocs, ol lv... Grant’ Professor “of Inorganic Chemistry in the University of Liverpool *. cr a recent paper {f the absorption system of sulphur dioxide was discussed, and it was shown that the whole of the absorption-bands shorn by this gas can se expressed in terms of three fundamental frequencies : ZA DSO, 8:19 x 104, and 1:°296x10". It was also aac that Hie frequency 2°4531x10" is characteristic of the atom of oxygen, and that the two frequencies 8:19 x 10! and 1-296 x 10!2 are characteristic of the atom of sulphur, since the infra-red absorption-bands of oxygen can be expressed in terms of the first and those of sulphur ‘and hydrogen sulphide can be expressed in terms of the last two constants. The combination of these three constants to give the frequencies characteristic of sulphur dioxide would seem * Communicated by the Author. + E. C. C. Baly and C. 8. Garrett, Phil. Mag. vol. xxxi. p.612 (1916), 566 Prof. E. C. C. Baly on Light to be of great importance, for it affords a very complete example of the least common multiple principle which forms the basis of the frequencies of absorption-bands, as has been pointed out in the earlier papers of this series. The least common multiple of the three frequencies of oxygen and sulphur given above is 2°89299x10!, and this number multiplied by 10, 12, 14, 18, 26, and 33 gives the exact central frequencies of all the absorption- bands which haye been observed for sulphur dioxide in the infra-red region between the wave-lengths 12 u and 3 p. Then, again, of these absorption-bands the one with the central “Frequency 2°89299 x 14x 10" has the greatest intensity, and this central frequency multiplied by 25 ives the exact central frequency of the less refrangible band in the ultra-violet, the central frequency of the more refrangible band not ‘having been observed. ° It is well known that the effect of cooling is to decrease the width of absorption-bands and that at very low tempe- ratures only the central frequency remains, this persisting at the lowest temperature yet reached. It is evident therefore that the central frequency i is the only one which is truly characteristic of the molecules, the subsidiary frequencies to which the breadth of the bands is due being connected in some way with the temperature of the mole- cules. These central frequencies are thus true molecular frequencies, and therefore it may be concluded that the fundamental molecular frequency of sulphur dioxide is the least common multiple of the frequencies characteristic of the sulphur and oxygen atoms it contains. . Tt was also shown that the subsidiary frequencies of sulphur dioxide to which the breadth of the absorption- bands is due are also derived from the three atomic fre- quencies of sulphur and oxygen. In the less refrangible absorption-band there are a number of sub-groups symme- trically distributed with respect to a central sub-group, and, further, each sub-group contains a central line of maximum absorptive power with a series of lines symmetrically arranged on either side of it. In the less refrangible absorption-band of sulphur dioxide the central lines of the sub-groups form a series with a constant frequency difference of 6°69696x10" which is the least common multiple of two of the atomic frequencies, namely 8:1) x 10” and 2°4531x 10". Again, in the more refrangible band there exist sub-groups the central frequencies of - which form a series with the constant frequency difference of Absorption and Fluorescence. 567 1:05972 x 10, which is the least common multiple of another pair of the atomic frequencies, namely 8°19 x 10" and 1:296x 10". lastly, in each sub-group of the less re- frangible band the component lines form a series with - constant frequency difference of 8°19 x10", which is one of those atomic frequencies compounded in the constant frequency difference of the series of the central lines of the sub-groups. The eomponent lines of the sub-groups of the more refrangible band of sulphur dioxide have not yet been accurately measured. It will thus be seen that, whilst the central frequencies of all the absorption-bands of sulphur dioxide are based on a molecular frequency which is the least common multiple of all three atomic frequencies, the sub-groups of the bands are due to the least common multiples of two out of the three atomic frequencies, and the component lines of the sub-groups are due to the atomic frequencies ‘ themselves. Hach complete absorption-band therefore consists of a central or true molecular frequency, com- pounded with intra-molecular frequencies in the central lines of the sub-groups and with atoniic frequencies in the component lines of the sub-groups. Sulphur dioxide was the first substance in which these relations were completely worked out, since it was the first instance for which accurate measurements had been made both for 'the infra-red and for the individual lines com- posing one of the absorption-bands. Recently, however, Sleator * has published highly accurate measurements of water vapour at 6m and 3y, and it becomes possible to test whether a structure similar to that found in sulphur dioxide exists in the absorption-spectrum of water vapour. Further, if the atomic frequency 2°4531 x 101! is truly characteristic of the oxygen atom, this value should form one of the atomic frequencies which is active in the case of the molecule of water. It is well known from the works of Bjerrum and of Miss von Bahr that the two great absorption-bands of water at 6 wu and 3 yw consist of sub-groups and that the central frequencies of these sub-groups can be arranged in two series with constant frequency differences of 7-5 x 101 and 1:7301 x10" respectivety. By analogy with sulphur dioxide these differences will be intra-molecular frequencies, and therefore will be the least common multiples of two * W. W. Sleator, Astrophys. Journ, vol. xlviii. p. 125 (1918). t 4 i Ry © z 4 iy ) : y s 568 Prof. BE. C. C. Baly on Light © out of the three atomic frequencies active in the water molecuie. We therefore have to find a solution satisfying the following conditions if the analogy with sulphur dioxide is complete. Three atomic frequencies must exist.of which one is 2°4531 x 10!!, the least common multiples of two pairs of these three must be 7°5x 10" and 1:7301 x 10, and the least common multiple of all three atomic frequencies must be a number of which exact multiples give the central _ frequencies of the great absorption-bands at 6 w and 3 p. The three atomic frequencies are at once found to be 150635 x 10", 21159 x LOM) and? 2-453) x10 ite mest common multiple of the first two is 75x10", the least common multiple of the last two is 1:°7301 x10”, whilst the least common multiple of all three is 6°1326 x 10”. ‘The first two conditions are therefore satisfied. As regards the last condition, 6°1326 x 8x 10" is the freauency corre- sponding to the wave-length 6°115 py, the value observed by Coblentz being 6°1 w, whilst 6°1326x 16x10” corre- sponds to the wave-length 3°057 mw, the values observed by Coblentz and by Paschen being 3°05 w and 3:07 p respectively. All three conditions therefore are exactly satished, and the fact that the atomic frequency of oxygen in the water and sulphur dioxide molecules is the same would seem to be of great importance. It now becomes possible to put this solution to a very severe test by calculating from the above frequencies the wave-lengths of the component absorption-lines in the two great absorption-bands, and comparing them with Sleator’s observed values. His measurements were made over the following regions: 1:35 to 1:45, 18ly to 1°92 p, 202 pw to 2:87, and. 9°02 4 to 6°83ju2) Whe wach sets deal with portions of the two bands with centres at 3°057 w and 6°115 w respectively, and since the third set contains the greater number of individual measurements it will afford the best opportunity of testing whether the atomic frequencies given above are correct. As stated, there are present in the band two series of sub-groups, the central lines of each of which form a series with constant frequency difference, the two constant frequency differences being 7°5x 10" and 1:7301 x 10” respectively. Now the central frequency of the band at 3y is 6:1326 x16 x 10” or 9°81216 x 10, and obviously the frequencies of the complete set of the central lines of the sub-groups will be given by 9°31216x 10+ m x 17200 < 102 sama 9°81216 x 108+nx75 x10", where m and n’ equal 1, 2, Absorption and Fluorescence. 569 3, ete. Sleator, however, only measured the lines in a portion of one side of the band, and all the central sub- group lines measured by him are given by 9°81216 x 10° +mx1:7301x10", where m equals 4, 5,...12, and 9°81216 x OP +nx 75x LOM, where n equals 9, oe Shs Then, again, there are the component lines in each sub- group, the fr equencies of which are given by compounding the central frequency of each sub-group with the atemic frequencies. There are four such series of lines possible, two in each series of sub-groups, but the number of lines in any one of the sub-groups must be small, since the sub- groups are very narrow and closely situated in the band. The maximum number of lines in any one sub-group without overlapping is seven—namely, the central line and three on each side of it. In the second series of sub- groups, the central frequencies of which are given by 9°81216 x 10%+4+nx 75x 10", there should be two series of ‘constant differences 1:0635 x 10!! and’ 2°1159 x 10", since 775 x10" is the least common multiple of these two atomic frequencies. But 2°1159 is very nearly twice 1:0635 and therefore it will be sufficient only to calculate the series with constant frequency difference 1:0635 x10", and the formula for all the lines in the sub-groups will obviously be 9°81216 x 108+n”x 79x nee x 1°0635 x 10", where f= lO... a, and. p= 0, 1,2 eo: Similarly, in the first series of sub- -oroups there should also be two series of lines with constant frequency dif- ference, but it happens that the frequencies calculated are very nearly the same as those calculated for the second series of sub-groups. For the present purpose therefore they may be neglected. In Table I. are given the calculated values of all the lines in the second series of sub- groups and the central lines only of the first series, together with Sleator’s measurements. In the first column are given the wave numbers (1/2) for all the lines in the second series of sub-groups calculated from the above formula. The sixth column contains the wave numbers of the central lines of the first series of subgroups calculated from the frequency formula 9°81216 x 10°4 mx 1°7301, where: m=4, 5,...12. ihe second and fourth columns contain the corresponding wave-lengths, and in the fifth column are given the wave- lengths as observed by Sleator. The agreement, on ‘the whole, hetween the calculated and observed values is remar kably e00d, the maximum difference being 0:0022 yw, whilst the general average 1s +0°00074 wu. Pil. Mag. 8. 6. Vols a9) No.-233, May 1920. - 2 P 570 Prof. H. C. C. Baly on Light TABLE I. SEconD SERIEs. First Series, gem. ea oe aN TRS Boa aS mm. 1/X. d eale. d obs. d eale. 1/A. n. 348°86 28664 2°8658 349°22 2°8635 9 349°57 28606 2°8590 349°93 28577 - 350°28 2°8548 2°8538 2°8560 300714 4 350°65 2°8520 35100 2°8490 2°8485 , 35164 28460 351°72 2°8432 10 352°07 2°8403 35243 2°8375 2°8373 352°78 2°8347 2°8336 353°13 28318 35350 2°8288 353°86 2°8259 2:8267 304/22 28231 li 354°57 2°8203 2:8197 354-93 28175 355°38 2°8147 - 2°8140 35563 ZO199 356°00 28089 2°8103 2°8097 30591 ®) 35636 2°8061 2°8077 306°72 2°8034 2°8029 12 397-07 28006 35743 27978 2°7972 307778 2°7949 27947 35815 2°7922 2°7926 358°50 2°7894 35886 2°7866 2°7867 359° 22 2°7838 27819 13 399°57 27811 2°7803 30993 2°7784 PTT 360°28 2°7756 27765 36063 2°7729 27712 361/00 27701 2°77 00 361°36 27673 2°7668 361-72 2°7646 2°7655 2°7690 36167 6 14 362°07 27619 2°7625 362°43 2°7592 27614 362°78 2°7565 363°13 2°7538 2°7545 363°50 27511 27511 363°86 2°7483 2747 364°22 2°7456 15 36457 2°7430 2°7445 36493 2°7402 2:7407 36528 27376 2°7386 365°63 27351 2°7340 365°80 2°7323 27314 366'36 2°7295 2 1204 366°72 2°7269 27218 16 36707 2°7243 2°7236 36743 27216 Jolt 367-44 f 367-78 27190 2°7198 36814 2-7 164 2°7168 368°50 3°7137 368°86 27111 27108 369°22 2 7084 27084 Absorption and Fluorescence, 571 Seconp Series. First SErizs. a ~——_ a es Se, m. 1/r. d eale, X obs. r eale. 1/r. n 17 369°57 2°7058 2°7044 369°93 2:7033 370°28 2°7007 2°7006 370°63 2°6980 26974 371:00 2:6954 a7 1°36 2-6927 2°6930 on (2 36902 18 37207 26876 26880 372743 26852 26853 37278 2°6826 2°6827 Solo 2-6800 2°6795 Siow 8 373 50 2-6774 2°6783 373°86 26748 2'6760 314°22 2°67 22 2°6720 19 37457 2°6697 2:6696 374:93 26672 26661 379°28 2-6647 92-6645 375°63 26620 26610 376 00 26595 26590 37636 2°6570 376°72 26545 9:6547 90 OT 2:6520 2°6515 37743 2°6495 ST 2°6469 ares 26445 2°6448 378°50 2°6420 26411 378°86 2°6394 2 6385 26387 37897 9 37922 2°6369 me AI | 379°57 26346 379°93 Da SPA! 2°6325 380°28 26297 2°6295 380°63 26272 381°00 2-6246 2 6256 38 1°36 92-6222 381°72 2°6198 2°6196 22 38207 26173 26161 382°43 2°6149 382°78 26125 2°6123 aon la 26101 2°6090 333 50 2:6075 26066 383 86 2°6051 2°6046 384 22 2:6027 23 38457 2-6003 26008 2°5991 384°74 10 384-93 2 5979 38)'20 2 3954 385°63 25931 2:5940 386°00 2°5907 386736 2°5882 25880 386:°72 2-589 92-5864 D4 38707 2-5835 2°5830 387:43 2-581 1 2°5803 387°78 2:5788 58813 25764 2°5760 888°50 2°5740 388°86 2°5716 LT at 389°22 2°5695 2°5688 o72 Frot. H.C. C. Baly on Light Sxeconp SERIEs. First Series. (a RE => | eee yar feb aa ia Me 1/d. dr eale. X obs. Xd eale. 1/X. Ne 25 389°57 2°5669 389°93 2°5646 25636 390°28 2:5622 2°5608 25608 390°51 11 390°63 2°5599 391:00 2°5575 391°36 252 391-72 2°5529 26 39207 2°5506 25520 392°43 2°5482 392-78 2°5460 2°5469 393°13 2°5437 393°50 2°5413 25425 : 393'86 2°5390 394°22 2°5367 2 394°57 275344 2°5352 394:93 D2 d2 i 2-ad1t 395:28 2°5299 2°5288 395'45 12 395'63 2°5276 396-00 2°5293 2°5262 39636 2°5230 2°5235 The justification for the correctness of the calculation does - not rest on this agreement so much as on the facts that Sleator has pleer cd no more lines than are obtained by the use of the formula, and that the average difference between the observed wave numbers of lnc consecutive lines is just equal to that given by the formula. In view of the fact that the whole calculation is based only on the frequency differences 75x 10! and 1:°7301-x10" first obtained by Eucken, and the atomic frequency of oxyg ven obtained from the absorption-system of sulphur dioxide, it may be claimed that these results afford a eoniconiay striking confirmation of the theory of the system of absorption-lines shown by a substance between the limits ionmeand, 02m Sleator’s measurements of the region between 5:02 pu and 6°85 «4 do not reveal such a complexity of absorption- lines, and it is not worth while therefore to calculate the frequencies of all the lines which should appear in this region. It will be sufficient if the central frequencies of the two series of sub-groups are calculated and compared with the observed values. The central molecular fre- quency of the band as already stated is 6°1326x 8x 10!" or 4:90608 x10, and therefore the frequencies of the central lines of the first series of sub-groups will be given by 4°90608 x 10% +m x 1°7301 x 10%, and those of the second series by 4 4:90608 x 108 +nx 7:5 x 10", where m and n equal 0,1, 2,3, ete. In Table II. are given the wave numbers 4 ‘ Absorption and Fluorescence. DY be and wave-lengths calculated from the different values of m and n, and in the fourth column are given Sleator’s measurements. The agreement between observed and calculated values is again very good, especially when the fact is considered that in these measurements Sleator estinrates his accuracy to be only one-tenth of that reached in the previous set set forth in Table I. TABLE II, Seconp SERIES. First SErirzEs. a ios are a fae ST IN Mm. I/A. d eale. r obs, d eale. 1/X. n. a 14604 6848 6828 6838 146°24 3 6 148°54 6-732 6°752 5 151-04 6621 6634 6°573 6579 152:00 2 4 leaps 6513 6520 3 156-04 6409 6418 6°344 6°338 Loa 1 2 158°54 6°308 6289 fy 16104 6210 6 216 0 163°54 6115 6114 6115 163°54 O it 166°04 6023 6-016 2 168°54 5934 5-938 5°895 5-907 169°3 il ae 171-04 5847 5864 4 173-54 3°763 5768 5-717 7 1 VOT, 2, a 176-04 5681 5°680 6 178°54 5601. 617 7 181-04 5524 5-525 5500 180°84 a 8 183°54 5-449 5447 9 186-04 5375 5°352 5359 186°60 4 10 188°54 5°304 5°309 il 191-04 5235 5: 242 12 193°54 5167 572 a Wl 195°37 By 13 196-04 5102 ELIT 14 198°54 5037 5022 5:022 199-14 6 The results now obtained for the system of frequencies possessed by the molecules of sulphur dioxide and water establish very definitely the existence of simple relationships between the three sets of frequencies—molecular, intra- molecular, and atomic. Whilst the atomic frequencies are characteristic of the individual atoms and the molecular frequencies are characteristic of the molecule as a whole, the intra-molecular frequencies are characteristic of groups of atoms within the molecule. Remarkable confirmatory evidence of the reality of these intra-molecular frequencies and their origin in atomic groupings within the molecule is afforded by naphthalene. In one of the early papers of this series I dealt with certain constant frequency 574 Prof. E. C. C. Baly on Light differences between the sub-groups in the solution phos- phorescence band of naphthalene. According to the theory of molecular rotational frequencies, this frequency difference of 1°4136x 10! should be the rotational frequency of naphthalene, and I therefore ventured to prophesy that this compound should exhibit a series of absorption-bands in the infra-red with frequencies given by 1:4136 x mx 10", where m=1, 2, 3, ete. The infra-red absorption-spectra of naphthalene had not then been observed, but it has recently been examined by Stang * with the specific aim of putting my prophecy to the test of experiment, and he found no evidence of absorption-bands at those positions. The experience gained from sulphur dioxide, and now from water, shows that the frequency differences exhibited between the compound sub-groups of an absorption-band are not molecular but intra-molecular. It does not neces- sarily follow therefore that they will be exhibited as infra-red frequencies of naphthalene itself, but we are brought to the conclusion that they are characteristic of a detinite atomic grouping present in naphthalene. Now the two principal atomic groupings in naphthalene are the benzene nucleus and the olefine linking. The intra- molecular origin of the frequency 1:4136 x 10% would lead to the belief that this number is characteristic of— and, indeed, is the true molecular frequency of—either benzene or the olefines. In other words, benzene or the oletines should exhibit absorption-bands the central lines of which have the frequencies 1°4136xXmx10' or the wave-lengths 21°22 yu, 10°61 wp, 707 pw, 5:31 yp, 4°24 p, etc. Moreover, of these the intensity of absorption will be a maximum at A=21:'22 and fall with decreasing wave- length. Now the absorption- spectrum of benzene shows impor faut bands at 9°78 pw, 6°75 pw, 5°5 pw, 3°25 w, and obviously therefore the above frequency is not associated with the atomic grouping of benzene. On the other hand, this frequency is absolutely characteristic of the olefines—a_ series of hydrocarbons which are almost identical in their absorptive power in the infra-red. Coblentz has investigated the region between 13pm and 2 yp, and he found bands with centres at 10: 5, 6°98 pw, 5°30 uw, and 4°32 un. These four bands are the only ones which the olefines show between 13 and 4m, and the amount of light absorbed at these maxima with the layer of olefine used was 98, 81, 47, and 14 per cent. respectively. Since Coblentz did not * Stang, Phys. Rev vol. ix. p. 542 (1917). 271 Absorption and Fluorescence. rhs extend his observations beyond 13m, he did not discover the great band which must exist with its centre at 21°22 w It is evident from the results given above that the obser- vations are in direct conflict with the theory that molecular rotational frequencies play a fundamentally important role in absorption-spectra. According to this theory the mole- cules possess definite velocities of rotation, and Bjerrum pointed out that on the energy quantum theory the fre- hn : quencies of rotation are given by Qn 27” where fh is the Planck constant, 6°56 x 10-2", I is the moment of inertia, and n=1, 2,3, etc. These so-called rotational frequencies exhibit themselves as a series of absorption-bands in the long-wave infra-red and also as frequency differences within the short-wave infra-red bands. The width of the rotational frequency bands is explained by saying that do] only represents a most probable value exhibited by the majority of the molecules. Two criticisms at once may be made of this theory as it stands. In the first place, the inexactness of the rotational frequencies is thoroughly unsatisfactory, for the variations that have to be allowed in these frequencies are very great indeed. In the case of water the so-called rotational frequency series of bands have actually been observed, their frequencies being given by 1:7301 xmx 10” and 75xnx10". Now in each of these two series tae consecutive bands overlap one another, and it follows therefore that the difference between the extreme values of any given molecular rotation must, at any rate, be equal to the: difference between any two con- secutive mean values. In the case, for example, of the : 6h molecular rotational frequency Fn2)? if # be the variation 6h the extreme values will be aay te: Now observation Qrr71 — 7 6h shows that ate must be equal to or greater than Th i , : 5 | —, a conclusion which deprives the theory of any exact basis. In the second place, it has been shown above and in previous papers that the sub-groups due to the so-called rotational frequencies have also a structure arising from 576 \ Prof, E, ©. C. Baly on Lnght the existence of frequencies far slower than the rotational frequencies. It is difficult to conceive of any vibrations exhibited by a molecule, as a whole, which are tar less rapid than its rotations. Kriiger * has proposed an alternative theory in that he anbaiines precessional motions for the rotational velocities. He gives the following expression for the frequency of precession : N I =e 2A cos d’ where N is the angular momentum of the electron spin, A is the moment of inertia of the atom nucieus perpendicular to the figure axis, and ¢ is the angle swept out by the figure axis as the result of the precession. By the absorption of one energy quantum the angle ¢ is increased and the value of aa doubled. Kruger himself saw the ditheulty connected pigh the width of the absorption-bands due to the precessional motions. He was led to the conclusion that for a given precession the values of @ cannot be exact, but must lie between broad limits. But, as stated above, the cogent criticism must be made that in order to explain the observed phenomena it is necessary to postulate a difference between Pie : 1 alee the extreme limits of a given mean value of ——., which is S equal to or greater than the difference between two con- secutive mean values of ——. Then, again, it is difficult to cos conceive of motions of a gyrostatic system far slower than its precessions. But the most 1 important fact to be considered in discussing these two theories is that the so-called molecular frequencies are due to small groups of atoms within the molecule, and that they are the least common multiples of the frequencies characteristic of those atoms. Further, the same atomic frequencies are shown by the same elementary atoms 1n different compounds —for example, oxygen in sulphur dioxide and water, and sulphur in hydrogen sulphide and sulphur dioxide. When two or more atoms enter into chemical combination, the resulting molecule is endowed with a * E. Kriiger, Ann. der Physik, vol. 1. p. 846, li. p. 450 (1916). Absorption and Fluorescence. BS i7 frequeney which is the least common multiple of the atomic trequency of all its atoms. The molecule also exhibits as subsidiary frequencies both the least common multiples of the frequencies of groups of atoms forming a component part of itself and also the atomic frequencies themselves. | am of opinion that development along these lines aftords much greater promise than either Bjerrum’s theory of rotational velocities or Kriiger’s theory of precessional motions. Before leaving the subject of the so-called rotational frequencies of water, attention may be drawn to a statement which is now becoming general in the literature—namely, that the central wave-length of the water band at 3°06 w is 3°26 w. It is nothing of the kind. The most accurate observations by Coblentz, Paschen, Rubens, and others show the centre to be at 3°06 p. The 3°26 w value is due to Miss von Bahr and is based not on measurements but on a misconception. In his early paper Bjerrum stated that the central line of an absorption-band should be the mean of the two central sub-groups. In this, he was wrong, because it is now well known from the study of some hundred absorption-bands that there is one central sub-group and that the central line of this group is the central line of the whole band. In the case of water the central sub-yroup has its central line at 3°06 yw and this is therefore the true centre of the whole band. Gastagion will undoubtedly arise if this error is not corrected, and I therefore venture to draw attention to it. Indeed, the fact the molecular rotation theory leads to an erroneous value for the molecular or true central fr equency of an absorption-band undoubtedly argues against the theory. In conclusion, it may be said that the criticism I have given of the Bjerrum and Kriiger theories must in no way be interpreted against the validity of the energy quantum theory or its application. It would, indeed, seem that the material now at hand forms an excellent opportunity for the application of the quantum theory, since a well-ordered discontinuity in the frequency shown. by a molecule has’ been established. This, however, is reserved for another paper. The University, Liverpool LV. On Turbulence in the Ocean. By Harotp JEFFREYS, M.A., D.Se., Fellow of St. John’s College, Cambridge * I. The Cause of Ocean Currents. fQ¥XHE main currents in the ocean were formerly attributed to the effects of expansion and contraction caused by differences in temperature between various parts of the water. In a heated area the surface would tend to be raised ; and an outward current would therefore commence at the surface, compensated by an inward return current below. This explanation has, however, ceased to be regarded asadequate. In the first place, the differences in temperature are so small in comparison with their great horizontal extent as to-be unable to produce the observed velocity of the surface water, on account of the low coefficient of expansion. In the second place, the heated areas of the sea would corre- spond approximately with those of the air above. Now heat tends to produce an area of high pressure at the top of a medium and one of low pressure at the bottom; thus, just below the surface of the water and on the same equipotential surface, we should have a high pressure in the heated area. In the air just above the pressure would be low. The hori- zontal forces on the air and the water would therefore always be in opposite directions, and hence the same would ee true of the velocities, even when the rotation of the earth taken into account. Actually this is not even approxi- aicie true; the ocean current and the trade wind agree roughly in direction. Accordingly the general circulation of the ocean has come to be regarded as cansed directly by the trade winds, the surface water being driven along by the wind above it. Here avain, however, there is a difficulty with regard to the direction of the current. The ocean currents vary so slowly with time that they can be regarded as phenomena of steady motion, and on a rotating globe a steady velocity is always perpendicular to the controlling force, in this case the skin friction of the wind on the surface of the water. Thus apparently the current driven by a trade wind should be perpendicular to the wind. The problem is however com- plicated by the existence of turbulence in the ocean, and a treatment based on the theory of eddy motion is therefore desirable. Let w and v be the components of the horizontal velocity * Communicated by the Author. On Turbulence in the Ocean. 579 of the water, m the component of the earth’s velocity of rotation about the vertical at the point considered, z the height above a standard equipotential surface, and k the coefticient of eddy viscosity. Then the equations of motion of the water are esos Quad aia kes cote l) re 2ou=0. (2) If we pnt w=u-+uv, these combine into the single equation pO! — ww =0. Tiara Vedi dc PON Suppose first that & is a constant. Then for a deep ocean the relevant solution of this equation is the one that remains finite when z tends to —#; and in the northern hemisphere this is w=wy exp{z(1+e),/(w/k)}, . . . (4) where w, is the value of w at the surface, and is to be deter- mined from the condition that the rate at which momentum is commuuicated to a vertical column of water shall be equal to the skin friction at its upper surface. Now a column of unit cross-section gains in unit time by friction a momentum (; we de, the integral being taken from the bottom of the A Ne ocean to the surface; and the skin friction on it is «pS?, where « is a constant equal to about 0-002, p being the density of the air, and S the wind velocity, supposed along the axis of w. Then the surface condition is wl l+s)(ok)F=00020S?. . . .-. (5) As all the other factors are real, it follows that wo(1 +.) is real. Hence in the northern hemisphere the surface current is always inclined at 45° to the wind, flowing. towards the side where the pressure is greater.. As the depth increases the velocity diminishes in magnitude and rotates clockwise. In the southern hemisphere the surface current is seen similarly to be inclined at 45° to the wind, flowing towards the side of greater pressure, but the current rotates counter-clockwise as the depth increases. The resultant momentum for all depths together is perpendicular to the wind, in accordance with the condition stated earlier. When ao, w, and 8 are known, equation (5) can be used 280) Dr. Harold Jeffreys on to calculate 4, on the assumption that it is independent of the time and ihe depth. Now if Q be the earth’s angular velocity of rotation, w is equal to QO sind, where 2X is the north latitude. Putting O73 x 10> */l'sec. and p—1-3 x 10s %ayemes we have | 5 St k=05 x 107-'—= 40-aN 2 (60° W: 8 S.W. 19 W.byS. 332° . 4: AOS IN 20° Ws 105 S85 We by We 3 N.W.byW. 673° 120 LOSING 40° Wes 10 N.E. 4 E. 45° 460 S. Atlantic ... 36° 8, 10° W. 8 N.W. 1 W. AHO 14 indian yes. NOS Sy ck ile 10 S.E. 3 E. 45° 200 ACIS em 40 Sie > 12, NE Weiby WE. 12 Wo 333° 30 INS Pacilie 2407 Na LOUCd:. 7 S.W. ] W. by S 355° 8 Sapeacitice.ees: NOD Sh NOG 135. 8 S.E. 1 E. 45° 45 40° §.. 120°. E. 10. N.W.by W. 3 W.’ 333° 120 The above localities were selected as giving some repre- sentation of each ocean and fairly determinate values of the mean annual wind and current. Many other localities were rejected on account of the large annual variation they showed. The prediction that the deviation in direction should be 45° is surprisingly well verified by these rough numerical data, for in no case does it differ from this by more than two points. The fundamental assumption that the vertical variation of kis small as far down as currents extend is therefore so far verified. The range of horizontal variation of k is however surprisingly great, the largest value found being over 100 times the smallest. Many factors may cooperate to cause this. The turbulence probably depends on the height of the waves produced, and this depends on the distance from land. Accordingly it was to be expected that the two smallest values of & would correspond to places near the American and Japanese coasts that the prevailing winds reach after traversing comparatively short distances is Turbulence in the Ocean. 581 over the sea. Accidental error in the determination of the wind or the current must also be a very important factor ; for in the calculation of & the wind velocity is raised to the fourth power and the current velocity to the second, so that any such error, evenif itself small, becomes greatly magnified in importance. Local Pons, such as cyclonic qis= turbance and the temperature lapse-rate, affect the turbulence in the air and perhaps indirectly that in the water. Vari- ations in the vertical distribution of salinity in the water near the surface may also affect the stability of the mass- distribution, and hence the turbulence ; though in mid- ocean mixing is probably sufficiently thorugh in the upper layers for such var lations to be insignificant. It is on the whole, therefore, not surprising that “considerable variations in the calculated turbulence occur. The relation of the surface current to the wind has previously been considered by Waltrid Kkman*, who ob- tained the result here given that they are mutually inclined at 45°, and showed that in the main Atlantic current in latitude 10° N. the velocity is reversed at a depth corre- sponding to a turbulence coefficient of 29°5 em.?/sec., about js of the value here found for a similar place, though well within the limits of real variation, He did not apparently determine / for comparison from the observed velocities of the wind and current, as has been done here; the possi- bility of this comparison is largely due to G. Xi, Taylor’s recent work on the skin friction of the wind on the surface of the earth. Tt remains to be seen whether any other evidence indicates that such turbulence as is here caleulated is possible. Direct observations of eddy currents in mid-ecean are lacking, but indirect evidence may be obtained from two other sources. Taylor showed that the value of & at any point is the average of the product of the vertical velocity in the eddies and the range through which they move vertically. Now near the surface the vertical range is presuma bly not very different from the wave-height h; while the vertical eddy velocity will not be greater than the maximum vertical velocity of the water in a wave, and is probably very much less, since wave-motion in mid ocean is fairly regular in character, and the eddy motion corre- sponds ‘only to the irregular part. Now D. W. Johnson mentions as a typical mid-Atlantic wave tT, one with wave- length 400 feet, height 15 feet, wave-velocity 45 ft./sec., * Arkiv for Matematik, §¢., Akad. Stockholm, Ba. 2, nr. 11 (1905). + ‘Shore Processes and Shore-line Development,’ pp. 25 & 52 (1919). 582 Dr. Harold Jeffreys on and maximum velocity of the water 52 ft./sec. Then the product of the height and the velocity of the particles is 80 ft.?/sec. or 72000 em.?/sec. This is of the order of 500 times the turbulence found, so that very little irregularity in the wave-motion is required to give the turbulence indicated by the current theory. Il. Vertical Transference of Heat. Turbulence facilitates the downward transfer of heat in much the same way as it does that of momentum ; this satisfies the equation ON 40 («22), Ot dc\ Oz where V is the temperature, ¢ the time, and / the thermo- metric coefficient of eddy-conductivity, which is numerically equal to the eddy-viscosity. The only indication I have found of the character of this transfer is given by Murray and Hjort *, in the following table showing the temperature variations in the Atlantic outside the Sognefiord in 1903. Temperatures are expressed in degrees centigrade. Depth (metres). February. May. August. November. 0 4°8 13 3°8 8-7 100 68 6-4 6:9 93 200 eo. 7:0 67 ae) : 300 63 65 6°4 ? It will be noticed that at the surface the highest temper- ature for the year occurs in September ; at 100 metres it is in November; at 200 metres in December; and at 300 metres in the spring. Thus the maximum occurs later and later as the depth increases, owing to the time needed for heat to penetrate downwards. It takes over six months to reach a depth of 300 metres, so that the phase of the annual variation is there reversed. -To determine the vertical distribution of turbulence from the few data available is a matter of some difficulty, and I finally adopted the following method. The annual parts of the temperature variation depending on the sine and cosine of /, the increase in the sun’s longitude since February 15th, were found for the four depths given. The coefficients of these were plotted against the depth on squared paper. If the annual part of the * ‘Depths of the Ocean,’ p. 228 (1918). ‘Turbulence in the Ocean. 583 temperature be Acos/+Bsin/, the equation of heat trans- ference can be written (Ge Os [EE = sin) )={- A sin +B eos) "ae where the lower. limit of the integ ralis z=—«, since A and B both tend to zero there. difdt i is the sun’s mean motion, nsay. Hquating coefiicients of cos/ and sin/ we have eee == ii) { Bdz, hk: - at —n| Adz. The two integrals and the differential coefficients were found roughly from the curves; the places where the integrals changed sign seemed fairly determinate, so that the maxima and minima of A and B could be placed moderately accurately. New curves were drawn to fit these, and from them the integrals and differential ccefticients were redetermined ; thus two estimates of /£ were obtained for each depth, and their agreement gave a test of the adequacy of the theory. The followi ing “table indicates the results of this rough interpolation. Depth A B dA © hs: GBB M2 ime, Paes, (in degrees C), de \ Bye: n- dz° ye a, n QO —45 —O7 —15-0 —248 16 50 ~—74 185) HON 055 —17 — 19 SAs2 95 Small Small ? 100 =—005) —15 — 06 —100 166 —0%5 65 13 150 =+0°3 —1:2 — 05 — 32 64 —O8 5d 6:9 200 = +0°6 —05 — 0-4 — l4 35 —2°5 35 14 250 +08 +0°2 Small 11 ? —0°3 Small ? 300 —-005 +01? Small 3? ? 0-3 — 30? ? In this table dA/dz and dB/dz are measured in degrees centigrade per hundred metres ; \Adz and \Bdz are in degrees multiplied by metres. Thus //n is in square deka- metres. Taking n to be 2x 1077/1 second, and adopting for each depth the average of the two values of & obtained, we have the following values of & in cm.? */sec. :— Depth (in metres)... 0 50 100 150 200-250 300 Gee cine ae U3. 1908-300 as 049? 2 The value of these results must not be exageerated ; but if the theory were seriously incorrect it is very unlikely that 084 Dr. Harold Jeffreys on _it would have proved possible to interpolate at all in such a way as to secure even approximate agreement between the two estimates of & obtained for any depth, and perhaps as it is a considerably different distribution of turbulence would satisfy the data equally well. It will be noted that & is comparatively small near the surface, increases downwards to 100 metres, and then diminishes again. Now the results obtained for oceanic currents suggest that in mid-ocean, so far down as they extend, the turbulence is comparatively constant. Their vertical extent is of order (//@)?, according to the theory already developed, which in this case is about 1-2 metres. For the greater turbulence in mid-ocean the vertical extent of the currents is, of course, greater. The variation in turbulence found does not therefore invalidate the assumption of its constancy in the theory of ocean currents ; but if the results just obtained are substantially correct they require an explanation. It is possible that near the Sdgnefjord the surface water is largely river water, and therefore lighter than that below. The greater density below would have a stabilising effect, tending to reduce turbulence in the surface layers. The Norwegian Oyster- basins, deseribed by Murray and Hyjort, afford an extreme example of such an effect; the ne temperature above hinders the heat absorbed in ihe salt water below from being conveyed away, with the result that the deep water there is several degrees hotter than elsewhere. If turbulence can be largely attributed to wave-motion, the way it dies down at oreat depths is readily accounted Gore for wave-motion diminishes as the depth increases, and able depth equal to the wave-length it has practically ceased. This suggestion is confirmed “by the fact that the depth where turbulence shows marked diminution is actually of the same order of magnitude as the length of Atlantic waves. Ill. Friction on the Ocean Bottom: The ocean currents driven by the trade winds are seen from the discussion in the first section of this paper to have a comparatively small vertical range. Hiven if the eddy- viscosity is as high as 1000 cm.?/sec. and the latitude as low as 10°, the depth through which the current is considerable, = being ee order (k/@)2, 1s only about 4000 centimetres, an insignificant fraction of the depth of the ocean. If the theory that the main ocean currents are caused by the skin friction of the wind on the surface is correct, they must therefore be practically confined to the upper layers.. The Turbulence in the Ocean. 585 atmosphere may, however, affect the movement of the ocean in another way. If the atmospheric pressure at a place A is greater than that at B, it will create a horizontal force tending to move the water from A’towards B. When such a pressure-distribution is steady, the horizontal force steers the water so as to keep it moving along the isobars. Thus a geostrophic distribution of velocity is maintained in the water, and extends to the bottom. This velocity is not, however, great ; for it is maintained by the same pressure differences as maintain the winds, and therefore, by the geostrophic condition 2wpv=dp/or, the velocities in the air and in the water are inversely pro- portional to the densities. Thus the current velocity is about gj, of the wind velocity, and is of the order of lem./sec. ‘These currents are therefore small. Temperature differences, again, produce little effect at great depths ; for the differences in, density produced by them are small. Salinity currents are probably more important, though they are not usually noticeable except in restricted areas, Another type of current that extends to the bottom of the ocean is the periodic tidal current, and the friction caused by it can be estimated. The two components of horizontal velocity u and v, except near the bottom, satisfy differential equations of the form dv On = ———e ) <= _—. — \ dt + ZOU 1 9y (¢ Cy where € is the elevation of the sea-level above its mean position and € the height of the equilibrium tide. Tor motions periodic in 12 hours, w and v must therefore be of order gf/@a, where a is the eats of the earth. The height of the lunar equilibrium tide is n?a?m/Mg, where m is the mass of the moon, » its mean motion, and M the mass of the earth. Thus in all the amplitude of € is 37 cm., and u=QO(1 cm./sec. ). The rate of dissipation of energy by skin friction is 0-002 u® per sq. cm., or 0-002 erg per sq. cm. per sec. Over the whole ocean this gives a dissipation of the order of 101° ergs per second. Phil. Mag. S. 6. Vola gano., 233, May 1920, 2Q 586 Dr. A. O. Rankine and Dr. L. Silberstein on This formula for the dissipation will, however, be correct only if the Osborne Reynolds epienion is calnetled. jer re denote the vertical dimensions of the motion and vy the viscosity, this criterion is that uh/v shall be greater than 1000. For semidiurnal motions in a viscous medium, lh? =v], so that the criterion becomes u/(v@)? or 800. Thus it is not satisfied in general, though it probably holds in many places. Where it fails, the dissipation is due to ordinary viscosity and not to turbulence, and is readily estimated to be of order 1°2x 107% erg per sq. em. per second or 0°6 x 10!° ergs per second in all. Thus friction in mid-ocean in any case is of the order of 10° ergs per second; whereas G. I. Taylor finds that the actual dissipation in the Irish Sea is about 3 x 10" ergs per second, and I find that the dissipation required to account for the moon’s secular acceleration is about 1'4 x 10” ergs per second. Thus the influence of tides in the open ocean on the moon is insignificant in comparison with that of the partly enclosed seas, and has no observable astronomical effect. LVI. Propagation of Light ina Gravitational Field. By A. O. Ranxine, D.Sc., and L. SILBERSTEIN, Ph.D. S far as we know it has always been taken for granted, witheut any experimental support, that the ‘velocity of light ina gravitational field, such as that of the earth, is independent of the orientation of the light oscillations. It seemed to us desirable to investigate, with as much precision as possible, whether, and how far, this is true; whether, for instance, the velocity of provagation c, of horizontal oscil- lations is, or is not, appreciably different: from c¢,, the velocity of propagation of vertical oscillations. Such an enterprise was the more attractive as a slab of space, so to speak, of considerable thickness is always readily available thus promising the possibility of reaching a very high degree of exactness. Before passing on to the description of our experiments and their simple theory, it may be well to notice that according to Hinstein’s generalized theory of relativity and gravitation, we should have, rigorously, Cy=Cy, more generally a velocity of propagation rigorously independent of the direction of the light vector relatively to the lines of force in any gravitational field. In fact, by Hinstein’s theory, * Communicated by the Authors. Propagation of Light in a Gravitational Field. 087 Maxwell’s electromagnetic equations im vaeno retain, in any ‘grayitational field, their familiar form mM omy +curl E=0, div 2#=0, c.0t ae (M) ~ oF —curl M=0, div ©=p. In absence of gravitation G= E, and S$=M. In any gravitational field G and Qf are certain linear vector functions of E and of M respectively. The relations between the two pairs of vectors are given, in the usual notation, by the six relativistic equations Fe OE Wegiiiint es, bk he a a) due to Kottler, taken over by Einstein. Now, in the system ain which the determinant | 1Gix| Ivequale—— le ando,,—05,—93,— 0; and which can be employed without loss to generality, we have, by (1), and writing ee M,= = F* = MY (y22¥23— 23" 23 ) + Mo(/23°%31— Ya1'7¥s3) . + Ms(yorys2—Ye2%a1). + (2) and two similar expressions for the second and the third component of M, and —©,= PYjqau(yn ky, + 12H, + ¥131"s4) = Yi (Yun, + y 2H. +m3Ks), asic eame eam) and two similar equations by cyclic permutation. Thus, it o be the three-dimensional linear vector operator iy oy - Yor Yoo Y23 ey 30n 933 or the matrix which belongs to y.4 as submatrix of the contravariant tensor y~, we have simply a a ay UG oE. Next, the coefficients of Mi, &e. in (2) are the minors of & belonging to YWus Nie» Ns» respectively. Thus, inverting (2) and writing | @ for the determinant of @, we have — : ~ oM. | ( 2Q2 588 Dr. A. O. Rankine and Dr. L. Silberstein ox But since |gx| = 2k and, therefore, yy)o,=—l, we have at once P= moe e aM. Thus, the relation of IY to M is exactly the same as ees a (© to E, or, if we write, in the usual way, G=KE, M= the permittivity operator Kis identical with the eee operator, \ KS yin Ww, . c ° C 5 (4) both operators having the principal axes of ® and y,, times its principal values. Now, such being the case, the velocity of propagation y can easily be proved to be independent of the orientation of the light vector. In fact ifn be the wave- normal, we have, from equations (M), with p=0, ” KE=VMn : 2 uM = VaE. ot) Gy and since the operator yu is identical with K, Y KE+Vn(K-!VnE) =0, A" being the inverse operator of K. If Ay, ete. be the principal values of A’*, and 7;, etc. the components of n along them (or the direction cosines of n), the last equation becomes jie a oT7 ow > , | y K,K.K; = Tl ia Taam K ne, | E $+ n(n AE) => 0, LG whatever the direction of the vector E. But since by the first of (5), nKE=0, we have ultimately a wee Kyn? —- Knee + Kynst (6 ce Knee Dale re Thus, as was announced, the velocity of propagation of light is, in the most general gravitational field, a function of n, but entirely mdependent of the orientation of the light vector Thus, also, we see that, according to Kinstein’s theory, no difference atoren 1s, to be ex pected between our Shere said two velocities c, and ¢,. As to other theories, none of them contemplates any connexion at all between light pro- pagation and gravity. But, apart from any theory, it * Of these three values K,, K., K; two are, as a matter of fact, equal in any gravitational field. But we can prove our statement even without profiting from this axial symmetry. Propagation of Light in a Gravitational Field. 389 seemed of great interest to investigate this matter experl- mentally. Consider two stations distant L apart on the same level. Let light rectilinearly polarized at 45° to the vertical be projected from the sending station. Then the phase dif- ference e of the vertical and horizontal components at the observing station will be given by ot ‘T=L(_—-). Ch Cy where T is the period of the light in question ; or if N=cT be the normal wave-length and c¢, and c, differ but little from e, li .¢,—c, * Seer Ee A as tik ih at CE eh ae (7) At the receiving station, therefore, we shall have elliptically polarized light, the principal axes of the ellipse making an angle of 45° with the vertical, and the coefficient of ellipticity, or ratio of minor to major axis being given by as ae 1— cos 27e (8) a 5) e e e . s y a 1+ cos 27 If 27re is a small] angle this can be written . (2 T7e \2 n=(—5-) = 76. Cy— Ch KN Bye m= ee c 7, This formula enables us to evaluate the difference of the two velocities, if any, by measuring the ellipticity «, the wave-length X and the distance L being known. Owing to the ease with which L can be made very great in comparison with 2, an exceptionally high degree of precision can be attained even if it be difficult to measure small ellipticities «. Let us now pass to the description of the experiments. For measuring optical ellipticity it is natural to turn first to the Babinet compensator, although it did not appear likely that the produced effect, if any, would be sufficiently great to be detected by means of this instrument. Preliminary experi- ments quickly established this view. Light, originally plane polarized at 45° to the vertical by means of a nicol, was examined at various distances up to 40 metres with a Babinet compensator, and no shift of the bands was observable. It was estimated that a shift of, perhaps, one-twentieth of a band could be detected with certainty, and these observations Thus by (7) 590 Dr. A. O. Rankine and Dr. L. Silberstein on therefore proved that no ellipticity amounting to about «=0'05 comes into existence in a range of L=40 Gomes In order to carry the measurements further, it was necessary to adopt a more sensitive method of devotion. namely, one based upon observations of change of light intensity. Two modes of procedure were ene leredl Hither one could! polarize the light at 45° to the vertical, and examine the extent of extinction obtainable by an analysing nicol, first, close up to the polarizer, and then at increasing distances ;, or, fixing the polarizer and analyser at a definite considerable distance apart, one could observe any possible effect on the: completeness of extinction caused by altering the orientation of original polarization from 0°* to 45°. For several reasons. the second alternative seemed preferable. Of course, if it were possible to construct a rigid framework upon which polarizer and analyser could be separated to a great distance trom one another without altering at all their relative orien-. =) = e tution, the first alternative might be used. Such an instal- lation would, however, be extremely costly, and, even so, it is unlikely that it would give results superior to those described in what follows. The arrangement finally adopted was this. The source of light, a 100 ‘candle- -power pointolite lamp, was placed at the: focus of a lens of 50 em. focal length which collimated the light. The beam then proceeded through a large nicol which rN Te igeeae served as polarizer. The edges of this nicol were about two nches long, but only the central portion of it was used, a stop of 2 2 inch aperture being interposed between the nicol and the collimating lens. A ‘small nicol, placed in the beam of light 40 metres away, was used as analyser at the receiving station. Its scale was graduated in degrees, and read to 0° 1. For the sake of greater accuracy, the levelling screw on the stand of the instrument was calibrated so as to enable us to measure much smaller rotations—down to 0°:003. It soon became evident that in order to get satisfactory extinction when the nicols were hominally crossed, lenses. and all glass must be dispensed with between the two nicols. Even chen there survived a small amount of light, which was. oO primarily due to scattering by dust particles at the interface. of the two halves of the polarizer. I was, however, very feeble. At 40 metres distance, viewed through the analyser, it needed careful watching, oh often disappeared owing to. the wandering of the eye Experiments were flow carried out after cessation of daylight in a long corridor in the * It will be obvious that even if cy ~ ¢, there will be no ellipticity produced provided the light vector at the source is either horizontal or: vertical, and a maximum effect if it is inclined at 45°. Propagation of Light in a Gravitational Field. — 59% Imperial College, most suitable for the purpose, for various orientations of the polarizer, the eye being focussed, through the analyser, on the light emerging from the polarizer. From this procedure there accrued two advantages. as compared with the usual method of observing the illumination, by means of an eyepiece, in a plane near the analyser. It was possible to separate the actual light under observation from stray unpolarized light not coming from the polarizer. Also, this stray light seen, as it was, close to, but not overlapping the polarized beam, and suffering no change of brightness. when the analyser was rotated, served as : standard oF com- parison for testing the completeness of extinction in relation. to the orientation of the light vector. No difference of extinction at all could be detected. A small speck of stray unpolarized light (referred to above) was fortuitously in existence by the side of the polarizer. For all orientations. of the polarizer it was possible to extinguish the polarized beam to such av extent that it was definitely less luminous than the stray speck. mentioned. Measurements were now made to establish an upper limit for the ellipticity of the light.. It was found that in order to bring about a revival of light equal in luminosity to a standard « ape a rotation of the analyser through 0°:012, about 0:0002 radian was necessary. Thus the Aes. produced by gravitation was ory distinetly smaller than 2x 1074. Substituting this result, 7.¢e.«<2.1074* with L=4000 cm. and A=5 x 107 cm., say, In equation (9), we have <5 10. é ’ In order to realize the meaning of this result, imagine a light wave travelling under similar conditions for one second.. Then if one component lags at all behind the other the difference would be smaller than three hundredths of a millimetre in the three hundred thousand kilometres covered in that time. There are, perhaps, few pairs of physical magnitudes whose equality could be stated with equally high precision. We take the opportunity of expressing our grateful thanks. to Messrs. Adam Hilger for the loan of the Babinet com- pensator and the analvsing nicol used in this investigation. London, 21 January, 1920. * It is perhaps worthy of notice that this corresponded to a relative: change of brightness 4x 10—°, which could be detected. by the eye f 592 4 LVI. Critical Velocities for Hlectrons in Helium. By Frank Horton, Professor of Physics in the University of London, and ANN CATHERINE Davies, Royal Holloway College, Linglefield (rreen”*. ie a recent paper ft the authors have shown that the first critical electron velocity for the normal helium atom is 20°4 volts, and that an electron with this velocity is able to produce radiation from the atom on collision with it. It was also shown that when the velocity of the impacting electron is Increased to 25°6 volts, it is able to produce ionization by collision. This value of the minimum ionization velocity is lower than that predicted for the normal helium atom by Bohr’s theory, according to which the electron velocity se for the removal of one electron from a helium atom is 2°13 times the electron velocity required to ionize an atom of hydrogen, while the electron velocity necessary to remove both electrons from a helinm atom ata single collision 1s 6°13 times the hydrogen ionization velocity. ‘Taking the values e=4°774 x 10°" E.8-U., e/m=1-767 x 107 H.M.U., and h=6547 x 10-7" om. em.7/sec., Bohr’s theory gives 13°54 volts as the hydrogen ionization velocity. The two critical ionization velocities for helium, calculated from Bohr’s theory, are therefore 28°84 volts ‘and 83°00 volts respectively, the difference between these two velocities, viz. 54°16 volts, being the electron velocity which, according to the same theory, would be required for the removal of the second electron from a helium atom which had already been ionized. After the conclusion of the experiments described in our earlier paper, we investigated the effects of collisions between helium atoms and electrons having considerably greater velocities, with a view to obtaining some turther evidence of the validity of Bohr’s assumptions, by determining accurately the velocity at which both electrons are removed from the ‘normal heliumatom ata single collision. This investigation presented considerable difficulty on account of the fact that, at the higher critical velocities, increases of radiation or of ionization have to be detected in the presence of the effects which occur at lower velocities. Our experiments showed that to obtain a degree of accuracy at all comparable with that attained in the earlier research, it would be necessary to use an electron stream in which all the electrons have the © same velocity, so that the breaks in the current curves, * Communicated by the Authors. ft Proc. Roy. Soc. A. vol. xev. p. 408 (1919). Critical Velocities for Electrons in Helium. 993 which indicate critical points, may be well marked. We hope to make further experiments using an apparatus in which the electrons in the primary stream are sorted out by means of a magnetic field, so that all those which enter the jonization chamber have the same velocity. The results -contained in the present paper, which can only be considered as approximate, are given for comparison w ith the recently published work of Franck and Knipping *. These authors have obtained values in agreement with those which we ‘found for the minimum electron velocities for the production -of radiation and of ionization in helium, and, in addition, have produced evidence that a third type of collision, ‘resulting in an increase of ionization, occurs when the velocity of the impacting electron is raised to 79°5 volts. In determining this latter point the difficulties we have mentioned, of obtaining really marked bends in the current curves with high electron velocities , were also experienced. The arrangements used in the present experiments have ‘been described in detail in earlier papers, but the method -ean be understood by reference to the diagrammatic repre- sentation in fig. J. Electrons from the glowing filament F were accelerated towards the surrounding electrode E by a -difference of potential, V1, which was constant during any particular series of observations. Those electrons which ‘passed through the small hole in the centre of the top of EK, were further accelerated by the potential difference i avhich was varied during the experiments. The potential * J. Franck and P. Knipping, Phys. Zeits. xx. p. 481 (1919). 594 - Prof. F. Horton and Miss A. C. Davies on difference V,, between the gauzes B and C, was in the reverse direction to V,; and V,, and was sufficiently great to. eee anv of the electrons from reaching B. The usual path of the electron stream is thus indicated ie the arrow in the diagram. A represents the collecting electrode which: was connected to a sensitive electrometer for measuring the currents due to ionization or radiation. Its distance from the gauze B could be varied over a range of several centi- metres. In most of the experiments the apparatus employed was that which we had used in our previous investigation with helium, but the gauze C had been added, the distance between © and D being about lem. The later experiments. were made with a slightly modified form of apparatus in which the distance between C and D was about 3 em., and this apparatus was afterwards used for experiments with argon; a complete description of it is given in the Pro-- ceedings of the Royal Society, A. vol. xcviu. p. 1. The helium used was carefully purified before being stored, and during the observations was slowly circulating through the: experimental tube, entering through a U-tube containing carbon and immersed in liquid air. The pressure of the gas. during the experiments was varied according to the part emer point ‘which was under inv estigation. In considering the points at which breaks might be. expected in the current-potential difference curves for high values of the accelerating potential difference, we started from the facts already es stablished, that, radiation! is produced from helium by 1 impacts of e leoinone with 20°4 volts velocity, and ionization by impacts of electrons with 25:6 volts velocity. Under suitable conditions of pressure it should be possible to obtain curves giving indications of the production of radiation and of ionization by the second, third, ete. inelastic collisions of eleetrons with helium atoms. ‘The particular points on the curve at which indications are to be expected will depend upon whether the distance over which the accelerating potential difference is applied is large or small compared with the mean free path of an electron in the gas. For instance, in order that a radiation curve should show increases of radiation at multiples of the resonance velocity (20 4 volts), the pressure must be so high that most of the electrons in the stream from the filament make several collisions in traversing the distance over which the accelerating potential difference is applied, so that for: the range of velocities used the potential fall along a free path is “not likely to exceed the resonance voltage. Under these conditions, very little, if any, ionization will take- Critical Velocities for, Electrons in Helium. 595» place at 25°6 volts, because so few of the electrons can acquire the necessary energy *. It would thus seem that the conditions necessary for the increases of radiation to. occur to any large extent at multiples of 20-4 volts, and the conditions necessary for the indication of ionization at 25°6 volts, are mutually exclusive. If the pressure is so adjusted that the distance over which the accelerating voltage is applied is about the same as, or is smaller than, the mean free path of an electron in the gas, : the points at which bends, other than those at 20°4 volts and 25°6 volts, occur in the current-potential difference curves. will depend upon the length of path traversed by the electrons before their energy is reduced by the retarding field to a value below that at which they can make inelastic collisions.. If this distance is Jarge compared with the mean free path, so that the electrons make several collisions with gas atoms, the current-potential difference curve may be expected to in- dicate an increased produetion of radiation at (25°6 + 20°4= ), 46°0 volts, and an increased production of ionization at (25°6 + 25°6 =) 51:2 volts. Increases of radiation and ioniza- tion should occur again at 71°6 volts and at 76°8 volts, respectively, and so on. If, however. this distance is about equal to the mean free path or is less than this, the current- potential difference curves would not be expected to show breaks at the velocities just mentioned, but would be ex- pected to do so when collisions between electrons and gas: atoms occurred which resulted in the transference, at a single impact, of a larger amount of energy than that corre-. sponding to 25°6 volts, such, for instance, as would result in both electrons being removed simultaneously from the helium atom, at which point the curve should indicate an increase: of ionization. Such collisions would not be prevented from occurring, even when the path traversed by the electrons. after acquiring the energy corresponding to the applied accelerating potential difference, and before they have had their velocity reduced below 20:4 volts, is several times the: mean free path (as in the case when bends at 46:0, 51-2, 71°6, and 76°8 volts would be expected) ; but it is probable: that under these conditions, the effect would be to produce a flattening of the curve, instead of an increase of slope, since. * Although, under the conditions mentioned, this statement is probably true of the greater proportion of the electrons in the stream, certain experiments which are referred to in Proc. Roy. Soe A. vol. xevii. p. 1,. seem to show that even when the distance traversed by an electron with a velocity above the critical value is several times the mean free path, some electrons can traverse this distance without suffering an inelastic collision. 596 Prof. F. Horton and Miss A. C. Davies on the double ionization of the helium atom at a single collision ‘which, according to Bohr’s theory, should occur at 83 volts, would result in the production of less Positive electricity than the expenditure of a smaller amount of energy in removing single electrons from three helium atoms, with the consequent loss of 25°6 volts velocity at each of the three ‘successive collisions. In most of the experiments described in this paper, the gauzes D and C were at the same potential, so that the electrons, after being accelerated by the fields V, and Vz, traversed the distance between D and C without suffering any reduction of velocity except that resulting from their eollisions with helium atoms. The electron stream was prevented from spreading laterally in this space by means of a magnetic field, parallel to the axis of the tube, applied by passing a current of about 2 amperes through a coil of many turns of wire wrapped round the experimental tube. Since there is no difference of potential between the gauzes D and C, the detection of ionization depends upon the diffusion of the positive ions out of this space. Those which pass through the gauze C will be accelerated by the field Vy (which retards the electron stream) and driven sowed the gauze B. Some of these will pass through B and reach the collecting electrode A, provided that the field between B and A does not oppose them, or is insufficient to turn them back. Experiments showed that this method gave satisfactory results, and that, with the gas-pressure suitably adjusted, ionization could inners be detected when the accelerating potential difference was increased beyond the ionization value. Some of the curves obtained, which illustrate the differ- ences in the effects occurring at different gas-pressures, are given in figs. 2, 3, and 4 —_In these curves (and in the others given in this paper) the electron velocities, expressed in equivalent volts, are those of the swiftest electrons present in the stream from the filament. The method of obtaining these velocities for given values of the applied accelerating potential difference is explained in our earlier paper”. The curve in fig. 2 was taken under such conditions that any positive ions produced would be prevented from reaching the collecting electrode, so that the current due to macleom was examined alone. The gas pressure during this experi- ment was.about 1°5 mm., which should cause the electrons to make several collisions with helium atoms while traversing the distance over which the accelerating potential difference was applied. It will be observed that the curve indicates an * Loe. cit. Current Critical Velocities for Electrons in Helium. SOF increased production of radiation at the second and third multiples of the resonance potential difference. Bro. 2. ee Sl ha Sees te Sees HP sreetanertae Fig. 3 Current aaeee . Be ee f eee SEn> (aes 60) | §: 70 Flectron 2a (volts) Fig. 3 also represents a series of observations of the variation with the velocity of the electrons, of the current | 598 Prof, F, Horton and Miss A. C. Davies on due to radiation alone. These observations were taken at -a pressure of 0°49 mm., so that most of the electrons would not make more than one collision while traversing the distance -over which the accelerating potential difference was applied. This curve indicates an increased production of radiation, not at the second multiple of the resonance velocity but at «(25:6 + 20°4=) 46-0 volts. | In the curve of fig. 4 the pressure is still lower, being only 0°17 mm., and at this pressure it is probable that a 110, 4 5 * S 5 ° y ore & R 4 S S) o S oS © pp 0; DUPgee @) == 0.020) 2 10 20 30 Electron velocity (vo/ts) ‘considerable proportion of the electrons pass through the “space between the gauzes D and C without making more than one collision with a helium atom, and it is certain that very few, if any, will be able to make three collisions while traversing this distance. Tor this series of observations the fields were so arranged that radiation and ionization would ‘both cause the electrometer to indicate a positive current. ‘The curve shows a bend at 20°5 volts, another at 25-6 volts, and another one in the neighbourhood of 80 volts. The -eritical velocity indicated at 80 volts is presumably that at which both electrons are removed from the atom at a single collision. Other series of observations agreed in fixing the critical velocity at which this occurs at about 80 volts, but the difficulty of obtaining the point accurately has already been mentioned. This value should be the sum of the ‘velocities required to remove the two electrons separately from the helium atom. We have shown experimentally that an electron velocity of 25-6 volts is required to remove the first electron, so that the difference between the two experimentally determined values, 7. e. about 54 volts, is the welocity required to remove the second electron. This Current 4 Critical Velocities for Electrons in Helium. 599 value agrees with the value deduced from Bohr’s theory. In the course of our investigation we obtained direct experi- mental evidence of the existence of a critical velocity at about this point. It has been mentioned that in most of our experiments there was no difference of potential between the gauzes D and ©, in the space between which most of the ionization occurred. In view of this fact, when the ionization velocity has been exceeded, positive ions will be distributed throughout the path of the primary electrons, and it is possible that at a suitable pressure and with a sufficiently intense electron stream, collisions take place between ionized helium atoms and electrons with considerable velocities. If a sufficient number of such collisions occurred, the current-potential difference curve would be expected to show a rise when further ionization resulted from these encounters. An example of a curve in J Current + ioe 20 30 40 Electron vélocity (volts) which an increase of current was obtained, which could noi be attributed to the production of radiation or ionization by the successive collisions of an electron with different helium atoms, is given in fig. 5. This curve was obtained 600 Prof. F. Horton and Miss A. C. Davies on at a pressure of 0°35 mm., and shows an increase of positive current at about 55 volts. It is therefore possible that this. bend does indicate the production of ionization as a result of encounters between ionized helium atoms and electrons. having this velocity, and that it thus affords direct experi- mental evidence of the correctness of Bohr’s assumptions as applied to the helium atom Which has lost one electron. If collisions occur hetween ionized atoms and electrons, it follows that the number of collisions between electrons and normal helium atoms will be reduced by the recombination which probably results when the velocity of the impacting electron is lower than some critical value. On this account, with the intense electron stream which must be used in. order that the electron collisions with ionized atoms may be sufficiently numerous for the removal of the second electrons from these ionized atoms to cause arise in the current curve, it is possible that there will be such a small proportion of the electrons making second or third inelastic collisions with normal helium atoms that the curve will show no bends at the electron velocities at which these would normally occur. Thus it was found that when an intense electron stream was used (as in obtaining the curve of fig. 5), the current curve showed a rise at about 55 volts but no bends at 46°0 volts or at 51°2 volts, whereas in curves taken at a similar pressure when a smaller electron current was employed, the former: point was not marked but bends at velocities corresponding to multiple collisions were obtained. The absence of the bends at 46°0 volts and at 51°2 volts under the conditions of experiment of fig. 5 may thus be an indication that a large number of collisions with ionized helinm atoms is occurring, a condition which is essential to the interpretation we havé given of the rise at 55 volts. If this interpretation is eerece it might be expected that there would be some: indication of an increase in the measured positive current at an electron velocity, lower than 55 volts, which would correspond to the production of radiation from the ionized helium atom by its collision with an electron. On Bohr’s. theory such radiation would require for its production an electron velocity of # of 54°16 volts = 40°62 volts. The radiation borrespondine to this potential difference would be of such high frequency as to be able to ionize some of the normal helium atoms present, and it is therefore unce1tain whether its production would show in the current-potential difference curve as an increase of radiation or as an increase of ionization, in a case where the arrangement of the electric con) fields made it possible to discriminate between these two or Critical Velocities for Electrons in Helium. 601 effects. The electron velocity at which the bend in the curve is to be expected (40:6 volts) is almost identical with the point at which, under suitable pressure conditions, an increase of radiation should be produced by the second collisions of electrons with normal helium atoms, but the two effects can be distinguished, since the conditions which favour the occurrence of ionization, and which therefore favour the production of the 40°62 volt-radiation, are those under which increases of radiation at multiples of the resonance velocity would not be likely to occur to any appreciable extent. In considering the possibility of detecting the production of a new type of radiation or of fresh ionization from the collisions of electrons with helium atoms which have already lost one electron, the effects of recombination must be taken into account. It is conceivable, for instance, that recom- bination takes place at all encounters between positively charged helium atoms and electrons with velocities below that necessary for the production of radiation from the charged atom—40°62 volts on Bohr’s theory. This being the case, an increased positive current should be measured by the electrometer at 40°62 volts on account of the reduction at this point in the amount of recombination occurring. Whether in this case an increased positive eurrent would also be obtained when the electron velocity is raised to that necessary to remove the second electron from the ionized atom is uncertain, and would depend upon whether the ionization then produced were more than that which the 40°62 volt radiation was producing from the normal helium atoms. Again, it is possible that an electron which gives rise to 40°62 volt radiation itself recombines with the positively charged helium atom with which it collides. In this case an increased positive current should be obtained when the critical velocity for ionization of the positively charged helium atom is reached, since the encounters would then result in the production of doubly charged helium atoms instead of neutral ones. As the precise conditions under which recom- bination ‘of a colliding electron and a positively charged helium atom on which it impinges occurs are unknown, it is impossible to predict how the existence of critical velocities at 40°62 volts and 54°16 volts for this atom should be indicated. The curve in fig. 5 shows a bend which we have attributed to the further ionization of the positively charged helium atom, but no bend at all in the neighbourhood of 40 volts. By altering the experimental arrangements we Phil. Mag. 8. 6. Vol, 39. Ne. 233. May 1920, 2R 602 Critical Velocities for Electrons in Helium. were able to make a bend show up at this latter point under conditions when it could not be due to a second radiating collision at twice the resonance yelocity. A series of oben vations in which this result was obtained are plotted i in the curve of fig. 6. In taking these observations a potential +t oe aus coo CP) leet ee 30 60 7Q Electron velocity (volts) Leocal difference of 7 volts was maintained between the gauzes D and C (which in most of the earlier experiments had been at the same potential) in such a direction as to drive positive ions towards the collecting electrode A, and the potential difference V.; was in the direction for radiation, as well as ionization, to give a positive charge to the electrometer. The gas-pressure (0°170 mm.) was too low for it to be likely that the bend at 41 volts is due to multiple collisions, and it may therefore be taken to indicate a new critical velocity at this point. An upward bend at about 41 volts was obtained in several curves taken under conditions similar to those stated. In some experiments at still lower pressures (about 0 01 mm.) taken with the final field V; reversed, a bend in the opposite direction was obtained, showing that radiation was produced at this point, and that at this low pressure its photoelectric effect on the cylinder was greater than the ionization it produced in the gas. The curves in figs. 5 and 6 may thus be interpreted as supporting the view “that at 41 volts and at 55 volts electron collisions with positively charged helium atoms occur, which result in an Increased positive current being measured by Method of solving Problems in Conduction of Heat. 603 the electrometer, as would be expected if radiation and further ionization, respectively, were produced from the ionized atoms at these velocities. Although these results are In agreement with the predictions of Bohr’s theory, it is clear that the method of experimenting affords no absolute proof that the observed increase of current does result from such collisions. | In connexion with the increase of current observed. at 80 volts, and interpreted as being due to the removal of both electrons from a helium atom ata single electron collision, it might be suggested here also that for some lower electron velocity a single collision should result in the removal of one electron and the displacement of the other to an orbit of greater radius, thus giving both ionization and radiation. Some experiments by Ran* seem to show that collisions producing an effect of this kind do oecur under certain conditions. On the basis of Bohr’s theory, the electron velocity at which such an effect would be expected is 25:6 volts + 2 of 54:16 volts = 66°22 volts. We were unable to obtain any evidence of this effect in our experiments. The results described in this paper agree with those of Franck and Knipping in showing that the double ionization of the helium atom results from collisions between helium atoms and electrons having about 80 volts velocity, but differ from theirs in indicating bends in the current curves at 41 volts and 55 volts which appear to be due to the pro- duction of radiation and of further ionization, respectively, from the helium atoms which have already lost one electron. LVIIL. Bromwich’s Method of solving Problems in the Con- duction of Heat. By Prof. H. 8. Carstaw, Se.D.T if + his paper—‘ Examples of Operational Methods in Matheinatical Physics,’ Philosophical Magazine, No. 220, April 1919—Bromwich advocates the use of -so- called operational methods, following Heaviside, in electrical _and other physical problems, and illustrates his method by the solution of various questiens, including some in the Conduction of Heat. In an earlier paper (Proc. London Math. Soc. (Ser-- 2)2-xv, ip 401, 1917) he discussed Heaviside’s work, and confirmed it by the aid of contour integrals. The object of the present paper is to illustrate by some * H. Rau, Wiirzburg Phys. Med. ges. Ber. Feb. 1914. + Communicated by the Author. 2R2 we CO e) 604 ' Prof. H.S. Carslaw on Bionniee Method of problems in linear flow the method which Bromwich’s work has led me to adopt in the discussion of a large class of pro-_ blems in the Conduction of Heat. It seems to me easier to build up the required solution by integrating a suitable solu- tion over a certain standard path in the plane of the complex variable. ‘The proper particular solution will be easily ob- tained after a little practice in the method. The problems in §§ 2, 3, and 4 have been chosen for their simplicity. They are, of course, elementary and solvable by the ordinary methods. The problem in §5 is here solved for the first time. The corresponding cases for the semi- infinite rod, and the sphere composed of two different materials, can be treated in the same way. These two questions are referred to by Heaviside* in his discussion of the Age of the Earth, following upon Perry’s criticism + of Kelvin’s classical treatment of this subject. 2. ited of length a. The ends #=0 and ve—o feo temperatures zero and wy respectively. The initial temperature zero. Let the temperature at the point wz at the time ¢ be v. Then the equations are as follows :— 2 aE ae O.< 2 2 ee) Fig. 1. i een at i The path (P) in the a-plane. Consider the value of v given by the integral Simian ex ke: =o | —* Ce Ci ean CO) over the path (P) of fig. 1 in the a-plane. In this path the * H ‘ oan eaviside, ‘ Klectromagnetic Theory, Vol. II, Chapter y. Cf. §§ 229, ‘li ‘ Nature,’ li. p, 224 et seg. (1895). +00 solving Problems in the Conduction of Heat. 605 argument of « on the right must lie between 0 and }7, and on the left between 2a and 7. We take the path (P) of fig. 1 as the standard path in the problems which follow. Since every element of the integral satisfies (1), the value of v in (5) satisfies (1). Also when z=0, we have v=0. Vo %e— Kart And when #v=a, v= — | itt dx, over the path (P). (6) a Since the integrand is an odd function of «, if we form the path (Q) by taking the image of the path (P) in the real Fig. 2. The path (Q) in the #-plane. axis of «, joining the ends of the path (P) and its image by ares of a circle, centre at the origin, whose radius tends to infinity, we have from (6) =a de, over the path (Q)), since the integrals over the circular arcs vanish in the limit. Therefore when =a, we have v=vw. Finally, when t=0, we have = 4 ce ele ae over the path (P). im \sin aa a Now the integrand has no infinities above the path (P), and if we complete the circuit by the arc of a circle, dotted in fig. 3, whose centre lies at the origin and radius tends to infinity, the integral over the complete path of fig. 3 is zero. 606. Prof. H.S. Carslaw on Bromwich’s Method of But the integral over the dotted path vanishes in the limit. Therefore pelt) when t=0. It follows that the value of 7 given by (5) satisfies all the conditions of our preblem. Fig. 5. er ™ — oo 0 | ae) The solution obtained in (5) as a contour integral is trans- formed into an infinite series by Cauchy’s Theorem, using the path (Q) of fig. 2. For we have Vo “ sin ax e — Kart da, over the path (Q), ~ Die sin aa a Ik dy y) n2 72 = Uo | - a Fuga 2otal ) sin — we7* 4? |. aa qo a 3. The same rod. The end «=0 kept at zero; radiation into a medium at vr at the end w=a. The ‘nna temperature the summation being taken over the positive roots of (9). 610 Method of solving Problems in Conduction of Heat. 6. It remains to discuss the roots of the equation F(a)=o cos aa sin wa(b—a) 4+ sin wa cos wa(b—a)=0. From the graphs of | y=o cotaa and...-_.. - y=—cotpa(b—a) it is clear that there are infinite number of real roots, and the position of the same can be determined. Also F(a) is an odd function of a and the real roots may be denoted by 5) pea Th Ay, oes By examining F(a), it will be seen that these roots are not repeated. 3 Also it is clear that F(«) has no pure imaginary root. We have now to show that it has no roots of the form &+77. Consider the functions U,, U, defined as follows :-— Us=siner, O veel. glis 2V2 ) a and 3 (ep) ‘ UF Vida + (VU, am ULV, \de z= 0, j 0 )0 Therefore J ca Gates ‘ e (eB) | | U.Va ey UV ide | ra Un = ( CUyV,!— VU, das & ( (U,V2"— V,U,")da a9 e/ 4 ‘ a i b Ely Me ii eee [ U.¥,—V.0,| tered 1 le : kK, 22 SSP) i It follows from (3), (4), and (5), that ra b (a? — 8”) ] U,V ida+ : ( U,Vede | ==), (0) Thus F(#)=0 cannot have imaginary roots of the form Em. LIX. The Mass-Spectra of Chemical Elements. By ¥. W. Aston, .A., D.Sc., Clerk Maxwell Student of the Uni- versity of Cambridge *. [Plate XV.] HE following paper is an account of some results obtained by the analyses of gases by means of the Positive Ray Spectrograph or, as it may be more con- veniently termed, Mass-Spectrograph. The principle of the method by which a focussed spectrum is obtained depending solely on the ratio of mass to charge has already been de&cribed f, but for the sake of others experimenting in this field it is now proposed to give an account of the actual apparatus in some detail. * Communicated by the Author. 7 ’. W.A., Phil. Mag. xxxviii. Dec. 1919, p. 707. 612 Dr. F. W. Aston on the The Discharge Tube. Fig. 1 is a rough diagram of the present arrangement. The discharge-tube B is an ordinary X-ray bulb 20 cm. in diameter. The anode A is of aluminium wire 3 mm. Tog Zh qe Fig. 1. thick surrounded concentrically by an insulated aluminium tube 7 mm. wide to protect the glass walls, as in the Lodge valve. The aluminium cathode C, 2°5 ecm. wide, is concave, about 8 cm. radius of curvature, and is placed just in the neck ot the bulb—this shape and position having been adopted after a short preliminary research*, In order to protect the opposite end of the bulb, which would be immediately melted by the very concentrated beam of cathode rays, a silica bulb D about 12 mm. diameter is mounted as indicated. The use of silica as an anticathode was suggested by Prof. Lindemann, and has the great advantage of cutting down the Prog ae of undesirable X rays to a minimum. The discharge is maintained by means of a large induction- coil actuated by a mercury coal-gas break; about 100 to 150 watts are passed through the primary, and the bulb is arranged to take from 0°5 to 1 milliampere at potentials ranging from 20,000 to 50,000 volts. Owing to the par- ticular shape and position of the electrodes, especially those of the anode, the bulb acts perfectly as its own rectifier. The method of mounting the cathode will be readily seen from fig. 2, which shows part of the apparatus in greater detail. The neck of the bulb is ground off short and cemented with wax to the flat brass collar EK, which forms the mouth of an annular space between a wide outer tube EF and the inner tube carrying the cathode. OE. WoA., Proc. @amb. Phil. Soc. xix, p.ol7, Mass-Spectra of Chemical Elements. 613 The concentric position of the neck is assured by three small ears of brass not shown. The wax joint is kept cool by circulating water through the copper pipe shown in section at G. K; y ws VID TY The gas to be analysed is admitted from the customary fine leak into the annular space and so to the discharge by means of the side-tube attached to F shown in dotted section at (. Exhaustion is perlormed by a Gaede mercury-pump through a similar tube on the opposite side. The reason for this arrangement is that the space behind the cathode is the only part of the discharge bulb in . which the gas is not raised to an extremely high potential. | If the inlet or outlet is anywhere in front of the cathode, failing special guards, the discharge is certain to strike to the pump or the gas reservoir. Such special guards have been made in the past by means of dummy cathodes in the bore of the tubes, but, notwithstanding the fact that the gas can only reach the bulb by diffusion, the present arrangement is far more satisfactory and has the | additional advantage of enabling the bulb to be dismounted | by breaking one joint only. ° The centre of the cathode is pierced with a 3 mm. hole, the back of which is coned out to fit one of the standard slits $,*. The back of the cathode is turned a gas-tight fit in the brass tube 2 cm. diameter carrying it, the other end of which bears the brass plug H which is also coned and fitted with the second slit S,. The two slits, which are ‘05 mm. wide by 2 mm. long, can be accurately adjusted parallel by means of their diffraction patterns. The space between the slits, which are about 10 cm. apart, is kept exhausted to the highest degree by the charcoal tube I,. The Slit System. | * F. W.A., Phil. Mag. xxxviii. Dec, 1919, p. 714, 614 Dr. F. W. Aston on the. By this arrangement it will be seen that not only is loss of rays by collision and neutralization reduced to a minimum, but any serious leak of gas from the bulk to the camera is eliminated altogether. The Electric Field. The spreading of the heterogeneous ribbon of rays formed by the slits into an electric spectrum takes place between two parallel flat brass surfaces, J), Jo, 5 cm. long, held 2°38 mm. apart by glass distance-pieces, the whole system being wedged immovably in the brass containing-tube in the position shown. The lower surface is cut from a solid cylinder fitting the tube and connected to it and earth. The upper surface is a thick brass plate, which can be raised to the desired potential by means of a set of small storage-cells. In order to have the plates as near together as possible, they are sloped at 1 in 20—7.e., half the angle of slope of the mean ray of the part of the spectrum which is to be selected by the diaphragms. Of these there are two: one, K,, an oblong aperture in a clean brass plate, is fixed just in front of the second movable one, K,, which is mounted in the bore of a carefully ground stopcock L. The function of the first diaphragm is to prevent any possibility of charged rays striking the greasy surface of the plug of the stopeock when the latter is in any working position. The variable diaphragm is in effect two square apertures sliding past each other as the plug of the stopcock is turned, the fact that they are not in the same plane being irrelevant. When the stopcock is fully open as sketched in fio. 2, the angle of rays passing is a maximum, and may be stopped Coren foe any desired extent by rotation vot the plug, becoming zero before any greasy surface is exposed to the rays. Incidentally the stopcock serves another and very convenient use, which is to cut off the camera from the discharge-tube, so that the latter need not be filled with air each time the former is opened to change the plate. The Magnetic Freld. After leaving the diaphragms the rays pass between the pole-pieces M ‘of a large Du Bois magnet of 2500 turns. The faces of these are Scleoters 8 cm. diameter, and held 3 mm. apart by brass distance- -pieces. The cylindrical pole-pieces themselves are soldered into a brass tube O, which forms part of the camera N. When the latter is built into position, the pole-pieces are drawn by screwed Mass-Spectra of Chemical Elements. 615 bolts into the arms of the magnet, and so form a structure of great weight and rigidity and provide an admirable foundation for the whole apparatus. Current for the magnet is provided by a special set of large accumulators. The hydrogen lines are brought on to the plate at about 0:2 ampere, and an increase to 5 amperes, which gives practical saturation, only just brings the singly-charged mercury lines into view. The discharge is protected from the strong field of the magnet by the usual soft iron plates, not shown. The Camera. The main body of the camera N is made of stout brass tube 6:4 cm. diameter, shaped to fit on to the transverse tube O containing the pole-pieces. The construction of the plate- holder is indicated by the side view in fig. 1 and an end-on view in fig.3. The rays after being magnetically deflected pass between two vertical brass plates Z, Z about 3 mm. apart, and finally reach the photographic plate through a narrow slot 2 mm. wide, 11°8 cm. long, cut in the horizontal metal plate X, X. ‘The three brass plates forming a T-shaped girder are adjusted and locked in position by a set of three levelling-screws at each end ; the right-hand upper one is omitted in fig. 3. The plates Z, Z serve to protect the rays completely from any stray electric field, even that caused by the photographic plate itself becomine charged, until within a few millimetres of their point aE impact. The photographic plate W, which is a % em. strip cut 616 Dr. F. W. Aston on the lengthwise from a 5x4 plate, is supported at its ends on two narrow transverse rails which raise it just clear of the plate X X. Normally it lies to the right of the slot as indicated, and to make an exposure it is moved parallel to itself over the slot by means of a sort of double lazy- tongs carrying wire claws which bracket the ends of the plate as shown. This mechanism, which is not shown in detail, is operated by means of a torque rod V working through a ground glass joint. Y is a small willemite screen. 3 The adjustment of the plate-holder so that the sensitized surface should be at the best focal plane was done by taking a series of exposures of the bright hydrogen lines with different magnetic fields on a large plate placed in the empty camera at a smal! inclination to the vertical. On developing this, the actual track of the rays could be seen and the locus of points of maximum concentration deter-. mined. The final adjustment was made by trial and error and was exceedingly tedious, as air had to be admitted and a new plate inserted after each tentative small alteration of the levelling-screws. Haperimental procedure. The plate having been dried in a high vacuum over- night, the whole apparatus is exhausted as completely as possible by the pump with the stopcock L open. JI, and I, are then cut oft from the pump by stopecocks and immersed in liquid air for an hour or so. The electric field, which may range from 200 to 500 volts, is then applied and a small current passed through the magnet sufficient to bring the bright hydrogen molecule spot on to the . willemite screen Y, where it can be inspected through — the plate-glass back of the cap P. In the meantime the _ leak, pump, and coil have all been started to get the bulb into the desired state. As soon as this is obtained and has become steady, J, is earthed to prevent any rays reaching the camera when the plate is moved over the slot to its first position, which is judged by inspection through P with a non-actinic lamp. ‘The magnet current having been set to the particular value desired and the diaphragm adjusted, the coil is momentarily interrupted while J, is raised to the desired potential, after which the exposure starts. During this, preferably both at the beginning and the end, light from a lamp T is admitted for a few seconds down the tube R (fig. 1) the ends of - Mass-Spectra of Chemical Elements. 617 which are pierced with two tiny circular holes. The lower hole is very close to the plate, so that a circular dot or register spot is formed from which the measurements of the lines may be made. . The exposures may range from 20 seconds in the case of hydrogen lines to 30 minutes or more, 15 minutes’ being usually enough. As soon as it is complete the above procedure is repeated, and the plate moved into the second position. In this way as many as six spectra can be taken on one plate, after which L is shut, I, warmed up, and air admitted to the camera. The cap P, which ison a ground joint, can now be removed, and the exposed plate seized and taken out with a special pair of forceps. A fresh plate is now immediately put in, P replaced, and the camera again exhausted, in which state it is left till the next operation. Form of the Spectrum Lines. As has been shown (Phil. Mag. Dec. 1919, plate ix.), the shape of the spot formed when undeflected rays from such a slit system strike a photograph surface normally, is somewhat as indicated at a (fig. 4). When they strike the plate obliquely the image would be spread out in one direction, as in 0. This would be the actual form in the apparatus, if the Fie. 5 4, a b Cc xX x deflexions of the mean and extreme rays (i. e., the rays forming the centre and the tips) were identical. This is true of the magnetic field since each cuts the same number of lines of force ; but it is not so in the case of the electric deflexion. Since the form of the plates, and therefore roughly of the boundaries of the field, is rectangular, the extreme rays passing diagonally will be deflected more than the mean rays and the spot bent into the form shown at c. The convex side will be in the direction of the magnetic deflexion, as this is opposed to the deflexion causing the Phil. Mag. 8. 6. Vol. 39. No. 233. May 1920. 25 618 '. Dr, F. W: Aston on the bend. The image on the plate will therefore be the part of this figure falling on the narrow slot in X, X; and as the apparatus is not exactly symmetrical, its shape in the spectra is the figure lying between the lines X, X in fig. 4, ¢. Measurement of the Lines. The plates are measured against a standard Zeiss scale on a comparator designed by the late Dr. Keith Lucas and kindly lent by the Physiological Department. Some of the very faint lines, although easily visible to the unaided eye, were lost even eich the lowest power eyepleces: obtainable. To measure these, an eyepiece giving a magnification of about 24 was designed Lohan Odi Hartridge of King’s College. The general method of deducing mass from position has already been described (Phil. Mag. April 1920, p. 453). Owing to some geometrical cause (probably analogous to a caustic in optics), the more deflected edge of the line is always the brighter and sharper, and it is the distance of this from the register spot which is found to give the most reliable values. For the highest accuracy, owing to halation, one must only compare lines of approximately equal intensity. As this edge is unfortunately not at right angles to the spectrum, measurements can never be regar ded as absolute, unless extreme care is taken in the levelling of the spectrum on the comparator. So although theoretically it is sufficient to know the mass of one line to determine (with the correction curve) those of all others, in practice every effort is made to bracket any unknown line by reference lines, and only to trust comparative measure- ments when the lines are fairly close together. Under these conditions the accuracy claimed for the instrument is about one part in a thousand. Order of Results and Nomenclature. The various elements studied will be considered as far as possible in the order in which the experiments were performed. This order is of considerable importance, as in most cases it was impossible to eliminate any element used before the following one was introduced. Evacuation and washing have little effect, as the gases appear to get. embedded in the surface of the discharge-bulb and are only released very gradually by subsequent discharge. The problem of nomenclature became serious when the very complex nature of the heavy elements was apparent, Mass-Spectra of Chemical Llements 619 After several possible systems had been discussed it was decided, for the present, to adopt the rather clumsy but definite and elastic one of using the chemical symbol of the mixed element with an index corresponding to its mass : e.g., Ne”, Kr*4. This system is made reasonable by the fact that the masses of constituents of mixed elements have all so far proved whole numbers on the scale used. In cases of particles carrying more than one charge it will be convenient to borrow the nomenclature of optics and reter to the lines given by singly, doubly, and multiply charged particles respectiv ely as lines of the first, second, and higher orders. Thus the molecule of oxygen gives a first order line at 32, and its atom first and second order lines at l6 and 8. The empirical rule that molecules only give first order lines (J. J. Thomson, ‘Rays of Positive Electricity,’ p. 54) is very useful in helping to differentiate between elementary atoms and compound molecules of the same mass. Some very recent results give indications that in certain ex- ceptional cases it may break down, so that inferences made from it must not be taken as being absolutely conclusive. Oxyeen (At. Wt. 16-00) and Carson (At. Wt. 12:00). On a mass-spectrum all measurements are relative, and so any known element could be taken as a standard. Oxygen is naturally selected. Its molecule, singly-charged atom, and doubly-charged atom give reference lines at 32, 16, and 8 respectively. ‘The extremely exact integral relation between the atomic weights of oxygen and carbon is itself strong evidence that both are “pure” elements, and so far no evidence appears to have arisen to throw any doubt on this point. Direct comparison of the © line (12) and the CO line (28) with the above standards shows that the expected whole number relation and additive law hold to the limit of accuracy, 7. e. one part in a thousand; and this provides standards C++ (6), C (12), CO (28), and CO, (44). In a similar manner, hydrocarbons give the ©, and ©, groups already mentioned (Phil. Mag. April 1920, pp. 452, 453) ; so that a fairly complete scale of reference is immediately available. NeEon (At. Wt. 20°20). The results obtained with this gas have already been fully dealt with (Phil. Mag. April 1920, p. 449). It has been shown to consist of two isotopes of masses 20 and 22 re- spectively, with the faint Boeeuihity of a third of mass 21. Zz 2 620 Dr. F. W. Aston on the Spectrum I, on Pl. XV. shows the singly-charged lines of neon, to the left of the C. group. It is reproduced here to show the condition of the dischar ge-tube immediately before compounds of chlorine were introduced. CHLORINE (At. Wt. 35°46). Spectra indicating that this element was a mixture of isotopes were first obtained by the use of hydrochloric acid gas, but as this was objectionable on account of its action on mercury, phosgene (COCI,) was substituted. Spectra IL., IIL., and IV. are reproduced from one of the plates anizen with this gas. It will be seen that chlorine is characterized by the appearance of four very definite lines in the previously unoccupied space to the right of O, (82): measurement shows these lines to correspond exactly to masses 35, 36, 37, and 38. There is nv indication whatever of a line at a point corresponding with the accepted atomic weight 35°46. On Spectrum IL., taken with a small magnetic field, faint lines will be seen at 17°5 and 18°5. These only appeared when chlorine was intro- duced, and are certainly second order lines corresponding to 35 and 37. These figures seem to leave no possible escape from the conclusion that chlorine 1s 4 unisture of isotopes and that two of these have masses 35 and 37. It might be argued that 36 and 38 are also elementary lines and at present there is no evidence to deny this, but it is much more probable that they are the hydrochloric acids HCl® and HCI”. Theline 18 is no indication of an element 36, as itis doubtless due to OHg. Corroborative evidence that Cl? and Cl®" are the main if not the only constituents is given by the strong lines 63 and 65 (Spectrum IV.) probably due to COC] and COCR. If chemical atomic weight is regarded as a statistical average, any lines due to C5 or its compounds should be considerably stronger than the corresponding ones due to Cl. This is actually found to be the case. In all spectra taken with chlorine present a faint line is distinguishable corresponding to 39. It is just possible that this is a third isotope. The unquestionable accuracy of its epmbining weight on the one hand and the striking whole-number masses given on its mass-spectra by its individual particles on the other, leave little doubt that chlorine is a mixed element, but much critical work will be necessary before its constituents and their relative proportions are decided with certainty. ARGON (At. Wt. 39°88 Ramsay, 39°91 Leduc). At the close of the experiments with phosgene the discharge- tube broke down and had to be cleaned and partially rebuilt, Mass-Spectra of Chemical Llements. 621 so that by the time it had reached suitable working conditions again, all traces of chlorine had disappeared. The tube was run with a mixture of CO, and CHy, and then about 20 per cent. of argonadded. The main constituent of the element was at once evident from a very strong line at 40 (Spectrum VI.) pee ced in the second and Siied orders at 20 and 13°33 (Spectrum V.). The third order line is exceedingly well placed for measurement, and from it the mass of the singly- charged atom is found to be 40°00+:-02. At first this was thought to be the only constituent, but later a faint companion was seen at 36, which further spectra showed to bear a very definite intensity relation to the 40 line. No evidence drawn from multiple charges is available ini this case owing to the probable presence of OH, and ©; but the above intensity relation and the absence of the line from spectra taken just before argon was introduced, make it extremely likely that it is a true isotope. The presence of about 3 per cent. would account for the fractional atomic weight determined from density. NirrocEn (At. Wt. 14:01). This element shows no abnormal characteristics : its atom cannot be distinguished, on the present apparatus, from CH, nor its molecule from (QO. Its second order line on careful measurement appears to be exactly 7, so it is evidently a pure element, as its chemical combining weight would lead one to expect. Hyprocen: (At. Wt. 1:008) and Hetium (At. Wt. 3:99). The determination of masses so far removed as these from the reference lines offers peculiar difficulties, but, as the lines were expected to approximate to the terms of the geometrical progression 1, 2, 4, 8, etc. the higher terms of which are known, a special method was adopted by which a two to one relation could be tested with some exactness. ‘l'wo sets of. accumulators were selected, each giving very nearly the same potential of about 250 aa The potentials were then made exactly equal by means of a subsidiary cell and a current- divider, the equality being tested to well within 1 in 1000 by means ot a null instrument. If exposures are made with such potentials applied to the electric plates first in parallel and thenin series, the magnetic field being kept constant, allmasses having an exact two to one relation will be brought into coin- cidence on the plate (Phil. Mag. April 1920, p. 453). Such 622 Dr. F. W. Aston on the coincidences cannot be detected on the same spectrum photo- graphically ; but if we first add and then subtract a small potential from one of the large potentials, two lines will be obtained which closely bracket the third. To take an actual Instance—with a constant current in the magnet of 0:2 ampere, three exposures were made with a gas containing hydrogen and helium at potentials of 250, "500-412, and 900—12 volts respectively. The hydrogen molecule line was found symmetrically bracketed by a pair of atomic lines (Spectram VII. a and c), showing that the mass of the molecule is exactly double the mass of the atom within experimental error. When after a suitable increase of the magnetic field the same procedure was applied to the helium line and that of the hydrogen molecule, the bracket was no longer symmetrical (Spectrum VII. 6), nor was it when the hydrogen molecule was bracketed by two helium lines (d). Both results show in an unmistakable manner that the mass of He is less than twice that of H,. In the same way He was compared with OT*, and H;, obtained from KOH by Sir J.J. Thomson’s bombardment method, with CTT. The method has some definite advantages and some dis- advantages. It is not proposea to discuss these in detail at present. The values obtained by its use can be checked in the ordinary way by comparing He with Ct* and H3; with He, these pairs being close enough together for the purpose. The following table gives the range of values obtained from the most reliable plates :—- Line Method. Mass assumed. Mass deduced. H f bracket ...... Ot+= 8 3°994—3'996 pene: (cineca eee Gtt= 6 =.) 40052010 a bracket ...... Ctt+t= 6 3'025—3:027 g direct si.) He = 4 3:021—3-030 EL its Saree bracket ...... He -— 4 2:012—2:018 rom these figures it is safe to conclude that hydrogen is “pure” element amtleutnteits atomie weight, determined anit such consistency and accuracy by chemical methods (1008), is the true mass of its atom. The above results incidentally appear to settle the nature of the molecule H; beyond doubt. Mass-Spectra of Chemical Elements. 623 Krypron (At. Wt. 82°92) and Xenon (At. Wt. 130-2). The results with these elements were particularly inter- esting. The only source available, for which the author is indebted to Sir J. J. Thomson, was the remains of two small samples of gas from evaporated liquid air kindly supplied by Sir James Dewar-some years ago for examination by the “parabola”? method. Both samples contained nitrogen, oxygen, argon, and krypton, but xenon was only detected in one and its percentage in that must have been quite minute. Krypton is characterized by a remarkable group of five strong lines at 80, 82, 83, 84, 86, anda faint sixth at 78. This group or cluster of isotopes is beautifully reproduced with the same relative values of intensity in the second, and fainter still in the third order. These multiply-charged clusters give most reliable values of mass, as the second order can be compared with A (40) and the third with CO or N, (28) with the highest accuracy. It will be noted that one member of each group is obliterated by the reference line, but not the same one. The singly and doubly charged krypton clusters can be seen to the right and left of Spectrum VIII. It will be noticed that krypton is the first element examined which shows unmistakable isotopes differing by one unit only. On the krypton plates taken with the greatest magnetic field faint, but unmistakable indications of lines in the region of 130 could just be detected. The richest sample was therefore fractionated over liquid air, and the last fraction, a few cubic millimetres, was just sufficient to produce the xenon lines in an unmistakable manner. These can be seen on Spectrum I[X., but are somewhat fuzzy owing to the wide diaphragm used to get maximum intensity. They are apparently five in number and appear to follow the integer rule. Until pure xenon is available no final figures can be given, but the values may be taken provisionally as 128, 130, 131, 133, and 135. Mercury (At. Wt. 200°6). . Owing to the presence of mercury vapour (which is generally beneficial to the smooth running of the discharge) the multiply-charged particles of this element appear on nearly all the plates taken. They appear as a series of blurred clusters of decreasing intensity around points cor- responding to 200, 100, 66°6, 50... ete., some of which are indicated in the spectra reproduced. It may be stated provisionally that they indicate a strong component 202, a weak one 204, and a strong band from 197 to 200 containing three or four more unresolvable at present. 624 Dr. F. W. Aston on the Table of Results. Minimum Element. Atomic Atomic number of Mass of isotopes _ number. weight. isotopes. in order of intensity. Hee: 1 1:008 1 1-008 Fie: haere 2 3-09 1 4 Os Seer 6 12-00 1 Zs ING meee: 7 14:01 1 14 eee Sane 8 16-00 za 16 IN eas oe 10 20:20 2 20522 ey) Clee 17 35°46 2 35, 87, (89) antes 18 39:9 (2) 40, (36) Ker sae 36 82-92 6 84, 86, 82, 83, 80, 78 Xo eee 54 130:2 5 (128, 131, 130, 133, 135) He = ok... 80 900°6 (5) * (197-200, 202, 204) [Numbers in brackets provisional only. | The Whole-number Rule. The most important generalization yielded by these experi- ments is the remarkable fact that (with the exception of Hy, H,, and H,) all masses atomic or molecular, element or compound, so far measured are whole numbers within the accuracy of experiment. It is naturally premature to state that this relation is true for all elements, but the number and variety of those already exhibiting it makes the probability of this extremely high. On the other hand, it must not be supposed that this would imply that the whole-number rule holds with mathematical exactness, but only that the approximation is of a higher order than that exhibited by the ordinary chemical com- bining weights and is quite close enough to allow of a theory of atomic structure far simpler than those put forward in the past ; for such theories were forced to attempt the explana- tion of fractions which now appear to be merely fortuitous statistical effects due to the relative quantities of the isotopic constituents. ‘Thus one may now suppose that an elementary atom of mass m may be changed to one of mass m+1 by fe addition of a positive particle and an electron. If both enter the nucleus an isotope results, for the nuclear charge is unaltered. If the positive particle only enters the ‘nucleus, an element of next higher atomic number is formed. In cases where both forms of addition give a stable configuration, the two elements will be isobares. The electromagnetic theory of mass asserts that mass is not generally additive but only becomes so when the charges Mass-Spectra of Chemical Elements. 625 are relatively distant from each other. This is certainly the ease when the molecules H, and H; are formed from H,, so that their masses will be two and three times the mass of H, with gret exactness. (It must be remembered here that the masses given by these experiments are those of positively charged particles, H, being presumably a single particle of positive electricity itself, and that the mass of an electron on the scale used is (00054 and too small to affect the results. ) In the case of helium, the standard oxygen, and all other elements, this is no longer the case; for the nuclei of these are composed of particles and electrons pecked exceedingly close together. The mass of these structures will not be exactly the sum of the masses of their constituents but probably less, so that the unit of mass on the scale chosen will be less than that of a single hydrogen atom. The Heavier Elements. The results hold out the probability of great complexity in elements of high atomic number, which has already been proved by entirely different methods in the case of lead. The present apparatus has a resolution factor too low to deal adequately with these; so attention is being given to elements within its scope and to which the analysis can be applied. Hesults are steadily accumulating, which will be published in due course. In conclusion the author wishes to express his indebtedness to the Government Grant Committee cf the Royal Society for defraying the cost of some of the apparatus employed. Summary. A positive ray spectrograph capable of giving a focussed mass-spectrum is fully described in detail and its technique explained. The results of a provisional analysis of eleven chemical elements—H, He, C, N, O, Ne, Cl, A, Kr, X, Hg—are given, showing that of fies ie fir st fig @ only are “pure,” the others being apparently composed of various numbers of isotopic constituents, krypton containing no less than six. With the exception of those due to H,, Hy, and Hi, all masses measured, allowing for multiple charges, are exactly whole numbers within the error of experiment OE 16). The lines due to hydrogen indicate that the mass of the atom of this element is vreater than unity on this seale and in good agreement with the chemical value 1008. Reasons for this are suggested. Cavendish Laboratory, March 1220. LX. On the Advance of the Perihelion of a Planet, and the Path of a Ray of Light i in the Gravitation Field of the Sun. By Prof. A. ANDERSON * 6 ee particular niga of Hinstein’s contracted tensor equations, which has been applied to the case of the motion of a planet in the sun’s gravitational field is ds? = —y hdr? — 1°d0? + dt’, neglecting the term in ¢, as the motion may be considered to take place in the plane ¢=0. The quantity denoted by y 1s 1—2m/r, where m is a constant. A contravariant vector is found that is suitable for the motion of a planet, and then it appears that 2m cone Dnihe es ee Fee 9 2 1B) r a where a is a constant and Lop. Thus m is identified with the astronomical mass of the sun, and a with the semi-axts major of the planet’s orbit. | 9 The term indicates an ee of the perihelion of the 2 m . e — in every revolution. The advance of the perihelion of Mercury (43” per century) is accounted for. In the case of the path of a ray of light ds=0, and we get, using the same particular integral, adr\2 dO? : (7; ) ee AG ae but this equation is unsuitable. We want an equation which will give us the velocity of light asa function of r. That is, we want an equation of the form oo + o(F) = some function of r lz ! open : planet ae an angle Now it appears that we may take any function of 7 instead of r and then call this function 7, So that we must get a function of » that, when substituted for 7, will give us an equation of the necessary form. If we make r=/(7;), it is easily proved that we must have 2 r , 2— Ay 2 {os giv / é'(r,) 2 es a) 7 * Communicated by the Author, . a ‘ ) Advance of the Perihelion of a Planet.: 627 and we get, remembering that =r when m=0, L < 1\9 47, = (7? +(r—2m)?)’, which gives = (27, +m)?/47,. 2 On Saas nee Writing » for 7, we now get Z , 3 : me \ m velocity of light = ae -) ee ) 5 27 or 2 m the index of refraction wp = (1 25 | a a) The p and + equation of the path of a ray of light is consequently 3 , \ : 3 mM 72 Me | Vr p(t i) HE 5,)= a(1+5°) (e- 9q)=% where a is the distance of the sun’s centre from the apse, and m is assumed to have the same meaning as before. | 2 If squares of = are neglected, w= 1+ =, and the equation of the path is 2m r(1 + —) =a+2m. We may remark, though perhaps the assumption is very violent, that if the mass of the sun were concentrated in a sphere of diameter 1°47 kilometres, the index of refraction near it would become infinitely great, and we should have a very powerful condensing lens, too powerful indeed, for the light emitted by the sun itself would have no velocity at its surface. ‘Thus if, In accordance with the suggestion of Helmholtz, the body of the sun should go on contracting, there will come a time when it will be shrouded in darkness, not because it has no light to emit, but because its gravita- tional field will become impermeable to light. Taking the p and rv equation of the path of a ray of light Pa . . we find that the angle between its asymptotes is : K , Where sin @== 2m(«? —2ma)— K(a’x? + 4max? —x*)? SF (dyn? 4+ KR? ; a{dm* + K*) m2 4° m afi 25 (1 *) ( 2a: i ZN. and «=c?—4m’?. If we neglect squares of m, this becomes In 4 - Zn Me a Sie Ol a ah 628 Advance of the Perihelion of a Planet. In obtaining the path of a ray of light, we have changed 7 in the original particular integral to Oa 2/4, and it must be legitimate to go back to the case of te planet and make the same change. We thus get QUE NA Sa Aal) =a m? m m a) +7(a.) =)(1-Gs) C5) where ¢ is a constant, and (2r+m)* dé G72 os cas Neglecting squares of m, these equations become ai “(ey Se 2m =/h, a constant. ds r yy : and m(1 + cue =h. ma kas It is clear that ? —1l=— = and dividing the first equation by the square of the second, we get, neglecting squares of m, : 9 6 e eee (is m 5 2mu d@ ah? Nie ® mM, 1 tus where u= = tf or jz the ordinary Newtonian equation for elliptic motion. So that Mercury, unfortunately, is left with the advance of his peribelion unexplained. But his rate of description of areas is not constant; it is least at perihelion and greatest at aphelion. The astronomical unit of mass used in the above is 9 x 10” times the ordinary astronomical unit of mass; the unit of length is the kilometre, and the unit of time the time required for light to travel one kilometre in a forceless field. University College, Galway, 15th February, 1920. 7 OM fee Oe od | ML oe ee. Oa ee ie oS ee r 629 J LXI. Notices respecting New Books. An Enquiry concerning the Principles of Natural Knowledge. By A. N. Waurrenean, Sc.D., F.R.S. Cambridge: The University Press, 1919. PROFESSOR WHIVEHEAD’S book has made its appearance at a very fortunate moment. He could not have anticipated that a few months after its publication the verification of a prediction in regard to an astronomical event would be astonishing the intel- lectual world and setting everyone trying to understand the revolution it implies in our fundamental concept of Nature. It is this fundamental concept and the principles on which we must, in view of recent developments of science, in future base cur physics, which are investigated in this most fascinating Enquiry. Professor Whitehead’s mathematical followers will no doubt complain that so large a part of the book deals with metaphysics ; his philosopher friends, on the other hand, may be appalled at the intricate mathematical maze into which they will seem to be unheedingly conducted. The conception that physics and meta- physics are one identical problem is, however, the dominating thought in science and philosophy to-day, and it finds clear expression in this book. The main thesis can be briefly stated. It is that the traditional concept which has hitherto subtended the structure of physical science, the concept of all nature at a durationless instant, is no longer possible. The simple attempt to realize it in thought inakes everything that is anything in science meaningless. There is no alternatiye in science but to abandon it, and replace it with a new fundamental concept of which duration is the essence. Nature does not consist of facts, that is, of material space- occupancy at definite moments of time, but of events. Nothing which is not event has any place whatever within the reality of nature. This has been clear from the first in biology but it has waited long for recognition in physics, and indeed all the attempts to base biology on physics have been conceived with the idea that the reality of physics is fundamentally different and more elemental. The theory of the electro-magnetic origin of mass has revolutionized the metaphysics of physics. “The modern theory of the molecule is destructive of the obviousness of the prejudgment in favour of the traditional concepts of ultimate material at an instant. Con- sider a molecule of iron. It is composed of a central core of positive electricity surrounded by annular clusters of electrons, composed of negative electricity and rotating round the core. No single characteristic property of iron as such can be manifested at an instant..... Tron and a biological organism are on a level in requiring time for functioning. There is no such thing as iron at an instant; to be iron isacharacter of an event.” (8. 4.) It is curious to those of-us who are old enough to remember the generation of the great Victorian men of science, to reflect on 630 Notices respecting New Books. the nature of the confidence which inspired them, and compare it with the present development of science and the quite different kind of eoufidence which inspires our men of science to-day. _ Tam thinking of the feelings with which that generation heard the British Association Addresses of Tyndall, Huxley, and Clifford, to take three leading names. There was in all they said a note of clear triumph, they proclaimed a victory achieved, all that remained to do was to reap the fruits. Beneath this con- fidence and its basis was the belief that Nature is extremely simple in its framework and universal laws, magnificent in its perspective, stupendous in its grandeur, but at bottom an arrangement of space and time and material in a vast chemical laboratory functioning automatically. It was too simple. The task of science has turned out very differently. New and unsuspected realms of reality have been disclosed, ever-growing complexities have destroved the simplicity of the first generalizations. Men of svience to-day have not lost confidence in science but the basis of that confidence is shifted. Nature does not present itself to us as self-revealing, as a school- inaster from whom like children we have obediently to learn. The first thmg on which Professor Whitehead insists in this Enquiry is the impossibility of any pure science of Nature which begins by ignoring the problem of the relation of the mind to its objects. We have to take into account that we perceive, and that the immediate objects of the mind are its perceptions. Physical science has been impatient of Berkeley’s problem, and has turned it over to speculative philosophy as a problem science can dispense with. ‘To-day we are realizing more and more clearly that this cannot be ignored without fatal consequences to science itself. The new principle of relativity is, in effect, the inclusion of the problem of perception in physical science itself. The Enquiry is divided into four parts. In Part I. Professor Whitehead criticizes the traditional concepts and shows how they have failed by reason of that very aloofness from the philosophical problem which has been their boast. The problem of movement then leads him to describe Newton’s laws of motion, Clerk Maxwell’s equations, and Hinstein’s scientific relativity. Part II. deals with the data of science, and here we have the exposition of the root concept which Professor Whitehead proposes for the new organi- zation of science, the event. Now, because an event seems analysable into factors and because these factors seem separable as well as distinguishable, it has always been taken to be com- posite, a synthesis of the factors into whichit is analysed. Science therefore has always sought to go behind the event to what have seemed to be its constituent elements, for its data. But the factors of an event are not isolable, and consequently there are no simple elements which constitute by their conjunction the event. In Part III. Professor Whitehead expounds his method. This part is, as we should expect, severely mathematical. Taking events as his data he shows how they must be coordinated to yield a Notices respecting New Books. 631 science of Nature. Instead of the traditional space, time, and material, we have now to deal with “durations” which we are asked to conceive as temporal thicknesses or slabs of nature, and ‘we are taken through the whole range of transformations which the new concept requires. In Part IV. we have the theory of objects. It is a most important section. For though we may hold with the idealist philosophers that there are no things, we cannot have physical science and no objects. How, then, do events take on the character of objects? The theory gives the answer. In his final chapter, entitled ‘‘ Rhythms,” Professor Whitehead shows how deeply he is in sympathy with the philosophers who, like Bergson, approach the great problem of the ultimate nature of physical reality from the side of spirit rather than from the side of matter. It will be seen then that this book is simply invaluable to anyone who wishes to bring himself into line with the new principle of relativity, whether his interest be scientific in the narrow sense or philosophical in the wide sense. John Stuart Mill tells us in his Autobiography that he was at times actually depressed by the thought that musical chords though practically infinite in the number of combinations they admitted were yet in reality finite and exhaustible. Our feeling as we close Professor Whitehead’s book is one almost of elation at the thought of how little we know, and how uncertain is the little we think we know, when we form our concepts of the framework of infinite Nature. Manual of Meteorology. Part IV.: The Relation of the Wind to the Distribution of Barometric Pressure. By Sir Napier Suaw, Se.D., F.RS. [Pp. xvi+166, with 3 plates.] Cambridge: At the University Press, 1919. Price 12s. 6d. net. Tue other Parts of the manual of which the volume under review forms Part LV. have not yet been published. The appearance of the last Part in advance, though it may at first appear strange, is due to excellent reasons. Whilst the subject-matter which will form the first three Parts of the manual is available to the student of meteorology in various sources of reference, the present volume represents mainly the progress made by those who have been associated with the work of the Meteorological Office during the past twenty years. The subject-matter was not, therefore, previously availabie in collected form. The author has, in fact, incorporated the results of several quite recent researches. Students of meteorology are indebted to him, not only for having collected together the subject-matter, the grcvater part of which has hitherto only been available in the scattered original publications, but also for having combined it into a homogeneous whole and, in doing so, setting out what is practically a general meteorological theory. As a working hypothesis, with which to test the complicated 632 Notices respecting New Books. and often apparently contradictory results of observation, the author assumes that the motion of the air is at right angles to the. direction of the pressure gradient and has the velocity de- duced from the gradient equation. The latter equation is derived ° by assuming that the motion is under balanced forces depending on the spin of the Earth and the spin ina small cirele on the Earth. This leads to a study of the relationship between the surface-wind aud the geostrophic wind at sea-level and of the variation of wind with height. A summary is given of Taylor’s theory of the diffusion of eddy-motion, and comparisons are made between the results calculated from it and those provided from observations of pilot-balloons. The detailed discussion of the geometry and mechanics of a travelling cyclone is very suggestive, and the dis- tinction drawn by the author between the instantaneous kine- matic centre, the tornado centre, and the dynamic curve as shown by the isobars is an important one. A summary is given of Rayleigh’s exposition of the properties of a stationary column of revolving fluid, and Aitken’s experimental illustrations of the dynamics of cyclones are critically discussed. Lord Rayleigh’s discussion neglected the effect of the rotation of the Harth, and it is somewhat surprising that the results agree as well as they do with observation. The reproduction of instrumental records relative to several notable storms and their discussion help to give a clear idea as to the extent to’which theory and observation agree. The volume is an_important contribution to meteorological literature. We hope that the other Parts will be published at an early date. inl Sed Stereochenistry: By Atrrep W. Stewart, D.Sc., Professor of Chemistry in the Queen’s University of Belfast. Second Edition. Pp. xvi+277. [Text Books of Physical Chemistry. | London : Longmans, Green & Co., 1919. Tue chief alterations in this second edition are the following: Recognition is made of the existence of optically active compounds containing no carbon atom, the activity of anilime oxides, and corresponding phosphorus derivatives, and of the preparation of allylene analogues in active forms. The Walden inversion phenomena are given the place for which they are worthy. On the other hand, we are glad to see that the space previously devoted to steric hindrance has been cut down. That branch of the subject seemed likely to spread as an obsession over the whole subject—to its detriment. A short chapter is devoted to the arrangement of atoms in erystals, in which the work of the Braggs (father and son) is outlined. It would have been well, however, if the author had submitted this’ chapter for revision to a friend competent to express an opinion on the subject. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND le. JOURNADL OF SCIENCE, Sas + SY {SIXTH SERIES.) > Ay a \ vet JF NE 20s LXII. The Specific Heat of Saturated Vapour and the Entropy-Temperature Diagrams of certain Fluids. By Sir J. A. Ewine, K.C.B., PAS. 7 FZ\HE “specific heat of saturated vapour,” is a thermo- dynamic quantity whose interest is mainly historical, for the phenomena associated with it are now usually stated in other ways. But it has given rise to a misconception which is still reflected in authoritative text-books. This is readily avoided if the subject be considered with reference to a type of diagram, familiar to engineers, in which the entropy of the saturated vapour is exhibited in relation to the temperature. Following Rankine, by whom the phrase was first used f, we shall represent the specific heat of a saturated vapour by K,. It means the quantity of heat required, per unit of mass, to increase the temperature by one degree, while the pressure and the volume are so altered that a state of saturation is maintained. In steam, as is now well known, K, is a negative quantity, It is also well known that when K, is negative a vapour becomes supersaturated or partially coudensed when it suffers adiabatic expansion, and, becomes yerheated when it suffers adiabatic compression ; con- su i aoe ba versely, when K, 1s positive a vapour is superheated by * Communicated by the Author. + In a paper read before the Royal Society of Edinburgh, Feb. 4, 1850, Trans. R.S. E. vol. xx., or Miscellaneous Scientific Papers, p. 259, Phil. Mag. 8. 6. Vol. 39. No. 234. June 1920. Zak 634 Sir J. A. Ewing on Specific Heat of Saturated Vapour adiabatic expansion and is supersaturated or partially con- densed by adiabatic compression. The fact that K, is negative in steam was pointed out independently and almost simultaneously by Rankine (in the paper cited) and by Clausius *, who also showed that in any fluid ie ik abe 1h = wt ap me 2 where L is the latent heat and K, is the specific heat of the liquid (at the same temperature) when heated under saturation pressure, Later, when Regnault’s data for various fluids became available, Clausius applied this relation to them, finding K, to be negative in most cases. In ether, however, — he found that Regnault’s data made K, positive “at least at ordinary temperatures,’ and in other fluids the value of K, calculated from them increased as the temperature rose (that is to say, its negative value decreased). He continues: ‘‘In the only case, that of ether, in which it is positive at ordinary temperatures its absolute value increases as the temperature rises. In the other cases, in which it is negative, its absolute value diminishes ; it thus approaches zero, and tt would appear that at some higher temperature it would attain the value zero and at still higher temperatures would become positive’? +. Thave italicised this passage because, although the statement is true of certain fluids, there are many for which it is not true. Other well-known writers have made the same suggestion, that in water and in fluids generally, the specific heat of the saturated vapour changes sign from negative to positive when the temperature is sufficiently raised t. From the form of the entropy-temperature diagram, however, it is clear that this does not happen in steam, nor in any of the fluids commonly used as working sub- stances in refrigerating machines (earbon dioxide, ammonia, sulphurous acid), nor in bisulphide of carbon, nor (as will presently be seen) in alcohol. On the other hand, it does happen in a considerabie number of other fluids, such as ether, chloroform, benzene, and many esters of the fatty acids, for which the researches of Ramsay and Young and * In a paper communicated to the Academy of Berlin, Feb. 1850, Pogg. Ann. vol. \xxix. p. 368 and p. 500; Phil. Mag. July 1851. See his ‘Mechanical Theory of Heat’ (Tr. Browne), p. 135. t Clausius, ‘ Mechanical Theory of Heat’ (Tr. Browne), p. 139. | Cf. Preston, ‘Theory of Heat,’ 3nd ed. p. 731; Peabody, ‘ Thermo- dynamics of the Steam-Engine,’ 5th ed. p. 94; Lewis, ‘Physical Chemistry, 2nd ed. vol. il. pp. 83-85; L. Natanson, Phil. Mag. Oct. . 1895, p. 277; C. Raveau, Jour. de Phys. (3) vol. i. p. 461 (1892). and Entropy-Temperature Diagrams of Fluids. 635 others have supplied data by help of which one may draw a first approximation to the entropy-temperature diagram. Irom that diagram it is easy, by inspection, to see how the sign and magnitude of K, are affected by changes of temperature. The diagram exhibits the entropy os the liquid, $,, at saturation pressure, and also the entropy of the saturated vapour, ¢,, in relation to the temperature, by means of a continuous curve whose summit is at the critical point. The outer limb of this curve, from the critical point onwards, is the saturation line, relating to saturated vapour, and is the part with which we are immediately concerned. Let the state of the substance change by a small s step along the saturation line, with the result that the entropy changes by dd, and the temperature by dT. The heat taken in is Ix wat, by definition of K,. It is also equal to Tdd,, since the s step is reversible. Hence and is negative under all conditions that make the entropy ofthe vapour increase with decreasing temperature. In. other words, it is negative so long as the entropy-temperature line for saturated vaponr slopes down to the right, as in fio. Jor fig. 2... This is what may be called the normal form of the curve, such as is found in steam, carbon dioxide, or ammonia. To make K, positive would require that a part of the ne for saturated vapour should slope the other way, as in fig. where positive values of K, would be found at any ee ature between A and B, with negative values at any tem- perature above A or below B. The point B corresponds to the réversal of the sign of K, which was, in certain sub- stances, predicted from Regnault’s data and verified by the experiments of Hirn * and Coan + on the effects of adiabatie expansion and compression. It is obvious that, however strongly positive KX, may be at an intermediate temperature, it necessarily becomes negative, in all substances, as the critical point is approached ; for the saturation line must then bend over to the left to become continuous with the liquid line which forms the other limb of the diagram. At the critical point the value of K, is —#. In any fluid the entropy increases during vaporizatior at constant temperature by the amount L/T. Thus L bs= x + my . * Hirn, Cosmos, vol. xxii. p. 415 (1868). + Cazin, Ann. de Chim. et de Phys. (4) vol. xiv. p. 374 (1868). eke 2 636 Sir J. A. Ewing on Specific Heat of Saturated Vapour On differentiating this with respect to T, and multiplying by T, the Clausius equation quoted above is at once obtained :— dds = (on dl 10 Tara att a el, a Coe or K,=K,+ dT aa ae ° But when it is practicable to draw the entropy diagram the changes of K, are made evident without recourse to this equation. To draw the entropy diagram one may proceed by first calculating , and then adding L/T to find ¢,. According to the usual convention @¢, is taken as zero when the tem- perature is 0°C. Then at any scale-temperature ¢, (or T— 273° 1) 'C a: — a bom ( T ) where Cp is the specific heat of the liquid at constant eq pressure, namely the pressure of saturation for the given temperature. When an empirical formula eonnecting Ce with the temperature is available, this allows ¢, to be readily ealeulated. The values so foamed cannot be regarded as more than approximate at temperatures beyond the experi- mental range from which the formula for C, has been deduced, and in general that range is rather narrow. But the method gives a first approximation to the entropy- temperature diagram, which at least suffices to distinguish what may be called normal cases, such as are represented by fig. 1, from others which clearly resemble fig. 3._ In these last there are unquestionable reversals of the sign of Ky. I have drawn the diagrams in this way for a number of fluids, using in most instances the values of L given by Young in his paper on the vapour-pressures and heats of vaporization of thirty pure substances*, along with such data for C, as I have been able to find. The following are representative examples. Alcohol—From the experiments of Bose ft, CU, for ethyl alcohol is taken as 075396 + 0:001698 ¢, from which hy=0'1747 logy) T+0-001698 T—0-8892. * S. Young, Scientific Proceedings of the Royal Dublin Society, vol. xii. p. 374 (1910). + Bose, Gott. Nachr. 1966, p. 278; Zeit. fiir phys. Chem. vol. lviii. p- 585 (1907). 7— . Temperaturé and Entropy-Temperature Diagrams of Fluids. 6387 Values of L/T are got from Ramsay and Young’s values of L*. With these data the following numbers have been obtained for the entropy of the liquid and of the saturated vapour, and the curve of fig. 1 has been drawn. Ethyl Alcohol. Entropy of i, Entropy of Temp. the liquid. ar the vapour. i) Po d.. 0° i) 0:8089 08089 20 00394 0°7528 0°7922 40 0:0785 06985 — 07770 60 O-1171 0:6406 O-7577 80 0°1555 - 05845 0-7400 100 0-1936 0°5283 0°7219 120 02315 0°4686 0:7001 140 02692 0:4142 0°6834 160 0°3068 0°3623 0°6691 180 03442 0:3072 0°6514 200 03814 02465 0°6279 220 0°4185 01789 05974 240 wat 00785 ane J (critical)... 0 0°51 (about) Fig. 1.—Ethyl Alcohol. nh 3 80 40 Entropy * Phil Deans. 18367 pt. 1, p. 153. 638 Sir Jd. A. Hwing on Specisic Heat of Saturated Vapour From 0° to 180° the saturation line is nearly straight, with a slight bend about 120° which may be due to irregular- ities in the determination of L. The entropy diagram for ethyl alcohol is clearly of the “‘normal” type, ‘showing negative values for the specific heat of the saturated vapour thre oughout the whole range of temperature. This is also true of methyl! alcohol, which has a diagram of the same type, but with a wider spread of the two limbs, on account of its possessing greater latent heat. -The diagram for propyl alcohol is shown in fig. 2. tis calculated in the same way from Ramsay and Young’s values of L, using Bose’s formula for the specific heat of the liquid, namely, 0:5279 + 0°001692 t, from which $o=0'1515 logy, T +0-001692 T—0-8312. The reoleie numbers are given below. Propyl Alcohol. Temp, Pye" L/T. o. 40° 00767 ae lee 80 . 0°1523 0:4899 0°6422 100 0°1898 04395 06293 120 02270 * 0°3892 0°6162 140 02642 0°3447 0:6089 160 03011 0:2979 0°5990 180 0°3379 0°2567 05946 200 0°3746 02160 05906 220 O-4111 0:1730 05841 240 04476 0:1236 05712 263°7 (critical)... 0 0°52 (about) 66 Here again the figure is of the is a digunet hen in the saturation line which, though insufficient to make the specific heat of the saturated vapour positive at any temperature, tends that way. Ether.—Turn now to a substance in which the specific. heat of the saturated vapour is known to be positive at some temperatures, namely ethyl ether. J have used Ramsay and Young’s values of L, along with Regnault’s formula for the specific heat of the liquid *, * namely, 0°529 + 0:0005917 ¢, * Rezgnault, Relation des Expériences, vol. ii. p. 175. normal” type, but there ae 5 vg bliat aie ° ———? ee Sl and Entropy- Temperature Diagrams of Fluids. Fig, 2.—Propy! Alcohol. 639 from which Entropy ,=0°8459 logy T+0°0005917 T— 22226. The results are shown below and in fig. 3. 20 40 60 80 100 120 140 160 180 193°8 (critical) Ethyl Ether. Ow 0 0°0377 0:0738 0°1084 0°1416 O-1737 0-2047 0°2348 02640 02924 1By/4 Me 0:3388 02987 02645 0:2355 0-2082 0185 0°1583 01339 01064 00703 0 5s 6°3388 6°3364 0:3383 0°3439 0:3A98 0°3571 "3630 0:3687 0-3 704 0°3607 0°33 (about) 640 Sir J. A. Ewing on Specific Heat of Saturated Vapour Here for the greater part of its course, from about 20° to 150° or thereabouts, the saturation line slopes up to the right, showing that within these limits the specific heat of Fig. 3.—Ether. Temperature Entropy the saturated vapour is positive. Below 20° it is negative, and it of course becomes negative again as the critical point is approached. 7 | Another substance which gives a diagram of the same type is Benzene (fig. 4). In drawing the diagram for Benzene I have taken Schiff’s formula for the specific heat of the liquid, namely *, | 0°3834 + 0:001043 ¢, from which b= 0°2270 logy T+0°001043 T—0°8380. Young’s values of L are used f, but as they do not extend below 70° I have determined L for 0°, 20°, 40°, and 60° by taking Regnault’s values of the total heat of formation of the vapour (under constant pressure, from liquid at 0° C.) and deducting from that the quantity - (Chae which repre- ; sents the heat required to warm the liquid up to the temperature of vaporization. The values of L below 70° so determined fit in pretty smoothly with those found by * R. Schiff, Liebiy’s Annalen, vol. 234. p. 319. + S. Young, doc. cit. p. 422. 4 i ‘ < " fi eee se OR ee oO ee ee and Entropy- Temperature Diagrams of Fluids. 641 Young at higher temperatures. In the table that follows they are distinguished by square brackets. Fig. 4. — Benzene. 320° : 260 aie = nN (2) (a) Temperature fon) OQ £0 L i; 40 s O oie ators 083 04 OB Entropy Benzene. ‘Temp. Oy: L. L/T. ?.: Qe 0 [109-0] 0:3991 0:3991 20 0°0278 {106-0} 63616 0°3894 40 0°0552 [1024] 0°3271 0:3823 60 0°0821 [98:3] 0-2951 0°3772 80 0:1088 95:45 0:2703 0:3791 100 0°1350 91°41 0:2450 0°3800 120 0°1610 86°58 0:22038 0°3813 140 0:1869 82°82 0:2005 0:3874 160 0:2123 78°94 0°1823 03946 180 0°2377 74:62 01647 0°4024 200 0°2628 68°81 0°1454 0°4082 220 0:2877 62°24 0°1262 0°4139 240 0°3125 54:11 0°1055 0:4180 260 0°3371 43°82 0:0822 04193 280 0°3616 27°43 Q:0496 0°4112 288°5 (critical) 0) 0 0°39 (about) 642 Sir J. A. Nwing on Specific Heat of Saturated Vapour ~The curve (fig. 4) changes the sign of its slope at about 80°, but the position of the minimum of @¢, is not well defined.. . Cazin *, using large amounts of sudden compression and sudden expansion, found thal a mist was produced in benzene: vapour by expansion at temperatures below 115° and by compression at temperatures above 130°. He places the point of inversion in the neighbourhood of 120°. The entropy diagram shows clearly that the specific heat of the saturated vapour of benzene is positive throughout a considerable range of temperature. In the paper cited, Young gives values of L for ten esters of the fatty acids. Of these esters seven are included among the substances whose specific heat, in the liquid ‘ ated by Schiff, namely, methyl and state, has been investig ethyl propionate, ethyl and propyl acetate, propyl formate, methyl butyrate, and methyl isobutyrate. From his mea- surements of specific heats in these seven esters (and many more) Schiff has concluded that their specific heat is expressed by the formula + 0-44.16 + 0:00088 ¢, which would make, for all of them, b= 0'4635 logy) T+0-00088 T — 1°3696. Using this formula along with the values of L given by Young, I have calculated ¢,, and ¢, and have drawn an entropy-temperature diagram for each of these substances.. The values so obtained are given in the table below, and one example of the diagrams is reproduced in fig. 9, namely the diagram for ethyl propionate, which in its veneral character’ is representative of the others. It closely resembles the diagram for benzene, and indicates a point of inversion somewhere about 40° ©. The table shows thay alle tne esters belong to the class of fluids in which the specific heat of the saturated vapour is positive throughout a certain: range of temperature. = 0G. Cit. Prous + RR. Schiff, doc: ent. p. 340: it Ce ee ee Rael eee Seen ey ee ‘pull SO ARIMIAE a a ia att I a el | ee ft ’ and Entropy-Temperature Diagrams of Fluids. 643: . Esters of the Fatty Acids. Temperature. 02 20 40 60 70 80 90 100 120 140 160 180 200 220 240, 260 Entropy of the Liquid. Pay: 0 (0):0518 0:0627 0:0927 0:1075 01221 01366 0:1508 01789 02065 02356 0°2605 0'2866 0°3125 |. 03381 03634 Critical Temperature... Entropy of the Saturated Vapour, ¢.. Methyl Propionate. 03667 03687 03712 03726 03780 03842 03910 +3972 04045 04081 0:4052 Ethyl Propionate. aoe \ 0°3588 03632 03701 03764 03840 03926 04021 0:4109 04165 04156 —_—>- 2729-9 Ethyl Acetate. 0°3623 0°3650 03681 ‘3710 0°3757 0 3814 0:3858 0°3924 03980 0:3990 0°3911 250°'1 Propyl Acetate. 03615 03647 03731 03804 03898 05989 04076 0°4155 0°4207 0:4210 Propyl Formate. 0:3670 0:3699 0:3706 0:3728 0:3777 6:3850 03913 0:3989 0:4069 0:4131 0:4150 04028 264°°8 Methyl Butyrate. 0°8593 03679 03720 O-3872 03971 04079 04166 0°4241 04280 281°°3 Methyl Isobutyrate. 0°3468 0°3512 0°3586 03682 03770 0°3865 03966 04041 04084 04113 O44 Sir J. A. Ewing on Specific Heat of Saturated Vapour | Fig. 5,—Ethyl Propionate. 250 200 o ° Tempera turé 420 Entropy Two other substances in Young’s list present points of interest. His values of L for Carbon Tetrachloride extend from 70° up to the critical point. Below tnat we have Regnault’s measurements of the total heat of formation, from which L is readily found as in the case of benzene. For the specific heat we have Regnault’s formula * 0:1980 + 0:0001812 ¢, from which h»=0°3419 logy, T+0:0001812 T—0°8826. ‘Combining these data we obtain :— * Relation des Expériences, vol. ii. p. 282. Lo | and Entropy-Temperature Inagrams of Fluids. 64 Carbon Tetrachloride. Temp. Pi L/T. $,. 0° 0 01904 01904 20 00141 0°1736 01877 40 0:0275 0°1582 0°1857 60 00403 01440 01848 80 00525 0°1306 071831 100 0:0643 01185 01828 120 0:0757 0°1070 01827 140 _ 0:0867 00966 0:1838 160 0:0974 00876 0°1850 * 180 01078 00781 0°1859 200 01178 00689 0°1867 220 0°1276 0:0597 0:1873 240 0°1371 60498 01869 260 01464 0:0376 01840 283°1 (critical) ... 0 0:17 (about) The figures therefore indicate the existence of a very slight double bend of the fig. 3 type; but the most note- Fig. 6.—Acetic Acid. 320° 280 240 200 Temperature oa) ° 0 ° go 40 | Hes | Qe 0-2 0-3 O-4 o:5 Q6 (Entropy URE PSE ene | ae ed Oe Aeon worthy feature is that the entropy of the saturated vapour is nearly constant except close to the critical point. 646 The Specific Heat of Saturated Vapour. Lastly, there is Acetic Acid, which gives the remarkable diagram shown in fig. 6. This is obtained by the use of Ramsay and Youneg’s values * of L, along with Schiff’s formula t according to which the specific heat of the liquid is | 0°4440 + 0:001418 ¢, making b= 0'1306 log;) T+0°001418 T—0°7054. The following are the calculated numbers for ¢, and ¢s. They show that in acetic acid the specific heat of the saturated vapour has an exceptionally high positive value, which it retains even at the lowest temperature to which the observations extend, namely 20° C. The slope of the curve there gives no promise of changing its sign at a lower temperature. Acetie Acid. Temp. Dire L/T. Pos 0° 0 ns ie 20 0:0323 0°2868 03191 40 0:0643 0:2779 03422 60 0:0963 0:2693 0:3656 80 0:1280 ():2594 0:3874 100 0:1595 0°2474 04069 120 0:1908 0-2401 04309 140 0:2220 0:22938 0:4443 160 0:2530 0°2069 0:4599 180 02839 0°19386 0:4775 200 0:3147 0°1808 0:4955. 220 03454 0°1663 O°5117 240 03760 071624 0:5284 260 04065 013835 0°5420 280 0°4370 01148 075518 300 0-4674 00854 0:5528 321°6 (critical) .., 0 : 0°53 (about) * Young, doc, cit, p. 448. + Schiff, doc. cxt. p. 322. meee LY Te ee ee ee ne ee ¢ ed ee en ee re his: ster eee ers LXUI. The Crystalline Structure of Zine Owide. By W. Lawrence Brace, M.A., Langworthy Professor of Physics, Manchester University * INC oxide crystallizes in the hexagonal system. The erystals are of the dihexagonal polar type, with an axial ratio a:c=1:1-608. Natural crystals of zine oxide, or zincite, are rare, and the material used for this investigation consisted ofa platy mass cf zincite of irregular shape. The direction of the cleavage and etching of the surface showed that the mass was composed of crystals which were ver nearly parallel in their orientation. By noting the direction of the facets produced by etching it was possible to grind surfaces on the material which were approximately parallel to the principal faces of the crystal. The basal plane (0001), the first-order prism face (1010), the second-order prism face (1120), and the pyramid face (1011) were prepared in this way. The reflexions from these faces of the X- -rays from an -anticathede of palladium were examined with the X-ray spectrometer +, and the results are shown in fig. 1. ) 6°45’ 10 20 20 Glancing angle, for weave length 0584 AL. The structure assigned to the crystal by these results is shown by fig. 2, the positions of the centres of the zinc and J oD 7 * Communicated by the Author. | W. H. Bragg and W. L. Bragg, Proc. Roy. Soc. A. vol. lxxxvii. April 1915. 648 Prof. W. L. Bragg on the oxygen atoms being indicated by the small circles in the figure. @ Zinc © Oxygen. The atoms of zine are arranged on two hexagonal space- lattices, their centres corresponding very closely with those of a set of equal spheres in hexagonal close- -packing, fora close-packed hexagonal arrangement of equal spheres, the axial ratio may be calculated to be sli : Ic632; The axial ratio in the case of zinc oxide is Oe C= AeOOG: The positions of the zine atoms are identical with those of the hexagonal close-packed arrangement of spheres, if the latter be supposed to contract in the direction parallel to the hexagonal axis so as to reduce the ratio c/a from 1°632 to 1:608. The oxygen atoms are light compared with the zine atoms, and the spectra cannot be held to determine their positions exactly. In so far as they do this, they are in agreement with the supposition that the oxygen atoms are on two hexagonal space-lattices identical with those on which the Crystalline Structure of Zine Oxrde. 649 zinc atoms are situated, and derived from these latter by a movement of translation parallel to the ¢ axis, which brings every oxygen atom into the centre of four zinc atoms arranged at the corners of what is very nearly a regular tetrahedron. The dimensions of the structure are given in fig. 2. They are calculated from the axial ratio c/a=1:608 and the density of zinc exide 5°78. That they agree with the dimen- sions measured by the angles of reflexion is shown by the following table :— Plane. Calculated Spacing. Observed Spacing. (0001) . =2°60 A.U. 2°58 ALU. (1010) IS Oro A. 2-80 AU, (1120) zihol ACI 161 AU. (1011) 2-46 A.U. 2°47 A.U. In the case of the basal plane (0001) the successive planes are arranged as shown in fig. 1, oxygen and zinc atoms being arranged on alternate planes. The oxygen planes divide the distance between the zine planes approximately in the ratio 1:3, thus explaining the small second-order refiexion and the large third-order reflexion. The first-order prism face shows an abnormally large third-order reflexion, corresponding to the spacing of the planes parallel to that face. The planes. represent the centres of equal numbers of zinc and oxygen atoms, and are so spaced that the distances between successive planes are alternately 0:93 A.U. and 1-87 A.U., thus reinforcing the third-order reflexion. Shed The symmetry of the crystal structure is dihexagonal polar. The polar nature of the hexagonal axis is shown by the arrangement of the planes (0001). Cadmium sulphide, CdS, and wurtzite, ZnS, are iso- morphous. Like zincite, their symmetry is of the dihexa- gonal polar type, and the axial ratio is very nearly the same for the three crystals. Din) ae eee OO? — NEOs Zi 4 one Ge = le eb ad CdSct ees AE SSI OAT Close packed structure :.. a@:c = 1:1°632 Phil. Mag. S. 6. Vol. 39. No. 234. June 1920. 2U) 650 The Crystalline Structure of Zine Oxide. The analysis of the structure of wurtzite is of especial interest, as zine sulphide crystallizes in another form, zine- blende, Such § is cubie. If CdS and ZnS possess the same structure as ZnO, calcu- lation shows that the reflexion from the basal plane should occur at the following angles:— Riane (0001) ZmSyae7 a « online ry) 399 C dS sig SB Ate on Bye ()’ Wurtzite occurs naturally only asa fibrous crystalline mass, and a crystal sufficiently large to measure was not available. Greenockite is a rare miner al occurring as very small crystals. Dr. Gordon, of King’s College, London, very kindly lent an unusually large crystal of greenockite on this investigation, and a well- deine Seilsciom Grow the basal plane was Bonar at an angle of 5° 2’, thus affording confirmation of the assumption that the structure is the same as that of zine oxide. It is hoped to make further measurements on this. crystal. Wurtzite shows a platy structure parallel to the fibres of the crystalline mass, and when this material was mounted in the spectrometer a faint reflexion at a glancing angle of 5° 20’ was obtained, indicating that the basal planes of the crystals were parallel to the fibres. The effect was so small that very little reliance can be placed on this measurement ; further observations are necessary to confirm it. It is of interest to compare these structures with that of zinc-blende. In zine-blende the zinc atoms are arranged. on a face-centred cubic lattice, of side 5°42 A.U. The oxygen atoms are on a similar lattice, derived from the former by a movement of translation which brings each oxygen atom into the centre of four zine atoms arranged on the corners of a regular tetrahedron. The trigonal axes in zine-blende are polar. If the structure of wurtzite is the sameas that of zine oxide, as would appear to be the ease, then in the crystals both of zinc-blende and wurtzite, every atom of sulphur is surrounded by four atoms of zine at the corners of a regular tetrahedron, and every atom of zine by four sulphur atoms similarly arranged. ‘The dimensions of the structures are also almost identical. In wurtzite, for example, the distance between neighbouring zinc atoms is 3°85 A.U., in zine-blende it 1S Ouse eal. The arrangement of the planes parailel to the plane (OO01) of wurtzite is the same as that of the planes (1il) in zine-blende, the axis perpendicular to the planes being in each case polar. A Fluid Analogue for the A’ther. 651 The experiments described in detail in this paper were for the main part carried out in 1914, and a short reference to the results was made in ‘‘ X Rays and Crystal Structure.” I wish to express my gratitude to Dr. Hutchinson, of the Mineralogical Laboratory, Cambridge, and Dr. Gordon, of King’s College, London, for their kindness in supplying the material used, and to Mr. W. R. James, who assisted me in making some of the observations. LXIV. A Fluid Analogue for the Ather. By Dr. G. Gregy, Lecturer on Natural Philosophy in the University of Glasgow™ ee purpose of the present paper is merely to illustrate by means of an analogy certain points of resemblance (especially with reference to wave-propagation) between the gether and ordinary fluids, to some of which particular attention has not hitherto been drawn. The conception of the xther as a fluid medium has already been very fully discussedf and it is known to be subject to important limitations. Nevertheless 1¢ is desirable at the present time that the relations of matter and ether should be examined in every possible aspect, and the analogy now to be con- sidered, though in itself incomplete, may be of interest as an illustrative system, and possibly also in its bearing upon some problems of the eether still requiring solution. Recent experimental observations have compelled us to modify certain ideas regarding the physical characteristics to be associated with the ether, in proving that the ether is capable of acting as a very slightly refracting medium in strong gravitational fields. This discovery has to some extent suggested the line of comparison between ether and ordinary matter which is followed in the present paper. One of the main functions to be fulfilled by the ether is the apparently two-fold function of conveying waves of light and of electric and magnetic force at a constant velocity and of propagating the forces, cf gravitation. There have been associated with the ether certain elastic qualities enabling it to transmit transversal vibrations and at the same time to transmit a stress analogous to tensional stress in elastic material; and it is natural to * Communicated by the Author. + See FitzGerald, Proc. Roy. Dublin Soe. vol. ix. (1899). ZU 2 652 Dr. G. Green on a Fluid suppose that all these different qualities will be concerned in the propagation of waves. We are accordingly led to look for an analogue to the ether in a material medium, in which the propagation of waves is governed by two apparently dissimilar motive influences. “This is of course the case with water in which the waves travel under the influence of gravity and of surface tension combined ; and we shall commence by showing that in water, under certain circumstances, all waves of a particular type are propagated at a very nearly uniform velocity. The well known expression for the velocity of a wave in water of any depth, /, is aT eae es uv =4/ (241m) tank mf, Mreerrt” (di) where 2a/m represents the wave-leneth under consideration, and T the value of the surface jeaione. This expression indicates that there is a minimum velocity of wave-pro- pagation corresponding to a certain wave- length for which the value of m is determined by the root of the equation— ame 4 lm? sinh 2m ga Tn? TS Doh eee (2) The wave-length which corresponds to the minimum velocity of wave e-propagation we shall refer to as the critical wave- length. For very deep water, the minimum velocity is associated with a wave- -length for which m *=9/1. As the depth is diminished the minimum velocity is asseciated with a greater and greater critical wave-length. Ultimately the critical -wave- length itself is greater than the depth when the water is very shallow. It is to be noticed, however, that when the depth is diminished below a certain value, not far different from half a centimetre, equation (2) has no positive real root and there is then no minimum wave-velocity. But in those cases where the depth is moderate, and a minimum velocity of wave-propagation obtains, Fhe value of the minimum velocity i is given by pr _ Sinhh Teh mh where m is again determined by equation (2). It is easy to verify from (3) that, when h is small, the speed of all waves whose length exceeds the critical wave- length is practically constant nant equal to the limiting speed, gh, of a long wave in water of depth 4. We have therefore Analogue for the ther. D3 in shallow water a medium which transmits waves whose_ length exceeds the critical wave-length at an almost constant velocity and also transmits a certain stress between solid bodies floating on its surface. According to the analogy suggested above, long waves in shallow water correspond to luminous or electromagnetic waves in ether, and the forces of attraction exhibited between bodies floating on the surface of water correspond to the forces of gravity propagated through the ether. From the standpoint of this analogy, it is significant that, though gravity and surface tension are both concerned in the propagation of waves, the influence of surface tension practically disappears in the case of the longer waves, while its influence predominates in the case of the shorter waves. The suggestion this contains with respect to sether waves is that the agencies which give rise to gravitational attraction between bodies in ether are also concerned in the pro- pagation cf luminous and electromagnetic waves, though they may play only a subordinate part in their propagation. The actual law of speed for different wave-lengths given in (1) may be taken as illustrating only certain possibilities of the case for ether gravitational waves. If, however, we adhere closely to the ‘analogy between ether oravitational waves and waves in water for which the motive influence governing the wave-motion is mainly surface tension, we would then expect to find waves in ether governed mainly by the agencies which give rise to gravitational attraction. These waves would be shorter than any hitherto observed, and-the law of velocity of propagation of the waves would be—velocity inversely proportional to square-root of wave- length. The comparison of the mode of action of gravitation with that of surface tension leads us on to an interesting similarity between the two systems compared. In the immediate neigh- bourhood of any body at rest or in moticn on the surface of a shallow fluid, there is owing to surface tension a region within which the depth of the fluid is sensibly different from the uniform depth obtaining at some distance from any solid body. Any portion of a plane wave, in passing close to the edge of a ‘solid body, would hneretone be subjected to an alteration in its speed of propagation corresponding to the alteration, in depth of the fluid. Thus the waves would suffer a certain amount of deviation from a rectilineal path. This effect produced by the presence of solid bodies upon waves In water is evidently analogous to the bending of luminous waves from their path in a strong gravitational 64 Dr. G. Green on a Fluid field, such as that found in the immediate neighbourhood of the sun, at the total Solar eclipse of May 29, 1919. Another important suggestion contained in the analogy between the longer waves in shallow water and luminous and electromag netic waves in the ether is—that the velocity of hight in wether is according to this analogy to be regarded as a minimum wave-velocity. Just as the minimum wave- velocity in water can be determined from equations (1) and (2) above, as a determinate function of g, h, and T, so the analogy leads us to expect to find some definite relation between the velocity of light, the elastic constant or the density of the ether and the constant of gravitation. The view that the velocity of light in ether is to be regarded as 2% minimum is of importance also later on when we come to consider the phenomena accompanying the motion of particles through the ether at speeds approaching the _ velocity of light in their reiation to analogous phenomena in ordinary fluids. ; The similarity existing between the two media, shallow water and eether, with respect to the propagation of waves is not confined to general relations between wave-length and speed of propagation. It can readily be verified tor example that the ordinary laws of reflexion and refraction are obeyed in the case of shallow-water waves travelling im fluid which is otherwise at rest. In addition, it is clear that all phenomena involving the Doppler effect would obtain in ordinary fluids. By suitable suppositions regarding the relations between matter and ether it is also possible to extend the analogy so as to inelude a representation of the refraction and dispersion of gether waves in passing through material bodies. Inasmuch as a sudden increase in the depth of the fluid would involve an alteration in the velocity of waves in such a way as to cause this to vary according to the wave-length—a region of greatly 1 increased depth might be regarded as corresponding in the analogy to a dispersive medium. It would be impossible, however, to represent the various forms of dispersive media by means of a variation in the depth of the fluid alone. A much more complete representation of dispersion and refraction of luminous waves on passing from ether into matter can be obtained by assuming the matter to be represented by a portion of space permeable by the fluid containing a very large number of vibrators having periods of their ‘own. In the number and values of the natural periods of the vibrators, and in variations of their density, and in the closeness or openness of their arrangement within the fluid, aaticiont bas been obtained in ao Analogue for the ther. 699 -a wide range of dispersive qualities is proyided for. If -.a system of wat eatons of the type suggested here were in steady motion through the fluid, it has been demonstrated * that there would he a ceniaalil convection of the fluid in the region occupied by the vibrators; and a convection greement with that required by the theory given by H. A. Lorentz 7 in 1895. We come now to consider the motion of solid bodies at various speeds through water in its bearing upon the effects observed in the case of motion of bodies at increasing speeds through the wether. Iu this connexion the analogy between the ether and an ordinary fluid has been very fully discussed, especially with reference to the theory advanced by Sir George Stokes in 1845 to account for aberration. It is not the intention meantime to discuss Stokes’s theory except in its direct relation to the parti- cular form of the fiuid analogue for ether with which we are at present concerned. Stokes assumed that the Earth and the planets in their motion would set the ether in motion, just as any solid in translational motion within an incompressible fluid would communicate motion to the fluid. Later writers have referred to Stokes’s theory as directly applicable to all solids in motion. But, from the standpoint of the present comparison between aather! and water, it does not appear to be consistent with the analogy to suppose that all actual solids would set the ether in motion in the manner described. In particular it seems a more natural comparison to make between the ether and an aes fluid, if we regard all solids which freely transmit luminous and electromagnetic waves as being more or less freely permeable also by the ether. This would appear to exclude from the application of Stokes’s theory all transparent bodies and certain dielectrics, and it would remove one of the outstanding difficulties in the way of the acceptance of Stokes’s theory ¢. An important objection to Stokes’s theory has been pointed out by H. A. Lorentz§. If the ether be an incompressible fluid, its irrotational motion would be com- pletely determined by the normal component velocity alone * See R. A. Houstoun, “Fizeau’s Experiment and the dither,’ Phil. Mag. vol. xxxvil. p. 214 (1919). + ‘Versuch einer Theorie der Electrischen und Optischen Er- ‘scheinungen in Bewegten Korpern,’ p. 101 (1895). {t The difficulty referred to is to account for the experimental result obtained by Rowiand, Ri ontgen, and Eichenwald. § Archives Néerl. xxi. p. 103 (1896). 656 Dr G. Green on a Flind at each point of a solid in motion within it. Unless the. ‘component of velocity of fluid at each point tangential to the solid were the same as the tangential velocity of the point of the solid in contact with it, the ether in contact with any solid would in general have a motion relative to the solid. The same writer, however, showed later, in discussing a suggestion put forward by Planck, that the condition that the ether should have no motion relative to the Earth would be fulfilled if the ether in contact with the Earth were compressible in accordance with Boyle’s law, and at the same time subject to gravity. It is remarkable that if these assumptions were correct there would be a condensation of ether around the heavenl bodies such as would account for the deflexion of light from its path as observed at the recent eclipse. Whether the above suggested explanation of the difficulty is correct or not, the essential point with respect to Stokes’s. theory is that it ‘accounts for aberration provided the steady irrotational motion of the ether arising from the translational motion of the Harth is very large in comparison with any other motion of the wether which 1 may depend on the rotational motion of the Earth, and which may exist along with the. irrotational motion. Arguing from the conditions pre- senting themselves in the analogous system which we are- considering *, we can admit at least the possibility that this. may be the case. In the analogy, the Sun and planets correspond to bodies in motion, rotational and translational, upen the surface of shallow water, the motion being very slow in comparison with the minimum wave- -speed 1 in the- fluid. The motion of the fluid which is produced by the motion of the. solids would in this case be irrotational except for a very small part depending on the motion of rotation of the solid bodies, and ultimately on the very small degree of viscosity within the fluid. An infinitesimal degree of viscosity within the fluid would give complete fulfilment of the condition that: the fluid in contact with each solid would have no motion relative to the solid, while the corresponding rotational motion within the fluid would be negligible in comparison with the irrotational motion of the fluid arising from the motion of translation of the solid. It thus appears that the fluid motion in the neighbourhood of each of the moving solids fulfils exactly the conditions required to be fulfilled in a fluid ether as indicated by the * These conditions resemble closely the conditions contemplated by Stokes in his papers— Phil. Mag. vol. xxix. p. 6 (1846), and vol. xxxiii. p>* 343 (1848). is When we come to consider the fluid motion arising from the motion of a solid on the surface of a fluid at a speed exactly equal to the minimum wave-speed in the fluid, we have, however, the theoretical guidance we require in the investi gation by the late Lord Rayleigh just referred to. The result Pe omed is that, in the entire absence of friction within the fluid, the amplitude of the resulting fluid motion is infinite; and when only a small retarding force pro- portional to the velocity, (av), acts at. each point of the fluid, the amplitude of the fluid motion arising from the uniform motion of the solid is proportional to Ga), an exceedingly large quantity. If a solid in steady motion at a speed equal to the minimum w ave-speed were suddenly brought to rest by an obstacle, a fluid pulse of large amplitude would be reflected from the obstacle into the * Lamb's ‘ Hydrodynamics,’ 3rd ed. § 331. T See, however, ‘‘ Standing Waves on the Surface of Running W ater,’ Proc. Lond. Math. Soc. vol, xv. no. 219. 698 A Fluid Analogue for the ther. surrounding region. Moreover the energy of wayve-motion constituting this pulse would be associated with the critical waye- length as a predominant wave-length ; that is to say, the greater part of its energy 1s associated with waves whose wave-lengths differ little from the critical wave-length *. It Fn) thus appears s that the apparent inertia of a solid mass moving steadily in, or on the surface of, a fluid would be greater and greater for increasing values of the speed of the mass for all speeds up to the minimum wave-speed. When a uniform speed equal to the minimum wave-speed in the fluid is attained by the solid, its apparent inertia then becomes infinite. The apparent increase in the inertia of a solid with speed, is in reality due to the reaction of the fluid set in motion by the solid which would act in a direction tending to stop the motion of the solid. Whether it would also tan to produce a contraction of the solid in the direction of its motion has not yet been definitely established by analysis, though this seems to be very probable. The ae considerations clearly indicate that there is- a very close general similarity between the apparent nce in the inertia of bodies moving in a fluid, and the corresponding apparent increase of inertia of electrons moving through ether at speeds increasing up to, and also equal to, the velocity of light. In addition to this general nesenniblemies between the ether and an ordinary fluid with respect to the motion of solid bodies within each, we have in the X-rays, evidence of the existence of wave-disturbances -of ver large amplitude in the ether arising from the motion Vy) I through it of particles moving at the speed of light, corre- sponding closely to the waves of large amplitude in water arising from the motion of solid bodies in water at a speed equal ‘to the minimum wave- speed. The analogy which we have been examining suggests other points of comparison between the interactions of matter and eether on the one hand and the interactions of ordinary solids and fluids on the other. With respect to the physical quantities—momentum, force, energy, associated with the bodies moving in an invisible fluid such as the ether, the interpretation of the analoyy is clear, or is virtually indicated in what precedes with reference to apparent increase of inertia. One additional point might, however, be mentioned in connexion with the motion of bodies in fluids. If the motion of a body on the surface of a fluid be an accelerated motion, this would evidently * Proc. Roy. Soc. A, vol. Ixxxix. pp. 583-4 (1914). Radioactivity and the Gravitational Field. 659 he accompanied by a change in the apparent magnitude of the force of attraction “between the -body and. other bodies in its neighbourhood arising from surtace tension. In the analogous system of the ether,—an apparent change in the gravitational force acting on a body would be Bhsehved in the case of a body moving with accelerated motion in the ether. In conclusion it should be remarked that there does not seem to be anything suggested by the analogy which we have been considering which is directly contrary to or inconsistent with the principle of relativity. At the same time it seems to encourage the view that it may yet be possible to detect absolute motion of the Earth in’ ether and that it may therefore also be possible to escape from -a principle involving so much indefiniteness with respect to fundamental units. In one sense therefore this analogy seems to present an avenue of escape from the principle -of relativity ; for if the velocity of light is really a minimum wave- velocity in ether, then in ‘lie apparent changes of inertia of electrons as the velocity of light is approached and in the behaviour of the pulses constituting X-rays, we are beginning to be able to detect velocity relative to ether. LAV. Radioactivity and the Gravitational Field. By ArrHur H. Compron, Ph.D)., National Research Fellow in Physics ™. T is well known that in order to account for the age and the present temperature of the earth, the average aclien activity of its component minerals must fall off rapidly a few miles below its surface. The high density of the radioactive minerals, however, makes it appear pr obable that they should vecur more abundantly in the earth’s interior than in the surface crust. Thus it appears that substances which at the earth’s surface are radioactive may have practically no radioactivity in the earth’s interior. These considerations suggest that the rate or energy of radioactive disintegration | may be a function of the intensity or potential Toh the earth's gravitational field. ‘This suggestion appears the more plausible since both radioactivity and gravitation are -essential attributes of the atomic nucleus. It has recently been pointed out by A. Donnan f that * Communicated by Prof. Sir FE. Rutheiford, F.R.S. + A. Donnan, ‘ Nature,’ Deemle 11:9: 660 Dr. A. H. Compton on Radioactivity thermodynamic reasoning predicts a change in the energy evolved in radioactive ie sintegration chet the potential of the gravitational field is varied. The following analysis. shows, however, that this change is by no means large enough to account, for the lack of radioactivity of the earth’s interior. The cycle considered by Professor Donnan consists of the following four steps :— 1. A system at Salata potential Z changes from state 1 to state 2, an amount of energy @ being liberated, which results in a change of mass from m, to mo. 2. The system in state 2 is raised from potential LO WoT. 3. At potential Z+6Z the system is changed back from state 2 to state 1. 4, The system in state 1 is lowered from potential Z+06Z. to potential Z. Being then in its original condition, the total energy evolved by the system is zero, and it possesses its original mass. If tive change 6Z in the gravitational potential is smali, the- total work loa by the system in performing te cycle is Q—m,84— (« + S700 \ +n dZ= 0, or Oot AW = My oo Mog, which is the expression obtained by Donnan. If now we consider as a function of Z only, we may write dQ m— my Putting R as the ratio between the energy evolved and the- mass which disappears, we have mM, — m,=QIR, whence dQ/O=adZih. / The difference in the gravitational potential between the surface and the centre of the earth is about 3 x 10™ em.’ sec. ~*, and the ratio R is of the order of the square of the velocity of ‘elit. jor 10" cm.7see. 57. Hence the decrease from this. cause in the energy of radioactive disintegration, being less than one part in a billion, is wholly inad quate to assert for the small amount of heat developed in the earth's interior. rs and the Gravitational Field. GOL This thermodynamic relation between the energy of radio- active disintegration and the gravitational potential does not, however, exclude the possibility of «a connexion of an intimate character between, for example, the intensity of the gravitational field and the rate of radioactive disintegra- tion. It is the latter type of relation which the present experiments have been designed to detect. According to Einstein’s generalized theory of relativity, gravitational accelerational field is essentially the same as a Feld of centrifuga] acceleration. We have therefore tested the effect due to a change in the gravitational field by subjecting the radioactive material to a strong centrifugal aeceleration. It was obvious that the maximum centrifugal acceleration which could be attained at the edge of a rotating wheel would fall far short of the mean acceleration to which the atomic nucleus is subject due to the thermal agitation of the atoms. There was a chance, however, that a+ com- paratively steady acceleration might have an effect different from that of the rapidly varying molecular accelerations. The Experiments.—A small tube of radium emanation was placed in a hole near the circumference of a brass disk of 10 cm. radius. The gamma radiation from the emanation was measured when the disk was rotating slowly and when turning at approximately 250 revolutions per second. The acceleration was thus varied from about 1°5 to about 20,000 times the acceleration of gravity. The gamma radiation was measured by a balance method. The ionization due to the gamma rays traversing a large ionization chamber was balanced against an adjustable current passing through a high resistance, a highly sensitive electro- meter being used to detect any difference between the two currents. For the high resistance a Bronson resistance was at first employed. This was later discarded in favour of a resistance consisting of lampblack on sulphur. The latter resistance, though more subject to variations over long periods, has the advantage that it introduces no short period probability variations such as those due to the ionization by discrete alpha particles when the Bronson resistance is used. ‘The probability variations in the ionization current due to the gamma rays, such as have been observed by Meyer * Laby + and others, were very noticeable in these ese ments, and were the cause of practically the whole of the differences between successive measurements. Altogether four extended series of measurements were * Meyer, Phys. Zeitschr. xi. p. 1022 (1910). “ + Laby, ‘ Nature,’ Ixxxvii. p. 144 (1911). 66: Mr. R. Hargreaves on the Difference between LN) made, comparing the intensity of the radiation when the emanation was subject to small and to large accelerations. The results of these tests were as follows :— 1. An inerease due to acceleration of (‘08+:07) per cent. 2. A decrease due to acceleration of ((17+°09) per cent. 3. A decrease due to acceleration of (-17+:07) per cent. 4. An increase due to acceleration of (‘02+:10) per cent. The average of all these tests indicates a decrease in the intensity of the gamma radiation of (‘06+°04) per cent. It is probable, therefore, that an acceleration of 20,000 times gravity produces no effect on the intensity of gamma radiation as large as one part in a thousand. In order to explain the small degree of radioactivity of the earth’s interior it would be necessary to assume a comparatively large change due to an increase in the gravi- tational acceleration. The negative result of this experiment therefore shows that we must look elsewhere for the cause of the confinement of the earth’s radioactivity to its surface erust. I wish to thank Professor Rutherford for proposing this. problem to me, and for his helpful suggestions and encourage- meni as the Ww ork progr essed. Cavendish Laboratory, Cambridge University. Feb. 3, 1920. LXVI. The Difference between Magnetic and Electric Energies asa Pressure.. By R. HArGReaves, VA.” N any electromagnetic field the pressure on a perfectly ] reflecting surface is normal, and is measured by the difference between the magnetic and electric energies (per unit volume) at the surface. This theorem, the subject of the present paper, may be viewed in conjunetion with an analogous theorem in Hydrody namics which appeared in this Maga zine j. There it was shown that Kelvin’s kinetic potential { or the motion of solids in infinite liquid, with circulation taken into account, could be got by taking a volume integral of the pressure. The analogy is at once apparent, and may be helpful in explaining the raison d’étre of the kinetie potential in Electromagnetics. For though * Communicated by the Author. + “A Pressure-integral as Kinetic Potential,” Sept. 1908. ra Magnetic and Electric Energies as a Pressure. — 663: magnetic and electric energies are contrasted forms, the contrast is not exactly that between potential and kinetic energy in dynamics, since the electric energy is not dependent on position alone when there is motion——i. ¢. the analogy with dynamics is imperfect. It is remarkable that the theorem has escaped attention in its application to the simple problem of reflexion of a plane wave at a fixed plane surface. Accordingly I take this case first, then deal with a moving plane surface, and finally take a general electromagnetic field near the surface of any perfect conductor in motion. § 1. If z constant is the surface on which a plane wave is incident, the conditiong at the surface are X=0, Y=0 ; and 1S TE RE Ce hogan Ox OY : these involve ¢=Q, since WaOee oy On: Hence the normal pressure Ze = 3(a?+b?—c?) +4(X?4+ Y?—-Z?) = re +l —Z’),| | (1) L = $(a@?+b?+ 07) —4(X?+ Y7+Z?) =4(21P—=7?),| have the same value at the surface: moreover X.=— ZX —ac vanishes, as also Yz, so that the pressure is purely normal. Consider the composition of these functions with reference to the separate waves. If (X, Y, Z,) refer to the incident wave, then een Nye Oe — iim 0 Vs) (2) For the reflected wave take the argument to be /sa+ may —inoz— Vt, so that (ly, m2, ng) = (l,, m, n,). At the surface then we have the relations (eee, 7) = — Ae a as (ats, bo, C2) = (4), 015 — 1). (3) The value of L at the surface is that of product terms, viz., L = ayag + byby + ecg —( XX + Y,Y,4+Z,Z,) ==" 0)" +b Oka VPRO : 2. (4a) On the other hand product terms in Z, vanish at the surface, since X,Not+ Y,:Y2—Z,Z, a A\A, SO bibs — eyes — —(X;? +Y,/? == Li + Cae + G2 > Gia . The pressure is shown in the Z: formula by the sum of ‘664 Mr. R. Hargreaves on the Difference between ae parts for incident and reflected waves; for the firs Z.(1) = $(ay?+b62—¢7) +4(X/7+Y"—-Z’), . (40) -and an equal value for the second. To evaluate this in terms of H, = a (ay” + b+ ¢,”) +4(X,74+ Y74 Z)7)=X74+ Y)°+Z,’, use ey +n? P= (LY,—m,X,)2 + (X44)? = (X24 Y)(1—n,?) = (E,—Z,2) (1—n2), or ef ZL? = Kyi 1—n,?),-and so, Z2(1) = 7,7 ae ee) The pressure 2K," of both waves is given by (4a). The flux of energy has a property like Z, as to its product “terms ; for xX 09 — Yja.4 NS 50, = Nese b,—Y,a,—X, by + Yaa => ‘The flux is therefore assessed as the sum for the ae ‘waves: for the incident wave V(Xq)1 — Yua;) = VX,(n,X,—1,Z,) —VY, (m,Z, —n, Y;) == 7b for the reflected wave —Vn,.H,, with a total flux vanishing ‘because there is no loss of energy. SN ith sa moving surface of reflexion there is loss ae energy, and the forme are sensibly modified. The boundary conditicns are now Mob) V 20, Y+wa/V=0; these involve c=0, . (6) -as before, for at the moving surface 1 ve rom Pe) ( 1 v)-2 SEONG) —+w )e= = —-{ X— —( Y + v} Oe edz Oy = The relative flux of energy, and - modified pressure formula, are shown in the equation 3 V(Xd— Ya)—wHh=w{Z,—w(Xb—-Ya)/Vi. 7 @) -valid because the difference between its members is V[(X—wb/V) (6—wX/V)— (Y + wa/V)(a+wY¥/V)], a quantity which vanishes in virtue of (6). The meaning -of (7) is that the relative flux of energy is equal to the work done on the moving face by the pressure. Since Z,—w(Xb—Ya)/V=L+ X(X-—wb/V) +Y(Y +wa/V)—e’, . (8) Magnetic and Electric Energies as a Pressure, 669 the pressure is still given by the value of [., as other terms on the right hand vanish. A component of tangential stress is now X,—w(Yc-Zb)/V or —Z(X—wb/V)—c(at+wY/V), which vanishes. To deal with the separate waves write X, +X, for X, and take (6) in conjunction with Prlr=polo, pr(V—wn,)= pol V + wre), Pim = poms, pi(Vny—w)= p2(Vrgtw), . . . (9) which bring the arguments p,(l)¢t+myy+njz—Vt) and Palloe + moy—ngz—Vt) into agreement for ¢—wt=constant. Using (6) with multipliers pol, poms, we get Z,;=Z,; then, quoting (5), H,(1l—n,’) = ¢° + Zf=c,.?+ Z, = b,(1—n,”), and Psy -tiey mses Ve 4 fee CLO) connects the energies of incident and reflected waves. With multipliers X, Y,, (6) yields >X X= — Hy pofp, + 2V?Z,/(V7—w’). . . (11) Then L = Sa,a,—>X1X5 = (4l,+mymg —nyng—1) 5X, X_q— (1X, + mY, —19Z1) (1, Xo + m,Yo+n,Ze) = {— EK, p,/p, + 2VZ?/(V?—w?)}{ —2(Vn—w)? x pilpaV? — w?) } + (prim + pots)? Ly" [pi ps- Writing pin; +pong= V(p; — pg)/w from (9), the Z, section vanishes through p.(V?—w?) = p,(V? +w?—2Vwn,), and finally Ty ei Woe — aA? =e?) x sh ce 2) If the calculation of pressure is made through the Z; formula (modified), the demonstration of the vanishing of product terms is the troublesome part. It follows from X,—whs/V + X,(1—wn,/V) + wl,Z,/V = 0, , Le ay 1 WY2/V + Y4(n;—w/V) —m,Z, =), ey and associated formule. ‘The parts of the pressure are given by square terms, viz , Eyny(m—w/V) and Hyno(ny+w/V), with ratio pin; : pons. Phil, Mag. Ser. 6. Vol. 39, No. 234, June 1920, 2X 666 Mr. R. Hargreaves on the Difference between Flux of energy and work done by pressure are shown very simply by interpreting B,(Vny — w)(1—wnfV) = E,(Vng+w)(1+wn,/V), 4) Which is an immediate consequence of (10) and (9), as difference of two fluxes giving work done by pressure. § 3. To deal with a conductor in a general electromagnetic field, the properties of the ideal conductor in virtue of which its interior is an electromagnetic blank need consideration. For this purpose the field equations are integrated through a layer in which a rapid transition is made from finite to zero values, and the volume of integration has a cross-section dS with thickness small in comparison with the linear dimensions of dS. Such integration gives | ay S =| or dpa ie ead {x dr = Ifd8, . (15a) / a direction-cosine of the normal, and fa value just outside the surface. To deal with ) we note that in respect to a, dt ot ee wo rapid change is obviated because the motion of an element is followed, and accordingly a © or . : : dt . zero value is given for the integral Of sim consequence dt of the small thickness of the layer. Thus (2 ar=—(( af of OA as | ny Sic a w 5.) ar= — (lu+mv+nw)d§ = —v,d5, say. (15 6) ox Applying these to the volume-integrals of p= re Se 0c 0b) I OaigoN o4 , we have result Vo Oy” Oc Wat Woe FOr. ‘ og = IX +mY +20, -- eG —uyX /V + ou,/V = me—nb, <3 2 ee (17) aut = mL — nV 2 ee reeny It is not assumed that the velocity (u.v.w.) altributed to charge at the surface agrees with the velocity (wvw) of the material surface. But 6) and (17) demand tle condition Magnetic and Electric ners asa Pressure. 667 Slue=t,= lu, or agreement of normal components, a condition which also follows from integrating oP 4 &. (pite) +...=Qasabove. The group (16) (17) (18) yields l—v,?7/V?)=a(l+u,v,/V"), af es —mw,)/V. pe GLO} We have therefore expressions for surface values in terms of o and (u.v.we). The formule for which we have immediate use are: o= ee ee = ol(1—SuZ/V2)/(l—v,2/V2), . (20) La?— TX? = —o7?(1— du 2/V*)/(1—v,7/V*), (21) Gee eave U2) Onis aise i wey ee te KD Sa ee FI Va ik no (23) From (20) and (21) we derive 5 (Sa? BX?) = spelen, + 7-7 5 (24) According to the well-known method of getting the force due to the transition layer, the right-hand member represents a component of the inward force. The force is therefore a pressure $(2a?—>X*) normal to the surface. This pressure is negative, as for the statical problem, with modifying factors due to the motion. The relation (22) is quoted as leading immediately to (23), from which Wek (ie Zb)/Ve noe. (2) If the value of L in (21) is differentiated with regard to u., the result is found to agree with the value of (Ye—Zb)/V as obtained from (19). Thus OL 7 ry OL ob ob ie (Ye—Zb)/V, and H = CaS ae re De —L; (26) and in respect to these surface values per unit volume we have the normal relation connecting energy with kinetic potential. 2X2 1 (668 24 LXVIT. The Diffraction of Wares by a Semi-infinite Screen with a Straight Edge. - By W. G. Bicxuny, B. Se., Assis- tant Lecturer in Mathematics, Battersea Polytechnic *. Sas SS Sommerfeld gave the first exact mathematical h solution of the problem of the diffraction ot waves by a semi-infinite screen with a straight edge, several other writers have attempted to obtain the same result by simpler analysis. Some have achieved “ sim- plicity ” by introducing assumptions not easily justified a priort, while others have used methods which are tantamount to assuming the known form of the solution. It is hoped that the following is free from these defects. It is an extension of the method used by Prof. Lamb f for perpendicular incidence, to the case of oblique incidence. § 2. The problem i is, in the first instance, two-dimensional. The trace of the screen will be taken as the positive half of the w axis. The incident waves will be considered to x! : re) ay advance in a direction parallel to IO (fig. 1), and ae angle IOX will be denoted by «. Then the “Incident waves will be given by i. sat ae cosa +y7 sin age) (V2 ke) db; =(0, . (1) where ¢ is the wave velocity, and k=27/(wave-length). It a be shown that the boundary conditions ean all be satished by expressions representing transmitted and reflected waves, d ahs wee cosa+ysin a+ ct) amides), aie, CC oo a Ye ee) respectively, u and v being functions of # and y alone. * * Communicated by the Author. + P. 1. M.S. (2) vol. iv. ; ‘Hydrodynamics ’ (4th 6p ry eS Diffraction of Waves by a Semi-infinite Screen. 669 Transferring to parabolic coordinates, &, n, given by L+1y = (E+in)?, the equation (V/;? + /*)¢ =) gives O7u Ores i 9B t D7? sik} (E cos +n sin a) So 9) Sn a uw +(£sina—7 cosa oul _o, es (2 (E n ) ae ( This equation is found to have a solution w= flagt+bn) if a= cos, b =sin5, in which case the equation to determine f becomes be ee a 4 WEIN f' Aik (Ecos 5 +1 in5) SS Oe ren 3) i cos +1 sing ke? é f= A+B URGE an Rane Rome ©) In the same way we find Ecos —7 sin 5 2 v=c+D| ee de. aa Therefore for the whole potential we have ¢ = ee A+B "2 arh 0 ee Dy acy CG) (; included in A), where for brevity we have written wxcosa+ysina+tct= BP, xCosa—ysina+ct = y, a 5 (42 ~— SIN ~ = Wp, =@, &cos 5 5 Bop ee E C08 5 7 sin 5 and the lower limit zero has been introduced, for definiteness, into the integrals. § 3. It now remains to satisfy the boundary and other 670 Mr. W. G. Bickley on the Diffraction of Waves conditions by finding suitable values of the constants A, B, C,and D. For points far to the left (@ negative), 7 is great compared with € Here ¢—>@,, so that, using the first terms of the asymptotic expansions of the integrals, we get Teen ap. / = jf 0 = 0-4D 4 / Zee. (8) For the boundary conditions on the plane, where »=0, we may have two cases : Caso vo—). Then A+C = 0, B+ D='0; oa be IG) Q at (8) above, giving, 200 epee pe (10) A-C+0, B—D=0, aes giving, with (8), ; 9 pees A=C=}], B=D=\/ me 4. 4807) So both cases may be included in —ik 2h 7% oyp92 pa ale Mf isay/ 2 | on ae 0 : ey >, [2k (e2 2ike ] | Fe 1424/7 |e de ae (11) or, since ae 2k °1 ke Ne “2 opine? Co) = yee t(. de + dg. . (13) by a Semi-injinite Screen with a Straight Edge. 671 Now é 2 at DOIN one by E coss ty sing) = Lir+ucosa+ysin «), “i.e. @,2 = 4r cos? 4(O—a) where atiy=re” ; and similarly w,” = trcos* (042). Using these results and also writing 2h¢?=u’, we get, | eal 7 TL ¢ CN 2hr cos $(0+a) a he N2kr cos 3(8+a) . ) = aN e@ du e due? Tt XJo 0 (14) giving the well-known solution for the diffraction of an harmonic wave train. §4. To deal with the diffraction of a disturbance of arbitrary wave form, it will be convenient to transform the integrals so that the limits are constants. Using (12), we have | 2k \". Dike? Ak ("? 2in2 ‘ “pdiku? i — a ae Cad a CLE gs (15) Now write u=C tan wW, and we get Ak 2 sec? es nae, (* 2 pike sc) Csec?pdwW. (w, +ve).. (16) Peeing the order of integration, this becomes Map [PMY gaescoyy? CAD II | Jie ox ae bola 2 ( p2ikw,’ sec? py 1)dw Greg dats itand posh tc LOy) = 5a [2 wv ay, bc 2 Ae 0 If w, is negative, the upper limit of integration becomes —> which is equivalent to changing the sign throughout (16)-(19). After this is applied to (14) the solution 672 = Diffraction of Waves by a Semi-infinite Screen. becomes discontinuous in form. (though, of course, con- oad in value) as we cross the lines @,=0, w.=0; the lines O0=r+a, 0=7—a. These lines, with the ‘iiteacties plane, divide space into three regions, A, B, and © (fig. ZA Fig. 2. A, @, and @, both positive. B B, w, positive, w, negative. C, w, and w, both negative. § 5. Now for an arbitrary incident wave we ce write bi = /() 2 =7 fed Wee PVA) provided the form of f is such that Fourier’s Double Integral may be applied. The solution for the region A then “becomes : ee a he mf a a {70 dn ihe Ss Qikw,? sec* ey ay at + similar term in‘y and @, . s)yeun 2 meum ala ons =1(8)-},| iy | if foyer Be ik(B—2o?seP? Y—d) Tee f(8)—- ve * (8 — 2eo,? sec? We) dw Fan { -2 LG f(y ~2002 sec? arp} Eis o1) + {f(y)—g9(y, @2) J. + eS eta (22a) Similarly for region B, b = f(B)—9(B, Or) FY, @2) 3 + + (226) @ = 9B; O19 (Y, @2)- ee c) The extension of the method and results to the case where the incident waves travel in a direction not penpendieivas to the diffracting edge is easy. London, Feb. 9th, 1920. and for C, Ra eee oar LXVIILI. Onthe Mathematical Relations Hee Magnetic Field. By G. H. Livens * T has already f been noticed that the usual conception of the mechanical relations of the magnetic field, which is based on the assumption that the magnetic force is the fundamental vector, is somewhat misleading, if not entirely erroneous, and that in order to obtaina more consistent theory of these relations it is necessary to invert the usual order of things and take the magnetic induction as the fundamental force vector and the magnetic force as the induced vector. The object of the present note is to present in a concise form the results obtained by following through the usual argu- ments but starting with the more consistent fundamental conception. In all physical theories it is usual to describe the relation between the inducing force and the induced effect by saying that the latter is an explicit function of the former. The inversion of sucha description, which would give the inducing force as an explicit function of the induced effect, may not necessarily be incorrect, but it is at least inexpedient, as it is very liable, as in the present case, to lead to a serious misapprehension of the physical processes involved. The mathematical relations of the magnetic field are always expressed in termsof three vectors: (1.) the magnetic force 1, which is defined at any point of the body as the vectorial ratio to a small magnetic pole of the force on it, if placed there ; (1i.) the magnetic polarization intensity I, which is the effective resultant bi-polar moment per unit volume of the medium at each place ; and (iii.) the magnetic induction vector B, which in the elementary theory is best regarded as a composite vector induced by the force H, and such that B= H+47I. In the more consistent form of these relations the magnetic induction B is taken as the fundamental force vector and the magnetic force H as the induced vector, and then it is better to write H = B—4yrlI. * Communicated by the Author. + The complete development of the usual argument, with complete references, will be found in my hook ‘The Theory of Electricity’ (COU. Press, 1918). 674 Mr. G. H. Livens on the Mathematical It appears, then, that the vector H is in reality the mechani- cally effective part of the total magnetic force from which the local part in the polarization has been rejected, and it will therefore be derived from a potential under the usual appropriate conditions. If the magnetization I is partly rigid (Ij) and partly induced (1,;) by the total force B, and if the induction follows a linear isotropic law, we should write I; == KB; d and then H = pB—47]), wherein a re The new “permeability” coefficient here introduced, which expresses the permeability of the medium to mechani- cally effective magnetic force, is practically equivalent to the reciprocal of the ordinary permeability, and it is actually equal to this reciprocal when there is no permanent magnetism in the medium concerned *. This inversion will naturally lead toa revision of our physical ideas on magnetic induction ; for it is now the induction of mechanical and not electro- motive force that is under review. In paramagnetic media « is positive and less than 47, so that w<1, and it is very small for the strongly ferro- magnetic media near their saturation point. Free space is thus the most permeable paramagnetic substance, and the ferromagnetic media are almost impermeable. In diamag- netic media « is negative, so that ~>1; so that here the permeability is still larger than it is in free space. Let us pursue these relations further. The energy required to establish the permanent polarity of intensity I) at each point of space is a Io a ( - ( (Bdl,), the volume integral being extended throughout all space. The usual argument proves that in the statical case 2 Io al; _ a) (BdIy)= gn. | Brae +f ao| (Bal). * The fact that the relation between the permeability as usually defined and that here employed is a function of the permanent mag- netism must not be regarded as detrimental to the present suggestion. It may be due entirely to the older definition. . Cr Relations of the Magnetic Field. 67 The second term on the right represents the internal elastic - energy stored in the media on account of the magnetic polarity inducedinthem. The first part therefore represents the true magnetic potential energy of the field, and on a tentative theory we could regard it as distributed throughout the field with the density at any place. In the case of the linear isotropic law of induction, we have He pba. l= B ; so that ol; = «OB, 1; 2 and thus ( (Baie = = The total work done in the field is thus aloes) : selene Seras Dh Ly eae = -| Ba dv. These relations are somewhat simpler than those obtained on the only consistent form of the older theory, as the permeability is again restored to the numerator of the expression for the energy and in addition the unknown local terms are entirely absent. When the field is statical and the magnetic force proper possesses a potential function, the characteristic equation satisfied by this function is slightly different from that usually given. The induction vector B is always circuital, so that EES OE, Be. div = ert SF a tear 0, and when the mechanical force H is derived from the potential @ we have B= —grad ¢+47l, 676 Mr. :G. H. Livens on the Mathematical and, further, if the polarization is induced according to a linear isotropic law, B= —egrad d+4Ip. Thus finall ius finally, div C orad ) a dor div Jy Ve is the characteristic equation satisfied by the potential ¢. In homogeneous media this is equivalent to Vd = Ar div ile The surface condition at a bounding interface, corre- sponding to this equation, involves the continuity of 'B: == (82 +4nl, i be on n It used to be assumed that a complete analogy exists between the electrostatic and magnetostatic fields, the electric force, complete electric displacement, and electric polarization corresponding to the magnetic force, magnetic induction, and polarization respectively ; but it has already been pointed out that no such complete analogy exists. With the suggested modification in the conception of the permeability the analogy again presents itself as a pos- sibility, but it is by no means perfect, although it is more complete than is otherwise possible. There is, of course, no reason why the two sets of relations should be analogous, as the two cases are quite distinet, and the similarity in the expressions for the energy is obtained only when the one is treated as potential energy and the other as kinetic energy. From another point of view it appears that a most unfortunate mistake has been committed in starting with magnetism in a statical theory with a potential function. The more correct procedure is to begin with the magnetic induction as the true ethereal magnetic force and then to separate out from this vector the mechanically effective part, the magnetic force proper, which is derived from a potential function under the appropriate circumstances. By proceeding in the other way we have been led to assume, among other things already indicated in sufficient detail, that the magnetic force is the proper vector to use in such expressions as the complete electromagnetic force on a moving charge element, and when we have then found that in ferromagnetic media Relations of the Magnetic Field. 677 such phenomena as the Hall effect or the various magneto- — optical effects turn out to be always proportional not to the magnetic force but to the induction (or polarization, the two are practically the same in such cases), we have coneluded that the behaviour of such media in this connexion is anomalous when, as a matter of fact, it is perfectly regular and provides one of the most powerful arguments in favour of our present contention. Finally, we may notice the way in which the present permeability coefficient j enters into the fundamental equation for wave propagation in the electromagnetic field. The fundamental equations in such cases are : fa Be 1 dD Corr curl E, ree curl H, where ee AB) pena These lead in the usual way to the wave equation e d’h ViPS Te ae which is satisfied by each cae of the two vectors defining the field. The velocity of propagation is now so that the magnetic permeability acts counter to the dielectric capacity, an increase in the one being negatived by a corresponding increase in the other. But we must remember tuat the Jarger values of p correspond to free space or diamagnetic media and that a “strongly mag- netic ’ medium in the ordinary sense has a small permeability coefficient according to the present definition—or in other words, in strongly magnetic media the mechanical effective- ness of the complete force (the induction) is: practically destroyed by the induced polarization which preduces a counterbalancing local forcive. The University, Manchester, Jan. 20th, 1920. [vocal LXIX. On the Cadmium- Vapour Are Lamp. To the Editors of the Philosophical Magazine. GENTLEMEN,— N the March number of the Philosophical Magazine Mr. F. Bates contributes a paper on a new cadmium- vapour arc lamp, in which a similar lamp described by me about four years ago (Proc. Phys. Soe. xxviii. p. 94, 1916) is subjected to adverse criticism. It is stated that “ the im- purities introduced into the lamp ”’ by my method of filling “effectively prevent obtaining a relatively pure intense cadmium spectrum.” It is not clear what is meant by this statement. The only impurity which I suggest introducing is zirconia, and zirconia sufficiently pure not to affect the spec- trum of a cadmium-vapour arc is neither diffieult to obtain nor to prepare. If the impurities are “‘ introduced ” in the process of filtering the metal in a vacuum, the fault obviously lies with the operator and not with the method. If, however, what is meant, is that metallic impurities such as zine are not removed, this is obviously true, but holds also for the method of distillation recommended by Bates. The only satisfactory method of avoiding such impurities is to start from pure cadmium. It is further stated that the zirconia does not sufficiently prevent adhesion of the metal to the class, and that “ vif the lamp does not crack upon the first solidification’... . “ upon cooling a second time the lamp was invariably cracked.” In reply to this, I must point out that Ido not remember a single case in-which a lamp which I have filled myself has failed from this cause. I have also obtained evidence from two independent users of my lamps. One user states “I have had one of your lamps in continuous use since 9/6/16... It is still working,—it has been in very frequent use, upon some occasions for 2 or 3 hours on end; the glass has now become somewhat opaque to the shorter wave-length radiation. Other lamps that we have used have not had the same length of life as the one I have used myself, but the breakdowns have not been due to ihe cause stated by Mr. Bates.” The second user states that his lamp has been in use for about three years. “It has been heated up at least 500 times, and has burnt for over 600 hours.’’ It is obviously not possible for me to state positively what were the reasons for Mr. Bates’s failures. Insuffieient thickness of the quartz-glass or too coarse zirconia might be On Mass, Energy, and Radiation. 679 suggested, but the balance of internal evidence in the paper points to faulty pumping as the cause. In filling lamps on a Toepler pump, I always leave the lamp evacuated at the pump in connexion with a large P,O; tube for at least 24 hours before running in the metal in order to be sure of the removal of water vapour. I also usually cast the cadmium into sticks by suction into a heated glass tube fitted with a tap in order to be sure of the absence of blow-holes. The process of filtration, if properly carried out, is quite sufficient to free the molten metal from solid extraneous matter such as oxide, and I should certainly have considered the operation faultily performed if I had ever noticed traces of oxide in the lamp such as Mr. Bates reports he occasionally observes. The peeling off of “thin sections of the quartz from the walls,” reported by Mr. Bates, certainly points to the for- mation of a cadmium-glass due to oxide as the result of water vapour, and not to the action of adhering metal. IT am, Gentlemen, The Sir John Cass Technical Yours faithfully, Institute, London. ; " March 20th, 1920. Henry J. 8. San. LXX. Mass, Energy, and Radiation. By). J. Lnomson, O.M., P.h.S.* HE object of this paper is to endeavour to supply a method of representing in terms of physical conceptions the processes occurring in physical phenomena, It is an attempt to help those who like to supplement a purely analytical treatment of physical problems by one which enables them to visualize physical processes as the working of a model ; who like in short to reason by means of images as well as by symbols. The ideas on which the method is based were suggested by the consideration, from the electrical point of view, of the origin of the mass of an electron. From this point of view this mass is distributed throughout the region sur- rounding the electron, and for an electron at rest the miss per unit volume at any point in this region is proportional to the square of the electric force at the point. The electro- static potential energy per unit volume at this point is also proportional to the square of the electric force and is thus proportional to the mass. In fact (see ‘ Electricity and Matter, J. J. Thomson, Chap. 2) the electrostatic potential * Communicated by the Author. 680 Sir J. J. Thomson on energy is equal to the kinetic energy which the mass would possess if it moved with the velocity of light. This result suggests that the potential energy in the electrostatic field is really the kiaetic energy possessed by the mass which is distributed thr oughout the field, the mass being regarded as an aggregate of equal particles each one of which moves with the velocity of light. In a stationary electric field we may suppose that these particles revolve with this velocity round the lines of electric force, much as the electrons from a hot wire can be made to revolve, though at a slower speed, round lines of magnetic force. It seems natural to generalize this result and to suppose that all mass, that of atoms as well as that of electrons, is distributed through space with a density determined by the electric field at the place where the mass is supposed to exist; and that energy of every kind, kinetic, potential, thermal, chemical or radiant, is of one and the same type, being the kinetic energy possessed by the particles which are supposed to constitute mass, these it is assumed always move with the velocity of light. On this view there is no such thing as the transformation of energy, if by that we mean a discontinuous change from something of one kind into something of another ; on our view the transformation of ener oy is mer rely the flow of the mass particles from one place to another. Thus for example, on this view when a body gains kinetic energy, it is not because any of its mass particles are moving faster; it is because the mass of the body has been increased and the increase in the mass implies a proportional increase in the ener gy. It will perhaps make it clearer if we follow out in detail this process in a special case—we will take that of a moving electron. When an electron is moving relatively to the bodies around it, the lines of electric force which start from it are no longer uniformly distributed in all directions, those running in directions at right angles to the direction of motion of the electron get more concentrated, and those running parallel to this direction more diffuse. The total number of lines starting from the electron is unaltered by the motion and depends only upon the charge on the electron. Since the mass per unit volume at any place in the neigh- bourhood of the electron is proportional to the square of the number of lines of force passing through uni! area at that place, the amount of mass between two spheres with their centres at the electron and whose radii differ by unity, will be proportional to | N’dS, where dS is an element of the area Mass, Energy, and Radiation. fe of surface of one of the spheres, and N the number of lines of force passing through unit area of the sphere. Since the charge is given, \|NdS is fixed, and when this is go, \N?*d5 will be least when N is unitormly distributed over 8, and a other cases the excess over the minimum value will crease with the amount by which the lines of force are Cee ated in definite directions. The greater the velocity of the electron the greater is this concentration and there- fore the greater the value of ee dS, i.e. the greater the value of the mass in the region close to the electron. Thus the moving electron has more mass in its immediate neigh- bourhood than an electron at rest, and as each unit of mass possesses, since the mass is moving with the velocity of light, a definite amount of energy, the energy of the moving electron will be greater ina that of an electron at rest. This increase in energy is what is usually called the Kinetic Hneregy of the moving electron. It is necessary to say a few warde about the detaidion and measurement of kinetic energy. When, as in ordinary dynamics, the Ane energy of a body is defined by the expression $mv?, it depends essentially upon the axes with respect to w ‘hich the velocity is measur red, the kinetic energy of the same body may be increasing w her measured with reference to one set of axes and decreasing when measured with reference to another. The changes, howe ever, of the total kinetic energy in a self- contained system, 2. e. one which is not acted upon by any external forces, il if action and reaction are equal and opposite, be independent of the axes used. What may be called the localization of energy, 7. ¢. the assignment of a certain amount of energy to each seat of a dynamical system, is a problem which, as far as rigid dynamics goes, has an unlimited number of solutions: any one of thes solutions will give the same chanves in ie configuration of the system as any other, so that Ae io eatou of energy could not be deduced without ambiguity from observations of the configuration of the system. On the method considered in this paper, the energy associated with an electron, for example, could be determined independently of any axes of reference if we hac the power of counting the individual mass particles in its vicinity. We know, however, of no physical phenomenon which will enable us to do this, all that with our present knowledge of physies we are able to do is to compare the number of mass particles in one region with that in another, and this will Phil. Mag. &. 6. Vol. 39. No. 234. June 1920. 2 ¥ 6 WD 32 Sir J. J. Thomson on f make the measurement of the mass of an electron, for exainple, depend upon the position of our measuring instru- ments.. We may illustrate this point in the following way.. Suppose we have a region A in which all the atoms and electrons were initially at rest relatively to each other. Now suppose that under electrostatic attraction an electron gets set in motion. Hrom our point of view this means that some of the mass particles which initially were remote from the electron have come much closer to it; this will produce an increase in its mass, and from the equations ot electro- dynamics we can calculate the ratio of the increased mass to the mass of the electron when it started from rest ; we can also, even if every constituent atom or electron of the: system gets set in motion under the electrostatic attraction and the mass of each is in consequence increased, calculate the ratio of the increased mass of each constituent to its original mass. Suppose, however, that the whole region A gets set in motion as a rigid body by the action of an external system B ; while the ve locity of A is increasing the mass particles will _be st rece into it, and while this is going on it is possible that the relative masses of the censtituents of A may be affected. But when the velocity of A has become steady and there is no longer any influx of mass particles into it from the outside, the particles which have come into it while this state was being reached will distribute themselves so that the number of new particles in any region is proportional to the number that were present before the influx. Thus - the relative masses of two constituents of A, say an electron and an atom, will be unaltered. Thus an observer in A will be unable to detect any effect due to a uniform motion of translation of this region, for though the mass of one of the constituents, as measured by the number of mass particles associated ath it, inay be altered, the mass of the unit by which that of the. constituents is measured will be altered in the same proportion, so that the alteration will not be detected. The argument is the same as that which applies to any changes which the motion may produce in the shape or size of the constituents of the region A; these escape. detection by an observer in A because his units are altered in the same proportion asthe quantities measured. If, how- ever, we had any method of counting the mass particles within the region A, an obseryer in dine region ought to be able to detect an aac due to changes in the velocity of translation. | Again, if an observer im a region © which didi mot Mass, Energy, and Radiation. 683 participate in the motion of A had the means of comparing the mass of an electron in his. region with that of one in A, he would find that the ratio depended on the velocity of translation of A. Following the ideas suggested by these illustrations we get what I think is a consistent scheme fer visualizing physical processes, if we assume the existence :— Of particles all of the same kind and with the same mass. ‘These particles all move with the velocity of light. Since. the mass particles are moving with the velocity of light they would on the Lorenztian transformation have this velocity whatever might be the axes to which their motion was referred. Any force on a particle due either to other particles or to the electric field is always at right angles to the direction of motion of the particle. Thus, though a particle may be deflected its velocity remains unaltered. The mass of one of these particles must, as we shall see, he exceedingly small compared with that of an electron. All mass, whether of electrons or atoms or radiant energy, arises from the presence of these particles, and inasmuch as each particle _possesses an invariable amount of energy, wherever there is mass there is an amount of energy pro- portional to it. The distribution of these particles and their movement from one place to another is determined by the distribution of the lines of electric force. For we assume that in addition to the mass particles we have in the universe : 2. Lines of electric force spreading through space. These lines may be closed or they may begin or end at definite points. These points are the seats of what we call electrical charges, the electron being at one end of a line of force and a unit of positive electricity at the other. Each electron and each unit of positive electricity forms the end of an invariable number of lines of electric force. The connexion between the distribution of the mass pau and the lines ‘of force is given by the rule that the mass per unit volume at any point *P is —— to frapsis Morea). which is also proportional to the energy per nnit volume. f, 9g, h are the number of lines of force passing through a unit area at Pat right angles to the axes of «, y, 2 respectiy rely : a, 8, y are the components of the magnetic force. ¢ is the velocity of light through a vacuum. ze? 684 Sir J. J. Thomson on ~ We regard magnetic force as due to the motion of the lines of electric force past the observer who is measuring the magnetic force. The relation between the electric and magnetic force when all the lines of electric force at P are moving with the same velocity is given by the equations a=An(qu—hv); B=4ar(hu—fw) ; y=4r(fo—gu) ; when w, v, ware the components of the velocity of the lines of Blociiee force relative to axes fixed with reference to the observer of the magnetic force. From this equation combined with the expressions for the energv per unit volume, we see that P, Q, R, the com- ponents of the momentum per unit volume at P are given by the equations 1 5 Laon ek ly be alg » eae iE : e Pee a 7) ; Q= , Ay La) ; R= 7 (Ya—X£) 3 where X, Y, Z are the components of the electric force. We can also, by the principle of varying action, deduce from the expression for the value of the energy the Maxwellian expressions for the stresses in the electric and magnetic fields which reproduce the mechanical forces existing in those fields. From the expression for the energy in the electric field, we see that the mass particles are concentrated in the places where the electric field is strongest. Thus when the electric charges are electrons or positively charged units of exceed- ingly small dimensions—when, in consequence, the electric force is exceedingly strong close to the charge—by far the greater part of the mass will be quite close to the charge. Thus, for example, if the radius of an electron is 10~™ em., only one thousandth part of its mass will be at a distance from the electron greater than 107" cm. Thus, though the mass particles are present wherever there is an electric field an enormous majority of them cluster close round the electrons and positive charges. The mass particles perform the functions both of zther and matter. They perform the function of matter by en- dowing the electrons and positive charges found in the atoms of the chemical elements with mass, and when they are moving through space and carrying energy with them with the velocity of light they are performing functions usually ascribed to the eether. By themselves the particles are not the whole, either of matter or of ether, for lines of electric force are an integral part both of ether al matter. We only get matter when we : ae Mass, Energy, and Radiation. 685 have lines of electric force anchored on to electrons or the units of positive electricity ; we only get radiation when we have along with the mass particles, closed lines of electric force. The distribution and movement of the lines of electric force determine the distribution and movement of the mass particles. Comparing the physical universe with a living organism we may regard the mass particles as the flesh, the lines of electric force as the nervous system. Muss ud energy are contributed by the mass particles, but the distribution, localization, and moyement of both mass and energy are determined by the lines of electric force. ‘he mass particles in a steady electrostatic field, though moving with the velocity of light, are constrained to follow closed paths round the lines of electric force. This produces i. tension along the lines of electric force, and these are only prevented from breaking away by being anchored to the electrons and positive char ges, and so being obliged to drag about with them wherever they may go the masses con- densed about these charges. In a steady electrostatic field all the lines of electric force have their ends either on electrons or positive charges, none of these lines form closed eurves. When, however, the electric field is changing, either by the motion of the positive and negative charges or otherwise, the lines of electric force may get looped, and some of them may form closed curves. These closed curves are not anchored to electrical charges, there is nothing to prevent the mass particles from dragging them away ; thus the mass particles will travel out through space with the velocity of light through a vacuum, dragging after them the closed lines of electric force. This, on the view we are considering, ts the way in which radiation is supposed to originate. Since both the energy and mass are due to the mass particles, we see that, on this view, radiation involves a- transference of mass proportional to the trans- ference of energy. ‘The speed with which the radiation travels is the speed of the mass particles, this speed is in- variable and equal to the velocity of light through a vacuum: it is independent of the medium through which the particles are travelling ; the velocity of light, however, depends upon the medium, and we have to show that an invariable velocity of the mass particles carrying the energy is consistent with the variation in the velocity of light with the medium through which it is travelling. When a wave of light passes through a refracting medium the electrons in the medium oO are edt in vibration and give out secondary waves ; the effect 686 Sir J. J. Thomson on - of these secondary waves is to make the apparent velocity of the light through the medium depend upon the number of electrons in that medium, though all the constituents which make up the resultant wave travel with the velocity of light through a vacuum. A detailed analytical investigation of this effect will be given in another paper, but the general principles on which the results depend may be illustrated by considering the special case of a pulse of electric force travelling through a slab of refracting matter bounded by planes at right angles to the direction of propagation of the pulse. Let_us suppose _that the electric force in the pulse is parallel to the axis of « and that the pulse is travelling parallel to z and bounded by two parallel planes at right angles to z. Let the thickness of the pulse be 2d, let the electric force in the pulse before | it strikes the slab be constant in the front half d and equal to X, while in the rear half it is also constant but equal to —X. Let us consider the effect produced by this pulse when it strikes the slab of electrons. When the force X strikes the electrons it will accelerate them, and in conse- quence they will emit secondary waves in which the electric _ force is in the opposite direction to X. Fig. 1. o Let AB be the slab containing the electrons and suppose the front of the pulse has reached P; the only secondary radiation which has had time to reach P is that coming from the electrons at B, the part of the slab nearest to P. When a little later a part of the pulse a little in the rear of the front reaches P, the secondary radiation carrying a negative X will have had time to come up from outlying places like C and D, and will diminish the electric field in this part of the pulse. Now consider what will happen when the H a : 7 Mass, Energy, and Radiation. : 687 . positive half of the pulse has just passed P and the negative half is just arriving. At first the secondaries which arrive at P will be those excited by the front and positive half of the pulse, and the force on them will be in the negative direction, 7. e. in the same direction as that in the part of the primary pulse which is now arriving at P, and thus the secondaries will increase the magnitude of the electric force in the pulse. After a time the secondary radiation excited by the negative part of the pulse will begin to come up; the force. on this will be in the positive direction, and will diminish the intensity of the force in the primary pulse. The secondary radiation from outlying regions will continue to arrive at P after the primary pulse has passed, so that the primary pulse will have developed a tail. Before passing through the slab the distribution of electric force in the pulse would be represented by a graph like @ (fig. 2), while after passing through the slab it will be represented by @. Wige2; ne | F We see that the result of passing through the slab has - been to diminish the energy in the front half of the pulse, , to inerease it in the rear half, and to develop a positive tail y. Now let this modified pulse go through a second slab ; the energy in the front half will be still further dimin- ished, the energy in the rear half and in the tail y will be ‘increased, and another tail of negative force 6 will be developed. ‘This process will go on as the pulse passes through other slabs until the energy in the front part is reduced to insignificance and the second half of the pulse will be the active front; this will in its turn be worn down by the same process and the tail y will take its place, this will be succeeded by the tail 6, and so on. In this way the virtual front of the pulse is continually falling behind the place which the pulse would have reached if it had not been passing through the slabs of electrons, and the amount by which it lags behind will depend on the density of the electrons 685 Sir J. J. Thomson on in the slab. Thus the velocity of the pulse through the medium containing the electrons will difter from that through empty space and will depend upon the nature of the medium, in spite of the fact that all the radiations which make up the pulse travel with the velocity of light through a vacvum. Hence we see that the constancy of the velocity of the mass particles which carry the energy and mass of light is con- sistent with heht travelling with quite a different velocity when passing through a refracting medium. On the view we are discussing the radiation as it were carries its ether along with it. The medium which carries. the radiation is not something uniformly distributed through space but fragments torn from matter, carrying along with them lines of electric force as an integral part of the radiation, Though this theory of radiation may be described as an emission one, yet since the velocity of the mass particles is invariable nie velocity of hght will not be affected by the motion of the source, or when the light is reflected, by the speed of rotation of the mirror. Experiments : -ecently made by Majorana are in accordance with this result. Since on the view we are discussing energy is made up of a number of equal units, the transference of energy from one body to another must take place by definite steps, and no transference is possible unless the amount to be trans- ferred exceeds a finite amount. This involves that the dynamics of processes involving very small transferences of energy must differ fundamentally from ordinary dy namics. We are not yet in the position to calculate the mass of one of these mass particles, but it is certain that it must be an exceedingly small fraction of that of an electron. Tor the energy of an electron is about 107’ erg, which can be represented by the fall of the atomic charge of electricity through about 6 x 10* volts. Now the average energy of a molecule of a gas at 0° C. corresponds to the fall au the atomic charge through a potential difference of about =), of a volt. Hence if the mass of a mass particle is w times the mass of an electron, the smallest amount of energy which eould be tri anshorred from one body to another would be about 1°8 x 10° x @ times the mean energy of a gas molecule at the temperature of 0° C. Now suppose a gas is raised from absolute zero to a higher temperature, if each molecule of the gas receives the minimum amount of energy pos-ible, co) the temperature of the gas would be raised to— 1:8 x 10° x w x 273 absolute; Mass, Energy, and Radiation. 689: when the temperature is less than this only a fraction of the molecules will acquire any additional energy from the rise in temperature. When a large number of molecules have acquired no additional energy at all, it would seem improbable that any large number “should have e aequir ed the extra energy corresponding to additional degrees of freedom, for example, for a diatomic molecule to have acquired the energy due to its notion round the centre of mass as well as that corr esponding to energy of translation, but unless it did this the el heats of diatomic gases at temperatures less than 1°8 x 10° x @ x 273 absolute, would approximate to those of monatomic Seer this consideration shows that c. must be less than 10~*. Again, we know from Michelson’s experiments on the green line of mercury that the source of this line can give out more than 400,000 vibrations without abrupt change of phase; from Planck’s rule, the energy in this radiation is that due to the fall of the atomic charge through a potential difference of 2°5 volts, 7. e. 1s about 1/(2: Ix 10*) of the energy of an electron. If there is only one mass particle per wave-length of the radiation, there will be more than 4X10’ mass particles in this amount of energy, so that the energy of one of these particles will be less than 1/(2° 4x 10!x 4x 10°) of that of any electron. Since the ratio of the masses is the same as that of the energies we conclude that at least 10!’ and probably many more mass particles are required to supply the mass of an electron. If energy is indivisible beyond’a certain limit, then the inverse square law of electrical attraction cannot hold at all distances. For when this law holds, the energy outside a sphere of radius 7 with its centre at an electr on, bears to the energy of the electron the ratio a/r, where a is the radius of the electron; hence if a/ris less than w the energy outside the sphere will be less than the energy possessed by one mass particle. Thus since the particles are indivisible there would be no particles and no force when r is greater than a/w, so that the law of electric force cannot be the inverse square law over more than a certain finite distance. F000 2] LAX. On the B-Recol, By ANTELA MuszKaT, Radiological Laboratory, Warsaw * fee experimental evidence concerning the 8-recoil radia- tion, as resulting from former publications on this subject, is in some points insufficient. Makower f and Russ established the existence of the phenomenon, but there are some doubts concerning the exceedingly small recoil efficieney observed, and the impurity of the recoil product. Makower and Russ have tried to explain their results by supposing that the active deposit forms agglomerations on the surface of the activated bodies,.these agolomerations exercising a strong absorption on the recoiling atoms. This hypothesis is not improbable in itself. The mechanism of the deposition of active matter may resemble the condensation of vapour on a cold surface, in which ease there are small droplets formed instead of an uniform layer. However, if the quoted authors are right, then the deficiency ought to depend upon the mean surface density, being greater for the less active surfaces. As this does not occur, one must search for another explanation. Mr. Wertenstein suggested to me that the absorption of the recoiling atoms may be due to two different causes: (1) an adsorption of a gas layer on the activated surface, the electric transport of the gaseous ions during the activation contributing greatly to this adsorption; (2) the penetration of the active deposit by diffusion into the mass of the activated body. The only possible way of eliminating these presumed causes of error seems to be in obtaining a layer of active matter by distillation in vacuo, and examining the recoil from this layer immediately after the distillation. In the present work I adopted a method based on this idea. A platinum wire, made active with RaB+C in the usual way, was placed very near the surface to be coated with distilled RaB. By the use of a ground-glass piece, this surface { could rotate through 180° incl; in this way direct its recoil stream ona receiver. The apparatus I used at first was very simple, but an unexpected difficulty compelled me to make it far more complicated. It was hoped that the “cold disk” would act asa perfect screen to the distilling matter, the atoms of RaB travelling in vacuo on straight lines and being retained—as I thought—by the disk, after they * Communicated by Dr. L. Silberstein. + Phil. Mag. vol. xix. p. 100 (1910). + Called in “this paper for the sake of briefness, “the cold disk.” eg phe P : 2 —; a —< ae” sl xX = . VS ‘ a ae ; Miss A. Muszkat on the B-Recoil. 691 had struck it. Experience showed, however, that the whole apparatus, in spite of the high vacuum obtained, vecame filled with the distilling RaB+C, which was present pro- bably in a gaseous form, and the receiver was contaminated with active matter before being exposed to the recoil stream. Diaphragms of different forms. proved useiess. This slows that at ordinary temperature a metallic surface does not possess the faculty of retaining an atom of RaB or RaC after a single shock. Similar Faerss were observed by Knudsen * in the case of the condensation of mercury and other metallic vapours on glass walls. No doubt, the results could be improved by cooling the surface to a very low temperature, but it would be a very troublesome operation, and so I tried to avoid this difficulty by disposing the receiver in a part of ; the vessel that could be separated hermetically from the other parts of the apparatus during the distillation, and then set in communication with them ‘during the recoil experiment. For this purpose a device was used similar to that adopted by Mr. Werteustein for his exposure vessel. The part of the apparatus A (cf. fig.) containing the active wire and the cold disk had at its top. a support on whieh reposed the bottom of the little cylindric vessel B, containing the receiver D. The two vessels had concentric openings of the same diameter (12 mm.), so that by displacing the upper vessel a perfect separation of A and B was obtained. The ‘‘ cold disk”? M was fixed on a cylinder rotating on a horizontal axis by means of the ground-cock 8. The wire Pt, 0'1 mm. thick, was fixed on a ground-cock §,, which also fitted well in the exposure vessel. It was stretched in the form of a loop between two thick platinum wires which passed through an ebonite plug. These wires enabled me to bring the thin wire to the high potential required for activating it in the exposure vessel and also to heat it by an electric current for distillation purposes, when in the recoil apparatus. The tube 7 connected the apparatus to the Gaede molecular pump and to a thermic manometer, described by Mr. Wer- tenstein f. The method of conducting the experiment consisted in heating at first thoroughly the wire in vacuo in order to clean the platinum and to remove the occluded gases. The wire was then made active by placing it for about 1} hours in the exposing vessel, containing several muillivcuries of radium * Ann. d. Physik, 1916, 1. p. 472. + L. Wertenstein, Thése, Paris, 1913. tak, Wertenstein, ‘Sur quelques procédés de la technique du vide.” C. R. de la Société Scientifique de Varsovie, 1917. 692 Miss A. Muszkat on the B- Recoil. emanation. After the activation, the wire was heated in a special apparatus in vacuo to remove the occluded emanation. The y-activity of the wire was then measured. Finally, after RaA had died off, the wire was introduced into the recoil vesse |. the upper part occupying the central position. The pump. was set in movement, until the highest obtainable vacuum (about 0-6 dyne/em 2) was reached. At this moment the upper vessel was displaced and an electric current sent through the platinum wire, heating it to a mean’ tempe- rature of 426°C., as measured by the variation of its resistance, controlled by means of an ampere- and voltmeter.. The current lasted for 1 minute, the cold disk (10 mm. in diameter) being maintained 1 mm. above the wire. It is worthy of notice that the temperature of the wire plays an important role in the subsequent recoil phenomena. If it is too low, the quantity of distilled matter is too small. On the other hand, it is useless to raise the temperature considerably above 426°, for although the disk B becomes more active, the: efficieacy of the recoil is diminished and falls to a very low value when the temperature is above 800°. This shows, as. oe ee ae ee Miss A. Muszkat on the B-Recoil. 693 I think, that even at these temperatures a slight volatilization of platinum takes place, being quite sufficient to cover the RaB atoms with an impenetr able layer. After the current is cut off, the disk M is turned to the receiver by a rotation of 180° ‘and the upper vessel is placed in the central position. ‘The exposure to the recoil lasts for five minutes. ‘The air is then introduced into the apparatus and the two disks M and D are removed. ‘Their activities are measured, the measurements being continued at regular intervals during an hour. From the analysis of the dis- integration curves I obtained the amount of RaB present on the cold disk at the beginning of the recoil experiment and the quantity of ItaC on the receiver at the end of it. If B be the former quantity, C the latter*, @ the solid angle subtended by the receiver (equal to 0 0336.47), then the efficiency E is given by ; ag Ca ee) S35 BB b(e7 oe ee It must be noticed that I obtained in all cases on the receiver almost pure RaC. The cold disk contained Rab} and RaC in proportions approaching those of the radioactive equilibrium between the two products, its activity amounted only to a small percentage of the activity of the active wire, thus showing the great dispersion, already mentioned, of the distilling matter. I found that the efficiency depended on the nature of the cold disk. It was equal to 0-2 for an ordinary brass disk, to 0-3 for the same disk after it was thoroughly polished, to O-4 for an aluminium disk, and to 0°5 for a platinum one. We see that in all cases the efficiency is of the order of unity, 7. e., that the conditions of my work are far more favourable to the study of the 8-recoil than the conditions used in previous work on this subject. he réle of the nature of the surface can easily be understood if one considers that the main obstacle to the expulsion of RaV is due to the penetration of the radioactive matter into the metal and probably also to chemical actions between this matter and the substance of the disk. That RaB penetrates into the metal can be proved by the fact that the efficiency of the recoil diminishes with time. In one case, when the recoil experiment was conducted 20 minutes after the distillation, the efficiency was reduced to half its value. The electric field, when established between the disk M * The quantities calculated from the curves are not B and C, but, as usual, Bé and Ce. 694 7 Prof. R. Whiddington on the and the receiver, does not influence the phenomenon, preving that the B- saci atoms are uncharged, as was already stated by Makower and Russ. It seems to result from my experiments that there is a limit to @-recoil efficiency, this limit being approximately equal to 0°5. The probable explanation of the existence of such a limit may be given by considering the thermal velocities of the expelled. RaC atoms in connexion with their recoil velocity, as calculated from the mean velocity of the B-rays of RaB. If vg be the recoil velocity, 4 the mean velocity of thermal agitation of RaC at ordinary jeniperaiar el then ip—o.. |0* em./sec. ; v= 1-64.. 10* ems/sec. ep — a The two components being quite independent of one another, it results that the effective velocity of the recoiling atoms oscillates between largely differing limits, being respec- tively “equal to 4¢; and 2u;.. We can speak of the equivalent temperature of the recoiling RaC-atom, meaning by it a temperature at which this atom would possess a mean thermal velocity equal to its effective velocity. aking 290° abs. as the temper ature of the experiment, we find that the equivalent temperature oscillates between : 16.290° abs. =4640° abs. and 4.290° abs. = 1160°abs. = 887°C. The lower limit being of the same order of magnitude as the temperature of volatilization of radioactive matter, one can easily imagine that the corresponding kinetic energy may not be suficient to remove the RaC from the “cold disk.” We see in this way a reason for the 8-recoil impulse being in some unfavourable cases not strong enough to tear off the radioactive matter from the surface of the activated body. Warsaw, June 1919. The Radiological Laboratory of the Scientific Society of Warsaw. LXXIT. Note on the X-ray Spectra of the Elements. By R. Wuippineron, J2.4., D.Se., Cavendish Professor of Physics in the University of Leeds * OON after Barkla’s discovery of the L series of cha- ‘acteristic X-radiations, I put forward an empirical formula of the type c-=C.w-+ D which approximately repre- sented the K and L series of radiations of the elements f. * Communicated by the Author. + ome, Nov. 20th, 1911; see also Proc. Roy. Soc. 1912, yol. lxxxvi. ser. A, p. 3/8 a ee ee Ee Se Pe eee ee ee ee ee ee ee nie el w 7 > t - Pep ee ee ee eee eee oOo ee oe ‘yo ah tee = - ak he J X-ray Spectra of the Elements. 695 Since at that time methods of wave-length measurement had not been developed, it was suggested that the quality of X-rays should be defined by this formula in terms of the speed (v) of the electron carrying the same amount of energy as the X-ray under definition. These speeds had been ‘measured for a number of elements between alu- minium and silver, from which it had been concluded that for the K series v=wx 10° cm./sec., and for the L series = (5-25 5) 10" em./sec., where w=the atomic weight of the element. With the advent of crystal spectrometry and the accurate measurement of a large number of the X-ray spectra, Moseley * in two well- Tnown papers showed that atomic number was more fundamental than atomic weight and proposed the relation y= A(N —6)?, where vy is the frequency of the characteristic radiation (strong « line in the K series, strong « line in the L series), N is the atomic number of the element, and A and 6 are the series constants. It is one object of this note to point out that the relation v=C.w+D becomes Moseley’s relation if we put = 2 Neer teu? == Ji, whieh is the usual quantum relation. For then 2hy y= = (2C.N4D), )L or a ee DL ec. N+Dy?, Ph v= a(N+d)’, 2m? D ae 3 : where a= ae and d=s », Which is Moseley’s relation. , ae Reverting to v=C.w+D, which is now revived in the form v=CN+D, the following table shows the values of v for the K series & line from tungsten to sodium, approxi- mately calculated from $mv?=hy f. * Phil. Mag. 1918, vol, xxvi. p. 1024, and 1914, vol. xxvii. p. 712. t See Bohr, Phil. Mag. 1913, vol. xxvi. p. 499. + The values of A were taken from Tables given by Uhler, Phys. Rey. vol. ix. No. 4 (1917). ‘696 On the X-ray Spectra of the Hlements. TaBLE I. (CK series.) vx 107 8* Element. N. x10°. vx10-5. 2(°N-—2). (determined experimentally). W eee Vi 0-210 145 144 aS Nd. aie 60 0:323 115 116 — ihe heen ed 50 0-489 95:1 96 ue Trecd (age aie 40 0-791 15:01 eno oy See crake ~ BE 1-107 63:4 64 73°8 Tis eed 30 1-435 556 56 63°2 Cre oF 24 2-286 44-0 44 50-9 Cae 20 3357 36:3 36 vas Naa ae a 11-951 18:5 18 mle cem./sec. for these calculated values of v. Column 6 gives the values of v determined experimentally for selenium, zinc, and chromium f. It appears that the experimental values of v in these three cases are about 16 per cent. greater than the calculated values—a result in general agreement with Webster’s { more recent experiments. Table II. is a repeat of Table I. for the L series from U to As, the 6th column being omitted owing to lack of data. TasLe II. (IL series.) Element. N. Ax 108, uxI0-*, (N= Io Wreeeen 92 0:720 78:4 a Rtg meeers ae 80 1049 65:0 65 SVaD an yacteea ss 70 1-474 54-9 5D ING ee nosam: 60 2:167 45'3 45 Sins oes 50 3:381 36:2 35 Tie ote ohare 40 5851 27°5 25 Asie cence 33 9:419 20°7 18 It will be seen from a comparison of columns 4 and 5 that for the L series v=(N—15) 10° em./sec. very approximately. The agreement is not nearly so close as for the K series. Summary. The speed of electron (v) carrying the same energy as the « lines in the K and L spectra of the elements is shown to be fairly well represented by the formula v=C.N+D, where N is the atomic number of the element, and C and D are constants determined by the series. For the K series es a: | vy = 2(N—2)10° cm./sec., and for the L series x Se eee : Ue NO" om see: * See Kaye’s ‘ X-Rays,’ p. 133, Srd edition (1918). + Whiddington, Proc. Roy. Soc. 1911. t Webster, Phys. Rev. 1916, vol. vil. p. 612. It appears that this 16 per cent. represents the difference between the a and y components. pO 4| LXXITI. Notices respecting New Books. Catalysis in Theory and Practice. By Eric k. Rrpean, Ph.D., and Hue S. Taytor, D.Sc. Pp. xv+496. London: Macmillan and Co., Ltd., 1919. 17s. net: : eu book comes in the fulness of time. Catalytic processes have played an important part in the development of chemical science for the last hundred years. The extent of the application of such processes in recent years is very remarkable. It is a pity that no satistactory explanation of this action is yet forthcoming. This is possibly, in part, owing to the wide scope of the term catalysis. Whenever a chemical change is accelerated (or the reverse) by the introduction of a foreign substance. the process is spoken of as catalytic (positive or negative, as the case may be). Now, in the multifarious cases that arise, itis not likely that the process is identicalin kind ; hencea corresponding multi- plicity of explanations is required. It is true that the authors attempt to restrict the field, and to lay down the requirement that the chemical composition of the catalytic agents is unchanged on completion of the reaction process. In some cases, however, of negative catalysis the retarding agents are not present at the beginfiing, and increase as time goes on; these hardly seem to fit in with the definition given. However, the definition covers what is denoted by the term positive catalysis very well. The authors do not commit themselves to any one theory, but outline several theories, each of which has something to be said for it. Concerning the thoria-ceria mantle which is still very much a conundrum, no explicit reference is made to Rubens’s theory that the poor general emissivity of thoria enables a high temperature to be re eached, at which the selective luminous radiation of ceria is extremely high (though not higher than that of a black body at the aforesaid high temperature). If this is right, the mantle is not an example of catalysis at all. Amongst the modern processes the hydrogenation of oils is of tremendous importance, due to the increase produced in our food- supply. Also during the war the necessity for ensuring a supply of nitrates tor explosives rendered imperative the creation of a fresh supply : just as in Germany the same imperative necessity had pre- sented itself before a war could be undertaken; and in both cases the utilization of the nitrogen from the air involved the use of catal ytic processes. These the authors are specially competent to deal w ith. We cannot here examine the book in detail. The writers deal in a masterly way with the problem. We conclude by quoting the final words of their own preface: “ Will the catalytic agencies be found which shall accelerate the velocity of atomic decay and render available the enormous stores of intra-atomic energy ? Such is the fitting problem for the years that lie ahead.” Ten British Physicists. By ALEXANDER MACFARLANE. Chapman & Hall. Price 7s. 6d. net. THE ten scientists whose lives are dealt with in the book before us Phil. Mag. 8. 6. Vol. 39. No. 234. June 1920. IT, 698 Geological Society :— are: J.Clerk Maxwell, W. J. M. Rankine, P. G. Tait, Lord Kelvin, Charles Babbage, Wiliam Whewell, Sir George Gabriel Stokes, Sir George Biddell Airy, J. C. Adams, and Sir John Herschel, The list does not, of course, include some of the most distinguished physicists of the nineteenth century, which is the period referred to on the title-page, and of those admitted most people would, perhaps, scarcely count Whewell, Babbage, Adams, or even Herschel, though he carried out well-known experiments on fluorescence, as physicists. This is, however, a mere matter of title, and the lives of the thinkers just mentioned are by no means the least interesting in the volume. Babbage, in particular, has sunk into ill-deserved oblivion: although his unfortunate tem- perament prevented him turning to full advantage his boundless ingenuity and great mathematical powers, his work on calculating machines was fundamental and has been of the utmost value to his successors. Probably, too, our present notation in the caleulus owes something for its establishment to Babbage’s “ Analytical Society,” which advocated “ the principles of pure d-ism in oppo- sition to the dot-age of the University.” Dr. Macfarlane’s short and sympathetic study tells us something of a stormy life, which is interesting to most readers. In the account of Rankine we are reminded ef the hypothesis of molecular vortices which the great engineer, so well-known for his work in thermodynamics, put forward with so much confidence, and of his poetical abilities. It is noteworthy that Whewell, Clerk Maxwell, and John Herschel were also accomplished versifiers. On the whole, these lines make very pleasant reading. They are, of course, short, and, being founded on lectures given to a mixed audience, do not devote much space to attempting to estimate in detail the value of the scientific achievements of the various men, or their precise place in scientific thought. The man who has read, for instance, Campbell’s ‘ Life of Maxwell’ will not find here anything new about that physicist. But the essays give a very readable im- pression of the life and personal character of the selected ten, and of the class of problems which they were engaged in solving. There is plenty of room for brief biographies of this kind, and the book should appeal to a large circle of readers. LXXIV. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from p. 384. | November 19th, 1919.—Mr. G. W. Lamplugh, F.R.S., President, in the Chair. peeve following communication was read :— ‘The Pleistocene Deposits around Cambridge.’ By Prof. John Edward Marr, Se.D.,- F.R.S., V.P.G.S. This paper deals with the deposits in the immediate vicinity of Cambridge, and contains new records of sections, fossils, and imple- ments. It is pointed out that, owing to alternating periods of erosion and aggradation, relative height above sea-level is not a Pleistocene Deposits around Cambridge. 699 trustworthy index of antiquity, and modifications of the classification proposed by W. Penning and A. J. Jukes-Browne are indicated. The author suggests the following chronological sequence, 1 descending order :— Feet, Gh) BarnwelllStatrommsedsvern. wy sss 20 (2) Newer Downing Site Beds ...............5........ 35 (3) Newer Barnwell Village Beds..................... 45 (4) Elumting-donemoadeClaysiescesastuwwcces ss estas. 70 @)- Obsenvatoryebedsi rm. ye W eek aids. 85 (6) Corbicula Gravels (Barnwell village, etc.) ... 30 The figures on the left give the approximate height above sea-level. It is believed that Nos. 6 and 5 were formed during a period of ageradation, and 4-1 during one of subsequent erosion with minor aggradation; but it cannot ine conclusively proved that 6 and 3 are of an ages, although the deposition of the beds 6 below those of series 3, W here they oceur together, and the occurrence of Hippopotamus and Belgrandia marginata with Corbicula suggest an early date for these Corbicula-bearing beds. Taking the beds inthe order of reputed age, the following observations are noted :— Chellean implements have been found at low levels at Barnwell and Chesterton, and may belong to the beds 1. The Observatory Beds have yielded abundant implements of Chellean, Acheulean, and early Mousterian types, the last-named apparently in deposits later than those containing the two first-named. Unfortunately mollusea and mammalia are very rare in these beds. The Hun- tingdon Road Clays require much further work, as only poor exposures have hitherto been found, and it is not clear that they are newer than the Observatory Beds. The beds referred to the Newer Barnwell Village Series contain abundant remains of the mammoth, woolly rhinoceros, and fairly numerous horse-bones. Implements associated with them suggest an Upper Paleolithic age. The Newer Downing Site Beds have yielded a cold molluscan fauna. They are probably somewhat earlier than the Barnwell Station Series, which has furnished a similar molluscan fauna, and also an Arctic flora, the plants of which were identified by the late Mr. Clement Reid. Reindeer occurs in these beds. The paper is chiefly a record of facts, but it is intended to be preliminary to a detailed survey of the Pleistocene deposits of the Great Ouse Basin, which are so important as throwing light upon the relationship of the Paleolithic beds to the glacial accumulations, and also to the marine beds of March and the Nar Valley. Appendix I, on the Non-Marine Mollusca, is supphed by Alfred Santer Kennard, F.G.S. and B. Barham Woodward, F.1L.S., F.G.S. Lists are given of the non-marine mollusca from the various sections, with their degrees of frequency. These lists are based on examination of old collections and on a large amount of new material. Notes are appended on some of the species, ‘and €on- clusions as to the ages of the Cambridge gravels are given, based on the molluscan evidence. Appendix II, on the Implements, is supplied by Miles C, Burkitt, 700 INDEX to VOL. XXXIX. a ABERRATION, on Stokes’s theory of, 164. Acoustical experiments, on a mecha- nical violin-player for, 535. Adeney (Prof. W. E.) cn the rate of solution of atmospheric nitrogen and oxygen by water, 385. AXther, on the recent eclipse results and Stokes-Planck’s, 161; on a possible structure for the, 170; on a fluid analogue for the, 651. Air, on the magnetic susceptibility of, 518; on the scattering of light by, 423. Alloys s, on the effect of centrifuging certain, 376. Anderson (Prof. A.) on Fresnel’s convection coefficient, 148; on the spheroidal electron, 175; on the advance of the perihelion of a planet and the path of a ray of light in the gvavitational field of the sun, 626. Anemometer, on the hot-wire, 505. Are lamp, on a cadmium vapour, 358, 678. Ayoon, on the mass-spectrum of, 620. Aston (Dr. F. W.) on the consti- tution of atmospheric neon, 449; on the mass spectra cf chemical elements, 611- Baly (Prof. E. C. ©.) on light ab- sorption and fluorescence, 565. Basu (M. N.) on the equation of state, 456. Bates (J*.) on a new cadmium vapour are lamp, 393. Beams, on the buckling of deep, 194. Becker (H. G.) on the1 rate of solution of atmospheric nitrogenand oxygen by water, 585. Bell (E. iD) on parametric solutions fer a fundamental equation in the general theory of relativity, 285, . Beta-recoil, on the, 690. Bickley (Ww. G.) on the diffraction of waves by a semi-infinite screen with a straight edge, 668. Books, new :—Hunter’s The Earth’s Axes and Triangulation, 239; Jeans’ Problems of Cosmogony and Stellar Dynamics, 377; Wood’s Researches in Physical Optics, 379; Karpinski, Benedict, and Calhoun’s Unified ‘Mathema- tics, 380; Lamb’s Hlementary Course of Infinitesimal Calculus, 381; Whitehead’s Enquiry con- cerning the Principles of Natural Knowledge, 629; Shaw’s Manual of Meteorology, 651 ; Stereochemistry, 632 ; ’ Rideal and Taylor’s Catalysis in ‘Theory and Practice, 697; Macfarlane’s Ten British Physicists, 697. Bragg (Prof.W. L.) on the crystalline structure of zinc oxide, 647. Buckling of deep beams, on the, 194. Cadmium vapour arc lamp, on a, 358, 678. Campbell (Dr. N. ) on the adjustment of observations, 177. Carbon, on the ‘excitation of the spectra of, by thermelectronic currents, 241; on the mass-spec- trum of, 619. dioxide, on the magnetic sus- ceptibility of, 322; on the specific | heat of, 551, Carrington (H.) on the buckling of deep beams, 220. Carslaw (Prof. H. 8.) on Bromwich’s method of solving problems in the conduction of heat, 603. Centrituge, on the separation of lead into isotopes by the, 372; on the action of the, on certain alloys, 376 Stewart’s a | | EN Dax: 701 Charged particles, on the dynamical motions of, 537. Chemical constants, on the deter- mination of, 1; on the significance of, 21. elements, on the mass-spectra of, 611. reactions, on theory and, 26. Chlorine, on the mass-spectrum of, 620. Coal-gas and air, on the radiation emitted in explosions of, 66, 84. Compton (A. H.) on radioactivity and the gravitational field, 659. Conduction of heat, on problems in the, 603. Contrasts, on the photographic ren- dering of, 15]. Convection coefficient, on Fresnel’s, 148. Cook (G.) on the inertia of a sphere moving in a fluid, 350. Copper, on the spectrum of, 457. Darwin (C. G.) on the dynamical motions of charged particles, 537. Das (Prof. A. B.) on the quantum theory of electric discharge, 233. David (Major W. T.) on the radia- tion emitted in gaseous explosions, 66, 84; on the specific heat of carbon dioxide and steam, 551. Davies (Miss A. C.) on critical velo- cities for electrons in helium, 592. Dey (A.) on the sounds cf splashes, 145. Diffraction of waves by a screen with a straight edge, on the, 668. Dynamical motions of charged par- ticles, on the, 537. Egerton (Capt. A. C.) on the deter- mination of chemical constants, 1. Electric current, on the equivalent ' shell of a, 1384; on the contri- butions to the, from the polari- zation and magnetization electrons, AO4. discharge, on the quantum theory of, 223. energy, on the difference between magnetic and, as a pres- sure, 662. Electrical circuit, on the period and decrement of an oscillatory, 553. conductivity, on the relation between illumination and, in sele- nium, 482, the quantum Electrical oscillations, on the mea- surement of the frequency of, 289. Hlectrons, on spheroidal, 175; on the contributions to the electric current from the polarization and magnetization, 404; on the dyna- mical motions of, 537 ; on critical velocities for, in helium, 592. Elements, on the X-ray spectra of the, 694. Energy, on mass, radiation, and, 679. Entropy-temperature diagrams of certain fluids, on the, 633. Equation of state, on the, 456. Everett (Miss A.) on a simple pro- perty of a refracted ray, 225. Hwing (Sir J. A.’ on the specific heat of saturated vapour and the entropy-temperature diagrams of certain fluids, 633. Explosions, on the radiation emitted in, 66, 84. Flint (H. T.) on applications of quaternions to the theory of rela- tivity, 439. Fluids, on the inertia of spheres moving in, 350; on the entropy- temperature diagrams of certain, 635. Fluorescence, on light absorption and, 565. Fokker (Dr. A. D.) on the contri- butions to the electric current from the polarization and magnet- ization electrons, 404. Gaseous explosions, on the radiation emitted in, 66, &4. Gases, on the relation of the che- mical constant to the behaviour of, at low temperatures, 21 ; on the magnetic susceptibilities of, 805; on the velocity of, in pipes, 005. Geological Society, the, 383, 698. Gravitation, on, 488. Gravitational field, on the propa- gation of light in a, 586; on the path of a ray of light in the, of the sun, 626; on radioactivity and the, 659. Green (Dr. G.) on a fluid analogue for the ether, 651. Hackh (Prof. I. W. D.) on a table of the radioactive elements, 155, proceedings of 702 Hammick (D. L.) on latent heat and surface energy, 32. Hargreaves (R.) on the difference between magnetic and _ electric energies as a pressure, 662. Heat, on latent, and surface energy, 02, 258; on the, radiated from a cylindrical wall, 359 ; on problems in the conduction of, 603. Helium, on critical velocities for electrons in, 592. Hemsalech (G. A.) on the excitation of the spectra of carbon, titanium, and vanadium by thermelectronic currents, 241. Hicks (Prof. W. M.) on the spectrum of copper, 457. Horton (Prof. I.) on critical velo- cities for electrons in helium, 592. Hot- wire 505. Hydrogen and air, on the radiation emitted in explosions of, 84; on the magnetic susceptibility of, 305; on the mass-spectrum of, 621. Illumination, on the relation between electrical conductivity and, in sele- nium, 482. Inertia of a sphere moving in a fluid, on the, 350. Jackson (L. C.) on variably coupled vibrations, 294. Jeffreys (Dr. H.) on turbulence in the ocean, 978. Joly (Prof, J.) on the separation of lead into isotopes by centrifuging, 372; on the effect of centrifuging certain alloys while in the liquid state, 376 Jones (I.) on the period and decre- ment of an oscillatory electric current, 552. Jones (L. A.) on the measurement of the visibility of objects, 96. Krypton, on the mass-specirum of, 623. Kunz (Prof. J.) on the scattering of heht by dielectric spheres, 416. Latent heat and surface energy, on, 32, 238. Lead, on the separation of, into isotopes, 372. Lewis (Prof. W. C. McC.) on the application of the quantum theory to chemical reactions, 26, anemometer, on the, INDEX. Light, on the scattering of, by di- electric spheres, 416; on the scat- tering of, by air, 423; on the pro- pagation of, in a gravitational field, 586; on the path of a ray of, in the gravitation field of the sun, 626. absorption and fluorescence, on, 565. Lindemann (Prof. F. A.) on the sig- nificance of the chemical constant, 21. Livens(G. H.) on the mathematical relations of the magnetic field, 673. Lodge (Sir O. J.) ona possible struc- ture for the ether, 170. Magnetic and electric energies, on the difference between, as a pres- sure, 662. -—— field, on the mathematical re- lations of the, 673. -—— induction of a circular current, on the, 154. susceptibilities of hydrogen and other gases, on the, 505. Majorana (Prof. Q.) on gravitation, 188 Malik (Prof. D. N.) on the quantum theory of electric discharge, 235. Marr (Prof. J. E.) on the Pleistocene deposits around Cambridge, 699. Mass, energy, and radiation, on, 679. Mass-spectra of chemical elements, on the, 611. Mercury, on the mass-spectrum of, - 623. Morten (Prof. W. B.) on the con- struction of a parabolic trajectory, 157. Muszkat (Miss A.) on the beta-recoil, 690. Neon, on the constitution of atmo- spheric, 449; on the mass-spectrum of, Ola Nitrogen, on the magnetic suscepti- bility of, 327; on the rate’ of solution of, by water, 385; on the mass-spectrum of, 621. Observations, on the adjustment of, Wie Ocean, on turbulence in the, 578. Oscillations, on the measurement of high-frequency, 289. Owen (8. P.) on radiation from a cylindrical wall, 359, INDEX. Oxygen, on the magnetic suscepti- bility of, 320; on the rate of solution of, by water, 385; on the mass-spectrum of, 519. Parabolic trajectory, on the con- struction of a, 157. Pendulums, on coupled, 294. Perihelion of a planet, on the ad- vance of the, 626. Photographic rendering of contrasts, on the, 151. Physical phenomena, on the processes — occurring in, 679. Pipes, on the velocity of gases in, 505. Poole (J. H. J.) on the separation of lead into isotopes by centrifuging, 372; on the effect of centrifuging certain alloys while in the liquid state, 376. : Prescott (Dr. J.) on the buckling of deep beams, 194. Quantum theory and chemical re- actions, on the, 26; of spectrum emission, on the, 46; of electric discharge, on the, 223. Quaternions, applications of, to the theory of relativity, 439. Radiation emitted in gaseous explo- sions, on the, 66, 84; from a cylindrical wall, on, 359 ; on mass, energy, and, 679; on the beta- recoil, 690. Radioactive elements, on a table of the, 155. Radioactivity and the gravitational field, on, 659. Raman (Prof. C. V.) on the sounds of splashes, 145; on a mechanical violin-player, 535. — Rankine (Prof. A. O.) on the rela- tion between illumination and conductivity in selenium, 482 ; on the propagation of light in a oravitational field, 586. Ray (8.) on the equivalent shell of a circular current, 134. Rayleigh (the late Lord) on resonant reflexion of sound from a perforated wall, 225. Refracted ray, on a simple propert of a, 223, °° oe Relativity, on parametric solutions for a fundamental equation in the theory of, 285; on an alternative view of, 433 ; applications of qua- ternions to the theory of, 489. 703 Renwick (F. F.) on the photographic ‘rendering of contrasts, 151. Resonant reflexion of sound, on the, 225. Rudorf (Dr. G.) on latent heat and surface energy, 238. Saha (5. N.) on the equation of state, 456. Sand (Dr. H. J. 8.) on the cadmium vapour arc lamp, 678. Selenium, on the relation between illumination and electric conduc- tivity in, 482. Silberstein (Dr. L.) on the quantum theory of spectrum emission, 46 ; on the recent eclipse results and Stokes-Planck’s ezther, 161; on the measurement of time, 366; on the propagation of light in a gravitational tield, 586. Silver-lead alloys, on the effect of centrifuging, 376. Sky, on the blue colour of the, 423. Slate (Prof. F.) on an alternative view of relativity, 433. Soné (T.) on the magnetic suscepti- bilities of hydrogen and _ other gases, 305. Sound, on resonant reftexion of, from a perforated wall, 225. Specific heat of gases, on the, 551; of saturated vapour, on the, 633. Spectra, on the excitation of the, of carbon, titanium, and vanadium by thermelectronic currents, 241’; on the mass-, of chemical elements, 611; onthe X-ray, of the elements, 694. Spectrum emission, on the quantum theory of, 46. of copper, on the, 457. Spheres, on the inertia of, moving in a fluid, 350; on the scattering of light by dielectric, 416. Splashes, on the sounds of, 145. Steam, on the specific heat of, 551. Sun, on the path ef a ray of light in the gravitation field of the, 626. Surface energy, on latent heat and, 32, 238. Thermelectronic currents, on the ex- citation of the spectra of carbon, titanium, and vanadium by, 241. Thomas (H. H.) on the geology of Palestine, 385. Thomas (J. 8S. G.) on the hot-wire anemometer, 505. 704 Thomson (Sir J. J.) on mass, energy, and radiation, 679. Time, on the measurement of, 366. Titanium, on the excitation of the spectrum of. by thermelectronic currents, 241. Tobin (T. C.) on the construction of a parabolic trajectory, 157. ‘Turbulence in the ocean, on, 578. Tykocinski-Tykociner (J.) on the Mandelstan method of measure- ment of frequency of electrical oscillations, 289. Vanadium, on the excitation of the spectrum of, by thermelectronic currents, 241. Vapour, on the specific heat of satu- rated, 638. Vapour-pressure formula, on the in- tegration constant of the, We alk Vibrations, on variably coupled, 294. INDEX. Violin-player, on a mechanical, 535. Visibility, on an instrument for the measurement of, 96. Water, on the rate of solution of nitrogen and oxygen by, 385; on the molecular rotational frequencies of, 565. Waves, on. the diffraction of, by a screen with a straight edge, 668. Whiddington (Prof. R.) on the X-ray spectra of the elements, 694. | Wood (Prof. R. W.) on light scat- tering by air and the blue ‘colour of the sky, 425. X-ray spectra of the elements, on the, 694, Xenon, on the mass-spectrum cf, 623. Zinc oxide, on the crystalline struc- ture of,-647, END OF THE THIRTY-NINTH VOLUME. Printed by Taytor and Francis, Red Lion Court, Fleet Street. Raman & Dey. Phil. Mag. Ser. 6, Vol. 39. Pl. I: Records of the Sounds of Splashes. -EMSALECH. Phil, Mag, Ser. 6, Wolk, 89), JPG OG a) SF © S Sa S} th Ne a) Sy mh 7 > & Ss 2.8 5 aS Q ny € a S san) S Q V G Q7 a Q Q x aye D> ae 33 3 a yy Q EMS aS S. S Sie YAS; aS) Ss 5 ane a SSeS ee S oe S ee © Go esta 2 ® eS ens Ea Pe a Re Ae ee ” + 3720 Fe. 4934 Ba, -3737 Fé, ~374/ Tr. 4983 Na, ~2746 Fe, 50/14 Tt -3753 Ti 5079 Al. 5/29 Baers ~3776 7/ te 4275 Cr. 5/65) : 52/0Th 4303Ca. 5269 4326 Fe. -3820 Fé. 537/ 5397 5405 -32860 Fe —-3862 5536 Ba, —387/ $C ~ 3883 5635 C, 5683 5686, )™ oo 3920 Fe hn -3934 Ca a. Zz -~ 3944 Al. 4607 Sr. 3962 Al —4648 Al. 3968 Ca 6/62 Ca, 3982 3990 2X 3999>Ti gee ca 47376. 4009 oS “a 4025 = 4 F031 Mn a pee, S j a 4047 SEE 4 4842 Al, 4058 Pb 7 i Sth 3 4072 Fe a Sa a a TY eer er a ~ ~ SS Se i. S een > Sig ee 5 as S 2 % ~ We Eee ae a5 im gee : > ae u t 1 ; ; (ey Q) 0d) oO Q Q Saas 9 ® St et = oO % es > ~ oo s % Say = a y 2 2 oO S a oR sae 3 & Q py (OP uty 5? o' 7 a oS ee cy a) Spectrum of Red Fringe and Luminous Vapours. Phil. Mag. Ser. 6. Vol. 39, Pl. IIT. HeEMSALECH, Influence of feeble magnetic fields on Visibility of Red Fringe Emission. Plate temperature: 2'700° C. Drop of potential alone plate: 5°8 volts. Se one Intensity of magnetic field: 175 C.G.S. units. q CS re ~ = ov ’ ry eee) uw Oo ec K aw G Oo & (I (Pe Sara 9G S TY NQAHO®D OD a \ eo) So TY a IN Me CON @ CP MS SG My ch S 6 ON Se) @) Ss Wy Ln DD © @ GQ) GC) g HAAAD SO SS LSS Ss WwW ty ee Pee ROR OD Rae Ne pp se NP RS NS | | | | Edge of plate \ Red fringe Without Bluish Viet OUTS magnetic Field Luminous vapours \ Edge of plate Red frin ge Magnetic force b, en LuMInous vapours @ecting downwards & spiral paths \Eage of plate Bluish vapour é Magnetic force acting upwards Lumfnous vapours Compression and Expansion of Red Fringe Emission by Magnetic Fields. : : : volts Plate temperature : 3000° C. Drop of potential along plate: 7 aa = in, Intensity of magnetic field: 590 C.G.S. units. o ) WW Gh eo ens i c xs) oh : ; WG ahaa ee ey eS oe ok Q OG Oy ~ So FY NO 99 ~ th’ NIN SOK O YO GCHOARRD VHAYYON So Oe Saw oS UO OD O 6 ODD 72mg O OO O05 S S212 ae a Hen See ae \Eage of plate ad } Red fringe Magnetic force Phil. Mag. Ser, 6, Vol. 39, Pl. IV, a. ‘ i acting upwards Luminous vapours Eage of plate F Red fringe b Magnetic force ee b Luminous vapours acting downwards & spiral paths HEMSALECH. a), Jell, VV. Mag. Ser. 6, Vol. Phil. HEMSALECH. Appearance of Red Fringe Emission above Plate under impulsion of an upwards acting Magnetic Force. Plate temperature: 8000° C. Drop of potential alone’ plate : 7 volts Cin. Intensity ot magnetic field: 1300 C.G.S. units. ; ; Bose cedar ic So 0 OP eo hoes dae OMG mete Omer eee S en Ale pee ae Oil BORE OuN a as oO + & NON ODA D Hr N@ Oy) SS I Gy ts Yt N Bas A) I Re Si C70 ia NN MM My» ) I99 9H Ft FPETTE F t+ Y YY Y Upper red fringe » £dge of plate ay SS ee Lower red fringe { Luminous vapours Ege thar [sete SS ES f Avila ¢ JACKSON AGKa ces of Lower Bo eS SES viol. 695 bly VAL. 6 . Mag. Ser. Phil Upper Bob struck JACKSON. Traces of Lower Bob when Upper Bob struck, PY ANALY Ag INA i) Ven 30%, Phil. Mag. Ser. 6, Vol. 39, Pl. VI. ‘ t i A Seren Be 4 0 arene teak PROAAAP SOA REARS hres hh a ROL EB HE TENEY ages ar B ~ i, 5 iw) Peo iy 1 i te KSON JAC Traces of Lower 22 21 Traces of Lower Bo Phil. Mag. Ser. 6, Vol. 39, Pl. VII. per Bob displaced. — JACKSON. } Phil. Mag. Ser. 6, Vol. 39, Pl. VII. Traces of Lower Bob y i upper Bob displaced. W of PE i . OR Ss cs TO Peery Serene ale i omer sven (Kt Aston, Phil. Mag. Ser. 6, Vol. 39. Pl. VIII. CO Ne® Ne* CO Ne® Ne2 ~ iN \ < - at GANS Phil. Mag. Ser. 6, Vol. 39, Pl. X. {. x $5 sili oral i; ir at | } iNpes! eet if i_|\ —<— a ] . 55 / ih ‘| a a ] a= 4 ; wi a By AI s as > i 2 —| ee 35 = : = fig.7 ‘0 4 2 = ee =e 7 : : Ao9 = | = 9 = + 25) 2 or Ae eS & Dae 2 ee Boe zis Pee eam | = = — “Sp 20 iF : a lo 4 Ist amp! al We Wiz x 4 2 amp 3 js} tt A — G41 3am p{@-_—_——] ‘ | O C 4 Ir4 am| ® C 4 15 om he} — a6 yee G=1019 ope ae HD C€ *Ji'l amp Ic © *{1'3) amb 0-S}— —_————_} © =| amp Bc [ns amp | Jena , J Q 10 20 30 40 50 bo 2» Bo cy 100 | Deflection: = + I Fe IL $0) 100 159 200 259 309 350 400 Ap? 709 fo J Ao - 50 60 7° £0 30 joe Anemometer Readin. Anemometer Reading, 3 4 = = [3 — on = o Po ao nm = = ry Deflection Log{Volme of dry air (cub ft per hove) af O° 8 TS0'mm. 5 5 a Br + by o . Re Sistan ce (ohms.) Phil. Mag, Ser. 6, Vol, 89, Pl. 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(amps) S18 o é e Sg a S 3 > c S | 4 ar rm Bare wire ater No2 2h Air. r 9 9 Ms : * ul { | 8 2 | i SS ee dou 400 400 509 of Shielded arm BIE: Resisfance of Resistance 0} 26 co 35 40 45 09 65 GO GS 70 15) 60 65 ~O 98 109 Ratio: ¥ d Defleetio Deflection { hl “5 6 7 i) Ey) ier WY i wae OFS UE ier Rey io 20 Current (amp) "SyusuTIOdxy [eosnosy OJ 19ACTq-uI[OTA, Tesruvyooa yy V Phil. Mag. Ser. 6, Vol. 39. Pl. XIV. RAMAN. a ASTON. Phil. Mag. Ser. 6, Vol. 39. Pl. XV. SSS Be ee arate tel 1 Sth ae i j Tae INSTITUTION LIBRARIES | ST EOE 3 9088 01202 5052 >